Abstract

We study theoretically and numerically the effect of rotation on resonant frequencies of microcavities in a rotating frame of reference. Cavity rotation causes the shifts of the resonant frequencies proportional to the rotation rate if it is larger than a certain value. Below the value, a region of rotation rate exists where there is no resulting the frequency shifts proportional to the rotation rate. We show that designing cavity symmetry as C nv (n≥3) can eliminate this region.

© 2007 Optical Society of America

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References

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  1. E. J. Post, "Sagnac effect," Rev. Mod. Phys. 39, 475-493 (1967).
    [CrossRef]
  2. W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, "The ring laser gyro," Rev. Mod. Phys. 57, 61-104 (1985).
    [CrossRef]
  3. F. Aronowitz, in Laser Applications, M. Ross, ed. (Academic, New York, 1971), Vol. 1, pp 133-200.
  4. L. N. Menegozzi and W. E. Lamb, Jr., "Theory of a ring laser," Phys. Rev. A 8, 2103-2125 (1971).
    [CrossRef]
  5. A. Kuriyagawa and S. Mori, "Ring laser and ring interferometer in accelerated systems," Phys. Rev. D 20, 1290-1293 (1979).
    [CrossRef]
  6. H. J. Arditty and H. C. Lefevre, "Sagnac effect in fiber gyroscopes," Opt. Lett. 6, 401-403 (1981).
    [CrossRef] [PubMed]
  7. W. M. Macek and D. T. M. Davis, "Rotation rate sensing with traveling-wave ring lasers," Appl. Phys. Lett. 2, 67-68 (1963).
    [CrossRef]
  8. J. L. Anderson and J. W. Ryon, "Electromagnetic radiation in accelerated systems," Phys. Rev. 181, 1765-1774 (1969).
    [CrossRef]
  9. E. Landau and E. Lifshits, The Classical Theory of Fields (Butterworth-Heinemann. Oxford, 1975).
  10. C. V. Heer, "Resonant frequencies of an electromagnetic cavity in an accelerated system of reference," Phys. Rev. 134, A799-804 (1964).
    [CrossRef]
  11. Y. Yamamoto and R. E. Slusher, "Optical processes in microcavities," Phys. Today 46, 66-73 (1993).
  12. R. K. Chang and A. J. Campillo, eds., Optical processes in microcavities (World Scientific Publishing, Singapore, New Jersey, Hong Kong, 1996).
    [CrossRef]
  13. <other>13. S. Sunada and T. Harayama, "Sagnac effect in resonant microcavities," Phys. Rev. A 74, 021801(R) (2006);T. Harayama, S. Sunada, and T. Miyasaka, "Wave chaos in rotating optical cavities," Phys. Rev. E 76, 016212 (2007).</other>
    [CrossRef]
  14. Equation (7) agrees with a conventional expression for frequency difference: (4A⌉)/(cnP)∧ by applying assumptions (i) and (ii) to Eq. (7). In the above, A is the area bounded by the optical path, P is the perimeter of the optical path, and ⌉is the eigen frequency of the non-rotating ring cavity approximated as ⌉¡Ö c(k1 +k2)/2.
  15. Here,CW - (CCW-) rotating waves mean those having mostly negative (positive) angular momentum components, as in the definition of Eq. (23) in Ref. [13].
  16. R. J. C. Spreeuw, R. Centeno Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, "Mode coupling in a He-Ne ring laser with backscattering," Phys. Rev. A 42, 4315-4324 (1990).
    [CrossRef] [PubMed]
  17. G. Hackenbroich, E. Narimanov, and A. D. Stone, "Quantum perturbation theory for the level splitting in billiards," Phys. Rev. E 57, R5-R8 (1998).
    [CrossRef]
  18. M. Choi, T. Fukushima, and T. Harayama, "Alternative osicillation with □ difference in quasi-stadium laser diodes," in CLEO/Pacific Rim 2007, WGI-3 (2007).
  19. Morton Hamermesh, Group Theory and Its Application to Physical Problems, (Dover New York, 1989).
  20. S. Sakanaka, "Classification of eigenmodes in rf-cavities using the group theory," Phys. Rev. ST Accel. Beams 8, 072002 (2005).
    [CrossRef]
  21. M. Sorel, P. J. Laybourn, G. Giuliani, and S. Donati, "Progress on the GaAlAs ring laser gyroscope," Alta Frequenza, Rivista Di Electonica,  1045-48, (1998);M. Sorel, G. Giuliani, A. Scire, R. Miglierina, S. Donati, and P. J. R. Laybourn, "Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model," IEEE J. Quantum Electron. 39, 1187-1195 (2003).
  22. H. Cao, C. Liu, H. Ling, H. Deug, M. Benavidez, G. M. Peake, G. A. Smolyakov, P. G. Eliseev, and M. Osinski, "Frequency beating between monolithically integrated semiconductor ring lasers," Appl. Phys. Lett. 86, 041101 (2005).
    [CrossRef]
  23. F. C. Rarnberg, R. Johnson, W. Ellerbusch, B. Schermer, and R. Gopinath, "Cavity element for resonant micro optical gyroscope," IEEE Trans. Aerosp. Electron. Syst. 15, 33-36 (2000).
    [CrossRef]
  24. M. N. Armenise, V. M. N. Passaro, F. Leonardis, and M. Armenise, " Modeling and design of a novel miniaturized integrated optical sensor for gyroscope systems," J. Lightwave Technol. 19, 1476 (2001).
    [CrossRef]
  25. Ben Zion Steinberg, "Rotating photonic crystals: a medium for compact optical gyroscopes," Phys. Rev. E 71, 056621, (2005).
    [CrossRef]

