WAVELENGTH MULTIPLEXING BY SPATIAL BEAM SHIFTING IN ... · of JDS Uniphase for taking the time in...

WAVELENGTH MULTIPLEXING BY SPATIAL BEAM SHIFTING

IN MULTILAYER THIN-FILM STRUCTURES

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF ELECTRICAL ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Martina Gerken

March 2003

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© Copyright by Martina Gerken 2003

All Rights Reserved

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I certify that I have read this dissertation and that, in my opinion, it is fully adequate in

scope and quality as a dissertation for the degree of Doctor of Philosophy.

____________________________________

David A. B. Miller, Principal Advisor



____________________________________

Olav Solgaard



____________________________________

Shanhui Fan

Approved for the University Committee on Graduate Studies.

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AbstractWavelength Division Multiplexing (WDM) systems allow the transmission of multiple

channels over a single fiber by encoding each channel with a different optical

wavelength. Compact and cost-effective wavelength multiplexing and demultiplexing

devices are needed for combining the different WDM channels on the transmitter side

and splitting them at the receiver. This dissertation investigates the use of a single

multilayer thin-film stack with high spatial dispersion for multiplexing or demultiplexing

multiple WDM channels by spatial beam shifting. The thin-film stack is designed such

that multiplexed light incident at an angle experiences a wavelength-dependent effective

group propagation angle in the stack. This translates to a wavelength-dependent spatial

beam shift and demultiplexing at the output surface.

We introduce four different types of thin-film stacks with high spatial dispersion:

Periodic stacks using the “superprism effect” in one-dimensional photonic crystals,

chirped stacks exploiting a wavelength-dependent penetration depth, resonator stacks

with dispersion due to stored energy, and numerically optimized non-periodic stacks

utilizing a mixture of the two previous dispersion effects. The experimental results of a

200-layer periodic stack and a 66-layer non-periodic stack are discussed and compared.

Because of its greater design freedom, the non-periodic stack gives both a linear shift

with wavelength, and a larger usable shift than the thicker periodic stack.

Multiple bounces off the stack can be performed to increase the spatial beam shift. Using

eight bounces off the 66-layer stack, a nearly linear 100-µm shift is achieved between 827

and 841 nm. This shift is sufficient to separate four channels by their Gaussian beam

widths. We discuss that the number of separable channels is proportional to the total

beam shift. Investigating over 600 different stacks, we develop a heuristic model

predicting the maximum shift for a given stack thickness, material system, and incidence

angle. From this model we find that the multiplexing of eight to sixteen WDM channels

using a single thin-film stack with high spatial dispersion seems well possible.

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AcknowledgementsFirst and foremost, I wish to thank my research advisor, Prof. David Miller, without

whom this work would never have happened or been possible. He got me started on

investigating the spatial dispersion of periodic thin-film stacks for wavelength

multiplexing. He extensively helped me during my first quarter at Stanford to figure out

the basic physics. In my further research he helped me whenever I asked, and gave me

the freedom to try out new ideas on my own. His continuous support as well as his

continuous challenge for more physical intuition helped me to advance in my research

and strive for more insight. Thank you very much!

Thanks to Prof. Rafael Piestun for discussing ideas with me and providing physical

insight during the initial research on the superprism effect in photonic crystals. Bianca

Keeler and I worked together on demonstrating the spatial dispersion of periodic stacks.

Her experimental expertise enabled us to quickly obtain first experimental proof of the

superprism effect. Prof. Franz Kärtner gave me the opportunity to spend the summer of

2001 in his research group at the University of Karlsruhe. I appreciated the discussions

on temporal and spatial dispersion, and gained more insight into the design of double-

chirped stacks. Thanks also to my reading committee, Prof. Shanhui Fan and Prof. Olav

Solgaard, as well as the chair of my defense committee, Prof. Brian Wandell, for their

interest in my work and their helpful questions and comments.

On the fabrication side, I am indebted to several people whose help allowed me to

transfer this work from a mere theoretical exercise to an actual device. I would like to

thank Chien-Chung Lin for quickly growing our first periodic stack that allowed us to

collect proof-of-principle experimental results. Many thanks to Petar Atanackovic and

Glenn Solomon for teaching me how to grow thin-film stacks in an MBE. Their vast

knowledge of fabrication issues allowed me to better judge what can be fabricated and

what is not reasonable. I wish to thank Andrew Clark, Anmin Zheng, and Phil Anthony

of JDS Uniphase for taking the time in their busy schedule to fabricate both the 200-layer

periodic stack and the 66-layer non-periodic stack presented in this thesis. I especially

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appreciated discussions with Andrew on how to design stacks with less stringent

fabrication tolerances. Finally, thanks to Helen Kung and Tom Carver for providing me

with reflective coatings for the thin-film stacks.

Thanks to everyone in the Miller group: Diwakar Agarwal, Hatice Altug, Sameer

Bhalotra, Aparna Bhatnagar, Ray Chen, Henry Chin, Christof Debaes, Volkan Demir,

Onur Fidaner, Noah Helman, Yang Jiao, Bianca Keeler, Gordon Keeler, Helen Kung, Jon

Roth, Vijit Sabnis, Liang Tang, Ryohei Urata, Michael Wiemer, Micah Yairi, and Wei

Zhou. I appreciated your help as well as our various discussions about research and life in

general. Thanks to each one of you for making research fun and bringing delicious food

to the group meetings! Thanks also to the “Mezzanine people” for the great work

environment. On the administrative side, I would like to thank Ingrid Tarien for

guaranteeing that everything runs smoothly in the Miller group.

I greatly appreciated the support of a Sequoia Capital Stanford Graduate Fellowship. This

fellowship not only provided the major part of my financial support at Stanford, but also

allowed me to meet students from other research groups and other departments as well as

the donors of the fellowship. This gave me a broader perspective and enriched my stay at

Stanford.

Before coming to Stanford, I had the fortune to work for Greg Faris at SRI International.

He taught me the art of optical experiments and how to perform successful research.

Much of my decision to pursue a doctoral degree was based on my great research

experience at SRI. Many thanks to Greg in particular as well as everyone else at the

Molecular Physics Lab.

I am grateful to my family and friends for their support in my research endeavors and for

reminding me that there are things in life other than research. Finally, I want to thank my

husband Ingo. He never tires of reading my research writings nor of listening to my

practice talks. He never complains about my strange work hours and always encouraged

me in my research. Thank you for your constant support!

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Table of Contents

Chapter 1 Introduction........................................................................................... 1

1.1 Current MUX/DEMUX Devices ............................................................................ 2

1.2 Thin-Film Spatial MUX/DEMUX.......................................................................... 5

References ...................................................................................................................... 6

Chapter 2 Designing Dispersive Structures......................................................... 8

2.1 “Superprism Effect” in Photonic Crystals .............................................................. 9

2.2 Periodic versus Non-Periodic Structures .............................................................. 10

2.3 Designs for Temporal Dispersion Compensation ................................................. 12

2.3 Thin-Film Filter Design Techniques..................................................................... 13

References .................................................................................................................... 14

Chapter 3 Superprism Effect in 1-D Photonic Crystals .................................. 18

3.1 Superprism Effect Theory..................................................................................... 18

3.2 Experimental Results ............................................................................................ 21

3.3 Finite Number of Periods and Finite Beam Width ............................................... 23

3.4 Improved Superprism Structures .......................................................................... 32

References .................................................................................................................... 36

Chapter 4 Physics of Spatial and Temporal Dispersion................................... 38

4.1 Relating Spatial and Temporal Dispersion ........................................................... 39

4.2 Relationship between Dispersion and Stored Energy........................................... 42

4.3 Sample Structures Verifying the Proportionality of Stored Energy,

Group Delay, and Spatial Shift ............................................................................. 44

4.4 Influence of Material Dispersion .......................................................................... 53

References .................................................................................................................... 59

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Chapter 5 Chirped Stacks .................................................................................. 61

References .................................................................................................................... 68

Chapter 6 Resonator Stacks............................................................................... 69

References .................................................................................................................... 80

Chapter 7 Numerically Optimized Stacks ........................................................ 82

7.1 Designing Stacks using Numerical Optimization ................................................. 83

7.2 Experimental Results for an Optimized Stack ...................................................... 88

7.3 Comparison of Periodic and Non-Periodic Designs ............................................. 90

7.4 Experiment: 4-Channel Wavelength Demultiplexer............................................. 92

References .................................................................................................................... 94

Chapter 8 Maximum Number of Channels ...................................................... 95

8.1 Number of Volume Modes ................................................................................. 100

8.2 Verification of Volume Mode Model ................................................................. 106

8.3 Number of Surface Modes .................................................................................. 108

8.4 Crosstalk between Modes ................................................................................... 111

8.5 Number of Surface Modes including Bounces ................................................... 114

8.6 Verification of Surface Mode Model.................................................................. 119

8.7 Designing for a Maximum Number of Modes.................................................... 123

8.8 Discussion of the 4-channel DEMUX in 7.4 ...................................................... 127

References .................................................................................................................. 132

Chapter 9 Maximum Shift ............................................................................... 133

9.1 Automatic Generation of Designs....................................................................... 136

9.1.1 Automating the Refinement Process........................................................... 137

9.1.2 Automatic Refinement of Fixed Start Designs ........................................... 138

9.1.3 Automatic Generation of Starting Designs ................................................. 141

9.2 Deriving a Heuristic Shift Model........................................................................ 143

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9.3 Constancy of Dispersion×Wavelength-Range-Product ...................................... 149

9.4 Model for the Maximum Channel Number ........................................................ 151

References .................................................................................................................. 152

Chapter 10 Applications of Spatially Dispersive Stacks.................................. 154

10.1Wavelength Multiplexing and Demultiplexing .................................................. 154

10.2Step-Like Beam Shifting..................................................................................... 160

10.3Temporal vs. Spatial Dispersion ......................................................................... 167

10.4Beam Steering..................................................................................................... 168

10.5Other Applications Using Beam Shifting ........................................................... 169

References .................................................................................................................. 170

Chapter 11 Conclusions...................................................................................... 171

Appendix A Bloch Calculation........................................................................... 175

References .................................................................................................................. 177

Appendix B Coordinate Transformation (K, β, ω)→(K, θ, ω)......................... 178

B.1 Group Propagation Angle in Terms of (K, β, ω) ................................................ 179

B.2 Group Propagation Angle in Terms of (K, θ, ω) ................................................ 181

References .................................................................................................................. 182

Appendix C Transfer Matrix Calculation......................................................... 183

References .................................................................................................................. 195

Appendix D Simulating Beams by Fourier Decomposition............................. 196

References .................................................................................................................. 205

Appendix E Numerical Optimization Methods ................................................ 207

References .................................................................................................................. 216

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Appendix F Beam Cone in a Dispersive Stack ................................................. 218

References .................................................................................................................. 223

Appendix G Composition of Structures ............................................................ 224

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List of Tables

Chapter 6

Table 6.1. Reflection coefficients for different λB/4-layer combinations

(λB=980nm) of the substrate S (ns=1.52), high index material H

(nH=2.06, dH= λB/4nH), and low index material L (nL=1.456,

dL= λB/4nL) at 860nm, 54° incidence angle, and s-polarization................ 76

Chapter 8

Table 8.1. Wavelength, low angle, and high angle of the different modes for the

example in Fig. 8.9.................................................................................. 107

Table 8.2. Wavelength, group propagation angle in the structure for the input

angle plus the half cone angle (Angle+), group propagation angle

minus the half cone angle (Angle-), position where the plus-angle

crosses the back interface (Position+), and position where the minus-

angle crosses the back interface (Position-) for the different modes for

the example in Fig. 8.16.......................................................................... 120

Table 8.3. Wavelength, group propagation angle in the structure for the input

angle plus the half cone angle (Angle+), group propagation angle

minus the half cone angle (Angle-), position where the plus-angle

crosses the back interface (Position+), and position where the minus-

angle crosses the back interface (Position-) for the different modes for

the example in Fig. 8.21.......................................................................... 125

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Chapter 9

Table 9.1. Comparison of linear shift characteristics obtained with the double-

chirped stacks in Chapter 5, the coupled-cavity stack in Chapter 6,

and the numerically optimized stack in Chapter 7.................................. 134

Chapter 10

Table 10.1. Relationship between spot size, angular range, and wavelength range

at 1540 nm, 45º incidence angle, and ∂ω/∂β=0.3c . ............................... 164

Appendix E

Table E.1. Comparison of refinement methods. ........................................................ 215

Appendix G

Table G.1. Layer Composition of the structures appearing in Chapters 3 to 5.......... 227

Table G.2. Layer Composition of the structures appearing in Chapters 6 to 10........ 230

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List of Illustrations

Chapter 1

Fig. 1.1. Schematic of a Wavelength Division Multiplexing (WDM) system............... 2

Fig. 1.2. Traditional MUX/DEMUX technologies. (a) Prism. (b) Grating.................... 3

Fig. 1.3. State-of-the-art MUX/DEMUX technologies. (a) Arrayed waveguide

grating (AWG). (b) Fiber Bragg grating (FBG). (c) Thin-film filter

(TFF). .......................................................................................................... 4

Fig. 1.4. Thin-film spatial MUX/DEMUX. ................................................................... 5

Chapter 2

Fig. 2.1. Operating schematics of four different types of thin-film structures that

can be used for demultiplexing multiple wavelengths by spatial beam

shifting. The structure in (a) is periodic; (b) – (d) are non-periodic

structures. (a) Superprism effect in a one-dimensional photonic crystal

(combined here with a simple reflection off of the right face). (b)

Wavelength-dependent penetration depth. (c) Wavelength-dependent

number of roundtrips in the structure. (d) Combination of wavelength-

dependent penetration depth and number of roundtrips. ............................ 9

Chapter 3

Fig. 3.1. Schematic of the superprism effect................................................................ 18

Fig. 3.2. 1-D photonic crystal operated in transmission (a) or reflection (b)............... 20

Fig. 3.3. Schematic of the experimental setup. ............................................................ 21

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Fig. 3.4. Experimentally observed intensity on a CCD trace as a function of

position and wavelength for a 100-period one-dimensional photonic

crystal........................................................................................................ 22

Fig. 3.5. Comparison of the theoretically expected and experimentally observed

shift of the beam center position as a function of wavelength for a

100-period one-dimensional photonic crystal........................................... 23

Fig. 3.6. Reflectance and shift as a function of wavelength predicted using the

transfer matrix method for plane waves.................................................... 24

Fig. 3.7. Group propagation angle as a function of wavelength calculated using

Bloch theory with incidence angles of 40° (solid), 36.5° (dash), and

43.5° (dash-dot). For a Gaussian beam with a spot size of 4.7 µm at

880 nm and a center incidence angle of 40°, the intensity has

decreased to 1/e2 for beam components at incidence angles of 36.5°

and 43.5°. .................................................................................................. 27

Fig. 3.8. Intensity reflected from the 100-period stack as a function of wavelength.

Red shows the experimental data and blue the simulation results

obtained using the transfer matrix method and Fourier decomposition

of a Gaussian beam. .................................................................................. 29

Fig. 3.9. Normalized intensity (a), spot size along interface (b), and beam shift (c)

for the stack reflection and the shifting beam as obtained from a

Gaussian beam fit to experiment and simultaion...................................... 31

Fig. 3.10. (a) Layer thickness as a function of layer number for a λ/4-Bragg stack

and for an impedance matched Bragg stack. (b) Reflectance for both

stacks having a backside gold coating applied. (c) Group propagation

angle with wavelength. Simulations are performed for plane waves

using the transfer matrix technique........................................................... 35

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Chapter 4

Fig. 4.1. Bragg Stack (App. G, 3-2) with Gold Coating on Backside: Reflectance,

group delay, group velocity in the x-direction, and spatial shift as a

function of wavelength (solid lines – exact calculations, dotted lines –

approximations). ....................................................................................... 46

Fig. 4.2. Impedance Matched Bragg Stack (App. G, 3-3) with Gold Coating on

Backside: Reflectance, group delay, group velocity in the x-direction,

and spatial shift as a function of wavelength (solid lines – exact

calculations, dotted lines – approximations)............................................. 47

Fig. 4.3. Double-Chirped Stack (App. G, 5-6): Reflectance, group delay, group

velocity in the x-direction, and spatial shift as a function of

wavelength (solid lines – exact calculations, dotted lines –

approximations). ....................................................................................... 48

Fig. 4.4. Gires-Tournois Resonator (App. G, 6-1): Reflectance, group delay,

group velocity in the x-direction, and spatial shift as a function of


approximations). ....................................................................................... 49

Fig. 4.5. Coupled-Cavity Stack (App. G, 6-2): Reflectance, group delay, group



approximations). ....................................................................................... 50

Fig. 4.6. Numerically Optimized Stack (App. G, 7-2) with Gold Coating on

Backside: Reflectance, group delay, group velocity in the x-direction,

and spatial shift as a function of wavelength (solid lines – exact

calculations, dotted lines – approximations)............................................. 51

Fig. 4.7. Four-Step Design (App. G, 10-1): Reflectance, group delay, group



approximations). ....................................................................................... 52

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Fig. 4.8. (a) Refractive index and (b) extinction coefficient of GaAs as a function

of wavelength............................................................................................ 54

Fig. 4.9. 10-µm layer of GaAs sandwiched between GaAs on the incidence side

and a backside gold coating: Reflectance, group delay, group velocity

in the x-direction, and spatial shift as a function of wavelength (solid

lines – exact calculations, dotted lines – approximations)........................ 55

Fig. 4.10. Change of the refractive index n, the group refractive index ng, the shift

along the x-direction sx and the group delay τ in GaAs. .......................... 57

Fig. 4.11. 500-nm layer of GaAs sandwiched between an incident material with

n=2 and a backside gold coating: Reflectance, group delay, group



approximations). ....................................................................................... 59

Chapter 5

Fig. 5.1. (a) shows the Bragg wavelength as a function of the position in the

structure for five different 60-layer SiO2/Ta2O5 double-chirped mirror

designs. In (b) the theoretical spatial shift as a function of wavelength

is plotted for an incidence angle of 45° and p-polarized light. An

approximately linear shift is observed for all five designs. The

dispersion increases with decreasing chirp in the Bragg wavelength.

The maximum dispersion is achieved with a single-chirped Bragg

stack (f=0). ................................................................................................ 64

Fig. 5.2. (a) Physical layer thicknesses for a 200-layer SiO2/Ta2O5 double-chirped

structure. (b) Theoretically calculated shift as a function of

wavelength at 40° incidence angle and p-polarization. The circles

indicate the wavelengths and shifts corresponding to the diagrams in

(c). (c) E-field parallel to the interface of the forward propagating

wave as a function of the position in the structure for four different

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wavelengths – 780 nm, 830 nm, 880nm, and 930 nm. The vertical

lines indicate the position of the interfaces between layers. Light is

incident from the left, and the structure extends from 0 µm to 28 µm. .... 67

Chapter 6

Fig. 6.1. (a) Physical layer thicknesses for a 33-layer SiO2/Ta2O5 Gires-Tournois

resonator structure. (b) Theoretically calculated shift as a function of

wavelength at 45° incidence angle and p-polarization. The circles

indicate the wavelengths and shifts corresponding to the diagrams in

(c). (c) E-field parallel to the interface of the forward propagating

wave as a function of the position in the structure for four different


lines indicate the position of the interfaces between layers. Light is

incident from the left, and the structure extends from 0 µm to 8 µm. ...... 70

Fig. 6.2. Results for approximating the desired phase characteristics by a fourth

order allpass polynomial. .......................................................................... 73

Fig. 6.3. (a) Periodic phase derivative of the transfer function as a function of the

normalized frequency. (b) The expected shift for a 4-cavity structure

is plotted as a function of wavelength for three different cavity optical

thicknesses. ............................................................................................... 74

Fig. 6.4. Shift as a function of wavelength at 54° incidence angle and s-

polarization for the design with approximate reflectors (blue) and the

final refined design (black). Below the graph, the quarter wave layer

sequences for both designs are given. Remember that the quarter

wave layers are for a wavelength of 980 nm and 0° incidence angle. ...... 77

Fig. 6.5. (a) Physical layer thicknesses for a 33-layer SiO2/Ta2O5 4-cavity

structure. (b) Theoretically calculated shift as a function of

wavelength at 54° incidence angle and s-polarization. The reflectance

of the structure is 100%. The circles indicate the wavelengths and

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shifts corresponding to the diagrams in (c). (c) E-field parallel to the

interface of the forward propagating wave as a function of the

position in the structure for four different wavelengths – 842 nm, 846

nm, 850nm, and 854 nm. The vertical lines indicate the position of the

interfaces between layers. Light is incident from the left, and the

structure extends from 0 µm to 15.4 µm................................................... 79

Chapter 7

Fig. 7.1. Generation of the starting design. (a) Quarter wave Bragg stack. (b)

Impedance matched stack. (c) Half-wave layers added to thinnest

layers. ........................................................................................................ 85

Fig. 7.2. (a) Shift as a function of wavelength and (b) reflectance as a function of

wavelength for the start design, the refined design, and the refined

design with backside gold coating at 45° incidence angle and p-

polarization. .............................................................................................. 86

Fig. 7.3. (a) Physical layer thicknesses for a 66-layer, numerically optimized

SiO2/Ta2O5 structure. (b) Theoretically calculated shift as a function

of wavelength at 54° incidence angle and p-polarization. The

reflectance of the structure is improved to nearly 100% by a gold

layer on the very right. The circles correspond to the diagrams in (c).

(c) E-field parallel to the interface of the forward propagating wave as

a function of the position in the structure for four different


lines indicate the ....................................................................................... 87


position and wavelength for a 66-layer numerically optimized stack

for an incidence angle of 54° and p-polarization. ..................................... 89

Fig. 7.5. Experimentally observed and theoretically calculated spatial dispersion

of a 66-layer SiO2/Ta2O5 dielectric stack with a total thickness of 13.4

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µm on a quartz substrate for an incidence angle of 54° and p-

polarization. .............................................................................................. 90

Fig. 7.6. Comparing the performance of a periodic and a non-periodic structure. ...... 91

Fig. 7.7. (a) Experimentally observed intensity on a CCD trace as a function of

position and wavelength for 8 bounces off a 66-layer numerically

optimized stack. (b) Experimentally observed and theoretically

calculated shift as a function of wavelength. ............................................ 92

Fig. 7.8. 4-channel wavelength demultiplexer. ............................................................ 93

Chapter 8

Fig. 8.1. (a) Volume modes. (b) Surface modes. ......................................................... 96

Fig. 8.2. Physical layer thicknesses of the periodic and non-periodic design and

the double-chirped mirror starting design................................................. 97

Fig. 8.3. Reflectance of the two designs at 40° incidence angle.................................. 99

Fig. 8.4. Zoomed-in reflectance of the two designs at 40° incidence angle. ............... 99

Fig. 8.5. Calculated group propagation angle. ........................................................... 100

Fig. 8.6. Calculated Shift along the interface for a single bounce. ............................ 100

Fig. 8.7. Schematic of three volume modes for focussing on the front surface......... 101

Fig. 8.8. Increase in the number of volume modes per 1 nm wavelength interval as

a function of the stack dispersion for an input half cone ∆θin =1°.......... 104

Fig. 8.9. Volume mode calculation for the non-periodic design for an incidence

angle of 40°, an input half cone of 0.6°, and a wavelength interval

from 1525 nm to 1565 nm. (a) shows the position of the obtained

modes in terms of wavelength and propagation angle within the

crystal. (b) depicts a cartoon of the obtained modes............................... 107

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Fig. 8.10. Number of volume modes as obtained from simulation and the expected

number of modes calculated using (8 - 6) as a function of the input

half cone in (a) and the input spot size in (b).......................................... 108

Fig. 8.11. Schematic of three non-overlapping surface modes on the back surface. . 109

Fig. 8.12. Schematic of beam shift after two bounces for two neighboring modes... 112

Fig. 8.13. Gaussian profile of two neighboring modes that are separated by 5 nm.

The vertical green lines represent the position of the blue channel.

The solid lines are for the center frequency and the dotted lines

represent the shift with 50 GHz signal modulation. ............................... 112

Fig. 8.14. Crosstalk as a function of signal bandwidth for the non-periodic design

and a channel spacing of 5 nm................................................................ 113

Fig. 8.15. Theoretical maximum number of modes in the interval from 1525 nm to

1565 nm as a function of incident angle assuming a dispersion of

60µm/40nm, ns=1.52 and a crosstalk of –40dB (c1=3.8). ...................... 119

Fig. 8.16. Surface mode calculation for the non-periodic design for an incidence

angle of 40°, focussing on the back side at 1545 nm with a spot size

of 20 µm (corresponding to ∆θin = 1.4°), a wavelength interval from

1525 nm to 1565 nm, and 13 bounces. To obtain a crosstalk around –

40dB, a spacing of 3.8*20 µm/cos(40°)=100 µm is chosen between

modes. (a) shows the position of the obtained modes in terms of

wavelength and propagation angle within the crystal. (b) depicts a

cartoon of the modes obtained graphing the two 1/e E-field rays for

each mode. .............................................................................................. 120

Fig. 8.17. Number of surface modes as obtained from simulation, the expected

number of modes calculated using (8 - 14), and the maximum number

of modes obtained from (8 - 23) for the non-periodic design as a

function of the number of bounces within the structure. ........................ 122

Fig. 8.18. Physical layer thicknesses of the optimal non-periodic design. ................ 123

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Fig. 8.19. Reflectance of the optimal designs at 40° incidence angle........................ 124

Fig. 8.20. Group propagation angle and shift of the optimal designs at 40°

incidence angle........................................................................................ 124

Fig. 8.21. Surface mode calculation for the optimal design for an incidence angle

of 40°, focussing on the back side at 1545 nm with a spot size of 20

µm (corresponding to ∆θin = 1.4°), a wavelength interval from 1525

nm to 1565 nm, and 13 bounces. To obtain a crosstalk around –40dB,

a spacing of 3.8*20 µm/cos(40°)=100 µm is chosen between modes.

(a) shows the position of the modes obtained, in terms of wavelength

and propagation angle within the crystal. (b) depicts a cartoon of the

obtained modes graphing the two 1/e-E-field rays for each mode. ........ 125


number of modes calculated using (8 - 14), and the maximum number

of modes obtained from (8 - 23) for the optimal design as a function

of the number of bounces within the structure........................................ 126

Fig. 8.23. (a) Schematic of a 4-channel demultiplexer. (b) Scaled drawing of the

same 4-channel demultiplexer. ............................................................... 128

Fig. 8.24. Increasing the number of channels by increasing the number of

bounces. .................................................................................................. 129

Fig. 8.25. Increasing the number of channels by decreasing the spot size. ............... 130

Fig. 8.26. Reducing the device size by reducing the substrate thickness. ................. 131

Chapter 9

Fig. 9.1. (a) Shift as a function of dispersion for the designs in Table 9.1. (b) Shift

divided by stack thickness for the different designs. .............................. 135

Fig. 9.2. 20-period SiO2-Ta2O5 design for a 40º incidence angle. The shift was

specified to increase from 5 µm to 25 µm over a 70 nm operating

range........................................................................................................ 139

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range........................................................................................................ 139



range........................................................................................................ 140

Fig. 9.5. Example of a 20-period starting design for refractive indices of 1.45 and

2.09, an incidence angle θ of 40º, and an operating wavelength of

1550nm. .................................................................................................. 142

Fig. 9.6. Dependency of the shift on the number of periods...................................... 144

Fig. 9.7. Dependency of the shift on the incidence angle. ......................................... 144

Fig. 9.8. Dependency of the shift on the average refractive index............................. 145

Fig. 9.9. Dependency of the shift on the refractive index contrast. ........................... 145

Fig. 9.10. Dependency of the shift on the physical stack thickness........................... 146

Fig. 9.11. Dependency of the shift on the optical stack thickness. ............................ 146

Fig. 9.12. Dependency of the shift on the relative error. ........................................... 147

Fig. 9.13. Dependency of the shift on the filling ratio. .............................................. 147

Fig. 9.14. Two poor models for comparison. (a) Assuming the shift to be

inversely proportional to the refractive index. (b) Assuming the shift

to be inversely proportional to the refractive index cubed. .................... 148

Fig. 9.15 Normalized wavelength range as a function of the specified dispersion.... 150

Chapter 10

Fig. 10.1. Operating principle of a frequency filter. .................................................. 156

Fig. 10.2. Demultiplexing architecture based on cascading filters. For 64 channels

63 filters are needed. ............................................................................... 156

xxv

Fig. 10.3. Operating principle of a wavelength interleaver........................................ 157

Fig. 10.4. Demultiplexing architecture based on cascading interleavers. For 64

channels 63 interleavers are needed........................................................ 157

Fig. 10.5. Demultiplexing architecture based on a combination of interleavers and

filters. For 64 channels 63 devices are needed. ...................................... 158


spatial MUXes. For 64 channels only 15 devices are needed................. 159

Fig. 10.7. Intensity as a function of position and wavelength for 8 bounces off a

66-layer numerically optimized stack at 54º incidence angle and p-

polarization. ............................................................................................ 160

Fig. 10.8. Schematic of a (a) Gaussian passband shape compared to a (b) flat-top

passband shape (transmission refers to the energy transfer from input

to output)................................................................................................. 161

Fig. 10.9. Flat-top passband shape (a) corresponds to a step-like beam shift with

wavelength (b). ....................................................................................... 162

Fig. 10.10. Shift (a) and reflectance (b) after 8 bounces off a 100-layer

numerically optimized stack at 45º incidence angle and p-polarization. 163

Fig. 10.11. Shift as a function of wavelength for a plane wave at 45º incidence

angle, and Gaussian beams with 15 µm, 30 µm, and 50 µm spot size. .. 165

Fig. 10.12. Shift as a function of wavelength for four different incidence angles

for a 66-layer stack.................................................................................. 166

Fig. 10.13. (a) Intensity as a function of position and wavelength for 8 bounces

off a 66-layer numerically optimized stack at 48º incidence angle and

p-polarization. (b) Comparison between experiment and plane wave

theory. ..................................................................................................... 167

Fig. 10.14. Systems using stacks with high spatial dispersion. (a) Combination of

two stacks with opposite dispersion can be used to obtain spatial

dispersion without temporal dispersion. (b) System providing

xxvi

temporal dispersion without spatial dispersion. This system could also

be used to manipulate channels of different wavelengths

independently.......................................................................................... 168

Fig. 10.15. Wavelength-dependent beam steering. .................................................... 169

Appendix A

Fig. A.1. Labeling of periodic stack........................................................................... 175

Appendix C

Fig. C.1. Naming conventions for a multilayer stack. ............................................... 186

Fig. C.2. Conventions for the positive directions of k, E, and H in the case of p-

polarization (TM) and s-polarization (TE). ............................................ 186

Appendix D

Fig. D.1. Gaussian beam with w0=4.7µm and λ=890nm. (a) Calculated from exact

formula (D - 4). (b) Obtained from Fourier decomposition (D - 22).

Note that the ξ- and the ζ-axes have different scales.............................. 198

Fig. D.2. Direction cosines α and γ for propagation in the x-z-plane. ....................... 200

Fig. D.3. Amplitude of Fourier components normalized by π00wE for a

Gaussian beam with w0=4.7µm and λ=890nm. ...................................... 202

Fig. D.4. Propagation of a Gaussian beam with w0=4.7µm and λ=890nm in free

space........................................................................................................ 202


space at an angle of 40º with respect to the z-axis.................................. 203

Fig. D.6. Intensity distribution for a Gaussian beam with w0=4.7µm and λ=890nm

incident onto a 40-µm slab of material with refractive index n=2.5 at

an angle of 40º with respect to the z-axis. .............................................. 205

xxvii

Appendix E

Fig. E.1. Target reflectance as a function of wavelength for an EDFA gain-

flattening filter. ....................................................................................... 208

Fig. E.2. Reflectance of two 200-layer start designs. The merit function and the

total thickness of both designs are given to the right.............................. 209

Fig. E.3. Golden section search refinement. .............................................................. 210

Fig. E.4. Secant method refinement. .......................................................................... 211

Fig. E.5. Conjugate gradient algorithm refinement.................................................... 212

Fig. E.6. BFGS algorithm refinement. ....................................................................... 212

Fig. E.7. Damped least squares refinement................................................................ 214

Fig. E.8. Hooke & Jeeves pattern search refinement. ................................................ 214

Fig. E.9. Refined design using a combination of the different methods.................... 215

Appendix F

Fig. F.1. The change in (∂β/∂ω)|K=const as a function of wavelength. ........................ 221

Fig. F.2. Angular range ∆θstruc for an input angular range of ∆θin = 0.5º directly

calculated and calculated from (8 – 17) as being proportional to the

dispersion. ............................................................................................... 222


calculated and calculated from (8 - 17) as a function of wavelength. .... 223

xxviii

xxix

… physical intuition is often just mathematical consequences

with which we have lived long enough

to make them part of our “world picture.”

– Amnon Yariv

A. Yariv, IEEE J. Select. Topics Quantum Electron., 6/6, 1486 (2000).

xxx

1

Chapter 1

Introduction

With the emergence of high-speed computer connections to the Internet, data traffic has

been increasing rapidly. To support this growth, transmission systems of higher and

higher capacity are needed. Today’s transmission systems use optical fiber to connect

different locations, since optical fiber has a much higher data capacity than electrical

wires. As the installation of new optical fiber is, however, very expensive, different

multiplexing techniques have been developed that allow the transmission of multiple

channels over a single fiber, thus increasing the data capacity that can be transmitted over

the existing fiber.1 These techniques include Time Division Multiplexing (TDM) and

Wavelength Division Multiplexing (WDM).

TDM systems interleave several lower bitrate datastreams for high bitrate transmission

over the optical fiber. In WDM systems each channel is encoded with a different optical

wavelength. Since light of different wavelengths does not interact (at least to first order),

the channels can be overlapped for transmission over a single fiber as shown

schematically in Fig. 1.1. Today Dense WDM (DWDM) – named so because of their

dense channel spacing of 100 GHz or less – with typically 64 channels is employed in

2

long-haul systems. Coarse WDM (CWDM), with channel spacings around 20 nm and

approximately four to sixteen channels, is used in metro systems.

λ1 λ8

λ1

λ8

λ1

λ8

MUX

DEMUX

Fig. 1.1. Schematic of a Wavelength Division Multiplexing (WDM) system.

Crucial components of a WDM system are the wavelength multiplexer (MUX),

performing the spatial overlap of the different wavelength channels on the transmission

side, and the wavelength demultiplexer (DEMUX), spatially separating the different

wavelength channels at the receiver. The MUX/DEMUX should have low loss, low

crosstalk between channels, low polarization dependent loss (PDL), as well as some

tolerance for the drift of the channel wavelengths. From a systems perspective it should

furthermore be compact, temperature-stable, and, especially in the case of CWDM, cost-

effective. In this thesis a novel wavelength MUX/DEMUX based on the spatial

dispersion of multilayer thin-film structures is introduced.

1.1 Current MUX/DEMUX DevicesPrisms2 and gratings3 are traditional MUX/DEMUX devices. Both prisms and gratings

allow for the separation of many different wavelengths using just a single device. Prisms

rely on the wavelength-dependent refractive index of the prism material as well as the

special prism geometry schematically shown in Fig. 1.2(a) to disperse different

wavelengths into different angular directions. Due to their insufficient dispersion, prisms

are not used in today’s WDM systems.

Gratings (Fig. 1.2(b)) use interference effects between the reflections off different grating

grooves to separate beams of different wavelengths. Depending on the number of

grooves, gratings can have sufficient dispersion for current WDM systems. Gratings have

3

low loss and crosstalk, are temperature-stable and cost-effective. As the beam steering of

gratings is continuous, a change in the center wavelength of a channel results in loss and

crosstalk. Current research is investigating the combination of gratings with micro-optics

to eliminate this effect.4 Low-frequency gratings (<400 lines/mm) are polarization

independent. Unfortunately, such gratings require quite large focussing optics leading to

large component packages. High-frequency gratings can be used in combination with

polarization control to reduce the component size.5

(a) (b)

Fig. 1.2. Traditional MUX/DEMUX technologies. (a) Prism. (b) Grating.

Besides diffraction gratings, state-of-the-art MUX/DEMUX devices include arrayed

waveguide gratings (AWG), fiber Bragg gratings (FBG), and thin-film filters (TFF)

schematically shown in Fig. 1.3.6 AWGs 7,8,9,10 (Fig. 1.3(a)) split the light of all

wavelengths onto an array of waveguides of different lengths. After propagation through

the waveguides, the exiting light interferes at a wavelength-dependent position resulting

in wavelength multiplexing or demultiplexing. AWGs are particularly interesting for high

channel count systems, as they are compact and allow the (de)multiplexing of many

channels using a single device. AWGs need to be temperature stabilized, and are

expensive in fabrication and operation. Their high cost is only justified for high channel

count systems.

For cost-effective lower channel count systems, thin-film filters or fiber Bragg gratings

are typically used, due to their lower initial cost and the possibility to upgrade the channel

count progressively. Fiber Bragg gratings10 (Fig. 1.3(b)) are optical fibers that have a

refractive index modulation imposed along the direction of propagation. This grating

structure is designed such that one channel is transmitted, while all other channels are

reflected. The reflected channels are separated from the input light with a circulator.

4

Fiber Bragg gratings are low loss and polarization independent. As each fiber Bragg

grating only demultiplexes one channel, a separate grating is needed for each channel

leading to cascading losses and a more complex module assembly as the number of

channels is increased.

(a)

(b)

Bragg grating(c)

L H L H L...

Fig. 1.3. State-of-the-art MUX/DEMUX technologies. (a) Arrayed waveguide

grating (AWG). (b) Fiber Bragg grating (FBG). (c) Thin-film filter (TFF).

Thin-film filters11 (Fig. 1.3(c)) are stacks of alternating high and low index materials.

They are designed such that one channel is transmitted and all other channels are

reflected. The typical size of a thin-film filter is 1 mm by 1 mm. Again a separate device

is needed for each channel. Thin-film filters are assembled with micro-optics and used

slightly off normal incidence to separate the incoming from the reflected light. They are

low loss, low crosstalk, polarization independent, tolerant towards a drift of the channel

wavelength, and very temperature-stable. Thin-film filters are manufactured cost-

effectively on a wafer-scale. Today thin-film filters have the largest market share for

applications in WDM systems. Their main drawback is that they can only demultiplex a

5

single channel per device, again resulting in larger modules and cascading losses as the

number of channels increases. In the next section I will introduce how a single thin-film

stack can be used for multiplexing or demultiplex multiple channels, overcoming this

drawback.

1.2 Thin-Film Spatial MUX/DEMUXHere a novel thin-film device is introduced that multiplexes or demultiplexes multiple

WDM channels simultaneously. While traditional thin-film filters use the amplitude

reflection and transmission properties of multilayer stacks, this device employs

wavelength-dependent spatial beam shifting. Fig. 1.4 shows a schematic of the proposed

device.

zy

x

Thin-FilmStack

Mirror

Mirror

Substrate

Out 1Out 2Out 3

In Focussing lens

Out 4

Fig. 1.4. Thin-film spatial MUX/DEMUX.

The multiplexed light is incident from the top left corner at an angle onto the thin-film

stack. The stack is designed such that different wavelengths propagate at different

effective group propagation angles. Thus, beams incur a wavelength-dependent shift in

the structure. Multiple bounces can be performed off or through the stack to increase the

6

spatial separation of beams of different wavelengths. This spatial MUX/DEMUX

provides the advantages of thin-film filters (except polarization independence), while at

the same time eliminating the need for a separate thin-film device for each channel.

The key component of this spatial MUX/DEMUX is the multilayer thin-film stack with

high spatial dispersion. This thesis investigates how to design thin-film structures with

high spatial dispersion and how they can be used in WDM systems. Chapter 2 suggests

four different approaches for obtaining highly dispersive thin-film stacks and reviews

relevant literature. Chapters 3, 5, 6, and 7 discuss the four different design approaches in

detail. Chapter 4 explores the relationship between spatial dispersion, temporal

dispersion, and stored energy in a structure. This is important for the design algorithms of

Chapters 5 and 6 and gives at the same time physical insights into the origins of spatial

dispersion. Chapter 8 investigates the maximum number of channels that can be obtained

using a given dispersive thin-film stack and a device architecture as depicted in Fig. 1.4.

Chapter 9 examines how much spatial dispersion can be obtained depending on the stack

materials, the stack thickness, and the incidence angle. Finally, Chapter 10 discusses how

thin-film stacks with high spatial dispersion can be used in DWDM and CWDM systems

as well as possible other applications. Chapter 11 summarizes the results obtained.

References

[1] R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective,

Morgan Kaufmann Publishers, San Francisco, CA (1998).

[2] E. Hecht, Optik, Addison-Wesley (Deutschland) GmbH, Bonn (1989).

[3] C. Palmer and E. Loewen, Diffraction Grating Handbook, Richadrson Grating

Laboratory, Rochester, New York (2000).

[4] C.X. Yu, D.T. Neilson, C.R. Doerr and M. Zirngibl, “Dispersion-Free (De)mux

with Very High Figure-of-Merit,” Optical Fiber Communication Conference 2002,

Anaheim, CA (2002). Talk WS1.

[5] W. T. Boord, T. L. Vanderwert, R. DeSalvo, “Bulk diffraction gratings play

increasing role in optical networking,” Lightwave (March 2001).

7

[6] “MUX-DEMUX components market to decline 22 %,” WDM Sol. (July 04, 2001).

[7] M. K. Smit, “New Focusing and Dispersive Planar Component Based on an Optical

Phased Array,” Electron. Lett., 24/7, 385-386 (1988).

[8] H. Takahashi, S. Suzuki, K. Kato, I Nishi, “Arrayed-Waveguide Grating for

Wavelength Division Multi/Demultiplexer With Nanometer Resolution,” Electron.

Lett., 26/2, 87-88 (1990).

[9] C. Dragone, “An N x N Optical Multiplexer Using A Planar Arrangement of Two

Star Couplers,” IEEE Photon. Techn. Lett., 3/9, 812-815 (1991).

[10] C.K. Madsen and J.H. Zhao, Optical Filter Design and Analysis - A Signal

Processing Approach, John Wiley & Sons, Inc. (1999).

[11] H. A. MacLeod, Thin-Film Optical Filters, Institute of Physics Publishing (2001).

8

Chapter 2

Designing Dispersive Structures

In this thesis four different approaches for designing thin-film multilayer structures with

high spatial dispersion are researched. Spatial dispersion refers to a change in the beam

exit position with wavelength. This chapter introduces the four different approaches

schematically shown in Fig. 2.1, and reviews relevant literature for each approach. The

different approaches are discussed in more detail in Chapters 3 to 7. For each structure,

polychromatic light is incident at an angle from the top left corner. All structures are

operated in reflection and demultiplex light by a wavelength-dependent shift along the x-

axis. After exiting the structures, beams of different wavelength propagate in parallel

once again. The stack in Fig. 2.1(a) is periodic; (b) – (d) are non-periodic stacks.

The first approach is to use the superprism effect in a one-dimensional photonic crystal as

shown in Fig. 2.1(a). Different wavelengths propagate at different group velocity angles

within the structure and are thus spatially shifted along the x-axis. The second approach

is to use a non-periodic structure that reflects different wavelengths at different positions

along the z-axis as depicted in Fig. 2.1(b). Since the structure is operated at an angle, this

wavelength-dependent penetration depth leads to a spatial shift along the x-axis. Thirdly,

9

a wavelength-dependent number of roundtrips, as in the structure in Fig. 2.1(c), also

results in a wavelength-dependent beam shift along the x-axis, because of the operation at

an angle. Finally, the structure in Fig. 2.1(d) utilizes a combination of wavelength-

dependent penetration depth and wavelength-dependent number of roundtrips to

demultiplex polychromatic light. The next sections review the relevant literature for each

approach.

Mirror

Periodic stack

θin θin

Resonator stack

θin

Chirped stack General stack

θin

zy

x

(a) (b) (c) (d)

Fig. 2.1. Operating schematics of four different types of thin-film structures that

can be used for demultiplexing multiple wavelengths by spatial beam shifting. The

structure in (a) is periodic; (b) – (d) are non-periodic structures. (a) Superprism

effect in a one-dimensional photonic crystal (combined here with a simple

reflection off of the right face). (b) Wavelength-dependent penetration depth. (c)

Wavelength-dependent number of roundtrips in the structure. (d) Combination of

wavelength-dependent penetration depth and number of roundtrips.

2.1 “Superprism Effect” in Photonic CrystalsPolychromatic light incident at an angle onto one of the surfaces of a prism is dispersed

within the prism, i.e., light rays of different wavelengths propagate at different angles in

the prism. Rays exiting the prism have a wavelength-dependent propagation angle due to

the prism geometry. Conventional prisms rely on material dispersion. Since the change in

refractive index with wavelength is rather weak for transparent materials, this limits the

obtainable dispersion.

10

Previous research has shown that photonic crystal structures can be used to obtain much

higher spatial dispersion.1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19 Photonic crystals are artificial

structures composed of one-dimensional, two-dimensional, or three-dimensional periodic

arrangements of different materials.20 Because of the wavelength scale feature sizes of

photonic crystals, these structures exhibit a behavior that is very distinct from that of bulk

materials. Wavelength regimes with high dispersion have been observed in theory and

experiment for one-dimensional, two-dimensional, and three-dimensional photonic

crystals. As these artificial structures exhibit much higher dispersion than the material

dispersion of conventional prisms, this phenomenon has been termed the “superprism

effect”.

Realizing that a periodic multilayer thin-film stack behaves as a one-dimensional

photonic crystal,9,21,22,23,24 we expect that such structures exhibit the superprism effect and

can be used for wavelength multiplexing and demultiplexing. In Chapter 3 the superprism

effect in a 100-period one-dimensional thin-film structure containing two dielectric layers

per period is explored. We operate the dielectric stack in reflection, performing two

passes through the stack as depicted in the schematic in Fig. 2.1(a). Seen from the side,

polychromatic light is incident from the top left corner onto the periodic dielectric stack.

