Abstract

The mode spectrum of a 3.8-cm-diameter fused-silica sphere has been studied in the vicinity of 1.06 μm. A single-frequency Nd:YAG laser was used to excite whispering gallery modes by means of evanescent wave coupling. The spectrum is in excellent agreement with predictions from Mie theory.

© 1991 Optical Society of America

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References

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  1. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  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).
  3. R. D. Richtmyer, J. Appl. Phys. 10, 391 (1939).
    [CrossRef]
  4. T. Baer, Opt. Lett. 12, 392 (1987).
    [CrossRef] [PubMed]
  5. C. G. B. Garrett, W. Kaiser, W. L. Bond, Phys. Rev. 124, 1807 (1961).
    [CrossRef]
  6. A. Ashkin, J. M. Dziedzic, Phys. Rev. Lett. 38, 1351 (1977).
    [CrossRef]
  7. R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, Phys. Rev. Lett. 44, 475 (1980).
    [CrossRef]
  8. V. B. Braginsky, M. L. Gorodetsky, V. S. Ilchenko, Phys. Lett. A 137, 393 (1989).
    [CrossRef]
  9. W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1962).
  10. P. Chylek, J. T. Kiehl, M. K. W. Ko, Phys. Rev. A 18, 2229 (1978).
    [CrossRef]
  11. For an(l),bn(l) ≪ n and an(l),bn(l) ≲ n/m, expressions (9.3.8), (9.3.12), and (9.3.23), (9.3.27) in Ref. 12 are appropriate.
  12. M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1973).
  13. The relative accuracy of the resonance frequencies computed from Eq. (2) is of order n−2 and decreases with increasing l. For extremely large l, see J. R. Probert-Jones, J. Opt. Soc. Am. A 1, 822 (1984).
    [CrossRef]
  14. Consider a plane wave propagating along a closed equatorial polygonal ring path with N total internal reflections. Addition of the individual phase shifts (given by the Fresnel formulas) that occur during one round trip leads to a relative frequency shift given by relation (4) in the limit of large N.
  15. R. L. Byer, Science 239, 742 (1988).
    [CrossRef] [PubMed]
  16. E. Garmire, T. McMahon, M. Bass, Appl. Phys. Lett. 29, 254 (1976); M. E. Marhic, L. I. Kwan, M. Epstein, IEEE J. Quantum Electron. QE-15, 487 (1979).
    [CrossRef]
  17. G. J. Milburn, M. D. Levenson, R. M. Shelby, S. H. Perlmutter, R. G. De Voe, D. F. Walls, J. Opt. Soc. Am. B 4, 1476 (1987).
    [CrossRef]

1989 (1)

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

1988 (1)

R. L. Byer, Science 239, 742 (1988).
[CrossRef] [PubMed]

1987 (2)

1984 (1)

1980 (1)

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, Phys. Rev. Lett. 44, 475 (1980).
[CrossRef]

1978 (1)

P. Chylek, J. T. Kiehl, M. K. W. Ko, Phys. Rev. A 18, 2229 (1978).
[CrossRef]

1977 (1)

A. Ashkin, J. M. Dziedzic, Phys. Rev. Lett. 38, 1351 (1977).
[CrossRef]

1976 (1)

E. Garmire, T. McMahon, M. Bass, Appl. Phys. Lett. 29, 254 (1976); M. E. Marhic, L. I. Kwan, M. Epstein, IEEE J. Quantum Electron. QE-15, 487 (1979).
[CrossRef]

1961 (1)

C. G. B. Garrett, W. Kaiser, W. L. Bond, Phys. Rev. 124, 1807 (1961).
[CrossRef]

1939 (1)

R. D. Richtmyer, J. Appl. Phys. 10, 391 (1939).
[CrossRef]

Abramovitz, M.

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1973).

Ashkin, A.

A. Ashkin, J. M. Dziedzic, Phys. Rev. Lett. 38, 1351 (1977).
[CrossRef]

Baer, T.

Barber, P. W.

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, Phys. Rev. Lett. 44, 475 (1980).
[CrossRef]

Bass, M.

E. Garmire, T. McMahon, M. Bass, Appl. Phys. Lett. 29, 254 (1976); M. E. Marhic, L. I. Kwan, M. Epstein, IEEE J. Quantum Electron. QE-15, 487 (1979).
[CrossRef]

Benner, R. E.

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, Phys. Rev. Lett. 44, 475 (1980).
[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).

Bond, W. L.

C. G. B. Garrett, W. Kaiser, W. L. Bond, Phys. Rev. 124, 1807 (1961).
[CrossRef]

Braginsky, V. B.

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

Byer, R. L.

R. L. Byer, Science 239, 742 (1988).
[CrossRef] [PubMed]

Chang, R. K.

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, Phys. Rev. Lett. 44, 475 (1980).
[CrossRef]

Chylek, P.

P. Chylek, J. T. Kiehl, M. K. W. Ko, Phys. Rev. A 18, 2229 (1978).
[CrossRef]

De Voe, R. G.

Dziedzic, J. M.

A. Ashkin, J. M. Dziedzic, Phys. Rev. Lett. 38, 1351 (1977).
[CrossRef]

Garmire, E.

