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Dissertations in Forestry and Natural Sciences

ISBN 978-952-61-2330-1ISSN 1798-5668


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245

GAURAV BOSE

DIFFRACTIVE OPTICS BASED ON V-SHAPED STRUCTURES AND ITS APPLICATIONS


This book provides a survey of V-shaped diffractive structures and its applications.

Both exact and approximate methods are used to model the near-field interaction with the

wavelength scale scatterers. Several techniques to encode phase function in the form of modulating the structures are discussed

followed by the experimental demonstration of a triplicator. Further, the transition from

anti-to retro-reflection is demonstrated experimentally by testing several gratings with

different periods.

GAURAV BOSE

GAURAV BOSE

Diffractive optics based on

V-shaped structures

and its applications

Publications of the University of Eastern Finland


No 245

Academic Dissertation

To be presented by permission of the Faculty of Science and Forestry for public

examination in the Auditorium E100 in Educa Building at the University of

Eastern Finland, Joensuu, on December, 13, 2016,

at 12 o’clock noon.

Institute of Photonics

Grano Oy

Jyvaskyla, 2016

Editors: Prof. Pertti Pasanen, Prof. Jukka Tuomela,

Prof. Pekka Toivanen, Prof. Matti Vornanen

Distribution:

University of Eastern Finland Library / Sales of publications

[email protected]

http://www.uef.fi/kirjasto

ISBN: 978-952-61-2330-1 (printed)

ISSNL: 1798-5668

ISSN: 1798-5668

ISBN: 978-952-61-2331-8 (pdf)

ISSNL: 1798-5668

ISSN: 1798-5676

Author’s address: University of Eastern Finland

Department of Physics and Mathematics

P.O. Box 111

80101 JOENSUU

FINLAND

email: [email protected]

Supervisors: Professor Jari Turunen, D.Sc.

University of Eastern Finland


P.O. Box 111

80101 JOENSUU

FINLAND


Associate Professor Jani Tervo, Ph.D.



P.O. Box 111

80101 JOENSUU

FINLAND

email:[email protected]

Professor Markku Kuittinen, Ph.D.



P.O. Box 111

80101 JOENSUU

FINLAND


Reviewers: Nicolas Passilly, Ph.D.

Institut FEMTO-ST

Department of Micro Nano Sciences and Systems

15 B avenue des Montboucons

F-25030 Besancon Cedex

FRANCE


Fredrik Nikolajeff, Ph.D.

Uppsala University

Department of Engineering Sciences

P.O. Box 534

S-75121 UPPSALA

SWEDEN


Opponent: Professor Thierry Grosjean, Ph.D.

Institut FEMTO-ST

Departement dOptique P. M. Duffieux

15 B avenue des Montboucons

F-25030 Besancon Cedex

FRANCE


ABSTRACT

This thesis contains numerical and experimental studies on electro-

magnetic properties of micro- and nano-structured optical systems.

Since structures with features in the wavelength scale are consid-

ered, the Fourier Modal Method (FMM) is applied in the rigorous

numerical studies. The potentiality of diffractive elements with V-

shaped features is examined for several different applications.

A geometrical model based on the local plane interface approach

is introduced to study near-field effects, which involve interactions

of evanescent and inhomogeneous waves with a V-shaped dielec-

tric wedge, and the results are compared with rigorous FMM anal-

ysis. This geometrical model is shown to provide physical insight

in understanding the interaction of such waves, including surface

plasmons, with wavelength-scale scatterers.

Different coding schemes of high-carrier-frequency diffractive

optical elements are examined, which involve carrier gratings with

V-shaped features. In particular, a new method to realize reflective

position-modulated V-ridge diffractive elements is introduced. This

coding scheme is demonstrated by fabricating and characterizing a

triplicator for visible light.

We also study V-ridge gratings, whose facets are inclined at 45◦

compared to the surface normal and realize such gratings by using

precise mask alignment in the anisotropic wet etching process. Nu-

merical investigations are carried out to show the transition from

antireflection to retroreflection behavior of such gratings in the vis-

ible and the near-infrared wavelengths. The measurement results

show good agreement with theoretical results given by FMM.

Universal Decimal Classification: 535.42, 537.87, 681.7.02, 681.7.063

INSPEC Thesaurus: optics; micro-optics; diffractive optical elements; diffrac-

tion gratings; electromagnetic wave diffraction; light diffraction; light

propagation; optical fabrication; microfabrication; nanofabrication; metals;

electron beam lithography; etching; surface plasmons; Fourier analysis;

numerical analysis

Yleinen suomalainen asiasanasto: optiikka; optiset laitteet; mikrorakenteet;

nanorakenteet; metallit; valmistustekniikka; litografia; etsaus; numeerinen

analyysi

Preface

The thesis summarizes the work I did as a researcher in the In-

stitute of Photonics, Joensuu. Most of the findings were done by

sitting in front of the computer by reading journals, e-books and

running simulations. On several occasions I engrossed in physical

understanding and ended up staring at window leaving behind the

brewing sound of the coffee maker or the whirring of a CPU fan.

I would like to express my deepest gratitude to my supervi-

sors Prof. Jari Turunen, Dr. Jani Tervo for introducing me with the

field of diffractive optics. Without their effort this work would

not have been possible. I am also grateful to my third supervi-

sor Prof. Markku Kuittinen for his constant support and patience

with me. Their cumulative effort and guidance has made me where

I stand now and look further.

My sincere gratitude to Heikki, Ismo, Matthieu, Ton and Toni

for their wonderful cooperation at times whenever I needed them

in the form of simulation, fabrication or building setups. I extend

my gratitude to Pertti Paakkonen, Pertti Silfsten and Tommi Itko-

nen for providing me with the optical accessories for building my

setup. I would also like to thank Prof. Pasi Vahimaa and Prof. Timo

Jaaskelinen for providing me the opportunity to work in this de-

partment. I would like to acknowledge reviewers Dr. Nicolas Pas-

silly and Associate Prof. Fredrik Nikolajeff for their reviews and

comments. I am equally grateful to Prof. Thierry Grosjean who has

accepted to be my opponent in a very short notice.

The quietude was felt more without their presence, to name but

a few Rahul, Somnath, Juha, Henri, Noora, Kimmo, Leila, Bisrat

and Vishal. I would like to thank all of you for encouraging me

with all kinds of technical support. Rahul has been on my side in

every taxing situations.

In addition to all the foregoing, I would like to thank my wife

Samriddhi. Her unwavering love, support and motivation were

undeniably the basis on which the past 4 years of my life have been

built. Her tolerance of my invariably variable moods is a testament

in itself of her devotion and love. Extending my love to my elder

brother, without his support I could have never went abroad for

study. Finally my loveliest mother who hasn’t prioritized anything

before my education and happiness, to my adorable father who

couldn’t see me finishing this work but I believe he must be proud

of me once again.

Joensuu November 15, 2016 Gaurav Bose

Contents

1 INTRODUCTION 1

2 FUNDAMENTALS OF THE ELECTROMAGNETIC

THEORY OF LIGHT 7

2.1 Complex field representation . . . . . . . . . . . . . . 7

2.2 Macroscopic Maxwell’s equations . . . . . . . . . . . 9

2.3 Constitutive relations . . . . . . . . . . . . . . . . . . . 10

2.4 Boundary conditions . . . . . . . . . . . . . . . . . . . 13

2.5 Wave equations and the TE/TM decomposition . . . 13

2.6 Simplest solution of Maxwell’s equations . . . . . . . 15

3 INTERACTION OF LIGHT WITH MICRO-

STRUCTURED SURFACES 17

3.1 Reflection and transmission . . . . . . . . . . . . . . . 17

3.2 Angular spectrum representation . . . . . . . . . . . . 20

3.3 Scattering from periodic structures . . . . . . . . . . . 22

3.3.1 Eigenvalue problem in non-conical mounting 23

3.3.2 Boundary condition solution in multilayered

gratings . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Effective medium theory . . . . . . . . . . . . . . . . . 32

3.5 Fundamentals of paraxial design methods . . . . . . 34

3.5.1 Thin element approximation . . . . . . . . . . 34

3.5.2 Local plane interface Approximation . . . . . 35

3.5.3 Iterative Fourier Transform Algorithm . . . . 36

4 LIGHT PROPAGATION IN WAVELENGTH

SCALE STRUCTURES 41

4.1 Near-field detection by the dielectric wedge . . . . . 41

4.1.1 Plane-wave incidence at high oblique angle . 42

4.1.2 Observation of evanescent-wave interference

patterns . . . . . . . . . . . . . . . . . . . . . . 47

4.2 Near field detection through a planar waveguide . . 49

4.2.1 Coupling efficiency from geometrical and rig-

orous models . . . . . . . . . . . . . . . . . . . 52

4.2.2 Detection of evanescent-wave interference pat-

terns . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3 Metallic gratings with sub-wavelength slits . . . . . . 56

4.3.1 Geometrical configuration and models . . . . 56

4.3.2 Coupling efficiencies by rigorous theory and

the geometrical phase matching model . . . . 58

4.3.3 Examples of field patterns within the structure 61

4.3.4 Observation of surface plasmon interference . 63

4.4 Detection of evanescent fields above binary subwave-

length gratings . . . . . . . . . . . . . . . . . . . . . . 64

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5 CODING OF HIGH-FREQUENCY CARRIER

V-SHAPE GRATINGS 71

5.1 V-groove width modulation . . . . . . . . . . . . . . . 73

5.1.1 Transmission-type groove width modulation . 74

5.1.2 Reflection-type groove width modulation . . . 75

5.2 V-ridge position modulation . . . . . . . . . . . . . . . 77

5.2.1 Reflection type ridge-position modulation . . 79

5.2.2 Coding of V-ridge structures . . . . . . . . . . 80

5.2.3 Numerical examples . . . . . . . . . . . . . . . 82

5.2.4 Effects of varying the wavelength and angle

of incidence . . . . . . . . . . . . . . . . . . . . 85

5.2.5 Transmission-type ridge position modulation 87

5.3 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.4 Experimental results . . . . . . . . . . . . . . . . . . . 90

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6 V-RIDGE GRATINGS: TRANSITION FROM

ANTIREFLECTION TO RETROREFLECTION 95

6.1 Geometry and principle . . . . . . . . . . . . . . . . . 96

6.2 Numerical simulations . . . . . . . . . . . . . . . . . . 96

6.2.1 Effect of varying the period . . . . . . . . . . . 97

6.2.2 Effect of varying the wavelength . . . . . . . . 98

6.2.3 Effect of finite substrate thickness . . . . . . . 100

6.2.4 Effect of varying flat bottom width . . . . . . 101

6.3 Experimental results . . . . . . . . . . . . . . . . . . . 102

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7 CONCLUSIONS AND OUTLOOK 107

REFERENCES 109

1 Introduction

In elementary physics and optics textbooks, interference and diffrac-

tion phenomena are approached by fairly elementary concepts and

techniques based on the scalar theory of light [1]. In that context,

the Helmholtz equation is satisfied in free space propagation, and

diffraction by gratings is also treated by elementary approaches

based on scalar theory. However, if a field propagating in free space

is non-paraxial, one can no longer ignore the electromagnetic na-

ture of light since also the longitudinal components of the electric

and magnetic fields become significant. The same is true in grating

diffraction if the period of the grating is of the same order of mag-

nitude as the wavelength of light [2–5]. In this case the grating gen-

erates diffraction orders propagating in greatly differing directions

and, in addition, inhomogeneous waves that propagate along the

grating surface can become important. In this so-called resonance

domain a multitude of unexpected effects emerge, which can only

be predicted by rigorous diffraction theory, i.e., by determination of

the diffracted field by means of exact solution of Maxwell’s equa-

tions. This thesis contains studies of phenomena that occur in the

resonance domain. In particular, gratings and non-periodic struc-

tures involving V-shaped structural details are considered, along

with applications of such structures.

The basic concepts of the electromagnetic theory of light are dis-

cussed in chapter 2, where Maxwell’s equations, constitutive rela-

tions, and boundary conditions are introduced. In chapter 3 we first

discuss Fresnel’s equations that describe the reflection and trans-

mission of a plane wave incident at a plane boundary. We then pro-

ceed to discuss the exact solution of macroscopic Maxwell’s equa-

tions in the presence of more complicated geometric configurations,

namely diffraction gratings. There is a plethora of appropriate algo-

rithms for grating analysis, which are based on solving Maxwell’s

equations numerically at a price of vastly increased memory con-

Gaurav Bose: Diffractive optics based on V-shaped structures and itsapplications

sumption and computational time compared to simple scalar mod-

els [6–10]. Of these methods, we consider in particular the Fourier

modal method (FMM) [4], which is the one used throughout the

thesis whenever rigorous solutions are needed.

Apart from the high computational complexity, rigorous solu-

tions of grating diffraction problems do not always provide intu-

itive understanding of what happens inside the structure and why

resonant phenomena occur. Therefore, it is physically appealing to

supplement rigorous solutions of Maxwell’s equations by heuristic

descriptions of wave propagation inside the structure. When suc-

cessful, such descriptions could alleviate the numerical modeling

burden and, at the same time, retain the essence of the pertinent

physical phenomena and further exploit them in applications. It

is of substantial interest to see how far these heuristic elementary

approaches can be pushed in the modeling of diffraction by fine

structures, and to investigate how close to the rigorous results one

may get by exploiting them.

In an attempt to fill the gap between the approximate and ex-

act methods, Swanson [11] showed that the standard scalar the-

ory can be extended by ray tracing: he considered thick blazed

gratings by taking into account shadowing effects near the vertical

boundary of the triangular profile. These effects were studied fur-

ther in Refs. [12, 13], and multiple scattering effects were treated in

Refs. [14–16]. In another development [17], the regions of validity

of the scalar diffraction theory were investigated for binary gratings

by using the rigorous coupled wave technique [6]. In Refs. [18, 19],

a computationally efficient refinement of the thin-element approx-

imation for the analysis and design of binary gratings in the non-

paraxial domain was introduced.

In most of the heuristic studies mentioned above, the field in-

side wavelength-scale structures was modeled by geometrical rays

associated with sections of homogeneous plane waves. This type

of models are known as local plane-wave and local plane-interface

approximations. In chapter 4 of this thesis, we proceed one step

further in the local-plane-interface approach, by adding evanescent

2 Dissertations in Forestry and Natural Sciences No 245

Introduction

fields in the analysis. In particular, we describe tunneling of in-

homogeneous plane waves into V-shape structures and their prop-

agation though such structures. We gradually introduce different

geometries for generating evanescent, inhomogeneous, and plas-

mon waves to be detected by such V-shaped probes, comparing the

results with those of the Fourier modal method at all appropriate

instances.

The phase of a light wave can be influenced in several ways, in-

cluding retardation when the wave travels through a dielectric, by

phase jumps on reflection, and by the detour-phase principle [20].

In an elementary picture, the detour-phase principle can be best

understood by considering two rays leaving any two adjacent grat-

ing slits and propagating into the direction of the first order. In

the case of a regular grating there will always be a path difference

of one wavelength between these two rays, and hence the detour

phase equals 2π. If the grating slits are not in their perfect regular

positions, the detour phases between two adjacent rays varies as a

function of position and the diffracted wavefront will be deformed.

Lohmann used this idea to his advantage by realizing complex fil-

ter functions that lead to predefined far-field diffraction patterns,

which lead to the birth of computer-synthesized diffractive optics.

Apart from fixed diffractive elements, which can be fabricated accu-

rately by lithographic techniques, real-time reconfigurable elements

can be realized by means of spatial light modulators [21, 22]. To

mention just a few of the vast number of applications of synthetic

diffractive optics, elements based on the detour-phase principle can

be used in, e.g., optical data storage [23, 24], and in designing dis-

crete and continuous photonic bandgaps in the form of a shifted

Bragg grating in the single mode fiber [25].

In chapter 5 we introduce a new method to realize high-carrier-

frequency diffractive elements on the basis of the detour-phase prin-

ciple. We employ carrier gratings with V-shaped profiles as an al-

ternative to previously considered binary resonance-domain grat-

ings [26–28]. Several general techniques are discussed for the re-

alization of diffractive structures by modulation of the width and

Dissertations in Forestry and Natural Sciences No 245 3


position of the V-grooves or V-ridges in both reflection and trans-

mission modes of operation. Methods based on width modula-

tion are found to have severe limitations, whereas the approaches

based on position modulation prove highly successful. The pro-

posed position-modulation coding technique is demonstrated ex-

perimentally in the reflection mode by fabricating and character-

izing triplicators, which divide one incident plane wave into three

diffraction orders of equal efficiency.

In chapter 6, we examine V-ridge gratings whose half tip apex

angle is 45◦. We first investigate numerically the zeroth-order trans-

mittance and reflectance for different ratios d/λ of the grating pe-

riod d and the wavelength λ. With d/λ < 1, the gratings behave

as antireflection layers, and at d/λ ≫ 1 they gradually become

retroreflectors. The transition from antireflection to retroreflection

is demonstrated experimentally by fabricating and testing several

gratings with different periods, and good agreements with theoret-

ical results are observed.

Some of the aforementioned results have been published in the

following original articles and presented in the following interna-

tional and national conferences:

1 G. Bose, H. J. Hyvarinen, J. Tervo, and J. Turunen, “Geomet-

rical optics in the near field: local plane-interface approach

with evanescent waves,” Opt. Express 23, 330–339 (2015).

2 G. Bose, A. Verhoeven, I. Vartiainen, M. Roussey, M. Kuit-

tinen, J. Tervo, and J. Turunen “Diffractive optics based on

modulated subwavelength-domain V-ridge gratings,” J. Opt.

18, 085602 (2016).

3 G. Bose, H. J. Hyvarinen, J. Tervo, and J. Turunen, “Probing

Surface Plasmons by Bare V-shaped Tips: Modeling by Geo-

metrical Optics and Rigorous Diffraction Theory,” Opt. Rev.

(submitted).

4 G. Bose, H. J. Hyvarinen, J. Rahomaki, S. Rehman, J. Tervo,

and J. Turunen, “Near-field microscopy by surface-wave-assist-


Introduction

ed extraordinary transmission of light,” Optics Days (Helsinki,

Finland, poster presentation, 2013).

5 G. Bose, A. Verhoeven, M. Kuittinen, J. Tervo, and J. Turunen,

“V-groove high frequency carrier diffractive optical elements,”

Optics in Engineering (Joensuu, Finland, poster presentation,

2015).

6 G. Bose, H. J. Hyvarinen, J. Tervo, and J. Turunen, “Analysis of

surface plasmons by scanning near-field optical microscopes:

Modeling by geometrical optics and rigorous diffraction the-

ory,” Optics-photonics Design and Fabrication (Weingarten, Ger-

many, oral presentation, 2016).

Many of the results presented in chapters 4 and 5, and all results

presented in chapter 6 are still unpublished. Several original papers

related to these subjects are currently under preparation.




2 Fundamentals of the

electromagnetic theory

of light

The thesis deals with studies of electromagnetic properties of micro-

and nano-structured optical systems. Thereby it is important to un-

derstand the basic principles of the electromagnetic theory of light

and its propagation through any medium or in free space. This

chapter introduces the basic electromagnetic equations for that pur-

pose, which are used throughout the thesis.

2.1 COMPLEX FIELD REPRESENTATION

The measurable field quantities used in classical optics must be

real functions of position vector r and time t, which often leads

to complicated mathematics. Hence, it is mathematically more suit-

able to use the complex representation of electromagnetic fields. So

the convenient form of a monochromatic stationary time-harmonic

field of frequency ω0 can be expressed as

Ure(r, t) = ℜ{U(r) exp(−iω0t)} , (2.1)

where U(r) represents the complex amplitude of the real-valued

function Ure(r, t) that may be replaced with any of the vectors E(r),

H(r), D(r), B(r), and J(r), which are the electric field, the mag-

netic field, the electric displacement, the magnetic induction, and

the electric current density, respectively.

In order to describe polychromatic light, the time-harmonic rep-

resentation of the field in Eq. (2.1) must be generalized. To this end

we again define a unique complex counterpart of the real field.



Considering any real physical field quantity, we assume it to be

square integrable with respect to time, i.e,

∫ ∞

−∞U2

re(r, t)dt < ∞. (2.2)

Then we can represent Ure(r, t) as a temporal Fourier integral

Ure(r, t) =∫ ∞

−∞Ure(r, ω) exp(−iωt)dω, (2.3)

where

Ure(r, ω) =1

2π

∫ ∞

−∞Ure(r, t) exp(iωt)dt, (2.4)

and Ure(r, ω) is the spectral amplitude of the real field in the space-

frequency domain. The Fourier-transform pair of Eqs. (2.3) and

(2.4) shows us that any space-time domain vector field Ure(r, t)

may be expressed as a superposition of spectral complex ampli-

tudes Ure(r, ω) of time-harmonic fields.