Just outside the stop-band (the high reflection spectral region), different wavelengths

propagate at different group propagation angles within the thin-film structure. Therefore,

beams of different wavelengths exit the dielectric stack at different positions along the x-

direction. In connection with ultrafast optics this effect has been called a “spatial chirp.”25

We experimentally observe this spatial shift with excellent agreement with theory. After

exiting the dielectric stack, the beams propagate parallel once again.

2.2 Periodic versus Non-Periodic StructuresFor application purposes, it is desirable to have the ability to design the spatial dispersion

with wavelength to given specifications. A linear spatial dispersion with wavelength or

frequency is, for example, of practical interest for multiplexing or demultiplexing

devices. Unfortunately, the design space of a periodic thin-film structure is very limited.

11

Only the period length, the materials, the distribution of the materials in a period, and the

incidence angle can be chosen. Considering this limited number of degrees of freedom, it

is not surprising that all periodic thin-film structures with two layers per period exhibit a

similar nonlinear spatial shift with wavelength.

To achieve a linear spatial shift, we need to increase the degrees of freedom. One

possible approach would be to increase the number of layers per period. For a sufficiently

large number of periods and a period length comparable to the wavelength, the optical

properties within such a structure do not depend on the position in the structure, and the

concept of effective optical properties is valid.2,4 Therefore, increasing the number of

layers per period is the method of choice if the application requires a constant phase

velocity or group velocity within the volume of the structure, as, for example, for phase

matching purposes. For other applications only the properties of the exiting light matter.

In the case of a wavelength demultiplexer only the properties of the light along the

exiting surface of the structure are important. As long as beams of different wavelengths

are spatially separated at the exit surface, it does not matter what happened to the beams

within the structure. This realization allows us to consider a whole new class of highly

dispersive structures – non-periodic structures with a high spatial dispersion along the

exit surface.19,26 Non-periodic structures have a much higher number of degrees of

freedom. In addition to the materials and the incidence angle, we can choose all the layer

thicknesses independently. With such a high number of degrees of freedom, we expect a

much higher design freedom as well.

The difficult task is now to devise a design that fulfills the desired specifications, such as,

e.g., a linear shift with wavelength and, ideally, to deduce design principles and physical

understanding that enable future designs. Considering that a 200-layer structure has more

than 200 degrees of freedom, searching the whole design space for an optimal structure is

not a feasible solution. The next two sections discuss three possible design approaches for

non-periodic structures with high spatial dispersion.

12

2.3 Designs for Temporal Dispersion CompensationWhile non-periodic thin-film structures with high spatial dispersion have not been

investigated previously, several thin-film design methods have been developed for

temporal dispersion compensation purposes in femtosecond-laser cavities27,28,29,30,31 and

optical fibers.32,33,34 In Chapter 4 we examine the relationship between spatial and

temporal dispersion, and show that methods used for designing structures with temporal

dispersion can be modified to obtain spatial dispersion.19 In Chapters 5 and 6 two

analytical approaches for designing thin-film stacks with high spatial dispersion based on

techniques developed for temporal dispersion compensation are introduced.

In Chapter 5 non-periodic thin-film structures are analyzed that exhibit a wavelength-

dependent penetration depth or turning point. An example of such a structure is a chirped

mirror, i.e., a dielectric stack with a position-dependent period length. Different

wavelengths are reflected at different positions along the z-axis. Operated at normal

incidence, such a chirped mirror exhibits temporal dispersion. An analytical algorithm for

designing structures with temporal dispersion has been developed previously based on

coupled-mode theory.27,28,29 Here we modify this algorithm for operating the structure at

an angle. At an angle the wavelength-dependent penetration depth corresponds to a

wavelength-dependent spatial shift along the x-axis at the exit surface as schematically

shown in Fig. 2.1(b). Therefore, such a structure can be used to demultiplex beams of

different wavelengths by spatial beam shifting.

In Chapter 6 resonator structures with one or multiple cavities are discussed. Examples of

single-resonator structures are Fabry-Perot and Gires-Tournois resonators.33 In a

resonator, light “bounces” back and forth. The effective number of bounces (or

roundtrips) differs significantly between wavelengths that are on-resonance and ones that

are off-resonance. If we operate such a resonator structure at an angle, this wavelength-

dependent number of roundtrips results in different amounts of beam shifting along the x-

axis as indicated figuratively in Fig. 2.1(c). Note that for a single resonator structure all

wavelengths are reflected at the same physical depth within the stack. Thus, the

maximum beam shift possible cannot be deduced from geometrical arguments. As the

13

incident light traverses a resonator multiple times before exiting, such structures exhibit

stored energy. In Chapter 4 we will analyze the relationship between stored energy and

beam shifting. More complex coupled-resonator structures with multiple cavities can be

designed to obtain superior dispersion characteristics. Coupled-cavity structures have

previously been designed for temporal dispersion compensation using digital lattice

techniques.32,33,34 Chapter 6 explains how this analytical technique can be modified to

design structures with spatial disperison.

Both analytical design procedures discussed in Chapters 5 and 6 are based on a reduction

of the degrees of freedom to a smaller subset. For the design of a 200-layer structure, for

example, not all possible 200-layer structures are considered, just a subset. It is shown

that both methods can be used to design thin-film structures exhibiting a linear spatial

shift along the x-axis with wavelength. Furthermore, the limitations of these two different

analytical design procedures are investigated.

2.3 Thin-Film Filter Design TechniquesThe fourth approach for designing multilayer thin-film stacks with high spatial dispersion

is based on modifying traditional thin-film filter design techniques. Thin-film filters with

a wide range of reflection and transmission properties have been designed for a long

time. The design process is typically divided into two stages – synthesis and

refinement.35,36,37,38 During the filter synthesis a multilayer stack is generated that closely

matches the desired performance.39 During refinement, the performance of a start design

is further improved.40 In Chapter 7 we show that thin-film filter design techniques can be

adapted to design thin-film stacks with high spatial dispersion. A synthesis procedure is

described to obtain a good starting design. Numerical optimization techniques are then

used to obtain the desired spatial dispersion. In this procedure all the degrees of freedom

are kept, and a locally optimal structure is achieved. We find that dispersive thin-film

structures obtained by numerical optimization combine effects based on a wavelength-

dependent turning point with effects based on a wavelength-dependent number of

roundtrips as depicted schematically in Fig. 2.1(d). We verify in theory and experiment

that a linear shift with wavelength can be obtained using such a structure and that it is

14

larger than can be obtained by a simple single reflection off of any particular depth in the

structure.

References

[1] R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,”

J. Mod. Opt., 34/12, 1589-1617 (1987).

[2] R. Zengerle and O. Leminger, “Frequency Demultiplexing Based on Beam Steering

in Periodic Planar Optical Waveguides,” J. Opt. Commun., 11/1, 11-12 (1990).

[3] J. P. Dowling and C. M. Bowden, “Anomalous index of refraction in photonic

bandgap materials,” J. Mod. Opt., 41/2, 345-351 (1994).

[4] S.-Y. Lin, V. M. Hietala, L. Wang, and E.D. Jones, “Highly dispersive photonic

band-gap prism,” Opt. Lett., 21/21, 1771-1773 (1996).

[5] H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S.

Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B, 58/16,

R10 096-R10 099 (1998).

[6] S. Enoch, G. Tayeb, D. Maystre, “Numerical evidence of ultrarefractive optics in

photonic crystals,” Opt. Comm., 161, 171-176 (1999).

[7] H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S.

Kawakami, “Superprism phenomena in photonic crystals: Toward microscale

lightwave circuits,” J. Lightw. Techn., 17/11, 2032-2038 (1999).

[8] B. Gralak, S. Enoch, G. Tayeb, “Anomalous refractive properties of photonic

crystals,” J. Opt. Soc. Am. A, 17/6, 1012-1020 (2000).

[9] B. E. Nelson, M. Gerken, D. A. B. Miller, R. Piestun, C.-C. Lin, J. S. Harris, Jr.,

“Use of a dielectric stack as a one-dimensional photonic crystal for wavelength

demultiplexing by beam shifting,” Opt. Lett. 25/20, 1502-1504 (2000).

[10] E. Silvestre, J. M. Pottage, P. St. J. Russell, P. J. Roberts, “Design of thin-film

photonic crystal waveguides,” Appl. Phys. Lett., 77/7, 942-944 (2000).

[11] A. N. Naumov, R. B. Miles, P. Barker, A. M. Zheltikov, “Ultradispersive prisms

and narrow-band tunable filters combining dispersion of atomic resonances and

photonic band-gap structures,” Laser Phys., 10/2, 622-626 (2000).

15

[12] M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, I. Yokohama,

“Extremely large group-velocity dispersion of line-defect waveguides in photonic

crystal slabs,” Phys. Rev. Lett., 87/25, 253902-1 - 253902-4 (2001).

[13] T. Ochiai and J. Sanchez-Dehesa, “Superprism effect in opal-based photonic

crystals,” Phys. Rev. B, 64, 245113-1 - 245113-7 (2001).

[14] W. Park and C. J. Summers, “Extraordinary refraction and dispersion in two-

dimensional photonic-crystal slabs,” Opt. Lett. 27/16, 1397-1399 (2002).

[15] T. Baba and M Nakamura, “Photonic Crystal Light Deflection Devices Using the

Superprism Effect,” IEEE J. Quantum Electron., 38/7, 909-914 (2002).

[16] L. Wu, M. Mazilu, T. Karle, T. F. Krauss, “Superprism Phenomena in Planar

Photonic Crystals,” IEEE J. Quantum Electron., 38/7, 915-918 (2002).

[17] K. B. Chung and S. W. Hong, “Wavelength demultiplexers based on the

superprism phenomena in photonic crystals,” Appl. Phys. Lett., 81/9, 1549-1551

(2002).

[18] T. Baba and T. Matsumoto, “Resolution of photonic crystal superprism,” Appl.

Phys. Lett., 81/13, 2325-2327 (2002).

[19] M. Gerken and D. A. B. Miller, “Multilayer thin-film structures with high spatial

dispersion,” Appl. Opt. 42/7 (2003), 1330-1345.

[20] J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals-Molding the

Flow of Light, Princeton University Press (1995).

[21] C. Elachi, “Waves in Active and Passive Periodic Structures: A Review,” Proc. of

the IEEE, 64/12, 1666-1698 (1976).

[22] P. Yeh, A. Yariv, C.-S. Hong, “Electromagnetic propagation in periodic stratified

media. I. General theory,” J. Opt. Soc. Am., 67/4, 423-438 (1977).

[23] A. Yariv and P. Yeh, “Electromagnetic propagation in periodic stratified media. II.

Birefringence, phase matching, and x-ray lasers,” J. Opt. Soc. Am., 67/4, 438-448

(1977).

[24] Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas,

“A Dielectric Omnidirectional Reflector,” Science, 282, 1679-1682 (1998).

[25] I. Walmsley, L. Waxer, C. Dorrer, “The role of dispersion in ultrafast optics,” Rev.

of Scientific Instr., 72/1, 1-29 (2001).

16

[26] M. Gerken, B. E. Nelson, and D. A. B. Miller, “Thin-Film Wavelength

Demultiplexer Based on Photonic Crystal and Group Velocity Effects,” OSA

Conference on Integrated Photonics Research 2002, Vancouver, BC (July 17-19,

2002).

[27] R. Szipöcs, K. Ferencz, C. Spielmann, F. Krausz, “Chirped multilayer coatings for

broadband dispersion control in femtosecond lasers,” Opt. Lett. 19/3, 201-203

(1994).

[28] P. Tournois and P. Hartemann, “Bulk chirped Bragg reflectors for light pulse

compression and expansion,” Opt.Commun. 119, 569-575 (1995).

[29] N. Matuschek, F.X. Kärtner, and U. Keller, “Exact Coupled-Mode Theories for

Multilayer Interference Coatings with Arbitrary Strong Index Modulations,” IEEE

J. Quantum Electron., 33/3, 295-302 (1997).

[30] N. Matuschek, F.X. Kärtner, and U. Keller, “Theory of Double-Chirped Mirrors,”

IEEE J. Select. Topics Quantum Electron., 4/2, 197-208 (1998).

[31] N. Matuschek, F.X. Kärtner, and U. Keller, “Analytical Design of Double-Chirped

Mirrors with Custom-Tailored Dispersion Characteristics,” IEEE J. Quantum

Electron., 35/2, 129-137 (1999).

[32] G. Lenz and C. K. Madsen, “General optical all-pass filter structures for dispersion

control in WDM systems,” J. Lightw. Techn., 17/7, 1248-1254 (1999).



[34] M. Jablonski, Y. Takushima, K. Kikuchi, “The realization of all-pass filters for

third-order dispersion compensation in ultrafast optical fiber transmission systems,”

J. Lightwave Techn., 19/8, 1194-1205 (2001).

[35] P. Baumeister, “Design of multilayer filters by successive approximations,” J. Opt.

Soc. Am., 48/12, 955-958 (1958).

[36] J. A. Dobrowolski, “Completely automatic synthesis of optical thin film systems,”

Appl. Opt., 4/8, 937-946 (1965).

[37] H. A. MacLeod, Thin-Film Optical Filters, Institute of Physics Publishing (2001).

[38] A. Thelen, Design of Optical Interference Coatings, McGraw-Hill, Inc. (1989).

17

[39] Li Li and J. A. Dobrowolski, “Computation speeds of different optical thin-film

synthesis methods,” Appl. Opt., 31/19, 3790-3799 (1992) and references herein.

[40] J. A. Dobrowolski and R.A. Kemp, “Refinement of optical multilayer systems with

different optimization procedures,” Appl. Opt., 29/19, 2876-2893 (1990) and

references herein.

18

Chapter 3

Superprism Effect in 1-D Photonic Crystals

The first section of this chapter discusses the theory of the superprism effect in one-

dimensional photonic crystals. The second section presents experimental results

demonstrating the superprism effect in a 200-layer periodic thin-film stack with excellent

agreement between theory and experiment. In the last two sections limitations of the

periodic design and possible improvements are discussed.

3.1 Superprism Effect TheoryIn this section we explain how to calculate the superprism effect, i.e. the wavelength-

dependent group propagation angle θgr and the exit position sx as depicted in Fig. 3.1.

zy

x

θin θgr

∆sx

λ1, λ2

λ1

λ2

sx1

Fig. 3.1. Schematic of the superprism effect.

19

Close to the transmission stop-band edge, one-dimensional periodic structures exhibit a

rapid change in the phase1 and group2,3 velocity properties with wavelength. As we are

here interested in the propagation direction of beams, we need to consider the group

propagation angle θgr, which corresponds to the direction of energy flow.4 First, we

consider the case of a wide beam whose behavior can be approximated by a plane wave

calculation. The propagation of a plane wave is described by a single wavevector k. We

can place the coordinate system such that this wavevector k only has components in the

x- and z-directions and is thus given by k = β x + K z. The effect of a finite beam size is

discussed in Section 3.3 and in Appendix D.

For a periodic structure with a sufficient number of periods, the dispersion properties of

the structure can be modeled by Bloch theory (see Appendix A).4,5 In this approach an

infinite structure is assumed and the dispersion relation between the wavevector K in the

z-direction, the wavevector β in the x-direction, and the frequency ω is obtained using

periodic boundary conditions. The group velocity in the photonic crystal is given by

(3 - 1).

( ) zxkv kg gzgx vv +=∇= ω (3 - 1)

The group velocities vgx in the x-direction and vgz in the z-direction are given by (3 - 2)

and (3 - 3).4

constKgxv

=∂∂

=βω (3 - 2)

constgz K

v=∂

∂=

β

ω (3 - 3)

The group propagation angle θgr is calculated using (3 - 4).

( )

= −

gz

gxgr v

vK 1tan,, βωθ (3 - 4)

All angles are taken with respect to the z-axis. Thus, θgr is 0° if the beam propagates

along the z-axis. In the case that K and β are given as functions of the frequency ω and

20

the incidence angle θ, we can transform (3 - 4) to (3 - 5) by performing a coordinate

transformation and carefully calculating the partial derivatives as shown in Appendix B.

( ) ( ) ( )

∂∂

∂∂

−= −

θωθβ

θωθ

ωθθ,,tan, 1 K

gr (3 - 5)

This group propagation angle is of course identical to the one obtained by determining

the normal n in a wavevector diagram plotting contours of constant frequency.2,3,4

( ) ( ) zxnθ

ωθβθ

ωθ∂

∂+

∂∂

−=,,K (3 - 6)

Finally, the exit position in reflection, sx, along the surface of the dielectric stack in the

x-direction is given by (3 - 7).

( )gz

gxgrx v

vLLs 2tan2 == θ (3 - 7)

L is the total stack thickness and θgr can be calculated either by (3 - 4) or by (3 - 5). In a

bulk material the group propagation angle θgr changes only slowly with wavelength.

Therefore, beams of different wavelengths exit the material at approximately the same

position. Close to the stop band in a periodic dielectric stack though, the group

propagation angle changes rapidly with wavelength. Due to this superprism effect, beams

of different wavelengths exit the material at different positions and are spatially

demultiplexed.3

Periodic thin-film stack

Operation in transmission Operation in reflection

Mirror

Periodic thin-film stack

θinθin

(a) (b)

Fig. 3.2. 1-D photonic crystal operated in transmission (a) or reflection (b).

21

Fig. 3.2 shows that the stack can be operated either in transmission or in reflection. All

stacks presented here are operated in reflection as this doubles the spatial shift and allows

easily for multiple bounces off the stack. The next section compares the theoretically

calculated and the experimentally obtained shift with wavelength for a 100-period

dielectric stack.

3.2 Experimental ResultsHere experimental results are discussed for a 100-period dielectric stack consisting of

alternating layers of SiO2 (n=1.456 at 880 nm) and Ta2O5 (n=2.06) with a total stack

thickness of 30 µm on a quartz substrate (n=1.52). All the layers have a physical

thickness of 150 nm. Appendix G lists the layer composition of this stack. The stack is

operated in reflection as shown in Fig. 3.2(b) except that no mirror is applied to the

backside and there is thus some transmission loss. A schematic of the experimental setup

is shown in Fig. 3.3.

Tunable Laser

CCDOscilloscope TVPC

Spherical lenses

Substrate

θin

Multilayer thin-film stack

Cylindrical lens

Fig. 3.3. Schematic of the experimental setup.

Light from a tunable laser is focussed onto the tilted sample under test. The tunable laser

beam is incident onto the dielectric stack through the substrate. In a telescope style setup

a magnified image of the exiting light is focussed onto a CCD camera. Due to the light

propagation through a tilted plate, the focal point in the plane of the light beams is

22

different from the one perpendicular to the plane. The cylindrical lens compensates for

this difference, such that both beam directions are focussed on the CCD and the beam

appears approximately circular. The CCD trace is observed on the TV and oscilloscope as

well as stored on a computer for further data evaluation.

Wavelength in nm

Posi

tion

in µ

m

Quartz substrate

100-period SiO2-Ta2O5thin-film stack

Shiftingbeam

Frontreflection


position and wavelength for a 100-period one-dimensional photonic crystal.

Fig. 3.4 shows the intensity observed along one CCD trace as a function of position and

wavelength for an incidence angle of 40°, p-polarization, and a spot size (Gaussian beam

radius) of 4.7 µm. The spot size is the incident Gaussian spot size in vacuum

perpendicular to the direction of beam propagation.6 Two exiting beams are observed (as

well as a weak reflection off the air-substrate-interface, which is not shown). One beam is

caused by a reflection off the front of the dielectric stack due to the impedance mismatch

and does not change position as a function of wavelength. The intensity of this beam

increases relative to the second beam as the reflectivity of the stack increases closer to the

stop-band edge. The second beam is the real signal beam that propagates through the

dielectric stack twice as seen in Fig. 3.2(b).

Fig. 3.5 plots the center position of the beams as a function of wavelength. The center

positions are obtained by fitting the data to a Gaussian beam shape, adjusting position,

width, and amplitude. Excellent agreement between the experimentally observed shift

23

compared to the theoretically expected shift along the x-direction is obtained for the

shifting beam. As expected, the front reflection is not shifting with wavelength.

870 875 880 885 890

0

5

10

15

Shifting beam (experiment)Front reflection (experiment)Shifting beam (Bloch theory)

Wavelength in nm

Wavelength in nm

Shift

in µ

m

Fig. 3.5. Comparison of the theoretically expected and experimentally observed

shift of the beam center position as a function of wavelength for a 100-period one-

dimensional photonic crystal.

3.3 Finite Number of Periods and Finite Beam WidthEvaluating the experimental results in section 3.2, we find two main differences between

the experimental results and the Bloch calculation modeling. The first difference is that

we observe two reflected beams instead of just one. The second difference is that Bloch

theory predicts an increase of the spatial shift up to the stop band edge at 901 nm with the

group propagation angle approaching 90° and the shift thus approaching infinity, while

we observe only an increase in the shift up to around 892 nm. At larger wavelengths the

shifting beam appears very distorted and no clear peak can by determined. Here it is

discussed how these two differences are based on the finite number of periods and the

finite beam width.

As Bloch theory models the propagation of plane waves in an infinite medium, it cannot

predict the front reflection off the interface between substrate and stack nor can it be used

to calculate the transmission loss. These are due to the finite extent of the real stack. In

24

order to correctly predict the reflected and the transmitted amplitudes, a different type of

calculation is needed. Here we use a transfer matrix method to calculate the relationship

between the incident, reflected, and transmitted field of plane waves for arbitrary thin-

film stacks. Appendix C describes the calculation details of this method. Using the

transfer matrix method, the reflectance and shift are calculated as a function of

wavelength for the 100-period stack discussed in the previous section. The plane wave

light is incident from the substrate side. Fig. 3.6 plots the results. Observe that the shift

calculated for very low reflected amplitudes is not correct due to numerical inaccuracies.

870 875 880 885 890 895 900 905 9100

0.2

0.4

0.6

0.8

1

Rp

λ2

nm

870 875 880 885 890 895 900 905 9100

20

40

60

80

sp

um

λ2 λ2,

Wavelength in nm

Ref

lect

ance

Wavelength in nm

Shift

in µ

m

Fig. 3.6. Reflectance and shift as a function of wavelength predicted using the

transfer matrix method for plane waves.

25

The obtained shift is very different from what is expected from Bloch theory. Instead of a

smoothly increasing shift with wavelength, we see oscillations that become more and

more rapid approaching the bandedge at 902 nm. These oscillations are due to a

combination of the finite nature of the stack and the plane wave calculation. Part of the

plane wave energy is reflected off the interface between substrate and stack. Another part

propagates through the stack, is reflected off the stack–air interface, propagates through

the stack again and is transmitted into the substrate. Other parts of the plane wave

perform even more bounces in the stack. Considering now at the total reflected light, we

see that it is composed of light that traveled different paths. Therefore, the interference of

these different parts causes oscillations in the reflectance and the shift. The stack behaves

like a Fabry-Perot resonator with transmission on resonance.

In order to separate out the part that experiences the superprism effect, we need to

eliminate the interference between parts of the light that have traveled different paths.

This can be done by reducing the spot size. Once the spot size is sufficiently small, the

different light paths lead to a splitting of the incident beam into different exiting beams as

seen in section 3.2. Before reducing the spot size, we observed strong interference effects

in our experiments. To correctly predict the transmitted and reflected beam parts of a

tightly focussed beam, we need to simulate the transfer of spatially limited beams through

thin-film stacks. In this research, I implemented two methods for calculating the

propagation of beams – the Finite Difference Time Domain (FDTD) technique7 and a

discrete Fourier decomposition technique.8

FDTD is based on replacing the derivatives in Maxwell’s equations by differences.7

Choosing the time and position steps correctly, the propagation of any type of

electromagnetic (EM) wave can be calculated. The advantage of FDTD is that it can be

used for calculating the propagation through arbitrary structures. The disadvantage is that

the calculation can become very time consuming. The position steps should be chosen at

approximately a tenth of the smallest feature size. If all features are larger than the

wavelength λ of the light, this corresponds to λ/10. But since the multilayer thin-film

stacks of interest here can have layer thicknesses smaller than λ /10 (e.g. as layers as thin

26

as 20 nm can be fabricated today) the position step size has to be chosen even smaller.

This also results in a smaller time step size. Such calculations therefore quickly become

time consuming and the accuracy requirements have to be stringent to prevent

propagation of calculation inaccuracies.

A multilayered stack is a linear system for sufficiently small intensities. Furthermore, it is

space-invariant along the layers, i.e. the fields only depend on the difference between

input and the exit position, not on the absolute position along the stack. The propagation

of arbitrary electromagnetic waves through such linear, space-invariant systems can be

calculated by decomposing the input field into elementary components, these can be

propagated individually, and the output field is obtained by summing the propagated

individual components.8 This technique is discussed in detail in Appendix D. By

performing a Fourier transform on the input field, the field is decomposed into plane

wave components that have identical wavelength, but different propagation directions.

These plane waves can be propagated using the transfer matrix method discussed in

Appendix C. Finally, superposing the individually propagated plane waves, the total field

is obtained.

Considering that a spatially limited beam can be described as a superposition of plane

waves with different incidence angles, we can understand why we only observe an

increase in the shift up to around 892 nm, even though Bloch theory predicts an increase

of the spatial shift up to the stop band edge at 901 nm with the group propagation angle

approaching 90° and the shift thus approaching infinity. The reason for this discrepancy

is that Bloch theory is only exactly correct for an infinitely wide beam as only one

incidence angle is included in the calculation.

For a Gaussian beam with a spot size (1/e2-intensity radius) of 4.7 µm at 880 nm and an

incidence angle of 40°, the intensity has decreased to 1/e2 for beam components at

incidence angles of 36.5° and 43.5° (see Appendix D). As seen in Fig. 3.7, the stop band

edge is a function of the incidence angle. Therefore, different parts of the beam are

shifted by different amounts leading to beam widening and distortion. These distortions

27

limit the usable portion of the theoretically predicted shift and clarify why we only

observed a beam shift up to 892 nm. At longer wavelengths part of the beam is already

within the stop band, distorting the beam shape severely.

860 870 880 890 900 91020

30

40

50

60

70

80

90

40 deg36.5 deg43.5 deg

Wavelength in nm

Gro

up p

ropa

gatio

n an

gle

in d

eg

Fig. 3.7. Group propagation angle as a function of wavelength calculated using

Bloch theory with incidence angles of 40° (solid), 36.5° (dash), and 43.5° (dash-

dot). For a Gaussian beam with a spot size of 4.7 µm at 880 nm and a center

incidence angle of 40°, the intensity has decreased to 1/e2 for beam components at

incidence angles of 36.5° and 43.5°.

This also explains why we will aim for a linear shift with wavelength in the following

chapters – for a linear shift with wavelength the beam shape at the exit surface is

independent of wavelength. The beam gets stretched at the output surface, but not

distorted. This stretching corresponds to a change in the focal point and is eliminated by

choosing the focusing lens correspondingly. In the case that the beam is focused in both

the x-direction and the y-direction, some polarization mixing occurs, since some of the

beam components have wavevector components in the y-direction. This has to be taken

into account for determining the exiting beam shape. Since the wavelength multiplexing

device discussed here does not require strong focusing in the y-direction, we limit

28

ourselves here to elliptical beams that are strongly focused in the x-direction and little

focused in the y-direction. In this case we can conclude that a constant dispersion for the

polarization of interest corresponds to an undistorted beam shape. Unfortunately, due to

the limited number of degrees of freedom available, a one-dimensional periodic dielectric

stack with two layers per period cannot be designed to exhibit a linear shift with

wavelength.

In order to correctly predict the front reflection off the 100-period stack and the reflected

shifting beam, the propagation of the incident Gaussian beam with w0=4.7µm is

calculated using the Fourier decomposition technique discussed in Appendix D. Fig. 3.8

plots the resulting reflected intensities and the experimental data for four different

wavelengths. The left beam is the front reflection and the right beam the shifting beam.

Very good agreement for both amplitude and position of the beams is obtained.

Furthermore, the broadening of the shifting beam as the bandedge is approached is

modeled correctly in the simulation. The extra oscillations in the experimental data are

Airy disks caused by beam clipping.

For a more detailed comparison, Fig. 3.9(a) plots the amplitudes of the stack reflection

and the shifted beams as a function of wavelength obtained by performing a Gaussian

beam fit to both experiment and simulation. Fig. 3.9(b) shows the spot size along the

interface. Finally, Fig. 3.9(c) graphs the shift obtained in experiment and theory. Both the

experimental and the theoretical spot sizes as well as the center positions of the shifting

beam are not calculated correctly for wavelengths longer than 892 nm as the intensity is

too low and the Gaussian beam fitting is therefore not working correctly. For

wavelengths shorter than 892 nm, we see that Bloch theory modeling predicts the same

shift as the transfer matrix method in combination with Fourier decomposition, but the

beam modeling is necessary to correctly predict amplitude and shape of the shifting beam

as seen in Fig. 3.8.

29

40 20 0 20 40 60 80

0

0.2

0.4

40 20 0 20 40 60 80

0

0.2

0.4

40 20 0 20 40 60 80

0

0.2

0.4

40 20 0 20 40 60 80

0

0.2

0.4

Position in µm

Nor

mal

ized

Inte

nsity

λ = 870 nm

λ = 880 nm

λ = 890 nm

λ = 894 nm

Experimental DataTransfer Matrix Simulation

Fig. 3.8. Intensity reflected from the 100-period stack as a function of wavelength.

Red shows the experimental data and blue the simulation results obtained using the

transfer matrix method and Fourier decomposition of a Gaussian beam.

30

870 875 880 885 890 895 9000

0.2

0.4

0.6

0.8

1

Wavelength in nm

Nor

mal

ized

inte

nsity

Wavelength in nm

Nor

mal

ized

Inte

nsity

(a)

870 875 880 885 890 895 9000

5

10

15

Wavelength in nm

Spot

size

in m

icro

n

Wavelength in nm

Spot

Siz

e in

µm

(b)

31

870 875 880 885 890 895 9000

5

10

15

20

25

Shifting beam (experiment)Stack reflection (experiment)Shifting beam (simulation)Stack reflection (simulation)Shift using Bloch calculation

Wavelength in nm

Shift

in m

icro

n

Wavelength in nm

Shift

in µ

m

(c)

Fig. 3.9. Normalized intensity (a), spot size along interface (b), and beam shift (c)

for the stack reflection and the shifting beam as obtained from a Gaussian beam fit

to experiment and simultaion.

In conclusion, we have seen in this section that the finite number of periods in the one-

dimensional photonic crystal results in reflections off the interfaces with the surrounding

media. These reflections result in interference effects. To see the superprism effect, the

incident beam has to be focussed tightly enough to prevent field overlap between the

different reflections. If this is achieved, the shifting beam portion is modeled correctly by

Bloch theory except for the wavelength region close to the bandedge. Since a focussed

beam can be seen as a superposition of plane waves with different propagation directions,

different parts of the beam see a different dispersion curve. This is particularly noticeable

close to the bandedge as some components are still propagating in the stack while others

32

are within the bandgap. Therefore, the finite beam width leads to beam distortions close

to the bandedge.

3.4 Improved Superprism StructuresIn the previous sections we have seen that we can use the photonic crystal superprism

effect to spatially offset beams of different wavelength. We also found, though, that part

of the incident light is lost by reflection off the front of the dielectric stack and by

transmission through the stack. The transmission loss can easily be eliminated by

applying a reflective coating to the backside of the periodic stack. This could for example

be a gold coating or a further dielectric mirror stack.

The reflection off the front of the stack is due to an impedance mismatch between the

substrate and the periodic dielectric stack that is caused mainly by the sudden periodicity

and not as much by the difference in refractive index.9 The same problem occurs in

corrugated waveguide structures and in fiber Bragg gratings. In the case of corrugated

waveguides the reflection can be eliminated by a tapering of the surface relief,2 while in

the case of apodized fiber Bragg gratings the refractive index contrast is slowly

increased.10,11 To prevent the reflection off the front of the thin-film stack, we could use a

“tapered” Bragg stack.9 In such a Bragg stack the periodicity is slowly “turned on” by

increasing the amount of high index material in each period.

Fig. 3.10(a) shows the layer thickness as a function of the layer number for a 60-layer

λ/4-Bragg stack and for a 60-layer impedance matched Bragg stack. The composition of

the stack is given in Appendix G. For the periodic λ/4-Bragg stack the layer thicknesses

dHB of the high index material and dLB of the low index material are calculated using

(3 - 8) and (3 - 9).

( ) 2sin14

−

=

HH

BHB

nn

dθ

λ (3 - 8)

33

( ) 2sin14

−

=

LL

BLB

nn

dθ

λ (3 - 9)

In these equations λB is the Bragg wavelength, nH and nL are the refractive indices, and θ

is the propagation angle in vacuum. In order to achieve impedance matching, the first PDC

periods of the stack are chirped as a function of the period p as given in (3 - 10) and

(3 - 11). In our example stack, the first 25 periods are chirped, while the layer thicknesses

of the last five periods are calculated using (3 - 8) and (3 - 9).

( )( )

05.1

2sin14

−

=DC

HH

BHBIM P

p

nn

pdθ

λ (3 - 10)

( )( ) 2

05.1

sin14

42

−

−

=

LL

DC

BB

LBIM

nn

Pp

pdθ

λλ

(3 - 11)

The Bragg wavelength is 850 nm for all periods for both stacks. The refractive indices are

1.45 and 2.09, the incidence angle in vacuum is 45°, and only p-polarization is

considered. The increasing ratio between high and low index material is clearly visible

for the impedance-matched stack. Fig. 3.10(b) shows the reflectance of the two different

stacks. Both stacks have a gold coating applied to the backside to eliminate transmission

loss. The reflectance is close to unity for both stacks. Fig. 3.10(c) shows the group

propagation angle as a function of wavelength for the two stacks and compares it to the

group propagation angle obtained from Bloch theory. Both reflectance and group

propagation angle are calculated using the transfer matrix method for plane waves.

Since the shifting beam and the reflection off the front of the stack interfere for the

periodic Bragg stack, we see large oscillations in the calculated group propagation angle.

The group propagation angle obtained for the impedance matched Bragg stack on the

other hand oscillates only slightly and follows the Bloch calculation curve well. Thus, we

34

see that the impedance matching eliminates the front reflection successfully. If no

oscillations occur in the plane wave calculation, this signifies that all of the intensity

propagates as a single beam through the structure. See Chapter 10 for a more detailed

discussion on this topic. The impedance matched stack is not ideally periodic anymore

and therefore does not track the Bloch curve close to the bandedge.

10 20 30 40 50 600

100

200

300

400

Bragg stackImpedance matched Bragg stack

Period Number

Bra

gg W

avel

engt

h in

nm

Layer number

Laye

r thi

ckne

ss in

nm

(a)

850 900 950 1000 1050 1100 1150 12000

0.2

0.4

0.6

0.8

1

Gold coated Bragg stackGold coated impedance matched Bragg stack

Wavelength in nm

Ref

lect

ance

(b)

35

850 900 950 1000 1050 1100 1150 12000

20

40

60

80

100

Bloch calculationGold coated Bragg stackGold coated impedance matched Bragg stack

Wavelength in nm

Gro

up p

ropa

gatio

n an

gle

in d

eg

(c)

Fig. 3.10. (a) Layer thickness as a function of layer number for a λ/4-Bragg stack

and for an impedance matched Bragg stack. (b) Reflectance for both stacks having

a backside gold coating applied. (c) Group propagation angle with wavelength.

Simulations are performed for plane waves using the transfer matrix technique.

We see that impedance matching can be used to eliminate front reflections and ensure

low loss for the shifting beam. Furthermore, the dispersion just outside the reflection

band is still due to the superprism effect, since it is well described by Bloch theory for

wavelengths larger than 950 nm. Interestingly, we note that the impedance matched stack

also has high dispersion between 910 nm and 950 nm. This dispersion is not due to the

superprism effect, but due to the wavelength-dependent penetration depth caused by the

chirping of the stack.

From Fig. 3.10(c) we see that this might be a very interesting effect as the total change in

the group propagation angle is larger than the one obtained with the superprism effect.

Furthermore, the group propagation angle changes approximately linearly with

wavelength between 910 nm and 950 nm. This is interesting for practical devices as it

36

allows for equidistant channel spacing. Such a constant dispersion cannot be obtained for

periodic one-dimensional photonic crystals. All periodic structures have a similar non-

linear shift with wavelength that cannot be changed much due to the limited number of

degrees of freedom that a periodic stack has. In Chapter 5 to 7 we will investigate non-

periodic stacks that have degrees of freedom and can be designed to have both low loss

and constant dispersion. Before turning to such structures, Chapter 4 investigates the

common features of multilayer thin-film stacks with spatial dispersion.

References

[1] J. P. Dowling and C. M. Bowden, “Anomalous index of refraction in photonic

bandgap materials,” J. Mod. Opt., 41/2, 345-351 (1994).

[2] R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,”

J. Mod. Opt., 34/12, 1589-1617 (1987).



demultiplexing by beam shifting,” Opt. Lett. 25/20, 1502-1504 (2000).



(1977).

[5] P. Yeh, A. Yariv, C.-S. Hong, “Electromagnetic propagation in periodic stratified

media. I. General theory,” J. Opt. Soc. Am., 67/4, 423-438 (1977).

[6] A. E. Siegman, Lasers, University Science Books, Sausalito, CA (1986).

[7] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-

Difference Time-Domain Method, Artech House (2000).

[8] J. W. Goodman, Introduction to Fourier Optics, The McGraw-Hill Companies, Inc.

(1996).



[10] F. Oullette, “Dispersion concellation using linearly chirped Bragg grating filters in

optical waveguides,” Opt. Lett. 12, 847-849 (1987).

37

[11] B. J. Eggleton, G. Lenz, N. Litchinitser, D. B. Patterson, R. E. Slusher,

“Implications of fiber grating dispersion for WDM communication systems,” IEEE

Phot. Techn. Lett., 9/10, 1403-1405 (1997).

38

Chapter 4

Physics of Spatial and Temporal Dispersion

In the Chapter 2 we introduced four physical effects resulting in high spatial dispersion in

multilayer thin-film stacks– the superprism effect in periodic structures, dispersion due to

a wavelength-dependent penetration depth, dispersion due to multiple effective roundtrips

in the structure, and a combination of penetration depth and roundtrips. In Chapter 3 we

investigated the superprism effect in one-dimensional photonic crystals in detail and

proved its existence in experiment. Before looking at non-periodic stacks with high

spatial dispersion in Chapters 5 to 7, we will derive here the relationship among spatial

dispersion, temporal dispersion, and stored energy.

These relationships show how all four different dispersion phenomena introduced in

Chapter 2 are related to a wavelength-dependent amount of stored energy in the structure.

This is very interesting as it allows the derivation of some general connections between

different structures with spatial dispersion. Before looking at stored energy dependencies,

the relationship between spatial and temporal dispersion is investigated. We will show

that spatial and temporal dispersion are approximately proportional. This is important

39

from both a systems point of view as well a for design considerations. We finish the

chapter with looking at the effects of material dispersion.

4.1 Relating Spatial and Temporal DispersionIn this section we will explore the relationship between spatial and temporal dispersion

characteristics, and explain why methods used for designing structures with temporal

dispersion can be applied in a modified form to obtain spatial dispersion. Spatial

dispersion manifests itself in a change of the beam exit position sx as a function of

wavelength as shown in Fig. 3.1. Temporal dispersion refers to the change in the group

delay τgroup as a function of wavelength. We are considering here the shift and the group

delay in reflection. A similar study can be performed for the case of transmission. The

group delay in reflection is calculated by dividing the shift sx along the x-direction by the

effective speed of light vgx in this direction. As shown in (4 - 1) we can rewrite this

expression in terms of the more commonly used dependency on the phase upon reflection

φrefl using (3 - 7) and (4 - 2), where L is the total thickness of the stack.

const

refl

constconstKx

gx

xgroup

KLsvs

=== ∂

∂=

∂∂

=∂∂

==ββ ω

φ

ωωβ

τ 2 (4 - 1)

KLrefl 2=φ (4 - 2)

In the case of a periodic structure, the relationship among the wavevector K in the z-

direction, the wavevector β in the x-direction, and the frequency ω is obtained from

Bloch theory as shown in Appendix A. For a non-periodic structure we can use the

transfer matrix technique to relate K, β, and ω. This is demonstrated in Appendix C. For

a non-periodic stack and a finite periodic stack, K is the effective wavevector for the

reflected light. Since the structure is not periodic, K is not constant throughout the stack.

The group velocities vgx and vgz are in this case also effective quantities, i.e., they

represent the total effect of the stack, but are not constant within the stack. Therefore, by

calculating the group delay τgroup, we obtain the total time elapsed from entering the stack

to exiting the stack, but we cannot determine how much delay the light incurred in each

part of the stack.

40

(4 - 1) shows that the shift experienced by a beam of light is related to the group delay in

reflection by the group velocity vgx along the layers. Here we are interested in the

dispersive properties of multilayer stacks, i.e. the change of the shift and the group delay

with wavelength. To relate spatial and temporal dispersion, we therefore need to

investigate the change of the group velocity vgx with wavelength. First, we will calculate

the group velocity in a WKB-type approximation.

The WKB-approximation (also called the semiclassical or quasiclassical approximation

in quantum mechanics) states that if the local wavelength λ(z), which is linked to the

local wavevector K(z), changes slowly with z, the accumulated phase change can be

calculated by integrating the wavevector K(z) from the start position z1 to the end position

z2.1,2 This result is exactly true for uniform media as well as infinite periodic media,

where the wavevector K is obtained from Bloch theory and independent of z as discussed

in Appendix A. The resulting phase is given by (4 - 2). The WKB-approximation has

previously been applied to calculate the accumulated phase change of a chirped Bragg

stack.3 Here we use the WKB-approximation to obtain an approximate phase upon

reflection φappr for a general multilayer stack.

As the local wavevector K(z) is not a priori known in this case, we will use the

wavevector corresponding to a uniform medium with refractive index ni for the ith layer.

Even though this is strictly only a good approximation for low index contrast stacks, we

will see in Section 4.3 that it is quite good for many structures of interest. Replacing the

WKB-integral by a sum, the approximated phase upon reflection φappr is then given by

(4 - 3), where di the layer thickness of the ith layer in the stack.

( ) ∑

−

−≈

iiiappr dn

c2

2

2, βω

ωβφ (4 - 3)

Using (4 - 2) and (4 - 3) we therefore obtain the approximate dispersion relation (4 - 4)

for the approximate wavevector Kappr in the z-direction.

41

( ) ∑

−

−=

iiiappr dn

cLK 2

21, βω

ωβ (4 - 4)

Again, this is an effective wavevector for the overall effect of the stack, not a local

wavevector. The derivatives of the approximate dispersion relation with respect to ω and

β are (4 - 5) and (4 - 6). Here it is assumed that the refractive indices ni are independent

of frequency, which is quite correct for the typical dielectrics used in multilayer thin-film

stacks. If the refractive indices do depend on frequency, the derivatives have to be

modified accordingly as discussed in Section 4.4.

( )∑

−

−=∂

∂

= i

i

ii

const

appr

cn

dc

n

LK

22

2

1,

ωβω

ωβ

β

(4 - 5)

( )∑

−

=∂

∂

= i

i

i

const

appr

cn

dc

LK

22

1,

ωβ

ωβ

β

ωβ

ω

(4 - 6)

Using the rules for taking implicit derivatives,4 we get expression (4 - 7) for vgx,appr.

∑

∑

−

−

≈∂∂

∂∂−=

∂∂

==

iii

iiii

appr

appr

constKapprapprgx cnd

cndn

cKK

vappr

22

222

2,

1

ωβ

ωβ

βω

β

ω

ωβ (4 - 7)

Substituting β = ω/c sin(θ), where θ is the incidence angle in vacuum, (4 - 7) is rewritten

as (4 - 8).

( )

( )

( )∑

∑

−

−

=

∂∂

==

iii

iiii

constKapprapprgx nd

ndn

cv 22

222

, sin

sin

sin11

θ

θ

θωβ (4 - 8)

42

We see that the resulting approximate expression for vgx,appr only depends on the

incidence angle and not on the frequency. Thus, vgx,appr is independent of frequency

within this approximation. We found numerically that vgx,appr is approximately constant

for many structures we evaluated, even though the constant value might be different from

the one calculated using (4 - 8). (4 - 8) can still be used to roughly approximate vgx

though. In section 4.3, vgx is plotted for different structures of interest validating that the

group velocity along the layers is approximately constant with frequency.

This is very interesting as we see from (4 - 1) that the change in the shift with frequency,

i.e. the spatial dispersion, and the change in the group delay with frequency, i.e. the

temporal dispersion, are proportional, if vgx is independent of frequency.5 This result

provides both physical insight and has practical consequences. As spatial and temporal

dispersion are approximately proportional, existing structures with temporal dispersion

can be modified to obtain structures with spatial dispersion. This is demonstrated in

Chapters 5 and 6. Furthermore, we see that a spatial shift with wavelength corresponds at

the same time to a temporal delay. If this is not desired, two structures with opposite

spatial dispersion can be used in series as discussed in more detail in Chapter 10. With

the correct arrangement this doubles the spatial shift and removes the temporal delay

between wavelengths. If on the other hand the temporal delay is desired, the light could

be backreflected through the same structure, canceling the spatial shift and doubling the

temporal delay.

4.2 Relationship between Dispersion and StoredEnergy

In section 4.1 we discussed that temporal dispersion and spatial dispersion are

approximately proportional for many structures of interest. Here we will investigate the

relationship among spatial dispersion, temporal dispersion, and stored energy in

multilayer thin-film stacks. As detailed in Appendix C, a multilayer thin-film stack with

uniform plane waves as incident light can be seen as a two-port system, one port on either

side of the stack. The transfer function r(β,ω) relates the reflected light to the incident

43

light, while the transfer function t(β,ω) relates the transmitted light to the incident light.

We are particularly interested here in multilayer stacks that reflect all of the light, such

that there is no transmission loss. For this case the magnitude of r(β,ω) is unity. As seen

from (4 - 1) the derivative of the phase change upon reflection φrefl with frequency

determines the group delay τgroup. Relationship (4 - 9) between stored energy and delay

has previously been derived for linear, time-invariant, lossless electrical networks6 and

microwave circuits7,8 from Tellegen’s theorem.

