Abstract

A generic perturbation theory for the morphology-dependent resonances (MDR’s) of dielectric spheres is developed based on the dyadic formulation of a completeness relation established previously [J. Opt. Soc. Am. B 16, 1409 (1999)]. Unlike other perturbation methods proposed previously, the formulation presented here takes full account of the vector nature of MDR’s and hence does not limit its validity to perturbations that preserve spherical symmetry. However, the second-order frequency correction obtained directly from the theory, which is expressed as a sum of contributions from individual MDR’s, converges slowly. An efficient scheme, based on the dyadic form of the completeness relation, is thus constructed to accelerate the rate of convergence. As an example illustrating our theory, we apply the perturbation method to study MDR’s of a dielectric sphere that contains another smaller spherical inclusion and compare the results with those obtained from an exact diagonalization method.

© 1999 Optical Society of America

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  1. See, e.g., M. Kerker, ed., Selected Papers on Light Scattering, Proc. SPIE 951, (1988), and references therein.
  2. P. W. Barber and R. K. Chang, eds., Optical Effects Associated with Small Particles (World Scientific, Singapore, 1988).
  3. R. K. Chang and A. J. Campillo, eds., Optical Processes in Microcavities (World Scientific, Singapore, 1996).
  4. R. E. Benner, P. W. Barber, J. F. Owen, and R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).
  5. J. B. Snow, S.-X. Qian, and R. K. Chang, “Stimulated Raman scattering from individual water and ethanol droplets at morphology-dependent resonances,” Opt. Lett. 10, 37–39 (1985).
  6. J.-Z. Zhang and R. K. Chang, “Generation and suppression of stimulated Brillouin scattering in single liquid droplets,” J. Opt. Soc. Am. B 6, 151–153 (1989).
  7. H. M. Tzeng, K. F. Wall, M. B. Long, and R. K. Chang, “Laser emission from individual droplets at wavelengths corresponding to morphology-dependent resonances,” Opt. Lett. 9, 499–501 (1984).
  8. L. M. Folan, S. Arnold, and D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
  9. See, e.g., P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).
  10. M. L. Goldberger and K. M. Watson, Collision Theory (Wiley, New York, 1964).
  11. L. Collot, V. Lefevre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).
  12. D. Brady, G. Papen, and J. E. Sipe, “Spherical distributed dielectric resonators,” J. Opt. Soc. Am. B 10, 644–657 (1993).
  13. F. Borghese, P. Denti, R. Saija, and O. I. Sindoni, “Optical properties of spheres containing a spherical eccentric inclusion,” J. Opt. Soc. Am. A 9, 1327–1335 (1992); R. L. Armstrong, J.-G. Xie, T. Ruekgauer, J. Gu, and R. G. Pinnick, “Effects of submicrometer-sized particles on microdroplet lasing,” Opt. Lett. 18, 119–121 (1993).
  14. M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
  15. K. M. Lee, P. T. Leung, and K. M. Pang, “Dyadic formulation of morphology-dependent resonances. I. Completeness relation,” J. Opt. Soc. Am. B 16, 1409–1417 (1999).
  16. We consider MDR’s of stable optical systems in the present paper; hence the imaginary part of the complex wave number of a MDR is always less than zero.
  17. H. M. Lai, P. T. Leung, K. Young, S. C. Hill, and P. W. Barber, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
  18. H. M. Lai, C. C. Lam, P. T. Leung, and K. Young, “The effect of perturbations on the widths of narrow morphology-dependent resonances in Mie scattering,” J. Opt. Soc. Am. B 8, 1962–1973 (1991).
  19. P. T. Leung and K. M. Pang, “Completeness and time-independent perturbation of morphology-dependent resonances in dielectric spheres,” J. Opt. Soc. Am. B 13, 805–817 (1996).
  20. K. M. Lee, P. T. Leung, and K. M. Pang, “Iterative perturbation scheme for morphology-dependent resonances in dielectric spheres,” J. Opt. Soc. Am. A 15, 1383–1393 (1998).
  21. C.-T. Tai, Dyadic Functions in Electromagnetic Theory, 2nd ed. (IEEE Press, New York, 1993).
  22. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, New York, 1995).
  23. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  24. M. M. Mazumder, S. C. Hill, and P. W. Barber, “Morphology-dependent resonances in inhomogeneous spheres: comparison of layered T-matrix and time-independent perturbation method,” J. Opt. Soc. Am. A 9, 1844–1853 (1992).
  25. A. L. Fetter and J. D. Walecka, Quantum Theory of Many Body Systems (McGraw-Hill, New York, 1971).
  26. In the present paper we assume that either the MDR’s of the unperturbed system are nondegenerate or the perturbation does not couple degenerate MDR’s by symmetry arguments. Otherwise a degenerate perturbation theory would have to be formulated, which is out of place here.
  27. S. C. Hill, H. I. Saleheen, and K. A. Fuller, “Volume current method for modeling light scattering by inhomogeneously perturbed spheres,” J. Opt. Soc. Am. A 12, 905–915 (1995).
  28. We notice that Qlmlm of Eq. (A12) is nonzero only if m≠ 0. Thus we concentrate on the m≠0 cases here.
  29. By mode order we mean, for TE modes, that the mode with the smallest positive Re(ωa) is of mode order 1, the second smallest is of mode order 2, and so on. The TM modes are similarly defined, except that the mode on the imaginary frequency axis is of mode order 0.
  30. See, e.g., D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum: Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols (World Scientific, Singapore, 1988).
  31. K. M. Pang, “Completeness and perturbation of morphology-dependent resonances in dielectric spheres,” Ph.D. thesis dissertation (Chinese University of Hong Kong, Hong Kong, 1999).