Since the aforementioned space-time field is a real-valued func-

tion, the corresponding space-frequency complex function should

satisfy the complex conjugate relation

Ure(r,−ω) = U∗re(r, ω), (2.5)

where the asterisk ∗ denotes the complex conjugate. This relation

shows that the negative frequency components contains no new in-

formation that is not already contained in the positive components.

We may therefore introduce a new field representation by writing

U(r, ω) =

{

0 if ω < 0

2Ure(r, ω) if ω ≥ 0.(2.6)

The space-time domain counterpart of this quantity has a Fourier

representation

U(r, t) =∫ ∞

0U(r, ω) exp(−iωt)dω. (2.7)


CHAPTER 2. FUNDAMENTALS OF ELECTROMAGNETIC THEORY

The positive part of the spectrum in Eq. (2.6) only differs by a con-

stant factor of 2 from that of the original real function. This prop-

erty of the complex space-time domain function connects its Fourier

spectrum

U(r, ω) =1

2π

∫ ∞

−∞U(r, t) exp(iωt)dt (2.8)

to physically observable phenomena. Hence the quantity in Eq. (2.8)

represents the complex analytic signal [29]. In order to analyze any

scalar field quantity, like the electric charge density, a similar ap-

proach could be used.

2.2 MACROSCOPIC MAXWELL’S EQUATIONS

The fundamental laws of electrodynamics were introduced by J. C.

Maxwell [30] and hence they are called Maxwell’s equations. They

are a set of partial differential equations for calculating fields from

currents and charges. These equations have two variants, one of

which is the “microscopic” set of Maxwell’s equations that uses to-

tal charge and total current. On the other hand, the “macroscopic”

formulation is based on free charges and currents. In the context of

this thesis, the macroscopic set of Maxwell’s equations is of main

interest.

For complex-valued space-time domain fields, Maxwell’s equa-

tions may be presented as a set of four partial differential equations

∇× E(r, t) = − ∂

∂tB(r, t), (2.9)

∇× H(r, t) = J(r, t) +∂

∂tD(r, t), (2.10)

∇ · D(r, t) = ρ(r, t), (2.11)

∇ · B(r, t) = 0. (2.12)

The above equations are valid in any continuous media as well as in

vacuum. Now, considering Eq. (2.8), the above space-time domain

representation of Maxwell’s equations can be transformed into the



space frequency domain using the uniqueness of the Fourier trans-

form [31]. This leads to a set

∇× E(r, ω) = iωB(r, ω), (2.13)

∇× H(r, ω) = J(r)− iωD(r, ω), (2.14)

∇ · D(r, ω) = ρ(r, ω), (2.15)

∇ · B(r, ω) = 0, (2.16)

which is as general as the set of space-time domain Maxwell’s equa-

tions.

2.3 CONSTITUTIVE RELATIONS

In the macroscopic Maxwell’s equations, it is necessary to spec-

ify relationships between different space-time and space-frequency

field vectors introduced in the previous section, such as the elec-

tric displacement D and the electric field E, or the magnetic field

H and the magnetic induction B. These equations specify the re-

sponse of bound charges and currents to the applied fields, and

they are called constitutive relations. The relationship between the

electric field and the electric displacement may be expressed in the

form

D(r, t) = ǫ0E(r, t) + P(r, t), (2.17)

where ǫ0 is the electric permittivity in vacuum and the vector P is

known as the electric polarization. Analogously, by introducing the

magnetization vector M(r, t) and the magnetic permeability µ0 of

vacuum, we may write

H(r, t) =1

µ0B(r, t)− M(r, t) (2.18)

to specify the relationship between the magnetic field and the mag-

netic induction. Since magnetization is very small at optical fre-

quencies, the magnetic response of the material can be neglected

and a linear dependence of H on B can be assumed. From a causal-

ity argument [32], the relationship between polarization and the



electric field is linear and can be written as

P(r, t) =ǫ0

2π

∫ ∞

0χ(r, t′)E(r, t − t′)dt′, (2.19)

where χ(r, t) is the real-valued time-domain dielectric susceptibility

tensor. The dipole response of the material is independent of the

external electric field vector in any axes if the medium is isotropic.

The susceptibility tensor can then be written as

χ(r, t) = χ(r, t)I, (2.20)

where χ(r, t) is the scalar susceptibility and I is the identity matrix.

In analogy with Eq. (2.19), the relation between the internal elec-

tric current density and the electric field can be expressed as

J(r, t) =1

2π

∫ ∞

0σ(r, t′)E(r, t − t′), (2.21)

where the real-valued electric conductivity tensor σ(r, t) in space-

time domain reduces to scalar conductivity σ(r, t) in isotropic me-

dia. So, in view of Eqs. (2.17) and (2.19), the space-time dependent

electric displacement and electric field may be written as

D(r, t) =ǫ0

2π

∫ ∞

0ǫ(r, t′)E(r, t − t′)dt′. (2.22)

The space-time domain Maxwell’s equations together with con-

stitutive relations provide the required relations between different

field quantities. For mathematical convenience, however, the space-

frequency domain representation is typically preferable. Thereby,

with the help of the convolution theorem [33], the Fourier trans-

form of Eq. (2.19) can be expressed as

P(r, ω) = ǫ0χ(r, ω)E(r, ω), (2.23)

where the convolution in Eq. (2.19) is transformed into multiplica-

tion in the space-frequency domain. By applying the same proce-

dure to Eqs. (2.21) and (2.22) in non-magnetic media, we get a set



of three equations

D(r, ω) = ǫ0ǫ(r, ω)E(r, ω), (2.24)

B(r, ω) = µ0H(r, ω), (2.25)

J(r, ω) = σ(r, ω)E(r, ω), (2.26)

which connect the field vectors in Maxwell’s equations. These are

the constitutive or material equations in the space-frequency do-

main, and ǫ(r, ω) is known as the relative permittivity tensor. Sub-

stituting Eq. (2.17) and Eq. (2.23) in Eq. (2.24) and applying the

Fourier transform, we get

ǫ(r, ω) = 1 + χ(r, ω) = 1 +1

2π

∫ ∞

0χ(r, t) exp(iωt)dt, (2.27)

which is known as the dispersion law of the electric permittivity

tensor.

Let us now define a new quantity, known as the complex rela-

tive permittivity tensor, which connects the real relative permittiv-

ity and conductivity tensors as

ǫ(r, ω) = ǫ(r, ω) +i

ǫ0ωσ(r, ω). (2.28)

In isotropic media this tensor can again be replaced with a scalar

quantity ǫ(ω). In general, for optical frequencies and isotropic me-

dia, the complex refractive index is defined as

n(ω) = n(ω) + iκ(ω) =√

ǫ(ω) =√

ǫ′(ω) + iǫ′′(ω), (2.29)

where n(ω), κ(ω), ǫ′(ω), and ǫ′′(ω) are real functions. The attenu-

ation index κ determines the damping of the propagating wave in

the medium. Making use of the constitutive relations and Eq. (2.28),

we may write Eq. (2.14) as

∇× H(r, ω) = −iωǫ0 ǫ(r, ω)E(r, ω). (2.30)

In optics the charge density in Eq. (2.30) can in practice be written

equal to zero in Eq. (2.15). Using Eq. (2.24) we then get

∇ · [ǫ(r, ω)E(r, ω)] = 0. (2.31)



In conclusion, using the constitutive relations, we have obtained the

set

∇× E(r, ω) = iωB(r, ω), (2.32)

∇× H(r, ω) = −iωǫ0ǫ(r, ω)E(r, ω), (2.33)

∇ · [ǫ(r, ω)E(r, ω)] = 0, (2.34)

∇ · B(r, ω) = 0 (2.35)

of Maxwell’s equations in the space-frequency domain.

2.4 BOUNDARY CONDITIONS

Maxwell’s equations at any point r are valid if the medium in the

immediate vicinity of r is continuous, but quite often we encounter

abrupt boundaries between two different media. Therefore we need

relationships between the field components across such boundaries

of discontinuity. These boundary conditions can be derived from

the space-frequency Maxwell’s equations [34].

By defining a unit normal vector n12 pointing into the medium

with index 2 from a medium with index 1, we may write the elec-

tromagnetic boundary conditions in the form

n12 · (B2 − B1) = 0, (2.36)

n12 · (D2 − D1) = 0, (2.37)

n12 × (E2 − E1) = 0, (2.38)

n12 × (H2 − H1) = 0. (2.39)

These equations are valid across any discontinuity between dielec-

tric or conducting materials. They imply that the normal compo-

nents of B and D, as well as the tangential components of E and H,

are continuous across any boundary in a non-magnetic medium.

2.5 WAVE EQUATIONS AND THE TE/TM DECOMPOSITION

In order to derive wave equations and the so-called TE/TM de-

composition of Maxwell’s equations, we will make the following

assumptions:



1. A y-invariant system (the permittivity distribution and the

field are independent on the y-coordinate).

2. The incident field propagates in the xz-plane.

3. A homogeneous medium (constant refractive index).

4. An isotropic medium (no birefringence).

Since the medium is homogenous and isotropic, the complex per-

mittivity ǫ(ω) is independent on the spatial position and the direc-

tion of the diffracted wave. It now follows from Eqs. (2.32), (2.33),

and (2.25) that

∇× [∇× E(r, ω)] = ω2ǫ0µ0ǫ(ω)E(r, ω). (2.40)

By defining the speed of light c = (ǫ0µ0)−1/2 and using the vector

identity ∇ × (∇ × U) ≡ ∇(∇ · U) − ∇2U we get the Helmholtz

wave equation for the electric field:

∇2E(r, ω) + k20ǫ(ω)E(r, ω) = 0. (2.41)

Here the wave number k0 in vacuum is defined as k0 = 2π/λ =

ω/c, and λ is the vacuum wavelength of the field. Repeating similar

steps we get

∇2H(r, ω) + k20ǫ(ω)H(r, ω) = 0, (2.42)

which is the Helmholtz wave equation for the magnetic field.

Next, based on the above assumptions, we consider the geom-

etry in which all the partial derivatives in y-direction vanish in

Maxwell’s equations (y-invariant system). In this case Maxwell’s

equations can be divided in component form into two independent

sets:

Hx(x, z) =i

k0

√

ǫ0

µ0

∂

∂zEy(x, z), (2.43)

Hz(x, z) = − i

k0

√

ǫ0

µ0

∂

∂xEy(x, z), (2.44)

∂

∂zHx(x, z)− ∂

∂xHz(x, z) = −ik0(ǫ(x, z)

√

ǫ0

µ0Ey(x, z), (2.45)



and

Ex(x, z) = − i

k0 ǫ(x, z)

√

µ0

ǫ0

∂

∂zHy(x, z), (2.46)

Ez(x, z) =i

k0 ǫ(x, z)

√

µ0

ǫ0

∂

∂xHy(x, z), (2.47)

∂

∂zEy(x, z)− ∂

∂xEz(x, z) = ik0

√

µ0

ǫ0Hy(x, z). (2.48)

Clearly, Eqs. (2.43)–(2.45) contain only field components Ey, Hx and

Hz. Since the electric field is now perpendicular to the xz-plane,

this set of equations describes the transverse electric (TE) polariza-

tion of incident light. Similarly, Eqs. (2.46)–(2.48) contain only field

components Hy, Ex and Ez. Now the magnetic field is perpendic-

ular to the xz-plane, and one talks about transverse magnetic (TM)

polarization of incident light. By substituting Eqs. (2.43) and (2.44)

into (2.45), we obtain a single partial differential equation for the

y-component of the electric field in the form

∂2

∂x2Ey(x, z) +

∂2

∂z2Ey(x, z) + k2

0ǫ(x, z)Ey(x, z) = 0. (2.49)

Analogously, for TM polarization, we obtain an equation

∂

∂x

[

1

ǫ(x, z)

∂

∂xHy(x, z)

]

+∂

∂z

[

1

ǫ(x, z)

∂

∂zHy(x, z)

]

+ k20Hy(x, z) = 0,

(2.50)

which is mathematically slightly less attractive than Eq. (2.49).

2.6 SIMPLEST SOLUTION OF MAXWELL’S EQUATIONS

The electromagnetic plane wave is the simplest solution of Maxwell’s

equations. The space-frequency domain representation of an elec-

tromagnetic plane wave is as follows:

E(r, ω) = E0(ω) exp(ik · r), (2.51)

H(r, ω) = H0(ω) exp(ik · r), (2.52)



where the quantities E0(ω) and H0(ω) denote the vectorial complex

amplitudes of the electric and magnetic fields, respectively. The

wave vector k = kx x + kyy + kz z, where |k| = k0n, is perpendicular

to the planar wavefront and it defines the propagation direction of

the plane wave.


3 Interaction of light with

microstructured surfaces

There is no generally applicable and numerically efficient way to

describe the interaction of light with microstructured matter; the

most appropriate method depends, in particular, on the dimen-

sions of the features in the microstructure compared to the wave-

length of light. When these dimensions are comparable to the

wavelength, rigorous solutions of Maxwell’s equations are usually

required. If the dimensions are far larger than the wavelength, sim-

plified diffraction models can be applied, and even the use of geo-

metrical optics (Snell’s law and Fresnel’s equations) can sometimes

be justified. On the other hand, if the dimensions are much smaller

than the wavelength, effective refractive index approximations are

useful. In this chapter the mathematical methods needed in this

thesis for analyzing the interaction of light with corrugated inter-

faces are discussed.

3.1 REFLECTION AND TRANSMISSION

We begin by considering a plane wave with electric-field ampli-

tude Ei incident on a dielectric interface at an angle θi as shown in

Fig. 3.1. The interface separates two dielectric media. The refractive

index of the half-space z < 0 is denoted by ni and that of the half-

space z > 0 is denoted by nt. After the interaction with the interface

the incident field splits into a transmitted field denoted by Et and

a reflected field denoted by Er. The amplitudes and the intensi-

ties of the transmitted and reflected plane waves can be calculated

by taking into consideration the boundary conditions, which state

that certain components of the electromagnetic field are continuous

across the interface. The appropriate boundary conditions are dif-



ferent for the two polarization states of light, TE and TM. In the

case of TM polarization illustrated in Fig. 3.1(a), the magnetic field

vector is perpendicular to the plane of incidence, whereas in the

case of TE polarization the electric field vector is perpendicular to

this plane as shown in Fig. 3.1(b).

On applying the boundary conditions one first arrives at the law

of refraction

ni sin θi = nt sin θt, (3.1)

known as Snell’s law, and at the law of reflection θr = θi. Fur-

ther, by applying the boundary conditions, one can determine the

complex-amplitude transmission and reflection coefficients, known

as Fresnel coefficients, in the form

rTE =ETE

r

ETEi

=ni cos θi − nt cos θt

ni cos θi + nt cos θt, (3.2)

rTM =ETM

r

ETMi

=ni cos θt − nt cos θi

ni cos θt + nt cos θi, (3.3)

tTE =ETE

t

ETEi

=2ni cos θi

ni cos θi + nt cos θt, (3.4)

tTM =ETM

t

ETMi

=2ni cos θi

ni cos θt + nt cos θi. (3.5)

The reflection and transmission efficiencies, which characterize

the amount of reflected and transmitted energy, require some fur-

ther investigation. First, we recall that for real relative permittivity

ǫ = n2, the time-averaged Poynting vector P may be written as

P =ǫ0ck

2k0|E0|2 . (3.6)

The energy flow towards the surface under investigation and out-

wards from it is characterized by the z-component of the time-

averaged (spectral) Poynting vector, i.e.,

Pz =ǫ0ckz

2k0|E0|2 . (3.7)


CHAPTER 3. INTERACTION OF LIGHT WITH MICROSTRUCTURES

ki

kr

kt

ETMi

BTMi

ETMt

BTMt

ETMr

BTMr

x

z

ni nt

(a)

θr

θi

θt

x

z

ni nt

(b)

ki

kr

kt

ETEi

BTEi

ETEt

BTEt

ETEr

BTEr

θr

θi

θt

Figure 3.1: Direction of field and wave vectors in (a) TM polarization and (b) TE polar-

ization of the incident light.



By comparing the values of this quantity for the reflected and inci-

dent fields we obtain reflection efficiencies (reflectances)

RTE,TM = |rTE,TM|2. (3.8)

Correspondingly we obtain the transmission efficiencies (transmit-

tances)

TTE,TM =nt cos θt

ni cos θi|tTE,TM|2. (3.9)

The Fresnel equations can be extended to boundaries between di-

electric and absorbing materials simply by replacing nt with a com-

plex refractive index nt. The transmittance is therefore a meaningful

quantity only for real values of nt because the field decays rapidly

absorbing media. The reflectance is always a meaningful quantity,

since we assume that ni is real.

3.2 ANGULAR SPECTRUM REPRESENTATION

In the previous chapter we dealt with the fundamental Maxwell’s

equations and arrived at the plane-wave solution of the wave equa-

tion. The beauty of a plane wave is that it is the most fundamental

and mathematically simplest wave form to deal with. Any com-

plex electromagnetic field can be thought of as a superposition of

a finite or an infinite number of plane waves propagating in differ-

ent directions. Such an angular spectrum of plane waves is a basic,

physically appealing, and completely rigorous tool to study wave

propagation and diffraction homogeneous media in, e.g., the fields

of electrodynamics, optics, and acoustics. In the angular spectrum

representation different plane-wave components of the field have

variable amplitudes and propagation directions as we will see for-

mally below.

In the angular spectrum representation we choose an arbitrary

plane z = z0 = constant, in which the field E(x, y, z0) is assumed

to be known. The goal is to determine the field E(r) at any point

defined by a position vector r = (x, y, z) in space. To this end we



first introduce the two-dimensional Fourier transform of the field E

at any plane z = constant:

E(kx, ky; z) =1

(2π)2

∫∫ ∞

−∞E(x, y, z) exp

[

−i(

kxx + kyy)]

dx dy,

(3.10)

where kx and ky are the spatial frequencies in the cartesian coordi-

nate system. The inverse of Eq. (3.10) reads as

E(x, y, z) =∫∫ ∞

−∞E(kx, ky; z) exp

[

i(

kxx + kyy)]

dkx dky. (3.11)

After inserting Eq. (3.10) into the Helmholtz equation (2.41) and

introducing the dispersion relation

kz =√

k2 − k2x − k2

y, (3.12)

where k = k0n, we find that the Fourier spectrum E evolves along

the z axis as

E(kx, ky; z) = E(kx, ky; z0) exp (±ikz∆z) , (3.13)

where ∆z = z − z0. Here the positive sign refers to forward propa-

gation towards the half-space z > z0 and the negative sign refers to

back-propagation into the half-space z < z0. We can conclude from

Eq. (3.13) that the angular spectrum at an arbitrary plane can be

obtained from the angular spectrum at z = z0 by multiplying with

the propagator exp (±ikzz) [35].

Inserting from Eq. (3.13) into Eq. (3.11), we finally arrive at the

angular spectrum representation

E(x, y, z) =∫∫ ∞

−∞E(kx, ky; z0) exp

[

i(

kxx + kyy ± kz∆z)]

dkx dky.

(3.14)

To propagate fields by the angular spectrum representation, we

therefore first evaluate E(kx, ky; z0) by Eq. (3.10), then use Eq. (3.13)

to propagate the angular spectrum, and finally return to the space



domain by means of Eq. (3.11). The direct and inverse Fourier trans-

forms involved in this process can be evaluated efficiently using the

Fast Fourier Transform algorithm.

In general, kz defined by Eq. (3.12) can have either real or imag-

inary values. If the only real-valued kz are non-zero, one speaks

about a free field. In general, however, the field contains also plane-

wave components with k2x + k2

y > k2, known as evanescent waves.

In this case kz becomes purely imaginary, i.e.,

kz = ±i√

k2x + k2

y − k2. (3.15)

Here the positive/negative sign is chosen when considering for-

ward/backward propagation. In either case, Eq. (3.15) implies an

exponential decay of the field in the propagation direction. Finally,

propagation in an absorbing medium can be governed simply by re-

placing k = k0n with a complex wave number k = k0n. In this case

all plane waves propagating in the medium are inhomogeneous and

the associated kz is a complex number.

