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Coupled Cavity Waveguides in Photonic Crystals:

Sensitivity Analysis, Discontinuities, and Matching

(and an application…) Ben Z. SteinbergAmir Boag

Adi ShamirOrli Hershkoviz Mark Perlson

A seminar given by Prof. Steinberg at Lund University, Sept. 2005

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Presentation Outline

• The CCW – brief overview

• Disorder (non-uniformity, randomness) Sensitivity analysis [1] : Micro-Cavity

CCW

• Matching to Free Space [2]

• Discontinuity between CCWs [3]

• Application: Sagnac Effect: All Optical Photonic Crystal Gyroscope [4]

[1] Steinberg, Boag, Lisitsin, “Sensitivity Analysis…”, JOSA A 20, 138 (2003)

[2] Steinberg, Boag, Hershkoviz, “Substructuring Approach to Optimization of Matching…”, JOSA A,

submitted

[3] Steinberg, Boag, “Propagation in PhC CCW with Property Discontinuity…”, JOSA B , submitted

[4] B.Z. Steinberg, ‘’Rotating Photonic Crystals:…”, Phys. Rev. E, May 31 2005

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The Coupled Cavity Waveguide (CCW)

A CCW (Known also as CROW):

• A Photonic Crystal waveguide with pre-scribed:

Center frequency

Narrow bandwidth

Extremely slow group velocity

Applications:

• Optical/Microwave routing or filtering devices

• Optical delay lines

• Parametric Optics

• Sensors (Rotation)

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Regular Photonic Crystal Waveguides

Large transmission bandwidth (in filtering/routing applications, required relative BW ) 310

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The Coupled Cavity Waveguide

a1

a2

Inter-cavity spacing vector:

b

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The Single Micro-Cavity

Localized Fields Line Spectrum at

Micro-Cavity geometry Micro-Cavity E-Field

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Widely spaced Micro-Cavities

Large inter-cavity spacing preserves localized fields

m1=2

m1=3

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Bandwidth of Micro-Cavity Waveguides

Transmission vs. wavelength Transmission bandwidth vs. inter-cavity spacing

Inter-cavity coupling via tunneling:

Large inter-cavity spacing weak coupling narrow bandwidth

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Tight Binding Theory

A propagation modal solution of the form:

where

Insert into the variational formulation:

The single cavity modal field resonates at frequency

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Where:

Tight Binding Theory (Cont.)

The result is a shift invariant equation for :

It has a solution of the form:

- Wave-number along cavity array

The operator , restricted to the k-th defect

Infinite Band-Diagonally dominant matrix equation:

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Variational Solution

kM

/|a1|/|b|

c

M

Wide spacing limit:

Bandwidth:

Central frequency – by the local defect nature; Bandwidth – by the inter cavity spacing.

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Center Frequency Tuning

Recall that:

Approach: Varying a defect parameter tuning of the cavity resonance

Example: Tuning by varying posts’ radius(nearest neighbors only)

Transmission vs. radius

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Structure Variation and Disorder:Cavity Perturbation + Tight Binding Theories

- Perfect micro-cavity

- Perturbed micro-cavity

Interested in:

Then (for small )

For radius variations

Modes of the unperturbed structure

[1] Steinberg, Boag, Lisitsin, “Sensitivity Analysis…”, JOSA A 20, 138

(2003)

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Disorder I: Single Cavity case

• Cavity perturbation theory gives:

Uncorrelated random variation - all posts in the crystal are varied

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

Variance of Resonant

Wavelength

• Perturbation theory:

Summation over 6 nearest neighbors

• Statistics results:

Exact numerical results of 40 realizations

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Disorder & Structure variation II: The CCW case

Mathematical model is based on the physical observations:

1. The micro-cavities are weakly coupled.

2. Cavity perturbation theory tells us that effect of disorder is local

(since it is weighted by the localized field ) therefore:

The resonance frequency of the -th microcavity is

where is a variable with the properties studied before.

Since depends essentially on the perturbations of the -th

microcavity closest neighbors, can be considered as

independent for .

3. Thus: tight binding theory can still be applied, with some

generalizations Modal field of the (isolated)

–th microcavity.

Its resonance is

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An equation for the coefficients

• Difference equation:

• In the limit (consistent with cavity perturbation theory)

Unperturbed system Manifestation of structure disorder

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Matrix Representation

Eigenvalue problem for the general heterogeneous CCW (Random or deterministic):

-a tridiagonal matrix of the previous form:

-And:

From Spectral Radius considerations :

CanonicalIndependent of specific

design/disorder parameters

Random inaccuracy has no effect if:

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Numerical Results – CCW with 7 cavities

n of perturbed microcavities

n of perturbed microcavities

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Sensitivity to structural variation & disorder

In the single micro-cavity the frequency standard deviation is proportional to geometry / standard deviation

In a complete CCW there is a threshold type behavior - if the frequency of one of the cavities exceeds the boundaries of the perfect CCW, the device “collapses”

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Substructuring Approach to Optimization of

Matching Structures for Photonic Crystal

Waveguides

Matching configuration

Computational aspects

– numerical model

Results

[2] Steinberg, Boag, Hershkoviz, “Substructuring Approach to Optimization of Matching…”, JOSA A, submitted

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Matching a CCW to Free Space

Matching Post

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Technical Difficulties

• Numerical size: Need to solve the entire problem:

~200 dielectric cylinders

~4 K unknowns (at least)

Solution by direct inverse is too slow for optimization

• Resonance of high Q structures Iterative solution converges slowly within cavities

• Optimization course requires many forward solutions

To circumvent the difficulties: Sub-structuring

approach

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Sub-Structuring approach

Main Structure

Unchanged during optimization

m Unknowns

Sub StructureUndergoes optimization

n Unknowns

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Sub-Structuring (cont.)

