Abstract

A ray-optics model is developed to describe the spoiling of the high-Q (whispering gallery) modes of ring-shaped cavities as they are deformed from perfect circularity. A sharp threshold is found for the onset of Q spoiling as predicted by the Kolmogorov-Arnol’d-Moser (KAM) theorem of nonlinear dynamics. Beyond the critical deformation bc, Q ~ (bbc)−α,α ≈ 2.4–2.6. The escaping light emerges in certain specific directions, which may be predicted.

© 1994 Optical Society of America

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  1. J. F. Owen, P. W. Barber, P. B. Dorain, R. K. Chang, Phys. Rev. Lett. 47, 1075 (1981).
    [CrossRef]
  2. S. C. Hill, R. E. Benner, in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), p. 3.
  3. L. Collot, V. Lefè-Seguin, M. Brune, J. M. Raimond, S. Haroche, Europhys. Lett. 23, 327 (1993).
    [CrossRef]
  4. S. L. McCall, A. F. Levi, R. E Slusher, S. J. Pearton, R. A. Logan, Appl. Phys. Lett. 60, 289 (1992).
    [CrossRef]
  5. B. R. Johnson, J. Opt. Soc. Am. A 10, 343 (1993).
    [CrossRef]
  6. L. E. Reichl, The Transition to Chaos (Springer-Verlag, Berlin, 1992), Chap. 2, p. 39.
  7. D. D. Holm, G. Kovacic, Physica D 51, 177 (1991).
    [CrossRef]
  8. M. Robnik, J. Phys. A 16,3971 (1983); M. Robnik, J. Phys. A 17,1049 (1983).
    [CrossRef] [PubMed]
  9. M. V. Berry, M. Robnik, J. Phys. A 19,649 (1986).
    [CrossRef]
  10. M. V. Berry, Eur. J. Phys. 2, 91 (1981).
    [CrossRef]
  11. D. Q. Chowdhury, P. W. Barber, S. C. Hill, Appl. Opt. 31, 3518 (1992).
    [CrossRef] [PubMed]
  12. B. Chirikov, Phys. Rep. 52, 263 (1979).
    [CrossRef]

1993 (2)

L. Collot, V. Lefè-Seguin, M. Brune, J. M. Raimond, S. Haroche, Europhys. Lett. 23, 327 (1993).
[CrossRef]

B. R. Johnson, J. Opt. Soc. Am. A 10, 343 (1993).
[CrossRef]

1992 (2)

S. L. McCall, A. F. Levi, R. E Slusher, S. J. Pearton, R. A. Logan, Appl. Phys. Lett. 60, 289 (1992).
[CrossRef]

D. Q. Chowdhury, P. W. Barber, S. C. Hill, Appl. Opt. 31, 3518 (1992).
[CrossRef] [PubMed]

1991 (1)

D. D. Holm, G. Kovacic, Physica D 51, 177 (1991).
[CrossRef]

1986 (1)

M. V. Berry, M. Robnik, J. Phys. A 19,649 (1986).
[CrossRef]

1983 (1)

M. Robnik, J. Phys. A 16,3971 (1983); M. Robnik, J. Phys. A 17,1049 (1983).
[CrossRef] [PubMed]

1981 (2)

M. V. Berry, Eur. J. Phys. 2, 91 (1981).
[CrossRef]

J. F. Owen, P. W. Barber, P. B. Dorain, R. K. Chang, Phys. Rev. Lett. 47, 1075 (1981).
[CrossRef]

1979 (1)

B. Chirikov, Phys. Rep. 52, 263 (1979).
[CrossRef]

Barber, P. W.

D. Q. Chowdhury, P. W. Barber, S. C. Hill, Appl. Opt. 31, 3518 (1992).
[CrossRef] [PubMed]

J. F. Owen, P. W. Barber, P. B. Dorain, R. K. Chang, Phys. Rev. Lett. 47, 1075 (1981).
[CrossRef]

Benner, R. E.

S. C. Hill, R. E. Benner, in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), p. 3.

Berry, M. V.

M. V. Berry, M. Robnik, J. Phys. A 19,649 (1986).
[CrossRef]

M. V. Berry, Eur. J. Phys. 2, 91 (1981).
[CrossRef]

Brune, M.

L. Collot, V. Lefè-Seguin, M. Brune, J. M. Raimond, S. Haroche, Europhys. Lett. 23, 327 (1993).
[CrossRef]

Chang, R. K.

J. F. Owen, P. W. Barber, P. B. Dorain, R. K. Chang, Phys. Rev. Lett. 47, 1075 (1981).
[CrossRef]

Chirikov, B.

B. Chirikov, Phys. Rep. 52, 263 (1979).
[CrossRef]

Chowdhury, D. Q.

Collot, L.

L. Collot, V. Lefè-Seguin, M. Brune, J. M. Raimond, S. Haroche, Europhys. Lett. 23, 327 (1993).
[CrossRef]

Dorain, P. B.

J. F. Owen, P. W. Barber, P. B. Dorain, R. K. Chang, Phys. Rev. Lett. 47, 1075 (1981).
[CrossRef]

Haroche, S.

L. Collot, V. Lefè-Seguin, M. Brune, J. M. Raimond, S. Haroche, Europhys. Lett. 23, 327 (1993).
[CrossRef]

Hill, S. C.