3.3 SCATTERING FROM PERIODIC STRUCTURES

Analysis methods of microstructured optical elements such as grat-

ings are well established in the paraxial domain, where the elec-

tromagnetic properties of light can usually be ignored unless the

element modulates the state of polarization of the incident field in

a spatially varying fashion. However, in the non-paraxial domain,

the polarization of light and multiple scattering must be taken into

consideration in order to predict the interaction of light with the

microstructure correctly [2]. On the other hand, going beyond the

limitations of the scalar theory paves the way to a wide range of in-

teresting designs, which rely on exact solutions of both Maxwell’s

equations and electromagnetic boundary conditions. Several rigor-

ous methods exist for solving Maxwell’s equations, including dif-

ferential [36–38], integral [3, 39, 40], finite element [7, 8], and finite

difference [9, 10] techniques. However, the modal method to be



described below is not only relatively easy to implement but also

trustworthy in most situations.

The Fourier Modal Method (FMM) is a widely used technique

to study the exact response of periodic scatterers (gratings) [4]. In

the homogeneous media on both sides of the modulated region the

rigorous solution of Maxwell’s equations is a Rayleigh expansion,

which is a discrete form of the angular spectrum representation. In-

side the grating the situation is more complicated, particularly if the

permittivity profile is z-dependent. In FMM we divide the modu-

lated region (slices) into layers, in each of which the refractive index

is invariant in the z-direction as illustrated in Fig. 3.2. The (gener-

ally complex) permittivity in each such slice is expressed as a trans-

verse Fourier series, and the fields inside the slices are expressed in

the form of pseudoperiodic (Floquet–Bloch) expansions [41]. These

expansions are transversely periodic apart from a common spatially

linear phase factor that depends on wave vector of the incident

plane wave. This leads to a solution of Maxwell’s equations in each

slice in terms of forward- and backward-propagating modal fields,

with z-dependence of the form exp(±iγz), where γ is the eigen-

value associated with the mode in question. Finally a set of bound-

ary value problems is solved numerically (by the so-called S-matrix

algorithm [42, 43]) to connect the solutions in each slice, and also

to the Rayleigh expansions outside the modulated region [44, 45].

The final result is a set complex amplitudes of the reflected and

transmitted diffraction orders. The method also allows the deter-

mination of field distributions inside the grating as superpositions

of the Bloch modes.

3.3.1 Eigenvalue problem in non-conical mounting

Let us next consider the FMM quantitatively under the following

assumptions (see Fig. 3.3):

1. The permittivity distribution ǫ is independent on the y coor-

dinate (linear grating).



replacements

x

z

slicing

w

d

α

I

II

III

Figure 3.2: Schematic representation of the way the cross section of the grating profile

is divided into z-invariant slices in FMM. Here a V-ridge grating with ridge width w,

half-angle α, and period d is considered as an example.

2. The incident plane wave arrives at non-conical mounting, i.e.,

it propagates in the xz-plane [46].

3. The permittivity distribution inside each layer z(j−1)< z <

z(j), j = 1, . . . , J, is independent on z.

4. The grating is periodic in the x-direction with period d.

In the case of a linear y-invariant grating, the complex permit-

tivity distribution inside the jth grating layer [2] is a function of x,

and it may be expressed in the form of a Fourier series

ǫ(j)(x) =∞

∑p=−∞

ǫ(j)p exp(i2πpx/d), (3.16)

where ǫ(j)p is the pth Fourier component of the complex permittivity,

given by

ǫ(j)p =

1

d

∫ d

0ǫ(j)(x) exp(−i2πpx/d)dx. (3.17)

In Sect. 2.5 we concluded that, in the case of a y-invariant grating,

Maxwell’s equations can be written in the component form for TE

and TM polarizations, respectively. For the TE polarization the only



. . . . . .

z(0) z(1) z(j−1) z(j)z(J−1) z(J)

ǫ(J)(x)ǫ(j)(x)ǫ(1)(x)

ǫ(0) ǫ(J+1)

A(0,−)−1

Ai

A(0,−)0

A(0,−)1

A(J+1,+)−2

A(J+1,+)−1

A(J+1,+)0

A(J+1,+)1

x

z0

d

Figure 3.3: Basic geometry and notation used in the FMM for linear gratings.

non-vanishing electric field component is Ey. Hence the incident

field with amplitude Ai (see Fig. 3.3) can be expressed as

Ey,i(x, z) = Ai exp(

i{

kx,0x + kz,0

[

z − z(0)]})

, (3.18)

where

kz,0 =√

k20ǫ(0) − k2

x,0. (3.19)

Similarly, for the reflected field in homogenous space z < z(0), the

Rayleigh expansion may be written as

E(0,−)y (x, z) =

∞

∑m=−∞

A(0,−)m exp

(

i{

kx,mx + k(0)z,m

[

z − z(0)]})

, (3.20)

where kx,m = kx,0 + m2π/dx define the propagation direction of the

diffracted orders,

k(0)z,m =

√

k20ǫ(0) − k2

x,m, (3.21)



and A(0,−)m are the unknown amplitudes of the reflected diffraction

orders. Correspondingly, the field in the region z > z(J) has the

Rayleigh expansion

E(J+1,+)y (x, z) =

∞

∑m=−∞

A(J+1,+)m exp

(

i{

kx,mx + k(J)z,m

[

z − z(J)]})

,

(3.22)

where

k(J)z,m =

√

k20ǫ(J) − k2

x,m (3.23)

and A(J+1,+)m are the unknown amplitudes of the transmitted diffrac-

tion orders.

Since the electric field inside the grating is pseudoperiodic, it

may be expressed as a pseudo-Fourier series [47]

E(j)y (x, z) =

∞

∑m=−∞

U(j)y,m(z) exp(ikx,mx), (3.24)

where U(j)y,m(z) denotes the amplitude of the mth space-harmonic

field and is given by

U(j)y,m(z) =

1

d

∫ d

0E(j)y (x, z) exp(−ikx,mx)dx. (3.25)

Substituting the Fourier series expansion (3.16) of the complex per-

mittivity and the pseudo-Fourier series Eq. (3.24) into Eq. (2.49), we

get

−∞

∑m=−∞

k2x,mU

(j)y,m(z) exp(ikx,mx) +

∞

∑m=−∞

∂2

∂z2U

(j)y,m(z) exp(ikx,mx)

+ k20

∞

∑m=−∞

∞

∑p=−∞

ǫp exp(i2πpx/d)U(j)y,m(z) exp(ikx,mx) = 0.

(3.26)

Multiplying Eq. (3.26) by (1/d) exp(−ikx,qx), where q may be any

integer, and then integrating over the grating period from x = 0 to

x = d, we get

−k2x,qU

(j)q (z) +

∂2

∂z2U

(j)q (z) + k2

0

∞

∑m=−∞

ǫq−mU(j)y,m(z) = 0. (3.27)



The general solution of Eq. (3.27) can now be written as

U(j)y,m(z) = U

(j)m exp

[

iγ(j)z]

, (3.28)

where γ(j) denotes the as-yet unknown eigenvalue of the mode.

Substituting Eq. (3.28) into Eq. (3.27) and rearranging terms, we

obtain

k20

∞

∑m=−∞

ǫq−mU(j)m − k2

x,qU(j)q = U

(j)q

[

γ(j)]2

. (3.29)

This equation has the form[

k20[[ǫ

(j)]]− kx

]

U(j) =[

Γ(j)]2

U(j) (3.30)

of a matrix eigenvalue problem for TE polarization, where the ele-

ments of the matrix kx are kx,q,m = k2x,qδq,m and δq,m denotes the Kro-

necker delta symbol [48]. We see at once that U(j) are the column

eigenvectors containing the Fourier components U(j)m,p, and the di-

agonal matrix [Γ(j)]2 contains the respective eigenvalues γ(j)p of the

matrix[

k20[[ǫ

(j)]]q−m − kx

]

. The eigenvectors and eigenvalues form

a discrete set such that the number of the eigenvectors and eigen-

values is same as one dimension of the matrix[

k20[[ǫ

(j)q−m]]− kx

]

.

It is of course not possible to solve numerically the eigenvalue

problem for a matrix with infinite dimensions, and therefore a trun-

cated set of eigenvectors and eigenvalues are computed up to a

finite index M, which also determines the number of diffraction or-

ders that are included in the analysis. By examining Eqs. (3.27) and

(2.38), we find that Ey is continuous over any discontinuities in the

x-direction. Therefore, the Fourier factorization product is of type

1 [49], i.e., Laurent’s rule can be applied to the truncated sum.

Once the matrix eigenvalue equation is solved, the general so-

lution for the field inside layer j can be written as

E(j)y,p(x, z) = exp

[

±iγ(j)p z

] M

∑m=−M

U(j)m,p exp(ikx,mx). (3.31)

The solution for[

Γ(j)]2

gives the propagation constants γ(j) in the

z-direction. The signs + and − denote the field modes propagating



in the positive and negative z-directions, respectively. To achieve

stability in the numerical solution it is important to choose the sign

of propagation constants properly. This is guaranteed if we choose

the sign according to the following rules:

1. If γ(j)p is complex, we choose its sign such that ℑ{γ(j)}p > 0.

2. If γ(j)p is real, we choose its sign such that ℜ{γ

(j)p } > 0.

Then, finally, we can write the general solution of the field inside

the layer region in the form

E(j)y (x, z) =

∞

∑p=1

(

a(j)p exp

{

iγ(j)p

[

z − z(j−1)]}

+ b(j)p exp

{

−iγ(j)p

[

z − z(j)]}) ∞

∑m=−∞

U(j)m,p exp (ikx,mx) ,

(3.32)

where a(j,±)p are yet undefined complex amplitudes of the forward

and backward propagating field components. These complex am-

plitudes are solved in the next section.

Finding the general solution for the field inside the modulated

in TM polarization is treated in a similar fashion. Now the only

non-vanishing magnetic field component is Hy, and therefore the

incident field is written as

Hy,i(x, z) = Ai exp(

i{

kx,0x + kz,0

[

z − z(0)]})

. (3.33)

The reflected and the transmitted TM Rayleigh fields in the homo-

geneous spaces may be represented in the form

H(0,−)y (x, z) =

M

∑m=−M

A(0,−)m exp

(

i{

kx,mx − k(0)z,m

[

z − z(0)]})

(3.34)

and

H(J+1,+)y (x, z) =

M

∑m=−M

A(J+1,+)m exp

(

i{

kx,mx + k0z,m

[

z − z(J+1)]})

.

(3.35)



The x-and z-components of the electric field are now solved from

Eqs. (2.46) and (2.47).

Special attention is needed in solving the boundary conditions

for Hy and its derivatives across the discontinuities along the x-

direction by following the rules of Fourier factorization given by

Li [49]. We first obtain the basic differential equation

ǫ(j)(x)

{

k20H

(j)y (x, z) +

∂

∂x

[

1

ǫ(j)(x)

∂

∂xH

(j)y (x, z)

]}

= − ∂2

∂z2H

(j)y (x, z).

(3.36)

We know from the boundary conditions that in TM polarization,

field Hy and its z-derivative have to be continuous across the bound-

aries of discontinuity along the x-direction. Hence, the left-hand-

side of Eq. (3.36) has to be continuous. Since ǫ(j)(x) is discontinu-

ous, the expression inside the curly brackets has to share the same

point of discontinuity so that their product is continuous (of type

2). Further, the product inside the square brackets has to be contin-

uous. According to the boundary conditions the x-derivative of Hy

and ξ(x) = 1/ǫ(x) are discontinuous, but their product is contin-

uous (again of type 2) and the inverse rule of Fourier factorization

must again be applied.

Applying both Laurent’s and inverse rule [49] as described above,

we can transfrom Eq. (3.36) into a matrix eigenvalue equation in TM

polarization:

[[ξ(j) ]]−1[k20 I − Kx[[ǫ

(j)]]−1Kx]U(j) =

[

Γ(j)]2

U(j). (3.37)

In the above eigenvalue equation we can interpret that a Toeplitz

matrix is generated from the Fourier coefficients of ǫp,q and ξ is

the inverse of ǫ based on the assumption that ǫ is non-zero. The

elements of the matrix Kx are kx,q,m = k2x,qδq,m and δq,m denotes

the Kroneckar delta [48]. The diagonal matrix[

Γ(j)]

contains the

propagation constants γ(j)p in the modulation layers.



3.3.2 Boundary condition solution in multilayered gratings

In order to solve the boundary-value problem in multilayered grat-

ings illustrated in Fig. 3.3 we use S-matrix algorithm. A single S-

matrix of the multilayer structure represents the entire scattering

properties. In each layer the field is represented as superposition

of upward and downward propagating and decaying waves as de-

scribed by Eq. (3.32). We modify this equation slightly to simplify

the S-matrix derivation by introducing the notation

c(j)p = a

(j)p exp

[

iγ(j)p h(j)

]

, (3.38)

which represents the forward-propagating mode in the layer j, and

h(j) = z(j) − z(j−1). By satisfying the continuity of the tangential

field components across the interface of discontinuity, we may write

the boundary value problem in the matrix form[

U(j−1) U(j−1)

Q(j−1) −Q(j−1)

] [

c(j−1)

b(j−1)

]

=

[

U(j) U(j)

Q(j) −Q(j)

] [

χj−c(j)

χj+b(j)

]

, (3.39)

where Q(j) = U(j)Γ(j) and the diagonal matrices χ

(j)± contain the

elements exp[

±iγ(j)p h(j)δm,p

]

. The same boundary conditions also

hold at the boundaries z = z(0) and z = z(J) between the homoge-

neous regions and the sliced grating region. The vector c(0) con-

tains the complex input-field amplitudes in Eq. (3.18). In the case

of a plane wave of unit amplitude, c(0) = 1 and other elements of

the vector becomes zero. c(J+1) represents the complex amplitudes

of the transmitted fields and b(0) shows the complex amplitude of

the reflected fields.

Our goal is to find the S-matrix connecting output field with the

input field in the sense of a relation

[

c(j+1)

b(0)

]

= S(J+1)↔(0)

[

c(0)

b(j+1)

]

=

[

S(j+1)↔(0)11 S

(j+1)↔(0)12

S(j+1)↔(0)21 S

(j+1)↔(0)22

] [

c(0)

b(j+1)

]

, (3.40)



where the notation (j + 1) ↔ (0) in the matrix S means that it

will be constructed from the plane J + 1 to the region 0 layer by

layer. Unfortunately, Eq. (3.40) can give rise to large numerical er-

rors in inverting the matrix S(j+1)↔(j)11 . For this reason we need to

reconsider Eq. (3.32) inside the jth grating layer and transform the

boundary value problem into the form

[

U(j) U(j)

Q(j) −Q(j)

] [

χ(j)+ a(j)

b(j)

]

=

[

U(j+1) U(j+1)

Q(j+1) −Q(j+1)

] [

a(j+1)

χ(j+1)+ b(j+1)

]

.

(3.41)

Now the S-matrix has been constructed starting from the region 0

and moving all the way to region J + 1 but this new matrix formula-

tion does not equal to S(J+1)↔(0) of the previous section. Therefore

a new matrix has been introduced as[

a(j)

b(0)

]

= W(0)↔(j)

[

a(0)

b(j)

]

=

[

W(0)↔(j)11 W

(0)↔(j)12

W(0)↔(j)21 W

(0)↔(j)22

] [

a(0)

b(j)

]

. (3.42)

Our goal is to find out W(0)↔(j+1) through the matrix W(0)↔(j). If

we again assume that there are no sources in the half-space z >

z(J), we can obtain the backward propagating field amplitudes from

Eqs. (3.38) and (3.40):

b(j) = S(J+1)↔(j)21 χ

(j)+ a(j). (3.43)

In view of Eqs. (3.41) and (3.42), we get the forward-propagating

field amplitudes from

a(j) =[

I − W(0)↔(j)12 S

(J+1)↔(j)21 χ

(j)+

]−1W

(0)↔(j)11 a(0). (3.44)

After some tedious calculations, we arrive at the matrix elements of

Eq. (3.42):

W(0)↔(j+1)11 = −

[

Y(0)↔(j+1)11 U(j) + Y

(0)↔(j+1)12 Q(j)

]

χ(j)+ W

(0)↔(j)11 ,

(3.45a)

W(0)↔(j+1)12 =

[

Y(0)↔(j+1)11 U(j+1) − Y

(0)↔(j+1)12 Q(j+1)

]

χ(j+1)+ , (3.45b)



where the matrix elements of Y(0) ↔ (j + 1) are defined as

Y(0)↔(j+1) =

[

Y(0)↔(j+1)11 Y

(0)↔(j+1)12

Y(0)↔(j+1)21 Y

(0)↔(j+1)22

]

=

[

−P(j+1) P(j)[χ(j)+ W

(0)↔(j)12 + I]

−Q(j+1) Q(j)[χ(j)+ W

(0)↔(j)12 − I]

]

. (3.46)

We can conclude from Eqs. (3.44) and (3.43) that S21 is the only ele-

ment in the whole S-matrix S(j+1)↔(j), and therefore we do not need

to solve the S-matrix element S(j+1)↔(j)11 . The complex amplitudes

of the transmitted field are finally obtained from Eq. (3.45a).

3.4 EFFECTIVE MEDIUM THEORY

When the grating period is significantly smaller than the wave-

length of incident light, d ≪ λ, only the zeroth reflected and trans-

mitted orders can propagate. In this case the modulated region

of the grating behaves as a homogenous effective medium hav-

ing a certain effective refractive index, and a simplified approach

known as the Effective Medium Theory (EMT) is available. The

treatment of the grating with the EMT approach makes the solu-

tion of the grating problem much faster because the transmittance

and the reflectance may be evaluated by thin film theory. This

type of permittivity-modulated media can show peculiar charac-

teristics such as form birefringence [50, 51]: light propagates with

different phase velocities depending on the state of polarization of

the incident field. This can be observed as a difference of refrac-

tive indices of different polarization modes, which further leads to

a polarization-dependent difference in the group velocity of light.

Furthermore, EMT provides clear physical insight into light propa-

gation in subwavelength-period gratings.

Let us consider the one-dimensional y-invariant grating illus-

trated in Fig. 3.4, where the triangular profile (other profiles could

be considered as well) is subdivided into J layers with equal thick-

ness h/J. Defining the fill factor in jth layer as f j = j/J, the dis-

tribution of refractive index within the modulated region may be



n1

dw

h

x

z

n2α J layers

Figure 3.4: One-dimensional y-invariant linear grating: n1 and n2 denote the refractive

indices of the input region and the grating material, h is the height of the modulated

grating region, and w and d are the groove width and period of the grating, respectively.

written as

n(x) =

n1 when 0 6 x < f jd/2

n1 when d − f jd/2 6 x < d

n2 otherwise.

(3.47)

The effective refractive index in layer j is defined as the ratio of

the propagation constant γ(j) of a local Floquet–Bloch mode to the

free-space propagation constant k0,

n(j)eff =

γ(j)

k0. (3.48)

There can be more than one effective index depending on the num-

ber Floquet–Bloch modes that are excited in the grating, as we saw

above when dealing with FMM. However, in EMT we assume that

only the lowest-order TE and TM modes are significant and all oth-

ers can be ignored.

One way of deriving expressions for the effective refractive in-

dices is to retain only the zeroth modes in the FMM analysis. Con-

sidering Eq. (3.48) and the TM eigenvalue equation, we arrive at the

approximation for the effective refractive index [52]

njeff,⊥ =

[

f jn−21 +

(

1 − f j

)

n−22

]− 12 . (3.49)

Similarly, starting from the TE eigenvalue equation,

n(j)eff,‖ =

[

f jn21 +

(

1 − f j

)

n22

]

12 . (3.50)



Although the EMT is in principle applicable also to crossed (two-

dimensionally periodic) gratings, it is not possible to derive unique

expressions for the effective refractive indices from the FMM for-

mulation [53, 54].

3.5 FUNDAMENTALS OF PARAXIAL DESIGN METHODS

Exact analysis of diffraction gratings is computationally time con-

suming and challenging, especially for two-dimensional gratings

when the grating period is much larger than the wavelength. To

avoid such difficulties, approximate analysis methods such as the

Thin Element Approximations (TEA) [34] and Local Plane Interface

Approximation (LPIA) [11,55] can be applied under certain circum-

stances. In these methods, to be presented below, the response of

the modulated region is described by means of geometrical optics.

3.5.1 Thin element approximation

The thin element approximation method is one of the traditional

approximate analysis methods. It is usually applicable to gratings

whose minimum transverse feature size is in the order of ∼ 10λ and

the maximum grating thickness H is of the order of the wavelength.

With the above assumptions we assure that no significant energy

redistribution takes place in the transverse direction and field in

the modulated region can be treated locally as a plane wave. Then

optical path length calculations yield the field distribution just after

the element with sufficient accuracy.

