Abstract

Highly directional emissions from whispering-gallery modes are demonstrated in deformed nonaxisymmetric fused-silica microspheres. Ray trajectory analysis indicates that ray dynamics in a three-dimensional resonator differs qualitatively from that of a two-dimensional system, with Arnol’d diffusion playing an essential role in the three-dimensional system.

© 2001 Optical Society of America

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References

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  1. A. Mekis, J. U. Noeckel, G. Chen, A. D. Stone, and R. K. Chang, Phys. Rev. Lett. 75, 2682 (1995).
    [CrossRef] [PubMed]
  2. J. U. Noeckel, A. D. Stone, G. Chen, H. L. Grossman, and R. K. Chang, Opt. Lett. 19, 1609 (1996).
    [CrossRef]
  3. J. U. Noeckel and A. D. Stone, Nature 385, 45 (1997).
    [CrossRef]
  4. See, for example, L. E. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics, 2nd ed. (Springer-Verlag, New York, 1992).
    [CrossRef]
  5. V. I. Arnol’d, Russ. Math. Surv. 18, 85 (1964).
    [CrossRef]
  6. For a more recent discussion, see, for example, C. Froeschle, M. Guzzo, and E. Lega, Science 289, 2108 (2000).
    [CrossRef] [PubMed]
  7. V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, Phys. Lett. A 137, 393 (1989).
    [CrossRef]
  8. S. C. Chang, J. U. Noeckel, R. K. Chang, and A. D. Stone, J. Opt. Soc. Am. B 17, 1828 (2000).
    [CrossRef]

2000 (2)

For a more recent discussion, see, for example, C. Froeschle, M. Guzzo, and E. Lega, Science 289, 2108 (2000).
[CrossRef] [PubMed]

S. C. Chang, J. U. Noeckel, R. K. Chang, and A. D. Stone, J. Opt. Soc. Am. B 17, 1828 (2000).
[CrossRef]

1997 (1)

J. U. Noeckel and A. D. Stone, Nature 385, 45 (1997).
[CrossRef]

1996 (1)

1995 (1)

A. Mekis, J. U. Noeckel, G. Chen, A. D. Stone, and R. K. Chang, Phys. Rev. Lett. 75, 2682 (1995).
[CrossRef] [PubMed]

1989 (1)

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, Phys. Lett. A 137, 393 (1989).
[CrossRef]

1964 (1)

V. I. Arnol’d, Russ. Math. Surv. 18, 85 (1964).
[CrossRef]

Arnol’d, V. I.

V. I. Arnol’d, Russ. Math. Surv. 18, 85 (1964).
[CrossRef]

Braginsky, V. B.

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, Phys. Lett. A 137, 393 (1989).
[CrossRef]

Chang, R. K.

Chang, S. C.

Chen, G.

J. U. Noeckel, A. D. Stone, G. Chen, H. L. Grossman, and R. K. Chang, Opt. Lett. 19, 1609 (1996).
[CrossRef]

A. Mekis, J. U. Noeckel, G. Chen, A. D. Stone, and R. K. Chang, Phys. Rev. Lett. 75, 2682 (1995).
[CrossRef] [PubMed]

Froeschle, C.

For a more recent discussion, see, for example, C. Froeschle, M. Guzzo, and E. Lega, Science 289, 2108 (2000).
[CrossRef] [PubMed]

Gorodetsky, M. L.

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, Phys. Lett. A 137, 393 (1989).
[CrossRef]

Grossman, H. L.

Guzzo, M.

For a more recent discussion, see, for example, C. Froeschle, M. Guzzo, and E. Lega, Science 289, 2108 (2000).
[CrossRef] [PubMed]

Ilchenko, V. S.

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, Phys. Lett. A 137, 393 (1989).
[CrossRef]

Lega, E.

For a more recent discussion, see, for example, C. Froeschle, M. Guzzo, and E. Lega, Science 289, 2108 (2000).
[CrossRef] [PubMed]

Lichtenberg, L. E.

See, for example, L. E. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics, 2nd ed. (Springer-Verlag, New York, 1992).
[CrossRef]

Lieberman, M. A.

See, for example, L. E. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics, 2nd ed. (Springer-Verlag, New York, 1992).
[CrossRef]

Mekis, A.

A. Mekis, J. U. Noeckel, G. Chen, A. D. Stone, and R. K. Chang, Phys. Rev. Lett. 75, 2682 (1995).
[CrossRef] [PubMed]

Noeckel, J. U.

Stone, A. D.

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

Nature (1)

J. U. Noeckel and A. D. Stone, Nature 385, 45 (1997).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. A (1)

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, Phys. Lett. A 137, 393 (1989).
[CrossRef]

Phys. Rev. Lett. (1)

A. Mekis, J. U. Noeckel, G. Chen, A. D. Stone, and R. K. Chang, Phys. Rev. Lett. 75, 2682 (1995).
[CrossRef] [PubMed]

Russ. Math. Surv. (1)

V. I. Arnol’d, Russ. Math. Surv. 18, 85 (1964).
[CrossRef]

Science (1)

For a more recent discussion, see, for example, C. Froeschle, M. Guzzo, and E. Lega, Science 289, 2108 (2000).
[CrossRef] [PubMed]

Other (1)

See, for example, L. E. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics, 2nd ed. (Springer-Verlag, New York, 1992).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Three orthogonal perspectives of a deformed fused-silica microsphere. The z axis lies along the line connecting the centers of the two original spheres, and the x axis is along the stem.

Fig. 2
Fig. 2

(a) Far-field emission pattern for WG modes excited near the yz plane, where θ is the polar angle with respect to the z axis. The microsphere used has a diameter near 180 μm. For the filled squares and the open circles, ϵ, the in-plane deformation defined in the text, is 0.7% and 0.5% respectively. The emission intensity for ϵ=0.5% is magnified by a factor of 10. Inset, schematics of the excitation geometry.

Fig. 3
Fig. 3

Ray trajectory in the SOS projected onto three different planes. θ and ϕ are the polar angle and the azimuthal angle, respectively. (a) Trajectories of a few hundred reflections for an axisymmetric spheroid. Arrow, the trajectory that has the same initial condition as the trajectories in (b)–(d). (b)–(d) Trajectories launched near the yz plane for a nonaxisymmetric spheroid with 1–5000, 310,000–315,000, and 582,500–587,500 reflections displayed in (b), (c), and (d), respectively. The horizontal line in (d) indicates sinχc for fused silica. The parameters of the spheroids are discussed in the text.

Equations (1)

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x=1+dcosusinv1-qcos2u×1+ecos2vcosusinv/1+q,y=1+dcosusinv1-qcos2u×1+ecos2vsinusinv/1+q,z=1+ecos2vcosv.

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