1 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 A seminar given by Prof. Steinberg at Lund University, Sept. 2005
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1 Coupled Cavity Waveguides in Photonic Crystals: Sensitivity Analysis, Discontinuities, and Matching (and an application…) Ben Z. Steinberg Amir Boag.
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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
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):