Let us assume that the input field Ei propagates parallel to the

z-axis. If the modulated structure does not affect the state of polar-

ization of the incident field, we can write the transmitted field just

after the grating as

Et(x, H) = t(x)Ei(x, 0), (3.51)

where t(x) is known as the amplitude transmission function de-



fined by

t(x) = exp

[

ik0

∫ H

0n(x, z)dz

]

(3.52)

(c.f. Fig. 3.5). If the modulated structure is a grating with period d,

the transmission function is periodic and may hence be expanded

in a Fourier series

t(x) =∞

∑m=−∞

Tm exp (i2πmx/d) , (3.53)

where

Tm =1

d

∫ d

0t(x) exp (−i2πmx/d) dx (3.54)

are the complex amplitudes of the transmitted diffraction orders.

Finally, the diffraction efficiencies of these orders are obtained from

ηm = |Tm|2 . (3.55)

The expressions given above can be readily extended to transversely

two-dimensionally modulated (periodic and non-periodic) struc-

tures.

3.5.2 Local plane interface Approximation

In the standard TEA, it is assumed that the optical field is propa-

gated through the modulated structure with the help of rays trav-

eling along straight lines [50, 56]. Further propagation beyond the

modulated region is then accomplished by means of wave optics

by applying, e.g., the Fresnel diffraction formula to the transmit-

ted field Et(x, H). While the limitations of TEA are well estab-

lished, its simplicity attracts one to search for extensions that would

retain some of its intuitiveness while improving its accuracy. A

straightforward extension is the local plane interface approxima-

tion. Beckmann studied LPIA, propagating light through the mod-

ulated structure using geometrical optics [55]. In his assumption

the structure is illuminated by considering a set of rays traced through

the grating and split according to Fresnel transmission and reflec-

tion coefficients when hitting boundaries. In Fig. 3.5, at plane



.

.

.

.

.

.

.

.

.

.

.

.

0 Hz

h(x)

x

Figure 3.5: Geometry assumed in the analysis of surface relief profiles h(x) separating

regions with constant refractive indices. More generally, the modulated region 0 < z < H

can have an arbitrary refractive-index profile n(x, z).

z = H, we construct the complex field from the path length and

the Fresnel coefficients. This approach is reliable, at least to a cer-

tain extent, in the non-paraxial domain [57, 58]. This method can

be used in a trustworthy matter for structures that have continuous

surface profiles free from abrupt transitions or deep slopes. An-

alyzing complicated surfaces can, however, yield unexpected and

sometimes incorrect results.

3.5.3 Iterative Fourier Transform Algorithm

The Iterative Fourier Transform Algorithm (IFTA) is one of the most

convenient and popular methods for designing both periodic and

non-periodic diffractive structures within the thin-element approx-

imation and in the paraxial domain. This method enables us to

design both phase and amplitude elements, but here we confine

our discussion to phase gratings.



x

y

z

u

v

initial plane

element plane

signal plane

Figure 3.6: Geometry assumed in generating an array of three equally bright diffraction

orders using a phase-modulated grating.

The fundamentals of IFTA were introduced by Gerchberg and

Saxton [59] to solve phase-retrieval problems. Later on Fienup [60],

Wyrowski and Bryngdahl [61,62], and others developed these ideas

to design diffractive structures that produce a predetermined sig-

nal. With time, the method has been refined to design highly

quantized diffractive optical elements [63–66]. Today, IFTA has

been established as extremely useful design approach that can be

adapted to different design problems, even including polarization-

modulating elements. The illumination wave, the number of quan-

tization levels, and the shape of the signal can be chosen freely.

Let us next proceed to the case where a plane wave is incident

on a grating and we need more than two diffraction orders with

equal efficiency (see Fig. 3.6). This type of elements are commonly

known as array illuminators. [67–70]. The design of such elements

can be performed easily by IFTA [36, 71]. As illustrated in Fig. 3.7,

the algorithm is started in the element, where the phase-only trans-

mission function is associated with a random phase and the result-

ing complex amplitudes Tm of the signal field are computed (in

practice by using the Fast Fourier Transform, FFT). Next the ampli-

tudes |Tm| are replaced with there target values, which the phases



Source intensity

|t(x)|2

φ(x)

|t| eiφ

Material constrains

FT

Amplitude in target plane

Approximation to target intensity

|Tm| eiΦ

|Tm|2

Φm

|T′m|2

Φm

|T′m| eiΦ

Target intensity

Approximation to desired target amplitude

IFT

Approximation to required source amplitude

|t′| eiφ

|t′(x)|2

φ(x)

phase

Figure 3.7: Block diagram of the IFTA algorithm.



are left untouched, Then the inverse FFT is applied to obtain the re-

sulting field in the element plane. Here the phase-only constraint is

used, by writing the amplitude of the transmission function equal

to unity and leaving its phase as it is. This completes the first iter-

ation round, and the same procedure is continued iteratively until

convergence is achieved. This is ensured only if one allows some

light to be contained in orders that do not belong to the desired

array (amplitude freedom), since the Fourier transform of a phase-

only function is band-limited only if the signal consists of a single

order. In the element plane, one can also quantize the phase to a

certain set of allowed values if the fabrication process requires such

quantization.

If correctly implemented (by proper use of the amplitude free-

dom), IFTA usually converges to a solution that is close to the global

optimum. Sometimes, however, stagnation into a local optimum

(such as the solution nearest to the starting point, which gives a

uniform array) may occur.




4 Light propagation in

wavelength scale structures

The Local Plane Interface Approximation (LPIA), which was briefly

introduced in the previous chapter, has been applied to a variety of

different structures, most notably to dielectric gratings. In the first

studies [11] this method was shown to provide insight into shadow-

ing effects in gratings with abrupt surface-profile transitions. How-

ever, if multiple reflections and refractions from the interfaces in-

side the modulated structure are taken into consideration, LPIA

can also provide intuitive understanding of some striking resonant

effects in gratings whose feature sizes are in the order of the wave-

length [12, 14–16].

In this chapter the LPIA is applied to even more extreme diffrac-

tion geometries. In particular, we associate a ray with an evanes-

cent wave propagating along an interface where Total Internal Re-

flection (TIR) occurs. This ray is perpendicular to the surfaces of

constant phase of the evanescent wave. The structure in close prox-

imity above the interface is a dielectric wedge, which can further be

connected to a planar waveguide. Hence, in a sense, we simulate a

scanning near-field microscope. We show that intuitive information

on the diffraction process can be obtained by adding the evanescent

field in the LPIA analysis to describe tunneling of light in the struc-

ture.

4.1 NEAR-FIELD DETECTION BY THE DIELECTRIC WEDGE

In this section we aim at a clear picture of the physics behind the

detection of evanescent waves by means of a triangular dielectric

wedge with wavelength-scale dimensions. We study two different

geometries for generating evanescent fields to be detected, in both



of which light is incident on a planar interface at angles above the

critical angle for TIR. We compare the diffraction efficiencies and

radiation intensities calculated from LPIA to the rigorous electro-

magnetic diffraction analysis based on FMM.

4.1.1 Plane-wave incidence at high oblique angle

The first geometry to be considered is illustrated in Fig. 4.1. Here

a homogeneous plane wave with a wavelength λ is incident from

a half-space z < 0 having refractive index n, at an angle θ that

exceeds the critical angle at the interface z = 0. Thus an evanescent

field propagating in the positive x-direction is generated above the

plane z = 0. The geometrical shape of the wedge is defined by

the half tip apex angle α and height L. The evanescent field is

probed using this dielectric wedge, which may be scanned in x and

z-directions. In the first case, the medium above the wedge is taken

to be homogeneous, and we examine the angular distribution of

light diffracted by the wedge.

We begin with the assumption that the incident plane wave is

TM polarized (similar treatment could be done for TE polarization)

light. If the half space z > 0 is empty then we can represent the

non-vanishing magnetic field component of the evanescent wave

by

Hy(x, z) = exp (ikxx) exp (−kzz) , (4.1)

where kx = k0n sin θ, kz=(

k2x − k2

0

)1/2, and k0 = 2π/λ have their

meanings as before. In the spirit of using homogeneous-plane-wave

analysis in LPIA and extending it to the evanescent waves, we treat

this non-propagating field as a non-uniform plane wave with sur-

faces of constant phase in the positive z-direction. Further this field

has exponentially decaying amplitude in z-direction and propagat-

ing in the positive x-direction. The part of the incident field that

will be refracted by the left-hand-side facet of the wedge has the

z-dependence and can be written as

Hy(z) = exp [−kz(z − h)] . (4.2)


CHAPTER 4. LIGHT PROPAGATION IN WAVELENGTH SCALESTRUCTURES

x

z

θ

Homogeneous medium

n = 1

n

L

0h

αα

n1

λ

γA B

C

D

β

φ

x0 a

x = 0

θd

Figure 4.1: The geometry involving the generation of an evanescent field on top of the

substrate and detection with a wedge.

where Eq. (4.2) is based on the assumption that the amplitude of

the evanescent wave is insignificant at the distance z = h + L. It

is a relatively simple task to show, using geometrical optics, the z-

dependent field in Eq. (4.2) is projected into the output plane z =

h + L of the wedge that has the form

H1(x) = H0T(α)R√

S exp [−κ (x − x0)] exp [iKx (x − x0)] , (4.3)

note that the phase at x = x0 equal to zero. In the above expression

H0 = exp (kzh), T(α) represents the Fresnel transmission coefficient

at point B, and R is the reflection coefficient at point C. Further, the

field amplitude has decay constant and is given by

κ = kzS = kzdz

dx= kz

cos α

cos γsin(3α − γ) (4.4)

where S is a scaling constant obtained by the intensity law of ge-

ometrical optics [50] and from the Snell’s law sin α = n1 sin γ we



obtained the angle γ. Moreover,

x0 = L tan φ, (4.5)

where

φ = 3α − γ − π/2. (4.6)

Finally, we represent the phase of the projected magnetic field by

the constant

Kx = −k0n1 cos(3α − γ) = k0n1 sin φ. (4.7)

We now have a complete representation of the geometrical field

at the exit plane of the wedge. Since the output region behind the

tip is homogeneous, the field in the half-space above the wedge can

be represented in terms of the an angular spectrum of plane waves

A(kx) =1

2π

∫ ∞

−∞Hy(x, h + L) exp (−ikxx) dx (4.8)

associated with Hy(x, h + L). If we denote by χ the imaginary part

of the wave-vector component kz, it follows quite straightforwardly

that

|A(kx)|2 =χ

2π2

∣

∣

∣

∣

∫ ∞

−∞H1(x) exp (−ikxx)dx

∣

∣

∣

∣

2

= |t(α)|2 χ

2π2ζ exp

{

−χ

[

2h + D cos(3α − γ)cos α cot α

cos γ

]}

×∣

∣

∣

∣

∫ a

x0

exp[(−ξ + iζ)x]dx

∣

∣

∣

∣

2

, (4.9)

where

x0 = −D

2cot α cot(3α − γ), (4.10)

a =D

2, (4.11)

ζ =χ cos α

cos γsin(3α − γ), (4.12)



and

ξ = −k0nd cos(3α − γ)− kx. (4.13)

Therefore we finally have an expression

|A(kx)|2 =|t(α)|2

(ξ2 + ζ2)

χ

2π2

cos α

cos γsin(3α − γ) exp(−2χh) (4.14)

for the squared absolute value of the angular spectrum of the mag-

netic field, which may be considered as a measure of the angular

distribution of the intensity of the diffracted field. The quantity

|A(kx)|2 determines the distribution of the radiant intensity (see,

e.g., Ref. [72], Sect. 5.3)

J(θd) = J0 cos2 θd |A(k0n1 sin θd)|2 (4.15)

of the scattered field.

The LPIA predictions provided above are compared to rigorous

results obtained by FMM with the S-matrix algorithm in Fig. 4.3.

In the FMM analysis we evaluate the diffraction efficiencies of the

orders created at the exit plane of the wedge when the geometry is

considered as periodic with a large enough period d to ensure that

the non-periodic structure is modeled correctly. As illustrated in

Fig. 4.2, in FMM the wedge region is divided into J layers, which are

thin enough to model the continuous facets with sufficient accuracy.

In the numerical FMM simulations we chose λ = 633 nm, n =

1.4569, L = 1300 nm, α = 40◦, D = 2L tan α, θ = 48◦, J = 256, d ∼25λ, and included ∼ 150 Floquet–Bloch modes in the calculations.

The predictions of LPIA and FMM are, in general, in reasonable

agreement considering the dimensions of the structure we are ana-

lyzing. However, the FMM results in Fig. 4.3 reveal a high radiant-

intensity peak near θd = 40◦, which is not predicted accurately by

LPIA. The reason for this peak is the propagating edge diffraction

wave generated at the tip end. This wave is studied with the aid of

FMM in Fig. 4.4, but it could in principle be added also to the LPIA

model using the geometrical theory of diffraction [73]. Figure 4.4(a)

illustrates the spatial dependence of the magnetic field amplitude



J layers

x

z

θ

Homogeneous region

λn

n = 1

d

0

Figure 4.2: The computational box showing the quantization of the wedge region into J

layers and the computational period d.

−20 −10 0 10 20 30 400

0.5

1

1.5

2x 10

−3

θd [deg]

ηF

MM

,J(

θ d)

Figure 4.3: Comparison of the rigorously calculated efficiencies ηFMM (blue bars) with

sampled values of the radiant intensity J(θd) (red bars) as a function of the diffraction

angle θd for a plane wave incident at 48◦ .



∣

∣Hy(x, z)∣

∣ within the structure, whereas Fig. 4.4(b) shows the real

part of the same field (this, of course, is the actual physical field as

discussed in Sect. 2.1). The edge diffraction pattern generated by

the apex of the wavelength-scale scatterer is seen particularly well

in Fig. 4.4(b), which also shows how this edge-diffracted contribu-

tion gains wavefront curvature as it propagates. The inclusion of

this edge-diffracted wave [74] in the geometrical LPIA analysis us-

ing methods introduced by Keller [73] is indeed a topic that will

deserve further attention.

1500 2000 2500 30000

500

1000

1500

2000

0.5

0.6

0.7

0.8

0.9

1

Edge di�raction

(a)

z[n

m]

x [nm]

1500 2000 2500 3000

500

1000

1500

2000

0

0.2

0.4

0.6

0.8

1

D

(b)

z[n

m]

x [nm]

Figure 4.4: A closer look at the interaction of the tip with the evanescent field, and the

generation of the boundary diffraction field at the tip end. Distribution of (a) the amplitude

and (b) the real part of the magnetic field.

4.1.2 Observation of evanescent-wave interference patterns

In this section we consider a similar geometry as above, except that

we assume two homogeneous plane waves (instead of one) being

incident from the half space z < 0 at angles ±θ that exceed the

critical angle at the interface z = 0. These mutually coherent plane

waves generate two counter-propagating evanescent waves propa-

gating along ±x directions, which interfere with each other to form

a standing-wave interference pattern in the near-field. By summing

Eq. (4.1) and its complex conjugate we find that this pattern is (in

TM polarization) of the form

Hy(x, z) ∝ cos (kxx) exp (−kzz) . (4.16)



The interference pattern∣

∣Hy(x, z)∣

∣

2is thus periodic with period

λ/2 sin θ, exhibiting maxima and minima corresponding to points

of constructive and destructive interference, respectively.

Figure 4.5 shows FMM-based plots of the magnetic-field inten-

sity∣

∣Hy(x, z)∣

∣

2in the presence of the tip when its edge is located

either at the position of constructive or destructive interference in

the evanescent-wave interference pattern. Evidently, inserting the

tip disturbs the interference pattern given by Eq. (4.16) significantly,

bending the wave fronts of the evanescent standing-wave pattern in

its close proximity due to the in-coupling of light. A distinct dif-

ference between the field patterns inside and above the tip is seen,

depending on its edge position. When located at a position of con-

structive interference, the in-coupled light propagates through the

tip forming a beam-like field in the upper half-space. However,

when the tip edge is located at a position of destructive interfer-

ence, the out-coupled field is split into two distinct sidelobes.

6000 7000 8000 9000

500

1000

1500

2000

2500

3000

3500

4000

0

0.05

0.1

0.15

0.2

0.25

0.3

(a)

x [nm]

z[n

m]

6000 7000 8000 90000

1000

2000

3000

4000

0

0.05

0.1

0.15

0.2

0.25

0.3

(b)

x [nm]

z[n

m]

Figure 4.5: Plots of the magnetic-field intensity in the structure shown in Fig. 4.1 when

two plane waves are symmetrically incident on it at θ = ±48◦ from the negative half-

space. (a) Tip edge at the point of constructive interference and (b) at a point of destructive

interference.

Figures 4.6 and 4.7 illustrate the comparison of the radiant inten-

sity given by LPIA and the diffraction efficiencies obtained by FMM

when the two input plane waves are incident at angles θ = ±48◦.

Here the tip position is again set at two different locations, in which

constructive and destructive interference take place in the near-

field. The geometrical (LPIA) analysis is based on the solution



given above for a single plane wave, but now a coherent superpo-

sition of the output fields generated by two symmetrically incident

plane waves is considered. The observations anticipated in view of

Fig. 4.5 are confirmed here: when the tip is located at a position of

constructive interference, the radiant intensity given both by FMM

and LPIA shows an axial peak, whereas an axial zero is seen when

the tip is located at a position of destructive interference. In analogy

with the case of one incident plane wave, edge diffraction waves are

seen by FMM, which now appear on both sides of the wedge.

−40 −30 −20 −10 0 10 20 30 40

2

4

6

8

10x 10

−5

θd [deg]

ηF

MM

,I r

ad

Figure 4.6: Same as Fig. 4.3, but for two plane waves incident at ±48◦ and the tip position

at a point of constructive interference.

4.2 NEAR FIELD DETECTION THROUGH A PLANAR WAVE-

GUIDE

Next we proceed to the case where the output plane of the dielec-

tric wedge is connected to a planar waveguide as shown in Fig. 4.8,

thus considering a y-invariant version of a scanning near-field mi-

croscope with a bare tip.

Let us denote by H2(x) the only existing component (in TM

polarization) of the magnetic field of an arbitrary waveguide mode

at the exit plane of wedge, z = h + L. The coupling efficiency from

an input field H1(x) into this particular mode is generally defined,



−40 −30 −20 −10 0 10 20 30 40

2

4

6

8

10

12

14

x 10−5

θd [deg]

ηF

MM

,I r

ad

Figure 4.7: Same as Fig. 4.6 but now the tip is positioned at a point of destructive inter-

ference.

x

z

θθ

n = 1

n

L

0h

n1n2 n2

λλ

A

B

C

D

Core: diameter 2aCladding Cladding

x0−x0

x = 0

H1(x)

H2(x)

Figure 4.8: The geometry involving the generation of an interference pattern over the

substrate and detection by tip attached to a planar waveguide.



by the overlap integral method [75], as

η =|∫ ∞

−∞H1(x)H∗

2 (x)dx|2∫ ∞

−∞|H1(x)|2dx

∫ ∞

−∞|H2(x)|2dx

. (4.17)

If only one incident plane wave is present, the input field and the

mode within the core of the waveguide overlap only in the interval

x0 ≤ x ≤ a. Since the first integral in the denominator of Eq. (4.17)

equals C2/(2kz), where C = H0T(α)R, we obtain

η =2kz

C2

|∫ a

x0H1(x)H∗

2 (x)dx|2∫ a−a

|H2(x)|2dx, (4.18)

which gives the coupling efficiency into this particular mode.

Figure 4.9: Distributions of magnetic field amplitude inside the structure for several com-

binations of the wedge angle α and core width D = 2a. Here λ = 633 nm, n = 1.4569,

θ = 44.91◦ , n1 = 1.52, and n2 = 1.49 (adapted from Ref. [76]).

The spatial dependence of the magnetic-field amplitude∣

∣Hy(x, z)∣

∣

is illustrated in Fig. 4.9 using FMM and assuming a single plane



wave input at an angle θ above the critical angle. Complicated

diffraction patterns within and around the wavelength-scale scat-

terer are seen, as expected. However, the rigorous results justifies

qualitatively the use of LPIA to study the phenomena related to

evanescent fields. It shows a beam-like field come up from the

wedge and is coupled smoothly into the (guided and radiation)

modes of the waveguide. The spatial profile differs from the LPIA

prediction because of diffraction due to propagation. Furthermore,

the direction of the beam propagation changes with the wedge an-

gle α, which is in qualitative agreement with Eq. (4.6). A closer

analysis (not shown here, see Media 1 in Ref. [76]) demonstrates

that the incoupled beam gains wavefront curvature as it propagates

upwards, but in the entrance plane of the waveguide the wavefronts

are still essentially planar as assumed in the LPIA model.

