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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

- 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.

Micro-cavities

[1] Steinberg B.Z., “Rotating Photonic Crystals: A medium for compact optical gyroscopes,” PRE71 056621 (2005).

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

- 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).

At rest

Rotating

w

|W0Q|

w (km ; W0)

Dw

w (-km ; W0)

w0

w (km ; W0 = 0 )

k

-km

km

W0Q

[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

Theoretical

Numerical

- Measure beats between Co-Rot and Counter-Rot modes:
- Rough estimate:
- For Gyros operating at FIR and CROW with :

Theoretical and

Numerical

- 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 most simple and familiar example: A ring resonator

- Degenerate modes in a Photonic Crystal Micro-Cavity

Local defect:

TM

How rotation affects this system ?

- 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)

- 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):

An M x M matrix eigenvalue problem for the frequency shift :

Then, is determined by the eigenvalues of the matrix :

where the matrix elements are expressed via the stationary cavity modes,

Frequency splitting due to rotation

Splitting depends on effective rotation radius, extracted byB

But recall:

- 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

For the PhC under study:

Full numerical simulation

Using rotating medium Green’s function theory

Extracting the peaks

- Mechanically Stationary system:
- Both modes resonate at
- “Good” coupling

- 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

- An -dependent gap in the CROW transmission curve

- 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.

- 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 !