Abstract

We have developed a finite-difference time-domain algorithm to simulate a wavelength-scale optical gyroscope based on a circular microdisk. In addition to the frequency shift, the rotation-induced changes in the quality factor and far-field emission pattern of the whispering gallery modes are studied. Compared to the closed cavity of same size and shape, an open cavity displays a larger frequency splitting by rotation, due to an increase of the mode size. When the disk dimension is on the order of the optical wavelength, the relative change in quality factor by rotation is over an order of magnitude larger than that in resonant frequency, due to enhanced evanescent tunneling of light. These results point to multiple methods for rotation sensing, monitoring the lasing threshold and the output power or the far-field emission pattern of a rotating microdisk laser, which can be much more sensitive than the Sagnac effect in ultrasmall optical gyroscopes.

© 2012 Optical Society of America

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References

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  1. C. Ciminelli, F. Dell’Olio, C. E. Campanella, and M. N. Armenise, “Photonic technologies for angular velocity sensing,” Adv. Opt. Photon. 2, 370–404 (2010).
    [CrossRef]
  2. B. Z. Steinberg, “Rotating photonic crystals: a medium for compact optical gyroscopes,” Phys. Rev. E 71, 056621 (2005).
    [CrossRef]
  3. 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]
  4. A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, “Optical gyroscope with whispering gallery mode optical cavities,” Opt. Commun. 233, 107–112 (2004).
    [CrossRef]
  5. J. Scheuer, and A. Yariv, “Sagnac effect in coupled resonator slow light waveguide structures,” Phys. Rev. Lett. 96, 053901 (2006).
    [CrossRef]
  6. C. Peng, Z. Li, and A. Xu, “Optical gyroscope based on a coupled resonator with the all-optical analogous property of electromagnetically induced transparency,” Opt. Express 15, 3864–3875 (2007).
    [CrossRef]
  7. C. Sorrentino, J. R. E. Toland, and C. P. Search, “Ultra-sensitive chip scale Sagnac gyroscope based on periodically modulated coupling of a coupled resonator optical waveguide,” Opt. Express 20, 354–363 (2012).
    [CrossRef]
  8. R. Novitski, B. Z. Steinberg, and J. Scheuer, “Losses in rotating degenerate cavities and a coupled-resonator optical-waveguide rotation sensor,” Phys. Rev. A 85, 023813 (2012).
    [CrossRef]
  9. B. Z. Steinberg, J. Scheuer, and A. Boag, “Rotation-induced superstructure in slow-light waveguides with mode-degeneracy: optical gyroscopes with exponential sensitivity,” J. Opt. Soc. Am. B 24, 1216–1224 (2007).
    [CrossRef]
  10. S. Sunada and T. Harayama, “Sagnac effect in resonant microcavities,” Phys. Rev. A 74, 021801(R) (2006).
    [CrossRef]
  11. T. Harayama, S. Sunada, and T. Miyasaka, “Wave chaos in rotating optical cavities,” Phys. Rev. E 76, 016212 (2007).
    [CrossRef]
  12. S. Sunada and T. Harayama, “Design of resonant microcavities: application to optical gyroscopes,” Opt. Express 15, 16245–16254 (2007).
    [CrossRef]
  13. J. Scheuer, “Direct rotation-induced intensity modulation in circular Bragg micro-lasers,” Opt. Express 15, 15053–15059 (2007).
    [CrossRef]
  14. M. Skorobogatiy and J. D. Joannopoulos, “Rigid vibrations of a photonic crystal and induced interband transitions,” Phys. Rev. B 61, 5293 (2000).
    [CrossRef]
  15. M. Skorobogatiy and J. D. Joannopoulos, “Photon modes in photonic crystals undergoing rigid vibrations and rotations,” Phys. Rev. B 61, 15554 (2000).
    [CrossRef]
  16. B. Z. Steinberg, “Two-Dimensional Green’s function theory for the electrodynamics of a rotating medium,” Phys. Rev. E 74, 016608 (2006).
    [CrossRef]
  17. C. Peng, R. Hui, X. Luo, Z. Li, and A. Xu, “Finite-difference time-domain algorithm for modeling Sagnac effect in rotating optical elements,” Opt. Express 16, 5227–5240 (2008).
    [CrossRef]
  18. T. Shiozawa, “Phenomenological and electron-theoretical study of the electrodynamics of rotating systems,” Proc. IEEE 61, 1694–1702 (1973).
    [CrossRef]
  19. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 1995).

