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

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

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)

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

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

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

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

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

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

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:

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.

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

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)

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

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

16 An equation for the coefficients Difference equation: In the limit (consistent with cavity perturbation theory) Unperturbed system Manifestation of structure disorder

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:

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

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

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

21 Matching a CCW to Free Space Matching Post

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

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

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

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

26 Search paths and Field optimum Matching 1st optimum Crystal Matching 7th 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.

27 Field 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.

28 Matching Bandwidth The entire CCW transmission Bandwidth

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

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

31 Basic Equations Difference Equation of Motion – general heterogeneous CCW In our case: Modal solution amplitudes:

32 Approach Due to the property discontinuity Substitute into the difference equation. The interesting physics takes place at Remote from discontinuity: Conventional CCWs dispersions

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

34 Interesting special case Both CCW s have the same central frequency And for a signal at the central frequency Fresnel – like expressions !

35 Reflection at Discontinuity Equal center frequencies

36 Different center frequencies Reflection at Discontinuity Reflection vs. wavelength

37 Quarter Wavelength Analog Matching by an intermediate CCW section Can we use a single micro-cavity as an intermediate matching section?

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: the central frequency:

39 Example

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

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

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):

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

44 The Gyro application Measure beats between Co-Rot and Counter-Rot modes: Rough estimate: For Gyro operating at optical frequency and CCW with :

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