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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 1 A seminar given by Prof. Steinberg at Lund University, Sept. 2005 Presentation Outline • • • • • 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 103 ) 4 The Coupled Cavity Waveguide a2 b a1 Inter-cavity spacing vector: 5 The Single Micro-Cavity Micro-Cavity geometry Localized Fields 6 Micro-Cavity E-Field Line Spectrum at Widely spaced Micro-Cavities Large inter-cavity spacing preserves localized fields m1=2 m1=3 7 Bandwidth of Micro-Cavity Waveguides Inter-cavity coupling via tunneling: Large inter-cavity spacing Transmission vs. wavelength 8 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: 9 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 10 Variational Solution w G M Wide spacing limit: wc Dw Bandwidth: M G 11 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) 12 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 13 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: 14 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. 15 Its resonance is An equation for the coefficients • Difference equation: • In the limit (consistent with cavity perturbation theory) Unperturbed system 16 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: 17 Canonical Independent of specific design/disorder parameters Numerical Results – CCW with 7 cavities n of perturbed microcavities n 18 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” 19 Substructuring Approach to Optimization of Matching Structures for Photonic Crystal Waveguides Matching configuration Computational aspects – numerical model Results 20 [2] Steinberg, Boag, Hershkoviz, “Substructuring Approach to Optimization of Matching…”, JOSA A, submitted Matching a CCW to Free Space Matching Post 21 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 22 Sub-Structuring approach Main Structure Unchanged during optimization m 23 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: • 24 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. 25 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 26 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 27 Matching Bandwidth The entire CCW transmission Bandwidth 28 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 29 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 ? 30 [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: 31 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 32 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 33 Interesting special case • Both CCW s have the same central frequency And for a signal at the central frequency Fresnel – like expressions ! 34 Reflection at Discontinuity Equal center frequencies 35 Reflection at Discontinuity Different center frequencies Reflection vs. wavelength 36 “Quarter Wavelength” Analog • Matching by an intermediate CCW section • Can we use a single micro-cavity as an intermediate matching section? 37 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: 38 Example 39 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 40 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: 41 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): 42 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 43 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 : 44 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 45 • Fast Optimization via Sub-Structuring Approach • Discontinuity Analysis - Link with CCW Bandwidth • Good Agreement with Numerical Simulations • Application of CCW to optical Gyros