Abstract

We study wave propagation in a rotating slow-light structure with mode degeneracy. The rotation, in conjunction with the mode degeneracy, effectively induces superstructure that significantly modifies the structure’s dispersion relation. It is shown that a rotation-dependent stop band is formed in the center of the slow-light waveguide transmission curve. A light signal of frequency within this stop band that is excited in a finite-length section of such a waveguide decays exponentially with the rotation speed and with the coupled resonator optical waveguide’s total length or total number of degenerate microcavities. This effect can be used for optical gyroscopes with exponential-type sensitivity to rotation.

© 2007 Optical Society of America

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References

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  1. B. Z. Steinberg, 'Rotating photonic crystals: a medium for compact optical gyroscopes,' Phys. Rev. E 71, 056621 (2005).
    [CrossRef]
  2. J. Scheuer and A. Yariv, 'Sagnac effect in coupled resonator slow light waveguide structures,' Phys. Rev. Lett. 96, 053901 (2006).
    [CrossRef] [PubMed]
  3. B. Z. Steinberg, A. Shamir, and A. Boag, 'Sagnac effect in rotating photonic crystal microcavities and miniature optical gyroscopes,' presented at the Conference on Lasers and Electro-Optics and Quantum Electronics and Laser Sciences (CLEO/QELS), Long Beach, Calif., May 21-26, 2006, paper CWL6.
  4. B. Z. Steinberg and A. Boag, 'Splitting of microcavity degenerate modes in rotating photonic crystals--the miniature optical gyroscopes,' J. Opt. Soc. Am. B 24, 142-151 (2007).
    [CrossRef]
  5. E. J. Post, 'Sagnac effect,' Rev. Mod. Phys. 39, 475-493 (1967).
    [CrossRef]
  6. O. Painter, J. Vuckovic, and A. Scherer, 'Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab,' J. Opt. Soc. Am. B 16, 275-285 (1999).
    [CrossRef]
  7. M. Loncar, M. Hochberg, A. Scherer, and Y. Qiu, 'High quality factors and room-temperature lasing in a modified single-defect photonic crystal cavity,' Opt. Lett. 29, 721-723 (2004).
    [CrossRef] [PubMed]
  8. T. Shiozawa, 'Phenomenological and electron-theoretical study of the electrodynamics of rotating systems,' Proc. IEEE 61, 1694-1702 (1973).
    [CrossRef]
  9. J. L. Anderson and J. W. Ryon, 'Electromagnetic radiation in accelerated systems,' Phys. Rev. 181, 1765-1775 (1969).
    [CrossRef]
  10. H. J. Arditty and H. C. Lefevre, 'Sagnac effect in fiber gyroscopes,' Opt. Lett. 6, 401-403 (1981).
    [CrossRef] [PubMed]
  11. A. Boag and B. Z. Steinberg, 'Narrow band microcavity waveguides in photonic crystals,' J. Opt. Soc. Am. A 18, 2799-2805 (2001).
    [CrossRef]
  12. B. Z. Steinberg and A. Boag, 'Propagation in photonic crystal coupled-cavity waveguides with discontinuities in their optical properties,' J. Opt. Soc. Am. B 23, 1442-1450 (2006).
    [CrossRef]
  13. P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, 1985).

2007 (1)

2006 (2)

2005 (1)

B. Z. Steinberg, 'Rotating photonic crystals: a medium for compact optical gyroscopes,' Phys. Rev. E 71, 056621 (2005).
[CrossRef]

2004 (1)

2001 (1)

1999 (1)

1981 (1)

1973 (1)

T. Shiozawa, 'Phenomenological and electron-theoretical study of the electrodynamics of rotating systems,' Proc. IEEE 61, 1694-1702 (1973).
[CrossRef]

1969 (1)

J. L. Anderson and J. W. Ryon, 'Electromagnetic radiation in accelerated systems,' Phys. Rev. 181, 1765-1775 (1969).
[CrossRef]

1967 (1)

E. J. Post, 'Sagnac effect,' Rev. Mod. Phys. 39, 475-493 (1967).
[CrossRef]

Anderson, J. L.