PW

PWW totme

group =+

=τ (4 - 9)

In this equation We is the electrical energy stored in the system, Wm is the stored magnetic

energy, and P is the incident power. Since multilayer dielectric stacks are lossless and

linear for sufficiently low energies, we expect the group delay to be proportional to the

stored energy divided by the incident power. Appendix C describes how to calculate the

time-averaged electromagnetic energy density wavg(z) as a function of the position z in the

layer. It is also detailed how the incident irradiance I⊥,inc perpendicular to the layer

interfaces is calculated. (4 - 10) and (4 - 11) give the equations that relate wavg(z) and I⊥,inc

to the total stored electromagnetic energy as well as to the incident power per unit area

Ax,y on the interface.

( )∫=L

avgxytot dzzwAW0

(4 - 10)

incxy IAP ,⊥= (4 - 11)

Neglecting material dispersion, we substitute the equations for wavg(z) and I⊥,inc from

Appendix C. The group delay is calculated from the E-field in the multilayer stack using

(4 - 12) for p-polarization and (4 - 13) for s-polarization.

( )( )[ ] ( )[ ]

( ) 0

2

0,,

0

2

,,,,,,20

2

,

cos21

expexpcos2

1

Z

nE

dzzzkjEzzkjEn

inc

incxf

L

iizibxiizifxi

i

TMgroup

θ

θ

ε

τ∫ −−−−

= (4 - 12)

44

( )[ ] ( )[ ]

( )0

2

0,,

0

2

,,,,,,02

,cos

21

expexp21

Z

nE

dzzzkjEzzkjEn

incincyf

L

iizibyiizifyi

TEgroupθ

ετ

∫ −+−−= (4 - 13)

Here, Ex,f,i, Ex,b,i, Ey,f,i, and Ey,b,i are the forward and backward propagating E-field

components along the interfaces at positions zi. The formulae differ between p- and s-

polarization due to the different relationship between E|| and the total E-field amplitude.

ninc is the refractive index and θinc is the propagation angle of the incident medium (in our

case mostly the substrate). Z0 is the vacuum impedance of 377Ω. (4 - 12) and (4 - 13) are

only exactly correct for lossless structures with zero transmission, but they are also good

approximations of the group delay for structures that have some transmission loss or

absorption. In this case (4 - 12) and (4 - 13) give an upper bound for the group delay. A

similar calculation can be performed for structures with unity transmission.6,7

Once the group delay is obtained from the stored energy, the expected shift can be

calculated using (4 - 1). Section 4.3 demonstrates the relationship between stored energy,

group delay, and shift for several example structures. This relationship is very interesting,

as it is common to all four types of dispersive structures introduced in Chapter 2 and

discussed in more detail in Chapters 3, 5, 6, and 7. We see that even though the design

algorithms and stack sequences are very different for the different structures, the origin of

the dispersion always lies in a changing amount of stored energy with frequency.9

Therefore, any linear, time-invariant, lossless stack that exhibits a change in the stored

energy with frequency will also exhibit temporal and spatial dispersion.

4.3 Sample Structures Verifying the Proportionality ofStored Energy, Group Delay, and Spatial Shift

In sections 4.1 and 4.2 we discussed that spatial shift, group delay, and stored energy are

all approximately proportional for lossless structures with unity reflectance. Here we

show that this is indeed a good approximation for many structures of interest. For all the

structures investigated in this section the layer composition is given in Appendix G. The

45

figure captions list the structure numbers for easy lookup in Appendix G. Except for the

two improved superprism structures, all structures are discussed in detail in later chapters.

Fig. 4.1 and Fig. 4.2 show results obtained for the two improved superprism structures

discussed in section 3.4. For both the gold coated Bragg stack and the impedance

matched Bragg stack the plane wave reflectance, the group delay, the group velocity in

the x-direction normalized by the vacuum speed of light c, and the spatial shift are

calculated for p- and s-polarization as a function of wavelength. The solid lines show the

exact results obtained from transfer matrix calculations for plane waves. We are not

considering focussed beams here. As discussed in Appendix D, the behavior of focussed

beams is obtained by a decomposition into plane wave components. Thus, if the

relationship among spatial shift, group delay, and stored energy is correct for plane

waves, they will also be applicable to focussed beams which are a superposition of plane

wave components.

The group delay is additionally calculated from the E-field using the relationship between

group delay and stored energy in the stack given in (4 - 12) and (4 - 13). As the

reflectance is close to unity for both stacks, we see that this approximation gives a very

good estimate of the group delay. The approximate group velocity vgx,appr is obtained

from (4 - 8). There is some deviation from the exact solution as the index contrast is not

low – here it is 2.06 to 1.456. Finally, the spatial shift is approximated by multiplying the

group delay obtained from the stored energy by vgx,appr. We see that for both structures the

spatial shift is estimated well verifying the approximate proportionality among group

delay, stored energy, and spatial shift. We conclude that the superprism effect is indeed

based on wavelength-dependent energy storage, i.e. for a larger change in the stored

energy with wavelength, a larger change in the propagation direction with wavelength is

obtained and vice versa.

46

700 800 900 10000.8

0.85

0.9

0.95

1

p-pols-pol

Wavelength in nm

Ref

lect

ance

700 800 900 10000

0.5

1

p-pol : exactp-pol : energy approx.s-pol : exacts-pol : energy approx.

Wavelength in nm

Gro

up d

elay

in p

s

700 800 900 10000.15

0.2

0.25

0.3

0.35

Wavelength in nm

vgx

/ c

700 800 900 10000

20

40

60

80

100

Wavelength in nm

Shift

in u

m

Fig. 4.1. Bragg Stack (App. G, 3-2) with Gold Coating on Backside: Reflectance,

group delay, group velocity in the x-direction, and spatial shift as a function of

wavelength (solid lines – exact calculations, dotted lines – approximations).

47

850 900 950 1000 10500.97

0.98

0.99

1

p-pols-pol

Wavelength in nm

Ref

lect

ance

850 900 950 1000 10500

0.05

0.1

0.15

0.2

0.25


Wavelength in nm

Gro

up d

elay

in p

s

850 900 950 1000 10500.22

0.24

0.26

0.28

0.3

0.32

0.34

Wavelength in nm

vgx

/ c

850 900 950 1000 10500

5

10

15

20

Wavelength in nm

Shift

in u

m

Fig. 4.2. Impedance Matched Bragg Stack (App. G, 3-3) with Gold Coating on

Backside: Reflectance, group delay, group velocity in the x-direction, and spatial

shift as a function of wavelength (solid lines – exact calculations, dotted lines –

approximations).

Fig. 4.3 shows the results obtained for a double-chirped stack. This type of stack is

explained in detail in Chapter 5. It uses a wavelength-dependent penetration depth to

generate spatial dispersion as introduced in Fig. 2.1(b). Assuming a constant E-field

amplitude up to the penetration depth and zero field beyond that depth, we see from the

integral (4 - 12) or (4 - 13) that a larger penetration depth corresponds to more stored

48

energy within the stack. Therefore, the proportionality between group delay and stored

energy is no surprise.

800 850 900 9500.8

0.85

0.9

0.95

1

p-pols-pol

Wavelength in nm

Ref

lect

ance

800 850 900 9500

0.1

0.2

0.3

0.4

0.5


Wavelength in nm

Gro

up d

elay

in p

s

800 850 900 9500.22

0.24

0.26

0.28

0.3

0.32

Wavelength in nm

vgx

/ c

800 850 900 9500

10

20

30

40

Wavelength in nm

Shift

in u

m

Fig. 4.3. Double-Chirped Stack (App. G, 5-6): Reflectance, group delay, group

velocity in the x-direction, and spatial shift as a function of wavelength (solid lines

– exact calculations, dotted lines – approximations).

Fig. 4.4 and Fig. 4.5 graph the results for two resonator stacks that directly use a

wavelength-dependent stored energy to obtain dispersion as introduced in Fig. 2.1(c).

These two stacks are discussed in detail in Chapter 6.

49

810 820 830 840 8500.94

0.96

0.98

1

p-pols-pol

Wavelength in nm

Ref

lect

ance

810 820 830 840 8500

0.2

0.4

0.6

0.8


Wavelength in nm

Gro

up d

elay

in p

s

810 820 830 840 8500.27

0.28

0.29

0.3

0.31

0.32

Wavelength in nm

vgx

/ c

810 820 830 840 8500

20

40

60

80

Wavelength in nm

Shift

in u

m

Fig. 4.4. Gires-Tournois Resonator (App. G, 6-1): Reflectance, group delay, group



50

845 850 8550.9

0.92

0.94

0.96

0.98

1

p-pols-pol

Wavelength in nm

Ref

lect

ance

845 850 8550

0.1

0.2

0.3

0.4

0.5


Wavelength in nm

Gro

up d

elay

in p

s

845 850 8550.195

0.2

0.205

0.21

0.215

0.22

Wavelength in nm

vgx

/ c

845 850 8550

10

20

30

40

Wavelength in nm

Shift

in u

m

Fig. 4.5. Coupled-Cavity Stack (App. G, 6-2): Reflectance, group delay, group



Fig. 4.6 plots the reflectance, the group delay, the group velocity in the x-direction, and

the spatial shift for a numerically optimized thin-film stack. This type of stack is

discussed in detail in Chapter 7. We argued with Fig. 2.1(d) that such a stack uses a

combination of wavelength-dependent penetration depth and stored energy to generate

spatial dispersion. Here we have discussed that a wavelength-dependent penetration

depth can also be seen as a type of energy storage effect. Indeed, we can also

approximate the spatial shift quite well from the stored energy for this structure.

51

810 820 830 840 8500.985

0.99

0.995

1

p-pols-pol

Wavelength in nm

Ref

lect

ance

810 820 830 840 8500

0.1

0.2

0.3

0.4


Wavelength in nm

Gro

up d

elay

in p

s

810 820 830 840 8500.24

0.26

0.28

0.3

0.32

0.34

Wavelength in nm

vgx

/ c

810 820 830 840 8500

10

20

30

40

Wavelength in nm

Shift

in u

m

Fig. 4.6. Numerically Optimized Stack (App. G, 7-2) with Gold Coating on

Backside: Reflectance, group delay, group velocity in the x-direction, and spatial

shift as a function of wavelength (solid lines – exact calculations, dotted lines –

approximations).

Finally, Fig. 4.7 shows the results obtained for a four-channel step design discussed in

Chapter 10. As in the other figures we see that the difference between the shift calculated

using the exact transfer matrix calculation and the approximated shift is mainly caused by

the error in the group velocity approximation. Transmission loss leads to an

underestimating of the group delay.

52

1500 1520 1540 1560 15800.96

0.97

0.98

0.99

1

p-pols-pol

Wavelength in nm

Ref

lect

ance

1500 1520 1540 1560 15800

0.2

0.4

0.6

0.8


Wavelength in nm

Gro

up d

elay

in p

s

1500 1520 1540 1560 15800.22

0.23

0.24

0.25

0.26

0.27

0.28

0.29

Wavelength in nm

vgx

/ c

1500 1520 1540 1560 15800

10

20

30

40

50

60

Wavelength in nm

Shift

in u

m

Fig. 4.7. Four-Step Design (App. G, 10-1): Reflectance, group delay, group



In this chapter we have discussed the proportionality among spatial shift, group delay,

and stored energy for many structures of interest. The key result is that all the different

types of dispersive structures discussed in this work ultimately rely on a wavelength-

dependent stored energy. Therefore, the dispersion in all the structures has the same

physical origin. The art is to develop design algorithms for generating multilayer thin-

film stacks with the desired dispersion characteristics, i.e. with the desired change in the

stored energy as a function of wavelength. Fig. 4.3 to Fig. 4.6 already give an impression

53

of the different types of dispersion characteristics that can be obtained. We see several

structures with a linear spatial shift over different wavelength intervals and almost unity

reflectance. As discussed in Chapter 3, these characteristics are very desirable from a

practical point of view. The next three chapters will discuss three different design

algorithms that were used to generate these structures. We will also emphasize that even

though the structures use very different layer combinations to generate the dispersion, the

dispersion is in each structure based on a changing amount of stored energy with

wavelength. Before proceeding, we will finish this chapter by looking at the influence of

material dispersion on spatial dispersion and temporal dispersion in Section 4.4.

4.4 Influence of Material DispersionIn sections 4.1 to 4.3 we investigated the relationship between spatial shift, group delay,

and stored energy neglecting material dispersion, i.e., assuming that the refractive indices

ni are independent of frequency. The dispersion in the stacks in section 4.3 is due to the

multilayer structure and we refer to it as “structural” dispersion to distinguish it from

material dispersion. Here we discuss the relationship between material dispersion,

temporal dispersion, and spatial dispersion.

For frequency-dependent refractive indices, we need to use the partial derivative (4 - 14)

instead of (4 - 5) in the derivation of the approximate group velocity vgx,appr along the

layers.

( ) ( ) ( ) ( )

( )∑

−

+

−=∂

∂

= i

i

iii

i

const

appr

cn

nnn

cd

LK

22

2

dd

1,

ωβ

ω

ωω

ωωω

ω

ωβ

β

(4 - 14)

This results in the approximate group velocity vgx,appr given in (4 - 15).

( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( )∑

∑

−

−

+

=

iii

iii

iii

apprgx nd

ndn

nn

cv 22

222

, sin

sind

d

sin1

,1

θω

θωωω

ωωω

θθω(4 - 15)

54

We see that vgx,appr is no longer independent of frequency. Therefore, the group delay

τgroup and the shift sx in (4 - 1) are no longer proportional.

Furthermore, we need to include the material dispersion in the calculation of the stored

energy as given in (4 - 16), which replaces (C - 42) in Appendix C.10

( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( )

∂∂

++=

∂∂

+=

2220

220

,,,2,,41

,,,41,,

zxEznznznzxH

zxEzzxHzxwavg

ωω

ωωωµ

ωωεω

µω

(4 - 16)

Let us consider the effect of the material dispersion on the stored energy, the group delay,

and the spatial shift for GaAs. The refractive index and the extinction coefficient of GaAs

are given in Fig. 4.8.11

500 1000 1500 20003

3.5

4

4.5

Wavelength in nm

Ref

ract

ive

Inde

x

500 1000 1500 20000

0.2

0.4

0.6

Wavelength in nm

Extin

ctio

n C

oeff

icie

nt

(a) (b)

Fig. 4.8. (a) Refractive index and (b) extinction coefficient of GaAs as a function of

wavelength.

Fig. 4.9 plots the reflectance, the group delay, the group velocity in the x-direction, and

the spatial shift as a function of wavelength for a 10-µm layer of GaAs for θ=45°. The

incidence material is also GaAs and the backside is assumed to be coated with a material

with constant refractive index nAu=0.16-5i (value for gold at λ=880nm). As there are no

material interfaces up to the reflective coating, there is no structural dispersion. All of the

dispersion is due to material dispersion. Fig. 4.9 plots the approximate group delay

55

calculated using (4 - 9) and (4 - 16) for the stored energy. To obtain a real result, the real

part of the group delay is taken at the end of the calculation. We see that the group delay

is still proportional to the stored energy for unity reflectance in the case of material

dispersion. For wavelengths smaller than 880 nm, GaAs has significant absorption and

the group delay cannot be approximated from the stored energy.

500 1000 1500 20000

0.5

1

p-pols-pol

Wavelength in nm

Ref

lect

ance

500 1000 1500 20000

0.2

0.4

0.6


Wavelength in nmG

roup

del

ay in

ps

500 1000 1500 20000.01

0.02

0.03

0.04

0.05

0.06

Wavelength in nm

vgx

/ c

500 1000 1500 20000

1

2

3

4

5

6

Wavelength in nm

Shift

in u

m

Fig. 4.9. 10-µm layer of GaAs sandwiched between GaAs on the incidence side and

a backside gold coating: Reflectance, group delay, group velocity in the x-direction,

and spatial shift as a function of wavelength (solid lines – exact calculations, dotted

lines – approximations).

56

The group velocity vgx,appr is approximated by the value of (4 - 15) at 1240 nm. Fig. 4.9

shows that this is not a good approximation and that the group velocity has significant

frequency dependency. The approximated shift does not agree with the exactly calculated

shift. While the exactly calculated shift increases with wavelength, the approximation

predicts a decreasing shift with wavelength. We conclude that temporal dispersion and

spatial dispersion are not proportional if the dispersion is due to a frequency-dependent

refractive index. In fact, we find that for a uniform dispersive material with refractive

index nu(ω), the group velocity in the x-direction vgx,u(ω,θ) is given by (4 - 17).

( ) ( )( ) ( ) ( )

( )( )( )ω

θωθ

ωω

ωωω

θθω

ug

u

uuu

ugx nc

nnn

cv,2

,,sin

dd

sin, =+

= (4 - 17)

The group refractive index ng,u(ω) is defined by (4 - 18).12

( ) ( ) ( )ωω

ωωωd

d,

uuug

nnn += (4 - 18)

The propagation angle in the uniform medium θu is calculated from the incidence angle

in vacuum θ using Snell’s law. The group velocity in the z-direction vgz,u(ω,θ) is given by

(4 - 19).

( ) ( )( )( )ω

θωθθω

ug

uugz n

cv

,,

,cos, = (4 - 19)

Therefore, we can calculate the group delay τgroup,u for propagation of light over a

distance of 2L in the uniform medium as shown in (4 - 20).

( ) ( )( )

( )( )

( )( )

2

,,

,,

sin1

2),cos(

2,

2,

−

===

ωθ

ω

θωθ

ω

θωθωτ

u

ug

u

ug

ugzugroup

nc

nLc

nLv

L (4 - 20)

Approximating the square root in the denominator of (4 - 20) by unity, we see that the

group delay τgroup,u changes proportionally to the group index ng,u(ω) and is thus a group

velocity effect. The spatial shift along the x-direction sx,u is calculated in (4 - 21).

57

( ) ( ) ( ) ( )( ) ( )

( ) ( )( )

2,,,

sin1

sin2,tan2,,,

−

===

ωθ

ω

θθωθθωθωτθω

uu

uugxugroupux

nn

LLvs (4 - 21)

Again approximating the square root in the denominator of (4 - 21) by unity, we find that

the spatial shift sx,u is inversely proportional to the refractive index nu(ω) and is thus a

phase velocity effect. Fig. 4.10 plots the relative change of the refractive index nu, the

group refractive index ng,u, the shift along the x-direction sx,u and the group delay τgroup,u

in GaAs. It is clearly visible that the group delay is a group effect, while the spatial shift

is a phase effect in the case of material dispersion.

600 800 1000 1200 1400 1600 1800 20000.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

n/n0ng/ng0s0/sGD/GD0

Wavelength in nm

Rat

ios

Re( n (λ) ) / Re( n (1300 nm) )Re( ng (λ) ) / Re( ng (1300 nm) )Re( sx (1300 nm) ) / Re( sx (λ) )Re( τ (λ) ) / Re( τ (1300 nm) )

Fig. 4.10. Change of the refractive index n, the group refractive index ng, the shift

along the x-direction sx and the group delay τ in GaAs.

For the case of structural dispersion, we found that the group delay and the spatial shift

are proportional and that both are group velocity effects. The difference between material

dispersion and structural dispersion lies in the fact that material dispersion is isotropic,

58

i.e., the refractive index changes equally with frequency for all propagation directions (of

course, there are also anisotropic materials, which we do not consider here13). Therefore,

the group velocities in the x-direction and in the z-direction change proportionally and the

changes cancel for the spatial shift. In the case of structural dispersion, on the other hand,

we introduce a rapid variation of the group velocity in the z-direction with frequency due

to the layered structure, but there is no variation of the refractive index along the layers in

the x-direction.

Fig. 4.11 shows an example of the combined effect of structural dispersion and material

dispersion. A 500-nm layer of GaAs is sandwiched between an incident material with

n=2 and a backside gold coating (assumed to have a constant refractive index of

nAu=0.16-5i) for θ=45°. Using (4 - 16) for the stored energy, good agreement between the

exact and the approximate group delay result. With part of the dispersion being due to

structural dispersion, the shift is again better approximated by multiplying the group

delay by a constant group velocity value.

In conclusion, we see that material dispersion and structural dispersion have a different

influence on the temporal dispersion and the spatial dispersion. In the case of structural

dispersion, spatial shift, group delay, and stored energy are proportional for unity

reflectance. In the case of material dispersion, on the other hand, stored energy and group

delay are proportional, but the spatial shift is a phase effect and only occurs due to the

frequency-dependent refraction at the incident surface. For the dielectric stacks used in

this thesis the material dispersion is very low and can be neglected. All of the observed

dispersion is due to structural dispersion.

59

1000 1200 1400 1600 1800 20000.85

0.9

0.95

1

p-pols-pol

Wavelength in nm

Ref

lect

ance

1000 1200 1400 1600 1800 20000

0.01

0.02

0.03


Wavelength in nm

Gro

up d

elay

in p

s

1000 1200 1400 1600 1800 20000.03

0.04

0.05

0.06

0.07

0.08

Wavelength in nm

vgx

/ c

1000 1200 1400 1600 1800 20000

0.1

0.2

0.3

0.4

0.5

Wavelength in nm

Shift

in u

m

Fig. 4.11. 500-nm layer of GaAs sandwiched between an incident material with

n=2 and a backside gold coating: Reflectance, group delay, group velocity in the x-

direction, and spatial shift as a function of wavelength (solid lines – exact

calculations, dotted lines – approximations).

References

[1] R. Shankar, Principles of Quantum Mechanics, New York (1994).

[2] A. B. Migdal, Qualitative Methods in Quantum Theory, Addison-Wesley

Publishing Co., Inc. Redwood City, CA (1989).

60



[4] I.N. Bronstein, K.A. Semendjajew, G. Musiol, H. Muehlig, Taschenbuch der

Mathematik; Verlag Harri Deutsch, Thun und Frankfurt am Main, pp. 232-237

(1993).

[5] M. Gerken and D. A. B. Miller, “Multilayer thin-film structures with high spatial

dispersion,” Appl. Opt. 42/7 (2003), 1330-1345.

[6] P. Penfield, Jr., R. Spence, and S. Duinker, Tellegen’s Theorem and Electrical

Networks, M.I.T. Press, Cambridge (1970).

[7] C. Ernst, V. Postoyalko, N. G. Khan, “Relationship Between Group Delay and

Stored Energy in Microwave Filters,” IEEE Trans. Microwave Theory and Techn.

49/1 (2001).

[8] C. Ernst and V. Postoyalko, “Comments on “Relationship Between Group Delay

and Stored Energy in Microwave Filters”,” IEEE Trans. Microwave Theory and

Techn. 49/9 (2001).

[9] M. Gerken and D. A. B. Miller, “The relationship between the superprism effect,

group delay, and stored energy in 1-D photonic crystals and photonic

nanostructures,” MRS Spring Meeting, San Francisco, CA (April 21-25, 2003).

Paper J2.7.

[10] H. A. Haus, Waves and Fields in Optoelectronics, Prentice-Hall, Inc., Englewood

Cliffs, New Jersey (1984).

[11] S. L. Chuang, Physics of Optoelectronic Devices, John Wiley & Sons, Inc., New

York (1995).

[12] G. Grau and W. Freude, Optische Nachrichtentechnik, Springer-Verlag, Berlin

(1991).

[13] A. Yariv and P. Yeh, Optical Waves in Crystals, John Wiley & Sons, New York

(1984).

61

Chapter 5

Chirped Stacks

Temporal dispersion compensation plays an important role in femtosecond lasers and in

fiber communications. It has previously been observed that double-chirped mirrors are

well suited for femtosecond laser cavities as they have a broad reflection bandwidth and

can at the same time be used to compensate for temporal dispersion in the cavity.1 In a

Bragg mirror, light is only strongly reflected for wavelengths near the Bragg wavelength.

In a simple-chirped mirror, the Bragg wavelength is slowly changed as a function of the

position in the stack. If the period length increases coming from the incidence side,

longer wavelengths penetrate deeper into the structure and thus accumulate more delay

upon reflection. For the opposite chirp, shorter wavelengths penetrate deeper and the

opposite dispersion is obtained.

As mentioned in Chapter 3 the sudden turn-on of the periodicity leads to a reflection off

the front of the dielectric stack. In section 3.4 it was shown that this reflection can be

suppressed by adding an impedance matching chirp to the stack. A stack that has both a

chirp in the wavelength to obtain dispersion and an impedance matching chirp to prevent

62

loss is called a double-chirped mirror. An analytical design of such mirrors is possible

using an exact coupled-mode theory2 and a WKB-type approximation.3

Remembering that temporal dispersion and spatial dispersion are approximately

proportional as discussed in Chapter 4, we expect that the same type of algorithm can be

used to design a structure with spatial dispersion. First a double-chirped stack with

temporal dispersion is designed as described in [3]. A standard Bragg reflector may

follow the double-chirped mirror section to increase the reflectivity to unity. This normal-

incidence design is then modified for oblique incidence by taking into account that the

Bragg wavelength λB is given by (5 - 1).

( ) ( )

−+

−=

22sin1sin12

LLL

HHHB n

dnn

dn θθλ (5 - 1)

nH and nL are the refractive indices, and dH and dL are the thicknesses of the high and low

index materials respectively. θ is the incidence angle in vacuum. Fig. 5.1 compares the

performance of five different 60-layer structures designed in this manner. The layer

sequences for these stacks are given in Appendix G. Fig. 5.1(a) shows the chirped Bragg

wavelength as a function of the period number p calculated using (5 – 2) and using five

different factors f. The chirp law (5 – 2) is chosen such that a linear shift as a function of

wavelength is obtained.3

( )pf

nmpB 02541.01800

−=λ (5 - 2)

Using this chirped Bragg wavelength, the layer thicknesses of the single-chirped

structures are calculated using (5 - 3) and (5 - 4).

( ) ( )( ) 2

sin14

−

=

HH

BHSC

nn

ppdθ

λ (5 - 3)

63

( ) ( )( ) 2

sin14

−

=

LL

BLSC

nn

ppdθ

λ (5 - 4)

In order to achieve impedance matching, the first PDC periods are double chirped as given

in (5 - 5) and (5 - 6). In our calculations, 25 of the 30 periods are double-chirped.

( ) ( )( )

05.1

2sin14

−

=DC

HH

DCBHDC P

p

nn

Ppdθ

λ (5 - 5)

( )

( ) ( ) ( )

( ) 2

2

sin14

sin12

−

−−

=

LL

HHHDC

B

LDC

nn

nnpdp

pdθ

θλ

(5 - 6)

In Fig. 5.1(b) the spatial shifts as a function of wavelength are plotted for the five

designed structures. For clarity the shift is only plotted up to the wavelength at which it

stops increasing. We see that all five structures have an approximately linear shift with

wavelength as desired.

Structures with a larger chirp in the Bragg wavelength exhibit less dispersion but show

that dispersion over a larger wavelength range. The maximum shift is approximately

constant for the given allowed ripple. There is a tradeoff between the maximum shift and

the ripple4 in the shift for a given number of layers. That is, we could design a structure

with less ripple and less shift, or a structure with more ripple and more shift depending on

the application requirements.

Because the dispersion increases with decreasing chirp in the Bragg wavelength, the

maximum dispersion is achieved with a simple Bragg stack (though the ratio between

high and low index material in a period is still chirped for impedance matching). Note

that this linear shift appears within the stop-band of the single-chirped Bragg stack. It is

64

not identical to the non-linear shift observed just outside the stop-band as discussed in

Chapter 3.

10 20 30750

800

850

900

950

1000

1050

Period Number

Bra

gg W

avel

engt

h in

nm

850 900 950 1000 10500

5

10

Wavelength in nm

Shift

in u

m

Wavelength in nmSh

ift in

µm

Period Number

Brag

g W

avel

engt

h in

nm f=0.5

f=0.33f=0.2f=0.1f=0

Period

(a) (b)

Fig. 5.1. (a) shows the Bragg wavelength as a function of the position in the

structure for five different 60-layer SiO2/Ta2O5 double-chirped mirror designs. In

(b) the theoretical spatial shift as a function of wavelength is plotted for an

incidence angle of 45° and p-polarized light. An approximately linear shift is

observed for all five designs. The dispersion increases with decreasing chirp in the

Bragg wavelength. The maximum dispersion is achieved with a single-chirped

Bragg stack (f=0).

The double-chirped mirror design algorithm cannot be used to obtain a higher dispersion

for the given number of layers. This limitation of the dispersion is not a fundamental

physical limit for layered dielectric structures. As we will see in Chapter 6, structures

with larger dispersion can be designed. The limitation is rather due to the fact that, in this

algorithm, the degrees of freedom are limited, and only a subset of all possible structures

is considered. Specifically, the algorithm only allows for a monotonic change in the

65

Bragg wavelength. We can easily imagine that relaxing this requirement would allow for

other structures that might have a larger dispersion.

The calculations in Fig. 5.1 are done using the transfer matrix method given in Appendix

C applied to the center ray of the beam. This plane-wave approximation correctly predicts

the behavior of a Gaussian beam if the shift does not exhibit any rapid changes with

wavelength. Rapid oscillations in the shift are indicative of interference effects between

different beams.

An example of a structure with strong oscillations is the periodic structure discussed in

Chapter 3. The plane-wave approximation predicts strong oscillations in the shift with

wavelength, but with sufficient focussing, we actually see two separate beams – one that

is stationary corresponding to a reflection off of the front of the stack and one that

changes position with wavelength. These distinct beams cannot be predicted by the

plane-wave approximation and the full beam-behavior has to be modeled using, e.g., the

Fourier decomposition technique in Appendix D.

For the double-chirped structures discussed in this section on the other hand, the plane-

wave approximation can be used because all reflections except the desired shifted beam

are suppressed sufficiently well and no rapid changes appear in the dispersion. In

particular, the reflection off the front surface of the stack has been suppressed by the

impedance matching chirp.

To visualize the origins of dispersion, we plot the E-field amplitude of the forward

propagating wave as a function of the position in the structure. Since the structures we are

discussing here are nearly 100% reflecting and lossless, the amplitudes of the forward

and backward propagating waves are approximately equal for any specific wavelength.

Interference effects between the forward and backward propagating waves form a rapidly

oscillating standing wave pattern. The amplitude of the forward propagating wave

outlines the envelope of this standing wave pattern.

66

In Fig. 5.2(a) an example of a 200-layer double-chirped structure is shown. Appendix G

gives the layer composition for this stack. The structure is again designed to exhibit a

linear shift with wavelength which is well achieved as seen in Fig. 5.2(b). In Fig. 5.2(c)

we see that the field penetrates deeper into the structure with increasing wavelength,

resulting in both temporal and spatial dispersion.

Note a surprising behavior shown by the shift with this stack. Suppose we propagated a

beam through a piece of material with the same thickness as our dielectic stack (28 µm),

and a refractive index of 1.6, corresponding to the average index of our dielectric stack

material, and reflected it off of the very back of that material, corresponding to the largest

penetration possible. Then, from a simple calculation based on Snell’s law, we would

obtain a shift of 25 µm. But our shift here is 35 µm. Thus, the structure exhibits

additional dispersion that cannot be explained by the penetration depth. This extra

dispersion is due to stored energy.

In Fig. 5.2(c), larger than unity E-field amplitudes correspond to energy buildup. We

observe that only the shortest wavelength plot does not exhibit any energy buildup. For

the other wavelengths the field forms an Airy-type standing wave pattern as is usual for a

linearly changing potential. As was explained in Chapter 4, stored energy results in

dispersion explaining the larger than expected total shift in Fig. 5.2(b). Fig. 4.3 shows

that the observed shift can be approximated well from the total stored energy in the

structure.

Note also that this structure stores energy without the use of a front mirror, but solely due

to the shape of the potential. In other words, the structure stores energy without requiring

a resonant cavity in the conventional sense; certainly we cannot describe the energy

storage in terms of a cavity with two fixed mirrors. The position of the energy buildup

changes continuously with wavelength. This effect might be interesting for future active

and passive devices, since it has the energy build-up of a simple resonant effect without a

narrow resonant response.

67

0

2

4

0

2

4

0

2

4

0 5 10 15 20 25 300

2

4

780 800 820 840 860 880 900 920 9405

0

5

10

15

20

25

30

35

40

Wavelength in nm

Shift

in u

m

50 100 150 2000

100

200

300

SiO2 layersTa2O5 layers

Layer Number

Phys

ical

Thi

ckne

ss in

nm

780 nm

830 nm

880 nm

930 nm

(a)

(b)

(c)

Layer numberPhys

ical

thic

knes

s in

nm

Wavelength in nm

Shift

in µ

m

Depth in µm

E-fie

ld p

aral

lel t

o in

terfa

ce in

a.u

.

Fig. 5.2. (a) Physical layer thicknesses for a 200-layer SiO2/Ta2O5 double-chirped

structure. (b) Theoretically calculated shift as a function of wavelength at 40°

incidence angle and p-polarization. The circles indicate the wavelengths and shifts

corresponding to the diagrams in (c). (c) E-field parallel to the interface of the

forward propagating wave as a function of the position in the structure for four

different wavelengths – 780 nm, 830 nm, 880nm, and 930 nm. The vertical lines

indicate the position of the interfaces between layers. Light is incident from the left,

and the structure extends from 0 µm to 28 µm.

In conclusion we demonstrated in this section that the analytical design method3

developed for generating double-chirped mirrors for temporal dispersion compensation in

femtosecond laser cavities can be modified for designing structures with spatial

dispersion. This method is very useful for broadband designs with rather low dispersion,

but it is not suitable for designing structures with high dispersion. In Chapter 6 we will

68

examine a different design method that is particularly useful for narrowband designs with

high dispersion.

References



[2] N. Matuschek, F.X. Kärtner, and U. Keller, “Exact Coupled-Mode Theories for

Multilayer Interference Coatings with Arbitrary Strong Index Modulations,” IEEE

J. Quantum Electron., 33/3, 295-302 (1997).



Electron., 35/2, 129-137 (1999).

[4] M. Sumetsky, B. J. Eggleton, C. M. de Sterke, “Theory of group delay ripple

generated by chirped fiber gratings,” Opt. Express, 10/7, 332-340 (2002).

69

Chapter 6

Resonator Stacks

In this chapter we investigate resonator type structures, i.e., structures with a wavelength-

dependent amount of stored energy. As first introduced in Fig. 2.1(c) the wavelength-

dependent stored energy corresponds to a wavelength-dependent number of effective

round-trips through the resonator cavity, which leads to a wavelength-dependent spatial

shift of the exiting light. This intuitive picture was confirmed by the calculations in

Chapter 4 showing that the change in stored energy with wavelength is proportional to

temporal and spatial dispersion.

Simple examples of resonator structures are Fabry-Perot and Gires-Tournois

resonators.1,2,3,4 A Fabry-Perot resonator consists of a cavity between two partial

reflectors usually (but not necessarily) of equal reflectivity. At the resonant wavelength

the transmittance of the filter is unity (for equal reflectivities) and a maximum amount of

energy is stored in the cavity. A Gires-Tournois structure consists of one partial reflector,

a cavity, and a 100% reflector. Technically, such a resonator is just a special case of a

Fabry-Perot resonator. For the Gires-Tournois resonator the reflectance is unity at all

wavelengths though again the amount of stored energy is maximum at the resonant

70

wavelength. Moving away from the resonant wavelength, the amount of stored energy

decreases. The transmittance and reflectance for the respective structures can only ever

be unity in the absence of absorption. For structures with absorption, the loss is highest at

the resonant wavelength because the beam executes the maximum number of roundtrips.

10 20 300

2000

4000


Layer Number

Phys

ical

Thi

ckne

ss in

nm

0

2

4

6

0

2

4

6

0

2

4

6

2 0 2 4 6 8 100

2

4

6

810 815 820 825 830 835 840 845 8500

5

10

15

20

25

30

35

Wavelength in nm

Shift

in u

m

Layer number

Phys

ical

thic

knes

s in

nm

Wavelength in nm

Shift

in µ

m

Depth in µm

E-fie

ld p

aral

lel t

o in

terfa

ce in

a.u

.

815 nm

827 nm

830 nm

832 nm

(a)

(b)

(c)

Fig. 6.1. (a) Physical layer thicknesses for a 33-layer SiO2/Ta2O5 Gires-Tournois

resonator structure. (b) Theoretically calculated shift as a function of wavelength at

45° incidence angle and p-polarization. The circles indicate the wavelengths and

shifts corresponding to the diagrams in (c). (c) E-field parallel to the interface of

the forward propagating wave as a function of the position in the structure for four

different wavelengths – 815 nm, 827 nm, 830nm, and 832 nm. The vertical lines

indicate the position of the interfaces between layers. Light is incident from the left,

and the structure extends from 0 µm to 8 µm.

Fig. 6.1 gives an example of a Gires-Tournois structure. This structure consists of 33

layers. The layer sequence is listed in Appendix G. A 3.6-µm SiO2 cavity is surrounded

71

by a four-period quarter wave stack on the incident side and a 12-period quarter wave

stack serving as a near unity reflector. The stack has interfaces to air on both sides. In

Fig. 6.1(a) the physical layer thicknesses are graphed. Fig. 6.1(b) plots the resulting shift

as a function of wavelength for this structure. The calculations in Fig. 6.1 are done using

the transfer matrix method given in Appendix C applied to the center ray of the beam.

This plane-wave approximation correctly predicts the behavior of a Gaussian beam if the

shift does not exhibit any rapid changes with wavelength as discussed in Chapter 5. The

non-linear shift as a function of wavelength in Fig. 6.1(b) leads to beam distortions as

observed in Section 3.3.

In Fig. 6.1(c) the E-field amplitude parallel to the interface of the forward propagating

wave is plotted as a function of the position in the structure. Comparing the shift in

Fig. 6.1(b) with the energy buildup in Fig. 6.1(c), we can clearly see that a larger energy

buildup corresponds to a larger spatial shift as expected. The field penetrates the stack to

the same distance for all wavelengths. Therefore, the shift cannot be explained by a

geometrical penetration depth argument. Fig. 6.1(b) reveals an impressive total shift of 34

µm, which is much larger than the total stack thickness of 8 µm.

Resonators allow for high dispersion over a narrow wavelength range. Unfortunately, the

degrees of freedom in the design of a single-cavity resonator structure are very limited.

For example, in the case of the Gires-Tournois structure we can only choose the

reflectivity of the partial reflector and the cavity thickness. Not surprisingly, the shape of

the dispersion for all Gires-Tournois structures is very similar to the one depicted in

Fig. 6.1. In order to match desired dispersion characteristics more closely, the number of

cavities in the structure can be increased. An analytical procedure for designing such

coupled-cavity filters using a digital lattice technique has previously been established

both for microwave filters5,6,7 and thin-film structures.3,8

We are particularly interested in the design of allpass structures, i.e., structures with

constant unity reflectance and wavelength-dependent phase properties. Allpass filters

have previously been designed for temporal dispersion compensation.2,9 Here we will

72

show with an example that the analytical procedure can be modified to design structures

with spatial dispersion. Our goal is again to design a structure exhibiting a linear spatial

shift along the x-direction with wavelength. The example structure consists of five

reflectors with four cavities in between. The last reflector has unity reflectance. Thus, the

reflectance of the structure is 100% neglecting any absorption. The design procedure has

three stages – first the desired transfer function is approximated by an allpass polynomial,

next the reflection coefficients of the reflectors between the cavities are derived using an

order reduction technique, and finally the reflectors and cavities are implemented in a

physical structure.

Initially, all cavity lengths are assumed to be equal with a cavity round-trip time T. The

transfer function of an allpass filter with N cavities is then given by (6 - 1), where

z-1 = e-jωT. ANR(z) is the reverse polynomial of AN(z) of order N.

)()(

)(

0

0

zAzA

za

zazzH

N

RN

N

n

nn

N

n

nn

NAP ==

∑

∑

=

−

=− (6 - 1)

In the first step of the design process, a polynomial AN(z) is determined such that the

phase of HAP(z) approximates the shape of the desired phase response. The allpass

polynomial coefficient a0 is set to unity, which corresponds to the last reflector having a

reflection coefficient of unity amplitude. The resulting transfer function HAP(z) exhibits a

periodic behavior with the period given by the free spectral range FSR=1/T.3 Here we

want to design a structure with a linear shift as a function of wavelength. In Chapter 3 it

is shown that the shift along the exit interface is calculated by (6 – 2).

const

constK

gz

gxx K

Lvv

Ls=

=

∂∂

∂∂==

βω

βω22 (6 - 2)

L is the total stack thickness, vgx the group velocity in the x-direction, vgz the group

velocity in the z-direction, K and β are the wavevectors in the z- and in the x-direction,

and ω the frequency. K is related to the phase upon reflection φrefl by (6 – 3).

73

reflLK φ

21

= (6 - 3)

From Chapter 4 we know that the group velocity vgx along the layers is approximately

constant. Therefore, it follows from (6 – 2) and (6 – 3) that the derivative of the phase

φrefl should change approximately linearly with respect to frequency in order to obtain a

shift that is linear with frequency. We will initially design for a linear change of the shift

with frequency instead of wavelength. Due to the narrow bandwidth of the generated

designs, we can easily change this to wavelength in the last optimization step of the

procedure.

0 0.1 0.2 0.3 0.4 0.5

Desired characteristicDesign after 1. least-squares runFinal design

Normalized frequency

Phas

e de

rivat

ive

in a

.u.

Normalized Frequency

Phas

e de

rivat

ive

in a

.u.

Fig. 6.2. Results for approximating the desired phase characteristics by a fourth

order allpass polynomial.

The crosses in Fig. 6.2 show the desired phase derivative as a function of frequency. As

the resulting function is periodic with the free spectral range, the frequency normalized to

the free spectral range FSR can be used for simplicity. The fourth order allpass

approximation shown in Fig. 6.2 is obtained using a least squares algorithm.9,10 This is a

numerical algorithm for finding an allpass polynomial of the form (6 – 1) to approximate

the desired phase characteristics. The final coefficients of the polynomial are given in

(6 – 4).

74

=

153.0661.0453.1779.11

a (6 - 4)

Alternatively, an analytic algorithm based on the calculation of the cepstral coefficients

can be used for finding the allpass polynomial coefficients.11 This technique essentially

uses a truncated Fourier decomposition of the desired transfer function. Here I used the

first, numerical technique as I did not find out about the second technique until after I had

completely designed this example. The second technique is probably superior as it is

analytical and always generates realizable reflection coefficients, i.e. reflection

coefficients with amplitudes smaller or equal to unity.

850 900 9500

5

10

15

20

25

30

Lc=1.7 umLc=2.9 umLc=5.8 um

Wavelength in nm

Shift

in u

m

Wavelength in nm

Shift

in µ

m

0 2 4Normalized frequency

Phas

e de

rivat

ive

in a

.u.

Normalized Frequency

Phas

e de

rivat

ive

in a

.u.

(a) (b)

Fig. 6.3. (a) Periodic phase derivative of the transfer function as a function of the

normalized frequency. (b) The expected shift for a 4-cavity structure is plotted as a

function of wavelength for three different cavity optical thicknesses.

75

Fig. 6.3(a) plots the phase derivative of the approximate transfer function with the

coefficients a given in (6 - 4). Once an approximate transfer function is obtained, the

expected shift can estimated from (6 - 2) after choosing a cavity round-trip time T. The

effective group velocity vgx in the cavity is approximated from (4 - 8) choosing an

incidence angle of 45° and a refractive index of 2.09 for the cavity. Fig. 6.3(b) shows the

influence of the optical thickness of the cavity on the expected spatial shift with

wavelength. As the same transfer function HAP(z) is used for all three cavity lengths, the

shape of the shift with wavelength is the same. The total shift increases proportionately

with the cavity length, while at the same time the operating range decreases inversely

with the length. Thus, the dispersion is proportional to the square of the cavity length.

After choosing an appropriate cavity length to obtain the desired dispersion, we need to

realize the partial reflectors as thin-film structures. The reflection coefficients ri (i=1 to 4,

r5=1) of the reflectors are deduced using an order reduction technique. Starting with order

m=N, the order m of the polynomial Am(z) in (6 – 2) is reduced in each step by 1. The

coefficients of the polynomial Am(z) are am,n (n=0 to m) as defined in (6 - 1).The

algorithm for each order reduction step is given in (6 - 5) and (6 - 6).12

mmmN ar ,1 −=−+ (6 - 5)

2,

1,

1 1)()(

)(mm

mmmmm

m azzAazA

zA−

−=

−−

− (6 - 6)

For our 4-cavity sample structure, we want to realize an allpass filter HAP(z) with the

polynomial A4(z)=1+1.779z-1+1.453z-2+0.661z-3+0.153z-4 as seen from (6 - 4). In the first

step of the order reduction we set m=N=4. From (6 - 5) we calculate the reflectivity of the

first partial reflector as r4+1-4=r1 =-a4,4 =-0.153. Then we use the order reduction (6 - 6) to

obtain the polynomial of order m=4-1=3 to be A3(z)=1+1.718z-1+1.26z-2+0.399z-3. Using

again (6 - 4) we find the second partial reflector to have r4+1-3=r2 =-a3,3 =-0.399.

Continuing the order reduction, we calculate the reflection coefficients for all four partial

reflectors to have the values given in (6 – 7).

76

−−−−−

=→

=

1859.0683.0399.0153.0

153.0661.0453.1779.11

ra (6 - 7)

All reflection coefficients have to have amplitudes smaller or equal to unity. Otherwise,

the stack cannot be realized with a physical structure. Once the necessary reflection

coefficients are determined, the reflectors can be designed using standard thin-film design

procedures.4 Another approach is to approximate the reflectors by available quarter-wave

structures and correct for the error in the reflectivity by modifying the individual cavity

lengths.9 This approach is chosen here.

Table 6.1 lists the reflection coefficients for different quarter wave layer combinations.

As typical deposition systems monitor the growth of quarter wave stacks at 0° incidence

angle, we choose 980 nm quarter wave layers. These stacks will be reflective at 54°

incidence angle around 860 nm. Light is incident from the quartz substrate and the cavity

material is the high index material Ta2O5. Using Table 6.1 the first reflection coefficient

r1=-0.153 is for example approximated by a single low index quarter wave layer between

the substrate and the high index material. Similarly the other reflectors are chosen.

