1. Propagation in Photonic Crystal Coupled Cavity Waveguides Possessing Discontinuities Ben Z. Steinberg Amir Boag Orli Hershkoviz Mark Perlson Tel Aviv University

2. Presentation Outline The CCW – brief overview  Basic Properties  Potential Applications  Problem Statement and Motivation Heterogeneous CCW [1]:  Cavity perturbation theory  General “equation of motion” for heterogeneous CCW Discontinuity between CCWs as a special case of heterogeneous CCW  Analytical Model  Numerical Simulations Conclusions [1] Steinberg, Boag, Lisitsin, JOSA A 20, 138 (2003)

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

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

5. Widely spaced Micro-Cavities Large inter-cavity spacing preserves localized fields; yet tunneling between micro-cavities (coupling) allows net propagation of energy m 1 =2 m 1 =3

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

7. Weak Coupling Perturbation Theory A propagation modal solution of the form: where Insert into the variational formulation: The single cavity modal field resonates at frequency

8. Variational Solution for CCW Boag, Steinberg, JOSA A, 18 2799 (2001) k M  /|a 1 |  /|b|  cc M  Wide spacing limit: Bandwidth: Central frequency – by the local defect nature; Bandwidth – by the inter cavity spacing. (define dispersion curve uniquely) The operator, restricted to the k-th defect

9. 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

10. Applications: An optical signal filter/router Use: Frequency tuning (nature of local defect) Bandwidth tuning (distance between local defects) Cavity modes symmetry: create perfect turns Wide Band CCW Fan of narrow band CCWs w. different central frequencies CCW properties discontinuity

11. Problem Statement: CCW Discontinuity Specifically:  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

12. Equations for CCW Structure Variation: Cavity Perturbation Theory + Strong Binding Theory - Perfect micro-cavity - Perturbed micro-cavity Interested in: Then (for small ) For radius variations Modes of the unperturbed structure

13. Disorder & Structure variation II: The CCW case Mathematical model is based on the following observations: 1.The microcavities 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: strong binding theory can still be applied, with some generalizations Modal field of the (isolated) – th microcavity. Its resonance is

14. In our case:   Modal solution amplitudes: Basic Equations Difference “Equation of Motion” – general heterogeneous CCW and characterizing the interface (the rest of the are exponentially small)

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

16. 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

17. Interesting special case Both CCW s have the same central frequency: A=1 For a signal at the central frequency Fresnel – like expression Perfect matching with an intermediate section: CCW Single Cavity “Quarter Wavelength plate”

18. Numerical Examples: Reflection at Discontinuity Equal center frequencies

19. Different center frequencies Numerical Examples (Cont.) Reflection vs. wavelength

20. Summary Filtering/guiding structure – the Coupled Cavity Waveguide (CCW)  Potential applications involve CCW discontinuities Analytical formulation for propagation in non uniform CCW:  Strong binding perturbation theory  Cavity perturbation theory CCW discontinuity analysis:  Link with CCW dispersion and bandwidth  Reflection / transmission coefficients Good agreement with numerical simulations