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Coupled Cavity Waveguides in Photonic Crystals:
Sensitivity Analysis, Discontinuities, and Matching
(and an application…)
Ben Z. Steinberg
Amir Boag
Adi Shamir
Orli Hershkoviz
Mark Perlson
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A seminar given by Prof. Steinberg at Lund University, Sept. 2005
Presentation Outline
•
•
•
•
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The CCW – brief overview
Disorder (non-uniformity, randomness) Sensitivity analysis

Micro-Cavity

CCW
Matching to Free Space
[1]
:
[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
2
[4] B.Z. Steinberg, ‘’Rotating Photonic Crystals:…”, Phys. Rev. E, May 31 2005
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:
3
•
•
•
•
Optical/Microwave routing or filtering devices
Optical delay lines
Parametric Optics
Sensors (Rotation)
Regular Photonic Crystal Waveguides
Large transmission bandwidth
(in filtering/routing applications, required relative BW  103 )
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The Coupled Cavity Waveguide
a2
b
a1
Inter-cavity spacing vector:
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The Single Micro-Cavity
Micro-Cavity geometry
Localized Fields
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Micro-Cavity E-Field

Line Spectrum at
Widely spaced Micro-Cavities
Large inter-cavity spacing preserves localized fields
m1=2
m1=3
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Bandwidth of Micro-Cavity Waveguides
Inter-cavity coupling via tunneling:
Large inter-cavity spacing
Transmission vs. wavelength
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weak coupling
narrow bandwidth
Transmission bandwidth vs. inter-cavity spacing
Tight Binding Theory
A propagation modal solution of the form:
where
The single cavity modal field
resonates at frequency
Insert into the variational formulation:
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Tight Binding Theory (Cont.)
The result is a shift invariant equation for
:
Infinite Band-Diagonally dominant
matrix equation:
Where:
It has a solution of the form:
The operator
, restricted
to the k-th defect
- Wave-number along cavity array
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Variational Solution
w
G
M
Wide spacing limit:
wc
Dw
Bandwidth:
M
G
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p/|b|
Central frequency – by the local defect nature; Bandwidth – by the inter cavity spacing.
p/|a1|
k
Center Frequency Tuning
Recall that:
Approach: Varying a defect parameter
Example: Tuning by varying posts’ radius
(nearest neighbors only)
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
tuning of the cavity resonance
Transmission vs. radius
Structure Variation and Disorder:
Cavity Perturbation + Tight Binding Theories
[1] Steinberg, Boag, Lisitsin, “Sensitivity Analysis…”, JOSA A 20, 138 (2003)
- Perfect micro-cavity
- Perturbed micro-cavity
Interested in:
Then (for small
)
For radius variations
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Modes of the unperturbed structure
Disorder I: Single Cavity case
Uncorrelated random variation
•
- all posts in the crystal are varied
Cavity perturbation theory gives:
Due to localization of cavity modes – summation can be restricted to N closest neighbors
• Perturbation theory:
Summation over 6 nearest neighbors
• Statistics results:
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Exact numerical results of 40 realizations
Variance of Resonant
Wavelength
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,
independent for
3.
can be considered as
.
Thus: tight binding theory can still be applied, with some generalizations
Modal field of the (isolated)
–th microcavity.
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Its resonance is
An equation for the coefficients
• Difference equation:
• In the limit
(consistent with cavity perturbation theory)
Unperturbed system
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Manifestation of structure disorder
Matrix Representation
Eigenvalue problem for the general heterogeneous CCW
(Random or deterministic):
-a tridiagonal matrix of the previous form:
-And:
From Spectral Radius considerations:
Random inaccuracy has no effect if:
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Canonical
Independent of specific
design/disorder parameters
Numerical Results – CCW with 7 cavities
n
of perturbed
microcavities
n
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of perturbed
microcavities
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
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[2] Steinberg, Boag, Hershkoviz, “Substructuring Approach to Optimization of Matching…”, JOSA A, submitted
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
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Unknowns
Sub Structure
Undergoes optimization
n
Unknowns
Sub-Structuring (cont.)
Solve formally for the master structure,
and use it for the sub-structure
•
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:
•
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Note that
Matching a CCW to Free Space
Two possibilities for Optimization in 2D domain (R,d):
• Full 2D search approach.
• Using series of alternating orthogonal 1D optimizations
 Fast
 Risk of “missing” the optimal point.
Additional important parameters to consider:
1.
2.
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Matching bandwidth
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
Optimal matching
Search paths and Field Structures @ optimum
Alternating 1D scannings approach: Good matching, but Radiation field is not well collimated.
@ R=1.2
.
Crystal
Matching Post
@ 1st optimum
Matching Post
@ 7th optimum
Full 2D search: Good matching, good collimation.
Achieved optimum
R=0.4, X=71.3
Starting point
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Improved beam collimation at the output
Field Structure @ Optimum (R=0.4, X=71.3)
Hexagonal lattice: a=4, r=0.6, =8.41.
Cavity: post removal.
Improved beam collimation at the output
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Matching Bandwidth
The entire CCW transmission Bandwidth
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Summary
Matching Optimization of Photonic Crystal CCWs

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

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dB
Good beam collimation achieved with 2D optimization
CCW Discontinuity
…
k=-2
k=-1
k=0
k=1
k=2
k=3
…
Problem Statement:
 Find reflection and transmission
 Match using intermediate sections
Deeper understanding of the
propagation physics in CCWs
 Find “Impedance” formulas ?
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[3] Steinberg, Boag, “Propagation in PhC CCW with Property Discontinuity…”, JOSA B , to appear
Basic Equations
• Difference “Equation of Motion”
– general heterogeneous CCW
• In our case:

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k=-2
k=-1
 Modal solution amplitudes:
k=0
k=1
k=2
k=3
Approach
• Due to the property discontinuity
• Substitute into the difference equation.
Remote from discontinuity:
Conventional CCWs
dispersions
• The interesting physics takes place at
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Approach (cont.)
• Two Eqs. , two unknowns
Solving for reflection and transmission, we get
Where
is a factor indicating the degree of which
mismatch
-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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Reflection at Discontinuity
Different center frequencies
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
A CCW folded back upon itself in a fashion that preserves symmetry
Micro-cavities
Stationary
Rotating at angular velocity
 C - wise and counter C - wise propag are identical.
 Co-Rotation and Counter - Rotation propag DIFFER.
 “Conventional” self-adjoint formulation.
 E-D in accelerating systems; non self-adjoint
 Dispersion is the same as that of a regular CCW
 Dispersion differ for Co-R and Counter-R:
except for additional requirement of periodicity:
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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:
At rest
w
Rotating
|W0 Q|
w (km ; W0 )
w (km ; W0 )
Dw
w0
w (km ; W0 = 0 )
Depends on system design
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km
W0 Q
km
k
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
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Waveguiding Structure – Micro-Cavity Array Waveguide
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Adjustable Narrow Bandwidth & Center Frequency
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Frequency tuning analysis via Cavity Perturbation Theory
•
Sensitivity to random inaccuracies via Cavity Perturbation Theory
and weak Coupling Theory – A novel threshold behavior
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•
Fast Optimization via Sub-Structuring Approach
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Discontinuity Analysis - Link with CCW Bandwidth
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Good Agreement with Numerical Simulations
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Application of CCW to optical Gyros