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Sensitivity Analysis of Narrow Band Photonic-

Crystal Waveguides and Filters

Ben Z. SteinbergAmir Boag

Ronen LisitsinSvetlana Bushmakin

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Presentation Outline

•Coupled Cavity Waveguide (CCW) (and micro-

cavities)

Filters/routers and waveguides – Optical comm.

Typical length-scale << λ (approaches today’s Fab

accuracy)

•Sensitivity analysis: (Random Structure inaccuracy)

Micro-Cavity

CCW

•Coupling (matching) to outer world

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The coupled micro-cavity photonic waveguide

Goals:

• Create photonic crystal waveguide with pre-scribed:

Narrow bandwidth

Center frequency

Applications:

• Optical/Microwave routing devices

• Wavelength Division Multiplexing components

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The Micro-Cavity Array Waveguides

a1

a2

Intercavity vector:

i ii

mb a

b

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The Single Micro-Cavity

Localized Fields Line Spectrum at ( , )o oE H 0

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Weak Coupling Perturbation Theory

A propagation modal solution of the form:

( ) ( )n nn

A

H r H r ( ) ( )n o n H r H r bwhere

( )oH r - The single cavity modal field

2 ,,

,

H H

c H H

Insert into the variational formulation:

0

1

( )

r

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The result is a shift invariant equation for :

2 2

0 0,m m m nm

h Ac c

mA

Where:

It has a solution of the form:

jkmmA e

00

, ,m m kk

H H

/k b - Wavenumber along cavity array

n

0,m mh H H

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Variational Solution

M

/|a1|/|b|

k

c

M

coso s k

s

c o

1 exp( )b

Wide spacing limit:

Bandwidth:

The isolated micro-cavity resonance

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Transmission & Bandwidh

Transmission vs. wavelength

Bandwidth vs. cavity spacing

Isolated micro-cavity resonance

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Varying a defect parameter tuning of the cavity resonance

Micro-Cavity Center Frequency Tuning

o

Example: Varying posts radius(nearest neighbors only, identically)

Transmission vs. radius

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Interested in: vs.

Then (can show for ! )

Cavity Perturbation Theory

2

0 0 0 ,H c H

2

0 0 0 ,H c H

0 0 0

0 002

0 0 0

,

2

E E

H

0 0 0H H H= + ,

- Perfect micro-cavity

- Perturbed micro-cavity

1

1

.

0H O 0

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Example: 2D crystal, with uncorrelated random variation - all

posts in the crystal are varied

Random Structure Inaccuracy

ia

Model: Treat radii variations as perturbations of the reference

cavity.

In a single realization different posts can have different radii.

Cavity perturbation theory gives:

Due to localization of cavity modes – summation can be restricted to closest neighbors

2( )10 0 02

1

( 1) ir i

i

N

oa a

E

1/ 2 1/ 222 2( )1

0 0 021

( 1) ir

N

o ii

a a

E

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Standard Deviation of Resonant Wavelength

• Perturbation theory:

Summation over 6 nearest neighbors

• Statistics results:

Exact numerical results of 40 realizations

All posts in the crystal are RANDOMLY varied

Hexagonal lattice, a=4, r=0.6, =8.41. Cavity: post removal. Resonance =9.06

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CCW with Random Structure Inaccuracy

Mathematical model is based on the physical observations:

1. The microcavities are weakly coupled.

2. The resonance frequency of the i -th microcavity is

where is a variable with the properties studied before.

3. Since depends essentially on the perturbations of the i

-th microcavity closest neighbors, can be

considered as independent for i ≠ j.

0 ,i

ii

i j

0( ) ( ),mm

m

H r A H r Modal field of the (isolated)

m –th microcavity.

Its resonance is 0 .m m

0 ( )mH r

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An equation for the coefficients:

2 2

0, , 0,m n m n m

m

h Ac c

, 0 0 0, 2 ,m nm n k n

k n

H H

, 02 ,m n m n n

Where:

In the limit we obtain 0 0mH H

Random inaccuracy has no effect if 1 0 perfect CCWn n

CanonicalIndependent of specific design

parameters

n

, 0 0,m nm nh H H

,m n m nh h

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Matrix Representation

220 ,A H A

Eigenvalue problem:

- a tridiagonal matrix of the form:

0 1

1 0 1

1 0 1

1 0

0

0

0

0

220 0diag[ ] 2 diag[ ]n kc H

220 0 12 cos ( 1)n nH n N

2 2 2 20c

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Numerical Results – CCW with 7 cavities

n of perturbed microcavities

n of perturbed microcavities

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Matching a CCW to Free Space

Matching Post

R

d

, cavit

perfect match n

y

i g1SWR max

11i

i jj

E

E

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SWR minimization results

0.1

0.05

Crystal ends here

Hexagonal lattice

a=4, r=0.6, =8.41.

Cavity: post removal.

m=2

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Field Structure @ Optimum (r=1.2)

Crystal Matching PostAt 1st optimum

Matching PostAt 2nd optimum

Matching PostAt 3rd optimum

Radiation field is not well

collimated.

Solutions:

• 2D optimization with more than a

single post

• Collect by a lens

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Summary

Cavity Perturbation Theory – effect on the isolated single

cavity

Linear relation between noise strength and frequency shift.

Weak Coupling Theory + above results – effect on the CCW

A novel threshold behavior : noise affects CCW only if it

exceeds certain level.

Matching to free space.

Sensitivity of micro-cavities and CCWs to random

inaccuracy :