1 sagnac effect in rotating photonic crystal micro-cavities and miniature optical gyroscopes tel...
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Sagnac Effect in Rotating Photonic Crystal Micro-Cavities and Miniature Optical Gyroscopes
Tel Aviv University
Ben Z. Steinberg
Ady Shamir
Amir 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 !