2006 (1)

<other>13. S. Sunada and T. Harayama, "Sagnac effect in resonant microcavities," Phys. Rev. A 74, 021801(R) (2006);T. Harayama, S. Sunada, and T. Miyasaka, "Wave chaos in rotating optical cavities," Phys. Rev. E 76, 016212 (2007).</other>
[CrossRef]

2005 (3)

S. Sakanaka, "Classification of eigenmodes in rf-cavities using the group theory," Phys. Rev. ST Accel. Beams 8, 072002 (2005).
[CrossRef]

H. Cao, C. Liu, H. Ling, H. Deug, M. Benavidez, G. M. Peake, G. A. Smolyakov, P. G. Eliseev, and M. Osinski, "Frequency beating between monolithically integrated semiconductor ring lasers," Appl. Phys. Lett. 86, 041101 (2005).
[CrossRef]

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

2001 (1)

2000 (1)

F. C. Rarnberg, R. Johnson, W. Ellerbusch, B. Schermer, and R. Gopinath, "Cavity element for resonant micro optical gyroscope," IEEE Trans. Aerosp. Electron. Syst. 15, 33-36 (2000).
[CrossRef]

1998 (2)

G. Hackenbroich, E. Narimanov, and A. D. Stone, "Quantum perturbation theory for the level splitting in billiards," Phys. Rev. E 57, R5-R8 (1998).
[CrossRef]

M. Sorel, P. J. Laybourn, G. Giuliani, and S. Donati, "Progress on the GaAlAs ring laser gyroscope," Alta Frequenza, Rivista Di Electonica,  1045-48, (1998);M. Sorel, G. Giuliani, A. Scire, R. Miglierina, S. Donati, and P. J. R. Laybourn, "Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model," IEEE J. Quantum Electron. 39, 1187-1195 (2003).

1993 (1)

Y. Yamamoto and R. E. Slusher, "Optical processes in microcavities," Phys. Today 46, 66-73 (1993).

1990 (1)

R. J. C. Spreeuw, R. Centeno Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, "Mode coupling in a He-Ne ring laser with backscattering," Phys. Rev. A 42, 4315-4324 (1990).
[CrossRef] [PubMed]

1985 (1)

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, "The ring laser gyro," Rev. Mod. Phys. 57, 61-104 (1985).
[CrossRef]

1981 (1)

1979 (1)

A. Kuriyagawa and S. Mori, "Ring laser and ring interferometer in accelerated systems," Phys. Rev. D 20, 1290-1293 (1979).
[CrossRef]

1971 (1)

L. N. Menegozzi and W. E. Lamb, Jr., "Theory of a ring laser," Phys. Rev. A 8, 2103-2125 (1971).
[CrossRef]

1969 (1)