J. L. Anderson and J. W. Ryon, 'Electromagnetic radiation in accelerated systems,' Phys. Rev. 181, 1765-1775 (1969).
[CrossRef]

Arditty, H. J.

Boag, A.

Hochberg, M.

Lancaster, P.

P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, 1985).

Lefevre, H. C.

Loncar, M.

Painter, O.

Post, E. J.

E. J. Post, 'Sagnac effect,' Rev. Mod. Phys. 39, 475-493 (1967).
[CrossRef]

Qiu, Y.

Ryon, J. W.

J. L. Anderson and J. W. Ryon, 'Electromagnetic radiation in accelerated systems,' Phys. Rev. 181, 1765-1775 (1969).
[CrossRef]

Scherer, A.

Scheuer, J.

J. Scheuer and A. Yariv, 'Sagnac effect in coupled resonator slow light waveguide structures,' Phys. Rev. Lett. 96, 053901 (2006).
[CrossRef] [PubMed]

Shamir, A.

B. Z. Steinberg, A. Shamir, and A. Boag, 'Sagnac effect in rotating photonic crystal microcavities and miniature optical gyroscopes,' presented at the Conference on Lasers and Electro-Optics and Quantum Electronics and Laser Sciences (CLEO/QELS), Long Beach, Calif., May 21-26, 2006, paper CWL6.

Shiozawa, T.

T. Shiozawa, 'Phenomenological and electron-theoretical study of the electrodynamics of rotating systems,' Proc. IEEE 61, 1694-1702 (1973).
[CrossRef]

Steinberg, B. Z.

B. Z. Steinberg and A. Boag, 'Splitting of microcavity degenerate modes in rotating photonic crystals--the miniature optical gyroscopes,' J. Opt. Soc. Am. B 24, 142-151 (2007).
[CrossRef]

B. Z. Steinberg and A. Boag, 'Propagation in photonic crystal coupled-cavity waveguides with discontinuities in their optical properties,' J. Opt. Soc. Am. B 23, 1442-1450 (2006).
[CrossRef]

B. Z. Steinberg, 'Rotating photonic crystals: a medium for compact optical gyroscopes,' Phys. Rev. E 71, 056621 (2005).
[CrossRef]

A. Boag and B. Z. Steinberg, 'Narrow band microcavity waveguides in photonic crystals,' J. Opt. Soc. Am. A 18, 2799-2805 (2001).
[CrossRef]

B. Z. Steinberg, A. Shamir, and A. Boag, 'Sagnac effect in rotating photonic crystal microcavities and miniature optical gyroscopes,' presented at the Conference on Lasers and Electro-Optics and Quantum Electronics and Laser Sciences (CLEO/QELS), Long Beach, Calif., May 21-26, 2006, paper CWL6.

Tismenetsky, M.

P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, 1985).

Vuckovic, J.

Yariv, A.

J. Scheuer and A. Yariv, 'Sagnac effect in coupled resonator slow light waveguide structures,' Phys. Rev. Lett. 96, 053901 (2006).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (3)

Opt. Lett. (2)

Phys. Rev. (1)

J. L. Anderson and J. W. Ryon, 'Electromagnetic radiation in accelerated systems,' Phys. Rev. 181, 1765-1775 (1969).
[CrossRef]

Phys. Rev. E (1)

B. Z. Steinberg, 'Rotating photonic crystals: a medium for compact optical gyroscopes,' Phys. Rev. E 71, 056621 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

J. Scheuer and A. Yariv, 'Sagnac effect in coupled resonator slow light waveguide structures,' Phys. Rev. Lett. 96, 053901 (2006).
[CrossRef] [PubMed]

Proc. IEEE (1)

T. Shiozawa, 'Phenomenological and electron-theoretical study of the electrodynamics of rotating systems,' Proc. IEEE 61, 1694-1702 (1973).
[CrossRef]

Rev. Mod. Phys. (1)

E. J. Post, 'Sagnac effect,' Rev. Mod. Phys. 39, 475-493 (1967).
[CrossRef]

Other (2)

B. Z. Steinberg, A. Shamir, and A. Boag, 'Sagnac effect in rotating photonic crystal microcavities and miniature optical gyroscopes,' presented at the Conference on Lasers and Electro-Optics and Quantum Electronics and Laser Sciences (CLEO/QELS), Long Beach, Calif., May 21-26, 2006, paper CWL6.