Layers r

SH 0.19

SLH -0.241

HLH -0.418

H(LH)2 -0.711

Layers r

H(LH)3 -0.869

H(LH)4 -0.943

H(LH)5 -0.974

H(LH)10 -0.997

Table 6.1. Reflection coefficients for different λB/4-layer combinations

(λB=980nm) of the substrate S (ns=1.52), high index material H (nH=2.06,

dH= λB/4nH), and low index material L (nL=1.456, dL= λB/4nL) at 860nm, 54°

incidence angle, and s-polarization.

For our 4-cavity example we choose an optical cavity round-trip length of 5.8 µm, Ta2O5

as the cavity material, s-polarized light, and an incidence angle of 54°. This corresponds

77

to a physical thickness of 2.6 µm for each cavity. After designing the reflectors it is

important to subtract the optical thickness of each reflector from the corresponding

optical cavity length, since they constitute part of the round-trip time. The necessity to

implement the reflector within the optical thickness of the cavity sets a lower limit on the

cavity thickness and thus the maximum achievable operating wavelength range. On the

other hand, the cavity length cannot be chosen too large, as field interference is necessary

for the operation of the device. If the distances between the reflectors are too large, a

pulse is split into several pulses and an optical rattler is obtained.13

842 844 846 848 850 852 8540

5

10

15

20

25

Desired shiftWavelength in nm

Shift

in u

m

Wavelength in nm

Shift

in µ

m

Approx. Sub L [H]24.88 L [H]24.88 LHL [H]22.98 (LH)2L [H]21.07 (LH)9L AirRefined Sub L [H]24.84 L [H]24.94 LHL [H]22.94 (LH)2L [H]20.98 (LH)9L Air

Fig. 6.4. Shift as a function of wavelength at 54° incidence angle and s-polarization

for the design with approximate reflectors (blue) and the final refined design

(black). Below the graph, the quarter wave layer sequences for both designs are

given. Remember that the quarter wave layers are for a wavelength of 980 nm and

0° incidence angle.

Fig. 6.4 plots the shift as a function of wavelength for the design with the approximate

reflectors in blue. Furthermore, the resulting layer sequence in terms of quarter wave

layers is listed as well. Note the different cavity lengths due to subtracting the optical

78

thicknesses of the reflectors from the cavity thicknesses. Due to the approximate

reflectors, the shift is not linear any more. As the final step of the design the cavity

thicknesses are numerically optimized to achieve a linear shift as a function of

wavelength. The optimization was performed using the conjugate gradient algorithm with

the Hestenes-Stiefel formula (see Chapter 7 and Appendix E for details on numerical

optimization techniques).14 Fig. 6.4 plots the resulting linear shift of the refined design

and gives the final layer sequence. The resulting SiO2/Ta2O5 thin-film structure has 33

layers and a total thickness of 15.4 µm. The layer composition is given in Appendix G.

Fig. 6.5 investigates the origin of the dispersion in more detail. In Fig. 6.5(a) the physical

layer thicknesses are graphed. The four cavities separated by quarter wave reflectors are

visible. The layer structure is also given in Appendix F. Fig. 6.5(b) plots the resulting

shift as a function of wavelength for this structure using a plane wave transfer matrix

calculation. A linear shift is obtained over a narrow wavelength range demonstrating that

this technique enables the design of narrowband structures with high dispersion.

In Fig. 6.5(c) the E-field amplitude parallel to the interface of the forward propagating

wave is plotted as a function of the position in the structure. For longer wavelengths a

larger amount of energy build-up occurs in the structure. As discussed for the single-

cavity resonators, this wavelength-dependent energy storage causes spatial and temporal

dispersion. We also see that for a structure with more than one cavity, part of the

dispersion can be attributed to a wavelength-dependent penetration depth.

79

(a)

(b)

(c)

842 844 846 848 850 852 8540

5

10

15

20

25

Wavelength in nm

Shift

in u

m

10 20 300

1000

2000

3000


Layer Number

Phys

ical

Thi

ckne

ss in

nm

Layer number

Phys

ical

thic

knes

s in

nm

0

2

4

0

2

4

0

2

4

0 5 10 150

2

4

842 nm

846 nm

850 nm

854 nm

Wavelength in nm

Shift

in µ

m

Depth in µm

E-fie

ld p

aral

lel t

o in

terfa

ce in

a.u

.

Fig. 6.5. (a) Physical layer thicknesses for a 33-layer SiO2/Ta2O5 4-cavity structure.

(b) Theoretically calculated shift as a function of wavelength at 54° incidence angle

and s-polarization. The reflectance of the structure is 100%. The circles indicate the

wavelengths and shifts corresponding to the diagrams in (c). (c) E-field parallel to

the interface of the forward propagating wave as a function of the position in the

structure for four different wavelengths – 842 nm, 846 nm, 850nm, and 854 nm.

The vertical lines indicate the position of the interfaces between layers. Light is

incident from the left, and the structure extends from 0 µm to 15.4 µm.

In conclusion, we demonstrated in this section that coupled-cavity allpass filters can be

designed to exhibit linear spatial dispersion using a digital lattice filter technique.

Furthermore, the technique can be employed to design structures with a non-monotonic

shift as a function of wavelength. The major practical limitation we found in this

approach is the necessity to design reflectors with very low reflectivity if a larger number

of cavities is desired. For example, a linear 8-cavity design calls for five reflectors with

80

less than 10% reflectivity, of which three should have less than 2% reflectivity.

Therefore, it becomes quite difficult to design appropriate thin-film reflectors for

structures with a higher number of cavities. As the only way to achieve a larger free

spectral range without decreasing the total spatial shift is to decrease the cavity length

and thus increase the number of cavities, this technique is in practice limited to the design

of narrowband structures. Again this is a limitation of this specific design technique;

using a different design technique, structures with dispersion over a broad wavelength

range can be designed as discussed in Chapter 5. Only a subset of all possible structures

is considered in this chapter – structures with large cavities separated by reflectors. In the

next chapter a third method for designing non-period multilayer stacks will be discussed

that is not limited to a particular type of structure, e.g., chirped or resonator. This more

general method based on numerical optimization permits more degrees of freedom and

therefore more flexibility in the design.

References

[1] I. Walmsley, L. Waxer, C. Dorrer, “The role of dispersion in ultrafast optics,” Rev.

of Scientific Instr., 72/1, 1-29 (2001).




Processing Approach, John Wiley & Sons, Inc., New York (1999).

[4] H. A. MacLeod, Thin-Film Optical Filters, Institute of Physics Publishing,

Philadelphia (2001).

[5] G. Matthaei, E. M. T. Jones, L. Young, Microwave filters, impedance-matching

networks, and coupling structures, Artech House (1980).

[6] I. Hunter, Theory and design of microwave filters, The Institution of Electrical

Engineers (2001).

[7] J.-S. Hong, M. J. Lancaster, Microstrip filters for RF/microwave applications,

Wiley-Interscience (2001).

81

[8] E. M. Dowling and D. L. MacFarlane, “Lightwave Lattice Filters for Optically

Multiplexed Communication Systems,” J. Lightwave Techn., 12/3, 471-486 (1994).

[9] M. Jablonski, Y. Takushima, K. Kikuchi, “The realization of all-pass filters for

third-order dispersion compensation in ultrafast optical fiber transmission systems,”

J. Lightwave Techn., 19/8, 1194-1205 (2001).

[10] M. Lang and T. I. Laakso, “Simple and Robust Method for the Design of Allpass

Filters Using Least-Squares Phase Error Criterion,” IEEE Trans. Circ. and Syst. II,

41/1, 40-48 (1994).

[11] K. Rajamani and Y.-S. Lai, “A novel method for designing allpass digital filters,”

IEEE Signal Proc. Lett., 6/8, 207-209 (1999).

[12] A. H. Gray, Jr. and J. D. Markel, “Digital Lattice and Ladder Filter Synthesis,”

IEEE Trans. Audio and Electroacoustics, AU-21/6, 491-500 (1973).

[13] V. Narayan, E. M. Dowling, D. L. MacFarlane, “Design of multimirror structures

for high-frquency bursts and codes of ultrashort pulses,” IEEE J. Quantum

Electron., 30/7, 1671-1680 (1994).

[14] E.K.P. Chong and S.H. Zak, An Introduction to Optimization, John Wiley & Sons,

Inc. (1996).

82

Chapter 7

Numerically Optimized Stacks

Chapter 4 demonstrates that the spatial dispersion of a multilayer stack is proportional to

the wavelength-dependent stored energy in that stack. Therefore, the task of designing the

spatial dispersion properties of a stack corresponds to designing for an appropriate

amount of stored energy at different wavelengths. Chapters 5 and 6 discussed two

methods for designing the dispersion characteristics of thin-film structures. The chirped

stacks in Chapter 5 result in a wavelength-dependent penetration depth with more energy

stored for further penetration. The resonator stacks in Chapter 6 store energy using

cavities that are separated by reflectors. Both types of stacks can be designed using a

mainly analytical design algorithm. Both algorithms allow for the design of a linear

spatial shift with wavelength in contrast to the periodic structures discussed in Chapter 3.

The algorithm for chirped stacks in Chapter 5 results in broadband designs, while the

coupled-cavity structures designed in Chapter 6 using digital signal processing techniques

are narrowband. The different bandwidths of the two structure types results from the fact

that each method considers another subset of all possible structures –chirped structures

and coupled-cavity structures respectively. This limitation to a particular type of structure

83

limits the available degrees of design freedom and thus the achievable dispersion

characteristics. In the case of the chirped stacks in Chapter 5, we limited ourselves, e.g.,

to stacks consisting of pairs of layers with a monotonically changing combined optical

thickness. It is clear that this limitation affects the dispersion characteristics that can be

achieved, limiting the algorithm for chirped stacks to broadband designs. A similar

argument holds for the resonator stacks in Chapter 6.

In this chapter we investigate how a thin-film structure can be designed without limiting

the degrees of freedom by just considering a specific type of structure. For fabrication

purposes it normally does not matter if the structure is double-chirped, coupled-cavity or

something completely different (though simple periodic structures are somewhat easier to

make because monitoring during growth is easier to interpret). More likely the total

thickness of the structure, the minimum and maximum thickness of individual layers, and

the choice of materials set the limits to what can be fabricated. Therefore, we would

ideally investigate all possible structures that can be fabricated, and find the one that most

closely matches our desired dispersion characteristics. Unfortunately, the computation

time of this approach becomes prohibitive for more than a couple of layers. This is a

common problem in the design of thin-film filters with specified reflectance and

transmittance characteristics.1,2,3,4,5,6 Here we explore how numerical techniques

developed for the design of thin-film filters can be applied to the design of thin-film

structures with spatial dispersion.

7.1 Designing Stacks using Numerical OptimizationThe design of thin-film filters is normally divided into two steps. In the first step a start

design is synthesized that approximately fulfills the required characteristics.5 In the

second step, numerical optimization procedures are used to gradually improve the

performance of the start design.6 The performance of a design is measured by a merit

function MF – a single number comparing the current design characteristics with the

desired design characteristics.7 The definition of the merit function we use is given in

(7 - 1). Qi is the current value of a quantity of interest, QiT the target value of that

84

quantity, ∆Qi the acceptable deviation, N is the number of sampling points, and p the p-

norm used.8 In the case of p = 2, the merit function is the root-mean-square difference

between the current values and the target values of the quantities of interest.

pN

i

p

i

iTi

QQQ

NMF

/1

1

1

∆−

= ∑=

(7 - 1)

The calculation of the merit function is by no means limited to reflectance or

transmittance values. In the same manner we can specify a desired spatial shift as a

function of wavelength and judge the performance of the current design by sampling the

shift at different wavelengths. Therefore, the same numerical refinement techniques used

to design thin-film filters can be applied to the design of thin-film structures with spatial

dispersion. Any one of the analytical methods discussed in Chapters 3, 5, and 6 can be

used to generate a starting design. In our experience the most important property of the

starting design is to eliminate strong ripple, i.e., to provide impedance matching. Rapid

oscillations in the shift with wavelength are difficult to remove by numerical

optimization. We had more success in starting with a low but constant dispersion and

increasing it during optimization. To ensure a constant dispersion, we normally calculate

the merit function MF employing a p-norm with p between 6 and 10. This large p-norm

ensures an approximately identical error in the shift for all wavelengths.8 Thus, the

resulting shift is, for example, linear but offset from the originally specified shift. The

sampling points have to be chosen close enough in wavelength to prevent oscillations

between the points. A spacing of around 2 nm at 850 nm appears to work well.

We implemented six different numerical optimization techniques – golden section

search,8 secant method,8 conjugate gradient algorithm,8 Broyden-Fletcher-Goldfarb-

Shanno (BFGS),8 damped least squares method,6 and Hooke&Jeeves pattern search.9 A

brief discussion of these techniques and a comparison of the results obtained is given in

Appendix E. The first two methods are one-dimensional search methods, i.e., the

different parameters are optimized sequentially. The other methods vary all parameters

simultaneously. All methods search for the local minimum of the merit function, where a

merit function with p=2 is used for the damped least squares method. A lower non-local

85

optimum may be found though by, e.g., choosing the interval size of the golden section

search large, or by taking large steps in the Hooke&Jeeves pattern search technique. See

Appendix E for more details on this topic. Our design algorithm uses the different

numerical optimization techniques sequentially. This is successful as a different design

method may find a lower minimum if one method is “stuck” in a shallow local optimum.

Numerical optimization allows the design of structures with dispersion characteristics

that cannot be achieved using the analytical techniques discussed in the previous sections.

Layer Number

Phy

sica

l Thi

ckne

ss in

nm 10 20 30 40 50 600

150

300

450

600

10 20 30 40 50 600

150

300

450

600

10 20 30 40 50 600

150

300

450

600

SiO2Ta2O5

(a)

(b)

(c)

Fig. 7.1. Generation of the starting design. (a) Quarter wave Bragg stack. (b)

Impedance matched stack. (c) Half-wave layers added to thinnest layers.

As an example we designed a 66-layer, 23.7-µm thick, thin-film structure with a linear

shift over a 40-nm wavelength range around 1550 nm (corresponding approximately to a

typical Er-fiber amplifier bandwidth for the C-band). The composition of this designed

stack is given in Appendix G. This type of medium-wide wavelength range is difficult to

86

achieve with the double-chirped structures in Chapter 5 or the coupled-cavity structures

in Chapter 6. We used an impedance matched Bragg stack as the start design. Fig. 7.1(a)

shows the initial quarter-wave Bragg stack, while the impedance matching can be seen in

Fig. 7.1(b). Light is incident from the left side onto the stack. Half-wave layers are added

to the thinnest layers of the structure to facilitate fabrication as seen in Fig. 7.1(c).

Fig. 7.2 plots the shift and reflectance as a function of wavelength for the start design

given in Fig. 7.1(c). The start design has unity reflectance and nearly no spatial

dispersion.

1500 1520 1540 1560 1580 16000

10

20

30

40

50

Wavelength in nm

Shift

in R

efle

ctio

n in

um

1500 1520 1540 1560 1580 16000

0.2

0.4

0.6

0.8

1

Wavelength in nm

Wavelength in nm

Shift

in µ

m

Wavelength in nm

Ref

lect

ance

Start designRefined designRefined design with gold mirrorTargeted linear shiftTargeted linear shift (shifted)

(a) (b)

Fig. 7.2. (a) Shift as a function of wavelength and (b) reflectance as a function of

wavelength for the start design, the refined design, and the refined design with

backside gold coating at 45° incidence angle and p-polarization.

We refined the design specifying a linearly increasing spatial shift for 21 different

wavelengths as seen in Fig. 7.2(a). Using a p-norm of 8 the linearly increasing shift is

obtained with only a small amount of ripple. Due to the high p-norm the shift of the

refined design is offset from the initially specified shift. This is not important, as only the

difference in the shift between wavelengths, i.e., the constant dispersion, matters. It can

be seen from the shifted target line that the desired dispersion is obtained. Fig. 7.2(b)

shows that the reflectance of the refined design drops around 1580 nm. In order to

achieve a high reflectance for all wavelengths, a gold coating can be applied to the last

87

layer. To prevent loss, the last layers of the structure could also be specified as a Bragg

stack. In our design algorithm we can set which layers are to be changed and what the

minimum and maximum layer thicknesses are. This guarantees that the design can be

fabricated. The performance of the stack with backside gold coating is also plotted in

Fig. 7.2. A linear shift with wavelength over the entire EDFA C-band is obtained with

high reflectance for all wavelengths.

810 815 820 825 830 835 840 8450

5

10

15

20

25

Wavelength in nm

Shift

in u

m

20 40 600

100

200

300

400

500


Layer Number

Phys

ical

Thi

ckne

ss in

nm

0

2

4

6

0

2

4

6

0

2

4

6

2 0 2 4 6 8 10 12 140

2

4

6

821 nm

828 nm

835 nm

842 nm

(a)

(b)

(c)

Layer number

Phys

ical

thic

knes

s in

nm

Wavelength in nm

Shift

in µ

m

Depth in µm

E-fie

ld p

aral

lel t

o in

terfa

ce in

a.u

.

Fig. 7.3. (a) Physical layer thicknesses for a 66-layer, numerically optimized

SiO2/Ta2O5 structure. (b) Theoretically calculated shift as a function of wavelength

at 54° incidence angle and p-polarization. The reflectance of the structure is

improved to nearly 100% by a gold layer on the very right. The circles correspond

to the diagrams in (c). (c) E-field parallel to the interface of the forward

propagating wave as a function of the position in the structure for four different

wavelengths – 821 nm, 828 nm, 835nm, and 842 nm. The vertical lines indicate the

88

position of the interfaces between layers. Light is incident from the left, and the

structure extends from 0 µm to 13.4 µm.

For experimental testing, this design was scaled from 1550 nm to 830 nm as we have a

tunable laser available around 830 nm. In the fabricated design the minimum layer

thickness is 57 nm, and the maximum thickness is 453 nm. Six calibration layers were

added in the front, which is the substrate side. The composition of stack is given in

Appendix G. In Fig. 7.3(a) the physical layer thicknesses are plotted. Fig. 7.3(b) shows

the shift as a function of wavelength for the 66-layer structure with a backside gold

coating. A highly linear shift is obtained. To investigate the origins of the observed

dispersion, in Fig. 7.3(c) the E-field parallel to the interfaces is plotted as a function of

the position in the stack for four different wavelengths. We see that the penetration depth

increases for longer wavelengths and more energy is stored at the same time. Thus, the

dispersion of this general structure is based on both phenomena – a wavelength-

dependent penetration depth and resonant energy storage.

7.2 Experimental Results for an Optimized StackFor the fabrication of the numerically optimized stack the deposition rate was determined

using the first six quarter-wave calibration layers and the remaining design was fabricated

by timed deposition. No active monitoring was used. Systematic deposition errors, i.e.

systematic relative errors in the layer thicknesses, occur if the calibration predicts the

wrong deposition rate. Such systematic errors in the layer thickness lead to a change in

the wavelength as well as the dispersion profile. First experiments with the fabricated

stack at 45° incidence angle did not show a linear shift as a function of wavelength. By

varying the layer thicknesses systematically and comparing the simulation results to the

experimental results, we found that the SiO2-layers of the fabricated stack were 4.3% too

thick, while the Ta2O5-layers were 0.3% too thick. Changing all the layer thickness of the

design by these factors, we obtain good agreement between experiment and simulation.

Appendix G gives the layer thicknesses for the scaled, fabricated stack. The fabricated

stack has a linear shift for an incidence angle of 54°. This is why all experimental results

are for 54° instead of the designed 45°. Random absolute errors, which are for example

89

caused by the shutter closing time, should not be larger than approximately ±1 nm in

order to obtain a working design. Random relative errors need to be less than ±0.3 % of

the layer thickness. Because we see good agreement between experiment and theory once

the systematic errors are taken into account, the random errors seem to be sufficiently

small.

The experiment is again conducted using the setup shown in Fig. 3.3. Fig. 7.4 shows the

intensity observed along a CCD trace as a function of wavelength. Contrary to the

experiment discussed in Chapter 3, only one beam exits the structure, which demonstrates

the successful suppression of all other reflections by impedance matching. The linear

beam shifting with wavelength is clearly visible.

Wavelength in nm

Posi

tion

in µ

m

Quartz substrate

66-layer SiO2-Ta2O5thin-film stack


position and wavelength for a 66-layer numerically optimized stack for an

incidence angle of 54° and p-polarization.

To analyze the spatial shift more closely, a Gaussian beam profile is fitted to the

experimental data at each wavelength adjusting the beam amplitude, beam center

position, and beam width. The experimentally observed shift of the peak as a function of

wavelength is plotted in Fig. 7.5. The linear shift between 820 nm and 840 nm is clearly

visible. As the shift does not exhibit any rapid changes in this wavelength range, the

plane-wave approximation predicts the experimentally observed shift accurately. For

90

wavelengths larger than 840 nm, only the beam simulation correctly predicts the

observed shift. The plane wave calculation is not accurate due to the rise and fall of the

shift at 843 nm. Different components observe a different shift as discussed in Chapter 3.

The slight difference between the shift in Fig. 7.3(b) and Fig. 7.5 is due to the fact that

the first calculation includes the backside gold coating and the second does not. The

overall excellent agreement between the experimentally observed shift and the

theoretically expected shift confirms the concepts discussed above for obtaining spatial

dispersion.

820 825 830 835 840 8450

5

10

15

20

25

Plane-Wave TheoryExperimentBeam Simulation

Wavelength in nm

Shift

in u

m

Fig. 7.5. Experimentally observed and theoretically calculated spatial dispersion of

a 66-layer SiO2/Ta2O5 dielectric stack with a total thickness of 13.4 µm on a quartz

substrate for an incidence angle of 54° and p-polarization.

7.3 Comparison of Periodic and Non-Periodic DesignsIn Chapters 3, 5, 6 and 7 we discuss four different ways of designing structures with high

spatial dispersion. In Chapter 3 we demonstrated the strong spatial dispersion of periodic

thin-film structures close to the stop-band edge both in theory and experiment.

Unfortunately, periodic structures with two layers per period do not offer enough degrees

of freedom to design structures with desired dispersion characteristics, e.g., a linear shift

91

with wavelength. In Chapter 5 we showed that the wavelength-dependent penetration

depth of double-chirped structures can be used to obtain spatial dispersion, and that these

structures are particularly useful for broadband designs. In this section we also found that

non-resonant energy storage has an important contribution to the shift, and that the

obtained shift cannot be explained from pure geometrical reasoning. The coupled-cavity

structures discussed in Chapter 6 employ a wavelength-dependent amount of stored

energy to obtain dispersion, and are very effective for narrowband designs. Finally, in

this chapter we demonstrated in theory and experiment that structures using a

combination of a wavelength-dependent turning point and stored energy can be designed

using numerical optimization methods.

1520 1525 1530 1535 1540 1545 1550 1555 1560 1565

0

10

20

30

40

Shift

in u

m

200-layer periodic (experiment)200-layer periodic (theory)66-layer non-periodic (experiment)66-layer non-periodic (theory)

Wavelength in nm

Shift

in µ

m

Fig. 7.6. Comparing the performance of a periodic and a non-periodic structure.

Fig. 7.6 compares the results obtained for the periodic structure from Chapter 3 and the

non-periodic structure from this chapter. The results are scaled to the 1550-nm

wavelength range for better comparison with current commercial interests for

telecommunication. The non-periodic structure exhibits a total shift comparable to that of

the periodic structure. But the same shift is obtained with just a third of layers, and the

observed shift is linear with wavelength, which is much more desirable for practical

applications. Furthermore, the non-periodic structure is essentially loss-less as all loss

92

reflections (e.g., front surface reflection) are suppressed. Fig. 7.6 clearly demonstrates the

superior dispersion characteristics of the non-periodic stack and shows why it is worth

the effort to explore new types of structures besides the photonic crystal superprism

effect.

7.4 Experiment: 4-Channel Wavelength DemultiplexerSo far we have discussed the spatial shift obtained after a single bounce off the stack. The

shift of approximately 20 µm in Fig. 7.5 is not sufficient though to separate multiple

channels of different wavelengths except if these are focussed very tightly. As discussed

in Chapter 1 several bounces can be performed off the stack in order to increase the

spatial separation between beams of different wavelengths. As we are operating the stack

through the substrate, a gold coating can easily be deposited on the opposite substrate

side allowing for multiple bounces as seen in Fig. 7.8.

Wavelength in nm

Posi

tion

in µ

m

828 830 832 834 836 838 840 8420

50

100

150

TheoryExperiment

Wavelength in nm

Shift

in u

m

Wavelength in nm

Shift

in µ

m

(a) (b)

Fig. 7.7. (a) Experimentally observed intensity on a CCD trace as a function of

position and wavelength for 8 bounces off a 66-layer numerically optimized stack.

(b) Experimentally observed and theoretically calculated shift as a function of

wavelength.

In Fig. 7.7(a) the intensity along one CCD trace is plotted as a function of wavelength for

eight bounces off the numerically optimized stack introduced in section 7.2 without the

93

backside gold coating applied. Fig. 7.7(b) compares the experimentally observed shift to

the shift calculated using the plane-wave calculation and multiplying by eight. We see

that the shift is indeed eight times as large for eight bounces off the stack as expected.

0 20 40 60 80 100 120 140 160

Position in µm

Am

plitu

de in

a.u

.

827 .3 nm

831.0 nm

834.7 nm

840.6 nm840.6 nm834.7 nm831.0 nm827.3 nm

zy

x

827.3 nm831.0 nm834.7 nm840.6 nm

Multiplexedlight

1 mm 13.4 µm

≈10

mm

…

Fig. 7.8. 4-channel wavelength demultiplexer.

Fig. 7.8 demonstrates that this shift can be used to separate four beams by their Gaussian

beam width.10 The first three beams have a channel spacing of 3.7 nm. The spacing

between the third and fourth channels, 5.9 nm, is chosen to be larger because of the

reduced dispersion. The amplitudes of the beams are decreasing with wavelength because

we have not yet applied the gold coating to the stack side. The dielectric stack itself has a

lower reflectance for larger wavelengths. The slight beam distortions visible in Fig. 7.8

are due to beam clipping in the experiment and would not appear in a properly designed

module. In Chapter 8 the question of how many channels can be separated using a given

stack with spatial dispersion is investigated, i.e., could the stack designed here separate

more than four channels and is there an ultimate limit to the number of channels that can

be separated.

94

References

[1] P. Baumeister, “Design of multilayer filters by successive approximations,” J. Opt.

Soc. Am., 48/12, 955-958 (1958).

[2] J. A. Dobrowolski, “Completely automatic synthesis of optical thin film systems,”

Appl. Opt., 4/8, 937-946 (1965).



[4] A. Thelen, Design of Optical Interference Coatings, McGraw-Hill, Inc., New York

(1989).





references herein.

[7] J. A. Dobrowolski, F.C. Ho, A. Belkind, V.A. Koss, “Merit functions for more

effective thin film calculations,” Appl. Opt., 28/14, 2824-2831 (1989).


Inc. (1996).

[9] T.E. Shoup and F. Mistree, Optimization Methods with Applications for Personal

Computers, Prentice-Hall, Inc. (1987).

[10] M. Gerken and D. A. B. Miller, “Thin-Film (DE)MUX based on group-velocity

effects,” ECOC 2002, Paper 11.3.3, Copenhagen, Denmark (September 8-12,

2002).

95

Chapter 8

Maximum Number of Channels

Chapters 3 to 7 discussed how to design multilayer stacks with high spatial dispersion.

This chapter now investigates how many channels can be multiplexed or demultiplexed

using a given multilayer stack. In this chapter, spatially separable channels are referred to

as “modes” of the structure. First we need to define what is meant by a mode of the

structure. As we are investigating structures with a continuous shift as a function of

wavelength, all wavelengths can propagate through the structure. A mode is therefore not

defined by the fact that it propagates through the structures whereas other modes do not.

Hence this definition is unlike that of modes in a waveguide. A mode is also not defined

by the fact that it can be physically distinguished from other modes. Since different

wavelengths are associated with the different modes, they can always be distinguished.

Our definition of a mode is related to the spatial extent of a channel without considering

its wavelength. Rigorously, perhaps, our modes are beam forms that are spatially

orthogonal. Beams that do not overlap at all are certainly spatially orthogonal and other

orthogonal patterns might also be possible, e.g., as in the communication modes of [1].

Beams that mostly do not overlap can be considered approximately orthogonal, and do

96

lead to the right overall counting of such modes as discussed in [1]. This definition is

based on the idea that we want to use the dispersion of multilayer structures to spatially

separate channels of different wavelengths, i.e. we want to measure the power of a

demultiplexed channel without considering its wavelength. The question of the number of

modes of a given stack is therefore equivalent to the question of how many separate

channels can we demultiplex assuming wavelength-insensitive detectors.

Two possible types of spatial modes are considered here as depicted in Fig. 8.1. “Volume

modes” are defined as being modes that have mutually exclusive propagation cones

within the volume of the stack as shown in Fig. 8.1(a). “Surface modes” may overlap

within the volume of the stack, but are separated on the exit surface as seen in Fig. 8.1(b).

(a) (b)

Fig. 8.1. (a) Volume modes. (b) Surface modes.

We chose these two mode definitions for their practical interest – the volume mode

picture corresponds to focusing the beam on the input surface, while the surface mode

picture corresponds to focusing on the exit surface. Other mode definitions could be

chosen equally well. Note that ultimately the number of resolvable modes does not

depend on the position of the focal point, but only on the structure.

The number of volume modes and the number of surface modes are calculated and

compared here for two different example designs – a periodic stack and a numerically

optimized stack that has a group propagation angle changing linearly with wavelength.

Both stacks are composed of 200 alternating layers of SiO2 (n=1.45) and Ta2O5 (n=2.09)

97

on a quartz substrate (n=1.52). The layer sequences are given in Appendix G. In the first

design all layers are 261 nm thick. The backside of the stack is gold-coated to increase

the reflectivity. This design uses the superprism effect discussed in Chapter 3 to obtain

spatial dispersion. This design will be called the “Periodic Design.” The second design is

a numerically optimized design. The method for designing double-chirped mirror

structures discussed in Chapter 5 was used to generate a start design. It was optimized

using the techniques described in Chapter 7 and Appendix E. This design will be called

the “Non-periodic Design.”

Fig. 8.2 plots the physical thickness of the layers for the two different designs. The blue

squares show the identical thickness of all layers for the periodic design. The cyan circles

are the layer thicknesses of the synthesized second design before refinement. It can be

seen how the thicknesses of the high refractive index layers increase gradually, while the

thicknesses of the low refractive index layers decrease. Furthermore, the thickness of a

period also increases with layer number. The magenta crosses represent the layer

thicknesses of the refined non-periodic design.

0 20 40 60 80 100 120 140 160 180 2000

200

400

600

Periodic DesignDouble-Chirped Mirror DesignRefined Non-periodic Design

Layer #

Phys

ical

Thi

ckne

ss in

nm

Fig. 8.2. Physical layer thicknesses of the periodic and non-periodic design and the

double-chirped mirror starting design.

98

Fig. 8.3 and Fig. 8.4 show the reflectance of the periodic and the non-periodic design at

40° incidence angle. It can be seen that the non-periodic structure is designed to have

unity reflectance without the need of a backside gold coating. For the periodic design

Fabry-Perot like oscillations are visible in the reflectance. They are due to the larger

absorption by the gold coating for wavelengths that are on resonance because theses have

a larger stored energy in the structure corresponding to multiple effective bounces off the

gold coating as discussed in Chapter 2.

Fig. 8.5 graphs the group propagation angle for the two different designs. Since the

periodic design has many periods it can be calculated using either the Bloch dispersion

relation (Appendix A) or the transfer matrix dispersion relation (Appendix C). It is quite

obvious that the results obtained from these calculations differ. The transfer matrix

calculation shows strong oscillations, which are caused by the interference of the shifting

beam with a front reflection as discussed in Chapter 3.

The Bloch calculation on the other hand results in a smooth curve, since an infinite

medium is assumed such that interface reflections are not included. Assuming that the

beam is focused sufficiently to prevent the interference effects, we can use the Bloch

calculation values for the group propagation angle. The non-periodic stack is designed to

suppress the front reflection and has a smooth change in the group propagation angle with

wavelength. Furthermore, this device structure was designed to have a linearly changing

group propagation angle with wavelength compared to the non-linear characteristics of

the periodic design. Fig. 8.6 graphs the calculated shift along the front interface assuming

a single bounce for both structures.

99

1520 1530 1540 1550 1560 15700

0.2

0.4

0.6

0.8

1

Transfer Matrix Calculation

Periodic Design

Wavelength in nm

Refle

ctan

ce

1520 1530 1540 1550 1560 15700

0.2

0.4

0.6

0.8

1


Non-periodic Design

Wavelength in nm

Refle

ctan

ce

Fig. 8.3. Reflectance of the two designs at 40° incidence angle.

1520 1530 1540 1550 1560 15700.95

0.96

0.97

0.98

0.99

1


Periodic Design

Wavelength in nm

Refle

ctan

ce

1520 1530 1540 1550 1560 15700.97

0.98

0.99

1


Non-periodic Design

Wavelength in nm

Refle

ctan

ce

Fig. 8.4. Zoomed-in reflectance of the two designs at 40° incidence angle.

100

1520 1530 1540 1550 1560 15700

20

40

60

80

Transfer Matrix CalculationBloch Calculation

Periodic Design

Wavelength in nm

Gro

up p

ropa

gatio

n an

gle

in d

eg

1520 1530 1540 1550 1560 15700

20

40

60

80


Non-periodic Design

Wavelength in nm

Gro

up p

ropa

gatio

n an

gle

in d

egFig. 8.5. Calculated group propagation angle.

1520 1530 1540 1550 1560 15700

200

400

600

800

Transfer Matrix CalculationBloch Calculation

Periodic Design

Wavelength in nm

Shift

in u

m

1520 1530 1540 1550 1560 15700

20

40

60

80

100


Non-periodic Design

Wavelength in nm

Shift

in u

m

Fig. 8.6. Calculated Shift along the interface for a single bounce.

8.1 Number of Volume ModesIn this section we are going to consider the number of volume modes that can be

separated by a structure. Here we do not consider what happens for multiple bounces and

101

intermediate propagation through the substrate. We define volume modes as modes that

propagate in mutually exclusive propagation cones. For simplicity we will assume a

device as shown in Fig. 8.7.

L

d

Fig. 8.7. Schematic of three volume modes for focussing on the front surface.

Plane waves of different wavelength propagate at different angles within the device. As

shown in Appendix D, a beam of light can be decomposed into plane-wave components

with different propagation directions. Thus, a beam of light consists of a range of

propagation angles. In principle a beam has components at all angles, but many of these

might have very small amplitudes and can thus be neglected. Here we consider the

angular range to be delimited by those components whose intensity has dropped to 1/e2

from the center component. For less crosstalk between channels a smaller value may be

chosen, which only results in a scaling of the results given here.

Assuming the input beam of multiplexed wavelengths is focused on the front of the

device and that the input diameter of the beam d is small compared to the width of the

device L, modes are non-overlapping in the volume and at the output if their angular

ranges are non-overlapping within the structure. If, for example, the first mode has an

angular content within the device of 10º to 20º and the next mode occupies the angular

range from 20º to 30º, these modes will not overlap at the output and are thus separate

modes.

Following this definition of separable modes, the number of modes within a small

frequency range ∆ω is given by (8 – 1).

102

( ) ( ) ( )( )instruc

groupgroupinesN

θωθθ

ωωθθωωθθθωωθ

∆∆

∆−−∆+=∆∆∆

,~,~2~,~2~,~

,,~,~mod (8 - 1)

In this equation ∆θstruc is the angular range of a mode within the structure for a given

angular range ∆θin of the input beam in vacuum. θ is the incidence angle of the beam,

θgroup is the group propagation angle in the stack, and ω is the frequency. As seen in

Appendix B, the dispersion relation can be expressed either in terms of the incidence

angle and the frequency ( )ωθ ~,~ or in terms of the wavevector along the layers β and the

frequency ( )ωβ , . In order to distinguish these two variable sets we use the tilde for the

first case. The variable transformation between ( )ωθ ~,~ and ( )ωβ , is discussed in more

detail in Appendix B. Defining the dispersion Dispω as shown in (8 – 2).

( ) ( )ω

ωθθωθω ~

~,~~,~

∂

∂= groupDisp (8 - 2)

The total number of modes between ω1 and ω2 is given by (8 – 3). Note there has to be

one mode even without any dispersion, since one beam can always simply propagate

through the structure.

( ) ( ) ( )( )∫∫ ∆∆

+=∆∆+=∆2

1

2

1

~,~,~~,~

1,~,~,~1,,,~mod21mod

ω

ω

ωω

ω

ωθωθθ

ωθθωωθθωωθ dDispdNN

instrucinesines (8 - 3)

To evaluate this number of modes, we need to find an expression for ∆θstruc. ∆θstruc can be

estimated from ∆θin using (8 – 4), where the difference is approximated by a differential.

( ) ( )in

groupinstruc θ

θ

ωθθθωθθ ∆

∂

∂=∆∆ ~

~,~,~,~ (8 - 4)

In order to relate ∆θstruc to the dispersion Dispω, the group velocity angle θgroup is

developed into a Taylor series as shown in Appendix F. This results in relationship (8 - 5)

between the angular range ∆θstruc of a beam within a dispersive stack, the input angular

range ∆θin, and the dispersion of the stack Dispω.

103

( ) ( ) in

avg

instruc

cDisp

nθ

ωβ

θ

θωωθ

θ

θθωθθ ω ∆

∂∂

−+

−=∆∆

~sin

~cos~~,~~sin

~cos,~,~22

(8 - 5)

As seen in Chapter 4 and Appendix F, ∂β/∂ω is approximately constant with wavelength.

The only rapidly varying term with wavelength in (8 - 5) is the dispersion Dispω. Thus,

the angular range of a mode ∆θstruc can be estimated as the input angular range ∆θin

multiplied by the sum of a constant term added to a term that is proportional to the

dispersion. The validity of this approximation is limited to stacks that have a constant

angular dispersion or to small input angular ranges as differences are replaced by

differentials. Substituting expression (8 - 5) for ∆θstruc into (8 - 3), we obtain the number

of volume modes within a given wavelength range as given in (8 - 6).

( )

( )

∫ ∆

∂∂

−+

−

+=∆2

1

~1

~sin

~cos~~sin~,~

~cos

11,,,~

22

21mod

ω

ω

ω

ωθ

ωβ

θ

θω

θωθ

θ

θωωθ d

cnDisp

Nin

avg

ines

(8 - 6)

(8 - 6) reveals that the number of modes is inversely proportional to the input angular

range. Thus, half the input angular range means double the number of modes, and double

the input range half the number of modes. This result was verified in simulations

presented in a section 8.2.

The interesting part of (8 - 6) is the denominator of the first fraction. It can be seen that

for small dispersions, ( )

− θωθθ ω

~sin~,~~cos 22avgnDisp dominates and the number of

modes increases linearly with increasing dispersion. For large dispersion on the other

hand,

∂∂

−ωβ

θθω c~sin~cos~ dominates and the number of modes is independent of the

dispersion. This result might be expected, since a larger dispersion not only leads to a

104

larger beam shift, but also to beam broadening. Thus within this model of volume modes,

above a certain dispersion nothing can be gained by increasing the dispersion further.

In Fig. 8.8 the number of volume modes per wavelength interval of 1 nm is graphed as a

function of the dispersion calculated using (8 - 6). An input half cone of ∆θin =1° is used.

In the left graph the group velocity vgx=∂ω/∂β of the periodic structure is used and in the

right graph vgx of the non-periodic design is used. The total number of volume modes is

obtained by multiplying the modes per nm by the wavelength range over which the

dispersion is achieved. As seen from (8 - 6) one more mode needs to be added. The non-

periodic design has, e.g., a dispersion of 0.7 deg/nm over a 40 nm wavelength range.

Thus, we expect from Fig. 8.8 a total of (0.093modes/nm)·40nm + 1 = 3.7 modes for

∆θin=1°. Section 8.2 verifies this estimate. It is clearly visible from Fig. 8.8 that the

number of volume modes does not increase significantly above a certain dispersion.

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2vgx of Periodic Design

Dispersion in deg/nm

Mod

es p

er n

m

0 0.5 1 1.5 20

0.02

0.04

0.06

0.08

0.1vgx of Non-periodic Design

Dispersion in deg/nm

Mod

es p

er n

m

Fig. 8.8. Increase in the number of volume modes per 1 nm wavelength interval as

a function of the stack dispersion for an input half cone ∆θin =1°.

∂β/∂ω can be estimated very roughly by the value obtained for a homogeneous layer of

refractive index navg. In this case βhom is given by (8 - 7).

22

hom Kcnavg −

=

ωβ (8 - 7)

105

The derivative with respect to ω for a constant K is then calculated as shown in (8 - 8).

θβ

ω

δωδβ

sin

2

hom2

2hom

cn

cn avgavg

constK

===

(8 - 8)

Thus, the higher average refractive index of the periodic design leads to a higher ratio

∂β/∂ω and to a higher number of modes per wavelength interval of 1 nm.

Knowing how to estimate the number of modes using (8 - 6), the question is how to

maximize the number of modes for a given stack. Here we will consider the case that we

have designed a structure that has a certain dispersion behavior and is operated at a given

incidence angle. Even though the incidence angle appears as a variable in (8 - 6), it

cannot really be changed easily as the dispersion curve is only designed and valid for a

certain angle. Thus, to change the input angle, we would have to design a new structure.

The same argument is true for the operating range of the structure, which is also

determined by the design. The only parameter we can change is the angular range of the

input beam (as long as we only probe a part of the dispersion curve that is well behaved).

In (8 - 6) the thickness of the structure does not appear except that the calculation is only

correct for d<<L as shown in Fig. 8.7. (8 - 6) predicts more modes for a structure with a

smaller input angular range. Thus, we would want to choose the input angular range as

small as possible to obtain the maximum number of volume modes. But a smaller input

angular range results in a larger beam waist, and the device has to have a larger thickness

for the condition d<<L to be true. Therefore, the only conclusion we can draw from

(8 - 6) is that the angular range should be as small as possible and the thickness has to

grow correspondingly. For a given physical thickness the optimal number of modes

cannot be determined using the volume-mode calculation as we leave the range of

validity d<<L of this calculation, when decreasing the angular range and thus increasing

the spot size d.

106

8.2 Verification of Volume Mode ModelIn section 8.1 we have derived (8 - 6) for calculating the number of volume modes. In

this section we will test this model by calculating the distinguishable volume modes for

the non-periodic stack in a numerical simulation and comparing the resulting number of

modes to the theoretical model. Assuming that the width d of the modes at the input

surface is negligible compared to the length of the structure L (d<<L), the number of

modes supported by the structure is numerically obtained as follows. First the angular

range of the mode with the lowest wavelength is calculated from the input angular range.

Fig. 8.9 gives an example of such a calculation.

The C-band of an Erbium-doped fiber amplifier (EDFA) from 1525 nm to 1565 nm is

chosen as wavelength interval. The lowest mode has a wavelength of 1525 nm. From

Table 8.1 we see that this mode occupies an angular range from 8.24° to 12.26°. Now we

know that the next higher mode has to have at least a low angle of 12.26°. Using a

numerical search algorithm, we find the next possible mode to have a wavelength of

1535 nm and an angular range in the structure of 12.26° to 19.74°. We continue to find

modes in this manner until the high wavelength of 1565 nm in this example is reached.

Table 8.1 gives the numerical data for the five modes found. In Fig. 8.9(b) a cartoon of

the modes is depicted.

Using the numerical simulation technique described above, we can now test the

theoretical number of modes expected from (8 - 6) against the number of modes actually

obtained. Using (8 - 6) the number of modes for the non-periodic design is calculated as a

function of the input angular range ∆θin. The integral is performed by calculating ten

intermediate points. Only the dispersion Dispω is assumed to be a function of frequency.

The resulting number of modes is plotted as the crosses in Fig. 8.10. The number of

modes is rounded off to the previous integer, since fractional modes are physically not

possible. The squares plotted in Fig. 8.10 represent the number of modes obtained from a

numerical simulation as described above.

107

1520 1530 1540 1550 1560 15705

10

15

20

25

30

35

Wavelength in nm

Prop

agat

ion

angl

e in

cry

stal

in d

eg

(a) (b)

Fig. 8.9. Volume mode calculation for the non-periodic design for an incidence

angle of 40°, an input half cone of 0.6°, and a wavelength interval from 1525 nm to

1565 nm. (a) shows the position of the obtained modes in terms of wavelength and

propagation angle within the crystal. (b) depicts a cartoon of the obtained modes.

Wavelength in nm Low angle in ° High angle in °15251535154415541565

8.2412.2619.7526.0034.50

12.2619.7425.9934.5043.79

Table 8.1. Wavelength, low angle, and high angle of the different modes for the

example in Fig. 8.9.

Fig. 8.10 nicely shows the agreement between simulation and theoretical number of

modes obtained for the volume mode model (8 - 6). Also we see again that the number of

modes is inversely proportional to the input angular range as predicted before. Thus, we

would want to choose a small angular range, e.g. 0.2°, which corresponds to a spot size

(Gaussian beam radius) of 140 µm. But if d is about twice the spot size, the thickness L

of the stack would have to be several times this value for the condition d<<L to be

fulfilled.

108

0 1 2 30123456789

101112131415

Simulated number of modesExpected number of modes

Input half cone in deg

Num

ber o

f mod

es

0 50 100 1500123456789

101112131415

Input spot size in um

Num

ber o

f mod

es

(a) (b)

Fig. 8.10. Number of volume modes as obtained from simulation and the expected

number of modes calculated using (8 - 6) as a function of the input half cone in (a)

and the input spot size in (b).

8.3 Number of Surface ModesIn the last section we have shown that the volume-mode calculation can be used to

estimate the number of volume modes for a given structure, if the condition d<<L is

fulfilled. The volume-mode calculation cannot, however, be used to estimate the

maximum number of modes, since we leave the range of validity of this calculation. In

this section, a different approach for obtaining the number of modes supported by a

structure is introduced. For wavelength demultiplexing, separable modes have to be

spatially non-overlapping at the output interface, but not necessarily within the structure.