D. Q. Chowdhury, P. W. Barber, S. C. Hill, Appl. Opt. 31, 3518 (1992).
[CrossRef] [PubMed]

S. C. Hill, R. E. Benner, in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), p. 3.

Holm, D. D.

D. D. Holm, G. Kovacic, Physica D 51, 177 (1991).
[CrossRef]

Johnson, B. R.

Kovacic, G.

D. D. Holm, G. Kovacic, Physica D 51, 177 (1991).
[CrossRef]

Lefè-Seguin, V.

L. Collot, V. Lefè-Seguin, M. Brune, J. M. Raimond, S. Haroche, Europhys. Lett. 23, 327 (1993).
[CrossRef]

Levi, A. F.

S. L. McCall, A. F. Levi, R. E Slusher, S. J. Pearton, R. A. Logan, Appl. Phys. Lett. 60, 289 (1992).
[CrossRef]

Logan, R. A.

S. L. McCall, A. F. Levi, R. E Slusher, S. J. Pearton, R. A. Logan, Appl. Phys. Lett. 60, 289 (1992).
[CrossRef]

McCall, S. L.

S. L. McCall, A. F. Levi, R. E Slusher, S. J. Pearton, R. A. Logan, Appl. Phys. Lett. 60, 289 (1992).
[CrossRef]

Owen, J. F.

J. F. Owen, P. W. Barber, P. B. Dorain, R. K. Chang, Phys. Rev. Lett. 47, 1075 (1981).
[CrossRef]

Pearton, S. J.

S. L. McCall, A. F. Levi, R. E Slusher, S. J. Pearton, R. A. Logan, Appl. Phys. Lett. 60, 289 (1992).
[CrossRef]

Raimond, J. M.

L. Collot, V. Lefè-Seguin, M. Brune, J. M. Raimond, S. Haroche, Europhys. Lett. 23, 327 (1993).
[CrossRef]

Reichl, L. E.

L. E. Reichl, The Transition to Chaos (Springer-Verlag, Berlin, 1992), Chap. 2, p. 39.

Robnik, M.

M. V. Berry, M. Robnik, J. Phys. A 19,649 (1986).
[CrossRef]

M. Robnik, J. Phys. A 16,3971 (1983); M. Robnik, J. Phys. A 17,1049 (1983).
[CrossRef] [PubMed]

Slusher, R. E

S. L. McCall, A. F. Levi, R. E Slusher, S. J. Pearton, R. A. Logan, Appl. Phys. Lett. 60, 289 (1992).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

S. L. McCall, A. F. Levi, R. E Slusher, S. J. Pearton, R. A. Logan, Appl. Phys. Lett. 60, 289 (1992).
[CrossRef]

Eur. J. Phys. (1)

M. V. Berry, Eur. J. Phys. 2, 91 (1981).
[CrossRef]

Europhys. Lett. (1)

L. Collot, V. Lefè-Seguin, M. Brune, J. M. Raimond, S. Haroche, Europhys. Lett. 23, 327 (1993).
[CrossRef]

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

J. Phys. A (2)

M. Robnik, J. Phys. A 16,3971 (1983); M. Robnik, J. Phys. A 17,1049 (1983).
[CrossRef] [PubMed]

M. V. Berry, M. Robnik, J. Phys. A 19,649 (1986).
[CrossRef]

Phys. Rep. (1)

B. Chirikov, Phys. Rep. 52, 263 (1979).
[CrossRef]

Phys. Rev. Lett. (1)

J. F. Owen, P. W. Barber, P. B. Dorain, R. K. Chang, Phys. Rev. Lett. 47, 1075 (1981).
[CrossRef]

Physica D (1)

D. D. Holm, G. Kovacic, Physica D 51, 177 (1991).
[CrossRef]

Other (2)

S. C. Hill, R. E. Benner, in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), p. 3.

L. E. Reichl, The Transition to Chaos (Springer-Verlag, Berlin, 1992), Chap. 2, p. 39.

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Figures (2)

Fig. 1
Fig. 1

SOS for three values of the deformation parameter: (a) b = 0, (b) b = 0.11, (c) b = 0.15. The real-space trajectories shown to the right correspond to a ray started with sin θ = 0.7, near the four-bounce periodic orbit. The starting value of φ in the lower two plots was chosen in the stochastic region between the four-island chain. The vertical arrow in (b) indicates the direction in which the elongation of the disk occurs when b > 0.

Fig. 2
Fig. 2

Q−1 versus deformation b obtained from the average escape rate (see text). The unperturbed circle has radius R, k is the vacuum wave number, and n is the refractive index of the dielectric. Escaping rays are started in a strip with 0.66 ≤ sin θ ≤ 0.71. A least-squares fit to the data gives a power law nkR/Q = 23.7(b–bc)α, with bc, = 0.124 and α = 2.5. Inset: polar plot of the escape probability versus angle in the far field for b = 0.15, showing strong directionality of emission.

Equations (2)

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w ( z ) = z + b z 2 + c z 3 ( 1 + 2 b 2 + 3 c 2 ) 1 / 2 ,
cot θ + θ = π ( p + 3 / 4 m + 1 2 ) .

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