1. Rotation Induced Super Structure in Slow-Light Waveguides w Mode Degeneracy Ben Z. Steinberg Adi Shamir Jacob Scheuer Amir Boag School of EE, Tel-Aviv University

2. Presentation Overview The effect of mechanical rotation on Slow-Light Structures –Previous studies: [1] Array of weakly coupled conventional resonators New manifestation of Sagnac Effect Present work: –What happens if the micro-resonators support mode-degeneracy ? Interesting NEW physical effects in Slow-Light Structures –Micro-resonators with mode degeneracy: two stages study Single resonator w mode-degeneracy: the smallest gyroscope in nature. [3,4] Set of coupled resonators: Emergence of rotation-induced superstructure No mode degeneracy [1] Steinberg B.Z., Rotating Photonic Crystals: A medium for compact optical gyroscopes, PRE 71 056621 (2005). [2] Scheuer J., Yariv A., Sagnac effect in coupled resonator slow light waveguide structure, PRL 96 053901 (2006). [3] Steinberg B.Z., Boag A., Splitting of micro-cavity degenerate modes in rotating PhC… submitted. [4] Steinberg B.Z., Shamir A., Boag A., CLEO 2006, Long Beach

3. Two waves having the same resonant frequency : Two different standing waves Or: (any linear combination of degenerate modes is a degenerate mode!) CW and CCW propagations Under rotation: (as seen in the rotating system rest frame!) Mode shapes are preserved Eigenvalues (resonant frequencies) SPLIT: classical Sagnac effect The single resonator with mode degeneracy The most simple and familiar example: A ring resonator Rotation eigenmodes:

4. The single resonator with mode degeneracy (Cont.) Degenerate modes in a Photonic Crystal Micro-Cavity (example, not limited to) Local defect: TM How rotation affects this system ? It turns out that: (slow rotation) The same general picture holds for ANY resonator w mode degeneracy: [3,4] Orthogonal Real Rotation eigenmodes: [3] Steinberg B.Z., Boag A., Splitting of micro-cavity degenerate modes in rotating PhC… submitted. [4] Steinberg B.Z., Shamir A., Boag A., CLEO 2006, Long Beach Rotation eigenmodes: specific LC of the degenerate modes

5. Rotation Eigenmodes Rotation eigenmodes:

6. … and the resonant frequency splits For the specific PhC under study: Full numerical simulation Using rotating medium Greens function theory Extracting the peaks

7. Interaction between micro-resonators w degenerate modes The basic principle: A CW rotating mode couples only to CCW rotating neighbor Mechanically Stationary system: Both modes resonate at Prescribed coupling A new concept: the miniature Sagnac Switch Mechanically Rotating system: Resonances split Coupling reduces PhC defects, Rings, Disks, etc..

8. cascade many of them… Periodic modulation of local relevant resonant frequency Periodic modulation of the CROW difference equation Mathematically rigorous derivation of the above physics by: –tight binding theory –applied to the wave equation in the rotating CROW rest frame!

9. Theory The wave equation in the rotating CROW rest frame: [1,5] [1] Steinberg B.Z., Rotating Photonic Crystals: A medium for compact optical gyroscopes, PRE 71 056621 (2005). [5] T. Shiozawa, Phenomenological and Electron-Theoretical Study of the Electrodynamics of Rotating Systems, Proc. IEEE 61 1694 (1973). Express the rotating system total field as a sum of the isolated resonator rotation eigenmodes Rotation operator: lost of self-adjointness Substitute into the wave equation, apply Galerkin method Tight-binding theory, adapted to mode degeneracy + rotation.

10. Theory (Cont.) The result is the difference (or matrix) equation for the CROWs excitation coefficients : Let and solve for An -dependent gap in the CROW transmission curve Size of gap: Periodic modulation of the CROW, by Coincides w the splitting of degenerate modes !!! Stationary CROW bandwidth

11. Example Micro-Ring based CROW: Transmission vs., 29 resonators Transmission at, vs Exponential decay rate as a function of, increases linearly with the number of resonators (splitting) Rotation induced stop-band BandWidth =

12. Conclusions Rotating crystals and SWS = Fun ! Rotation of degenerate modes CROW – new physical effects The added flexibility and the new physical effects offered by micro- cavities and slow-light structures a potential for –New generation of Gyroscopes –Exponential type sensitivity to rotation. Thank You !