P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, 1985).

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Figures (9)

Fig. 1
Fig. 1

Splitting of the ring degenerate modes due to rotation.

Fig. 2
Fig. 2

Stationary and rotating CROW consisting of ring resonators.

Fig. 3
Fig. 3

Electric field magnitudes in decibel scale of a doubly degenerate TM microcavity ( M = 2 ) , in a 2D hexagonal PhC. The crystal is made of dielectric cylinders, outlined by the black circles. (a) E 0 ( 1 ) . (b) E 0 ( 2 ) . These modes are nonorthogonal, and E 0 ( 2 ) is a π 3 -rotated replica of E 0 ( 1 ) . (c) The linear combination E 0 ( 1 ) E 0 ( 1 ) + E 0 ( 2 ) . (d) The linear combination E 0 ( 2 ) E 0 ( 1 ) E 0 ( 2 ) . These modes are orthogonal.

Fig. 4
Fig. 4

Electric field of the rotation eigenmodes associated with the PhC microcavity of Fig. 3. The two modes have exactly the same spatial form shown here. E Ω + , H Ω + rotate CW and E Ω , H Ω rotate CCW. They are obtained by the linear combination in Eq. (2.3), using the orthogonal pair shown in Figs. 3c, 3d. All quantities are shown on linear scale. (a) The instantaneous field Re ( E Ω e i ω t ) for ω t = π 4 . (b) The same as (a), but for E Ω + . (c) E Ω . The rotating field forms a ring along which the field power propagates. (d) E Ω + .

Fig. 5
Fig. 5

Instantaneous field of a degenerate modes CROW, made of a set of equally spaced PhC microcavities such as the one discussed in Figs. 3, 4. Counting from left, the modes in the odd- (even-) numbered cavities are rotating in a CCW (CW) direction. A rigid mechanical rotation of the entire structure in the CW direction causes the alternating of the resonance shift δ ω that depends linearly on the rotation rate as shown in Eq. (2.5). Thus, rotation induces a modulation of the local resonance frequency, similar to the phenomenon described in Fig. 2.

Fig. 6
Fig. 6

Normalized dispersion relation for stationary and rotating CROWs. For the rotating CROW we used δ ω ( Ω ) = 0.1 Δ ω . This value is chosen in order to get a good graphical resolution of the effect.

Fig. 7
Fig. 7

Normalized transmission for a ring resonator CROW. The rotation induces a stop band in the center of the CROW transmission curve.

Fig. 8
Fig. 8

Transmission in decibels at the center of the rotating CROW stop band (i.e., at ω = 12.16 × 10 14 in Fig. 7), as a function of rotation speed, and for different CROW lengths. Exponential dependence of the transmission value on Ω is evident.

Fig. 9
Fig. 9

Exponential sensitivity of the CROW to rotation, as a function of the CROW length (number of resonators).

Equations (44)