J. L. Anderson and J. W. Ryon, "Electromagnetic radiation in accelerated systems," Phys. Rev. 181, 1765-1774 (1969).
[CrossRef]

1967 (1)

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

1964 (1)

C. V. Heer, "Resonant frequencies of an electromagnetic cavity in an accelerated system of reference," Phys. Rev. 134, A799-804 (1964).
[CrossRef]

1963 (1)

W. M. Macek and D. T. M. Davis, "Rotation rate sensing with traveling-wave ring lasers," Appl. Phys. Lett. 2, 67-68 (1963).
[CrossRef]

Appl. Phys. Lett. (2)

W. M. Macek and D. T. M. Davis, "Rotation rate sensing with traveling-wave ring lasers," Appl. Phys. Lett. 2, 67-68 (1963).
[CrossRef]

H. Cao, C. Liu, H. Ling, H. Deug, M. Benavidez, G. M. Peake, G. A. Smolyakov, P. G. Eliseev, and M. Osinski, "Frequency beating between monolithically integrated semiconductor ring lasers," Appl. Phys. Lett. 86, 041101 (2005).
[CrossRef]

IEEE Trans. Aerosp. Electron. Syst. (1)

F. C. Rarnberg, R. Johnson, W. Ellerbusch, B. Schermer, and R. Gopinath, "Cavity element for resonant micro optical gyroscope," IEEE Trans. Aerosp. Electron. Syst. 15, 33-36 (2000).
[CrossRef]

J. Lightwave Technol. (1)

Opt. Lett. (1)

Phys. Rev. (2)

J. L. Anderson and J. W. Ryon, "Electromagnetic radiation in accelerated systems," Phys. Rev. 181, 1765-1774 (1969).
[CrossRef]

C. V. Heer, "Resonant frequencies of an electromagnetic cavity in an accelerated system of reference," Phys. Rev. 134, A799-804 (1964).
[CrossRef]

Phys. Rev. A (2)

R. J. C. Spreeuw, R. Centeno Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, "Mode coupling in a He-Ne ring laser with backscattering," Phys. Rev. A 42, 4315-4324 (1990).
[CrossRef] [PubMed]

L. N. Menegozzi and W. E. Lamb, Jr., "Theory of a ring laser," Phys. Rev. A 8, 2103-2125 (1971).
[CrossRef]

Phys. Rev. D (1)

A. Kuriyagawa and S. Mori, "Ring laser and ring interferometer in accelerated systems," Phys. Rev. D 20, 1290-1293 (1979).
[CrossRef]

Phys. Rev. E (3)

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

G. Hackenbroich, E. Narimanov, and A. D. Stone, "Quantum perturbation theory for the level splitting in billiards," Phys. Rev. E 57, R5-R8 (1998).
[CrossRef]

<other>13. S. Sunada and T. Harayama, "Sagnac effect in resonant microcavities," Phys. Rev. A 74, 021801(R) (2006);T. Harayama, S. Sunada, and T. Miyasaka, "Wave chaos in rotating optical cavities," Phys. Rev. E 76, 016212 (2007).</other>
[CrossRef]

Phys. Rev. ST Accel. Beams (1)

S. Sakanaka, "Classification of eigenmodes in rf-cavities using the group theory," Phys. Rev. ST Accel. Beams 8, 072002 (2005).
[CrossRef]

Phys. Today (1)

Y. Yamamoto and R. E. Slusher, "Optical processes in microcavities," Phys. Today 46, 66-73 (1993).

Rev. Mod. Phys. (2)

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

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, "The ring laser gyro," Rev. Mod. Phys. 57, 61-104 (1985).
[CrossRef]

Rivista Di Electonica (1)

M. Sorel, P. J. Laybourn, G. Giuliani, and S. Donati, "Progress on the GaAlAs ring laser gyroscope," Alta Frequenza, Rivista Di Electonica,  1045-48, (1998);M. Sorel, G. Giuliani, A. Scire, R. Miglierina, S. Donati, and P. J. R. Laybourn, "Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model," IEEE J. Quantum Electron. 39, 1187-1195 (2003).

Other (7)

Equation (7) agrees with a conventional expression for frequency difference: (4A⌉)/(cnP)∧ by applying assumptions (i) and (ii) to Eq. (7). In the above, A is the area bounded by the optical path, P is the perimeter of the optical path, and ⌉is the eigen frequency of the non-rotating ring cavity approximated as ⌉¡Ö c(k1 +k2)/2.