4.2.1 Coupling efficiency from geometrical and rigorous models

In this section we compare the coupling efficiencies predicted by

FMM and LPIA as a function of the wedge angle α. In all examples

we choose λ = 633 nm, n = 1.4569, and θ = 44.91◦. In each ex-

ample we keep the width 2a of the waveguide constant, and hence

the wedge height varies as L = a/ tan α. The results for the single

mode waveguide is shown in Fig. 4.10 with parameters n1 = 1.52,

n2 = 1.49, and 2a = 1000 nm. For the sake of comparison the

coupling efficiency curves evaluated by FMM and LPIA are both

normalized to their respective maximum values. Although the two

curves are not identical, their general behavior is similar. The max-

imum coupling of light is obtain at a certain optimum wedge angle

α, which is nearly the same according to FMM and LPIA calcula-

tions. The upward-pointing arrow (shown in black) in Fig. 4.10 cor-

responds to the value of α for which the angle φ in Fig. 4.1 matches

the geometrical propagation angle θ0 of the fundamental waveguide

mode according to Eqs. (7.2)–(7.5) in Ref. [77].

Next, we proceed to find out similar results for a two-mode

waveguide in Fig. 4.11 with 2a = 1600 nm. Again we obtain fairly



30 35 40 45 50

0.6

0.7

0.8

0.9

1

η/

ηm

ax

α [deg]

Figure 4.10: Normalized coupling efficiency into the fundamental mode of a single-mode

waveguide with 2a = 1000 nm as a function of the wedge angle α. Black line: FMM

calculation. Red line: overlap integral method based on LPIA. Arrow: optimum angle

given by the phase matching condition (adapted from Ref. [76]).

good agreement between the FMM and LPIA calculations around

the optimum angle, which is now different for the fundamental

mode m = 0 and the antisymmetric mode m = 1. From the phase

matching conditions φ = θ0 and φ = θ1 we find out the position

of arrows that corresponds to the values of α. The total coupling

efficiency is the sum of the coupling efficiencies for the fundamental

and antisymmetric modes. The maximum coupling efficiency into

the antisymmetric mode, compared to that into the fundamental

mode, is ∼ 88 % according to FMM and ∼ 59 % according to LPIA.

30 35 40 45 50

0.4

0.5

0.6

0.7

0.8

0.9

1

η/

ηm

ax

α [deg]

(a)

30 35 40 45 50

0.4

0.5

0.6

0.7

0.8

0.9

1

η/

ηm

ax

α [deg]

(b)

Figure 4.11: Same as Fig. 4.10, but for a two-mode waveguide with 2a = 1600 nm.

(a) Fundamental mode m = 0. (b) First antisymmetric mode m = 1 (adapted from

Ref. [76]).



Finally, we proceed to consider the similar results shown in

Fig. 4.12 but for a waveguide with 2a = 2200 nm, which supports

three guided modes. Again we obtain reasonably good agreement

between the FMM and LPIA calculations for the fundamental mode

m = 0 and the antisymmetric mode m = 1. However, the results

do not agree for the second symmetric mode m = 2, for which the

geometrical propagation angle θ2 = 11.33◦ is already rather close to

the cut-off θc = 11.40◦. The cause of disagreement is as the prop-

agation angle approaches cut-off, the field extends much into the

cladding region, resulting in a poor confinement of energy in the

core than in the cladding. According to FMM, the maximum cou-

pling into fundamental mode m = 0, compared to m = 1 is ∼ 90%.

According to LPIA, the maximum coupling efficiency in m = 1 is

∼ 55% of m = 0. The corresponding ratios for m = 2 is ∼ 76% of

m = 0 calculated from LPIA and ∼ 11% of m = 1 calculated rig-

orously, which shows that rigorously evaluated coupling efficiency

into m = 2 mode is relatively weak.

4.2.2 Detection of evanescent-wave interference patterns

As a further illustration of the LPIA model we now consider the two

incident plane waves arriving from the half-space z < 0 at symmet-

rical angles ±θ as in Sect. 4.1.2 but assuming the observation ge-

ometry of Fig. 4.8. Hence, in view of Eq. (4.16), the magnetic-field

intensity of this pattern is

∣

∣Hy(x, z)∣

∣

2∝ cos2 (kxx) exp (−2kzz) (4.19)

with kx = k0 sin θ and kz = k0 cos θ. We assume that the wedge is

scanned at a constant height h and the coupling efficiency is deter-

mined as a function of tip position x.

Fig. 4.13 illustrates the results for a two mode waveguide con-

sidered in Fig. 4.11 with 2a = 1600 nm. The coupling efficiencies of

the fundamental and antisymmetric mode obtained by FMM and

LPIA are compared with magnetic field intensity distribution given

by Eq. (4.19). The two models give indistinguishable results. The



30 35 40 45 50

0.2

0.4

0.6

0.8

1η

/η

max

α [deg]

(a)

30 35 40 45 50

0.2

0.4

0.6

0.8

1

η/

ηm

ax

α [deg]

(b)

30 35 40 45 50

0.2

0.4

0.6

0.8

1

η/

ηm

ax

α [deg]

(c)

Figure 4.12: Same as Fig. 4.10, but for a three-mode waveguide with 2a = 2200 nm.

(a) Fundamental mode m = 0. (b) First antisymmetric mode m = 1. (c) Second symmetric

mode m = 2 (adapted from Ref. [76]).

−300 −200 −100 0 100 200 300

0.2

0.4

0.6

0.8

1(a)

x [nm]

η/

ηm

ax,

I/I m

ax

−300 −200 −100 0 100 200 300

0.2

0.4

0.6

0.8

1(b)

x [nm]

η/

ηm

ax,

I/I m

ax

Figure 4.13: Observation of evanescent-wave interference patterns. Solid blue : magnetic-

field intensity given by Eq. (4.19). Normalized coupling efficiencies given by FMM (black

crosses) and LPIA (red circles) into (a) the fundamental mode m = 0 with α = 39.5◦ and

(b) the antisymmetric mode m = 1 with α = 41.8◦ (adapted from Ref. [76]).



results for the fundamental mode agree with Eq. (4.19). However,

since the planar waveguide is symmetric about the plane x = 0

(see Fig. 4.8) the mode solution are either even or odd in x. For

odd modes Hy(x, z) = −Hy(−x, z) is a solution, consequently in

the Fig. 4.13(b) there is a half a period phase shift between coupling

efficiency curves and the undisturbed interference pattern.

4.3 METALLIC GRATINGS WITH SUB-WAVELENGTH SLITS

Now we extend the discussion to plasmon waves and near-field op-

tics in the presence of metallic nanostructures. Similarly as above,

in the spirit of LPIA, diffraction effects in the incoupling process are

ignored. The geometrical approach is applied until the rays reach

the step-index waveguide, where its coupling to various guided

modes can be treated by the phase-matching approach (overlap in-

tegral). Hence, as usual in LPIA models, we move from the geomet-

rical optics picture to a wave optical model at the wedge-waveguide

interface. Making use of this quasi-geometrical approach as before,

we determine the optimum tip opening angle and compare the re-

sults of the geometrical model to rigorous simulations by the FMM.

4.3.1 Geometrical configuration and models

Figure 4.14 illustrates the geometrical configuration to be studied.

A homogeneous plane wave with wavelength λ is now assumed to

be normally incident at the interface z = 0 from the half space z < 0,

filled with a refractive index n. We assume a metallic (Al) grating

of height l, with very narrow slits, above the interface z = 0. Only

a TM-polarized wave can be transmitted efficiently through such

narrow slits [78]. Such a wave excites collective surface electron os-

cillations, known as plasmon oscillations, on both the entrance and

exit interfaces of the grating. Because of the geometrical symmetry,

two counter-propagating plasmonic waves are generated. Surface

plasmon interference then takes place in the wavelength-scale re-

gion above the grating [79–81], and we consider the observation



x

ββαα

γ

φ

Al AlAl

z

n1

n2 n2

TM incidenceD

d

2a

l

x = 0z = 0

w

L

Figure 4.14: A schematic diagram of the y-invariant system considered here, showing ray

propagation into and inside a wedge with tip half-angle α connected to a waveguide of

width 2a.

of this interference pattern by inserting the tip-waveguide probe in

close proximity to the grating.

Let us first consider the inhomogeneous field associated with

just one of the two excited counter-propagating plasmons. Its sur-

faces of constant phase are planes close to (but coinciding with) the

direction of the surface normal. Hence one may associate a local

ray direction to the field, denoted by angle β in Fig. 4.14 [82]. In the

LPIA model we assume that one of the two plasmon waves arrives

from the left-hand-side of the wedge and the other from the right-

hand-side, these two waves being mutually fully correlated. The

inhomogeneous field arriving from the left is first refracted inside

the wedge through its left facet, and after total internal reflection

from the right-hand-side facet, coupled to the planar waveguide.

The plasmon field arriving from the right is treated similarly. The

theoretical LPIA analysis is a simple extension of that presented in

Sect. 4.1.1, the only difference being the inclusion of the angle β.

In the numerical simulations that follow, we consider wave-

length λ = 671.4 nm and assume that the metal is (bulk) aluminium

with complex refractive index 1.6231 + 8.0261i, and having a thick-



ness l = 70 nm. The grating period and slit width are taken as

d ∼ 12λ and w = 200 nm, and we choose a superperiod D ∼ 24λ

large enough to ensure that the non-periodic structure (including

the wedge and the waveguide) is modeled correctly. The rigor-

ous results are obtained using FMM with the S-matrix algorithm to

evaluate the coupling efficiencies into the waveguide modes [83].

Figure 4.15(a) illustrates the magnetic-field intensity in the grat-

ing region and the homogeneous spaces around it, without the

wedge being present. In order to see the surface wave intensity

pattern on the front side of the grating (z < 0) we have subtracted

the incident zeroth-order field before computing the intensity pat-

tern. Figure 4.15(b) shows the time-average Poynting vector for the

same configuration. Surface plasmons are seen on both the entrance

and the exit sides of the metallic grating, but our point of interest is

to detect the surface plasmons on the exit side of the grating, which

finally couple to the planar waveguide through the wedge.

4.3.2 Coupling efficiencies by rigorous theory and the geometri-

cal phase matching model

Let us first compare the coupling efficiencies predicted by LPIA

and by FMM as a function of the wedge angle α. In the examples

we keep the core width 2a of the waveguide constant, considering

0 200 400 600

−8000

−6000

−4000

−2000

0

2000

4000

6000

8000 0

0.2

0.4

0.6

0.8

1(a)

z [nm]

D[n

m]

−100 −50 0 50 100 150 200

3800

4000

4200

0

0.1

0.2

0.3

0.4

0.5(b)

z [nm]

D[n

m]

Figure 4.15: (a) Magnetic field intensity∣

∣Hy

∣

∣

2in and around the grating. (b) A closer

look at the direction of energy flow in the slit region, given by the Poynting vector.



two different cases in which two and three propagating modes are

present, respectively. Hence, in each case, the wedge height varies

38 39 40 41 42

0

2

4

6

8

α13 α12 α23

α33α22

α[deg]

φ[d

eg]

Figure 4.16: Plot of the propagation angle φ as a function of the half tip angle α for different

guided modes. Red: fundamental mode. Blue: lowest antisymmetric mode. Magenta:

first higher symmetric mode. Dashed line: two-mode waveguide. Solid line: three-mode

waveguide.

as L = a/ tan α.

We continue using the same phase matching considerations as

in Sect. 4.1.1. However, due to the metal-air interface, the tilt angle

β should also be taken into consideration and then we have

γ = sin−1 [sin(α + β)/n1] . (4.20)

In Fig. 4.16 we illustrate the dependence of the propagation angle

on the wedge angle (sloped line) and the geometrical propagation

angles of the guided modes [77] shown by the horizontal lines of

different colors. The intersections of the horizontal lines with the

sloped line with the horizontal lines give the optimum wedge an-

gles for different modes, denoted by α12, α22 and α13, α23, α33, where

the first subscript denotes the mode number and the second sub-

script denotes the total number of guided modes supported by the

waveguide.



Figure 4.17 renders the results for a two-mode waveguide with

parameters n1 = 1.52, n2 = 1.51, and 2a = 2800 nm. The coupling

efficiency curves evaluated by FMM are normalized to their respec-

tive maximum values. In view of FMM, we obtain maximum cou-

pling at a certain optimum wedge angle α. The upward-pointing

arrow in Fig. 4.17 corresponds to the values of α for which the an-

gle φ in Fig. 4.14 matches the geometrical propagation angle of the

fundamental mode. These optimum wedge angles are nearly the

same according to FMM and the geometrical phase matching con-

dition.

32 34 36 38 40 42 44 46 48

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

α12

(a)

α [deg]

η/

ηm

ax

32 34 36 38 40 42 44 46 48

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

α22

(b)

α [deg]

η/

ηm

ax

Figure 4.17: Rigorously evaluated coupling efficiencies into a two-mode waveguide as

a function of wedge half-angle α and comparison with the geometrical phase matching

condition (shown by arrows). (a) Fundamental mode m = 0. (b) Lowest antisymmetric

mode m = 1.

Figure 4.18 illustrates corresponding results for a waveguide

with 2a = 5000 nm and three propagating modes. The agreement

between FMM and phase matching condition (shown by arrows) is

good for the fundamental mode and antisymmetric mode as shown

in Fig. 4.18(a) and Fig. 4.18(b), respectively. As in Sect. 4.2.1, we ob-

serve a disagreement for the second symmetric mode m = 2, for

which the geometrical propagation angle φ = 6.0622◦ is again close

to the cutoff at φc = 7.25◦.



32 34 36 38 40 42 44 46 48

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

α13

(a)

α [deg]

η/

ηm

ax

32 34 36 38 40 42 44 46 48

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

α23

(b)

α [deg]

η/

ηm

ax

32 34 36 38 40 42 44 46 480.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

α33

(c)

α [deg]

η/

ηm

ax

Figure 4.18: Same as Fig. 4.17 but for a three-mode waveguide. (a) Fundamental mode

m = 0. (b) Lowest antisymmetric mode m = 1. (c) Symmetric mode m = 2.

4.3.3 Examples of field patterns within the structure

Let us now take a brief detour by considering, instead of a nor-

mally incident plane wave, obliquely incident illumination of a thin

Al film (thickness 20 nm) at an angle θ = 48◦ and wavelength

λ = 671 nm. In this case the plasmon field propagates only in

the positive x-direction and no plasmon interference pattern is gen-

erated. However, the incoupling process can be illustrated quite

clearly in this geometry.

Figure 4.19 shows the distribution of the real part of the mag-

netic field for several different wedge angles of the waveguide tip

for a fixed L = 2000 nm. The height of the wedge is kept constant

in each case. So, with increase in α, the diameter 2a increases and



0 2000 4000 6000 8000 10000 120000

2000

4000

6000

8000

10000

12000

0 2000 4000 6000 8000 10000 120000

2000

4000

6000

8000

10000

12000

0 2000 4000 6000 8000 10000 120000

2000

4000

6000

8000

10000

12000

0 2000 4000 6000 8000 10000 120000

2000

4000

6000

8000

10000

12000

0 2000 4000 6000 8000 10000 120000

2000

4000

6000

8000

10000

12000

0 2000 4000 6000 8000 10000 120000

2000

4000

6000

8000

10000

12000

0 2000 4000 6000 8000 10000 120000

2000

4000

6000

8000

10000

12000

0 2000 4000 6000 8000 10000 120000

2000

4000

6000

8000

10000

12000

0 2000 4000 6000 8000 10000 120000

2000

4000

6000

8000

10000

12000

α = 32◦ α = 34◦ α = 36◦

α = 38◦ α = 40◦ α = 42◦

α = 44◦ α = 46◦ α = 48◦

x [nm]x [nm]x [nm]

x [nm]x [nm]x [nm]

x [nm]x [nm]x [nm]

z[n

m]

z[n

m]

z[n

m]

z[n

m]

z[n

m]

z[n

m]

z[n

m]

z[n

m]

z[n

m]

Figure 4.19: Distributions of the real part of the magnetic field inside the structure for

several values of the wedge angle α for oblique illumination of a thin metallic film.

hence more modes appear. The effect of the wedge angle in the pre-

dominant propagation direction of the incoupled field is prominent

in these figures: if α is either too small or two large, the light beam

generated by the wedge tends to couple into radiation modes of the

planar waveguide. Also, since the real field is now considered, the

bending of the wave fronts is clearly seen.



2000 3000 4000 5000 6000

2

4

6

8

10

x 10−3

x [nm]

η(x),

I(x)

(a)

2000 3000 4000 5000 6000

1

2

3

4

5

6

7

x 10−3

x [nm]

η(x),

I(x)

(b)

2000 3000 4000 5000 6000

2

4

6

8

10

x 10−4

x [nm]

η(x),

I(x)

(c)

2000 3000 4000 5000 6000

1

2

3

4

x 10−3

x [nm]

η(x),

I(x)

(d)

Figure 4.20: Observation of surface plasmon interference. Red: magnetic field intensity

in the absence of the wedge. Black: coupling efficiency into the waveguide. (a) Sum of all

modes. Coupling efficiency into (b) the fundamental mode m = 0, (c) lowest antisymmetric

mode m = 1, and (d) symmetric mode m = 2.

4.3.4 Observation of surface plasmon interference

We are now returning to consider normally incident illumination

of the grating from the half-space z < 0. Due to the counter-

propagating surface plasmons, an interference pattern is formed

above the grating surface in the half-space z > l. We assume that

the wedge probes this pattern at h = 10 nm above the grating sur-

face and determine the coupling efficiency into the waveguide as a

function of the tip position x using FMM. The waveguide parame-

ters are those of the three-mode case in Fig. 4.18.

Figure 4.20(a) shows a comparison of the coupling efficiency

η(x) into the waveguide with the field intensity I(x) =∣

∣Hy(x)∣

∣

2in

the absence of the tip at a distance of 10 nm from the exit side of

the grating. The black line illustrates the total coupling efficiency



as the sum of all the propagating modes and the red line gives

the undisturbed field intensity. The maxima of the two curves are

normalized equal for easy comparison. The overall coupling effi-

ciency gives a rough estimate of the free-field intensity, although

the central maximum (due to the directly transmitted field through

the slit) appears remarkably high in proportion to that of the undis-

turbed plasmon-wave interference pattern. However, the locations

of the minima and maxima are not in perfect agreement. To clarify

the reason for this effect we investigate the modal contributions to

the total coupling efficiency, assuming wedge angles optimized for

the mode in question as discussed in Sect. 4.3.2. Figure 4.20(b)–(d)

show the results for the modes m = 0, m = 1, and m = 2, respec-

tively. Since the planar waveguide is symmetric about the plane

x = 0 as considered in Fig. 4.14, modes m = 0 and m = 2 are even

and mode m = 1 is odd in x. For odd modes Hy(x, z) = −Hy(−x, z)

is a solution, which results in half a period phase shift between the

coupling efficiency curve and the undisturbed interference pattern

as shown in Fig. 4.20(c). The contribution from the antisymmetric

mode thus gives rise to the distortion in the interference pattern in

Fig. 4.20(a).

4.4 DETECTION OF EVANESCENT FIELDS ABOVE BINARY

SUBWAVELENGTH GRATINGS

As a final example, we examine a case similar to that of Sect. 4.3.1,

assuming however a subwavelength-period grating of height l above

the interface z = 0. We consider both dielectric and metallic grat-

ings, and in all examples choose the parameters λ = 671 nm,

n = 1.456, n1 = 1.52, n2 = 1.51, and 2a = 2200 nm. The refrac-

tive index of the grating material is chosen as 1.456 in the dielectric

case and as 1.6231 + 8.0261i in the metallic case. The illumination

from the substrate is at an angle of incidence θ = 48◦ that exceeds

the critical angle at the interface z = 0. Moreover, the grating period

d is chosen small enough to prevent the generation of any propagat-

ing diffraction orders above the grating. Thus the field generated



Scanning

λθ

n = n1

Weak field

x

z

d

L

0l

Cladding Cladding2a

α

Figure 4.21: Schematic diagram illustrating the evanescent-field detection above a

subwavelength-period grating by the SNOM tip.

above the plane z = l is a superposition of many evanescent diffrac-

tion orders and shows lateral structure in the absence of the tip. We

detect this evanescent field at the exit side of the grating at a fixed

scanning height h = 10 nm and compare it with the undisturbed

field intensity at the same height. Figure 4.21 shows a schematic

diagram of the geometry.

