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

Tel Aviv University

Ben Z. SteinbergAdy ShamirAmir Boag

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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

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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).

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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).

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• Procedure:– Tight binding theory– Non self-adjoint formulation (Galerkin)

• Results:– Dispersion:

Solution

Q

mm

Q|

m ; )

m ; )

m ; )

At rest Rotating

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

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The Gyro application

• Measure beats between Co-Rot and Counter-Rot modes:

• Rough estimate:

• For Gyros operating at FIR and CROW with :

Theoretical andNumerical

TheoreticalNumerical

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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 ?

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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)

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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 rotationSplitting depends on effective rotation radius, extracted by B

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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:

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Specific results

For the PhC under study:

Full numerical simulationUsing rotating medium Green’s function theory

Extracting the peaks

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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

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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

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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 !