Here,CW - (CCW-) rotating waves mean those having mostly negative (positive) angular momentum components, as in the definition of Eq. (23) in Ref. [13].

F. Aronowitz, in Laser Applications, M. Ross, ed. (Academic, New York, 1971), Vol. 1, pp 133-200.

R. K. Chang and A. J. Campillo, eds., Optical processes in microcavities (World Scientific Publishing, Singapore, New Jersey, Hong Kong, 1996).
[CrossRef]

E. Landau and E. Lifshits, The Classical Theory of Fields (Butterworth-Heinemann. Oxford, 1975).

M. Choi, T. Fukushima, and T. Harayama, "Alternative osicillation with □ difference in quasi-stadium laser diodes," in CLEO/Pacific Rim 2007, WGI-3 (2007).

Morton Hamermesh, Group Theory and Its Application to Physical Problems, (Dover New York, 1989).

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Figures (7)

Fig. 1.
Fig. 1.

(Solid line) Optical cavity defined by boundary R(θ)=R 0(1+εcos2θ), where (r,θ) denotes cylindrical coordinates. The parameters of the cavity are set as R 0=6.2866µm, ε=0.12, and n=1. Dashed line denotes a ring trajectory.

Fig. 2.
Fig. 2.

Wave functions of eigenstates localized around ring trajectory shown in Fig. 1. (a) (Dimensionless) eigen-wavenumber nkAR 0 is 49.3380585 and the wave function has odd (odd) parity with respect to horizontal (vertical) axis. (b) Eigen-wavenumber nkBR 0=49.3380615 and even (even) parity with respect to horizontal (vertical) axis. White curves denote cavity boundary.

Fig. 3.
Fig. 3.

(a) (Dimensionless) frequency difference R 0Δω/c versus (dimensionless) angular velocity R 0Ω/c. Frequency difference does not change for R 0Ω/c<R 0Ω th /c(~5×10-8). For R 0Ω/c>R 0Ω th /c, it becomes proportional to angular velocity Ω.

Fig. 4.
Fig. 4.

Wave functions of rotating cavity with (dimensionless) angular velocity R 0Ω/c≈6.28×10-5(>R 0Ω th /c) respectively corresponding to (a) mode A and (b) mode B.

Fig. 5.
Fig. 5.

Four parity symmetry classes of a C 3v symmetric cavity defined by the boundary R(θ)=R 0(1+εcos3θ), where ε=0.065. Even (odd) symmetry is marked by dashed (solid) lines.

Fig. 6.
Fig. 6.

(a-b) Degenerate standing-wavefunctions of (dimensionless) eigen-wavenumber nkR 0=50.220063 in non-rotating cavity. (c–d) Wavefunctions of rotating cavity with R 0Ω/c≈6.28×10-11. White curves denote cavity boundary.

Fig. 7.
Fig. 7.

(Dimensionless) frequency difference R 0Δω/c v.s. (dimensionless) angular velocity R 0Ω/c.

Equations (12)

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( xy 2 + n 2 k 2 ) ψ 2 ik ( h · ) ψ = 0 ,
Ω th = n 2 D [ ψ 1 ( y x x y ) ψ 2 ] dxdy 1 c Δ k 0 ,
δ k = 1 n 2 k 1 D [ ψ 1 ( h · ) ψ 1 ] dxdy = 0
c i = 2 i k 1 D [ ψ i ( h · ) ψ 1 ] dxdy n 2 ( k 1 2 k i 2 ) .
ψ ± = 1 2 ( ψ 1 ± i ψ 2 ) ,
k ± = k 1 + k 2 2 ± 1 n 2 D [ ψ 1 ( h · ) ψ 2 ] dxdy .
Δ ω = 2 D [ ψ 1 ( y x x y ) ψ 2 ] dxdy Ω n 2 ,
Δ k 0 = 0 .
σ ψ ± = ± ψ ± .
R n k ψ ± R n k ψ ± .
σ ( R n k ψ R n k ψ ) = R n k σ ψ R n k σ ψ
= + ( R n k ψ R n k ψ ) ,

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