This results in a mode picture as given in Fig. 8.11.

109

L

∆x

Fig. 8.11. Schematic of three non-overlapping surface modes on the back surface.

Here we assume that the backside is the output surface of interest, i.e. the position where

we place the detectors or output waveguides. Any other plane could be chosen equally

well, as this is a linear system. A separate lens may be necessary to focus the light onto

the output devices in that case. Assuming a monotonically increasing group propagation

angle θgroup with frequency, the number of modes within a frequency range ω1 to ω2 is

given by the total shift in this range divided by the surface ∆x occupied by one mode.

Furthermore, we have already one mode without any dispersion, which is added in

(8 - 9).

( ) ( )( ) ( )( )( )1

~,~tan~,~tan,~,~,~ 12

21mod +∆

−=∆

xL

xN groupgroupes

ωθθωθθωωθ (8 - 9)

In order to calculate the number of modes within a frequency range ∆ω, we start from

expression (8 - 10) for the number of modes within an angular range ∆θgroup.

( )( ) ( )

x

LxN

groupgroup

groupgroup

groupes ∆

∆−−

∆+

=∆∆∆2

~,~tan2

~,~tan,,~,~

mod

θωθθ

θωθθ

θωθ

(8 - 10)

Using identity (8 - 11), (8 – 10) can be approximated by (8 - 12).

( )βα

βαβα

coscossintantan ±

=± (8 - 11)

110

( ) ( )( )ωθθ

θθωθ ~,~cos

,,~,~2mod

group

groupgroupes x

LxN

∆

∆=∆∆∆ (8 - 12)

Finally, plugging in the definition of the dispersion Dispω as defined in (8 - 2), we obtain

the number of modes within a frequency range ∆ω as given in (8 - 13).

( ) ( )( )( )ωθθ

ωωθωωθ ω

~,~cos

~,~,,~,~

2modgroup

es xDispLxN

∆∆

=∆∆∆ (8 - 13)

If ∆x or Dispω are not the same for all the modes, but a function of frequency, (8 - 13) can

be integrated to obtain the number of modes within a frequency range given in (8 - 14).

( ) ( )( )( )∫ ∆

+=∆2

1

~~,~cos

~,~1,,,~

221mod

ω

ω

ω ωωθθ

ωθωωθ d

xDispLxN

groupes (8 - 14)

Now we consider again the question, how to maximize the number of modes obtained for

a given dispersion curve and given incidence angle. We see that we get more modes the

larger the thickness L of the stack is and the smaller the surface ∆x occupied by one mode

is. In the volume-mode calculation, we assumed that the input beam is focused on the

front surface of the structure. But this is not necessary. From (8 - 14) we see that we

obtain a larger number of modes if the beam is focused on the output surface.

For a general structure, only one mode will be focused exactly at the backside of the

structure occupying a surface depending on the spot size w0. Other modes will be out of

focus and require a larger area. Given a certain structure, the focus has to be chosen in

such a way that the largest number of modes can be fitted on the back surface. In

principle, a structure could be designed in such a way that all modes focus at the same

distance L from the front surface. This will allow for the largest number of modes

possible, since moving any mode out of focus will increase its size ∆x and therefore

lower the total number of modes.

Concluding this argument, we see from (8 - 14) that a larger thickness L of the stack and

a smaller surface ∆x occupied by each mode lead to a larger number of modes. Again we

did not find a real optimal beam size. The number of modes is larger the smaller the beam

111

waist is. One limit to decreasing the beam waist further is given by the fact that a smaller

beam waist means a larger input angular range and thus a larger area of the dispersion

curve is probed. Since the dispersion curve is only valid over a certain wavelength range

and thus over a certain angular range, this limits the size of the input beam. In the next

section ∆x is related to the spot size w0 using crosstalk considerations. From w0 we can

calculate what angular range a beam has and what range of the dispersion curve will be

probed. In section 8.5 we investigate how performing several bounces within a given

structure influences the number of modes.

8.4 Crosstalk between ModesIn this section we will calculate the crosstalk between modes as a function of the channel

spacing ∆x and the Gaussian spot size w0. The crosstalk is largest between two

neighboring modes and is determined by the overlap of the mode fields. Fig. 8.12 shows

a schematic of two neighboring modes with two bounces in the device, introducing the

nomenclature used. In the following calculations it is assumed that the beams have a

Gaussian profile along the direction of the beam shift and are elongated in the other

direction. Fig. 8.13 shows the Gaussian profile of two neighboring modes as it is seen

along the exit interface in Fig. 8.12. Here the dispersion profile of the non-periodic

design, as given in Fig. 8.5, is assumed. The solid lines are for the center wavelengths,

which are separated by 5 nm and the dotted lines show how far the beams would shift due

to a modulation bandwidth of 50 GHz. The vertical green lines represent the spatial width

∆x of the blue mode.

112

Ls

2wt

2w0

2w1

d1

ns

Fig. 8.12. Schematic of beam shift after two bounces for two neighboring modes.

120 140 160 180 200 220 240 2600

0.2

0.4

0.6

0.8

1

Position in um

Pow

er in

a.u

.

Fig. 8.13. Gaussian profile of two neighboring modes that are separated by 5 nm.

The vertical green lines represent the position of the blue channel. The solid lines

are for the center frequency and the dotted lines represent the shift with 50 GHz

signal modulation.

113

The crosstalk can be calculated by integrating the power of the red mode between the

green lines and dividing it by the power of the blue mode within this spatial section. For

the case that all beams are focused on the exit interface and have a spot size w0 at that

position, all channels have an equal spatial width ∆x and the center positions xc1 and xc2

of two channels are also separated by ∆x. For a negligible shift due to signal modulation,

the crosstalk can be calculated using (8 - 15).

( )

( )5.0

5.02erf2

5.12erf

2exp

2exp

1

12

22

1

12

21

−

∆

∆

=

−−

−−

=

∫

∫∆+

∆+

t

txx

x t

c

xx

x t

c

wx

wx

dxw

xx

dxw

xx

Crosstalk (8 - 15)

In order to incorporate the additional crosstalk due to the beam shift with signal

modulation, integration over frequency as shown in (8 - 16) is performed.

( )( )

( )( )∫ ∫

∫ ∫∆+

∆−

∆+

∆+

∆−

∆+

−−

−−

=mod

mod

mod

mod

5.0

5.0

1

12

22

5.0

5.0

1

12

21

2exp

2exp

λλ

λλ

λλ

λλ

λλ

λλ

c

c

c

c

ddxwxx

ddxwxx

Crosstalkxx

x t

c

xx

x t

c

(8 - 16)

Fig. 8.14 graphs the crosstalk as a function of the modulation frequency. As expected the

crosstalk increases with the signal modulation frequency, but up to 100 GHz we can

expect a quite low crosstalk of around –40 dB.

0 100 200 300 400 50050

40

30

20

10

Signal bandwidth in GHz

Cro

ssta

lk in

dB

Fig. 8.14. Crosstalk as a function of signal bandwidth for the non-periodic design

and a channel spacing of 5 nm.

114

Using these crosstalk considerations, we can thus determine the necessary spatial

separation ∆x of two modes. In the next section the maximum number of modes possible

is determined.

8.5 Number of Surface Modes including BouncesAs discussed in section 8.3 the highest number of modes is obtained if all modes are

focused on the output surface. It turns out that the condition of all modes focussing on the

output surface is approximately equivalent to having a shift that changes linearly with

wavelength. This can be seen as follows. First we redefine the dispersion as given in

(8 - 17), where s is the shift along the interface upon reflection defined in (8 - 18) with

respect to the total thickness of the multilayer stack L.

( ) ( )( )ωλ

ωθωθλ ~

~,~~,~

∂∂

=sDisps (8 - 17)

The definition of the dispersion with respect to the wavelength λ has the advantage that

the dispersion is without dimension in this case and does not change upon scaling the

device. Taking the derivative of the shift in (8 - 18) we obtain (8 - 19).

( )groupLs θtan2= (8 - 18)

( )groupgroup

Ld

dsθθ 2cos

2= (8 - 19)

Thus, Dispω in (8 - 2) and Dispsλ in (8 - 17) are related by (8 - 20).

( ) ( ) ( ) ( )

−=

cDisp

LDisp s

group

πωλ

ωθθ

ωθ λω 2

~~,~2

cos~,~ 22

(8 - 20)

A beam with an angular range of 2∆θstruc for a given input angular range 2∆θ focuses

approximately at a distance Lf as given in (8 - 21). w1 is the spot size at the input interface

as seen in Fig. 8.12, which is identical for all modes.

( )struc

groupf wL

θ

θ

∆≈

2

1

cos(8 - 21)

115

The focal distance has to be equal to the thickness of the structure if the modes are to

focus at the output surface. Therefore, we obtain condition (8 - 22) for ∆θstruc.

( )L

w groupstruc 2

cos2

1

θθ =∆ (8 - 22)

If ∆θstruc is to fulfill (8 - 22) for the given input angular range, we see from (8 - 19) by

replacing the differential by a difference that ∆s has to be constant for that same input

range. In Appendix F we derived in (F - 14) that a change in the input angular range ∆θ is

approximately proportional to a change in the frequency ∆ω. Therefore we deduce that

the change in the shift should be constant over the given frequency interval, which will at

the same time result in a constant shift for the given input angular range. This condition is

equivalent to having a constant spatial dispersion. As the frequency ranges considered are

small compared to the absolute frequency, a change in frequency is approximately

proportional to a change in wavelength. The simplest type of structure that fulfills

condition (8 - 22) for focussing all modes at the same position is a structure that has a

linear shift as a function of wavelength. In Chapter 10 we will see, though, that other

structures can fulfill the equation as well. The change in the shift only has to be constant

over the input angular range ∆θ corresponding to a specific frequency range ∆ω at the

position of the different channels. It is still fulfilled if there are discontinuities between

the center positions of the different channels such as in the step-design in Chapter 10.

Next let us consider the maximum number of channels we obtain if all channels are

focussed at the output plane and therefore have an equal spatial extent ∆x and are equally

spaced in wavelength. For the following calculations we will consider structures with a

linear shift as a function of wavelength and thus a constant dispersion Dispsλ = cDisp. For

operation in transmission, cDisp is the dispersion in transmission, while for operation upon

reflection, it is the dispersion after one bounce. Thus, to obtain the total dispersion we

need to multiply by the number of bounces Nb. With these considerations the number of

surface modes (8 - 14) is rewritten as (8 - 23).

( ) ( )∫ −∆

+=∆

+=∆2

1

1221mod 11,,,λ

λ

λλλλλx

cNd

xcN

xcN DispbDispbDispes (8 - 23)

116

In this calculation the channel spacing ∆λ is given by (8 - 24).

Dispb cNx∆

=∆λ (8 - 24)

Also, the number of modes agrees with the picture of dividing the total bandwidth by the

bandwidth of one mode as given in (8 - 25).

λλλ

∆−

+= 12mod 1esN (8 - 25)

So far we only considered the propagation through the multilayer stack and not the

propagation through the substrate. For the broadening of Gaussian beams the total

propagation length is relevant. In the following it is assumed that the substrate is thick

compared to the stack. Thus, the total propagation distance Lp is determined by the

substrate thickness Ls and is given by (8 - 26), where θs is the propagation angle in the

substrate.

( )s

sbp

LNL

θcos2

= (8 - 26)

All beams have the same spot size w1 at the input and ∆x is a constant as discussed above.

While the propagation through the substrate determines the beam size, the layers only

cause the beam shift and may offset the focal plane or lead to distortions as discussed

previously. As seen in section 8.4, ∆x is determined by crosstalk considerations. It

follows from (8 - 15) that for a constant ratio c1=∆x/wt the crosstalk is constant. c1=3.2

corresponds to approximately -30dB crosstalk and c1=3.8 to -40dB. ∆x is related to ∆λ as

given in (8 - 27), where cDisp is the dispersion of a single bounce and Nb is the number of

bounces.

t

bDisp

t wNc

wxc

λ∆=

∆=1 (8 - 27)

The beam propagates as a Gaussian beam within the structure. The beam broadening

limits the number of bounces. In order to keep the field of the different bounces

separated, (8 - 28) follows from Fig. 8.12.

117

21

1 ≥wd (8 - 28)

In (8 - 28) d1 is the separation between the center of the different bounces and w1 is the

spot size along the interface at the entering surface. d1 is calculated with (8 - 29), where

θs is the propagation angle in the substrate.

( )ssLd θtan21 = (8 - 29)

The beam size of a Gaussian beam is given by (8 - 30) and (8 - 31).2

( )2

0 1

+=

Rzzwzw (8 - 30)

λπ nwzR

20= (8 - 31)

Using these equations, w1 can be approximated for z>> zR as in (8 - 32). The subscript

“s“ refers to the values within the substrate.

( )( ) ( )

( )( ) ( )sss

sb

s

s

ss

ssb

s

s

s

s

sb

nwLNw

nwLNw

LNww

θπλ

θλ

πθ

θθθ

cos2

coscos/21

coscoscos2

2

2

21 ≈

+=

= (8 - 32)

Substituting (8 - 29) and (8 - 32) into (8 - 28) we obtain (8 - 33).

( )

( ) ( )

( ) ( ) 2cossin

cos2

cos

tan2 2

2

≥=λ

πθθ

θπλ

θ

θ

b

stss

sss

sb

s

s

ss

Nnw

nwLNw

L (8 - 33)

Thus, we get as condition (8 - 34) for the spot size.

( ) ( ) sss

bt n

Nwπθθ

λ2cossin

2≥ (8 - 34)

This is a very important equation, since it defines how far we can reduce the spot size. In

section 8.2 we saw that we get more modes the smaller we choose the spot size. Now we

have a condition for the smallest spot size. Thus, the maximum number of modes is

118

obtained if the equal sign in (8 - 34) is fulfilled. Substituting (8 - 34) into (8 - 27), we get

condition (8 - 35) for ∆λ.

( ) ( ) sssDispbDisp

t

ncc

Ncwc

πθθλ

λ 211

cossin2

≥=∆ (8 - 35)

Finally, using (8 - 25) we obtain the maximum number of surface modes possible on the

wavelength interval from λ1 to λ2 as shown in (8 - 36).

( ) ( ) ( )λ

πθθλλ

λλλ

1

21212

2cossin

11c

ncN sssDisp

m

−+≤

∆−

+= (8 - 36)

Or, rewriting (8 - 36) in terms of the incident angle in air, we reach (8 - 37).

( )( ) ( )

1

2

2

12

2

sin1sin1

cn

cN s

Disp

m

−

−+≤

θθπ

λλλ for z>> zR (8 - 37)

(8 - 37) assumes that the channels are focussed at the output surface. In principle we

could perform double the number of bounces such that the channels are focussed after

half the bounces and the exit beam size is identical to the incident beam size. In this case

we obtain twice as many channels, but we need to use a second lens on the output side to

refocus the channels.

It is interesting to note in (8 - 37) that increasing the number of bounces or decreasing the

spot size cannot increase the number of surface modes. For a given crosstalk, the beam

size determines the number of bounces to be performed. The first factor in expression

(8 - 37) shows that only the relative wavelength interval matters. All designs are scalable

to a different wavelength range and have, after the scaling, the same number of modes as

before. As expected, either a higher dispersion or a larger crosstalk (smaller c1) leads to a

larger number of modes. Furthermore, a larger index of refraction in the substrate also

increases the number of modes. Fig. 8.15 graphs the dependence of the maximum number

of modes on the incident angle. It can be seen that the number of modes is approximately

independent of the angle, if the incident angle is larger than 40°.

119

0 20 40 60 800

5

10

15

Incidence angle in deg

Max

imum

num

ber o

f mod

es

Fig. 8.15. Theoretical maximum number of modes in the interval from 1525 nm to

1565 nm as a function of incident angle assuming a dispersion of 60µm/40nm,

ns=1.52 and a crosstalk of –40dB (c1=3.8).

8.6 Verification of Surface Mode ModelSimilarly to the verification of the volume mode model in section 8.2 we will here test

the surface mode model by numerical simulation. The number of surface modes can be

calculated numerically by determining the number of non-overlapping channels along the

exit surface of the structure. First we determine, which wavelength is going to be focused

at the exit interface in distance L. In the example of Fig. 8.16 the center wavelength 1545

nm of the interval is chosen to be focused with a spot size of 20µm. Now we calculate

how large the spot size w1 at the input interface has to be for that beam to be focused.

Together with the incidence angle and the focal spot size, this completely determines the

Gaussian beam at the input interface.

The first mode is chosen to be at the lowest wavelength, here at 1525 nm. For this mode

the 1/e E-field rays are calculated and the cross points with the exit interface are

determined. Table 8.2 lists the calculated angles for the two 1/e-E-field rays and the

position where they cross the back interface. The spacing separating the first mode from

the next mode depends on the desired crosstalk characteristics. To obtain a crosstalk of

around –40dB between neighboring modes, c1=3.8 is chosen. For the definition of c1 see

(8 - 27).

120

1520 1530 1540 1550 1560 157010

20

30

40

50

Wavelength in nm

Prop

agat

ion

angl

e in

cry

stal

in d

eg

(a) (b)

Fig. 8.16. Surface mode calculation for the non-periodic design for an incidence

angle of 40°, focussing on the back side at 1545 nm with a spot size of 20 µm

(corresponding to ∆θin = 1.4°), a wavelength interval from 1525 nm to 1565 nm,

and 13 bounces. To obtain a crosstalk around –40dB, a spacing of 3.8*20

µm/cos(40°)=100 µm is chosen between modes. (a) shows the position of the

obtained modes in terms of wavelength and propagation angle within the crystal.

(b) depicts a cartoon of the modes obtained graphing the two 1/e E-field rays for

each mode.

Wavelength in nm Angle+ in ° Angle- in ° Position+ in µm Position- in µm15251542154815541563

16.9629.5834.5938.7044.91

6.8112.9517.2621.9527.23

382.3 711.6 864.51004.01250.0

610.4 748.7 849.9 965.71106.0

Table 8.2. Wavelength, group propagation angle in the structure for the input angle

plus the half cone angle (Angle+), group propagation angle minus the half cone

angle (Angle-), position where the plus-angle crosses the back interface (Position+),

and position where the minus-angle crosses the back interface (Position-) for the

different modes for the example in Fig. 8.16.

If a mode is focused on the backside, the rays cross at that position and the spot size

along the interface is wt=w0/cos(θ) resulting in a separation between modes given by

121

∆x = c1* wt. For a mode that is not focused on the backside, a larger surface area ∆x is

needed. Here the surface area is increased by the amount between the two rays. This is

not exactly right, but a sufficiently good approximation.

The second mode can now be calculated by searching numerically for a mode that is

separated by the spacing width c1* wt from the first mode. In our example the next mode

is found to have a wavelength of 1542 nm. Table 8.2 lists the numerical data for the five

modes obtained. The distance between the high point of one mode and the low point of

the next mode is always separated by the spacing of 100 µm. In Fig. 8.16 (b) the modes

are shown graphically. Even though the structure might be operated in reflection or with

multiple bounces, the beam path is unfolded in the forward direction. As different

wavelengths propagate in parallel in the substrate, the substrate changes the focal position

approximately equally for all beams. Thus, it does not need to be considered. It is clearly

visible that the three center modes in Fig. 8.16 (b) are quite well focused while the outer

modes are running out of focus.

In (8 - 14) we obtained for the number of surface modes the formula repeated below.

( ) ( )( )( )∫ ∆

+=∆2

1

~~,~cos

~,~1,,,~

221mod

ω

ω

ω ωωθθ

ωθωωθ d

xDispLxN

groupes (8 - 14)

Now we can compare the number of modes predicted by (8 - 14) against the number of

modes obtained by simulation. For the calculation of the integral in (8 - 14) ten

intermediate points are used and a sum is formed. ∆x is calculated using the focal

distance given in (8 - 21) and the spot size at the input w1. Since the input angular range

∆θin =1.4° is rather large, the difference is not replaced by a differential to determine the

angular range within the structure ∆θstruc. The maximum number of surface modes for a

given dispersion is calculated using (8 - 37), also repeated below, where ∆x is assumed to

be the constant ∆x = c1* wt independent of wavelength.

( )( ) ( )

1

2

2

12

2

sin1sin1

cn

cN s

Disp

m

−

−+≤

θθπ

λλλ (8 - 37)

122

0 1 2 3 4 5 6 7 8 9 10 11 12 13 140

1

2

3

4

5

6

7

8

9

Simulated number of modesExpected number of modesMaximum number of modes

Bounces

Num

ber o

f mod

es


number of modes calculated using (8 - 14), and the maximum number of modes

obtained from (8 - 23) for the non-periodic design as a function of the number of

bounces within the structure.

Fig. 8.17 graphs the expected number of modes, the maximum number of modes, and the

simulated number of modes as a function of the number of bounces. The expected

number of modes agrees well with the simulated number of modes verifying (8 - 14) for

modeling the number of surface modes. The number of modes obtained with this design

remains significantly below the maximum number of modes though. This is due to the

fact that not all the modes are focused. Some of the surface space is used for unfocused

modes instead of for new modes as visible in Fig. 8.16. In order to obtain the maximum

number of modes, all the modes have to be focused on the backside, i.e. the beam exit

position should change linearly as a function of wavelength. Such a design is introduced

in the next section. This is one of the reasons we were especially interested in designing

stacks with a linear shift as a function of wavelength in Chapters 3 to 7.

123

8.7 Designing for a Maximum Number of ModesIn the preceding sections we concluded that the surface mode picture is valid to calculate

the number of Gaussian modes supported by a structure. The total number of modes is

limited by the condition that the field of consecutive bounces should not overlap and is

given by (8 - 37) repeated below.

( )( ) ( )

1

2

2

12

2

sin1sin1

cn

cN s

Disp

m

−

−+≤

θθπ

λλλ (8 - 37)

To obtain the maximum number of modes, the shift of the structure has to be linear with

wavelength. In this case all modes can be focused on the exit surface of the structure. An

example structure with a linear shift was designed by refining a double-chirped start

design. The new structure has again 200 layers and a thickness of 49 µm. Fig. 8.18 shows

the physical thickness of the layers as a function of the layer position within the stack. As

this design has the optimal linear shift as a function of wavelength, we will call it the

“optimal design.” Its layer composition is given in Appendix G.

0 20 40 60 80 100 120 140 160 180 2000

200

400

600

Layer #

Phys

ical

Thi

ckne

ss in

nm

Fig. 8.18. Physical layer thicknesses of the optimal non-periodic design.

Fig. 8.19 gives the reflectance of the new design as a function of wavelength at 40°

incidence angle. The reflectance drops somewhat for the longer wavelengths and might

be increased with a gold coating. Fig. 8.20 graphs the group propagation angle and the

124

shift as a function of wavelength. The calculations were performed using the transfer-

matrix method. The shift of this design is nicely linear with a dispersion of

cDisp = 1.4 µm/nm.

1520 1540 15600

0.2

0.4

0.6

0.8

1

Wavelength in nm

Ref

lect

ance

Fig. 8.19. Reflectance of the optimal designs at 40° incidence angle.

1520 1540 156010

20

30

40

50

Wavelength in nm

Gro

up p

ropa

gatio

n an

gle

in d

eg

1520 1540 15600

20

40

60

80

100

120

Wavelength in nm

Shift

in u

m

Fig. 8.20. Group propagation angle and shift of the optimal designs at 40°

incidence angle.

125

1520 1530 1540 1550 1560 157025

30

35

40

45

50

Wavelength in nm

Prop

agat

ion

angl

e in

cry

stal

in d

eg

(a) (b)

Fig. 8.21. Surface mode calculation for the optimal design for an incidence angle of

40°, focussing on the back side at 1545 nm with a spot size of 20 µm

(corresponding to ∆θin = 1.4°), a wavelength interval from 1525 nm to 1565 nm,

and 13 bounces. To obtain a crosstalk around –40dB, a spacing of 3.8*20

µm/cos(40°)=100 µm is chosen between modes. (a) shows the position of the

modes obtained, in terms of wavelength and propagation angle within the crystal.

(b) depicts a cartoon of the obtained modes graphing the two 1/e-E-field rays for

each mode.

Wavelength in nm Angle+ in ° Angle- in ° Position+ in µm Position- in µm1525153215381544154915561563

28.3332.6935.8138.9141.6944.1446.68

12.5816.8521.3025.1528.9232.8836.80

684.9 815.3 916.51025.01132.01233.01347.0

708.0 809.2 919.71021.01126.01246.01375.0

Table 8.3. Wavelength, group propagation angle in the structure for the input angle

plus the half cone angle (Angle+), group propagation angle minus the half cone

angle (Angle-), position where the plus-angle crosses the back interface (Position+),

and position where the minus-angle crosses the back interface (Position-) for the

different modes for the example in Fig. 8.21.

126

Using (8 - 37) the maximum number of modes possible with this design in the

wavelength interval from 1525 nm to 1565 nm is obtained to be Nm=8. For a spot size of

20 µm at the focus, this requires 13.7 bounces within the structure. Since only an integer

number of bounces can be performed, we will use 13 bounces. Fig. 8.21 shows the result

of the numerical simulation of this structure and Table 8.3 gives the data corresponding

to the modes. The cartoon in Fig. 8.21(b) depicts that all modes are focused on the back

interface of the structure. Instead of the theoretical limit of eight modes, seven modes are

obtained with this structure. This is due to the reduction of the number of bounces from

13.7 to 13.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 140

1

2

3

4

5

6

7

8

9

Simulated number of modesExpected number of modesMaximum number of modes

Bounces

Num

ber o

f mod

es


number of modes calculated using (8 - 14), and the maximum number of modes

obtained from (8 - 23) for the optimal design as a function of the number of

bounces within the structure.

In Fig. 8.22 the simulated, the expected, and the maximum number of modes are given as

a function of the number of bounces within the structure. It can be seen that for 13

bounces only seven modes are predicted. The quality of the design can also be seen in the

fact that the expected and the maximum number of modes are nearly identical.

Furthermore, the number of modes obtained in the simulation also agrees with the

127

maximum number of modes. Thus, we accomplished our goal of designing a structure

that supports the maximum number of modes for the given dispersion. Chapter 9 will

discuss if it is possible to design a structure with the same bandwidth that has a higher

dispersion and therefore supports more modes. This structure has not been fabricated and

we conclude this chapter with a discussion of the experimental results obtained for the

66-layer design with a linear shift as a function of wavelength introduced in Chapter 7.

8.8 Discussion of the 4-channel DEMUX in 7.4After the rather abstract discussion of the factors limiting the number of channels in the

last sections, we will here repeat some key results using the 66-layer non-periodic stack

from section 7.4 as an example. In section 7.4 we experimentally demonstrated the

demultiplexing of four channels by their Gaussian beam width using a stack with a linear

shift as a function of wavelength. In the experiment 8 bounces and a spot size of

w0=10µm were used. In Chapters 1 to 7 only schematic drawings of the device have been

considered such as the one shown in Fig. 8.23(a) depicting the center ray of the beam. In

reality the beam propagates as Gaussian beams through the stack as shown in the scaled

drawing in Fig. 8.23(b). In the drawing the beams are delimited by the angular

components of 1/e2 intensity. As shown in the previous sections, the widening of the

Gaussian beam limits the number of bounces that can be performed.

From (8 - 37) we obtain that we should be able to demultiplex 7 channels using the given

66-layer stack. In order to obtain all seven channels, we could modify the experiment in

two ways – increase the number of bounces of decrease the incident spot size. In section

8.5 we have seen that we can deduce the number of bounces necessary to obtain the

maximum number of channels from the spot size. We calculate that for a spot size of

w0=10µm, 18 bounces should be performed.

128

zy

x

827.3 nm831.0 nm834.7 nm840.6 nm

Multiplexedlight

1 mm 13.4 µm

≈10

mm

…

240 µm

400 µm

Zoomed-in:

10 mm

1.013 mm

Gold coating

Substrate

Dielectric stack(13.4 µm thick)

Lens

(a) (b)

Fig. 8.23. (a) Schematic of a 4-channel demultiplexer. (b) Scaled drawing of the

same 4-channel demultiplexer.

Fig. 8.24 shows on the left the beam propagation with 8 bounces and on the right with 18

bounces. The beams are depicted by their 1/e2-intensity rays. The top ray is shown as a

dashed line and the bottom ray as a solid line. The center ray is not depicted. Due to the

larger number of bounces the device size is increased, but we can now demultiplex 7

channels. We see that the number of bounces is limited by field overlap between bounces

at the input side. In order to prevent loss through the input window in the gold coating,

the bounces should not overlap. A second method for demultiplexing seven channels with

the given stack is to reduce the input spot size to w0=4.5µm as shown in Fig. 8.25. Again

the number of bounces is limited by field overlap at the input window in the gold coating.

Due to the smaller focus size, the Gaussian beam broadens quicker and therefore only 8

bounces are possible. But on the other hand a smaller total shift is required to separate the

channels due to their smaller spatial extend.

129

240 µm

400 µm

Zoomed-in:

10 mm

1.013 mm

w0=10 µm, 8 bounces,4 channels


240 µm

400 µm

Zoomed-in:

25 mm

1.013 mm

Rays of1/e2-intensity

Overlap limitsnumber of bounces

Fig. 8.24. Increasing the number of channels by increasing the number of bounces.

Fig. 8.24 and Fig. 8.25 clearly show that the field of different bounces shouldn’t overlap

to prevent loss and that the number of bounces and the input spot size are coupled. As

discussed theoretically in section 8.5 from a given spot size follows the maximum

number of bounces that can be performed without field overlap. Equivalently, for a given

number of bounces, the smallest possible spot size follows. As long as an optimal pair of

spot size and number of bounces is chosen, the maximum number of channels is

obtained. This is the reason that the number of bounces and the spot size do not appear in

the model (8 - 37) for the maximum number of channels possible. Note that we could

double the number of bounces for the same spot size if we use a second focusing lens at

the output, thus doubling the number of channels possible.

130

240 µm

400 µm

Zoomed-in:

10 mm

1.013 mm


10 mm

1.013 mm

w0=4.5 µm, 8 bounces,7 channels

240 µm

400 µm

Zoomed-in:


Overlap limitsreduction inspot size

Fig. 8.25. Increasing the number of channels by decreasing the spot size.

One might think that it is possible to increase the number of bounces if the substrate

thickness is reduced, because this reduces the total propagation distance. As seen from

Fig. 8.26 that is not true because the distance between the bounces is reduced by the same

factor. Therefore the substrate thickness has no influence on the number of separable

channels. A thinner substrate has the advantage, though, that the total device size shrinks

without decreasing the number of separable channels.

Fig. 8.26 demonstrates that reducing the substrate thickness from Ls=1mm to Ls=80µm

reduces the device length from 10 mm to 1.2 mm. The substrate thickness is limited by

overlap of the exiting light of the least shifted channel with the last bounce of the most

shifted channel as seen in Fig. 8.26. For a substrate thinner than 80µm either the red

channel would experience loss if we extended the gold coating or the blue channel would

have loss as part of the light already exits the structure before the last bounce.

Furthermore, this “blue” light would appear as crosstalk in the “red” channel. Therefore,

131

for the most compact device the substrate thickness should be chosen just thick enough to

prevent overlap (mechanical stability permitting).

70 µm

150 µm

Zoomed-in:

10 mm

1.013 mm

w0=4.5 µm, Ls =1 mm,7 channels

240 µm

Zoomed-in:

400 µm

1.2 mm

93.4 µm

w0=4.5 µm, Ls=80 µm,7 channels


Overlap limitsreduction insubstrate thickness

Fig. 8.26. Reducing the device size by reducing the substrate thickness.

In conclusion, Chapter 8 has derived two models for calculating the number of modes

that a given multilayer stack can demultiplex. The volume mode model (8 - 6) is

concerned with modes that are separated throughout the whole volume of the structure by

mutually exclusive angular content of the modes. More interesting for the case of a

wavelength demultiplexing device is the number of possible surface modes, i.e. the

maximum number of channels we can spatially separate on one output surface. Using

(8 - 14) we can calculate how many modes we can expect at best for a given dispersion

profile. Furthermore, we derived that the best dispersion profile is given by a linear shift

as a function of wavelength, since in this case all modes are focussed on the same output

plane. Assuming that we have such an ideal linear shift as a function of wavelength, the

optimal number of channels can be calculated using (8 - 37). We see that (8 - 37) does

not depend on the number of bounces or the spot size as long as an optimal combination

132

of the two is chosen. In order to obtain a larger number of modes, either the dispersion or

the bandwidth of the stack has to be increased as the number of modes is proportional to

the dispersion-bandwidth product. In Chapter 9 we will investigate how large of a

dispersion-bandwidth product – and therefore number of channels – can be obtained for a

given material system, stack thickness, and incidence angle.

References

[1] D. A. B. Miller, "Communicating with waves between volumes - evaluating

orthogonal spatial channels and limits on coupling strengths," Appl. Opt. 39, 1681

– 1699 (2000).


133

Chapter 9

Maximum Shift

In Chapter 8 we discussed how many wavelength channels we can expect to spatially

demultiplex using a given thin-film stack. We found that the largest number of channels

is obtained if the stack has a linear shift as a function of wavelength. In this case all

channels focus at the same output plane and the maximum number of channels Nchannels is

given by (9 - 1).

( ) ( )

crosstalk

schannels c

nsN2

sin1sin1

2

2

−

∆+≤

θθπ

λ(9 - 1)

Here we replaced the dispersion×wavelength-range product by the total shift ∆s as given

by (9 - 2).

λ∆=∆ Dispcs (9 - 2)

In (9 - 1) θ is the incidence angle in vacuum, ns the refractive index of the substrate, and

ccrosstalk a constant fixed by the crosstalk between channels. ccrosstalk=3.2 corresponds to

approximately -30dB and ccrosstalk=3.8 to -40dB crosstalk between adjacent channels as

derived in section 8.4. A linear shift corresponds to a constant dispersion cDisp=∂s/∂λ. λ is

134

the center wavelength of the device and ∆λ is the wavelength range over which the

device has a linear shift.

Since a given design typically only works for a particular wavelength λ, incidence angle

θ, and substrate refractive index ns, and shows a specific total shift ∆s, the only way we

can influence the number of channels given in (9 - 1) is by trading off the crosstalk.

Allowing for a larger crosstalk, we obtain more channels, for less crosstalk, less channels.

Thus, if we want to demultiplex a particular number of channels, we need to design a

stack with a minimum total shift ∆s that can be calculated from (9 - 1). A larger total shift

∆s corresponds to more channels. Table 9.1 compares the structures we have obtained

using the different design methods discussed in Chapters 3 to 7. As we could not obtain a

linear shift as a function of wavelength for the superprism effect, it is not listed here.

Double-Chirped Coupled-Cavity Optimized

Number of Layers 200 60 33 66Stack Thickness L 28 µm 9 µm 16 µm 13 µm

Wavelength Range ∆λ 170 nm 40→200nm 12 nm 22 nm

Total Shift ∆s 35 µm 10 µm 22 µm 22 µm

Spatial DispersioncDisp=∆s/∆λ 0.2 µm/nm 0.05→0.25

µm/nm 1.8 µm/nm 1.0 µm/nm

Table 9.1. Comparison of linear shift characteristics obtained with the double-

chirped stacks in Chapter 5, the coupled-cavity stack in Chapter 6, and the

numerically optimized stack in Chapter 7.

As discussed previously, the different design methods result in different spatial dispersion

values. It is clear from Table 9.1 that the double-chirped designs have a low dispersion,

the coupled-cavity design has a high dispersion, and the optimized design has an

intermediate dispersion. Due to the different bandwidth of the designs, the dispersion is

not a good figure of merit. The number of channels that can be demultiplexed depends on

the total shift, i.e. the product of the dispersion with the wavelength range of the design.

Therefore, we consider the total shift as the figure of merit for how good a design is. In

135

Table 9.1 we observe that the 200-layer double-chirped design has the largest total shift

and would therefore demultiplex the largest number of channels. The five 60-layer

double-chirped designs have on the other hand the lowest total shift, while the coupled-

cavity design and the optimized design exhibit an intermediate amount of shift. Checking

now the stack thickness values for the different design, we see that the 200-layer double-

chirped design is also the thickest design, while the 60-layer double-chirped stacks are

the thinnest designs. This might explain why the 200-layer design exhibits the largest

shift. To test this hypothesis, we divide the shift by the stack thickness for each design.

Fig. 9.1(a) shows the shift as a function of the dispersion for the different designs and

Fig. 9.1(b) plots the shift normalized by the stack thickness for the different designs.

0.01 0.1 1 100

10

20

30

40

Chirped DesignsCavity DesignOptimized Design

Dispersion [um/nm]

Wav

elen

gth

Ran

ge/T

hick

ness

[nm

/um

]

0.01 0.1 1 100

0.5

1

1.5

2

Dispersion [um/nm]

Wav

elen

gth

Ran

ge/T

hick

ness

[nm

/um

]

Dispersion in µm/nm

Shift

in µ

m

Dispersion in µm/nm

Shift

/ St

ack

Thic

knes

s

(a) (b)

Fig. 9.1. (a) Shift as a function of dispersion for the designs in Table 9.1. (b) Shift

divided by stack thickness for the different designs.

Inspecting Fig. 9.1(b) we see that all double-chirped designs perform equally well if the

varying thickness is taken into account. Therefore, we can now hypothesize that the total

shift is proportional to the stack thickness.

Suppose now we are given the task to design a structure that demultiplexes 100 channels

with a wavelength spacing of 100 GHz or 0.8 nm around 1550 nm. From Fig. 9.1(b) we

conclude that we probably need to have a specific minimum stack thickness to achieve

136

this goal. The interesting questions now are what materials should we use for the design,

how many layers are necessary, and is such a design possible at all.

In the next sections I will try to answer these questions by developing a heuristic model

for estimating the achievable total shift ∆s of a structure with certain material and

operational properties. The approach used will be to generate large numbers of designs

with different parameters and then compare the results to extract common features. In

section 9.1 I will discuss how to generate appropriate designs automatically. In section

9.2 the model derived from the more than 700 generated designs will be presented.

Section 9.3 discusses the numerical results. In section 9.4 the maximum number of

channels is calculated as a function of the material refractive indices, the stack thickness,

the incidence angle, the crosstalk, and the center wavelength. Using this heuristic model

our design question of a 100-channel demultiplexer will be analyzed.

9.1 Automatic Generation of DesignsIn Chapter 7 we discussed that numerical optimization techniques allow the design of the

most general stacks, as they are not a priori limited to certain types of structures such as

chirped stacks or resonator stacks. Therefore, we will use numerical techniques in this

chapter. As shown in Chapter 7 the procedure for numerically obtaining a stack with the

desired dispersion properties is to synthesize a starting design followed by repeated

iterations of numerical refinement. An impedance matched Bragg stack is for example a

good starting design as discussed in Chapter 7. This starting design was then optimized

by numerical refinement techniques. The implemented refinement techniques include the

Golden-Section-Search (GS) and the Secant-Method (SEC), as well as the Conjugate-

Gradient-Algorithm (CG), the Broyden-Fletcher-Goldfarb-Shanno-Algorithm (BFGS),

the Damped-Least-Squares-Method (DLSQ), and the Hooke-&-Jeeves-Pattern-Search

(HJ) technique.1,2,3,4,5,6,7 While the first two are one-dimensional search algorithms, the

last four methods use techniques varying all design parameters simultaneously. All

techniques except DLSQ use merit functions to measure the performance of a design and

achieve performance improvements. During the refinement process the various methods

137

are used in varying order and for different numbers of iterations. The resulting designs

are manually tested every couple of iterations to ensure a good progress. Such a design

process takes on the order of four to five days as a single iteration of a specific algorithm

has a run time on the order of hours.

9.1.1 Automating the Refinement Process

It is obvious that this design process cannot be used to generate a sufficient number of

designs to derive a heuristic model within a couple of months. To accomplish this, we

have to both decrease the design time and automate the design process. Decreasing the

design time leads to less optimized designs. Thus, it is important to find a good balance

between optimization time and design error. By trial and error I found that using the

algorithm given in (9 - 3) results in reasonable designs.

Refinement algorithm (9 - 3)

1. 2 iterations HJ

2. 2 iterations CG

3. 1 iteration SEC

4. 2 iterations HJ

5. 1 iteration GS

6. 2 iterations BFGS

The algorithm is performed either one or two times for a given set of parameters using a

p-norm of six (see Chapter 7). The DLSQ method is not used as it is not based on a merit

function and only allows for second-degree i.e. square error correction.

The runtime trefine in minutes needed to refine one design can be estimated by (9 - 4),

where the proportionality constant is specific to the type of computer used. Here we

implemented all algorithms as MathCad programs and used a Pentium III 750 MHz

computer with 256 MB RAM. In (9 - 4), i is the number of times the algorithm in (9 - 3)

is performed, m is the number of periods, and p the number of points used for the merit

function.

138

[ ] pmitrefine2

6501min = (9 - 4)

From (9 - 4) we can calculate that refining a design with 100 periods using 15 points in

the merit function and one run of algorithm (9 - 3) will take approximately 230 minutes.

For 50 periods, we would need 60 minutes and, for 20 periods, 10 minutes. The number

of points necessary for the merit function depends on the wavelength interval of interest.

The spacing of the points has to be fine enough in wavelength to prevent resonances in

the design. For a 1550-nm design the spacing between points should be at least 5 nm or

0.3% of the wavelength. Thus, for a 75 nm operating range, 15 points are needed.

9.1.2 Automatic Refinement of Fixed Start Designs

For approximately the first half of the designs I manually generated a starting design

using the theory of double-chirped mirror structures.8 Then I used the automated

refinement process described in the previous section to refine the starting design to

different performances. Mostly, I fixed the operating range and specified a different total

shift for the different designs. Fig. 9.2 to Fig. 9.4 show three designs generated in such a

manner. The blue crosses represent the start design, the black circles the specified shift,

and the red crosses the refined design. The solid black line is the actual operating range

of the designs. On the right the actual operating parameters of each design as well as the

fitting error are given.

For the design in Fig. 9.2 the merit function was set such that the shift increases from 5

µm to 25 µm over a wavelength range of 70 nm. We see that the refined design agrees

well with desired characteristics.

For Fig. 9.3 and Fig. 9.4 the same starting design was used, but the shift was specified to

increase to 35 µm and 45 µm respectively. The final designs obtained do not agree well

with the desired characteristics. They start out with a shallower slope than desired, but do

show the right slope for higher wavelengths. Therefore, only part of the wavelength range

is actually usable.

139

1520 1540 1560 15800

5

10

15

20

25

30

Start designRefined designSpecified shiftvalid shift

Wavelength in nm

Shift

in u

m

∆λ 63= nm

cDisp 0.286= um/nm

shift 18= um

errorlsq 0.68= um

errorrel 3.779%=


specified to increase from 5 µm to 25 µm over a 70 nm operating range.

1520 1540 1560 15800

10

20

30

40


Wavelength in nm

Shift

in u

m

∆λ 50.167= nm

cDisp 0.429= um/nm

shift 21.5= um

errorlsq 2.604= um

errorrel 12.112%=



140

1520 1540 1560 15800

10

20

30

40

50


Wavelength in nm

Shift

in u

m

∆λ 32.667= nm

cDisp 0.571= um/nm

shift 18.667= um

errorlsq 1.59= um

errorrel 8.518%=



Algorithm for finding the usable wavelength range of a design (9 - 5)

1. Assume that the maximum specified wavelength is the end

wavelength of the range.

2. Assume that the design has the specified dispersion in the usable

range.

3. Calculate the shift at 60 points in the specified wavelength range.

4. Calculate the standard deviation from the specified line, weighting

the result by the inverse of the square root of the number of points.

5. Repeat 4, sequentially neglecting one more point at the beginning

and shifting the specified line to the new starting point until only 6

points are left.

6. Find the starting point that results in the smallest weighted standard

deviation. This is the minimum wavelength of the range.

7. Plot the resulting fit for the wavelength range found.

141

For most of the designs obtained in the way described here, the usable wavelength range

lies on the high wavelength side of the specified operating range. Under the assumption

that the usable wavelength range extends all the way to the maximum specified

wavelength, I implemented a search algorithm to find the smallest wavelength for which

the design still performs sufficiently well. This algorithm is given in (9 - 5).

The weighing by the number of points in steps 4 and 5 is done to favor larger wavelength

intervals. Otherwise the algorithm finds only parts of the actual wavelength range. That is

also the reason why at least 6 points are used for the evaluation. Fewer points can lie on a

perfect line by chance leading to a too small found wavelength range. This algorithm has

been applied to the designs in Fig. 9.2 to Fig. 9.4. The best weighted fit is given by the

black solid line clearly visible in Fig. 9.3 and Fig. 9.4. The determined wavelength

interval ∆λ, as well as the specified dispersion cDisp and the resulting shift are given to the

right of the graphs. Also given are the standard deviation errorlsq and the relative standard

deviation errorrel, i.e. the error divided by the total shift. The relative error represents the

quality of the design.

9.1.3 Automatic Generation of Starting Designs

In the previous paragraphs it has been demonstrated that we can automatically generate

different designs from a single starting design and find the usable wavelength ranges of

the designs. To test the performance of a more general type of design, it is necessary to

vary also the starting design. We want to be able to specify the refractive indices of the

two materials used, the number of layers, the incidence angle, the polarization, and the

operating wavelength and then automatically generate an appropriate starting design. It

turns out that an impedance matched Bragg stack is a good starting design. The Bragg

wavelength under a non-zero incidence angle is given by (9 - 6) and (9 - 7).

( )HHHLLLB dndn θθλ coscos2 += (9 - 6)

=

HLHL n ,

,sinarcsin θ

θ (9 - 7)

142

A single chirp of the structure performs the impedance matching. Thus, the layer

thicknesses dL and dH for the low and high refractive index layers can be calculated as a

function of the period number i using (9 - 8) and (9 - 9).

