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

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

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:

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

Rotation Eigenmodes

Rotation eigenmodes:

… and the resonant frequency splits

For the specific PhC under study:

Full numerical simulationUsing rotating medium Green’s function theory

Extracting the peaks

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..

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!

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.

Theory (Cont.)

• The result is the difference (or matrix) equation for the CROW’s excitation coefficients :

• Let and solve for

• An -dependent gap in the CROW transmission curve

• Size of gap:

Periodic modulation of the CROW, byCoincides w the splitting of degenerate modes !!!

Stationary CROW bandwidth

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-bandBandWidth =

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 !