Sagnac Effect in Rotating Photonic Crystal Micro-Cavities and Miniature Optical Gyroscopes

1. Tel Aviv University Sagnac Effect in Rotating Photonic Crystal Micro-Cavities and Miniature Optical Gyroscopes Ben Z. Steinberg Ady Shamir Amir Boag

2. No mode degeneracy Presentation Overview • The PhC CROW – based Gyro • New manifestation of Sagnac Effect • Array of weakly coupled “conventional” micro-cavities • What happens if the micro-cavities support mode-degeneracy ? • Micro-cavities with mode degeneracy • Single micro-cavity: the smallest gyroscope in nature. • Set of micro-cavities: interesting physics Steinberg, Scheuer, Boag, Slow and Fast Light topical meeting, Washington DC July 06.

3. Micro-cavities [1] Steinberg B.Z., “Rotating Photonic Crystals: A medium for compact optical gyroscopes,” PRE71 056621 (2005). CROW-based Gyro: Basic Principles A CROW folded back upon itself in a fashion that preserves symmetry Stationary Rotating at angular velocity • C - wise and counter C - wise propag are identical. • Dispersion: same as regular CROW except for additional requirement of periodicity: • Co-Rotation and Counter - Rotation propag DIFFER. • Dispersion differ for Co-R and Counter-R: Two different directions

4. Formulation • E-D in the rotating system frame of reference:non-inertial • We have the same form of Maxwell’s equations: • But constitutive relations differ: • The resulting wave equation is (first order in velocity): [2] T. Shiozawa, “Phenomenological and Electron-Theoretical Study of the Electrodynamics of Rotating Systems,” Proc. IEEE61 1694 (1973).

5. At rest Rotating w |W0Q| w (km ; W0) Dw w (-km ; W0) w0 w (km ; W0 = 0 ) k -km km W0Q Solution [1] Steinberg B.Z., “Rotating Photonic Crystals: A medium for compact optical gyroscopes,” PRE71 056621 (2005). • Procedure: • Tight binding theory • Non self-adjoint formulation (Galerkin) • Results: • Dispersion: Depends on system design ! =Stationary micro-cavity mode

6. Theoretical Numerical The Gyro application • Measure beats between Co-Rot and Counter-Rot modes: • Rough estimate: • For Gyros operating at FIR and CROW with : Theoretical and Numerical

7. 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 • Rotationaffects these two waves differently:Sagnac effect The single micro-cavity with mode degeneracy • The most simple and familiar example: A ring resonator • Degenerate modes in a Photonic Crystal Micro-Cavity Local defect: TM How rotation affects this system ?

8. The rotating system field satisfies the wave equations: Reasonable approximation because: Rotation has a negligible effect on mode shapes. It essentially affects phases and resonances. H.J. Arditty and H.C. Lefevre, Optics Letters, 6(8) 401 (1981) FormulationRotating micro-cavity w M -th order degeneracy • M - stationary system degenerate modes resonate at : • After standard manipulations (no approximations): • Express the rotating system field as a sum of the stationary system degenerate modes (first approximation):

9. An M x M matrix eigenvalue problem for the frequency shift : Then, is determined by the eigenvalues of the matrix : Formulation (Cont.) where the matrix elements are expressed via the stationary cavity modes, Frequency splitting due to rotation Splitting depends on effective rotation radius, extracted byB

10. But recall: More on Splitting: Symmetries • The matrix C is skew symmetric , thus • M even: are real and always come in symmetric pairs around the origin • Modd: The rule above still applies, with the addition of a single eigenvalue at 0. • For M=2, the coefficients (eigenvector) satisfy: The eigen-modes in the rotating system rest-frame are rotating fields

11. Specific results For the PhC under study: Full numerical simulation Using rotating medium Green’s function theory Extracting the peaks

12. Mechanically Stationary system: • Both modes resonate at • “Good” coupling Interaction between micro-cavities • The basic principle: A CW rotating mode couples only to CCW rotating neighbor • Mechanically Rotating system: • Resonances split • Coupling reduces A new concept: the miniature Sagnac Switch

13. An -dependent gap in the CROW transmission curve cascade many of them… • Periodic modulation of local resonant frequency • Periodic modulation of the CROW difference equation, by Excitation coefficient of the m-th cavity Steinberg, Scheuer, Boag, Slow and Fast Light topical meeting, Washington DC July 06.

14. Conclusions • Rotating crystals = Fun ! • New insights and deeper understanding of Sagnac effect • The added flexibility offered by PhC (micro-cavities, slow-light structures, etc) a potential for • Increased immunity to environmental conditions (miniature footprint) • Increased sensitivity to rotation. Thank You !