Sagnac effect in rotating photonic crystal micro cavities and miniature optical gyroscopes
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Tel Aviv University. Sagnac Effect in Rotating Photonic Crystal Micro-Cavities and Miniature Optical Gyroscopes. Ben Z. Steinberg Ady Shamir Amir Boag. No mode degeneracy. Presentation Overview. The PhC CROW – based Gyro New manifestation of Sagnac Effect

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

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Sagnac effect in rotating photonic crystal micro cavities and miniature optical gyroscopes

Tel Aviv University

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

Ben Z. Steinberg

Ady Shamir

Amir Boag


Presentation overview

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.


Crow based gyro basic principles

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


Formulation

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).


Solution

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


The gyro application

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


The single micro cavity with mode degeneracy

  • 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 ?


Formulation rotating micro cavity w m th order degeneracy

  • 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):


Formulation cont

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


More on splitting symmetries

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


Specific results

Specific results

For the PhC under study:

Full numerical simulation

Using rotating medium Green’s function theory

Extracting the peaks


Interaction between micro cavities

  • 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


Cascade many of them

  • 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.


Conclusions

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 !


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