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

2. 2 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. No mode degeneracy

3. 3 CROW-based Gyro: Basic Principles Stationary Rotating at angular velocity A CROW folded back upon itself in a fashion that preserves symmetry  C - wise and counter C - wise propag are identical.  Dispersion: same as regular CROW except for additional requirement of periodicity: Micro-cavities  Co-Rotation and Counter - Rotation propag DIFFER.  Dispersion differ for Co-R and Counter-R: Two different directions [1] Steinberg B.Z., “Rotating Photonic Crystals: A medium for compact optical gyroscopes,” PRE 71 056621 (2005).

4. 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. IEEE 61 1694 (1973).

5. 5 Procedure: –Tight binding theory –Non self-adjoint formulation (Galerkin) Results: –Dispersion: Solution QQ   mm  m Q|Q|   m ;    )   m ;    )  m ;    ) At restRotating Depends on system design ! [1] Steinberg B.Z., “Rotating Photonic Crystals: A medium for compact optical gyroscopes,” PRE 71 056621 (2005). = Stationary micro-cavity mode

6. 6 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 Theoretical Numerical

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

8. 8 Formulation Rotating micro-cavity w M -th order degeneracy M - stationary system degenerate modes resonate at : The rotating system field satisfies the wave equations: After standard manipulations (no approximations): Express the rotating system field as a sum of the stationary system degenerate modes (first approximation): 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)

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

10. 10 More on Splitting: Symmetries The matrix C is skew symmetric, thus –M even: are real and always come in symmetric pairs around the origin –M odd: 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 But recall:

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

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

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

14. 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 !