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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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2 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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3 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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4 Regular Photonic Crystal Waveguides Large transmission bandwidth (in filtering/routing applications, required relative BW )

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5 The Coupled Cavity Waveguide a1a1 a2a2 Inter-cavity spacing vector: b

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6 The Single Micro-Cavity Localized Fields Line Spectrum at Micro-Cavity geometryMicro-Cavity E-Field

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7 Widely spaced Micro-Cavities Large inter-cavity spacing preserves localized fields m 1 =2 m 1 =3

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8 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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9 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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10 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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11 Variational Solution k M /|a 1 | /|b| c M Wide spacing limit: Bandwidth: Central frequency – by the local defect nature; Bandwidth – by the inter cavity spacing.

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12 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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13 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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14 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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15 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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16 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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17 Matrix Representation Eigenvalue problem for the general heterogeneous CCW (Random or deterministic): -a tridiagonal matrix of the previous form: -And: From Spectral Radius considerations: Canonical Independent of specific design/disorder parameters Random inaccuracy has no effect if:

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18 Numerical Results – CCW with 7 cavities of perturbed microcavities of perturbed microcavities

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19 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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20 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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21 Matching a CCW to Free Space Matching Post

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22 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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23 Sub-Structuring approach Main Structure Unchanged during optimization m Unknowns Sub Structure Undergoes optimization n Unknowns

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24 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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25 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 optimizations Fast Risk of missing the optimal point. Additional important parameters to consider: 1.Matching bandwidth 2.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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26 Search paths and Field Structures @ optimum Matching Post @ 1st optimum Crystal 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 optimum R=0.4, X=71.3 Starting point Full 2D search: Good matching, good collimation.

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27 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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28 Matching Bandwidth The entire CCW transmission Bandwidth

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29 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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30 CCW Discontinuity Problem Statement: Find reflection and transmission Match using intermediate sections Find Impedance formulas ? … k=0k=-1 k=1 k=2k=3 k=-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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31 Basic Equations Difference Equation of Motion – general heterogeneous CCW In our case: Modal solution amplitudes:

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32 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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33 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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34 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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35 Reflection at Discontinuity Equal center frequencies

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36 Different center frequencies Reflection at Discontinuity Reflection vs. wavelength

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37 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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38 Intermediate section w/one micro-cavity Matching w single micro-cavity? Yes! Note: electric length of a single cavity = –If all CCWs possess the same central frequency –Matching for that central frequency –Requirement for R=0 yields: and, @ the central frequency:

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39 Example

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40 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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41 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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42 Formulation E-D in the rotating system frame of reference: –We have the same form of Maxwells equations: –But constitutive relations differ: –The resulting wave equation is (first order in velocity):

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43 Solution Procedure: –Tight binding theory –Non self-adjoint formulation (Galerkin) Results: –Dispersion: Q m m Q| m ; ) At restRotating Depends on system design

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44 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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45 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

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