Sensitivity Analysis of Narrow Band PhotonicCrystal Waveguides and
Sensitivity Analysis of Narrow Band Photonic-Crystal Waveguides and Filters Ben Z. Steinberg Amir Boag Ronen Lisitsin Svetlana Bushmakin 1
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: q Micro-Cavity q CCW (Random Structure inaccuracy) • Coupling (matching) to outer world 2
The coupled micro-cavity photonic waveguide Goals: • Create photonic crystal waveguide with pre-scribed: q Narrow bandwidth q Center frequency Applications: • • 3 Optical/Microwave routing devices Wavelength Division Multiplexing components
The Micro-Cavity Array Waveguides b a 2 a 1 Intercavity vector: 4
The Single Micro-Cavity Localized Fields 5 Line Spectrum at
Weak Coupling Perturbation Theory A propagation modal solution of the form: where - The single cavity modal field Insert into the variational formulation: 6
The result is a shift invariant equation for : Where: It has a solution of the form: - Wavenumber along cavity array 7
Variational Solution w G M Wide spacing limit: wc Dw The isolated micro-cavity resonance M Bandwidth: G 8 p/|b | p/|a 1| k
Transmission & Bandwidh Transmission vs. wavelength Bandwidth vs. cavity spacing Isolated micro-cavity resonance 9
Micro-Cavity Center Frequency Tuning Varying a defect parameter Example: Varying posts radius (nearest neighbors only, identically) 10 tuning of the cavity resonance Transmission vs. radius
Cavity Perturbation Theory - Perfect micro-cavity - Perturbed micro-cavity Interested in: Then (can show 11 vs. for ! )
Random Structure Inaccuracy Example: 2 D crystal, with uncorrelated random variation - all posts in the crystal are varied 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 12
Standard Deviation of Resonant Wavelength Hexagonal lattice, a=4, r=0. 6, e=8. 41. Cavity: post removal. Resonance All posts in the crystal are RANDOMLY varied • Perturbation theory: Summation over 6 nearest neighbors • Statistics results: Exact numerical results of 40 realizations 13 l=9. 06
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 3. 3. where Since is a variable with the properties studied before. 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. 14 Its resonance is
An equation for the coefficients: Where: In the limit we obtain Random inaccuracy has no effect if 15 Canonical Independent of specific design parameters
Matrix Representation Eigenvalue problem: - a tridiagonal matrix of the form: 16
Numerical Results – CCW with 7 cavities of perturbed microcavities 17
Matching a CCW to Free Space Matching Post R d 18
SWR minimization results Hexagonal lattice a=4, r=0. 6, e=8. 41. Cavity: post removal. m=2 Crystal ends here 19
Field Structure @ Optimum (r=1. 2) Crystal Matching Post At 1 st optimum Radiation field is not well collimated. Solutions: • 2 D optimization with more than a single post • Collect by a lens 20 Matching Post At 2 nd optimum Matching Post At 3 rd optimum
Summary Sensitivity of micro-cavities and CCWs to random inaccuracy : q Cavity Perturbation Theory – effect on the isolated single cavity Linear relation between noise strength and frequency shift. q Weak Coupling Theory + above results – effect on the CCW A novel threshold behavior : noise affects CCW only if it exceeds certain level. q Matching to free space. 21
- Slides: 21