2012

R. Novitski, B. Z. Steinberg, and J. Scheuer, “Losses in rotating degenerate cavities and a coupled-resonator optical-waveguide rotation sensor,” Phys. Rev. A 85, 023813 (2012).
[CrossRef]

C. Sorrentino, J. R. E. Toland, and C. P. Search, “Ultra-sensitive chip scale Sagnac gyroscope based on periodically modulated coupling of a coupled resonator optical waveguide,” Opt. Express 20, 354–363 (2012).
[CrossRef]

2010

2008

2007

2006

B. Z. Steinberg, “Two-Dimensional Green’s function theory for the electrodynamics of a rotating medium,” Phys. Rev. E 74, 016608 (2006).
[CrossRef]

S. Sunada and T. Harayama, “Sagnac effect in resonant microcavities,” Phys. Rev. A 74, 021801(R) (2006).
[CrossRef]

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

2005

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

2004

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, “Optical gyroscope with whispering gallery mode optical cavities,” Opt. Commun. 233, 107–112 (2004).
[CrossRef]

2000

M. Skorobogatiy and J. D. Joannopoulos, “Rigid vibrations of a photonic crystal and induced interband transitions,” Phys. Rev. B 61, 5293 (2000).
[CrossRef]

M. Skorobogatiy and J. D. Joannopoulos, “Photon modes in photonic crystals undergoing rigid vibrations and rotations,” Phys. Rev. B 61, 15554 (2000).
[CrossRef]

1973

T. Shiozawa, “Phenomenological and electron-theoretical study of the electrodynamics of rotating systems,” Proc. IEEE 61, 1694–1702 (1973).
[CrossRef]

Armenise, M. N.

Boag, A.

Campanella, C. E.

Ciminelli, C.

Dell’Olio, F.

Harayama, T.

S. Sunada and T. Harayama, “Design of resonant microcavities: application to optical gyroscopes,” Opt. Express 15, 16245–16254 (2007).
[CrossRef]

T. Harayama, S. Sunada, and T. Miyasaka, “Wave chaos in rotating optical cavities,” Phys. Rev. E 76, 016212 (2007).
[CrossRef]

S. Sunada and T. Harayama, “Sagnac effect in resonant microcavities,” Phys. Rev. A 74, 021801(R) (2006).
[CrossRef]

Hui, R.

Ilchenko, V. S.

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, “Optical gyroscope with whispering gallery mode optical cavities,” Opt. Commun. 233, 107–112 (2004).
[CrossRef]

Joannopoulos, J. D.

M. Skorobogatiy and J. D. Joannopoulos, “Rigid vibrations of a photonic crystal and induced interband transitions,” Phys. Rev. B 61, 5293 (2000).
[CrossRef]

M. Skorobogatiy and J. D. Joannopoulos, “Photon modes in photonic crystals undergoing rigid vibrations and rotations,” Phys. Rev. B 61, 15554 (2000).
[CrossRef]

Li, Z.

Luo, X.

Maleki, L.

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, “Optical gyroscope with whispering gallery mode optical cavities,” Opt. Commun. 233, 107–112 (2004).
[CrossRef]

Matsko, A. B.

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, “Optical gyroscope with whispering gallery mode optical cavities,” Opt. Commun. 233, 107–112 (2004).
[CrossRef]

Miyasaka, T.

T. Harayama, S. Sunada, and T. Miyasaka, “Wave chaos in rotating optical cavities,” Phys. Rev. E 76, 016212 (2007).
[CrossRef]

Novitski, R.

R. Novitski, B. Z. Steinberg, and J. Scheuer, “Losses in rotating degenerate cavities and a coupled-resonator optical-waveguide rotation sensor,” Phys. Rev. A 85, 023813 (2012).
[CrossRef]

Peng, C.

Savchenkov, A. A.

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, “Optical gyroscope with whispering gallery mode optical cavities,” Opt. Commun. 233, 107–112 (2004).
[CrossRef]

Scheuer, J.

R. Novitski, B. Z. Steinberg, and J. Scheuer, “Losses in rotating degenerate cavities and a coupled-resonator optical-waveguide rotation sensor,” Phys. Rev. A 85, 023813 (2012).
[CrossRef]

B. Z. Steinberg, J. Scheuer, and A. Boag, “Rotation-induced superstructure in slow-light waveguides with mode-degeneracy: optical gyroscopes with exponential sensitivity,” J. Opt. Soc. Am. B 24, 1216–1224 (2007).
[CrossRef]

J. Scheuer, “Direct rotation-induced intensity modulation in circular Bragg micro-lasers,” Opt. Express 15, 15053–15059 (2007).
[CrossRef]

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

Search, C. P.

Shiozawa, T.

T. Shiozawa, “Phenomenological and electron-theoretical study of the electrodynamics of rotating systems,” Proc. IEEE 61, 1694–1702 (1973).
[CrossRef]

Skorobogatiy, M.