<≤−

−<≤

−−

=

mimforn

miforn

mi

d

LL

B

LL

B

Li

4cos4

41cos4

52

05.1

θλ

θ

λ

(9 - 8)

<≤−

−<≤

−=

mimforn

miforn

mi

d

HH

B

HH

B

Hi

4cos4

41cos4

5

05.1

θλ

θ

λ

(9 - 9)

Fig. 9.5 gives an example of a 20-period starting design for refractive indices of 1.45 and

2.09, an incidence angle θ of 40º, and an operating wavelength of 1550nm.

Fig. 9.5. Example of a 20-period starting design for refractive indices of 1.45 and

2.09, an incidence angle θ of 40º, and an operating wavelength of 1550nm.

Now we have completely automated the design process including the generation of the

starting design and the refinement to a final design.

0 5 10 15 20 25 30 35 400

200

400

600

Layer #

Phys

ical

Thi

ckne

ss [n

m]

143

9.2 Deriving a Heuristic Shift ModelUsing the automated design process described in section 9.1 a total of 760 different

designs was generated including 190 designs with 20 periods, 49 with 30 periods, 379

with 50 periods, 22 with 60 periods, and 120 with 100 periods. Each design had a

different set of design and material parameters. The refractive indices, the incidence

angle, the number of layers, and the specified dispersion characteristics were varied. All

designs assume a center wavelength of 1550 nm and p-polarization. Different center

wavelengths could be obtained by simple scaling, while the effect of polarization still

needs to be analyzed at a later time. The pure computation time for these designs was

approximately 1100 hours or 45 days. In this section I will present the results obtained

from these designs. 137 designs were filtered out as they had a relative error larger than

25%, leaving 623 results to be analyzed.

As discussed in the introduction to this chapter, we want to determine the total shift ∆s as

a function of the design and material parameters. Fig. 9.6(a) to Fig. 9.11(a) plot the

resulting shift as a function of the different design parameters. In all the graphs red points

represent designs of 20 or 30 periods, blue points 50 or 60 periods, and magenta points

100 periods, as seen from Fig. 9.6. By considering the resulting shift of all these different

structures, the heuristic model (9 - 10) is obtained for the total shift. In this equation ∆n is

the refractive index difference between the two used material, navg is the average

refractive index, L the total stack thickness and θ the incidence angle in vacuum.

( )θλ sin16 2avg

Disp nnLcs ∆

=∆=∆ (9 - 10)

Intuitively, it seems reasonable that the maximum shift possible should be proportional to

the relative width of the Bragg stopband for the given parameters and to the distance

traveled in the design. These assumptions were used to find a model fitting the data. If

this model is correct, normalizing the observed shift by the right hand side of (9 – 10)

should lead to a unity result independent of the material and design parameters.

Fig. 9.6(b) to Fig. 9.11(b) plot the normalized shift as a function of the different design

parameters.

144

0 20 40 60 80 1000

200

400

600

800

m=20/30m=50/60m=100

Average Refractive Index

Shift

[um

]

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

m=20/30m=50/60m=100


Nor

mal

ized

Shi

ft [a

.u.]

Number of Periods

Shift

in µ

m

Number of Periods

Shift

/ M

odel

Fig. 9.6. Dependency of the shift on the number of periods.

20 40 600

200

400

600

800

m=20/30m=50/60m=100


Shift

[um

]

20 40 600

0.5

1

1.5

2

2.5

m=20/30m=50/60m=100


Nor

mal

ized

Shi

ft [a

.u.]

Incidence Angle in deg

Shift

in µ

m

Incidence Angle in deg

Shift

/ M

odel

Fig. 9.7. Dependency of the shift on the incidence angle.

From Fig. 9.7 we see that the maximum obtainable shift increases with incidence angle.

The maximum shift decreases with the average refractive index as shown in Fig. 9.8. It

increases with the refractive index contrast ratio and the stack thickness as seen from

Fig. 9.9 to Fig. 9.11.

145

1 1.5 2 2.5 3 3.5 40

200

400

600

800

m=20/30m=50/60m=100


Shift

[um

]

1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

m=20/30m=50/60m=100


Nor

mal

ized

Shi

ft [a

.u.]

Average refractive index

Shift

in µ

m


Shift

/ M

odel

Fig. 9.8. Dependency of the shift on the average refractive index.

0 10 20 30 40 50 60 700

200

400

600

800

m=20/30m=50/60m=100


Shift

[um

]

0 10 20 30 40 50 60 700

0.5

1

1.5

2

2.5

m=20/30m=50/60m=100


Nor

mal

ized

Shi

ft [a

.u.]

Refractive Index Contrast in %

Shift

in µ

m

Refractive Index Contrast in %

Shift

/ M

odel

Fig. 9.9. Dependency of the shift on the refractive index contrast.

146

0 20 40 600

200

400

600

800

m=20/30m=50/60m=100


Shift

[um

]

0 20 40 600

0.5

1

1.5

2

2.5

m=20/30m=50/60m=100


Nor

mal

ized

Shi

ft [a

.u.]

Physical Stack Thickness in µm

Shift

in µ

m

Physical Stack Thickness in µm

Shift

/ M

odel

Fig. 9.10. Dependency of the shift on the physical stack thickness.

0 10 20 30 40 50 600

200

400

600

800

m=20/30m=50/60m=100


Shift

[um

]

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

m=20/30m=50/60m=100


Nor

mal

ized

Shi

ft [a

.u.]

Optical Stack Thickness in µm

Shift

in µ

m

Optical Stack Thickness in µm

Shift

/ M

odel

Fig. 9.11. Dependency of the shift on the optical stack thickness.

Fig. 9.12 shows the shift as a function of the relative fitting error. It can be seen that there

is no particular dependency of the shift on the fitting error.

147

0 10 20 300

200

400

600

800

m=20/30m=50/60m=100


Shift

[um

]

0 10 20 300

0.5

1

1.5

2

2.5

m=20/30m=50/60m=100


Nor

mal

ized

Shi

ft [a

.u.]

Relative Fitting Error in %

Shift

in µ

m

Relative Fitting Error in %

Shift

/ M

odel

Fig. 9.12. Dependency of the shift on the relative error.

0 20 40 60 80 1000

200

400

600

800

m=20/30m=50/60m=100


Shift

[um

]

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

m=20/30m=50/60m=100


Nor

mal

ized

Shi

ft [a

.u.]

Filling Ratio in %

Shift

in µ

m

Filling Ratio in %

Shift

/ M

odel

Fig. 9.13. Dependency of the shift on the filling ratio.

Fig. 9.13 plots the shift as a function of the wavelength interval filling ratio, i.e. the ratio

of the usable wavelength interval to the total specified wavelength interval. The large

number of designs with a filling ratio of approximately 50 % is due to the fact that I

actually specified the filling ratio to be 50 % after guessing a shift model. The 50 %

148

filling ratio is desirable for this type of test as small filling ratios have a larger error and

large filling ratios do not test the design limits as well.

( )inavg

Disp nnLcs θλ sin16 1

∆=∆=∆

1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

m=20/30m=50/60m=100


Nor

mal

ized

Shi

ft [a

.u.]


Shift

/ M

odel

1 1.5 2 2.5 3 3.5 40

2

4

6

8

m=20/30m=50/60m=100

Average Refractive IndexN

orm

aliz

ed S

hift

[a.u

.]

( )inavg

Disp nnLcs θλ sin16 3

∆=∆=∆

Average refractive indexSh

ift /

Mod

el

(a) (b)

Fig. 9.14. Two poor models for comparison. (a) Assuming the shift to be inversely

proportional to the refractive index. (b) Assuming the shift to be inversely

proportional to the refractive index cubed.

After looking at Fig. 9.6(b) to Fig. 9.11(b) the reader may not be fully convinced that

(9 - 10) is indeed a good model for the shift. As a comparison Fig. 9.14 plots the

normalized shift for two wrong modes. Fig. 9.14(a) assumes the shift to be inversely

proportional to the refractive index, while Fig. 9.14(b) assumes the shift to be inversely

proportional to the refractive index cubed instead of the correct inverse proportionality to

the refractive index squared. In Fig. 9.14(a) a decrease in the shift as a function of the

average refractive index is clearly visible. In Fig. 9.14(b) an increase of the shift with

average refractive index is seen. Thus, a shift that is inversely proportional to the square

of the average refractive index seems indeed to be the best model. As we are basically

guessing a model here based on a numerical data set, there is room for error and

149

improvement. Maybe the average refractive index is really inversely proportional to the

refractive index to the power 2.1. As this result is close to the guessed one, we cannot

conclusively decide either way from the given data. An analytical model is needed to

answer this question satisfactorily. But since no analytical model is available to date, we

will use the derived heuristic model in the mean time.

9.3 Constancy of Dispersion×Wavelength-Range-Product

In 9.2 we numerically derived a model for the maximum total shift that can be obtained

for a given parameter set. The heuristic model in (9 - 10) tells us that for a given set of

refractive indices, a given stack thickness, and a given incidence angle, the total shift ∆s,

which is equal to the dispersion×wavelength-range-product cDisp∆λ, is constant.

Therefore, if we specify a specific dispersion, the wavelength range over which we can

achieve this dispersion is limited and vice versa. This fact can be seen very impressively

if we plot the wavelength range normalized by the expected cDisp∆λ-product as a function

of the dispersion for all the generated designs. In Fig. 9.15 this normalized wavelength

range is plotted as a function of the dispersion. The red diamonds represent designs with

less than 10 % fitting error, the pink plusses 10 % to 25 % fitting error, and the blue

points show the performance of the designs listed in Table 9.1. If our model is correct

that the dispersion×wavelength-range-product cDisp∆λ is constant, dividing the observed

wavelength range by the theoretical value of cDisp∆λ should result in 1/cDisp., which is just

inversely proportional to the dispersion itself. The expected curve is plotted as the solid

black line in Fig. 9.15.

Fig. 9.15 clearly shows that there seems to be a physical limit to the size of the

dispersion×wavelength-range-product. It cannot be pure chance that we did not find any

design with a higher normalized wavelength range. Designs can always be worse as the

automated design process may have failed. The high number of designs close to the

maximum dispersion×wavelength-range-product shows though that our design procedure

works quite well. The designs discussed in Chapters 5 to 7 all lie slightly below the

150

model. This may be due to a tradeoff between ripple in the shift and total shift. As the

designs in this chapter are generated automatically, they may not be as flat as the designs

of the previous chapters. Fig. 9.15 also shows very impressively that all three design

algorithms discussed in Chapters 5 to 7 yield structures of similar performance.

Therefore, the algorithm that most easily generates a good design should be chosen. For

broadband designs this is most likely the double-chirped mirror design technique, while

for narrowband designs digital lattice techniques resulting in coupled-cavity structures

are interesting. If none of these analytical techniques is applicable as for structures with

an intermediate bandwidth, numerical optimization can be used.

0.01 0.1 1 10 1001 .10 3

0.01

0.1

1

10

100

Dispersion [um/nm]

Nor

mal

ized

Wav

elen

gth

Ran

ge [a

.u.]

Dispersion [µm/nm]

Wav

elen

gth

Ran

ge /

(cD

isp∆

λ )th

eory

[nm

/µm

]

Model10 % < relative error < 25 %relative error < 10 %Chirped DesignsCoupled-Cavity DesignOptimized Design

Fig. 9.15 Normalized wavelength range as a function of the specified dispersion.

151

9.4 Model for the Maximum Channel NumberNow we can use the heuristic model (9 - 10) for the maximum shift to estimate the

maximum number of channels given the refractive indices of the two materials used, the

stack thickness, the incidence angle, the polarization, and the operating wavelength.

Substituting (9 - 10) into (9 - 1), we obtain (9 - 11).

( ) ( )

−

∆+≤ 2

22

2sin1sin81

s

inin

avgcrosstalkchannels nn

nc

LN θθπ

λ(9 - 11)

It is interesting to note that for a given set of materials, incidence angle, and adjacent

channel crosstalk, everything on the right hand side except for the ratio of the stack

thickness L to the wavelength λ is fixed. For a crosstalk –40dB, a 45º incidence angle,

and alternating SiO2 and Ta2O5 layers, the number of channels can be estimated by

(9 - 12).

λLN OTaSiO 53.0145,522 +≤°− (9 - 12)

For a 50 µm stack and an operating wavelength of 1550 nm, we obtain a maximum of 18

channels from (9 - 12). For a 13 µm stack and an operating wavelength of 850 nm, we

expect 9 channels. The 13-µm stack discussed in Chapter 8 had a theoretical limit of 7

channels, which is not far off from the nine channels estimated here. Keep in mind that

this shift model is an approximation and thus serves as a rough guide to how many

channels we might expect. For comparison we consider the number of channels obtained

for a –40dB crosstalk, a 45º incidence angle, and alternating GaAs and AlAs layers in

(9 - 13).

λLN AlAsGaAs 21.0145, +≤°− (9 - 13)

We see that we expect 2.5 times fewer channels using GaAs-AlAs as compared to SiO2-

Ta2O5 for the same stack thickness. If a design is transferred from SiO2-Ta2O5 to GaAs-

AlAs though, the thickness scales by the inverse of the refractive index. Therefore, the

same design would be a factor of two thinner in GaAs-AlAs, resulting in about 5 times

fewer channels.

152

It is interesting to note that the number of channels scales as the thickness divided by the

wavelength. Thus, scaling a design to a different wavelength in the same material system

does not change the number of channels, just as we would expect. In this model the

thickness not the number of layers seems to matter. That is not correct though, since the

layer thickness should be on the order of a quarter wavelength. Therefore, fixing a stack

thickness also fixes the number of layers needed.

Finally, let us consider the design question posed at the beginning. Can we design a

structure that demultiplexes 100 channels with a wavelength spacing of 100 GHz or

0.8 nm around 1550 nm? Setting the number of channels in (9 – 12) equal to 100, we see

that we would need a stack that is approximately 270 µm thick. This corresponds to

approximately 1000 layers, which is well out of the range of what is possible today. If we

have the freedom to change the material system from SiO2-Ta2O5 to something different,

we see from (9 - 11) that we would want a high contrast ratio and a low average

refractive index. Furthermore, changing to a larger incidence angle could also help to

obtain more channels.

References

[1] Li Li and J. A. Dobrowolski, Appl. Opt., 31/19, pp.3790-3799 (1992).

[2] A. Thelen, Design of Optical Interference Coatings, McGraw-Hill, Inc. (1989).



[4] J. A. Dobrowolski, F.C. Ho, A. Belkind, and V.A. Koss, Appl. Opt., 28/14,

pp.2824-2831 (1989).

[5] J. A. Dobrowolski and R.A. Kemp, Appl. Opt., 29/19, pp.2876-2893 (1990).


Inc. (1996).



153



Electron., 35/2, 129-137 (1999).

154

Chapter 10

Applications of Spatially Dispersive Stacks

In this chapter we discuss possible applications for thin-film stacks with high spatial

dispersion. Section 10.1 considers using such stacks for dense wavelength division

multiplexing (DWDM) or coarse wavelength division multiplexing (CWDM). While in

the previous chapters we have targeted a linearly changing beam position as a function of

wavelength, section 10.2 investigates the use of step-like beam shifting. While a linear

beam shift with wavelength corresponds to a Gaussian passband shape, this step-like

beam shifting results in a passband shape that is flat-top. This is interesting for system

considerations. Section 10.3 suggests the use of spatially dispersive stacks for

wavelength-dependent beam steering.

10.1 Wavelength Multiplexing and DemultiplexingIn Chapter 1 we discussed that wavelength division multiplexing (WDM) can be

employed to increase the data capacity of optical fibers. In WDM systems each channel is

encoded with a different optical wavelength. The different wavelength channels are

spatially overlapped using a wavelength multiplexer (MUX) for transmission over a

single fiber as shown schematically in Fig. 1.1. On the receiver side a wavelength

155

demultiplexer (DEMUX) is used to spatially separate the wavelength channels. Today

Dense WDM (DWDM) with typically hundreds of channels is employed in long-haul

systems, and Coarse WDM (CWDM) with approximately four to sixteen channels is used

in metro systems. Here we will first consider the use of the spatial MUX/DEMUX

discussed in this work for DWDM systems and afterwards for CWDM systems.

The International Telecommunications Union (ITU) has standardized a frequency grid for

DWDM. It is centered at 193.1 THz (1552.52 nm) with a channels spacing of 100 GHz

(0.8 nm).1 The center frequency is chosen at 1.55 µm as optical fibers have minimum

propagation loss at this wavelength. Erbium-doped fiber amplifiers (EDFA) can be used

to simultaneously amplify all wavelength channels between approximately 1520 nm to

1570 nm in the optical domain. Therefore, the so-called conventional band (C-band) from

1520 nm to 1570 nm has initially been used for the transmission of up to 64 channels

over a single fiber. Recently, the number of DWDM channels has been increased further

by using smaller channel spacings of 50 GHz or even 25 GHz and by extending the

wavelength range to the short band (S-band) from 1450 nm to 1525 nm and the long band

(L-band) from 1560 nm to 1610 nm. As there is no standard definition for the position of

these bands, all the intervals given are only approximate.

Here we will consider a typical DWDM system with 64 100-GHz channels in the C-band.

Today the most common choice of a multiplexing/demultiplexing device for such a

system is probably an arrayed waveguide grating (AWG). As discussed in Chapter 1

AWGs allow a high channel count at low loss in a compact device. A single AWG can be

used to multiplex or demultiplex all 64 channels, justifying the rather high initial cost of

the AWG. As seen in the experimental results in Chapter 7 as well as the discussion on

the maximum number of expected channels in Chapters 8 and 9, it is not likely that the

spatial MUX discussed in this work can be used to multiplex or demultiplex 64 channels.

A different fabrication technique or device architecture (such as an in-plane device)

would be needed to achieve a sufficient stack thickness and refractive index contrast for

demultiplexing 64 channels. Likely, the thin-film spatial MUX can demultiplex around 8

channels using today’s fabrication technology. Therefore, a single such device is not

156

sufficient for a 64-channel system. As shown below several devices can be combined

though for a 64-channel system.

The main disadvantage of AWGs is that they cannot be upgraded modularly. Therefore, a

company setting up a DWDM system with eight channels to be upgraded to 64 channels

gradually, might choose thin-film filters (TFF) or fiber Bragg gratings (FBG) instead of

an AWG. Both these devices are essentially filters and operate by transmitting one

channels while reflecting all other channels as depicted in Fig. 10.1. Therefore, in order

to multiplex or demultiplex 64 channels, 63 different TFFs or FBGs need to be cascaded

as shown in Fig. 10.2.

Frequency100 GHz

Frequency

Frequency

100 GHzFilter

Fig. 10.1. Operating principle of a frequency filter.

Channel 1

…

100 GHzFilter

64 channels with100 GHz spacing

Channel 2100 GHzFilter


100 GHzFilter

…

Fig. 10.2. Demultiplexing architecture based on cascading filters. For 64 channels

63 filters are needed.

157

The architecture as shown in Fig. 10.2 has two main disadvantages. First, all filters need

to be different, as they are targeted for a different wavelength. This increases the

fabrication cost. It is less expensive to fabricate many identical devices. Second, the last

channel has to pass through all the filters and is thus heavily attenuated, while the first

channel is hardly attenuated at all. Both these disadvantages can be alleviated by using a

log2N-architecture, where N is the number of channels.2 This architecture is based on the

use of interleavers instead of filters. As shown in Fig. 10.3, an interleaver separates out

every other channel. Thus, every channel passes through a maximum of log2N stages

distributing the attenuation evenly. An example of a log2N-architecture is given in

Fig. 10.4. Even though this architecture is advantageous compared to the one in Fig. 10.2,

we see that still 63 devices are needed to separate out 64 channels.

Frequency100 GHz

100 GHzInterleaver

Frequency

Frequency200 GHz

200 GHz

Fig. 10.3. Operating principle of a wavelength interleaver.

100 GHzInterleaver

200 GHzInterleaver

200 GHzInterleaver

400 GHzInterleaver

400 GHzInterleaver

800 GHzInterleaver

800 GHzInterleaver

1600 GHzInterleaver

1600 GHzInterleaver

3200 GHzInterleaver

3200 GHzInterleaver

Channel 1

Channel 2

…

…

…

… …

64 channels with100 GHz spacing Channel 3

Channel 4

Fig. 10.4. Demultiplexing architecture based on cascading interleavers. For 64

channels 63 interleavers are needed.

158

100 GHzInterleaver

200 GHzInterleaver

200 GHzInterleaver

400 GHzInterleaver

400 GHzInterleaver

…

…

Channel 1…

800 GHzFilter



800 GHzFilter

…


Channel 2

…

800 GHzFilter



800 GHzFilter

…


filters. For 64 channels 63 devices are needed.

As shown in Fig. 10.5, any combination of the architectures introduced in Fig. 10.2 and

Fig. 10.4 may also be used for demultiplexing, but any architecture based on filters or

interleavers will need 63 devices to separate out 64 channels. The reason why any of the

N-channel architectures using filters or interleavers needs N-1 devices is that both these

devices have only two outputs – the reflected and the transmitted beam. The big

advantage of the spatial MUX discussed in this work is that multiple channels can be

demultiplexed using a single device. Instead of being a 3-port device like filters or

interleavers (one input, two outputs), the spatial MUX can, for example, have nine ports

(one input, eight outputs). Using it in an architecture as shown in Fig. 10.6, we see that

only 15 instead of 63 devices are needed for a 64-channel system. Since only 15 devices

are necessary, the optical assembly is less complicated than using filters or interleavers

and the overall module size is reduced. The size of a single spatial MUX is larger though

than the size of a single TFF. Since the number of stages is reduced using a spatial MUX,

the loss may be lowered compared to other architectures. The typical loss for commercial

8-channel 100-GHz multiplexers based on thin-film filters is around 4 to 5 dB. Our

device could probably do better than that.

159

100 GHzInterleaver

200 GHzInterleaver

200 GHzInterleaver

400 GHzInterleaver

400 GHzInterleaver

…

…


Channel 3Channel 4Channel 5

SpatialMUXwith

6.2 nmchannelspacing Channel 6

Channel 7Channel 8

Channel 1Channel 2

Channel 3Channel 4Channel 5

SpatialMUXwith

6.2 nmchannelspacing Channel 6

Channel 7Channel 8

Channel 1Channel 2


spatial MUXes. For 64 channels only 15 devices are needed.

Concluding, we see that the spatial MUX could be used beneficially for DWDM

applications if modularity of the system is desired. Due to the similarity with a TFF, it

can be expected that the cost of a spatial MUX is substantially lower than that of an

AWG, while the simultaneous multiplexing of several channels decreases the module

complexity and cost compared to TFFs of FBGs. The only disadvantage of the spatial

MUX is that it is polarization sensitive. A combination of two spatial MUXs can be used

to spatially shift the two different polarizations separately. One spatial MUX could be

designed to shift the p-polarization and not effect the s-polarization and the second MUX

could work vice versa. The incoming light does not need to be split into the two

polarizations, but can be propagated through the two stacks in series. Alternatively,

multilayer coatings could be deposited on both sides of the substrate, each side affecting

only one polarization.

CWDM systems are just emerging for increasing the data capacity of local or metro

optical networks. CWDM systems have typically four to sixteen channels with channel

spacings around 20 nm in the wavelength ranges around 0.8 µm, 1.3 µm, or 1.55 µm. For

these systems cost is the most important factor. As the cost of multiplexing and

160

demultiplexing devices is distributed over only a couple of channels, the MUX/DEMUX

devices have to be cost-effective both in initial cost as well as in operation. Typically,

TFFs or FBGs are used in an architecture as the one shown in Fig. 10.2. For CWDM

systems the spatial MUX described in this thesis is particularly interesting, as a single

spatial MUX is sufficient for multiplexing or demultiplexing all CWDM channels.

Furthermore, the spatial MUX can be fabricated cost-effectively using well-known thin-

film fabrication technology. As shown in the previous chapters, it is no problem to design

a spatial MUX with a 20-nm channel spacing. A factor important for CWDM that we

have not discussed so far is the passband shape. In the next section we will discuss this

topic and introduce an improved spatial MUX for CWDM applications.

10.2 Step-Like Beam ShiftingIn the previous chapters we have discussed stacks that have a linear shift as a function of

wavelength. These stacks allow for the highest number of channels, as all the different

channels focus on the same output plane. Fig. 10.7 repeats the experimental result for 8

bounces off a 66-layer numerically optimized stack discussed in section 7.4.

Posi

tion

alon

g x-

axis

in µ

m

Out 4

Out 3

Out 2

Out 1

Wavelength in nm

λc1=827.3 nmλc2= 831.0 nm

λc3= 834.7 nmλc4= 840.6 nm

Fig. 10.7. Intensity as a function of position and wavelength for 8 bounces off a 66-

layer numerically optimized stack at 54º incidence angle and p-polarization.

161

Fig. 10.7 also shows the spatial position of the four wavelength channels that can be

spatially demultiplexed using this structure. The field of the channels at 827.3 nm,

831.0 nm, 834.7 nm, and 840.6 nm is spatially separated and these channels are therefore

demultiplexed. Even though the field at 831.0 nm is exactly within the output Out 2, as

soon as the wavelength of the channel drifts to longer or shorter wavelengths, part of the

field is in Out 3 or Out 1 respectively. This leads to loss in Out 2 and to crosstalk in Out 3

or Out 1. Since the beams have a Gaussian beam shape, a linear beam shift as a function

of wavelength corresponds to a Gaussian passband shape as plotted schematically in

Fig. 10.8(a). In a transmission system the laser wavelengths may drift, e.g., due to

temperature changes. In DWDM systems channel wavelengths are normally stabilized

and do not drift much. In CWDM systems on the other hand, the lasers are not stabilized

for cost reasons and may drift significantly. Therefore, it is very important for good

system performance that the multiplexing devices have a flat-top passband shape as

shown in Fig. 10.8(b).

1500 1520 1540 1560 1580

Channel 1Channel 2Channel 3Channel 4

Wavelength in nm1500 1520 1540 1560 1580

Wavelength in nm

Wavelength in nm

0

-30

-60

Tran

smis

sion

in d

B

Wavelength in nm

0

-30

-60

Tran

smis

sion

in d

B

Gaussian passband Flat-top passband

(a) (b)

Fig. 10.8. Schematic of a (a) Gaussian passband shape compared to a (b) flat-top

passband shape (transmission refers to the energy transfer from input to output).

162

A flat-top passband shape allows for laser drift, as there is no loss over a range of

wavelengths not just a single wavelength as in the case of Gaussian passbands. Fig. 10.9

shows that in order to implement a flat-top passband shape for the spatial MUX, we need

to design a structure that has a step-like shift as a function of wavelength. Such a step like

shift results in a range of wavelengths to be shifted to the same exit position. Therefore, if

the output is located at this position, there will be no loss for any wavelength within the

range.

1500 1520 1540 1560 1580

0

50

100

150

200

Desired shiftWavelength in nm

Shift

in u

m a

fter 8

bou

nces

Wavelength in nm

Shift

in µ

m

1500 1520 1540 1560 1580

Channel 1Channel 2Channel 3Channel 4

Wavelength in nm

Wavelength in nm

0

-30

-60

Tran

smis

sion

in d

B

(a) (b)

Fig. 10.9. Flat-top passband shape (a) corresponds to a step-like beam shift with

wavelength (b).

Fig. 10.10(a) demonstrates that we can indeed design a multilayer thin-film stack that has

a step-like shift as a function of wavelength.3 The results are for a 100-layer SiO2/Ta2O5

stack at 45º incidence angle and p-polarization. The stack composition is given in

Appendix G. The stack was designed using the numerical optimization techniques

described in Chapter 7. The last 20 layers were fixed as a quarter-wave dielectric mirror,

while the other 80 layers were varied to approximate the desired step-like shift shown in

Fig. 10.10(a). The reflectance seen in Fig. 10.10(b) could be increased further by adding

more mirror layers. The resulting stack has a total thickness of 33.1 µm.

163

1500 1520 1540 1560 15800

0.2

0.4

0.6

0.8

1

Wavelength in nm

Ref

lect

ance

afte

r 8 b

ounc

es

1500 1520 1540 1560 1580

0

50

100

150

200

DesignDesired shift

Wavelength in nm

Shift

in u

m a

fter 8

bou

nces

Wavelength in nm

Shift

in µ

m a

fter 8

bou

nces

Ref

lect

ance

afte

r 8 b

ounc

es

Wavelength in nm

(a) (b)

Fig. 10.10. Shift (a) and reflectance (b) after 8 bounces off a 100-layer numerically

optimized stack at 45º incidence angle and p-polarization.

Fig. 10.10 proves that we can use the techniques developed in the earlier chapters for a

varied range of dispersion profiles. The resulting 4-channel step-like design with 20-nm

channel spacing is particularly interesting for CWDM applications.

It is also advantageous for cascading multiple devices to prevent a shrinking of the low

loss passband range. In Chapter 8 we discussed that a linear shift as a function of

wavelength results in all beams being focussed on the same output plane. The channels

are also all focussed at the same position if for each channel wavelength and for all

angular components of the incident beam the spatial dispersion has the same constant

value. From Fig. 10.10 we see that the dispersion is zero around the center wavelength of

each channel. Therefore, if the input angular range is sufficiently small, all beams will

focus at the same position. Appendix D discusses that a beam can be seen as a

composition of plane waves with different incidence angles. Only a limited range of these

angular components has a significant intensity. Thus, the input angular range of ±∆θ can

be approximated by the drop of the intensity to 1/e2 as given in (10 - 1).

0wπλ

θ ≈∆ (10 - 1)

164

As shown in Appendix F, a change in the input angle θ is approximately proportional to a

change in frequency ω, since the right hand side of (10 - 2) is approximately constant.

ωβ

θ

θωθω

∂∂

−−=

∆∆

c~sin

~cos~(10 - 2)

Table 10.1 gives three examples of the relationship between spot size, angular range, and

wavelength range. The group velocity along the layers ∂β/∂ω is set to 0.3 times the speed

of light. This is approximately the right value as seen from different investigated

structures in Chapter 4.

Spot Size w0 Angular Range ∆θ Wavelength Range ∆ω

15 µm

30 µm

50 µm

1.9º

0.9º

0.6º

14 nm

7 nm

4 nm

Table 10.1. Relationship between spot size, angular range, and wavelength range at

1540 nm, 45º incidence angle, and ∂ω/∂β=0.3c .

From Table 10.1 we see that for a beam with a spot size of 50 µm, the different beam

components probe approximately ±4 nm of the design around the incident frequency.

Remember that the incident light is assumed to be monochromatic and that this probing is

due to the different angular components. As the design in Fig. 10.10 is quite flat over a

±4 nm wavelength range around the center wavelength, we expect the beams to be

focussed at the same output plane. Towards the edge of the band of zero dispersion, they

will become broadened though. Therefore, not the whole range that is flat for the plane

wave calculation in Fig. 10.10 may be usable. Due to the different angular components, a

beam is distorted if the dispersion is not constant. This again shows why stacks with a

linear shift as a function of wavelength are so interesting. Features of the shift in addition

to linear dispersion will be averaged out depending on the angular range of the beam.

Depending of the shape of the dispersion curve, this averaging can have very distorting

consequences for the beam profile. You can basically imagine that for a strong ripple in

the dispersion, the different beam components are not transferred to the correct position

at the output, resulting in beam distortion. In a very extreme case the ripple of the Bragg

165

stack depicted in section 3.3 corresponds to half of the beam components being shifted,

while the other components are reflected off the front of the stack. This splitting of the

angular content of the beam into two sets, one shifted and one not shifted results in two

separate output beams as discussed in Chapter 3 and broadening of these output beams,

as their angular content is reduced. Careful beam simulations for the Bragg stack show

that the reflected beam is broader below the stop band than within the stop band. A

dispersion profile that has a constant component, and additionally only components that

vary slowly with wavelength, will transfer the input beam to the output with very little

distortion.

1500 1510 1520 1530 1540 1550 1560 1570 1580

25

30

35

40

45

50

Plane WaveBeam 15 umBeam 30 umBeam 50 um

Wavelength in nm

Shift

in u

m a

fter 8

bou

nces

Wavelength in nm

Shift

afte

r one

bou

nce

in µ

m

Fig. 10.11. Shift as a function of wavelength for a plane wave at 45º incidence

angle, and Gaussian beams with 15 µm, 30 µm, and 50 µm spot size.

166

As a 50-µm beam probes a ±4 nm wavelength range, features smaller than this

wavelength range will be averaged out. For a 15-µm beam features smaller than ±14 nm

will be averaged out. Considering this, we cannot expect the design in Fig. 10.10 to work

well with a 15-µm beam. Fig. 10.11 plots the shift as a function of wavelength for plane

waves and three different beam sizes. Indeed we see that for a 15-µm beam the step-like

shift is completely averaged out, while it is well visible for the 30-µm and 50-µm beam

size. This shows that in constructing a spatial MUX with a step-like shift we have to take

care to use a sufficiently large spot size.

800 820 840 860 880 900 9200

5

10

15

20

25

40 deg45 deg50 deg55 deg

Wavelength in nm

Spat

ial S

hift

alon

g x-

dire

ctio

n in

um

Wavelength in nm

Shift

in µ

m

Fig. 10.12. Shift as a function of wavelength for four different incidence angles for

a 66-layer stack.

As the 4-channel step design has not yet been fabricated, we conclude this section with

some experimental results obtained from the 66-layer stack discussed in Chapter 7. In

Chapter 7 we used the stack at 54º incidence angle and observed a linear shift as a

function of wavelength. By luck (and consideration, of course) we found that the same

design can serve as a 3-channel step design. As a change in incidence angle and a change

in wavelength are only approximately proportional, we observe a change in the shift

profile as well as the wavelength position if we change the incidence angle. Fig. 10.12

167

plots the shift as a function of wavelength for four different incidence angles. At 55º we

see the linear shift, but at around 50º the shift appears step-like.

Indeed, performing the experiment we clearly see in Fig. 10.13(a) the step-like spatial

shift at an incidence angle of 48º demultiplexing three channels.3 The intensity at each

wavelength is normalized to unity, as this structure was not designed for high reflectance

at 48º and shows significant loss. Nevertheless, the agreement between the

experimentally observed shift and the theoretically calculated shift as seen in

Fig. 10.13(b) is promising.

Wavelength in nm

Posi

tion

in µ

m

830 840 850 860 870 88050

0

50

100

150

200

ExperimentTheory

Wavelength in nm

Shift

in u

m

Wavelength in nm

Shift

in µ

mOut 3

Out 2

Out 1

(a) (b)

Fig. 10.13. (a) Intensity as a function of position and wavelength for 8 bounces off

a 66-layer numerically optimized stack at 48º incidence angle and p-polarization.

(b) Comparison between experiment and plane wave theory.

10.3 Temporal vs. Spatial DispersionAs discussed in Chapter 4, stacks with spatial dispersion exhibit temporal dispersion at

the same time. For short pulse operation, it may be important to eliminate this temporal

dispersion. Fig. 10.14(a) shows how a combination of two stacks with opposite

dispersion can be used in series to cancel the temporal dispersion, while doubling the

spatial dispersion. On the other hand, backreflection through the same stack can be used

168

if a system with temporal dispersion, but no spatial dispersion, is desired as shown in

Fig. 10.14(b). Replacing the mirror by a modulator array or another optical element, the

different wavelength components can be modulated independently. This system can

easily be extended to a 2-D array by using the dimension out of the plane of the page. In

fact, all of the designs here have the advantage that multiple beams in the y-direction can

be multiplexed in parallel – a potentially significant benefit.

Stack 2Stack 1 Stack 1 Mirror,modulatorarray, or otheroptical element

(a) (b)

Fig. 10.14. Systems using stacks with high spatial dispersion. (a) Combination of

two stacks with opposite dispersion can be used to obtain spatial dispersion without

temporal dispersion. (b) System providing temporal dispersion without spatial

dispersion. This system could also be used to manipulate channels of different

wavelengths independently.

10.4 Beam SteeringAnother potential application for stacks with high spatial dispersion is wavelength-

dependent beam steering. Beams of different wavelength leave the device at different exit

positions, but propagate in parallel contrary to the case of a prism or a grating. The spatial

dispersion of the stack may be converted to an angular dispersion by use of a simple lens

as shown in Fig. 10.15. From Fig. 10.15 we see that the dispersive stack could therefore

be used for wavelength-dependent angular beam steering, i.e. by changing the

wavelength of the beam, the propagation direction is changed. Since the stack can be

designed to exhibit a large spatial shift over a narrow wavelength range, in principle very

little tuning of the laser wavelength may be necessary. Thus, we obtain a beam steering

device that is both compact and can be tuned rapidly, as it is tuned electrically not

mechanically. The experimental setup discussed in section 3.2 uses this beam steering

169

effect as the focal plane is imagined onto a CCD camera approximately 1m away. As

long as there is no beam clipping on the front side gold coating, the focal plane does not

have to be exactly at the output plane, since the device is linear. The same is true for

alignment along the x-direction. Since the y-direction anyway has no influence on the

device operation, the device is therefore tolerant to an offset in any direction.

Furthermore, since a change in the incident angle has, to first order, an equivalent effect

to a change in the incident wavelength, a deviation from the specified incidence angle is

also not critical. It can be corrected by wavelength tuning without resulting in loss. The

high tolerance to alignment errors is a big advantage of this type of device, compared,

e.g., to waveguide devices that have to fulfill mode-matching criteria to prevent loss.

zy

x

Lens

Fig. 10.15. Wavelength-dependent beam steering.

10.5 Other Applications Using Beam ShiftingIn contrast to thin-film filters or fiber Bragg gratings, the stacks with spatial dispersion

discussed here have continuous dispersion. Therefore, their application is not limited to

discrete channels. As seen in section 10.3 the spatial dispersion of the device can be

converted to an angular dispersion using a lens and the continuous spatial shift is

170

converted to a continuous angular sweep. Therefore, this device can in principal replace

prisms or gratings used to obtain angular dispersion in many applications including many

other spectrometer applications. The advantage of stacks with spatial dispersion is that

their dispersion profile can be designed to best suit the application. In this work we have

seen that a wide range of dispersion profiles can be obtained using the design techniques

described in Chapters 5 to 7. This unprecedented flexibility in designing the spatial

dispersion properties is interesting for existing systems and might lead the way to other

novel systems.

References

[1] R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective,

Morgan Kaufmann Publishers, San Francisco, CA (1998).



[3] M. Gerken et al, IEEE LEOS 2002 Annual Meeting, Glasgow, Scotland (November

10-14, 2002). Paper ThV 3.

171

Chapter 11

Conclusions

This work investigated multilayer thin-film stacks with high spatial dispersion in terms of

the physical origin of spatial dispersion, methods for designing stacks with spatial

dispersion, and applications of stacks with spatial dispersion. We found that the spatial

shift a beam experiences upon reflection off a multilayer stack is proportional to the

stored energy within the stack. Therefore, designing a stack with spatial dispersion is

equivalent to designing a structure that has a wavelength-dependent amount of stored

energy. Four different types of structures that have a wavelength-dependent amount of

stored energy and thus exhibit spatial dispersion have been discussed – one-dimensional

photonic crystal structures, chirped stacks, resonator stacks, and numerically optimized

stacks.

Periodic Bragg stacks are one-dimensional photonic crystals and exhibit the superprism

effect, i.e. a spatial beam shifting as a function of wavelength, just outside a reflection

band. Unfortunately, periodic structures with two layers per period do not offer enough

degrees of freedom to design structures with desired dispersion characteristics, e.g., a

linear shift with wavelength. One way to overcome this limitation is to increase the

172

number of layers per period. This method should be used if a structure with constant

effective properties is desired. Here, on the other hand, we were more interested in the

aggregate transfer function from entering to exiting the structure. That is we wanted to

design a structure exhibiting a spatial beam shift along the exit surface with wavelength.

In that case, it does not matter what happens to the beam within the structure. Taking this

into account, we are no longer limited to periodic structures.

Three different algorithms were developed for designing the spatial dispersion

characteristics of non-periodic stacks. Considering that spatial and temporal dispersion

are approximately proportional, we modified two analytical methods used for temporal

dispersion compensation to obtain spatial dispersion. We showed that a coupled-mode

theory approach could be modified to design double-chirped structures with spatial

dispersion over a broad wavelength range. Narrowband coupled-cavity structures can be

designed using digital lattice techniques. Both these techniques can be used to design, for

example, stacks exhibiting a linear shift as a function of wavelength. Depending on the

desired wavelength range the best algorithm is chosen. Both chirped stacks and resonator

stacks are still limited to a specific stack configuration though. Chirped stacks always

have a monotonically increasing period length, while resonator stacks consist of cavities

separated by reflectors. This limits the available degrees of freedom and explains why

chirped stacks result in useful broadband designs, but are not good for narrowband

designs, and resonator stacks vice versa.

To make full use of all available degrees of freedom, we should not limit ourselves to a

particular type of stack, but consider all possible stack configurations. The problem in

this case is to find the best configuration out of all possibilities. For more than a couple of

layers, not all possible stack configurations can be investigated due to the prohibitively

large number of parameters. This is a problem well known from the design of thin-film

filters with desired reflectance and transmittance characteristics. As in the case of thin-

film filter design, numerical optimization methods can be used to find a good design,

even though this may not be the “best” possible design. Since numerically optimized

thin-film stacks are not limited to a particular stack configuration, more complex

173

dispersion profiles can be obtained than in the case of chirped or resonator stacks. The

drawback is the increased computation time necessary for the numerical optimization

procedure compared to the analytical design used for chirped and resonator stacks. In

general we conclude that a larger number of degrees of freedom leads to more design

flexibility and therefore superior dispersion characteristics.

We discussed in detail the use of multilayer stacks with spatial dispersion as wavelength

multiplexing and demultiplexing devices. Considering a device geometry with the

dispersive stack on one side of the substrate and a reflective gold coating on the other

side, several bounces can be performed through the substrate. Each bounce off the stack

increases the spatial beam shift and therefore the spatial separation between different

wavelength channels. Gaussian beam broadening limits the number of bounces. We find

that the largest number of channels is obtained if the stack has constant dispersion, since

in this case all beams focus on the output plane.

The number of channels that can be demultiplexed with a given stack is proportional to

the total shift obtained with one bounce off the stack. Therefore, the total shift of

different stacks should be compared to judge their performance. The dispersion is not a

good figure of merit, since the total number of channels is proportional to the dispersion

multiplied by the wavelength range over which the dispersion is observed. Thus, low

dispersion over a broad wavelength range may be better than high dispersion over a very

narrow wavelength range. The dispersion×wavelength-range-product is equal to the total

shift. Investigating over 600 different structures, we found that there seems to be a

physical limit to how much shift can be obtained for a given stack thickness, material

system, and incidence angle. The total shift appears proportional to the stack thickness,

the refractive index contrast, and the sine of the incident angle, while it is inversely

proportional to the square of the average refractive index. Interestingly, all three design

methods for non-periodic stacks result in a similar amount of total shift that is close to the

observed physical limit.

174

Thin-film stacks are not limited in their use to wavelength multiplexing or demultiplexing

components in WDM systems. As they have a continuous dispersion, dispersive thin-film

stacks could for example replace prisms or gratings in applications requiring a

wavelength-dependent element. In contrast to prisms or gratings, their dispersion

characteristics can be freely designed, which is a potential benefit for current applications

and may pave the way for new system architectures. The spatial dispersion of multilayer

stacks can also be converted to an angular dispersion by a lens in sequence with the stack.

Therefore, the device can, for example, be used for wavelength-dependent beam steering.

Future integrated photonic systems might use the concepts discussed here to obtain

spatial dispersion. The ideas considered here for one-dimensional thin-film structures can

also be transferred to two- or three-dimensional structures. Two-dimensional photonic

crystal structures fabricated by lithography, for example, are by no means limited to

periodic structures. As seen here for one-dimensional structures, breaking the periodicity

actually offers more design freedom and interesting new physics. The difficult task is to

design two- or three-dimensional structures with the desired characteristics. The methods

discussed here for designing non-periodic stacks may be a good starting point for

investigating the design of higher dimensional systems.

175

Appendix A

Bloch Calculation

For an infinite periodic medium with two layers per period the dispersion relation is

given by (A - 1).1 In this equation, K is the amplitude of the wavevector in the z-direction

of the periodicity, while β is the amplitude of the wavevector parallel to the layers in the

x-direction. c in the speed of light in vacuum and ω is the angular frequency of the light.

na and nb are the refractive indices, and la and are lb the layer thicknesses of the two

different materials as shown in Fig. A.1.

na nb na nb... na

la lb

Λz

y

x...nb

Fig. A.1. Labeling of periodic stack.

176

( )

( )

−

−

∆−

−

−

−

=+

bbaa

bbaaba

lnc

lnc

lnc

lnc

llK

22

22

22

22

sinsin,

coscos)(cos

βω

βω

ωβ

βω

βω

(A - 1)

β is calculated by (A - 2) and is the same in all layers. nin is the refractive index of the

incidence medium and θin is the incidence angle of the light.

( )ininnc

θω

β sin= (A - 2)

∆(β,ω) is given by (A - 3) for TE-polarization and by (A - 4) for TM-polarization.

( )

−

−

+

−

−

=∆2

2

22

22

22

21,

βω

βω

βω

βω

ωβ

b

a

a

b

TE

nc

nc

nc

nc

(A - 3)

( )

−

−

+

−

−

=∆2

22

22

2

22

2

22

2

21,

βω

βω

βω

βω

ωβ

ba

ab

ab

ba

TM

nc

n

nc

n

nc

n

nc

n(A - 4)

For plotting a K-β wavevector diagram or other calculations requiring the absolute value

of K, the dispersion relation (A - 1) has to be solved for K as a function of ω and β. This

is problematic due to the periodicity of the cosine. To determine the absolute value of K,

we have to consider carefully how many times π should be added. This can be done by

first calculating K far way from any stopband using the angle obtained from Snell’s law

as the group propagation angle. Slowly approaching the stopband, the correct number of

π is obtained by guaranteeing that there are no discontinuities in K. For many calculations

though, only the derivative of K with respect to ω or β is of interest and the constant term

can be neglected.

177

The Bloch calculation can also be used to model the dispersion properties of a finite bulk

periodic medium neglecting all boundary effects.