Figure 4.22 illustrates the fields inside and near the gratings,

without the tip being present. A dielectric grating with period

d = 250 nm is considered in Fig. 4.22(a) for TE polarized illumi-

nation and in Fig. 4.22(b) for TM polarized illumination. Inside the

grating, the electric-field amplitude is concentrated within the di-

electric ridges and in TM polarization in the air grooves, and the

evanescent fields appears somewhat stronger in TE than in TM po-



0 100 200 300 400 500

0

100

200

300

400

500

600

7000

0.5

1

1.5

2

2.5

z [nm]

x[n

m]

(a)

0 100 200 300 400 500

0

100

200

300

400

500

600

7000

0.5

1

1.5

2

2.5

z [nm]

x[n

m]

(b)

0 100 200 300 400 500

0

100

200

300

400

500

600

7000

0.2

0.4

0.6

0.8

1

z [nm]

x[n

m]

(c)

0 100 200 300 400 500

0

100

200

300

400

500

600

7000

0.2

0.4

0.6

0.8

1

z [nm]

x[n

m]

(d)

Figure 4.22: Distributions of (a) the electric field amplitude∣

∣Ey(x, z)∣

∣ in TE polarization

and (b) the magnetic field amplitude∣

∣Hy(x, z)∣

∣ in TM polarization inside and near the

binary dielectric grating. Correspondingly, (c) and (d) show the distributions of the same

quantities for a metallic binary grating.

larization (however, it should be stressed that we are plotting the

same quantity in these two cases). In the case of metallic gratings,

the polarization dependence is far more obvious. For TE illumina-

tion in the metallic case, the effective refractive index of the lowest-

order Bloch mode in the grating region is well known to have a large

imaginary part and therefore light can not penetrate deep into the

air grooves. This is seen clearly in Fig. 4.22(c). In the case of TM

polarization, however, the imaginary part of the Bloch mode is sub-

stantially smaller. Extraordinary transmission of light though the

grating takes place as seen in Fig. 4.22(d). Thus the metallic grating

becomes nearly transparent and a strong inhomogeneous field is

generated on its exit side.

Figure 4.23 shows a closer look at the magnitude and the local



direction of energy flow in the binary metallic grating for TM illu-

mination given by the time-averaged Poynting vector. We observe

a counter-clockwise rotation of energy flow around the metallic

block, though one should be rather careful with such interpreta-

tions in the subwavelength scale. Nevertheless, the Poynting vector

in the half-space above the grating points consistently in the neg-

ative z-direction, indicating a plasmon wave propagating in this

direction for the parameters chosen here. Let us now introduce

the tip-waveguide probe and compare the coupling efficiency with

the intensity of the undisturbed field. Figure 4.24 shows such a

comparison for TM illumination of a dielectric grating at a large

angle of incidence at θ = 70◦. In the FMM analysis we repeated

the subwavelength period of 250 nm by 80 times to create a compu-

tation window with superperiod 20000 nm, which is large enough

to contain also the probe. Figure 4.24(a) shows the comparison for

the fundamental mode and Fig. 4.24(b) for the first anti-symmetric

mode. In the latter case we again see a half-phase shift between

the observed and the undisturbed patterns. The results of a similar

0 100 200 300

400

500

600

700

800

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

z[nm]

x[n

m]

Figure 4.23: Distribution of the magnitude and local direction of the time-averaged Poynt-

ing vector in and near a metallic binary grating. Here λ = 671.4 nm, d = 620 nm, and

θ = 50◦ .



300 400 500 600 700 800

2

4

6

8

10

x 10−5 (a)

x [nm]

η(x),

I(x)

300 400 500 600 700 8000

2

4

6

8

x 10−5 (b)

x [nm]

η(x),

I(x)

Figure 4.24: Observation of fields above a binary dielectric subwavelength grating. Red:

Magnetic field intensity. Black: coupling efficiency. (a) Fundamental mode m = 0.

(b) Anti-symmetric mode m = 1.

analysis for a binary metallic grating are shown in Fig. 4.25.

200 300 400 500 600 700 800

2

4

6

8

10

x 10−5 (a)

x [nm]

η(x),

I(x)

200 300 400 500 600 700 800

2

4

6

8

10x 10

−5 (b)

η(x),

I(x)

x [nm]

Figure 4.25: Same as Fig. 4.24 but for a binary metallic subwavelength grating.

4.5 SUMMARY

We have introduced a geometrical optics based local-plane-interface

approach to model near field phenomena involving interactions of

evanescent, inhomogeneous, and plasmon waves with wavelength-

scale scatterers. In particular, a model for wedge-based scanning

near-field microscope with a bare tip was developed and applied

to observation of near fields behind different types of gratings. The

results were compared to a rigorous model based on the Fourier



modal method, and fair agreement was reached in many cases. The

coupling efficiencies did not match perfectly since the geometri-

cal model ignores diffraction effects within the wedge region. The

results obtained (especially the diffraction patterns in the absence

of the waveguide) could probably be improved by including the

boundary diffraction wave created by the tip edge. Additionally,

the inclusion of the evanescent field on the second (reflecting) sur-

face of the wedge in the geometrical analysis could also improve

significantly the diffraction pattern at the output plane of the wedge

but these remains the topic of future research.




5 Coding of high-frequency

carrier V-shape gratings

Diffractive optics is a versatile technology to create optical elements

with complex functionality. The first computational coding scheme

to create diffractive elements from mathematically defined signals

(computer-generated holography) was introduced by Lohmann [84].

Other schemes soon followed, such as the phase-only (kinoform)

coding technique put forward by Lesem, Hirsch, and Jordan [85].

Yet another approach emerged, which is based on modulation of

high-frequency carrier gratings and has its origin in off-axis op-

tical holography [86]. By the early 1990’s, the synthesis methods

in diffractive optics had been developed to a level, which allowed

the design of high-quality, high-efficiency elements using iterative

techniques [87]. By this era the development of microlithographic

fabrication techniques also permitted the encoding of diffractive

elements in different types of wavelength-scale carrier structures.

Coding methods that employ wavelength-scale nanostructures de-

signed by rigorous diffraction theory [5] are now commonplace,

and their fabrication is feasible using modern nanolithography [88,

89].

Coding methods that employ high-frequency carrier gratings

and involve modulation of the larger-scale structure of the element

are of substantial interest. In early studies [26–28] binary grat-

ings were used as carriers, providing high transmission-mode or

reflection-mode efficiency when the carrier period, groove depth

and width, and the angle of incidence are chosen properly. In

the transmission mode such elements feature rather high input-

angle selectivity of the diffraction efficiency, but in the reflection

mode high efficiency can be maintained over a much wider angular

range [27].



In this chapter, we propose a new method to realize high-carrier-

frequency diffractive elements, which is based on using V-shaped

carrier gratings as an alternative to binary resonance-domain grat-

ings. In terms of optical performance, these two approaches are

comparable especially in the reflection mode. However, the sil-

icon wet etching technique [90] of V-shaped structures facilitates

precise control of the groove shape and easy replication. Regular

V-gratings have been used in, e.g., spectroscopy [91–93], infrared

beam splitting [94, 95], reflection suppression [96, 97], and optical

interconnection [98]. Recently, wet etching of silicon has triggered

a new era in the production of photovoltaic devices [16, 99, 100].

At first we introduce several general techniques for the real-

ization of diffractive structures by modulation of the high-carrier-

frequency V-gratings. Corresponding methods based on modula-

tion of the local groove position or width have been introduced for

binary gratings in the past; see, e.g., Refs. [101, 102]. We also dis-

cuss the limitations of the new coding schemes. Then we proceed

to design profiles that are capable of splitting an incident beam

into several output beams of equal efficiency. Such array illumi-

nators are among the most important diffractive elements, and in

addition they are easy to characterize both theoretically and exper-

imentally. We apply rigorous diffraction theory to study conditions

under which good signal quality can be achieved. An experimen-

tal demonstration is provided by patterning the modulated grating

structure using electron beam lithography, wet etching it in silicon,

replicating it in polymer, and coating the replicated structure with

a thin metal layer.

High-frequency carrier techniques used in diffractive optics can

be divided into two main categories: on-axis and off-axis coding

techniques. In the on-axis schemes discussed in Sect. 5.1 one makes

use of the zeroth diffraction order of a subwavelength-period car-

rier grating. The local material volume fill factor is varied from

one period of the carrier grating to another, which results in mod-

ulation of the effective refractive index of the grating and thereby

in phase modulation of the output field. In off-axis schemes one


CHAPTER 5. CODING OF HIGH-FREQUENCY CARRIER V-SHAPEGRATINGS

uses a higher order (typically first) of the carrier grating and mod-

ulates the phase of the incident field by varying the groove/ridge

positions. This scheme is discussed in Sect. 5.2.

5.1 V-GROOVE WIDTH MODULATION

Let us start by considering the encoding of the phase information of

the wavefront into a high-frequency carrier, making use of groove-

width variations. This scheme was introduced for binary gratings

independently by Farn [103] and Stork et al. [104]. It is based on

modulating the fill factor of a binary subwavelength-period grating

as a function of position to encode the desired continuous phase-

delay profile φ(x) of a paraxial-domain diffractive element (we con-

sider, for simplicity of illustration, only y-invariant optical functions

in this chapter).

Figure 5.1 illustrates the principle of groove width modulation

of a high-frequency-carrier V-grating. Here the simplest example of

a linear phase function φ(x) = cx, where c is a constant, is consid-

ered. The period of the carrier grating is taken as d < λ, and the

volume fraction of the material is controlled by varying the groove

width w as a function of position. Since α is constant, varying w also

leads to groove-depth modulation. In order for this coding method

to work, the values of φ(x) must be reduced to the interval [0, 2π),

which is always possible (for monochromatic light). Now the idea

is to map the values of φ(x) to values of w at an equally spaced

grid of sampling points separated by d, which results in a profile of

the type shown in Fig. 5.1(b). Note that the higher the value of c is,

the smaller is the number of sampling points of φ(x) in the [0, 2π)

interval. In order to keep the local diffraction efficiency of the mod-

ulated element at a sufficiently high level, this number should be at

least four (as it is in Fig. 5.1).

Rigorous FMM analysis is available to determine the mapping

between the groove width and the resulting phase delay. Because

of fabrication constraints in Si wet etching, one should ensure that

the widths d − w of all flat regions are at least ∼ 50 nm. With



2π

x

z

d

dw

αAir

COC

(a)

(b)

λ

φ

0

0

Figure 5.1: Groove width modulation. (a) Phase modulation showing four-level quanti-

zation between phase values 0 and 2π (b) Schematic diagram showing the groove width

modulated structure with four different phase values.

this constrain in mind, we aim to optimize w to obtain maximum

efficiency in the zeroth order and, at the same time, achieve a max-

imum phase delay of at least 2π radians. To this end we scan w

by using FMM with the S-matrix propagation algorithm to evalu-

ate the diffraction efficiencies. As illustrated in Fig. 3.2, the wedge

region is again divided into J layers thin enough to model the con-

tinuous facets with sufficient accuracy. Typically we now choose

J = 50 and include ∼ 100 Floquet–Bloch modes in the calculation

of carrier-grating efficiencies.

5.1.1 Transmission-type groove width modulation

We consider first the optimization of the carrier-grating structure

in the transmission case. The V-groove apex half-angle is assumed

to be α = 35.26◦ and we consider two design wavelengths, λ0 =

457 nm and λ0 = 533 nm. The material on the incident side is

Cyclo Olefin Copolymer (COC) with refractive index n = 1.52. In



0 200 4000

0.2

0.4

0.6

0.8

1

w [nm]

η0

(a)

0 200 400 6000

1

2

3

4

5

(b)

w [nm]

φ[r

ad]

Figure 5.2: Groove width modulation for transmission type gratings. (a) Plots of zeroth

order efficiency and (b) the phase delay as a function of groove width w for periods d =

457 nm (blue) and d = 533 nm (red). The solid lines show the results for TM incident

light and the dashed lines for TE incident light.

Fig. 5.2(a) we plot the zeroth-order efficiency as a function of w up

to w = d. The efficiency for TM-polarized incident light (shown by

solid lines) is remarkably high, but in TE polarization the results

are far less satisfactory. On the other hand, as shown in Fig. 5.2(b),

the maximum phase delay that can be obtained is 1.65π radians,

which is almost 0.35π radians less than what is ideally desired.

Due to the insufficient phase delay, the full phase function cannot

be coded perfectly into the carrier grating profile. Equally-spaced

four-level coding still appears possible at first sight since it only

requires phase delays up to 1.5π radians. However, in transmission-

type optics the phase delay is evaluated at the deepest apex level

of the entire modulated structure, not only between the apex and

the top level of the carrier grating. This implies an extra phase due

to propagation through the homogeneous section of the material,

which severely reduces the available phase coding range. Hence, in

practise, it is not possible to code arbitrary phase profiles properly

in transmission-type V-carrier gratings.

5.1.2 Reflection-type groove width modulation

After the failure of transmission-type gratings we proceed to con-

sider the case of optimizing reflection-type gratings. We consider a



design wavelength λ0 = 457 nm and assume that the reflecting ma-

terial is aluminium with complex refractive index 0.6402 + 5.5505i.

In view of Fig. 5.1(b), instead of air in the output medium we now

have an Al substrate, and the input medium is air instead of COC.

In Fig. 5.3(a) we plot the efficiency for three different values of the

carrier period d = 320 nm, d = 330 nm, and d = 340 nm, for a

normally incident plane wave. In each plot, at w/d ∼ 0.35 there is

a huge drop in the efficiency for TM polarized light. In the case of

TE incidence, the efficiency is maintained at a high level and it is al-

most invariant over the entire scanning range of w. In Fig. 5.3(b) the

corresponding phase delays are shown as a function of w for differ-

ent periods and polarizations. The desired phase delay (in excess

of 2π radians) is achieved in TM polarization, but the phase mod-

ulation is far too weak in TE polarization. Because a large range of

phase values is effectively lost in TM polarization due to the drop

of efficiency, the groove-width modulation scheme effectively fails

also for reflection gratings.

0 50 100 150 200 250 3000.2

0.4

0.6

0.8

1

w [nm]

η0

(a)

0 50 100 150 200 250 300 3500

1

2

3

4

5

6

7(b)

w [nm]

φ[r

ad]

Figure 5.3: Groove width modulation for reflection-type gratings. (a) Plot of zeroth

order efficiency and (b) the phase delay as a function of groove width w for periods

d=320 nm (black), d=330 nm (blue) and d=340 nm (red), respectively. The solid lines

show the result for TM polarized incident light and the dashed line represent TE polarized

incident light.

Next we investigate the reason for the low efficiencies by eval-

uating the energy distributions within the groove for different po-

larization states of light. In doing so for TM-polarized light, we fix

the groove width and the period within the region where the drop



in efficiency occurs. We illustrate in Fig. 5.4(a) distributions of the

time-averaged Poynting vector, with arrows for direction and colors

for magnitude as before. It seems evident from these results that, in

the case of TM incidence, two counter-propagating plasmon waves

propagating in ±x directions are excited, which results in the drop

of the TM efficiency for the zeroth reflected order due to absorption

in the grating material. Qualitatively similar behavior, confirmed

by our calculations, is observed also for binary reflection-type grat-

ings [105].

The interpretation of Fig. 5.4(b), which illustrates the case of

TE polarized incident light, is even simpler: light cannot penetrate

deep inside the grooves of the metallic subwavelength-period grat-

ing, which causes the insufficient phase delay seen in Fig. 5.3(b).

This explains, at least qualitatively, the nearly flat efficiency curve

in Fig. 5.3(a). Having again arrived at discouraging results, we pro-

ceed look for another coding scheme, which we discuss in the fol-

lowing section.

−50 0 50 100

−100

−50

0

50

100 0

0.1

0.2

0.3

0.4

0.5

z [nm]

x[n

m]

(a)

−50 0 50

−100

−50

0

50

100 0

0.02

0.04

0.06

0.08

0.1

z [nm]

x[n

m]

(b)

Figure 5.4: The direction of energy flow given by the time-averaged Poynting vector for

(a) TM polarization and (b) TE polarization of the incident light. Here λ = 457 nm,

w = 120 nm, and d = 340 nm. Only the reflected part of the field is considered here.

5.2 V-RIDGE POSITION MODULATION

The idea behind off-axis coding of high-frequency carrier gratings

is essentially similar to Lohmann’s detour-phase principle. If the

entire grating is moved sideways in the x-direction with respect to



z d

air

metal

metal w

w

Sn

θ

air

W

D x

(a)

(b)

Figure 5.5: Geometry and notation. (a) Structure of the carrier V-ridge grating. (b) Ridge

position modulated gratings: Sn denotes the detour-phase shift of the center position of the

nth ridge and W represents the signal window (adapted from Ref. [106]).

a fixed frame, the phase of the first order is modulated in direct

proportion to the lateral shift, and a shift over one carrier period

produces a detour-phase shift of 2π radians. If we now modulate

the lateral shift as a function of position according to the phase

function φ(x) that we wish to encode, the desired optical effect is

generated around the first diffraction order of the carrier grating. In

this scheme there is, in fact, no need to reduce φ(x) into the interval

[0, 2π) as long as the detour-phase shift is a slowly varying function

compared to the carrier-grating period. And this, in turn, is the case

when a carrier grating with wavelength-scale period is used and the

signal generated by the phase function φ(x) is paraxial.

We continue by considering Fig. 5.5, which illustrates both the

structure of the V-ridge carrier grating and the principle of ridge-

position modulation. The carrier grating of period d now has sym-

metric V-ridges with an apex half-angle α = 35.26◦ and a width w as

discussed earlier, separated by flat surface sections of width d − w.

Here the ridge width is kept constant but the center positions Sn are

chosen according to the coding scheme. To achieve high efficiency

in the (minus) first order of the carrier grating, we choose the angle



of incidence θ and the carrier period d in such a way that the only

propagating reflected orders are m = −1 and m = 0. We then op-

timize w to obtain maximum efficiency η−1 in order m = −1. We

further assume that

sin θ =λ0

2d. (5.1)

This is in fact the Bragg condition (or the condition for Littrow

mounting) for the first diffraction order at design wavelength λ0.

Again we scan w, using FMM with the S-matrix propagation algo-

rithm to evaluate the diffraction efficiencies.

5.2.1 Reflection type ridge-position modulation

In what follows, we consider reflection-type elements at a design

wavelength λ0 = 457 nm, assuming that the metal is (bulk) alu-

minium with complex refractive index n = 0.6402 + i5.5505. In the

present coding scheme we have some freedom to choose the carrier

period d and the angle of incidence θ. In Fig. 5.6 we plot carrier-

grating design results for three different values of the angle of in-

cidence, namely θ = 30◦, θ = 42◦, and θ = 60◦. The carrier period

is kept at a fixed value d = 340 nm. Remarkably, for all values of θ

considered, the optimum solution with η−1 ≈ 87% is achieved with

a ridge width w ≈ 220 nm and a fill factor w/d ≈ 0.64. This effi-

ciency is high since the reflectance of a flat air-aluminum interface

at these angles of incidence is in the range 86 − 91%. In fact, the

zeroth-order efficiency reduces close to zero at the optimum value

of η−1. In view of Fig. 5.6(b), the value of d is not critical: we obtain

essentially the same optimum η−1 at the same value of w as above

for all carrier periods d considered in Fig. 5.6.

The dashed lines in Fig. 5.6 show results for TE polarized light

with θ = 42◦ in Fig. 5.6(a) and d = 340 nm in Fig. 5.6(b). Now the

maximum efficiency is limited to ∼ 50%, at least if we leave some

room for position modulation. The reason for the low efficiency is

again that TE polarized light cannot penetrate deep into the grooves

of metallic subwavelength-period gratings, and this prevents us

from achieving sufficient phase modulation for high diffraction effi-



0 100 200 3000

0.2

0.4

0.6

0.8

1

θ=30°

θ=42°

θ=60°

θ=42° (TE)

w [nm]

η−

1

(a)

0 100 200 3000

0.2

0.4

0.6

0.8

1

d=330 nm d=340 nmd=350 nm d=340 nm (TE)

w [nm]

η−

1

(b)

Figure 5.6: Efficiency of carrier order m = −1 for (a) different incident angles and (b) dif-

ferent carrier periods when we vary the ridge width w. The solid and dashed lines refer to

TM and TE polarization, respectively (adapted from Ref. [106]).

ciency. At the wavelength and period considered here, Al is a good

plasmonic material. However, now the optimized grating period is

too small for plasmon resonance to be excited, and this one reason

why the coding method works.

Figure 5.7 illustrates the time-averaged Poynting vector for TM

polarized light in the structure. The fill factor w/d ≈ 0.64 of the

ridge and the angle of incidence θ = 42◦ are fixed to show the field

amplitude and the direction of the energy flow (arrows). Unlike

in groove-width modulation at normal incidence, here we do not

observe any evidence of surface plasmon resonance.