M. Skorobogatiy and J. D. Joannopoulos, “Photon modes in photonic crystals undergoing rigid vibrations and rotations,” Phys. Rev. B 61, 15554 (2000).
[CrossRef]

M. Skorobogatiy and J. D. Joannopoulos, “Rigid vibrations of a photonic crystal and induced interband transitions,” Phys. Rev. B 61, 5293 (2000).
[CrossRef]

Sorrentino, C.

Steinberg, B. Z.

R. Novitski, B. Z. Steinberg, and J. Scheuer, “Losses in rotating degenerate cavities and a coupled-resonator optical-waveguide rotation sensor,” Phys. Rev. A 85, 023813 (2012).
[CrossRef]

B. Z. Steinberg, J. Scheuer, and A. Boag, “Rotation-induced superstructure in slow-light waveguides with mode-degeneracy: optical gyroscopes with exponential sensitivity,” J. Opt. Soc. Am. B 24, 1216–1224 (2007).
[CrossRef]

B. Z. Steinberg, “Two-Dimensional Green’s function theory for the electrodynamics of a rotating medium,” Phys. Rev. E 74, 016608 (2006).
[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]

Sunada, S.

T. Harayama, S. Sunada, and T. Miyasaka, “Wave chaos in rotating optical cavities,” Phys. Rev. E 76, 016212 (2007).
[CrossRef]

S. Sunada and T. Harayama, “Design of resonant microcavities: application to optical gyroscopes,” Opt. Express 15, 16245–16254 (2007).
[CrossRef]

S. Sunada and T. Harayama, “Sagnac effect in resonant microcavities,” Phys. Rev. A 74, 021801(R) (2006).
[CrossRef]

Taflove, A.

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 1995).

Toland, J. R. E.

Xu, A.

Yariv, A.

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

Adv. Opt. Photon.

J. Opt. Soc. Am. B

Opt. Commun.

A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, “Optical gyroscope with whispering gallery mode optical cavities,” Opt. Commun. 233, 107–112 (2004).
[CrossRef]

Opt. Express

Phys. Rev. A

R. Novitski, B. Z. Steinberg, and J. Scheuer, “Losses in rotating degenerate cavities and a coupled-resonator optical-waveguide rotation sensor,” Phys. Rev. A 85, 023813 (2012).
[CrossRef]

S. Sunada and T. Harayama, “Sagnac effect in resonant microcavities,” Phys. Rev. A 74, 021801(R) (2006).
[CrossRef]

Phys. Rev. B

M. Skorobogatiy and J. D. Joannopoulos, “Rigid vibrations of a photonic crystal and induced interband transitions,” Phys. Rev. B 61, 5293 (2000).
[CrossRef]

M. Skorobogatiy and J. D. Joannopoulos, “Photon modes in photonic crystals undergoing rigid vibrations and rotations,” Phys. Rev. B 61, 15554 (2000).
[CrossRef]

Phys. Rev. E

B. Z. Steinberg, “Two-Dimensional Green’s function theory for the electrodynamics of a rotating medium,” Phys. Rev. E 74, 016608 (2006).
[CrossRef]

T. Harayama, S. Sunada, and T. Miyasaka, “Wave chaos in rotating optical cavities,” Phys. Rev. E 76, 016212 (2007).
[CrossRef]

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

Phys. Rev. Lett.

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

Proc. IEEE

T. Shiozawa, “Phenomenological and electron-theoretical study of the electrodynamics of rotating systems,” Proc. IEEE 61, 1694–1702 (1973).
[CrossRef]

Other

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 1995).

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Figures (6)

Fig. 1.
Fig. 1.

Frequency splitting between the CW and CCW modes in a circular dielectric microdisk of radius R=590nm and n=3 in free space (n=1) as a function of the rotation speed Ω. The circles are the FDTD simulation results for a rotating closed cavity for a WG mode (l=1, m=7, λ=1009.8nm) and the solid line is the analytical result from Eq. (20). Squares are the FDTD simulation results for the same mode (l=1, m=7, λ=1131.48nm) in the rotating microdisk with open boundary. Dotted line represents the frequency shifts obtained from the FDTD simulation of the stationary microdisk with the effective indices of refraction (inside and outside the disk) that include the rotation-induced changes.

Fig. 2.
Fig. 2.

Mode size s as a function of rotation speed Ω. The circles (connected by solid line) are for the mode (l=1, m=7, λ=1009.8nm) in a closed cavity of R=590nm and n=3 in free space. The squares (connected by dotted line) are for the same mode (l=1, m=7, λ=1131.48nm) in the open cavity. The inset shows the electric field distribution of this mode in the open cavity (left half circle) and the closed cavity (right half circle).