References



(1977).

178

Appendix B

Coordinate Transformation (K, β, ω)→(K, θ, ω)

In this appendix, the details of the coordinate transformation from variables (K, β, ω) to

(K, θ, ω) are given, where K is the wavevector in the z-direction perpendicular to the

layers, β is the wavevector in the x-direction along the layers, ω is the frequency, and θ is

the incidence angle in vacuum. Two main equations often used in this thesis are (B - 1)

and (B - 2) for calculating the group velocities vgx in the x-direction and vgz in the z-

direction.

constKgxv

=∂∂

=βω (B - 1)

constgz K

v=∂

∂=

β

ω (B - 2)

The calculation of these equations is straigthforward using the variables (K, β, ω), since it

is clear how to hold the third variable constant when taking the partial derivatives. On the

other hand, we often want to know how the group velocities change with θ and ω. As β

is the same in all layers, we can easily calculate it from (B - 3).

179

( ) ( )θω

ωθβ sin,c

= (B - 3)

c is the speed of light in vacuum. Remember that θ is in vacuum and the corresponding

refractive index is therefore unity. K(θ, ω) is calculated by replacing the independent

variables (β, ω) by (θ, ω) in the dispersion relation. It is then straightforward to plot

K-β-wavevector diagrams1,2 using K(θ, ω) and β(θ, ω) as parametric functions and

matching pairs. It is much more challenging though to calculate the partial derivatives

(B - 1) and (B - 2) correctly for given values of θ and ω. As the coordinate transform

from (K, β, ω) to (K, θ, ω) has caused quite some confusion and calculation mistakes, I

am giving my derivation for calculating the group propagation angle in detail here.

Section B.1 shows how to calculate the group propagation angle using equations in terms

of (K, β, ω). Section B.2 looks at the case using the independent variables (K, θ, ω).

B.1 Group Propagation Angle in Terms of (K, β, ω)First let us assume that we have the dispersion relation given as an implicit function

relating (K, β, ω) as shown in (B - 4). This relationship can, e.g., be obtained by

subtracting one side of (A - 1) from the other side.

0),,( =ωβKf (B - 4)

Now we take the derivative of f(K,β,ω) with respect to ω as shown in (B - 5).3,4

( ) ( ) ( ) 0,,,,,,=

∂∂

∂∂

+∂∂

∂∂

+∂

∂ωβ

βωβ

ωωβ

ωωβ KfK

KKfKf (B - 5)

Partial derivatives (indicated by ∂) always assume that all other variables are held

constant, e.g., ∂f(K,β,ω)/∂ω assumes that K and β are held constant. The constant

variables are sometimes indicated after a vertical line as shown in (B - 1) and (B - 2).

This is not necessary if we know what the variables of the function are. It is very helpful

though, if equations are shown without a context. If (B - 2) were given without the

clarification β=const, we might think that we could take the partial derivative of our

180

function K(θ,ω) with respect to ω to obtain the group velocity in the z-direction. But in

fact, this would give a completely wrong result, since we held θ constant instead of β.

To calculate ∂β/∂ω, we set K constant in (B - 5) and obtain (B - 6).1 Then (B - 1) is just

the reciprocal of (B - 6).

( )( ) βωβ

ωωβωβ

∂∂∂∂−

=∂∂

= ,,,,

KfKf

constK

(B - 6)

Note here that we have explicitly justified that ( )constKconstK ==

∂∂=∂∂ ωββω 1 .

Similarly, ∂Κ/∂ω in (B - 7) is obtained by setting β constant in (B - 5). (B - 2) is the

reciprocal of (B - 7).

( )( ) KKf

KfK

const ∂∂∂∂−

=∂∂

= ωβωωβ

ω β ,,,, (B - 7)

Finally, the group propagation angle θgr is calculated by (B - 8).

( )

( )( )( )

( )

( )( )

∂∂∂∂

=

∂∂∂∂−

∂∂∂∂−

=

=

−−

−

KKfKf

KfKf

KKfKf

vv

Kgz

gxgr

ωββωβ

βωβωωβ

ωβωωβ

ωβθ

,,,,tan

,,,,

,,,,

tan

tan,,

11

1

(B - 8)

Solving f(K,β,ω) for K, we can rewrite (B - 4) as given in (B - 9).

0),(),,( =−= ωβωβ gKKf (B - 9)

The function g(β,ω) only depends on β and ω. In the case of a periodic stack as discussed

in Appendix A, this separation is for example achieved by taking the inverse cosine of

(A - 1). Now we can calculate the partial derivatives in (B - 8) as shown in (B - 10) and

(B - 11).

( ) ( )β

ωββ

ωβ∂

∂−=

∂∂ ,,, gKf (B - 10)

181

( ) 1,,=

∂∂

KKf ωβ (B - 11)

Substituting (B - 10) and (B - 11) into (B - 8) and writing g(β,ω)≡K(β,ω) since

g(β,ω)=K(β,ω) from (B - 9) we obtain (B - 12).

( ) ( )

∂

∂−= −

βωβ

ωβθ,tan, 1 K

group (B - 12)

Thus, if we have solved the dispersion relation to obtain K(β,ω) we can use (B - 12) to

calculate the group propagation angle θgr.

B.2 Group Propagation Angle in Terms of (K, θ, ω)In the last section we discussed the calculation of the group propagation angle θgr in terms

of the independent variables (K, β, ω). Here we will perform the coordinate

transformation to (K, θ, ω). We assume that K(β,ω) is given and we want to transform it

to θ and ω as shown in (B - 13).

( ) ( )ωθωβ ~,~~, KK → (B - 13)

To distinguish the new variables from the old ones, the new variables are given with a

tilde on top. The quantities K and K~ will be identical at the end of any calculation in a

given physical situation, but K and K~ are functions of different variables, and similarly

for θ, θ~ and ω, ω~ . The new independent variables are calculated in terms of the old

variables as given in (B - 14) and (B - 15).

( )

= −

ωβ

ωβθc1sin,~ (B - 14)

( ) ωωβω =,~ (B - 15)

We use the chain rule to calculate the group propagation angle θgr after the coordinate

transformation.4 Thus, expression (B - 12) for the group velocity angle becomes (B - 16).

182

( ) ( )( )

( )( )

∂∂

∂∂

+∂

∂∂

∂−= −

ωθβω

ωωθ

ωθβθ

θωθ

ωθθ ~,~~

~~,~~

~,~~

~~,~~

tan~,~~ 1 KKgroup (B - 16)

From (B - 15) it is clear that the second term in the parantheses is zero. Thus the group

propagation angle is given by (B - 17).

( ) ( )( )

( )( )

∂∂

−=

∂∂

∂∂

−= −−

θωθωθ

ωθβθ

θωθ

ωθθ ~cos~~~,~~

tan~,~~

~~,~~

tan~,~~ 11 cKKgroup (B - 17)

We see that we actually have to take the derivative of K with respect to θ not to ω as we

might first expect after looking at (B - 1) and (B - 2).

References



(1977).



demultiplexing by beam shifting,” Opt. Lett., 25/20, 1502-1504 (2000).



(1993).

[4] G. Merziger and T. Wirth, Repetitorium der Höheren Mathematik, Binomi,

Springe, pp. 389-391, p.531 (1993).

183

Appendix C


Here we explain how to calculate the propagation of plane waves through arbitrary

multilayer stacks using a transfer matrix approach.1 Appendix D discusses the

propagation of beams of light. The propagation of electromagnetic waves is governed by

Maxwell’s equations given in (C - 1) to (C - 4).1,2,3

( ) ( )tt ,, rBrE &−=×∇ (C - 1)

( ) ( ) ( )ttt ,,, rDrJrH &+=×∇ (C - 2)

( ) ( )tt ,, rrD ρ=∇ (C - 3)

( ) 0, =∇ trB (C - 4)

E is the electric field strength, D is the electric displacement, J is the electric current

density, H is the magnetic field strength, B is the magnetic flux density, and ρ is the

electric charge density. r indicates the position in space and t the time. Bold letters

indicate vectors and matrices, italic letters are scalars. Assuming a source-free medium,

the electric charge density is zero everywhere and there are no source currents.

184

Furthermore, we assume simple media,3 i.e., linear, time-invariant, isotropic and

homogeneous media. In this case, the material equations are given by (C - 5) to (C - 7),

with ε, µ, and κ being simple scalars.

( ) ( ) ( )tt ,, rErrD ε= (C - 5)

( ) ( ) ( )tt ,1, rBr

rHµ

= (C - 6)

( ) ( ) ( )tt ,, rErrJ κ= (C - 7)

Taking the curl of (C - 1) and (C - 2) and substituting the other two Maxwell’s equations

as well as the material equations, we obtain (C - 8) and (C - 9).

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )tttt ,,,, rErrrErrrBrE &&&& εµκµ −−=×−∇=×∇×∇ (C - 8)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ttttt ,,,,, rHrrrHrrrDrJrH &&&& εµκµ −−=+×∇=×∇×∇ (C - 9)

With the vector identity (C - 10)

( ) AAA 2∇−∇∇=×∇×∇ (C - 10)

as well as (C - 3), (C - 4), and ρ =0, (C - 8) and (C - 9) are rewritten as (C - 11) and

(C - 12).

( ) ( ) ( ) ( ) ( ) ( ) ( )ttt ,,,2 rErrrErrrE &&& εµκµ +=∇ (C - 11)

( ) ( ) ( ) ( ) ( ) ( ) ( )ttt ,,,2 rHrrrHrrrH &&& εµκµ +=∇ (C - 12)

Equations (C - 11) and (C - 12) are called the telegraph equations or wave equations.

Equations (C - 11) and (C - 12) are not independent. Solving for the E-field, the H-field

can be obtained from (C - 2) and vice versa. Therefore, we limit ourselves to solving for

the E-field and relate it to the H-field at the end.

As we are interested in the propagation of light, we now consider waves with harmonic

time dependencies. Such waves are most easily dealt with using vector phasors as given

185

in (C - 13). Remember, though, that the real part of the phasor represents the actual

physical field.

( ) ( ) ( )tt ωjexp, rErE = (C - 13)

Substituting (C - 13) into (C - 11), we obtain (C - 14).

( ) ( ) ( ) ( ) ( ) ( ) 0,j, 22 =−+∇ tt rErrrrrE ωκµωεµ (C - 14)

The amplitude of the complex wavevector k is defined by (C - 15).

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )cn

ck r

rrrrrrrrrrrr ω

εωκµ

εµω

ωκµωεµ =−=−=0

2 jj (C - 15)

In this reformulation, c is the speed of light in vacuum and n is the complex refractive

index. The wavevector itself is defined pointing in the direction of propagation, which is

indicated by the unitvector ek in (C - 16).

( ) ( ) ( )rkerrk k= (C - 16)

We obtain the differential equation (C - 17) for the E-field.

( ) ( ) ( ) 0,, 22 =+∇ tkt rErrE (C - 17)

For the case that k is not a function of r, (C - 17) has two solutions, a forward

propagating plane wave given in (C - 18) and a backward propagating plane wave given

in (C - 19). The E-field is pointing perpendicular to the direction of propagation for both

solutions.

( ) ( )[ ] kekrrE ⊥−= tjEt ff ωexp, (C - 18)

( ) ( )[ ] kekrrE ⊥+= tjEt bb ωexp, (C - 19)

These solutions are uniform plane waves, as their amplitude does not change along the

planes of constant phase.

Now we discuss the case of a multilayered stack as the one shown in Fig. C.1. Within

each layer i the wavevector ki is constant and (C - 18) and (C - 19) are therefore solutions

describing the propagation of light in that layer. We will call the forward propagating

186

amplitude in layer i Ef,i and the backward propagating amplitude Eb,i. As shown in

Fig. C.1 the coordinate system is oriented such that the layer interfaces are perpendicular

to the z-axis. Without any loss of generality, we now assume that the propagation

direction of the light lies in the x-z-plane and there is thus no y-dependency of the fields.

z

x

n0 n1 n2 n3 nN+1nNnN-1

…

d1 d2 d3 dNdN-1 E||b,N+1=0

E||f,N+1=Etrans=1

E||f,0= Einc

E||b,0= Erefl

E||b,1

E||f,1E||b,2

E||f,2E||b,N

E||f,NE||b,3

E||f,3

z1=0

z2 z3 z4 zN-1 zN zN+1=L

Fig. C.1. Naming conventions for a multilayer stack.

ni ni+1

kf,i

z

x

kb,i

Ef,i

Eb,i

kf,i+1

Ef,i+1

Hf,i

kb,i+1

Eb,i+1

Hb,i+1

Hb,i

Hf,i+1

p-polarization (TM)

θi θi+1

ni ni+1

kf,i

z

x

kb,ikf,i+1

Ef,i

Eb,i

Ef,i+1

Hf,i

Hb,i

Hf,i+1

s-polarization (TE)

θi

kb,i+1

Eb,i+1

Hb,i+1θi+1

Vector pointing out of pageVector pointing into page

Fig. C.2. Conventions for the positive directions of k, E, and H in the case of p-

polarization (TM) and s-polarization (TE).

The E-field can have components in any direction perpendicular to the propagation

direction. The mathematics is most easily performed by splitting the field into the

components in the plane of incidence (p-polarization, TM) and perpendicular to the plane

of incidence (s-polarization, TE). Since, we are only considering linear media here, the

187

two components can be treated separately and the total field is obtained in the end by

adding the two components again. The H-field is perpendicular to both the E-field and

the direction of propagation. Fig. C.2 gives the conventions for the positive directions for

k, E, and H for both TE- and TM-polarization.1 Any other convention could be chosen

equally well as long as the same convention is used throughout.4

The forward and backward propagating E-field in layer i are related to each other and to

the forward and backward propagating E-field in layer i+1 by the boundary conditions at

the interface between i and i+1. Following Maxwell’s equations, the tangential

components of the E-field and the H-field have to be continuous across the interface.3

This condition can only be fulfilled over the whole interface if the wavevector component

along the interface is constant. This constant wavevector component is called β. As we

confined the wavevector to the x-z-plane, β points in the x-direction and only the z-

component changes as a function of the position. As β is the same everywhere in the

structure, it is convenient to cast equations in terms of β. β can easily be calculated by

(C - 20)

( )iinc

θω

β sin= (C - 20)

using any known refractive index and propagation angle combination. Normally, the

refractive index n and the propagation angle θ of the incident medium or vacuum are

used. Snell’s law given in (C - 21)

( ) ( )11 sinsin ++= iiii nn θθ (C - 21)

follows immediately from the fact that β is constant across all interfaces. As discussed

previously, we will perform all calculations for the E-field. The H-field amplitude is

calculated from the E-field amplitude by (C - 22).

( ) ( )( )

( ) ( )0ZnE

ZEH rr

rrr == (C - 22)

In this equation Z(r) is the impedance at position r, Z0=376.7Ω is the impedance in

vacuum, and n(r) the refractive index at position r. The direction of the H-field can be

188

determined from Fig. C.2. Since the E-field component E|| parallel to the interface is

constant across the interface, we express the total E-fields in terms of E||. (C - 23) to

(C - 26) give the equations for the forward and backward propagating E- and H-fields in a

multilayer stack for the case of TM-polarization. Ex,f,i and Ex,b,i are the E-field

components parallel to the interface at the left interface position zi as shown in Fig. C.1.

They are complex numbers with the phase representing the phase of the E-field at

position zi. The index i is determined from the position z in the stack.

( ) ( ) ( )( )[ ] ( ) ( ) zx eeE iiiizi

ifxfTM zzkxtj

Etzx θθβω

θsincosexp

cos,, ,

,, −−−−= (C - 23)

( ) ( ) ( )( )[ ] ( ) ( ) zx eeE iiiizi

ibxbTM zzkxtj

Etzx θθβω

θsincosexp

cos,, ,

,, +−+−= (C - 24)

( ) ( ) ( )( )[ ] yeH iizii

ifxfTM zzkxtj

ZE

tzx −−−= ,,

, expcos

,, βωθ

(C - 25)

( ) ( ) ( )( )[ ] ( )yeH −−+−= iizii

ibxbTM zzkxtj

ZE

tzx ,,

, expcos

,, βωθ

(C - 26)

The total E-field and H-field are calculated by adding the forward and the backward

propagating fields. As the H-field has only a y-component, it is easiest to calculate the

total H-field and obtain the E-field from (C - 22). The total H-field is given in (C - 27).

( ) ( ) ( )( )[ ]( ) ( )[ ] ( )[ ] ye

HHH

iizibxiizifxii

bTMfTMTM

zzkjEzzkjEZ

xtj

tzxtzxtzx

−−−−−

=

=+=

,,,,,,

,,

expexpcos

exp

,,,,,,

θβω (C - 27)

We see from (C - 27) that the H- and E-field are separable into a part that only depends

on x and one that only depends on z. Thus, we can consider the x-dependency and the z-

dependency separately and multiply the results. This is NOT the same though as

calculating the field for normal incidence and multiplying it by the part with x-

dependency. This approach would not fulfill the boundary conditions. Ex,f,i and Ex,b,i

depend on the angle of incidence and the frequency as seen below.

189

For the case of TE-polarization Ey,f,i and Ey,b,i are the E-field components parallel to the

interface. (C - 28) to (C - 31) give the equations for the forward and backward

propagating E- and H-fields for TE-polarization.

( ) ( )( )[ ] yeE iizifyfTE zzkxtjEtzx −−−= ,,, exp,, βω (C - 28)

( ) ( )( )[ ] yeE iizibybTE zzkxtjEtzx −+−= ,,, exp,, βω (C - 29)

( ) ( )( )[ ] ( ) ( ) zx eeH iiiizi

ifyfTE zzkxtj

ZE

tzx θθβω sincosexp,, ,,

, +−−−−= (C - 30)

( ) ( )( )[ ] ( ) ( ) zx eeH iiiizi

ibybTE zzkxtj

ZE

tzx θθβω sincosexp,, ,,

, +−+−= (C - 31)

Again, the total E-field is calculated by adding the forward and backward propagating

fields as shown in (C - 32).

( ) ( ) ( )( )[ ] ( )[ ] ( )[ ] ye

EEE

iizibyiizify

bTEfTETE

zzkjEzzkjExtjtzxtzxtzx

−+−−−=

=+=

,,,,,,

,,

expexpexp,,,,,,

βω(C - 32)

Equations (C - 27) and (C - 32) together with (C - 22) allow us to calculate the E-field

and the H-field at any position in the multilayer stack assuming that the fields parallel to

the interface E||,f,i and E||,b,i are known for each layer i. Next we discuss how to calculate

these fields using using a standard transfer matrix method.1 Assuming that no light is

incident from the right side onto the N-layer stack shown in Fig. C – 1 and that the

transmitted field has unity magnitude, the incident and reflected amplitudes on the left

side of the stack can be calculated using (C - 33).

=

−− 0

1EE

,113,222,111,0refl||,

inc||,NNN DPDPDPD K (C - 33)

In (C - 33) Pi and Di,i+1 are the propagation and interface matrices as obtained from the

interface boundary conditions from Maxwell’s equations. Pi(β,ω) is given in (C - 34)

with ni being the refractive index of layer i, di the layer thickness, β is the amplitude of

190

the wavevector parallel to the layers in the x-direction, ω is the angular frequency of the

light, and c the speed of light in vacuum.

( )

−

−

−

=

ii

ii

i

dnc

j

dnc

j

22

22

exp0

0exp

,

βω

βω

ωβP (C - 34)

The interface matrices Di,i+1(β,ω) given in (C - 35) depend on the reflection coefficients

ri,i+1(β,ω) and the transmission coefficients ti,i+1(β,ω) between adjacent layers which are

given by equations (C - 36) and (C - 37). These formulae differ from the Fresnel

formulae, as these are the reflection and transmission coefficients for the component of

the field parallel to the boundary, not for the total field.1,5

( ) ( )( )

( )

=

+

+

++ 1,

,1,

1,1,

1,

1,1, ωβ

ωβ

ωβωβ

ii

ii

iiii r

rt

D (C - 35)

( ) ( ) ( )( ) ( )ωβωβ

ωβωβωβ

,,,,

,1,,

1,,1,

+

++ +

−=

ieffieff

ieffieffii nn

nnr (C - 36)

( ) ( )( ) ( )ωβωβ

ωβωβ

,,,2

,1,,

,1,

++ +

=ieffieff

ieffii nn

nt (C - 37)

The effective refractive indices for the cases of TE- and TM-polarization are given by

(C - 38) and (C - 39) respectively.1

( )2

,, 1,

−=

iiiTEeff n

cnnω

βωβ (C - 38)

( )2,,

1

,

−

=

i

iiTMeff

nc

nn

ωβ

ωβ (C - 39)

If the expression under the square root becomes negative as, e.g., in the case of total

internal reflection, the sign of the effective refractive indices has to be chosen carefully to

191

prevent unphysical exponentially growing field amplitudes, though the method does then

model such situations correctly.

The fields parallel to the interface E\\,f,i and E\\,b,i in intermediate layers i are calculated by

just performing a subset of the matrix multiplications in (C - 33) as shown in (C - 40).

=

−−++++ 0

1EE

,112,111,ib,||,

if,||,NNNiiiiii DPDPDP K (C - 40)

Alternatively, the intermediate fields can be stored while performing calculation (C - 33)

from right to left. Now the E-field and the H-field at any position in the multilayer stack

can be calculated using equations (C - 27) and (C - 32) together with (C - 22). The

electromagnetic energy density w is given by (C - 41).3

( ) ( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )22

22

,,Re,,Re

,,Re21,,Re

21,,

tzxHztzxEz

tzxHztzxEztzxw

µε

µε

==

+=(C - 41)

Normally, we are not interested in the instantaneous variations of the energy density, but

instead in the time-averaged values. The time-averaged energy density wavg is given by

(C - 42). Note that the extra factor of ½ is due to the time averaging of a harmonic wave.

( ) ( ) ( ) ( ) ( ) 22 ,21,

21, zxHzzxEzzxwavg µε == (C - 42)

In the case of TM-polarization, wavg is given by (C - 43), where we have used (C - 27) for

the total H-field.

( )( )

( )[ ] ( )[ ] 2

,,,,,,20

2

, expexpcos2

1, iizibxiizifxi

iTMavg zzkjEzzkjEnzxw −−−−=

θε (C - 43)

For TE-polarization (C - 44) is obtained by substituting the total E-field from (C - 32)

into (C - 42).

( ) ( )[ ] ( )[ ] 2

,,,,,,02

, expexp21, iizibyiizifyiTEavg zzkjEzzkjEnzxw −+−−= ε (C - 44)

192

Next we will consider the energy transport across interfaces as this gives the reflectance

and the transmittance of the stack. The Poynting vector S in (C - 45) gives the

instantaneous rate of energy flow across a unit area, where the direction of propagation is

the normal vector to this unit area. As multiplying two fields is a nonlinear operation, we

cannot do the multiplication in phasor notation, but have to take the real parts.1

( ) ( )( ) ( )( )tzxtzxtzx ,,Re,,Re,, HES ×= (C - 45)

Again, we are more interested in the time-averaged value of S, which is called the

irradiance I. For a harmonic wave, the irradiance I can be calculated using (C - 46).1

( ) ( ) ( )( )∗×= tzxtzxzx ,,,,Re21, HEI (C - 46)

In order to calculate the transmittance of a multilayer stack, the ratio of the transmitted

irradiance to the incident irradiance has to be taken. In general, this CANNOT be done by

simply calculating the irradiances of the forward and backward propagating fields from

(C - 46) and taking the ratios, since these irradiances are not parallel if the refractive

indices of the incident and transmitting media are different. The tilt of the irradiances

changes the unit area.4 The correct transmittance could be calculated by obtaining the

irradiances from (C - 46) and accounting for the changed unit area at the end in the

calculation of the transmittance.5 Alternatively, the unit area can be fixed along the

interfaces and all irradiances can be calculated with respect to this unit area, i.e., the

irradiances I⊥ perpendicular to the interfaces are used.1 Here we use the second approach.

I⊥ is calculated from the fields along the layers as given in (C - 47).

( ) ( ) ( )( )∗⊥ = tzxHtzxEzxI ,,,,Re

21, |||| (C - 47)

Using (C - 23) to (C - 26), the incident irradiance I⊥,TM,inc for TM-polarization is

calculated to be (C - 48).

( ) ( ) 0

2

0,,,, cos2

1,Z

nEzxI

inc

incxfincTM θ

=⊥ (C - 48)

193

ninc is the refractive index in the incident medium and θinc the incidence angle. The

reflected irradiance I⊥,TM,refl and the transmitted irradiance I⊥,TM,trans are given by (C - 49)

and (C - 50).

( ) ( ) 0

20,,

,, cos21,

ZnE

zxIinc

incxbreflTM θ

=⊥ (C - 49)

( ) ( ) 0

2

1,,,, cos2

1,Z

nEzxI

trans

transNxftransTM θ

+⊥ = (C - 50)

ntrans is the refractive index in the transmitting medium and θtrans the propagation angle of

the transmitted light. For the case of TE-polarization, the incident irradiance I⊥,TE,inc, the

reflected irradiance I⊥,TE,refl, and the transmitted irradiance I⊥,TE,trans are given by (C - 51)

to (C - 53). These equations are obtained using (C - 28) to (C - 31).

( )( )

0

2

0,,,,

cos21,

ZnE

zxI incincyfincTE

θ=⊥ (C - 51)

( )( )

0

2

0,,,,

cos21,

ZnE

zxI incincybreflTE

θ=⊥ (C - 52)

( )( )

0

2

1,,,,

cos21,

ZnE

zxI transtransNyftransTE

θ+⊥ = (C - 53)

To calculate the transmittance, TTM for the case of TM-polarization and TTE for the case

of TE-polarization, we take the ratio of the transmitted and the incident irradiances. The

results are given in (C - 54) and (C - 55).

( ) ( )( )

( )( )transinc

inctrans

xf

NxfTM n

nE

ET

θθ

ωβ

ωβωβ

coscos

,,

,2

0,,

1,, += (C - 54)

( ) ( )( )

( )( )incinc

transtrans

yf

NyfTE n

nE

ET

θθ

ωβ

ωβωβ

coscos

,,

,2

0,,

1,, += (C - 55)

194

Finally, taking the ratio of the reflected and incident irradiances, we see that the

reflectance is given by (C - 56) for both polarization states.

( ) ( )( )

2

0,||,

0,||,

,,

,ωβ

ωβωβ

f

b

EE

R = (C - 56)

For the calculation of beams, the amplitude reflection coefficient r(β,ω) and transmission

coefficient t(β,ω) of the entire stack are also of interest. r(β,ω) and t(β,ω) are also called

the E-field transfer functions in reflection and transmission. Viewing the complete stack

as a system, these functions describe how the system acts on a given incident field, i.e.

r(β,ω) describes how the stack changes the amplitude and phase of an incident plane

wave at position z=0 upon reflection, while t(β,ω) characterizes how the amplitude and

phase of the transmitted field at position z=L compare to the incident field at z=0. As is

detailed in Appendix D, an incident beam of light can be decomposed into its plane wave

components. Since a multilayer dielectric stack is a linear, space-invariant (in the x- and

the y-direction) system, these components can be propagated through the stack

individually with their propagation determined by r(β,ω) and t(β,ω). Finally, the reflected

and transmitted total field is obtained by recombining the individually propagated

components. r(β,ω) and t(β,ω) are obtained by dividing the total reflected and

respectively transmitted E-field amplitude by the total incident E-field amplitude as

shown in (C - 57) to (C - 59). Again TM- and TE-polarization have to be distinguished

for the transmission coefficient.

( ) ( )( )ωβ

ωβωβ

,,

,0,||,

0,||,

f

b

EE

r = (C - 57)

( ) ( )( )

( )( )trans

inc

xf

NxfTM E

Et

θθ

ωβ

ωβωβ

coscos

,,

,0,,

1,, += (C - 58)

( ) ( )( )ωβ

ωβωβ

,,

,0,,

1,,

yf

NyfTE E

Et += (C - 59)

Finally, in order to calculate the spatial shift upon reflection, we are interested in the

wavevector K(β,ω) within the stack in the z-direction. This then determines the effective

195

group propagation angle. Since the stack is not periodic, K(β,ω) is only an effective

wavevector. The phase change upon reflection can be calculated using (C - 60).

( ) ( )( ) ( )( )ωβωβωβφ ,arg,arg, 0,||,0,||, fbrefl EE −= (C - 60)

The sign of the phase change depends on the positive vector directions chosen in

Fig. C.2. This has to be taken into account in determining if there is a maximum or a

minimum of the standing wave at the interface. Assuming a total stack thickness L, the

wavevector K(β,ω) in the z-direction is related to the phase change upon reflection as

given in (C - 61).

( ) ( )L

K refl

2,

,ωβφ

ωβ = (C - 61)

(C - 61) is the dispersion relation among K, β, and ω for a finite periodic or non-periodic

stack calculated using the transfer matrix method.

References



[2] D. A. Mlynski, Elektrodynamik, Skript der Vorlesung Elektrodynamik, Karlsruhe

(1994).

[3] U. S. Inan and A. S. Inan, Engineering Electromagnetics, Addison-Wesley

Longman, Inc., Menlo Park, CA (1999).

[4] E. Hecht, Optik, Addison-Wesley (Deutschland) GmbH, Bonn (1989).

[5] M. Born and E. Wolf, Principles of Optics, Cambridge University Press,

Cambridge, UK (1999).

196

Appendix D

Simulating Beams by Fourier Decomposition

In this appendix we discuss how to calculate the propagation of beams through linear,

space-invariant systems using a Fourier transform technique. The E-field in space is

given by the complex function E(x,y,z)=Ef(x,y,z)+Eb(x,y,z), where Ef(x,y,z) is the

complex amplitude of the forward propagating wave and Eb(x,y,z) is the complex

amplitude of the backward propagating wave at position (x,y,z).1 The goal is to calculate

E(x,y,z) at any point in space for a given incident field Ef(x,y,0) at position z=0 (any other

constant z could be chosen as well). We are assuming here for simplicity that a scalar

theory can be used to solve Maxwell’s equations and that the H-field can be calculated

from the E-field.1,2 For simple media a direct analytical relationship might exist between

E(x,y,z) and Ef(x,y,0). Let us, for example, consider the case of a uniform plane wave

propagating in the z-direction in free space. Since the plane wave is uniform, i.e., its

amplitude does not change along the plane of constant phase, Ef(x,y,0) is a constant as

given in (D - 1).

( ) 00,, EyxE fPW = (D - 1)

The forward propagating field at any point is then given by (D - 2).

197

( ) ( ) ( )zkiyxEzyxE fPWfPW 0exp0,,,, = (D - 2)

k0 is the wavevector as given in (D - 3).

ck πω2

0 = (D - 3)

Since no reflections occur in free space, EbPW(x,y,z)=0 everywhere and the total E-field is

equal to the forward propagating E-field. In Appendix C it is shown how the forward and

backward propagating waves can be calculated for an arbitrary layered stack assuming

that the incident field is a uniform plane wave.

Now we consider the propagation of beams of light, i.e. incident fields Ef(x,y,0) that are

not plane waves. In particular we will discuss the propagation of the fundamental

Gaussian beam. The Gaussian beam is the ideal diffraction limited wave and often

approximates laser beams well.3 It has rotational symmetry around the axis of

propagation. (D - 4) gives the field equation for the Gaussian beam in polar coordinates.

The direction of propagation is renamed ζ, since in this thesis the z–axis is the direction

perpendicular to the layers of a multilayer stack and we shortly will want to look at

Gaussian beams incident on such stacks with non-normal incidence such that ζ and z are

not the same direction.

( ) ( ) ( ) ( ) ( )( ) ( )( )tkR

rkw

rwwEtrEGB ωζζη

ζζζζ −

−

−= 0

20

2

20

0 iexpiexp2

iexpexp,, (D - 4)

(D - 5) to (D - 8) define the functions used in the definition of the Gaussian beam. z0 is

the Rayleigh range.

2000 2

1 wkz = (D - 5)

( ) 20

2

0 1z

ww ζζ += (D - 6)

( )

+= 2

201

ζζζ

zR (D - 7)

198

( )

=

0

arctanzζ

ζη (D - 8)

ξ-po

sitio

n in

µm

ζ-position in µm

ξ-po

sitio

n in

µm

ζ-position in µm

0 0.25 0.5 0.75 1Intensity color code

(a) (b)

Fig. D.1. Gaussian beam with w0=4.7µm and λ=890nm. (a) Calculated from exact

formula (D - 4). (b) Obtained from Fourier decomposition (D - 22). Note that the ξ-

and the ζ-axes have different scales.

Fig. D.1(a) shows a plot of a Gaussian beam calculated using (D - 4). Now let us assume

the incident field at a plane z=0 has the Gaussian beam profile EfGB(r,0,0). A multilayered

stack extends from z=0 to z=L. We could calculate the field EfGB(x,y,z) directly at all

positions and times using a Maxwell solver, such as the finite difference time domain

(FDTD) technique.4 Since this approach is, however, quite computationally intensive, we

are choosing a different approach that uses the fact that a multilayered stack is a linear,

space-invariant system.2 For sufficiently small intensities, the system is linear. The

system is space-invariant, since the z-direction transfer function only depends on the

difference between the input position (xin,yin) and the exit position (xout,yout), not on the

absolute position along the stack. For linear, space-invariant systems, the input field can

be decomposed into elementary components, these can be propagated individually, and

the output field is obtained by summing the propagated individual components.

199

One possible decomposition of the incident field E(x,y,z=0) is obtained by performing the

two-dimensional spatial Fourier transform given by (D - 9).2

( ) ( ) ( )( ) yxyfxfzyxEzffA yxyxFxy dd2iexp0,,0;, ∫ ∫∞

∞−

∞

∞−

+−=== π (D - 9)

The inverse Fourier transform (D - 10) expresses the field as a function of the Fourier

components AFxy(fx,fy;z=0).

( ) ( ) ( )( ) yxyxyxFxy ffyfxfzffAzyxE dd2iexp0;,0,, ∫ ∫∞

∞−

∞

∞−

+=== π (D - 10)

Comparing the integrand of (D - 10) with the equation for plane waves (D - 2), we see

that the Fourier transform decomposes the E-field into plane waves with amplitudes

AFxy(fx,fy;z=0) and wavevectors kx in the x-direction and ky in the y-direction given by

(D - 11) and (D - 12).

xx fk π2= (D - 11)

yy fk π2= (D - 12)

Here we limit ourselves to propagation of the beam in the x-z-plane and to calculations

for y=0. The one-dimensional Fourier transform and inverse transform are then given by

(D - 13) and (D - 14).

( ) ( ) ( ) xxfzxEzfA xxFx d2iexp0,0; ∫∞

∞−

−=== π (D - 13)

( ) ( ) ( ) xxxFx fxfzfAzxE d2iexp0;0, ∫∞

∞−

=== π (D - 14)

The spatial frequency fx is related to the directional cosine α, depicted in Fig. D.2, by

(D - 15).

λα

=xf (D - 15)

200

z

xk

θin=acos(γ)

acos(α)

Fig. D.2. Direction cosines α and γ for propagation in the x-z-plane.

Thus, we see that the integration in (D - 14) corresponds to the summation of plane

waves with amplitudes AFx(fx;z=0) propagating in different directions in the x-z-plane.

We can calculate the change of the plane wave amplitudes AFx(fx;z) as a function of z

using e.g. the transfer matrix method. The transfer function relating AFx(fx;z) to AFx(fx;0)

is called H(fx;z) as shown in (D - 16).

( ) ( ) ( )0;;; xFxxxFx fAzfHzfA = (D - 16)

Therefore, we can now calculate E(x, z) at any point in the x-z-plane by propagating the

individual Fourier components and summing them up after propagation as given in

(D - 17).

( ) ( ) ( )λα

αλπ

λα

π d2iexp;d2iexp;, ∫∫∞

∞−

∞

∞−

== xzAfxfzfAzxE FxxxxFx (D - 17)

For practical purposes we will replace the continuous Fourier transform in (D - 17) by a

discrete Fourier transform as shown in (D - 18), where αi is the directional cosine of the

ith-component and ∆α is the angular separation of the different components.

( )λα

αλπ

λα ∆

= ∑

ii

iFx xzAzxE 2iexp;, (D - 18)

Due to the discrete nature of the Fourier transform (D - 18), the resulting E-field is

repetitive in space. The repetition distance ∆x is given by (D - 19).

αλ

∆=∆x (D - 19)

If the E-field of interest is limited in space or can be approximated by a spatially limited

function extending over a spatial distance smaller than ∆x, the discrete Fourier transform

201

(D - 18) combined with a spatial filter of width ∆x correctly reconstructs the original

field. Solving (D - 19) for ∆α the minimum required angular sampling distance is

obtained. This is the Whittaker-Shannon sampling theorem well known from information

theory.2,5

Next we will apply the Fourier decomposition technique to calculating the propagation of

Gaussian beams. Instead of considering a rotationally symmetric Gaussian beam as given

in (D - 4), we will limit ourselves to a beam that has a Gaussian beam profile

perpendicular to the direction of propagation in the x-z-plane and is infinite in the y-

direction. Such an elongated beam is interesting for our application because spatial

wavelength demultiplexing using a multilayer thin-film stack only requires focusing in

the x-direction. Simultaneous focusing in the y-direction leads to unwanted polarization

mixing effects. Using again the ξ-ζ-coordinate system, where ζ is again the propagation

direction of the beam and ξ is the direction perpendicular to the propagation direction,

(D - 20) gives the E-field at position ζ=0.

( )

−=== 2

0

2

0 exp0,0,w

EtEGBapprξ

ζξ (D - 20)

Taking the Fourier transform of (D - 20), the Fourier decomposition amplitudes (D - 21)

are calculated.

( ) ( )222000 exp0,0, xxFGBappr fwwEtfA ππζ −=== (D - 21)

Using relation (D - 15) and the definition of the incident angle given in Fig. D.2, Fig. D.3

plots the Fourier amplitudes as a function of the plane wave propagation angle for the

components of a Gaussian beam with w0=4.7µm and λ=890nm. It can be seen that the

amplitude has decreased to 1/e for beam components at incidence angles of ± 3.5°.

Substituting (D - 16) and (D - 21) into (D - 18), we obtain the propagation of the

approximate Gaussian beams to be given by (D - 22).

( )λα

ξαλπ

λα

ππζλα

ζξ∆

−

= ∑

ii

iiGBappr wwEHE 2iexpexp;,

222

000 (D - 22)

202

H(fx;ζ) is the transfer function along the ζ-axis. As seen from (D - 2) the transfer function

of plane waves in free space is given by (D - 23).

( ) ( )zkizfH xfPW 0exp; = (D - 23)

10 8 6 4 2 0 2 4 6 8 100

0.10.2

0.30.40.5

0.60.7

0.80.9

1

Propagation Angle in deg

Nor

mal

ized

Am

plitu

de

Fig. D.3. Amplitude of Fourier components normalized by π00wE for a Gaussian

beam with w0=4.7µm and λ=890nm.

0 0.5 120

0

20

40

60

Incident (zeta=0)Position in microns


Normalized Intensity

ξ -po

sitio

n in

µm

ζ-position in µm


space.

203

Fig. D.4 plots the propagation of a Gaussian beam with w0=4.7µm and λ=890nm in free

space using 41 Fourier components spaced by dα =0.0075. This corresponds to a spatial

repeat distance of ∆x=119µm. The right hand graph in Fig. D.4 shows the intensity

distribution of the forward propagating light along the ξ–axis for ζ=0. Fig. D.1(b) shows

a more extended plot of the same beam. Comparing the beam in Fig. D.1(b) obtained

from the Fourier decomposition to the one in Fig. D.1(a) calculated from the exact

formula (D - 4), we see that they agree well within the Rayleigh range. Outside the

Rayleigh range, the beam calculated from the exact formula has a lower amplitude. This

is due to the widening of the beam in both the x- and the y-direction as the rotationally

symmetric widening beam approaches a spherical wave. The intensity of the beam is

constant for integration over the entire x-y-plane at a fixed z-value. (D - 22) on the other

hand considers a beam that is only focussed in the x-direction and infinite in the y-

direction. In this case the intensity is constant for integration along the x-axis for a fixed

y- and z-value.

F t

0 0.5 120

0

20

40

60

Incident (z=0)Position in microns


Normalized Intensity

x-po

sitio

n in

µm

z-position in µm

z

x

ζ

ξ


space at an angle of 40º with respect to the z-axis.

204

Next let us perform a coordinate transformation to calculate the field of a Gaussian beam

propagating at an angle θin with respect to the z-axis as depicted in Fig. D.5. ξ and ζ are

related to x and z as given in (D - 24) and (D - 25).

( ) ( )zx inin θθξ sincos −= (D - 24)

( ) ( )zx inin θθζ cossin += (D - 25)

Substituting (D - 24) and (D - 25) into (D - 20) and taking the Fourier transform at z=0,

we obtain the Fourier amplitudes given in (D - 26).

( ) ( )( )

−

−=

==

22

2

000

sincos

expcos

0,0,λ

θαπ

θπ

θλα in

ininFGBappr

wwEtzA (D - 26)

Substituting (D - 26) into (D - 18), we finally obtain (D - 27) for the propagation of a

Gaussian beam at an angle.

( ) ( ) ( )( )

λα

ξαλπ

λθα

πθ

πθλ

α d2iexpsincos

expcos

;,2

22

000∑

−

−

=

ii

ini

inin

iGBappr

wwEzHzxE

(D - 27)

Fig. D.5 plots the propagation of a Gaussian beam with w0=4.7µm, λ=890nm, and

θin=40º in free space using 41 Fourier components spaced by dα =0.0075. Fig. D.6 shows

the propagation of the same beam incident onto a 40-µm slab of material with refractive

index n=2.5. The total field is plotted, summing the forward and backward propagating

E-fields at each position. The transfer function H(fx;z) relating the incident plane waves at

z=0 to the forward and the backward propagating waves at each position z is obtained

using the transfer matrix method discussed in Appendix C. In the left hand plot of

Fig. D.6, both the refracted beam within the slab and the transmitted beam are clearly

seen. The reflections off the first air – slab interface and off the second slab – air interface

are dimly visible. We can also see the standing wave interference pattern between

forward and backward propagating light at the interfaces. On the right hand side of

Fig. D.6, the reflected light, i.e., the backward propagating intensity at z=0, is plotted

along the x-axis for a unity amplitude incident beam. The first and second reflections can

205

be clearly distinguished and it can be seen that they are approximately equal in

amplitude. Also shown is the transmitted light, i.e., the forward propagating intensity at

z=L=40µm.

0 0.5 1 1.5 2Intensity color code

0 0.5 120

0

20

40

60

Incident (z=0)Reflected (z=0)Transmitted (z=L)

Position in microns

Normalized Intensityz-position in µm

x-po

sitio

n in

µm

n=2.5n=1 n=1

Fig. D.6. Intensity distribution for a Gaussian beam with w0=4.7µm and λ=890nm

incident onto a 40-µm slab of material with refractive index n=2.5 at an angle of

40º with respect to the z-axis.

In conclusion we have described how to calculate the propagation of a beam of light

through an arbitrary multilayered stack by employing a Fourier decomposition of the

beam into plane wave components, propagating these components individually, and

obtaining the total field by summing the individual components.

References

[1] U. S. Inan and A. S. Inan, Engineering Electromagnetics, Addison Wesley

Longman, Inc., Menlo Park, CA (1999).

[2] J. W. Goodman, Introduction to Fourier Optics, The McGraw-Hill Companies, Inc.

(1996).

206


[4] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-

Difference Time-Domain Method, Artech House (2000).

[5] A. V. Oppenheim, R. W. Schafer, J. R. Buck, Discrete-Time Signal Processing,

Prentice Hall, New Jersey (1999).

207

Appendix E

Numerical Optimization Methods

The design of thin-film filters is typically divided into two stages – (i) synthesis of a start

design with characteristics that somewhat match the desired ones, and (ii) refinement of

this start design to gradually improve its performance. The performance of a structure is

measured by a merit function MF – a single number comparing the current design

characteristics with the desired design characteristics.1 The definition of the merit

function used is given in (E - 1).

pN

i

p

i

iTi

QQQ

NMF

/1

1

1

∆−

= ∑=

(E - 1)

Qi is the current value of a quantity of interest, QiT the target value of that quantity, ∆Qi

the acceptable deviation, N is the number of sampling points, and p the p-norm used.2 In

the case of p = 2, the merit function is, e.g., the root-mean-square difference between the

current values and the target values of the quantities of interest.

Several synthesis methods are available to obtain a good start design. Graphical methods

are possible if the design has only a few layers.3,4 Analytical techniques are particularly

desirable as they normally allow the fast generation of a start design. Such techniques

208

include the synthesis based on coupled-mode theory5 as discussed in Chapter 5, digital

signal processing techniques6 introduced in Chapter 6, as well as techniques based on an

inverse Fourier transform.3,7 As the Fourier transform technique is only applicable to the

synthesis of transmittance profiles, it was not implemented here. A numerical synthesis

method that can be used for any type of design is the flip-flop method.8 This method is

based on the idea that thin layers of just two materials can approximate any index profile.

The desired total thickness of the stack is divided into a large number of thin layers with

alternating refractive indices. The refractive index of these layers is sequentially flipped,

keeping the index flipped if this improves the performance. At the end neighboring layers

of the same index are combined. Finally, intuition and experience are very important in

generating a good start design.

As an example we discuss here the design of an erbium doped fiber amplifier (EDFA)

gain equalizing filter. Since an EDFA does not amplify all wavelengths equally, a gain-

equalizing filter is needed to achieve a flat gain spectrum. Operating the filter in

reflection, the desired reflectance as a function of wavelength is plotted in Fig. E.1.

Fig. E.2 shows the performance of two 200-layer start designs – one generated using the

flip-flop technique and one a guessed start structure. In the following the performance of

different refinement techniques in improving these structures will be compared.

1530 1535 1540 1545 1550 1555 1560 15650

0.2

0.4

0.6

0.8

1

Target reflectanceWavelength in nm

Ref

lect

ance

Fig. E.1. Target reflectance as a function of wavelength for an EDFA gain-

flattening filter.