1999 (1)

1998 (1)

1996 (2)

1995 (1)

1993 (2)

D. Brady, G. Papen, and J. E. Sipe, “Spherical distributed dielectric resonators,” J. Opt. Soc. Am. B 10, 644–657 (1993).

L. Collot, V. Lefevre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).

1992 (1)

1991 (1)

1990 (1)

H. M. Lai, P. T. Leung, K. Young, S. C. Hill, and P. W. Barber, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).

1989 (1)

1985 (2)

L. M. Folan, S. Arnold, and D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).

J. B. Snow, S.-X. Qian, and R. K. Chang, “Stimulated Raman scattering from individual water and ethanol droplets at morphology-dependent resonances,” Opt. Lett. 10, 37–39 (1985).

1984 (1)

1980 (1)

R. E. Benner, P. W. Barber, J. F. Owen, and R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).

Arnold, S.

L. M. Folan, S. Arnold, and D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).

Barber, P. W.

M. M. Mazumder, S. C. Hill, and P. W. Barber, “Morphology-dependent resonances in inhomogeneous spheres: comparison of layered T-matrix and time-independent perturbation method,” J. Opt. Soc. Am. A 9, 1844–1853 (1992).

H. M. Lai, P. T. Leung, K. Young, S. C. Hill, and P. W. Barber, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).

R. E. Benner, P. W. Barber, J. F. Owen, and R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).

Benner, R. E.

R. E. Benner, P. W. Barber, J. F. Owen, and R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).

Brady, D.

Brune, M.

L. Collot, V. Lefevre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).

Chang, R. K.

Collot, L.

L. Collot, V. Lefevre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).

Druger, D.

L. M. Folan, S. Arnold, and D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).

Folan, L. M.

L. M. Folan, S. Arnold, and D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).

Fuller, K. A.

Gorodetsky, M. L.

Haroche, S.

L. Collot, V. Lefevre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).

Hill, S. C.

Ilchenko, V. S.

Lai, H. M.

H. M. Lai, C. C. Lam, P. T. Leung, and K. Young, “The effect of perturbations on the widths of narrow morphology-dependent resonances in Mie scattering,” J. Opt. Soc. Am. B 8, 1962–1973 (1991).