5.2.2 Coding of V-ridge structures

Let us again refer to Fig. 5.5, in which the modulation of the carrier

is determined by the shifts Sn of the ridge tips with respect to their

undisturbed positions xn = (n − 1/2)d of the carrier grating. Using

the detour-phase principle [84] these shifts are determined by the

phase-only modulation function φ(x) to be encoded in the element.

Denoting the sampled values of φ(x) at xn by φn, we have

Sn =φn · d

2π. (5.2)

The above φ(x) represents the modulation function, which is de-

signed to convert an incident plane wave into a predefined angu-

lar spectrum of plane waves within some on-axis signal window



−100 0 100

−150

−100

−50

0

50

100

1500

0.02

0.04

0.06

0.08

0.1

z [nm]

x[n

m]

(a)

Figure 5.7: The direction of energy flow for Bragg’s incidence (shown in blue) and re-

flectance (shown in black arrows) in case of V ridge carrier grating given by the Poynting

vector for TM polarization of incident light. Here λ = 457 nm, w = 220 nm, and

d = 340 nm.

W [107]. After encoding, this W is shifted off-axis around the cho-

sen diffraction order of the carrier grating.

In the spirit of the position modulation scheme, φ(x) should

vary slowly relative to the carrier period in the sense that the con-

dition

|sn − sn+1| ≪ d (5.3)

holds for all n. If this condition holds, the adjacent ridges do not

overlap, except perhaps at points where the phase of φ(x) changes

abruptly. Such points occur inevitably if φ(x) is wrapped into the

[0, 2π] interval. However, in many designs one also encounters

phase jumps less than 2π radians, which cannot be unwrapped.

Nevertheless, we assume φ(x) to vary slowly enough to ensure that

these isolated phase jump points are sufficiently far apart to cause

only local disturbances in the coded profile.

In what follows we will consider periodic modulation functions

with period D, so that φ(x + D) = φ(x). We choose D to be an in-

teger multiple of the carrier period d, i.e., D = Nd as illustrated in

Fig. 5.5. Then the carrier order m = −1 splits into a discrete angular



spectrum of plane waves in the signal window W and correspond

to the diffraction orders of an on-axis element with phase function

φ(x). In order to satisfy the condition (5.3), the effective spread of

the angular spectrum within W must be small compared with the

diffraction angle of the carrier order m = −1. In other words, the

diffraction geometry should be parabasal. In the next section we

proceed to consider this case more quantitatively by giving numer-

ical examples.

5.2.3 Numerical examples

In this section we consider diffractive elements that divide the in-

cident plane wave into an array of Q plane waves with equal effi-

ciencies, propagating in different directions. Such elements are fre-

quently called beam splitters or array illuminators. We refer to the

index of the orders of the modulation grating by symbol q. Thereby

ηq denotes the diffraction efficiency of order q and the total effi-

ciency within the set W of signal orders is

η = ∑q∈W

ηq. (5.4)

Furthermore, the array uniformity error is defined as

E =max ηq − min ηq

max ηq + min ηq, (5.5)

with q ∈ W. Note that, after modulation of the carrier grating,

order q in the axial design corresponds to order q − N of the super-

periodic grating with period D.

With the help of FMM, we investigate the effect of N in the

performance of ridge-position modulated gratings. In all examples

considered here we take λ = 457 nm, θ = 42◦, and d = 340 nm. In

order to obtain good convergence, the number of diffraction orders

included in the FMM analysis is at least ∼ 30N. The bulk refractive

index of Al was used to model the metal substrate, assumed semi-

infinite as in Fig. 5.5.



A triplicator is an element with Q = 3 equal-efficiency orders.

The optimum continuous profile φ(x) of a triplicator is known an-

alytically [108] and has a total efficiency η = 0.92556 into orders

q = −1, 0,+1 contained in W. Figure 5.8 illustrates the discrete

N = 16 bit design of φ(x), obtained by IFTA, and the resulting

on-axis diffraction pattern calculated within the thin-element ap-

proximation. The efficiency is η ≈ 0.926 and the array uniformity is

virtually perfect. The rest of the incident light goes to higher orders

since amplitude freedom must be allowed in the design.

5 10 150

1

2

3

n

φn

[rad

]

(a)

−5 0 50

0.1

0.2

0.3

q

ηq

(b)

Figure 5.8: (a) The continuous phase profile φ(x) of the triplicator and its discrete form

with N = 16 bits, shown by the sampled dots and vertical lines. (b) The efficiencies ηq

of the central orders q in the paraxial domain predicted by the thin-element approximation

(adapted from Ref. [106]).

Similarly, the continuous φ(x) for the case Q = 5 is shown in

Fig. 5.9(a). This design has a total efficiency of η = 0.921 into orders

q = −2, . . . ,+2. The diffraction pattern including some higher or-

ders is shown in Fig. 5.9(b). The average uniformity error predicted

by IFTA is E = 9.945× 10−7. As third example, the continuous φ(x)

for Q = 8 is shown in Fig. 5.9(c) and produces a total efficiency of

η = 0.96512 into orders q = −3, . . . ,+4. The diffraction pattern

is shown in Fig. 5.9(d), and the uniformity error given by IFTA is

E = 6.12 × 10−7.

Let us next consider an encoded triplicator with N = 16 bits.

We assume λ = 457 nm, θ = 42◦, and choose the carrier period

as d = 340 nm. The result shown in Fig. 5.10(a) is based on FMM

analysis. On comparing this result with Fig. 5.8(b) we see that the

array is somewhat distorted and there is some residual light in the



10 20 300

1

2

3

4

5

n

φn

[rad

]

(a)

−5 0 50

0.05

0.1

0.15

q

ηq

(b)

10 20 30−4

−2

0

2

4

n

φn

[rad

]

(c)

−5 0 50

0.02

0.04

0.06

0.08

0.1

0.12

q

ηq

(d)

Figure 5.9: (a) The continuous form of the phase profile φ(x) of the Q = 5 array illumi-

nator and its discrete N = 32 bit form. (b) The efficiencies ηq of the central orders q in the

paraxial domain for the 5-beam case. Similar plots for Q = 8 are shown in (c) and (d).

central region of the pattern. This axial noise is due to order m = 0

of the carrier grating, which is also the zeroth generalized order of

the modulated grating. It may be concluded that, at N = 16, the

geometry is not yet sufficiently parabasal to satisfy the condition

(5.3), which explains the large uniformity error. The array unifor-

mity is indeed improved if we increase N, as shown in Figs. 5.10(c)

and 5.10(d). This behaviour is analogous to observations for binary

carrier gratings [27]. Table 5.1 illustrates the results of the FMM

analysis more quantitatively. The diffraction efficiencies of the en-

coded elements are close to the expected value ∼ 80.6%, which is

the efficiency of the axial design (92.6%) given by IFTA, multiplied

by the rigorously calculated efficiency (87%) of the carrier grating.

A similar analysis is performed also for array illuminators that

generate five beams in orders q = −2, . . . ,+2 and eight beams in

orders q = −3, . . . , 4. IFTA predicts η ≈ 0.92 in the 5-beam case

and η ≈ 0.96 in the 8-beam case. In the case of triplicator, φ(x)

was limited within the [0, 2π) interval. In the case Q = 8 this in-



−20 −15 −10 −5 00

0.05

0.1

0.15

0.2

0.25

q − N

ηq−

N

(a)

−18 −16 −140

0.05

0.1

0.15

0.2

0.25

q − N

ηq−

N

(b)

−34 −32 −300

0.05

0.1

0.15

0.2

0.25

q − N

ηq−

N

(c)

−66 −64 −620

0.05

0.1

0.15

0.2

0.25

q − N

ηq−

N

(d)

Figure 5.10: Rigorous analysis of encoded triplicators. (a) Efficiencies ηq of diffraction

orders of the N = 16 bit design. (b) A closer look around W. (c) Efficiencies of the

N = 32 bit design. (d) Efficiencies of the N = 64 bit design (adapted from Ref. [106]).

terval is exceeded but we first unwrapped φ(x), then shifted its

minimum to −π, and finally wrapped the profile again. In doing

so we minimize the number 2π phase jump to two. An unwrapped

profile contains one phase jump of π radians, which cannot be un-

wrapped. Tables 5.2 and 5.3 show quantitative results correspond-

ing to Table 5.1 for the Q = 5 and Q = 8 designs, respectively. As

expected, if we increase Q while keeping N constant, the uniformity

error increases. However, the uniformity again improves when N is

increased. We noticed that for the Q = 8 design, rewrapping φ(x)

somewhat unexpectedly improved the array uniformity.

5.2.4 Effects of varying the wavelength and angle of incidence

It is of substantial interest to examine the effects of the angle of inci-

dence of the input plane wave, as well as the wavelength sensitivity

of the designs at Bragg incidence. We concentrate here on the tripli-

cator case, but the other designs behave analogously. Figure 5.11(a)



Table 5.1: Rigorously calculated performance of the triplicator for different values of N,

showing the efficiencies ηq of the signal orders, the total diffraction efficiency η, the unifor-

mity error E, the number DO of orders included in the FMM analysis, and the superperiod

D of the modulated element (adapted from Ref. [106]).

Signal orders

N η1 η0 η−1 η E [%] DO D (µm)

16 0.271 0.256 0.256 0.783 3.0 1500 5.44

32 0.270 0.264 0.261 0.795 1.6 2500 10.88

64 0.268 0.265 0.262 0.795 1.1 3500 21.76

Table 5.2: The same as Table 5.1 but for 5-beam designs.

Signal orders

N η2 η1 η0 η−1 η−2 η E [%] DO D (µm)

32 0.148 0.167 0.158 0.141 0.165 0.779 8.4 2500 10.88

64 0.153 0.163 0.158 0.151 0.161 0.786 3.8 3500 21.76

128 0.149 0.153 0.150 0.147 0.152 0.751 2.0 4000 43.52

Table 5.3: The same as Table 5.1 but for 8-beam designs. The values of DO and D (not

shown) are the same as in Table 5.2.

Signal orders

N η4 η3 η2 η1 η0 η−1 η−2 η−3 η E[%]

32 0.103 0.109 0.090 0.108 0.082 0.109 0.098 0.082 0.782 14

64 0.101 0.105 0.102 0.103 0.092 0.106 0.101 0.095 0.806 6.8

128 0.094 0.096 0.100 0.093 0.096 0.100 0.100 0.097 0.776 3.6



shows the effect of varying the wavelength λ with a sampling inter-

val of 0.6 nm. The efficiency of the signal orders q = 1 (blue), q = 0

(green), and q = −1 (red) changes little over the spectral region

extending from 400 nm to 500 nm. At λ = 535 nm, order q = −1

vanishes because of a Rayleigh anomaly. At this point, of course,

the design principle breaks down and the efficiency of the zeroth

carrier order increases rapidly. On the other hand, angular toler-

ance of the triplicator with N = 16 is shown in Fig 5.11(b). Between

the incident range of 36◦ − 64◦, the signal orders are weakly depen-

dent implying huge angular tolerance for the blue light. However,

if the angular spread of W is reduced by considering large N, then

the tolerance might become even better.

400 450 500 5500

0.1

0.2

0.3

0.4

0th

−15th

−16th

−17th

λ [nm]

ηq−

N

(a)

40 45 50 55 600

0.1

0.2

0.3

0.4

−15th

−16th

−17th

0th

(b)η

q−N

θ [deg]

Figure 5.11: Simulation results for the pulse-position-modulated triplicator with N = 16.

Efficiencies of orders q − N = −17 (red), q − N = −16 (green), q − N = −15 (blue),

and q − N = 0 (black) as a function of (a) wavelength and (b) angle of incidence (adapted

from Ref. [106]).

5.2.5 Transmission-type ridge position modulation

It is also worth analyzing the behavior of transmission-type V-ridge

gratings. Here we keep the same geometry as in Fig. 5.5 but, in-

stead of Aluminium, we have COC as a grating material. The inci-

dent side is COC and the transmission side is air. Moving forward

with a similar approach as earlier, we aim to achieve a high effi-

ciency η−1 of carrier order m = −1. We consider thee different

wavelengths λ = 457 nm, λ = 520 nm and λ = 633 nm. To de-

sign the carrier grating, we first optimize d, θ, and w using a stan-



0 500 1000 15000

0.2

0.4

0.6

0.8

replacements

w [nm]

η−

1(a)

−19 −18 −17 −16 −15 −14 −130

0.05

0.1

0.15

0.2

0.25(b)

q − N

ηq−

N

0 500 1000 15000

0.2

0.4

0.6

0.8

w [nm]

η−

1

(c)

−18 −16 −140

0.05

0.1

0.15

0.2

(d)

q − N

ηq−

N

Figure 5.12: Theoretical results for transmission-type V-ridge carrier and position-

modulated gratings. (a) TM-mode efficiencies of carrier order m = −1 for different carrier

periods when the wedge width w is varied. (b) Rigorously calculated efficiencies ηq with

N = 16 bit for TM polarization. (c) Same as (a), but for TE polarization. (d) Same as (b),

but for TE polarization. Red: λ = 633 nm. Green: λ = 520 nm. Blue: λ = 457 nm.

dard procedure known as the unconstrained nonlinear optimiza-

tion algorithm. This leads to solutions with θ ≈ 27◦, d ≈ 3λ, and

w/d ≈ 0.67. Because of the relatively large value of d, the propaga-

tion direction of transmitted order m = −1 is ∼ 21◦. Hence the

transmission-type designs based on V-gratings differ fundamen-

tally from those based on binary gratings [26,27], in which the angle

of incidence must satisfy the Bragg condition for a high diffraction

efficiency to be achieved.

To study the solution obtained above, we scan w and evaluate

η−1 using FMM. As illustrated in Fig. 5.12(a), the maximum η−1 in

TM polarization is ≈ 63% for all wavelengths considered. We en-

code the triplicator modulation function φ(x) into the carrier grat-



ing in a similar way as in Sect. 5.2.2 and Sect. 5.2.3. Figure 5.12(b)

illustrates the results of FMM analysis of a modulated triplicator

with N = 16 bits for TM polarization. The array is somewhat dis-

torted for all values of λ, i.e., the efficiencies depend nearly linearly

on the order index within W. We have established that this distor-

tion, which arises because the design with N = 16 is not yet suf-

ficiently parabasal, can be compensated for by designing φ(x) by

IFTA in such a way that the paraxial design has opposite distortion.

Figure 5.12(c) shows the optimization results of the carrier grat-

ing in TE polarization, and now η−1 ≈ 67% is achieved with nearly

the same values of w and as in the TE case. Hence the designs also

work for unpolarized light. Finally, Fig. 5.12(d) illustrates the re-

sults of FMM analysis of the triplicator with N = 16 bits in the TE

case.

5.3 FABRICATION

Figure 5.13(a) illustrates the process flow for the fabrication of the

V-ridges into COC. For the demonstration we have fabricated a trip-

licator with N = 16. Initially a thermal oxide layer with 250 nm of

thickness was grown on a standard silicon 〈100〉 wafer. Above that a

200 nm thick electron beam resist layer (ZEP 7000) was spun on the

substrate. Then using a Vistec EBPG 5000+ ES electron beam pat-

tern generator at 100 keV the exposures are made on the resist layer

followed by reactive ion etching of the oxide layer shown in step (3).

In step (4), this oxide layer was then used as etching mask in silicon

wet etching using 30% KOH at 70◦C. This resulted in grooves with

α = 35.26◦ defined by the wet etching characteristics of 〈100〉 silicon

wafer and groove widths defined by the e-beam exposure. Finally,

in step (5), the thermal oxide was removed in buffered oxide etch.

The fabricated silicon master element was utilized in hot-embossing

of COC using an Obducat Eitre 3 nanoimprinter. As a last step, the

developed triangular ridges were coated with h = 70 nm aluminum

layer using sputtering, resulting in a type of profile illustrated in

Fig. 5.14.



1)

2)

3)

4)

5)

6)

ZEPSiO2

Si

COC

AlCOC

(a)(b)

Figure 5.13: (a) Process flow showing the formation of anisotropic etch profiles, nano-

imprint lithography and metal deposition. (b) An SEM image of the V-grooves in silicon

(inset) and the COC-grooves with aluminium (adapted from Ref. [106]).

x

z

d

h

θ air

AlCOC

m = −1

w

Figure 5.14: A schematic picture of the aluminium coated carrier grating (adapted from

Ref. [106]).

5.4 EXPERIMENTAL RESULTS

Here we discuss the experimental setup and the experimental re-

sults on the fabricated triplicator element. The setup shown in

Fig. 5.15 has two parts, one for aligning the tilt orientation of the

beam using mirrors M1 and M2, and the other is for grating mea-

surements. In the first part we separate the input laser beam from

the array of reflected beams to permit all measurements. As a

source we use a diode pumped solid state laser operating at central

wavelength λ = 457 nm and having beam width of 3 mm, with a

maximum output power of 29 mW. Horizontally polarized light is



incident on the grating having size 4 mm × 5 mm after successive

reflections from the mirrors.

In this section the experimental results were compared with

FMM simulations. We proceed to measure the fabricated triplicator

at λ = 457 nm, incident at an angle of θ = 38◦. In the numerical

simulation the finite metal layer thickness is taken into an account.

Further, in modeling the complex refractive index of the thin metal-

lic films, we used the data for the sputtered aluminium from the

Ref. [109]. The diffraction patterns are scanned along the circular

periphery shown in Fig. 5.15 (dotted lines), and we recorded the

efficiencies of orders q − N = −20 . . . , 5. Fig. 5.16(a) shows the

comparison of the experimental and theoretical results. The mea-

sured efficiencies (red bars) of the signal orders were 23.3%, 23.6%

and 25%, resulting in a uniformity error E = 4%. whereas, the sim-

ulated values shows E = 3% for an ideal grating profile. This shows

a remarkable agreement between the simulation and the measured

values.

To proceed further, we compared the measured efficiencies (stars)

with the simulated (solid lines) signal orders as a function of the in-

x

z

y

Source

M1M2

Power meter

grating

Rotatable mount

Figure 5.15: Schematic diagram of the experimental setup of the alignment mirrors and

the grating measurement arrangement (adapted from Ref. [106]).



cident angle as shown in Fig. 5.16(b). Here also we achieved good

correlation between the theoretical and the experimental findings.

The zeroth order reflection in the measurement is just ∼ 2% higher

than the theoretical prediction whereas the variation in signal order

efficiencies within the incident-angle range of 36◦ to 64◦ being only

1.5%.

−20 −15 −10 −5 0 50

0.05

0.1

0.15

0.2

0.25

0.3

SimulatedExperimental

q

ηq−

N

(a)

40 45 50 55 600

0.05

0.1

0.15

0.2

0.25

0.3

θ [deg]

ηq−

N

(b)

Figure 5.16: Comparison of the theoretical (solid lines) and experimental (dots) results for a

triplicator. (a) Simulated (blue) and measured (red) diffraction efficiencies for λ = 457 nm

and θ = 38◦. (b) Angular dependence of the diffraction efficiencies for λ = 457 nm and

diffraction orders q − N = −17 (red), q − N = −16 (green), q − N = −15 (blue), and

q − N = 0 (black) (adapted from Ref. [106]).

Finally, Table 5.4 shows the quantitative comparative results at

wavelengths λ = 406, 457, 520, and 633 nm at θ = 42◦. The theoreti-

cal and experimental results are again in good agreement. Note that

the design fails at λ = 633 nm since at this wavelength all signal or-



Table 5.4: Comparison of simulated (sim.) and experimental efficiencies (meas.) of the

triplicator at different wavelengths (adapted from Ref. [106]).

λ [nm] q − N sim. meas.

406 −17 0.185 0.165

−16 0.197 0.190

−15 0.232 0.231

0 0.011 0.041

457 −17 0.250 0.231

−16 0.237 0.240

−15 0.240 0.250

0 0.005 0.021

520 −17 0.268 0.250

−16 0.243 0.240

−15 0.220 0.205

0 0.001 0.013

633 −17 0 0

−16 0 0

−15 0 0

0 0.596 0.610

ders become evanescent. Nevertheless, the measured zeroth-order

efficiency is in good agreement with theory.