• The large matrix has to be computed & inverted only

once;

unchanged during optimization

• At each optimization cycle:

invert only matrix

• Major cost of a cycle scales as:

• Note that

Solve formally for the master structure, and use it for the sub-structure

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Two possibilities for Optimization in 2D domain (R,d):

Optimal matching

Matching a CCW to Free Space

•Full 2D search approach.

•Using series of alternating orthogonal 1D

optimizationsFast

Risk of “missing” the optimal point.

Additional important parameters to consider:

1. Matching bandwidth2. Output beam collimation/quality

Tests performed on the CCW:Hexagonal lattice: a=4, r=0.6,

=8.41. Cavity: post removal.Central wavelength: =9.06

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Search paths and Field Structures @ optimum

Matching Post@ 1st optimumCrystal Matching Post

@ 7th optimum

@ R=1.2

.

Alternating 1D scannings approach: Good matching, but Radiation field is not well collimated.

Improved beam collimation at the output

Achieved optimumR=0.4, X=71.3

Starting point

Full 2D search: Good matching, good collimation.

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Field Structure @ Optimum (R=0.4, X=71.3)

Improved beam collimation at the output

Hexagonal lattice: a=4, r=0.6,

=8.41. Cavity: post removal.

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Matching Bandwidth

The entire CCW transmission Bandwidth

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Summary

Simple matching structure – consists of a single dielectric

cylinder.

Sub-structuring methodology used to reduce computational

load.

Good ( ) matching to free space.

Insertion loss is better than dB

Good beam collimation achieved with 2D optimization

Matching Optimization of Photonic Crystal CCWs

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CCW Discontinuity

Problem Statement:

Find reflection and transmission

Match using intermediate sections

Find “Impedance” formulas ?

…k=0k=-1 k=1 k=2 k=3k=-2…

Deeper understanding of the propagation physics in CCWs

[3] Steinberg, Boag, “Propagation in PhC CCW with Property Discontinuity…”, JOSA B , to appear

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Basic Equations

• Difference “Equation of Motion” – general heterogeneous CCW

• In our case:

Modal solution amplitudes:

k=0k=-1 k=1 k=2 k=3k=-2

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Approach

• Due to the property discontinuity

• Substitute into the difference equation.

• The interesting physics takes place at

Remote from discontinuity:Conventional CCWs dispersions

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Approach (cont.)

• Two Eqs. , two unknowns

Where is a factor indicating the degree of which mismatch

Solving for reflection and transmission, we get

-Characterizes the interface between two different CCWs

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Interesting special case

• Both CCW s have the same central frequency

And for a signal at the central frequency

Fresnel – like expressions !

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Reflection at Discontinuity

Equal center frequencies

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Different center frequencies

Reflection at Discontinuity

Reflection vs. wavelength

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“Quarter Wavelength” Analog

• Matching by an intermediate CCW section

• Can we use a single micro-cavity as an intermediate matching section?

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Intermediate section w/one micro-

cavity

• Matching w single micro-cavity? Yes! Note: electric length

of a single cavity = – If all CCW’s possess the same central frequency– Matching for that central frequency – Requirement for R=0 yields:

and, @ the central frequency:

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Example

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CCW application:

All Optical Gyroscope Based on Sagnac Effect in

Photonic Crystal Coupled- (micro) - Cavity

Waveguide

[4] B.Z. Steinberg, ‘’Rotating Photonic Crystals:…”, Phys. Rev. E, May 31 2005

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Basic Principles

Stationary Rotating at angular velocity

A CCW folded back upon itself in a fashion that preserves symmetry

C - wise and counter C - wise propag are

identical. “Conventional” self-adjoint formulation. Dispersion is the same as that of a regular

CCW except for additional requirement of

periodicity:

Micro-cavities

Co-Rotation and Counter - Rotation propag DIFFER.

E-D in accelerating systems; non self-adjoint Dispersion differ for Co-R and Counter-R:

Two different directions

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Formulation

• E-D in the rotating system frame of reference:

– We have the same form of Maxwell’s equations:

– But constitutive relations differ:

– The resulting wave equation is (first order in velocity):

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Solution

• Procedure:

– Tight binding theory

– Non self-adjoint formulation (Galerkin)

• Results:

– Dispersion:

Q

mm

Q|

m ; )

m ; )

m ; )

At rest Rotating

Depends on system design

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The Gyro application

• Measure beats between Co-Rot and Counter-Rot modes:

• Rough estimate:

• For Gyro operating at optical frequency and CCW with :

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Summary

• Waveguiding Structure – Micro-Cavity Array Waveguide

• Adjustable Narrow Bandwidth & Center Frequency

• Frequency tuning analysis via Cavity Perturbation Theory

• Sensitivity to random inaccuracies via Cavity

Perturbation Theory

and weak Coupling Theory – A novel threshold

behavior

• Fast Optimization via Sub-Structuring Approach

• Discontinuity Analysis - Link with CCW Bandwidth

• Good Agreement with Numerical Simulations

• Application of CCW to optical Gyros