H. M. Lai, P. T. Leung, K. Young, S. C. Hill, and P. W. Barber, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).

Lam, C. C.

Lee, K. M.

Lefevre-Seguin, V.

L. Collot, V. Lefevre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).

Leung, P. T.

Long, M. B.

Mazumder, M. M.

Owen, J. F.

R. E. Benner, P. W. Barber, J. F. Owen, and R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).

Pang, K. M.

Papen, G.

Qian, S.-X.

Raimond, J. M.

L. Collot, V. Lefevre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).

Saleheen, H. I.

Savchenkov, A. A.

Sipe, J. E.

Snow, J. B.

Tzeng, H. M.

Wall, K. F.

Young, K.

H. M. Lai, C. C. Lam, P. T. Leung, and K. Young, “The effect of perturbations on the widths of narrow morphology-dependent resonances in Mie scattering,” J. Opt. Soc. Am. B 8, 1962–1973 (1991).

H. M. Lai, P. T. Leung, K. Young, S. C. Hill, and P. W. Barber, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).

Zhang, J.-Z.

Chem. Phys. Lett. (1)

L. M. Folan, S. Arnold, and D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).

Europhys. Lett. (1)

L. Collot, V. Lefevre-Seguin, M. Brune, J. M. Raimond, and S. Haroche, “Very high-Q whispering-gallery mode resonances observed on fused silica microspheres,” Europhys. Lett. 23, 327–334 (1993).

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (5)

Opt. Lett. (3)

Phys. Rev. A (1)

H. M. Lai, P. T. Leung, K. Young, S. C. Hill, and P. W. Barber, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).

Phys. Rev. Lett. (1)

R. E. Benner, P. W. Barber, J. F. Owen, and R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phys. Rev. Lett. 44, 475–478 (1980).

Other (16)

F. Borghese, P. Denti, R. Saija, and O. I. Sindoni, “Optical properties of spheres containing a spherical eccentric inclusion,” J. Opt. Soc. Am. A 9, 1327–1335 (1992); R. L. Armstrong, J.-G. Xie, T. Ruekgauer, J. Gu, and R. G. Pinnick, “Effects of submicrometer-sized particles on microdroplet lasing,” Opt. Lett. 18, 119–121 (1993).

We consider MDR’s of stable optical systems in the present paper; hence the imaginary part of the complex wave number of a MDR is always less than zero.

C.-T. Tai, Dyadic Functions in Electromagnetic Theory, 2nd ed. (IEEE Press, New York, 1993).

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, New York, 1995).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

A. L. Fetter and J. D. Walecka, Quantum Theory of Many Body Systems (McGraw-Hill, New York, 1971).

In the present paper we assume that either the MDR’s of the unperturbed system are nondegenerate or the perturbation does not couple degenerate MDR’s by symmetry arguments. Otherwise a degenerate perturbation theory would have to be formulated, which is out of place here.

We notice that Qlmlm of Eq. (A12) is nonzero only if m≠ 0. Thus we concentrate on the m≠0 cases here.

By mode order we mean, for TE modes, that the mode with the smallest positive Re(ωa) is of mode order 1, the second smallest is of mode order 2, and so on. The TM modes are similarly defined, except that the mode on the imaginary frequency axis is of mode order 0.

See, e.g., D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum: Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols (World Scientific, Singapore, 1988).

K. M. Pang, “Completeness and perturbation of morphology-dependent resonances in dielectric spheres,” Ph.D. thesis dissertation (Chinese University of Hong Kong, Hong Kong, 1999).

See, e.g., P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

M. L. Goldberger and K. M. Watson, Collision Theory (Wiley, New York, 1964).

See, e.g., M. Kerker, ed., Selected Papers on Light Scattering, Proc. SPIE 951, (1988), and references therein.

P. W. Barber and R. K. Chang, eds., Optical Effects Associated with Small Particles (World Scientific, Singapore, 1988).

R. K. Chang and A. J. Campillo, eds., Optical Processes in Microcavities (World Scientific, Singapore, 1996).

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