5.5 SUMMARY

In this chapter we introduced and analyzed several schemes for

encoding arbitrary phase functions in the form of modulated V-

gratings. Some of the schemes failed, but those based on ridge-

position modulation proved remarkably successful. We analyzed in

detail reflection-type position-modulated diffractive elements based

on high-frequency V-ridge carrier gratings. We demonstrated the

position modulation scheme by making a triplicator to work on

the visible wavelength. The triplicator was fabricated using wet

etching of silicon and nanoimprint lithography followed by metal



deposition. The precise positioning of the grooves in the master

element was done using Electron beam lithography, which on the

other hand ensures large scale production of the diffractive struc-

tures. Remarkable angular tolerance between input range 36◦ − 64◦

and spectral tolerance between 406− 520 nm for the designed wave-

length λ = 457 nm were shown. Hence this technique could pro-

vide an alliance between high angular/spectral tolerance and large

scale production with the simplicity of the structure. We also estab-

lished the potential of transmission-type V-ridge gratings and their

experimental demonstration is one subject for future work.


6 V-ridge gratings:

transition from antireflection

to retroreflection

Optical losses from the interfaces of optical components are a ma-

jor issue. Thus several approaches, based on multilayer dielectric

coatings and subwavelength-structured surfaces, exist for suppres-

sion of surface reflectance from visible wavelengths to the infrared

region [51, 71, 96, 110–112]. A general interest lies on those struc-

tures where, in addition to low reflectance, also other beneficial op-

tical characteristics are present. These include, for example, low re-

flectance with highly hydrophobic and oleophobic properties [113],

low reflectance along with color perception of the surface in some

display applications [51, 114], and along with wide-angle illumina-

tion applications [115]. On the other hand, retroreflecting struc-

tures with microscopic and macroscopic dimensions are crucial in

laser communication systems [116], accurate measurement of dis-

tances [117, 118], cavity mirrors [119, 120], and safety devices in-

cluding road signs [121, 122]. In the past, design and applications

of retroreflectors have been studied widely [121–124]. Polarization

properties of retroreflectors have also been investigated [125, 126].

In this chapter we introduce a surface consisting of periodi-

cally arranged V-ridges, showing a transition from antireflection

to retroreflection when the dimensions of the V-ridge increase from

the subwavelength to the superwavelength domain. The gratings

are analyzed by FMM and demonstrated by wet etching of silicon

and subsequent replication in polymer using nanoimprint lithogra-

phy.



6.1 GEOMETRY AND PRINCIPLE

Let us start by reconsidering Fig. 3.2. The modulated region II (grat-

ing) separates the homogeneous dielectric regions I (COC substrate)

and III (air) with refractive indices n1 = 1.52 and n3 = 1, respec-

tively. The modulated region consists of symmetric V-ridges (made

of COC) with apex half-angle α = 45◦ and width w, separated by

flat surface sections of width d − w. As before, the wedge region is

divided into J layers. Our main interest is in the zero-order trans-

mittance and reflectance of V-grating with different values of the ra-

tio d/λ for a normally incident plane wave arriving from substrate

side to the grating. In the subwavelength domain (d < λ/n1), only

the zeroth transmitted and reflected orders propagate and all others

are evanescent. In our case we describe this region as an antireflec-

tion region, and here the effective-index approximation yields good

results provided that d ≪ λ. When the grating period is of the order

of the wavelength, one speaks about the resonance domain. Here

the effective-index approximation is no more valid, the polariza-

tion sensitivity is significant [106], and the zeroth order reflectance

and transmittance change rapidly with structural parameters since

higher orders are also present. With increasing values of d/λ, up to

10, higher orders have significant efficiencies and thus the structure

still needs to be considered as a grating. Finally, sufficiently far in

the superwavelength domain (d ≫ λ), the geometry can be well

described by geometrical optics. Now the incident plane wave can

be described in term of rays, which are totally reflected by ridge

facets and turned back in the direction of the incident light. We re-

fer to this domain region as the retroreflection region. The behavior

of perfectly retroreflectors in the superwavelength region has been

studied by Ichikawa [126].

6.2 NUMERICAL SIMULATIONS

In the following examples we employ FMM to investigate the effect

of varying d while keeping λ fixed, and vice versa, on the zeroth


CHAPTER 6. V-RIDGE GRATINGS: FROM ANTIREFLECTION TORETROREFLECTION

order transmittance and reflectance. We further investigate the ef-

fect of adding a substrate of finite thickness. Finally, we examine

the effect of varying the width d − w of the flat area.

6.2.1 Effect of varying the period

In what follows, we scan d from 0.4λ0 to 22λ0, considering a fixed

design wavelength λ0 = 457 nm and keeping d − w fixed at the

value of 50 nm dictated by the fabrication constraints as discussed

above. Note again that, with the change in d, the height h of the

grating varies according to the relation

h =d − w

2. (6.1)

The effect of such d-scanning is shown in Fig. 6.1 for the two dif-

ferent polarization states of light. The entire curve can be subdi-

vided into three parts, as anticipated in the discussion presented

above. In the subwavelength domain the grating transmits ∼ 97%

of light as no higher diffraction orders exist, and the rest of the en-

ergy goes to the zeroth reflected order. In the resonance domain

the efficiency varies rapidly, as expected, since more and more re-

flected and transmitted orders emerge as the period increases. Nev-

ertheless, the transition from high transmittance to high reflectance

begins to take place in this domain for both TE and TM polarized

illumination. Fast oscillations in the efficiency curves are seen, and

it is hard to predict the behavior of the light in this region by any

heuristic arguments. At the largest values of d considered here, the

zero-order reflectance is on average ∼ 85% for TE polarized inci-

dent light and ∼ 79% for TM polarized light.

Figure 6.2 illustrates the electric-field amplitude |Ey(x, z)| calcu-

lated by FMM within the structure for a normally incident plane

wave. As expected, highly structured diffraction patterns are seen

within the scatterer. Note that, in order to see the reflected ze-

roth order on the substrate side of the grating (z < 0), we have

subtracted the incident field before computing the amplitude dis-

tribution. High transmission is evident for small periods, as shown



0.5 1 2 3 5 10 200

0.2

0.4

0.6

0.8

1

d/λ0

η0

Figure 6.1: Efficiency of a V-ridge grating as a function of the period d for a fixed wave-

length λ. Black lines: zero-order transmittance. Blue lines: zero-order reflectance. Solid

lines: TE polarization. Dashed lines: TM polarization. Note the logarithmic horizontal

scale.

in Fig. 6.2(a). For larger periods, the transmittance decreases grad-

ually and light starts reflecting from both facets of the scatterer, cre-

ating a regular interference pattern inside the ridge. Figure 6.2(d)

shows the high reflected field amplitude in the input medium.

6.2.2 Effect of varying the wavelength

In anticipation of an experimental verification to be presented in

Sect. 6.3, we scan λ over a range 400 nm < λ < 800 nm for several

fixed gratings periods d = 400 nm, d = 2000 nm, d = 5000 nm, and

d = 10000 nm. Figure 6.3 shows the results for TE and TM polar-

ization, and we again fix d − w at 50 nm. The results are plotted at

a sampling interval of 2 nm and the variation of the refractive index

of COC over the broad spectral range is taken into account.

Next we proceed to analyze the rapid fluctuations that are seen

in the efficiencies when the ratio of d/λ is large. In Fig. 6.4(a) we

vary the period from 5 µm to 10 µm at λ0 = 457 nm for both

polarization states of light. A close observation of those variations



0 200 400

−400

−200

0

200

400

600

0.4

0.6

0.8

1

x [nm]

z[n

m]

(a)

0 1000 2000−1000

−500

0

500

1000

1500

0.4

0.6

0.8

1

x [nm]

z[n

m]

(b)

0 2000 4000−2000

−1000

0

1000

2000

0.4

0.6

0.8

1

x [nm]

z[n

m]

(c)

0 5000 10000

−2000

0

2000

4000

0.4

0.6

0.8

1

x [nm]

z[n

m]

(d)

Figure 6.2: Distribution of the electric-field amplitude inside the structure for several

periods. (a) d = 500 nm, (b) d = 2000 nm, (c) d = 5000 nm, (d) d = 10000 nm. Here

λ0 = 457 nm. Note that the horizontal and vertical scales not the same in all subfigures.

For smaller periods the output region (towards positive z-direction) are kept long showing

transmitted field, whereas for bigger periods the input region (negative z-direction) are

kept long showing reflected field. White indicates high and black indicates low amplitude.

shows a periodicity in the pattern with a period of λ/n1. Earlier,

in Fig. 6.2(d), we observed that in the grating region light interferes

after TIR from both the facets of the grating. Similarly, Fig. 6.4(b)

shows the same as Fig. 6.4(a), but here we scan the wavelength from

400 nm to 800 nm over the fixed period of 10 µm.

Although not perfectly regular, the observed fluctuations can be

explained qualitatively by a geometrical argument. The optical path

of a retroreflected geometrical ray inside the ridge is w irrespective

of the point of incidence. Hence the phase experienced by the ray

is φ = 2πn1w/λ. A phase range of 2π radians is covered when w

changes by λ/n1. The reason why the pattern is somewhat irregu-



0.5 1 2 3 5 10 200

0.2

0.4

0.6

0.8

1

400 nm

10000 nm

5000 nm

2000 nm

d/λ

η0

Figure 6.3: Transmittance and reflectance as a function of wavelength for different periods.

Black lines: transmittance. Blue lines: reflectance. Solid lines: TE polarization. Dashed

lines: TM polarization.

12 14 16 18 200.5

0.6

0.7

0.8

0.9

(a)

d/λ0

η

14 16 18 20 220.5

0.6

0.7

0.8

0.9

(b)

d/λ

η

Figure 6.4: A closer look at the fluctuations. (a) Reflectance as a function of period.

(b) Reflectance as a function of wavelength. The solid lines refers to TE polarized light and

the dashed line refers to TM polarized light.

lar is that many higher diffraction orders are present, though their

efficiencies are low in the superwavelength domain.

6.2.3 Effect of finite substrate thickness

In this section we analyze the effect of adding the finite substrate

of thickness ∼ 0.3 mm on the zeroth order reflectance. So now,

instead of incidence from a semi-infinite COC region, we have in-



cidence from air into the COC substrate and then into the ridge

structure. This is the physical situation in the experiments to be de-

scribed below. Fluctuations due to interference effects are expected

when the substrate thickness is changed slightly, and to show a few

oscillation we cover a thickness range of one wavelength in air. The

fluctuations are quite strong. Nevertheless, we may take the aver-

age reflectance in the case of finite substrate thickness to the results

for a semi-infinite input medium. Figure 6.5(a) shows the average

reflectance (red line) of the zeroth order (∼ 74%) for the TE po-

larized incident light, as compared to the reflectance of ∼ 73% in

semi-infinite input medium (COC). Similarly, Fig. 6.5(b) shows the

results for TM polarized light. In this case the average zeroth-order

reflectance is ∼ 65%, while the reflectance in the semi-infinite case

is ∼ 63%. Thus reasonable agreement is obtained and we may con-

clude that the effect of finite substrate thickness is probably small

compared to fabrication and measurement errors.

0 100 200 300 4000.5

0.6

0.7

0.8

0.9

Substrate thickness [nm]

ηre

f

(a)

0 100 200 300 4000.4

0.5

0.6

0.7

0.8

Substrate thickness [nm]

ηre

f

(b)

Figure 6.5: Reflectance as a function of varying the thickness of the substrate in (a) TE

polarization and (b) TM polarization. The solid red lines show the averages over one λ.

6.2.4 Effect of varying flat bottom width

In a view of fabrication constraints (a mask is needed in anisotropic

wet etching of silicon) we need a narrow flat area between adjacent

ridges. In practice, it is difficult to keep the ratio (d − w)/d below

10%, and for small periods the lower bound is d − w ∼ 50 nm as

mentioned in several instances above. Figure 6.6(a) shows a plot of



1000 1500 20000.4

0.5

0.6

0.7

0.8

d − w [nm]

ηre

f(a)

1000 1500 20000.45

0.5

0.55

0.6

0.65

0.7

d − w [nm]

ηre

f

(b)

Figure 6.6: Reflectance as a function of varying the flat bottom area d − w for (a) TE

polarization and (b) TM polarization.

the zeroth-order reflectance as a function of d − w. Here we fix the

period at d = 10 µm and vary d−w over a large range from 600 nm

to 2000 nm. The reflectance decreases because d is now fixed and

hence the net area of the V ridges decreases with increasing d − w.

Figure 6.6(b) illustrates the same for TM polarized incident light.

On the other hand, oscillations with period λ/n1 result due to the

interference as discussed in Sect. 6.2.

6.3 EXPERIMENTAL RESULTS

We fabricated and measured gratings with four different periods.

The fabrication steps are essentially the same as in the process for

position-modulated ridge gratings described in Sect. 5.3. However,

the electron beam exposure is now made on a 〈100〉 wafer at an an-

gle of 45◦, compared to the primary 〈110〉 flat. For the wet etching

process we used 10% KOH (instead of 30% KOH) at a temperature

of 70◦C. The solution needs to be saturated with IPA to make the

surface tension lower during the wet etching process [90]. This pro-

cess allows us to obtain sidewalls oriented at 〈110〉 direction with a

slope angle of 45◦ compared to the surface normal. A top view of

the fabricated sample with the largest period is shown in Fig. 6.7(a).

It is clear from the top-view that the value d − w is ∼ 1.010 µm.

Consider next the experimental setup shown in Fig. 6.8 and the



1.010 µm

(a) (b)

Figure 6.7: SEM images of a grating with d = 10 µm. (a) Top view. (b) Cross section.

Courtesy of Ismo Vartiainen

xy

z

S2

S1

Pin hole

Grating

Integrating sphere

Mirror

Reflection standard

Holder mount

Spectrophotometer

Figure 6.8: Schematic diagram showing the working principle of spectrophotometer.

corresponding measurements. The setup has two parts, one for the

reference beam measurement from S1 and other for grating mea-

surements from S2. In the first part we scale the reference light

coming from S1 with the incident light coming from S2; the second

part is further divided into two sub-parts depending on whether we

measure transmittance (direct measurement) or reflectance (indirect

measurement). In the transmittance measurement we keep the grat-



0.5 1 2 3 5 10 200

0.2

0.4

0.6

0.8

1

400 nm

2000 nm

5000 nm

10000 nm

d/λ

η0

Figure 6.9: Zeroth-order efficiency as a function of wavelength for several fixed periods.

Numerical results: black line is transmittance and blue line is reflectance. Experimental

results: Red line is transmittance and green line is reflectance.

ing in front of the source S2 and allow only the transmitted zeroth

order to enter the integrating sphere. The other higher orders are

blocked by the card, preventing them from entering the integrat-

ing sphere. Measurement of reflectance is an example of an indi-

rect measurement. For this we hang the grating using the holder

mount so that the grating is located inside the integrating sphere

and measure everything except the zeroth order, which comes out

of the sphere in the direction of incoming light. Subtracting the

measured transmittance from the 100%, we obtain the values of the

retro-reflected zeroth order. Throughout the experiment we used

unpolarized light at the sampling interval of 2 nm from 400 nm to

800 nm.

The experimental results are compared with FMM simulations.

Note that Fresnel reflections at the bottom of the substrate have

been taken into account for comparison with the theoretical simu-

lation. Both in modeling and in fabrication the flat bottom width

d−w for d = 400 nm is considered as 50 nm. For the higher periods

such as d = 2000 nm, d = 5000 nm and d = 10 µm, d − w is taken

as 10% of the period. By scanning the wavelength over the fixed

periods with a spectrophotometer we determined the transmitted



and reflected zeroth-order efficiencies, shown in Fig. 6.9 together

with the simulation results. The measured efficiency of the zeroth

transmitted orders (red) is in very good agreement with the sim-

ulated results. The measured efficiencies of the zeroth reflected

order (green) are in good agreement with the simulated results as

the measurement on is an average only ∼ 6% lower than the theo-

retical prediction for d = 10 µm, and ∼ 4% lower for d = 2000 nm

and d = 5000 nm. There is an overall 3% − 5% variation in the

measured signal due to noise.

6.4 SUMMARY

In this chapter we have discussed geometrical retroreflection from

V-ridge gratings, which is obviously based on total internal reflec-

tion. This functionality is unaltered as long as the grating period

is sufficiently large compared to the wavelength of light. Our rig-

orous FMM analysis shows that, as the period becomes smaller,

diffraction effects become apparent. At subwavelength periods they

again disappear and the gratings behave as antireflection layers,

thus exhibiting properties that are completely opposite than in the

retroreflection domain. A gradual transition from antireflectance to

retroreflectance is clearly seen when the ratio of the grating period

and the wavelength increases.

We demonstrated the above-described elements by fabricating

gratings with four different periods using wet etching of silicon

and nanoimprint lithography. These gratings were characterized at

visible and near-infrared wavelengths. As in the previous chapter,

electron beam lithography is only required for master fabrication,

where the writing is now done along the 〈010〉 crystal orientation

with high precision. Good agreement between the numerical simu-

lations and the experimental results was achieved over the 400 nm

to 800 nm wavelength range considered.




7 Conclusions and outlook

In this thesis the principles of electromagnetic theory of light have

been presented. The Fourier modal method has also been discussed

in detail, since the rigorous numerical studies presented in this the-

sis are based on it.

The geometrical-optics-based local-plane-interface approach was

extended to model the near-field interactions of an evanescent field

with a dielectric wedge. Particularly, we have considered analyti-

cally, the projection of the incident evanescent field from the near-

field to the exit-plane of the wedge scatterer (with wavelength-scale

dimensions) and compared the evaluated radiant intensities with

rigorously calculated diffraction efficiencies. It was noticed that

the edge-diffracted wave from the apex of the scatterer, which was

not included in the geometrical model, disturbs the intensity pro-

file at the exit-plane of the wedge. However, if the exit plane of

the wedge is connected to a step-index waveguide, the effect of the

edge-diffracted wave becomes insignificant and we may compute

the coupling efficiency into the planar waveguide using the overlap-

integral method. We compared the intensities evaluated from the

quasi-geometrical approach with the modal efficiencies obtained

rigorously with reasonable good results. We further discovered

that the possibilities of using the geometrical approach in model-

ing the electromagnetic field is not limited to evanescent waves, but

also allow the analysis of the interaction of the wedge with surface

plasmons.

In view of the results obtained in Sect. 5.1.2, we have concluded

that the groove-width modulation scheme for coding high-carrier-

frequency diffractive elements did not work well. Groove width

modulation along with simultaneous height modulation causes an

insufficient delay in phase (by only few radians), which severely

restricts the applicability of this coding scheme. In striking con-

trast, coding in the form of ridge position modulation worked well.



Particularly, the coding scheme based on position modulation and

reflection geometry was demonstrated by fabricating a triplicator

for visible light. The fabricated grating was shown to posses a re-

markable angular and spectral operation range. The uniformity

error of the three beams of the triplicator was measured to be 4%,

which corresponds well to the theoretically calculated 3% error for

the blue laser, when measured at Bragg angle.

The transition of V-ridge gratings from subwavelength to super-

wavelength domain via the resonance-domain was studied by nu-

merical examples and an experimental verification was provided.

The side facets of the fabricated gratings were inclined at 45◦ com-

pared to the surface normal, which was achieved by using precise

mask alignment in the anisotropic wet etching process. The fab-

ricated subwavelength gratings were measured, and the observed

zeroth-order transmittance was, on average, only ∼ 5% lower than

the theoretical prediction. On the other hand, superwavelength

gratings were also fabricated and characterized, and the measured

zeroth order reflectances were, on average, ∼ 6% lower than the

theoretical predictions.

The results presented in this thesis suggest several possible ex-

tensions. For example, the inclusion of the boundary diffraction

emanating from the wedge apex appears possible. In future, the

correct implementation of this edge-diffracted wave could lead to

the better correspondence between the geometrical model and rig-

orous results even for wavelength-scale wedges in the near fields of

plane interfaces or scatterers. In view of Sect. 5.1.2, if we could

make a set of nanogrooves narrower than the metal skin depth

by forming a subwavelength grating on the metal surface, then in

TM polarized illumination, surface plasmon excitations along the

walls [similar to Fig. 5.4 (a)] may yield absorption peaks (dips in

reflectance). These could perhaps be utilized in plasmonic light

trapping on a metal surface whose absorption could be higher than

that of plain metals. However, fabrication of such nano V-grooves

in metals is a challenging task.


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ISBN 978-952-61-2330-1ISSN 1798-5668


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GAURAV BOSE

DIFFRACTIVE OPTICS BASED ON V-SHAPED STRUCTURES AND ITS APPLICATIONS


This book provides a survey of V-shaped diffractive structures and its applications.

Both exact and approximate methods are used to model the near-field interaction with the

wavelength scale scatterers. Several techniques to encode phase function in the form of modulating the structures are discussed

followed by the experimental demonstration of a triplicator. Further, the transition from

anti-to retro-reflection is demonstrated experimentally by testing several gratings with

different periods.

GAURAV BOSE

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