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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

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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

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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:

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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

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Rotation Eigenmodes Rotation eigenmodes:

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

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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..

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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!

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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.

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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

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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 =

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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 !

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