1 Sensitivity Analysis of Narrow Band Photonic-Crystal Waveguides and Filters Ben Z. Steinberg Amir Boag Ronen Lisitsin Svetlana Bushmakin

2 Presentation Outline Coupled Cavity Waveguide (CCW) (and micro-cavities) Filters/routers and waveguides – Optical comm. Typical length-scale << λ ( approaches today ’ s Fab accuracy ) Sensitivity analysis: ( Random Structure inaccuracy )  Micro-Cavity  CCW Coupling (matching) to outer world

3 The coupled micro-cavity photonic waveguide Goals: Create photonic crystal waveguide with pre-scribed:  Narrow bandwidth  Center frequency Applications: Optical/Microwave routing devices Wavelength Division Multiplexing components

4 The Micro-Cavity Array Waveguides a1a1 a2a2 Intercavity vector: b

5 The Single Micro-Cavity Localized Fields Line Spectrum at

6 Weak Coupling Perturbation Theory A propagation modal solution of the form: where - The single cavity modal field Insert into the variational formulation:

7 The result is a shift invariant equation for : Where: It has a solution of the form: - Wavenumber along cavity array

8 Variational Solution M  /|a 1 |  /|b | k  cc M  Wide spacing limit: Bandwidth: The isolated micro-cavity resonance

9 Transmission & Bandwidh Transmission vs. wavelength Bandwidth vs. cavity spacing Isolated micro-cavity resonance

10 Varying a defect parameter tuning of the cavity resonance Micro-Cavity Center Frequency Tuning Example: Varying posts radius (nearest neighbors only, identically) Transmission vs. radius

11 Interested in: vs. Then (can show for ! ) Cavity Perturbation Theory - Perfect micro-cavity - Perturbed micro-cavity

12 Example: 2D crystal, with uncorrelated random variation - all posts in the crystal are varied Random Structure Inaccuracy Model: Treat radii variations as perturbations of the reference cavity. In a single realization different posts can have different radii. Cavity perturbation theory gives: Due to localization of cavity modes – summation can be restricted to closest neighbors

13 Standard Deviation of Resonant Wavelength Perturbation theory: Summation over 6 nearest neighbors Statistics results: Exact numerical results of 40 realizations All posts in the crystal are RANDOMLY varied Hexagonal lattice, a=4, r=0.6,  =8.41. Cavity: post removal. Resonance =9.06

14 CCW with Random Structure Inaccuracy Mathematical model is based on the physical observations: 1.The microcavities are weakly coupled. 2.The resonance frequency of the i -th microcavity is where is a variable with the properties studied before. 3.Since depends essentially on the perturbations of the i -th microcavity closest neighbors, can be considered as independent for i ≠ j. Modal field of the (isolated) m – th microcavity. Its resonance is

15 An equation for the coefficients: Where: In the limit we obtain Random inaccuracy has no effect if Canonical Independent of specific design parameters

16 Matrix Representation Eigenvalue problem: - a tridiagonal matrix of the form:

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

18 Matching a CCW to Free Space Matching Post R d

19 SWR minimization results Crystal ends here Hexagonal lattice a=4, r=0.6,  =8.41. Cavity: post removal. m=2

20 Field Optimum (r=1.2) Crystal Matching Post At 1st optimum Matching Post At 2nd optimum Matching Post At 3rd optimum Radiation field is not well collimated. Solutions: 2D optimization with more than a single post Collect by a lens

21 Summary  Cavity Perturbation Theory – effect on the isolated single cavity Linear relation between noise strength and frequency shift.  Weak Coupling Theory + above results – effect on the CCW A novel threshold behavior : noise affects CCW only if it exceeds certain level.  Matching to free space. Sensitivity of micro-cavities and CCWs to random inaccuracy :