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![Page 1: 1 Coupled Cavity Waveguides in Photonic Crystals: Sensitivity Analysis, Discontinuities, and Matching (and an application…) Ben Z. Steinberg Amir Boag.](https://reader036.fdocuments.us/reader036/viewer/2022062318/5517c4df5503461b658b48e8/html5/thumbnails/1.jpg)
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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