1 / 45

Ben Z. Steinberg Amir Boag Adi Shamir Orli Hershkoviz Mark Perlson

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. Presentation Outline.

elmo
Download Presentation

Ben Z. Steinberg Amir Boag Adi Shamir Orli Hershkoviz Mark Perlson

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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 31 2005

  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 b a2 a1 Inter-cavity spacing vector:

  6. The Single Micro-Cavity Micro-Cavity geometry Micro-Cavity E-Field Localized FieldsLine Spectrum at

  7. Widely spaced Micro-CavitiesLarge inter-cavity spacing preserves localized fields m1=2 m1=3

  8. Bandwidth of Micro-Cavity Waveguides Large inter-cavity spacing weak coupling narrow bandwidth Inter-cavity coupling via tunneling: Transmission vs. wavelength Transmission bandwidth vs. inter-cavity spacing

  9. Tight Binding Theory The single cavity modal field resonates at frequency A propagation modal solution of the form: where Insert into the variational formulation:

  10. The result is a shift invariant equation for : Infinite Band-Diagonally dominant matrix equation: The operator , restricted to the k-th defect - Wave-number along cavity array Tight Binding Theory (Cont.) Where: It has a solution of the form:

  11. Variational Solution w G M wc Dw M G p/|b| p/|a1| Wide spacing limit: Bandwidth: k 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 Interested in: [1] Steinberg, Boag, Lisitsin, “Sensitivity Analysis…”, JOSA A 20, 138 (2003) • Perfect micro-cavity • Perturbed micro-cavity Then (for small ) For radius variations Modes of the unperturbed structure

  14. Disorder I: Single Cavity case Uncorrelated random variation - all posts in the crystal are varied Variance of Resonant Wavelength • 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: • Exact numerical results of 40 realizations

  15. Disorder & Structure variation II: The CCW case Modal field of the (isolated) –th microcavity. Its resonance is • Mathematical model is based on the physical observations: • The micro-cavities are weakly coupled. • 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 . • Thus: tight binding theory can still be applied, with some generalizations

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

  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: Random inaccuracy has no effect if: Canonical Independent of specific design/disorder parameters

  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 Sub Structure Undergoes optimization n Unknowns Main Structure Unchanged during optimization mUnknowns

  24. 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: • Note that

  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: • Matching bandwidth • Output beam collimation/quality Tests performed on the CCW: Hexagonal lattice: a=4, r=0.6, e=8.41. Cavity: post removal. Central wavelength: l=9.06

  26. Search paths and Field Structures @ optimum @ R=1.2 Matching Post @ 7th optimum Matching Post @ 1st optimum Crystal Achieved optimum R=0.4, X=71.3 Improved beam collimation at the output Starting point Alternating 1D scannings approach: Good matching, but Radiation field is not well collimated. . Full 2D search: Good matching, good collimation.

  27. Field Structure @ Optimum (R=0.4, X=71.3) Improved beam collimation at the output Hexagonal lattice: a=4, r=0.6, e=8.41. Cavity: post removal.

  28. Matching Bandwidth The entire CCW transmission Bandwidth

  29. 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 dB • Good beam collimation achieved with 2D optimization

  30. k=-2 k=-1 k=0 … k=1 k=2 k=3 … CCW Discontinuity Problem Statement: Find reflection and transmission Match using intermediate sections Find “Impedance” formulas ? 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 k=-1 k=-2 k=0 k=1 k=2 k=3 • 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.) Where is a factor indicating the degree of which mismatch • Two Eqs. , two unknowns Solving for reflection and transmission, we get -Characterizes the interface between two different CCWs

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

  35. Reflection at Discontinuity Equal center frequencies

  36. Reflection at Discontinuity Different center frequencies 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 CCW’s possess the same central frequency • Matching for that central frequency • Requirement for R=0 yields: and, @ 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 31 2005

  41. Basic Principles Micro-cavities A CCW folded back upon itself in a fashion that preserves symmetry Rotating at angular velocity Stationary • 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: • 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 Maxwell’s equations: • But constitutive relations differ: • The resulting wave equation is (first order in velocity):

  43. Solution At rest Rotating w |W0Q| w (km ; W0) Dw w (-km ; W0) w0 w (km ; W0 = 0 ) k -km km W0Q • Procedure: • Tight binding theory • Non self-adjoint formulation (Galerkin) • Results: • Dispersion: 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

More Related