209

1530 1540 1550 15600

0.2

0.4

0.6

0.8

1

Target reflectanceMartina's guessFlipflop synthesis

Wavelength in nm

Ref

lect

ance

Merit function

MFMG 54.5=

MFFF 17.4=

Thickness

dMG 36.8um=

dFF 36.0um=

Fig. E.2. Reflectance of two 200-layer start designs. The merit function and the

total thickness of both designs are given to the right.

We implemented six different numerical optimization techniques – golden section

search,2 secant method,2 conjugate gradient algorithm,2 Broyden-Fletcher-Goldfarb-

Shanno (BFGS),2 damped least squares method,9 and Hooke&Jeeves pattern search.10 All

six techniques are local optimization techniques, i.e., if used in their range of validity

they will find the closest local minimum in the merit function. Due to the large number of

degrees of freedom available in the design of a 200-layer structure, this next local

minimum may be very shallow and several minima may be located close together. Lower

minima can therefore be found by using the numerical techniques somewhat outside their

range of validity.

Let us look at the example of the golden section search technique. This technique is a

one-dimensional search method that uses an golden section interval reduction to reduce

the number of points that have to be calculated in each search step.2 The golden section

search technique only finds the local minimum if it is guaranteed that there is only one

local minimum within the initial interval. Using the interval reduction, this local

minimum is found. If we now use this technique on an interval that has several local

210

minima, there is no guarantee which one it will find. But it is likely that a lower

minimum is found than if we choose a small interval. Therefore, using local optimization

techniques out of their original range of validity often results in better designs.

Choosing for example a large initial interval for the golden section search can result in

structures that are quite different from and superior to the original structure. As the

golden section search is a one-dimensional technique, the different layer thicknesses are

optimized sequentially keeping the refractive indices constant. Fig. E.3 plots the

performance of the two start designs refined using the golden section search technique.

Figs. E.3 to E.8 are obtained by running the refinement technique in question for one

iteration on each start design. As different methods are programmed differently, the

length of the calculation time is given in addition to the final merit function. Longer

runtimes of some optimization techniques might further improve the performance.

1530 1540 1550 15600

0.2

0.4

0.6

0.8

1


Wavelength in nm

Ref

lect

ance

Merit function

MFMG 13.5=

MFFF 14.9=

Thickness

dMG 67.7um=

dFF 41.5um=

Calculation time

tMG 101min=

tFF 132min=

Fig. E.3. Golden section search refinement.

Newton’s method2 is an important technique in numerical optimization. In one

dimension, this method approximates the function at the given position by a parabola.

The next estimate for the local minimum is then the minimum of the parabola. This

method allows fast optimization, but may not converge if the second derivative of the

211

function is positive for some values. The secant method is basically Newton’s method

with the second derivative necessary for the calculation of the parabola replaced by a

difference of first derivatives.2 This technique is also a one-dimensional search method

and the layers are optimized sequentially. Fig. E.4 plots the performance improvements

achieved using the secant method.

1530 1540 1550 15600

0.2

0.4

0.6

0.8

1


Wavelength in nm

Ref

lect

ance

Merit function

MFMG 12.4=

MFFF 13.2=

Thickness

dMG 36.4um=

dFF 35.9um=

Calculation time

tMG 66min=

tFF 74min=

Fig. E.4. Secant method refinement.

Next we turn to multivariable search techniques. These techniques optimize all layer

thicknesses simultaneously. One such method is the steepest descent method.2 At a given

point of the function, the gradient of steepest descent is calculated. Then a one-

dimensional line search is performed to find the minimum in this direction and the

procedure is repeated from this point on. Due to the nature of the steepest descent

algorithm, the directions of search are always perpendicular leading to a slow

approximation of the local minimum. A much quicker convergence is obtained using the

multidimensional Newton’s method.2 Here a quadratic Taylor expansion of the function

is used and the minimum of the expansion is taken as the next point. As for the one-

dimensional case, this algorithm does not guarantee convergence. The conjugate gradient

algorithm is an intermediate between the steepest descent and Newton’s method. The

search direction is calculated as a linear combination of the previous direction and the

212

current gradient such that all directions are mutually conjugate with the Hessian.2 For this

algorithm convergence is guaranteed and the rate is faster than for the steepest descent

method. Fig. E.5 plots the results obtained using the conjugate gradient algorithm.

1530 1540 1550 15600

0.2

0.4

0.6

0.8

1


Wavelength in nm

Ref

lect

ance

Merit function

MFMG 40.7=

MFFF 15.3=

Thickness

dMG 36.8um=

dFF 35.9um=

Calculation time

tMG 17min=

tFF 50min=

Fig. E.5. Conjugate gradient algorithm refinement.

1530 1540 1550 15600

0.2

0.4

0.6

0.8

1


Wavelength in nm

Ref

lect

ance

Merit function

MFMG 13.6=

MFFF 11.5=

Thickness

dMG 36.7 um=

dFF 35.8 um=

Calculation time

tMG 76min=

tFF 127 min=

Fig. E.6. BFGS algorithm refinement.

213

The Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm is a quasi-Newton method

based on the idea of modifying the update formula such that descent is guaranteed.2

Fig. E.6 graphs the performance improvements obtained using the BFGS algorithm.

All the algorithms discussed so far – golden section search, secant method, conjugate

gradient algorithm, and BFGS algorithm – use a merit function to judge the performance

of a design. For the design of a specific reflectance profile, the performance of the design

is calculated using the merit function in (E - 1) and the design is improved such that this

merit function value decreases. In contrast to this, the damped least squares algorithm

directly calculates the influence of the layer thicknesses on the reflectance values. By

calculating the derivative of each reflectance value with respect to each layer thickness,

the best (in the least squares sense) combination of layers is obtained.9 In order to prevent

unphysical layer thicknesses, dampening vectors are introduced that “damp” the change

of the layer thickness in the undesired direction. As seen in Fig. E.7 the results obtained

with this algorithm here are very unsatisfactory. Keep in mind that we are comparing

here the implementation of algorithms as well as the algorithms themselves. Therefore,

the implementation of the damped least squares algorithm might not be very good. It did

perform better though for simpler structures than the example chosen here.

All prior techniques except for the one-dimensional golden section search required the

calculation of derivatives. As these derivatives are often not available analytically, they

have to be calculated numerically, adding to the total computation time. The Hooke &

Jeeves pattern search technique is a very powerful multivariable search algorithm that

does not require the calculation of derivatives.10 The algorithm is divided into two phases

– an exploration phase and a pattern move. During the exploration phase the parameters

are sequentially increased and decreased by a small step. If an improvement in the merit

function is achieved, the step is recorded in a pattern vector. For the pattern move the

pattern vector is multiplied by an acceleration factor and is added to the parameter vector.

If there is no improvement after a pattern move and a subsequent exploration, the step

size is reduced. Fig. E.8 plots performance of both start designs after the application of

one iteration of the Hooke & Jeeves pattern search technique.

214

1530 1540 1550 15600

0.2

0.4

0.6

0.8

1


Wavelength in nm

Ref

lect

ance

Merit function

MFMG 55.7=

MFFF 17.9=

Thickness

dMG 36.7um=

dFF 36.0um=

Calculation time

tMG 10min=

tFF 18min=

Fig. E.7. Damped least squares refinement.

1530 1540 1550 15600

0.2

0.4

0.6

0.8

1


Wavelength in nm

Ref

lect

ance

Merit function

MFMG 8.8=

MFFF 13.7=

Thickness

dMG 36.5um=

dFF 35.9um=

Calculation time

tMG 96min=

tFF 67min=

Fig. E.8. Hooke & Jeeves pattern search refinement.

Table E.1 compares the performance of the different implemented designs in terms of the

final merit function, the final thickness, and the calculation time for each refinement

algorithm and for both start designs. The green boxes show the best results, while the red

215

numbers indicate poor performance. It can be seen that all methods except for the

damped least squares algorithm improved the performance of the start designs. The larger

thicknesses of the golden section search designs are due to the large intervals used in the

implementation. A connection between calculation time and merit function is visible as

well. Interestingly, both start designs result in equally good refined designs.

MFMG MFFFdMG

(µm)dFF

(µm)tMG

(min)tFF

(min)

Starting designs 54.5 17.4 36.8 36.0

Golden Section 13.5 14.9 67.7 41.5 101 132

Secant Method 12.4 13.1 36.4 35.9 66 74

Conjugate Gradient 40.7 15.3 36.8 35.9 17 50

BFGS 13.6 11.5 36.7 35.8 76 127

Damped Least Sq. 55.7 17.9 36.7 36.0 10 18

Hooke & Jeeves 8.8 13.7 36.5 35.9 96 67

Table E.1. Comparison of refinement methods.

1530 1540 1550 15600

0.2

0.4

0.6

0.8

1

Target reflectanceCombination of methods

Wavelength in nm

Ref

lect

ance

Merit function

MFMG 3.6=

Thickness

dMG 42.4um=

Fig. E.9. Refined design using a combination of the different methods.

216

In conclusion we find that no one algorithm is the clear winner. All algorithms have their

merits and we actually achieved the best performance improvements using the different

optimization algorithms sequentially. Fig. E.9 shows the performance of a gain flattening

filter designed using a combination of the different techniques.

For this work numerical optimization is a means to achieve multilayer thin-film stacks

with superior dispersion characteristics. To this end, I programmed this selection of

standard optimization procedures. As seen in Chapter 9, the resulting designs seem to be

close to the limit of what is physically achievable. Therefore, there was no need to

implement further, maybe more powerful, optimization techniques such as the needle

technique or genetic algorithms.2 In the pursuit of the best design, it might be well worth

the effort to implement these or other modern techniques.

References

[1] J. A. Dobrowolski, F.C. Ho, A. Belkind, V.A. Koss, “Merit functions for more

effective thin film calculations,” Appl. Opt., 28/14, 2824-2831 (1989).


Inc. (1996).







Electron., 35/2, 129-137 (1999).



[7] P.G. Verly, Appl. Opt., 35/25, pp.5148-5153 (1996).

[8] W. H. Southwell, Appl. Opt., 24/4, pp.457-460 (1985).

217



references herein.



218

Appendix F

Beam Cone in a Dispersive Stack

Here the angular range ∆θstruc of a beam within a dispersive stack is discussed as a

function of the input angular range ∆θin. As shown in Appendix D, a beam of light can be

decomposed into plane-wave components with different propagation directions. Thus, an

incident beam of light consists of a range of incidence angles. Here we will show that a

change in the incidence angle has a similar effect on the group propagation angle θgroup as

a change in the incident frequency. Therefore, we expect that the angular range of a beam

in a multilayer stack depends on the dispersion of the stack. Here we will quantify this

dependency.

In principle, a beam has components at all angles, but many of these might have very

small amplitudes and can thus be neglected. Here we choose a beam to be delimited by

the 1/e2 intensity components. All calculations are equally correct for a different

delimiting value, as long as the same value is chosen for the input angular range ∆θin and

the angular range ∆θstruc in the stack. Approximating differences by a differentials, ∆θstruc

can be estimated for a given ∆θin using (F – 1).

219

( ) ( )in

groupinstruc θ

θ

ωθθθωθθ ∆

∂

∂=∆∆ ~

~,~,~,~ (F - 1)

We see that the angular range ∆θstruc depends on the change of the group propagation

angle θgroup with the incident angle θ. Without dispersion, i.e., for a frequency-

independent constant average refractive index navg, the group propagation angle θgroup is

given by Snell’s law as seen in (F – 2).

( )

= −

=avg

Dispgroup nθ

ωθθ~sinsin~,~ 1

0, (F - 2)

An approximation of navg can be obtained by calculating the total optical thickness of the

structure and dividing the result by the physical thickness of the structure. For this case

the angular range ∆θstruc is calculated as.

( ) ( )in

avg

inDispgroup

inDispstrucn

θθ

θθ

θ

ωθθθωθθ ∆

−=∆

∂

∂=∆∆ =

= ~sin

~cos~

~,~,~,~

22

0,0, (F - 3)

We see that that the size of the beam cone in the stack is proportional to the incident

beam cone for a given center incidence angle θ.

Next we will consider the beam cone in a dispersive stack with the dispersion of the stack

Dispω defined in (F – 4).

( ) ( )ω

ωθθωθω ~

~,~~,~

∂

∂= groupDisp (F - 4)

Intergrating (F – 4), we obtain the group propagation angle in the dispersive stack as

given in (F – 5).

( ) ( )∫ ∂= ωωθωθθ ω~~,~~,~ Dispgroup (F - 5)

In order to calculate (F – 1) from (F – 5), we remember that the partial derivatives ω~∂

and θ~

∂ are related by (F – 6).1

220

( )( ) θ

ωωωθθ

θωθθ

θω

θ

~~

~~,~

~~,~

~~

∆∆

≈∂∂

∂∂−=

∂∂

= group

group

constgroup

(F - 6)

Therefore, we can rewrite (F – 1) as (F – 7).

( ) ( ) ( )ω

ω

ωθθθ

θ

ωθθθωθθ ~

~~,~

~~,~

,~,~∆

∂

∂−=∆

∂

∂=∆∆ group

ingroup

instruc (F - 7)

Substituting (F – 5) into (F – 7) we obtain (F – 8).

( ) ( ) ( ) ωωθωω

ωωθθωθθ ω

ω ~~,~~~

~~,~,~,~

∆+−=∆∂

∂∂−=∆∆ ∫ CDisp

Dispinstruc (F - 8)

Since (F – 8) has to reduce to (F – 3) in the case of Disp=0, we can calculate the value of

the integration constant C. Substituting C and (F – 6) into (F – 8), we obtain the

relationship between the beam cone ∆θstruc in a dispersive stack and the incident beam

cone ∆θin as (F – 9).

( ) ( ) ( ) ( )( ) in

group

groupinDispstrucinstruc Disp θ

ωωθθ

θωθθωθθωθθθωθθ ω ∆

∂∂

∂∂+∆∆=∆∆ = ~~,~

~~,~~,~,~,~,~,~

0, (F - 9)

For a given dispersive stack, the beam cone Disp=0 can be estimated by calculating the

beam cone in a wavelength regime far away from any dispersion. Next we will examine

the second part of the sum in (F – 9) closer. Using the variable transformation (F – 10)

and (F – 11), we obtain (F – 12) and (F – 13).

( )θω

β~sin

~

c= (F - 10)

ωω ~= (F - 11)

( ) ( ) ( )

( )θ

ωβ

ωβθθω

ω

ωβθ

θβ

β

ωβθ

θ

ωθθ

~cos~,

~,

~,

~~,~

cgroup

groupgroupgroup

∂

∂=

∂∂

∂

∂+

∂∂

∂

∂=

∂

∂

(F - 12)

221

( ) ( ) ( )

( ) ( )ω

ωβθθβ

ωβθωω

ω

ωβθ

ωβ

β

ωβθ

ω

ωθθ

∂

∂+

∂

∂=

∂∂

∂

∂+

∂∂

∂

∂=

∂

∂

,~sin,

~,

~,

~~,~

groupgroup

groupgroupgroup

c

(F - 13)

Substituting (F - 12) and (F - 13) into (F - 6) we obtain (F - 14).

( )( ) ( )

( ) constgroup

groupgroup

group

group

cc

c

=∂∂

−=

∂∂

∂∂+

=∂∂

∂∂

θωβ

θ

θω

βωβθ

ωωβθθ

θω

ωωθθ

θωθθ

~sin

~cos~

,,~sin

~cos~

~~,~

~~,~(F - 14)

In Chapter 4 we have seen that (∂β/∂ω)|K=const is approximately constant with wavelength.

Fig. F.1 plots the change in (∂β/∂ω)|K=const for the two structures discussed in Chapter 8.

1520 1530 1540 1550 1560 15702

0

2

4

6

8

10


Periodic Design

Wavelength in nm

Chan

ge in

%

1520 1530 1540 1550 1560 15700

1

2

3


Non-periodic Design

Wavelength in nm

Chan

ge in

%

Fig. F.1. The change in (∂β/∂ω)|K=const as a function of wavelength.

Without proof, we are assuming now that relation (F - 15) is a good approximation.

( )( )

( )( ) βωβ

ωωβωβ

βωββωωβ

ωβ

θ ∂∂∂∂

−=∂∂

≈∂∂

∂∂∂−=

∂∂

== ,,

,,

22

2

KK

KK

constKconstgroup

(F - 15)

As seen in Chapter 4, a change in the group propagation angle is tightly linked to a

change in the wavevector K in the case of structural dispersion. This could be the

222

underlying reason that (F - 15) is a good approximation for many structures of interest. A

rigorous proof still remains to be performed.

Substituting (F - 3) and (F - 14) into (F - 9), we finally obtain an interesting

approximation for ∆θstruc as a function of frequency. The result is given in (F - 16).

( ) ( ) in

K

avg

instruc

cDisp

nθ

ωβ

θ

θωωθ

θ

θθωθθ ω ∆

∂∂

−+

−≈∆∆

~sin

~cos~~,~~sin

~cos,~,~22

(F - 16)

The only rapidly varying term with wavelength in (F - 15) is the dispersion Dispω. Thus,

the angular range of a mode ∆θstruc can be estimated as the input angular range ∆θin

multiplied by the sum of a constant term added to a term that is proportional to the

dispersion. The validity of this approximation is shown with Fig. F.2.

1520 1530 1540 1550 1560 15700

2

4

6

Bloch CalculationApproximation ~Dispersion + const

Periodic Design

Wavelength in nm

Ang

ular

rang

e in

deg

1520 1530 1540 1550 1560 15701

2

3

4

5

6

Transfer Matrix CalculationApproximation ~Dispersion + const

Non-periodic Design

Wavelength in nm

Ang

ular

rang

e in

deg


calculated and calculated from (8 – 17) as being proportional to the dispersion.

In Fig. F.2 the angular range within the structure is graphed as a function of wavelength

for an input angular range of 0.5º. The black and pink curves are obtained by directly

calculating the group propagation angles for the two extreme input angles. The cyan

223

curves on the other hand are calculated using (F - 15). The deviations of the two curves

are mainly due to the fact that (F - 15) estimates the angular range by just looking at the

central incidence angle. Fig. F.3 shows the same calculation for an input angular range of

0.05º. In this case the agreement between the curves is very good even for extreme

points, thus confirming the hypothesis that the error in Fig. F.2 is mainly due to replacing

differences by differentials.

1520 1530 1540 1550 1560 15700

0.2

0.4

0.6

Bloch CalculationApproximation ~Dispersion + const

Periodic Design

Wavelength in nm

Ang

ular

rang

e in

deg

1520 1530 1540 1550 1560 15700

0.1

0.2

0.3

0.4

0.5

0.6

Transfer Matrix CalculationApproximation ~Dispersion + const

Non-periodic Design

Wavelength in nm

Ang

ular

rang

e in

deg


calculated and calculated from (8 - 17) as a function of wavelength.

References



(1993).

224

Appendix G

Composition of Structures

Here the layer compositions of the discussed structures are given. Table G.1 lists the

layer thicknesses as a function of the layer number and the layer material for the stacks

discussed in Chapters 3 to 5. The first row names the structures in the format

Chapter-Stack number. The stacks are numbered by order of appearance. The second row

describes the type of structure. The third row lists the wavelength range of interest. As

many structures have a periodic behavior with frequency, a different wavelength range

could be chosen just as well. The wavelength range given here corresponds to the one

discussed in the text and can be seen as a rough guide of where to look if plotting the

dispersion characteristics of the different stacks. Next the incidence angle in vacuum and

the polarization used in designing the stack, as well as the total stack thickness, are given.

The remaining rows list the stack composition starting with the substrate material. All

layer thicknesses are in nanometers. The substrate material for the fabricated stacks is

quartz (n=1.52 at 880 nm). Some of the simulations have been performed assuming an

incidence material such as SiO2 or air. All stacks considered have alternating layers of

SiO2 (n=1.456 at 880 nm) and Ta2O5 (n=2.06 at 880 nm). As in some stacks the first

layer is Ta2O5, while it is SiO2 in other stacks, designs that have SiO2 as the first layer are

225

shifted down by one row. Table G.2 gives the composition of the stacks discussed in

Chapters 6 to 10.

3-1 3-2 3-3 5-1 5-2 5-3 5-4 5-5 5-6200-layerPeriodic

60-layerPeriodic

60-layerImpedance

MatchedPeriodic

60-layerDouble-Chirped

f=0.5

60-layerDouble-Chirpedf=0.33


f=0.2


f=0.1

60-layerSingle-Chirped

f=0


860 - 901 935 - 1050 900 - 1050 800-1100 800-1010 800-950 800-900 800-880 780-950

40° 45° 45° 45° 45° 45° 45° 45° 40°

p-pol. p-pol. p-pol. p-pol. p-pol. p-pol. p-pol. p-pol. p-pol.

30.0 8.3 9.0 9.4 9.1 8.8 8.7 8.5 28.1

Quartz SiO2 SiO2 Quartz Quartz Quartz Quartz Quartz Quartz1 Ta2O5 150.0 108.0 3.72 SiO2 150.0 167.9 330.0 312.2 312.0 311.9 311.8 311.7 291.83 Ta2O5 150.0 108.0 7.6 2.1 1.9 1.8 1.8 1.7 0.54 SiO2 150.0 167.9 323.9 307.4 307.1 306.7 306.4 306.1 290.65 Ta2O5 150.0 108.0 11.7 6.5 6.1 5.8 5.6 5.4 1.56 SiO2 150.0 167.9 317.6 302.4 301.9 301.3 300.8 300.3 289.47 Ta2O5 150.0 108.0 15.8 11.2 10.4 9.9 9.5 9.2 2.58 SiO2 150.0 167.9 311.2 297.4 296.6 295.8 295.1 294.3 288.19 Ta2O5 150.0 108.0 19.9 15.9 14.8 14.0 13.6 13.1 3.510 SiO2 150.0 167.9 304.8 292.2 291.2 290.1 289.3 288.3 286.811 Ta2O5 150.0 108.0 24.2 20.7 19.2 18.3 17.7 17.1 4.612 SiO2 150.0 167.9 298.2 287.1 285.8 284.5 283.4 282.2 285.513 Ta2O5 150.0 108.0 28.4 25.5 23.8 22.6 21.8 21.1 5.714 SiO2 150.0 167.9 291.6 281.9 280.3 278.7 277.4 276.1 284.215 Ta2O5 150.0 108.0 32.7 30.4 28.3 26.9 26.0 25.1 6.816 SiO2 150.0 167.9 285.0 276.7 274.8 272.9 271.4 269.9 282.817 Ta2O5 150.0 108.0 37.0 35.4 32.9 31.3 30.2 29.2 7.918 SiO2 150.0 167.9 278.3 271.5 269.2 267.1 265.4 263.7 281.519 Ta2O5 150.0 108.0 41.3 40.3 37.5 35.6 34.4 33.3 9.020 SiO2 150.0 167.9 271.6 266.3 263.7 261.3 259.3 257.4 280.121 Ta2O5 150.0 108.0 45.6 45.3 42.2 40.1 38.7 37.4 10.122 SiO2 150.0 167.9 264.8 261.1 258.1 255.4 253.2 251.1 278.823 Ta2O5 150.0 108.0 50.0 50.3 46.8 44.5 43.0 41.6 11.324 SiO2 150.0 167.9 258.1 255.9 252.5 249.5 247.1 244.7 277.425 Ta2O5 150.0 108.0 54.4 55.4 51.5 49.0 47.3 45.7 12.426 SiO2 150.0 167.9 251.2 250.8 246.9 243.5 241.0 238.4 276.027 Ta2O5 150.0 108.0 58.8 60.4 56.2 53.4 51.6 49.9 13.528 SiO2 150.0 167.9 244.4 245.7 241.3 237.6 234.8 232.0 274.629 Ta2O5 150.0 108.0 63.2 65.5 61.0 57.9 55.9 54.1 14.630 SiO2 150.0 167.9 237.6 240.6 235.7 231.6 228.6 225.6 273.231 Ta2O5 150.0 108.0 67.6 70.6 65.7 62.4 60.3 58.3 15.832 SiO2 150.0 167.9 230.7 235.5 230.0 225.6 222.4 219.2 271.833 Ta2O5 150.0 108.0 72.1 75.8 70.5 67.0 64.7 62.6 16.934 SiO2 150.0 167.9 223.8 230.5 224.4 219.6 216.1 212.7 270.435 Ta2O5 150.0 108.0 76.5 80.9 75.3 71.5 69.1 66.8 18.136 SiO2 150.0 167.9 216.8 225.6 218.8 213.6 209.8 206.2 269.037 Ta2O5 150.0 108.0 81.0 86.1 80.1 76.1 73.5 71.1 19.238 SiO2 150.0 167.9 209.9 220.7 213.2 207.6 203.6 199.7 267.639 Ta2O5 150.0 108.0 85.5 91.2 84.9 80.7 77.9 75.4 20.440 SiO2 150.0 167.9 202.9 215.8 207.6 201.5 197.3 193.2 266.241 Ta2O5 150.0 108.0 90.0 96.4 89.7 85.2 82.3 79.6 21.542 SiO2 150.0 167.9 195.9 211.0 202.0 195.5 191.0 186.7 264.843 Ta2O5 150.0 108.0 94.5 101.6 94.5 89.8 86.7 83.9 22.744 SiO2 150.0 167.9 188.9 206.3 196.4 189.4 184.6 180.2 263.445 Ta2O5 150.0 108.0 99.0 106.8 99.4 94.4 91.2 88.2 23.946 SiO2 150.0 167.9 181.9 201.6 190.9 183.4 178.3 173.6 262.047 Ta2O5 150.0 108.0 103.5 112.0 104.2 99.1 95.6 92.5 25.048 SiO2 150.0 167.9 174.9 197.0 185.3 177.3 171.9 167.0 260.549 Ta2O5 150.0 108.0 108.0 117.3 109.1 103.7 100.1 96.9 26.250 SiO2 150.0 167.9 167.9 192.5 179.8 171.2 165.6 160.4 259.151 Ta2O5 150.0 108.0 108.0 122.5 114.0 108.3 104.6 101.2 27.452 SiO2 150.0 167.9 167.9 191.1 177.5 168.4 162.5 157.1 257.753 Ta2O5 150.0 108.0 108.0 125.7 116.7 110.8 106.9 103.4 28.654 SiO2 150.0 167.9 167.9 192.9 178.4 168.9 162.7 157.1 256.355 Ta2O5 150.0 108.0 108.0 126.9 117.4 111.1 107.0 103.4 29.756 SiO2 150.0 167.9 167.9 194.8 179.4 169.4 162.9 157.1 254.8

Substrate

Description

Incidence Angle in Vacuum

PolarizationTotal Stack Thickness

in µm

Operatingwavelengthrange in nm

226

3-1 3-2 3-3 5-1 5-2 5-3 5-4 5-5 5-657 Ta2O5 150.0 108.0 108.0 128.2 118.0 111.5 107.2 103.4 30.958 SiO2 150.0 167.9 167.9 196.8 180.4 169.9 163.2 157.1 253.459 Ta2O5 150.0 108.0 108.0 129.4 118.7 111.8 107.3 103.4 32.160 SiO2 150.0 167.9 167.9 198.7 181.4 170.4 163.4 157.1 251.961 Ta2O5 150.0 130.7 119.4 112.1 107.5 103.4 33.362 SiO2 150.0 250.563 Ta2O5 150.0 34.564 SiO2 150.0 249.165 Ta2O5 150.0 35.766 SiO2 150.0 247.667 Ta2O5 150.0 36.868 SiO2 150.0 246.269 Ta2O5 150.0 38.070 SiO2 150.0 244.771 Ta2O5 150.0 39.272 SiO2 150.0 243.373 Ta2O5 150.0 40.474 SiO2 150.0 241.975 Ta2O5 150.0 41.676 SiO2 150.0 240.477 Ta2O5 150.0 42.878 SiO2 150.0 239.079 Ta2O5 150.0 44.080 SiO2 150.0 237.581 Ta2O5 150.0 45.282 SiO2 150.0 236.183 Ta2O5 150.0 46.484 SiO2 150.0 234.685 Ta2O5 150.0 47.686 SiO2 150.0 233.287 Ta2O5 150.0 48.888 SiO2 150.0 231.789 Ta2O5 150.0 50.090 SiO2 150.0 230.391 Ta2O5 150.0 51.292 SiO2 150.0 228.893 Ta2O5 150.0 52.594 SiO2 150.0 227.395 Ta2O5 150.0 53.796 SiO2 150.0 225.997 Ta2O5 150.0 54.998 SiO2 150.0 224.499 Ta2O5 150.0 56.1100 SiO2 150.0 223.0101 Ta2O5 150.0 57.3102 SiO2 150.0 221.5103 Ta2O5 150.0 58.5104 SiO2 150.0 220.1105 Ta2O5 150.0 59.7106 SiO2 150.0 218.6107 Ta2O5 150.0 61.0108 SiO2 150.0 217.2109 Ta2O5 150.0 62.2110 SiO2 150.0 215.7111 Ta2O5 150.0 63.4112 SiO2 150.0 214.3113 Ta2O5 150.0 64.6114 SiO2 150.0 212.8115 Ta2O5 150.0 65.8116 SiO2 150.0 211.3117 Ta2O5 150.0 67.1118 SiO2 150.0 209.9119 Ta2O5 150.0 68.3120 SiO2 150.0 208.4121 Ta2O5 150.0 69.5122 SiO2 150.0 207.0123 Ta2O5 150.0 70.7124 SiO2 150.0 205.5125 Ta2O5 150.0 72.0126 SiO2 150.0 204.1127 Ta2O5 150.0 73.2128 SiO2 150.0 202.6129 Ta2O5 150.0 74.4

227

3-1 3-2 3-3 5-1 5-2 5-3 5-4 5-5 5-6130 SiO2 150.0 201.2131 Ta2O5 150.0 75.7132 SiO2 150.0 199.7133 Ta2O5 150.0 76.9134 SiO2 150.0 198.2135 Ta2O5 150.0 78.1136 SiO2 150.0 196.8137 Ta2O5 150.0 79.4138 SiO2 150.0 195.3139 Ta2O5 150.0 80.6140 SiO2 150.0 193.9141 Ta2O5 150.0 81.8142 SiO2 150.0 192.4143 Ta2O5 150.0 83.1144 SiO2 150.0 191.0145 Ta2O5 150.0 84.3146 SiO2 150.0 189.5147 Ta2O5 150.0 85.5148 SiO2 150.0 188.1149 Ta2O5 150.0 86.8150 SiO2 150.0 186.6151 Ta2O5 150.0 88.0152 SiO2 150.0 185.2153 Ta2O5 150.0 89.3154 SiO2 150.0 183.7155 Ta2O5 150.0 90.5156 SiO2 150.0 182.3157 Ta2O5 150.0 91.7158 SiO2 150.0 180.8159 Ta2O5 150.0 93.0160 SiO2 150.0 179.4161 Ta2O5 150.0 94.2162 SiO2 150.0 177.9163 Ta2O5 150.0 95.5164 SiO2 150.0 176.5165 Ta2O5 150.0 96.7166 SiO2 150.0 175.0167 Ta2O5 150.0 98.0168 SiO2 150.0 173.6169 Ta2O5 150.0 99.2170 SiO2 150.0 172.2171 Ta2O5 150.0 100.5172 SiO2 150.0 170.7173 Ta2O5 150.0 101.7174 SiO2 150.0 169.3175 Ta2O5 150.0 103.0176 SiO2 150.0 167.8177 Ta2O5 150.0 104.2178 SiO2 150.0 166.4179 Ta2O5 150.0 105.5180 SiO2 150.0 165.0181 Ta2O5 150.0 106.7182 SiO2 150.0 164.4183 Ta2O5 150.0 107.4184 SiO2 150.0 164.6185 Ta2O5 150.0 107.6186 SiO2 150.0 164.8187 Ta2O5 150.0 107.7188 SiO2 150.0 165.1189 Ta2O5 150.0 107.9190 SiO2 150.0 165.3191 Ta2O5 150.0 108.1192 SiO2 150.0 165.6193 Ta2O5 150.0 108.2194 SiO2 150.0 165.8195 Ta2O5 150.0 108.4196 SiO2 150.0 166.1197 Ta2O5 150.0 108.5198 SiO2 150.0 166.3199 Ta2O5 150.0 108.7200 SiO2 150.0 166.6201 Ta2O5 108.9

Table G.1. Layer Composition of the structures appearing in Chapters 3 to 5.

228

6-1 6-2 7-1 7-2 8-1 8-2 8-3 10-1 10-133-layerGires-

Tournois

33-layerCoupled-

Cavity

66-layerNumericallyOptimized(Designed)

66-layerNumericallyOptimized,

(Fabricated)

200-layerPeriodic

200-layerNon-PeriodicLinear Angle

200-layerNon-PeriodicLinear Shift

100-layerFour-Step

Design

66-layerThree-Step

Design

810-850 842-854 1510-1580 815-845 1520-1570 1520-1570 1520-1570 1500-1580 810-850

45° 54° 45° 54° 40° 40° 40° 45° 48°

p-pol. s-pol. p-pol. p-pol. p-pol. p-pol. p-pol. p-pol. p-pol.

8.0 15.4 23.7 13.4 52.2 48.2 48.9 33.1 13.4

Air Quartz Quartz Quartz Quartz Quartz Quartz Quartz Quartz1 Ta2O5 337.0 188.0 261.2 360.0 404.8 188.02 SiO2 167.0 168.3 485.8 276.6 261.2 0.8 473.7 656.8 276.63 Ta2O5 109.8 2954.0 337.0 188.0 261.2 415.5 20.3 837.2 188.04 SiO2 167.0 168.3 485.8 276.6 261.2 20.0 465.0 643.9 276.65 Ta2O5 109.8 2967.0 337.0 188.0 261.2 378.6 11.5 439.5 188.06 SiO2 167.0 168.3 485.8 276.6 261.2 40.9 453.6 672.6 276.67 Ta2O5 109.8 118.9 372.7 208.0 261.2 411.2 14.5 492.8 208.08 SiO2 167.0 168.3 668.3 380.6 261.2 6.2 482.4 673.5 380.69 Ta2O5 109.8 2728.0 346.9 194.0 261.2 482.2 27.5 456.9 194.010 SiO2 3577.7 168.3 668.5 380.6 261.2 25.5 449.0 556.9 380.611 Ta2O5 109.8 118.9 391.0 218.0 261.2 568.6 19.0 418.2 218.012 SiO2 167.0 168.3 562.1 320.3 261.2 9.9 483.2 531.8 320.313 Ta2O5 109.8 118.9 447.7 250.0 261.2 442.7 21.0 357.3 250.014 SiO2 167.0 168.3 518.7 295.4 261.2 11.8 476.7 576.5 295.415 Ta2O5 109.8 2495.0 497.3 278.0 261.2 548.2 22.0 409.2 278.016 SiO2 167.0 168.3 484.8 276.6 261.2 13.8 502.1 462.0 276.617 Ta2O5 109.8 118.9 812.1 453.0 261.2 361.2 22.6 404.8 453.018 SiO2 167.0 168.3 502.2 286.0 261.2 15.7 511.8 560.1 286.019 Ta2O5 109.8 118.9 810.7 453.0 261.2 578.4 24.0 420.4 453.020 SiO2 167.0 168.3 462.3 263.1 261.2 0.2 520.1 462.1 263.121 Ta2O5 109.8 118.9 421.1 235.0 261.2 487.3 15.9 416.0 235.022 SiO2 167.0 168.3 546.2 311.0 261.2 56.2 496.9 504.3 311.023 Ta2O5 109.8 118.9 439.0 245.0 261.2 577.3 18.6 441.7 245.024 SiO2 167.0 168.3 507.9 289.1 261.2 21.5 494.3 485.7 289.125 Ta2O5 109.8 118.9 446.8 249.0 261.2 480.7 21.5 437.4 249.026 SiO2 167.0 168.3 485.6 276.6 261.2 23.5 471.1 408.3 276.627 Ta2O5 109.8 118.9 454.7 254.0 261.2 478.2 25.5 453.0 254.028 SiO2 167.0 168.3 459.4 262.1 261.2 25.5 541.5 453.0 262.129 Ta2O5 109.8 118.9 462.7 258.0 261.2 475.8 37.5 457.7 258.030 SiO2 167.0 168.3 452.1 257.9 261.2 27.5 539.9 432.0 257.931 Ta2O5 109.8 118.9 470.6 263.0 261.2 509.9 48.8 475.4 263.032 SiO2 167.0 168.3 440.8 251.7 261.2 29.5 526.0 392.5 251.733 Ta2O5 109.8 118.9 106.8 60.0 261.2 544.1 38.7 491.1 60.034 SiO2 167.0 168.3 422.6 241.3 261.2 68.1 502.8 463.2 241.335 Ta2O5 103.9 58.0 261.2 505.1 28.8 475.9 58.036 SiO2 401.1 228.8 261.2 33.5 478.1 387.1 228.837 Ta2O5 101.9 57.0 261.2 502.6 30.0 83.8 57.038 SiO2 387.9 221.5 261.2 35.5 466.3 415.1 221.539 Ta2O5 110.0 61.0 261.2 463.6 32.4 89.6 61.040 SiO2 385.7 219.4 261.2 37.5 474.2 405.9 219.441 Ta2O5 118.1 66.0 261.2 461.2 36.1 95.5 66.042 SiO2 363.1 207.0 261.2 39.6 472.4 393.8 207.043 Ta2O5 126.2 70.0 261.2 493.6 40.2 101.9 70.044 SiO2 350.4 199.7 261.2 41.6 491.5 409.3 199.745 Ta2O5 134.4 75.0 261.2 438.9 54.3 107.1 75.046 SiO2 348.2 198.6 261.2 61.0 509.1 422.5 198.647 Ta2O5 142.5 80.0 261.2 558.0 67.7 113.0 80.048 SiO2 325.2 185.1 261.2 82.2 466.6 334.1 185.149 Ta2O5 150.7 84.0 261.2 578.4 47.5 118.8 84.050 SiO2 286.1 163.3 261.2 84.3 442.4 371.1 163.351 Ta2O5 158.9 89.0 261.2 448.8 46.9 124.6 89.052 SiO2 301.1 171.6 261.2 49.7 459.3 362.6 171.653 Ta2O5 167.2 93.0 261.2 428.9 46.9 130.5 93.054 SiO2 344.6 196.6 261.2 51.8 456.4 307.2 196.655 Ta2O5 175.4 98.0 261.2 443.8 47.9 136.3 98.056 SiO2 289.0 164.3 261.2 53.8 433.9 365.6 164.3

Total Stack Thickness

in µmSubstrate

Description

Operatingwavelengthrange in nm

Incidence Angle in Vacuum

Polarization

229

6-1 6-2 7-1 7-2 8-1 8-2 8-3 10-1 10-157 Ta2O5 175.4 98.0 261.2 441.3 50.8 142.2 98.058 SiO2 253.5 144.6 261.2 55.9 452.3 345.6 144.659 Ta2O5 175.4 98.0 261.2 457.9 53.7 148.1 98.060 SiO2 361.3 205.9 261.2 57.9 450.2 347.6 205.961 Ta2O5 175.4 98.0 261.2 455.4 67.6 154.0 98.062 SiO2 346.7 197.6 261.2 79.2 429.6 317.6 197.663 Ta2O5 195.4 109.0 261.2 433.8 71.4 160.0 109.064 SiO2 307.6 175.8 261.2 81.2 456.0 308.6 175.865 Ta2O5 186.3 104.0 261.2 319.6 65.2 165.9 104.066 SiO2 100.3 57.2 261.2 81.6 433.2 299.5 57.267 Ta2O5 261.2 488.2 66.9 171.968 SiO2 261.2 66.2 431.1 290.569 Ta2O5 261.2 426.2 68.2 177.970 SiO2 261.2 68.3 418.1 281.471 Ta2O5 261.2 368.0 69.0 177.972 SiO2 261.2 53.0 425.0 248.373 Ta2O5 261.2 384.7 70.1 177.974 SiO2 261.2 55.1 422.1 281.575 Ta2O5 261.2 436.1 71.7 178.076 SiO2 261.2 92.0 439.7 291.577 Ta2O5 261.2 487.6 84.0 178.078 SiO2 261.2 41.8 428.3 248.279 Ta2O5 261.2 377.1 76.3 178.080 SiO2 261.2 96.2 426.9 305.781 Ta2O5 261.2 482.6 89.0 200.382 SiO2 261.2 98.2 434.6 304.583 Ta2O5 261.2 426.0 81.6 200.384 SiO2 261.2 82.9 410.8 304.585 Ta2O5 261.2 406.1 84.1 200.386 SiO2 261.2 85.0 408.2 304.587 Ta2O5 261.2 403.5 86.2 200.388 SiO2 261.2 87.1 405.2 304.589 Ta2O5 261.2 401.1 88.0 200.390 SiO2 261.2 89.2 413.1 304.591 Ta2O5 261.2 398.5 89.4 200.392 SiO2 261.2 91.3 400.4 304.593 Ta2O5 261.2 395.9 90.8 200.394 SiO2 261.2 93.5 406.5 304.595 Ta2O5 261.2 376.0 92.1 200.396 SiO2 261.2 95.6 394.7 304.597 Ta2O5 261.2 390.8 93.8 200.398 SiO2 261.2 97.7 400.4 304.599 Ta2O5 261.2 459.8 105.7 200.3100 SiO2 261.2 82.4 389.1 304.5101 Ta2O5 261.2 424.1 107.8102 SiO2 261.2 121.1 385.7103 Ta2O5 261.2 419.9 90.2104 SiO2 261.2 123.2 394.0105 Ta2O5 261.2 382.4 113.0106 SiO2 261.2 125.3 371.6107 Ta2O5 261.2 380.0 105.9108 SiO2 261.2 90.9 389.2109 Ta2O5 261.2 394.8 109.1110 SiO2 261.2 93.0 358.3111 Ta2O5 261.2 373.2 112.1112 SiO2 261.2 112.5 365.3113 Ta2O5 261.2 353.2 114.9114 SiO2 261.2 114.6 363.2115 Ta2O5 261.2 368.0 117.2116 SiO2 261.2 116.8 370.9117 Ta2O5 261.2 365.6 119.4118 SiO2 261.2 118.9 368.0119 Ta2O5 261.2 380.4 121.3120 SiO2 261.2 121.1 365.0121 Ta2O5 261.2 360.5 122.9122 SiO2 261.2 123.2 352.2123 Ta2O5 261.2 357.9 124.1124 SiO2 261.2 125.3 349.2125 Ta2O5 261.2 355.5 125.3126 SiO2 261.2 127.5 345.8127 Ta2O5 261.2 352.8 126.6128 SiO2 261.2 129.6 333.0129 Ta2O5 261.2 404.4 128.1

230

6-1 6-2 7-1 7-2 8-1 8-2 8-3 10-1 10-1130 SiO2 261.2 149.2 329.3131 Ta2O5 261.2 328.6 130.1132 SiO2 261.2 133.9 327.6133 Ta2O5 261.2 273.8 132.2134 SiO2 261.2 136.1 354.9135 Ta2O5 261.2 396.7 134.6136 SiO2 261.2 138.2 342.7137 Ta2O5 261.2 286.1 147.3138 SiO2 261.2 122.9 327.3139 Ta2O5 261.2 374.3 159.8140 SiO2 261.2 142.5 326.7141 Ta2O5 261.2 317.7 151.9142 SiO2 261.2 144.7 304.1143 Ta2O5 261.2 278.7 144.4144 SiO2 261.2 129.4 323.2145 Ta2O5 261.2 403.2 146.7146 SiO2 261.2 149.0 320.8147 Ta2O5 261.2 310.2 139.2148 SiO2 261.2 151.1 318.6149 Ta2O5 261.2 290.1 141.8150 SiO2 261.2 153.3 316.2151 Ta2O5 261.2 468.8 144.5152 SiO2 261.2 155.5 314.0153 Ta2O5 261.2 283.4 146.9154 SiO2 261.2 157.6 311.5155 Ta2O5 261.2 300.1 149.1156 SiO2 261.2 142.4 278.0157 Ta2O5 261.2 314.9 151.5158 SiO2 261.2 161.9 347.1159 Ta2O5 261.2 275.9 163.1160 SiO2 261.2 200.6 272.7161 Ta2O5 261.2 273.3 155.2162 SiO2 261.2 166.3 333.1163 Ta2O5 261.2 270.8 157.2164 SiO2 261.2 168.4 298.6165 Ta2O5 261.2 268.2 168.9166 SiO2 261.2 170.6 296.0167 Ta2O5 261.2 265.8 192.4168 SiO2 261.2 172.8 291.7169 Ta2O5 261.2 263.3 173.6170 SiO2 261.2 138.5 279.8171 Ta2O5 261.2 260.7 175.6172 SiO2 261.2 177.1 288.2173 Ta2O5 261.2 223.4 177.6174 SiO2 261.2 142.8 285.6175 Ta2O5 261.2 255.7 169.9176 SiO2 261.2 144.9 261.9177 Ta2O5 261.2 253.2 171.7178 SiO2 261.2 147.2 280.5179 Ta2O5 261.2 250.8 163.8180 SiO2 261.2 149.3 246.9181 Ta2O5 261.2 286.3 176.0182 SiO2 261.2 187.0 267.0183 Ta2O5 261.2 286.7 177.3184 SiO2 261.2 187.4 277.5185 Ta2O5 261.2 304.4 177.8186 SiO2 261.2 187.6 277.7187 Ta2O5 261.2 324.1 178.1188 SiO2 261.2 187.9 278.3189 Ta2O5 261.2 230.2 178.4190 SiO2 261.2 188.3 278.7191 Ta2O5 261.2 271.0 188.8192 SiO2 261.2 188.4 290.3193 Ta2O5 261.2 290.7 178.9194 SiO2 261.2 188.8 289.8195 Ta2O5 261.2 212.5 179.4196 SiO2 261.2 187.2 267.3197 Ta2O5 261.2 52.6 169.6198 SiO2 261.2 29.1 273.9199 Ta2O5 261.2 164.8 201.0200 SiO2 261.2 170.5 277.5201 Ta2O5 161.5

Table G.2. Layer Composition of the structures appearing in Chapters 6 to 10.

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