E. Garmire, T. McMahon, M. Bass, Appl. Phys. Lett. 29, 254 (1976); M. E. Marhic, L. I. Kwan, M. Epstein, IEEE J. Quantum Electron. QE-15, 487 (1979).
[CrossRef]

Garrett, C. G. B.

C. G. B. Garrett, W. Kaiser, W. L. Bond, Phys. Rev. 124, 1807 (1961).
[CrossRef]

Gorodetsky, M. L.

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

Hill, S. C.

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

Ilchenko, V. S.

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

Kaiser, W.

C. G. B. Garrett, W. Kaiser, W. L. Bond, Phys. Rev. 124, 1807 (1961).
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Kiehl, J. T.

P. Chylek, J. T. Kiehl, M. K. W. Ko, Phys. Rev. A 18, 2229 (1978).
[CrossRef]

Ko, M. K. W.

P. Chylek, J. T. Kiehl, M. K. W. Ko, Phys. Rev. A 18, 2229 (1978).
[CrossRef]

Levenson, M. D.

McMahon, T.

E. Garmire, T. McMahon, M. Bass, Appl. Phys. Lett. 29, 254 (1976); M. E. Marhic, L. I. Kwan, M. Epstein, IEEE J. Quantum Electron. QE-15, 487 (1979).
[CrossRef]

Milburn, G. J.

Owen, J. F.

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, Phys. Rev. Lett. 44, 475 (1980).
[CrossRef]

Panofsky, W. K. H.

W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1962).

Perlmutter, S. H.

Phillips, M.

W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1962).

Probert-Jones, J. R.

Richtmyer, R. D.

R. D. Richtmyer, J. Appl. Phys. 10, 391 (1939).
[CrossRef]

Shelby, R. M.

Stegun, I. A.

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1973).

Walls, D. F.

Appl. Phys. Lett. (1)

E. Garmire, T. McMahon, M. Bass, Appl. Phys. Lett. 29, 254 (1976); M. E. Marhic, L. I. Kwan, M. Epstein, IEEE J. Quantum Electron. QE-15, 487 (1979).
[CrossRef]

J. Appl. Phys. (1)

R. D. Richtmyer, J. Appl. Phys. 10, 391 (1939).
[CrossRef]

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

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

Opt. Lett. (1)

Phys. Lett. A (1)

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

Phys. Rev. (1)

C. G. B. Garrett, W. Kaiser, W. L. Bond, Phys. Rev. 124, 1807 (1961).
[CrossRef]

Phys. Rev. A (1)

P. Chylek, J. T. Kiehl, M. K. W. Ko, Phys. Rev. A 18, 2229 (1978).
[CrossRef]

Phys. Rev. Lett. (2)

A. Ashkin, J. M. Dziedzic, Phys. Rev. Lett. 38, 1351 (1977).
[CrossRef]

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, Phys. Rev. Lett. 44, 475 (1980).
[CrossRef]

Science (1)

R. L. Byer, Science 239, 742 (1988).
[CrossRef] [PubMed]

Other (6)

For an(l),bn(l) ≪ n and an(l),bn(l) ≲ n/m, expressions (9.3.8), (9.3.12), and (9.3.23), (9.3.27) in Ref. 12 are appropriate.

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1973).

Consider a plane wave propagating along a closed equatorial polygonal ring path with N total internal reflections. Addition of the individual phase shifts (given by the Fresnel formulas) that occur during one round trip leads to a relative frequency shift given by relation (4) in the limit of large N.

W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1962).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

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

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Figures (2)

Fig. 1
Fig. 1

Setup for the study of modes of spherical resonators. FC1, FC2, thermal and piezoelectric frequency control inputs. D1, D2, alternate positions for the photodetectors that register scattering and outcoupling of the mode field, respectively.

Fig. 2
Fig. 2

Experimental and theoretical whispering gallery mode spectra, (a), (b) The intensity of the light scattered from the equator, obtained by lock-in detection, versus the laser frequency together with Fabry–Perot resonances. The scan time was 20 s each, (a) Parallel (in-plane) laser polarization excites TM resonances, (b) TE modes excited by perpendicular laser polarization, (c) The line positions according to Mie theory [Eq. (2)] for l = 1−22, using the best-fit mode number n = 162554. For clarity the heights of the bars were scaled by 1 / l. The center frequency is approximately 9394 cm−1.

Equations (5)

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E r ( r , θ , ϕ ) n ( n + 1 ) k r j n ( k r ) Y n q ( θ , ϕ ) TM modes , E θ ( r , θ , ϕ ) q sin θ j n ( k r ) Y n q ( θ , ϕ ) TE modes ,
n s x n ( l ) = n + 1 2 ζ l ( n + 1 / 2 2 ) 1 / 3 m p m 2 1 + 3 ζ l 2 2 2 / 3 10 ( n + 1 / 2 ) 1 / 3 m 3 p ( 2 p 2 / 3 1 ) ζ l 2 1 / 3 ( m 2 1 ) 3 / 2 ( n + 1 / 2 ) 2 / 3 ,
ν f = c 2 π R [ x n + 1 ( l ) x n ( l ) ] c 2 π n s R .
ν ˆ p = a n ( l ) b n ( l ) ν f m 2 1 m ,
E ( r ) A i [ ζ l + ( 2 k 2 R ) 1 / 3 ( R r ) ] .

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