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Θ d H 0 ( m ) ( r ) = k 0 2 H 0 ( m ) ( r ) , k 0 = ω 0 c , m = 1 , 2 , , M ,
Θ d × 1 ϵ d ( r ) × .
H Ω ± = m = 1 2 a m ± H 0 ( m ) , a 2 ± = ± i a 1 ± ,
Θ d H Ω ± ( r ) = k 0 2 H Ω ± ( r ) ,
ω ± = ω 0 ± δ ω ( Ω ) , δ ω ( Ω ) = ω 0 Ω Λ .
Ω = z ̂ Ω .
× E = i ω B , B = 0 ,
× H = i ω D , D = 0 .
D = ϵ E c 2 Ω × r × H ,
B = μ H + c 2 Ω × r × E .
D × E = i ω μ H ,
D × H = i ω ϵ E ,
D i k β ( r ) , k = ω c , β ( r ) = c 1 Ω × r .
Θ H Ω ( r ) = k 2 H Ω ( r ) + i k L Ω H Ω ( r ) .
Θ × 1 ϵ r ( r ) × ,
L Ω H = × β ( r ) ϵ r ( r ) × H + β ( r ) ϵ r ( r ) × × H , β ( r ) = Ω × r c .
H Ω = m A m H m ( r ) , H m ( r ) = { H Ω + ( r r m ) m even H Ω ( r r m ) m odd ,
1 ϵ r ( r ) = 1 ϵ b ( r ) + k d ( r , r k ) ,
d ( r , r k ) = 1 ϵ d ( r r k ) 1 ϵ b ( r ) .
Θ = Θ b + k Θ k ,
Θ b = × 1 ϵ b ( r ) × , Θ k = × d ( r , r k ) × .
( Θ b + Θ m ) H m = ( ω 0 c ) 2 H m .
m A m Θ H m , H n = k 2 m A m H m , H n + i k m A m L Ω H m , H n .
F , G F G ¯ d 3 r , F , F = F 2 ,
( k 0 2 k 2 ) H Ω + 2 A n + m τ m n A m i k m L Ω H m , H n A m = 0 .
τ m n = k m Θ k H m , H n .
L Ω H m , H n = i c 1 Ω ω 0 ϵ 0 ρ ϕ ̂ , H ¯ m × E n + H n × E ¯ m .
L Ω H m , H n { 0 m n i 2 Ω k 0 ϵ 0 ρ ϕ ̂ , Re S m m = n ,
S m E m × H ¯ m = S m 1 = ( 1 ) m S 0 .
L Ω H m , H m = 2 i c 1 H Ω + 2 ( 1 ) m δ ω ( Ω ) , δ ω ( Ω ) = Ω ω 0 ϵ 0 ρ ϕ ̂ , Re S 0 ,
Δ ω 2 ( A n 1 + A n + 1 ) + ( 1 ) n δ ω ( Ω ) A n = ( ω ω 0 ) A n ,
Δ ω = c 2 τ 1 ω 0 H Ω + 2 .
[ 0 0.5 δ ω ( Ω ) Δ ω 0.5 0 0 0.5 δ ω ( Ω ) Δ ω 0.5 0 0 0.5 δ ω ( Ω ) Δ ω 0.5 0 0 0.5 δ ω ( Ω ) Δ ω 0.5 0 ] A = ω ω 0 Δ ω A .
A n = A 0 e i β n ,
ω = ω 0 + Δ ω cos ( β ) .
Δ ω r = δ ω ( Ω ) .
T ( ω = ω 0 , Ω ) = e α ( Ω ω 0 ) N ,
[ × β ϵ r × H m ] H ¯ n = [ ( β ϵ r × H m ) × H ¯ n ] + ( β ϵ r × H m ) ( × H ¯ n ) ,
[ β ϵ r × × H m ] H ¯ n = ( β ϵ r × H ¯ n ) ( × H m ) .
V [ ( β ϵ r × H m ) × H ¯ n ] d 3 x = S = V [ ( β ϵ r × H m ) × H ¯ n ] d s 0 .
L Ω H m , H n = β ϵ r × H m , × H n × H m , β ϵ r × H n .
L Ω H m , H n = β ϵ r , H ¯ m × × H n β ϵ r , H n × × H ¯ m .
L Ω H m , H n = c 1 Ω ρ ϵ r ϕ ̂ , H ¯ m × × H n H n × × H ¯ m .
L Ω H m , H n = i c 1 Ω ω 0 ϵ 0 ρ ϕ ̂ , H ¯ m × E n + H n × E ¯ m .

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