Fig. 3.
Fig. 3.

Calculated quality factors for the l=1, m=±7 modes as a function of Ω. The squares are obtained from the FDTD simulation of a rotating microdisk of R=590nm and n=3 in free space, and the crosses from the stationary microdisk with the effective indices of refraction (inside and outside the disk) that include the rotation-induced changes. The dotted lines are linear fits showing that the Q changes exponentially with rotation speed.

Fig. 4.
Fig. 4.

Relative change in Q factor ΔQ/Q0 (squares) and the normalized frequency splitting Δω/ω0 (circles) as a function of the rotation speed Ω for the WG mode of l=1, m=7 in the dielectric microdisk of R=590nm and n=3. The dotted lines are the linear fits. The slope of ΔQ/Q0 versus Ω is 2.16×1012 and for Δω/ω0 is 1.12×1013.

Fig. 5.
Fig. 5.

(a) Normalized frequency splitting Δω/ω0 due to rotation for different refractive indices nodisk of the dielectric microdisk (R=590nm, l=1, m=±7). The lower inset shows the slopes from the linear fits for different nodisk; (b) The relative change of Q factor ΔQ/Q0 due to rotation for the CW mode (m=7) for different refractive indices nodisk of the same disk. The lower inset shows the slopes from the linear fits for different nodisk.

Fig. 6.
Fig. 6.

(a) Far field pattern for the WG mode of l=1 and m=±7 in the stationary microdisk of R=590nm and n=3; (b) far field pattern of the same disk which rotates at Ω=1.9×1012rad/s; (c) the far-field intensity modulation depth γ as a function of Ω; (d) Bessel decomposition of the intracavity field for the same mode as in (a); (e) Bessel decomposition of field inside the rotating cavity for the mode in (b); (f) ratio of the amplitudes of the Bessel coefficients as a function of Ω.

Equations (25)

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εE⃗=D⃗+c2Ω⃗×r⃗×H⃗,
μH⃗=B⃗c2Ω⃗×r⃗×E⃗.
εEz=Dzc2(ΩyHy+ΩxHx),
μHx=Bxc2(ΩxEz),
μHy=Byc2(ΩyEz).
Dzt=HyxHxy,
εEz=Dzc2(ΩyHy+ΩxHx),
Bxt=Ezy,
Byt=Ezx,
μHx=Bxc2(ΩxEz),
μHy=Byc2(ΩyEz).
εi1/2,j+1/2Ez|i1/2,j+1/2n=Dz|i1/2,j+1/2nc2(ΩyHy|i1/2,j+1/2n+ΩxHx|i1/2,j+1/2n).
εi1/2,j+1/2Ez|i1/2,j+1/2n=εi1/2,j+1/2(Ez|i1/2,j+1/2n+1/2+Ez|i1/2,j+1/2n1/2)/2.
Hy|i1/2,j+1/2n=(Hy|i1,j+1/2n+Hy|i,j+1/2n)/2.
εi1/2,j+1/2Ez|i1/2,j+1/2n+1/2=εi1/2,j+1/2Ez|i1/2,j+1/2n1/2+(Dz|i1/2,j+1/2n+1/2+Dz|i1/2,j+1/2n1/2)c2(Ωy(Hy|i1,j+1/2n+Hy|i,j+1/2n)+Ωx(Hx|i1/2,j+1n+Hx|i1/2,jn)).
μi1/2,j+1Hx|i1/2,j+1n+1=μi1/2,j+1Hx|i1/2,j+1n+(Bx|i1/2,j+1n+1+Bx|i1/2,j+1n)c2(Ωx(Ez|i1/2,j+1/2n+1/2+Ez|i1/2,j+3/2n+1/2)),
μi,j+1/2Hy|i,j+1/2n+1=μi,j+1/2Hy|i,j+1/2n+(By|i,j+1/2n+1+By|i,j+1/2n)c2(Ωy(Ez|i+1/2,j+1/2n+1/2+Ez|i1/2,j+1/2n+1/2)).
eglobal|n=ij|Ez,T|i,jnEz,B|i,jn|2
[2r2+(1r)(r)+1r22θ2+2ikΩcθ+n2k2]Ez=0.
[2r2+(1r)(r)m2r2+Km2]f(r)=0,
Km2=k2[n2+2m(Ωω)].
Δω=2(m/n2)Ω.
s=(|Ez(x,y)|2dxdy)2A(|Ez(x,y)|4dxdy),
nΩdisknΩoutside(n0diskn0outside)+(mΩω)[1nodisk1nooutside].
nΩdisknΩoutside(n0disk1)+(mΩω)[1nodisk1].

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