Abstract

Morphology-dependent resonances commonly observed in both the elastic and the inelastic scattering of light waves from dielectric spheres are in fact direct manifestations of complex frequency poles of the scattering matrix. Time-independent solutions to the wave equation at these poles are termed quasi-normal modes, which are characterized by the outgoing wave boundary condition at infinity and cannot be normalized in the usual sense. These resonances (or quasi-normal modes) are shown to form a complete set inside the dielectric sphere, provided that there is a spatial discontinuity in the refractive index, say, at the edge of the sphere. Novel definitions of norm and inner product are introduced. In addition, a time-independent perturbation method based on this completeness relation is developed to evaluate shifts in resonance frequencies when the refractive index is changed.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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 (1980).
    [Crossref]
  2. 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 (1985).
    [Crossref] [PubMed]
  3. 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 (1989).
    [Crossref]
  4. 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 (1984).
    [Crossref] [PubMed]
  5. L. M. Folan, S. Arnold, and D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322 (1987).
    [Crossref]
  6. D. Brady, G. Papen, and J. E. Sipe, “Spherical distributed dielectric resonators,” J. Opt. Soc. Am. B 10, 644 (1993).
    [Crossref]
  7. See, e.g., S. C. Hill and R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber and R. K. Chang, eds. (World Scientific, Singapore, 1988), p. 3.
  8. M. L. Goldberger and K. M. Watson, Collision Theory (Wiley, New York, 1964).
  9. P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057 (1994); P. T. Leung, S. Y. Liu, S. S. Tong, and K. Young, “Time-independent perturbation theory for quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3068 (1994); P. T. Leung, S. Y. Liu, and K. Young, “Completeness and time-independent perturbation of the quasinormal modes of an absorptive and leaky cavity,” Phys. Rev. A 49, 3982 (1994).
    [Crossref] [PubMed]
  10. For problems defined on a half-line x ∈ [0, ∞), the origin is counted as a discontinuity so that only one more discontinuity in the dielectric constant distribution is required for establishment of the completeness relation.
  11. E. S. C. Emily, P. T. Leung, W. M. Suen, and K. Young, “Quasinormal mode expansion for linearized waves in gravitational systems,” Phys. Rev. Lett. 74, 4588 (1995).
    [Crossref]
  12. See, e.g., P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).
  13. 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 (1991).
    [Crossref]
  14. R. L. Armstrong, J.-G. Xie, T. E. Ruekgauer, J. Gu, and R. G. Pinnick, “Effects of submicrometer-sized particles on microdroplet lasing,” Opt. Lett. 18, 119 (1993).
    [Crossref] [PubMed]
  15. A. L. Huston, H.-B. Lin, J. D. Eversole, and A. J. Campillo, Naval Research Laboratory, Washington D.C. 20375 (private communication, 1992).
  16. 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 (1993).
    [Crossref]
  17. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  18. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  19. R. G. Newton, “Analytic properties of radial wave functions,” J. Math. Phys. 1, 319 (1960).
    [Crossref]
  20. E. S. C. Ching, P. T. Leung, W. M. Suen, and K. Young, “Late-time tail of wave propagation on curved spacetime,” Phys. Rev. Lett. 74, 2414 (1995).
    [Crossref] [PubMed]
  21. M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Progr. Phys. 35, 315 (1972).
    [Crossref]
  22. R. E. Langer, “On the connection formulas and solutions of the wave equation,” Phys. Rev. 51, 669 (1937).
    [Crossref]
  23. A similar calculation has been carried out previously. See P. R. Conwell, P. W. Barber, and C. K. Rushforth, “Resonant spectra of dielectric spheres,” J. Opt. Soc. Am. A 1, 62 (1984).
    [Crossref]
  24. C. C. Lam, P. T. Leung, and K. Young, “Explicit asymptotic formulas for the positions, widths, and strengths of resonances in Mie scattering,” J. Opt. Soc. Am. B 9, 1585 (1992).
    [Crossref]
  25. P. T. Leung and S. T. Ng, “Determination of quasinormal modes in leaky cavities by diagonalization,” J. Phys. A 29, 143 (1996).
    [Crossref]
  26. See, e.g., W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1966).
  27. A. L. Fetter and J. D. Walecka, Quantum Theory of Many Body Systems (McGraw-Hill, New York, 1971).
  28. 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 (1990).
    [Crossref] [PubMed]
  29. D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Morphology-dependent resonances in radially inhomogeneous spheres,” J. Opt. Soc. Am. A 8, 1702 (1991).
    [Crossref]
  30. O. B. Toon and T. P. Ackerman, “Algorithms for the calculation of scattering by stratified spheres,” Appl. Opt. 20, 3657 (1981).
    [Crossref] [PubMed]
  31. A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242 (1951); R. Bhandari, “Scattering coefficients for a multi-layered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1960 (1985).
    [Crossref]
  32. 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 (1992); 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 (1992).
    [Crossref]

1996 (1)

P. T. Leung and S. T. Ng, “Determination of quasinormal modes in leaky cavities by diagonalization,” J. Phys. A 29, 143 (1996).
[Crossref]

1995 (2)

E. S. C. Ching, P. T. Leung, W. M. Suen, and K. Young, “Late-time tail of wave propagation on curved spacetime,” Phys. Rev. Lett. 74, 2414 (1995).
[Crossref] [PubMed]

E. S. C. Emily, P. T. Leung, W. M. Suen, and K. Young, “Quasinormal mode expansion for linearized waves in gravitational systems,” Phys. Rev. Lett. 74, 4588 (1995).
[Crossref]

1994 (1)

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057 (1994); P. T. Leung, S. Y. Liu, S. S. Tong, and K. Young, “Time-independent perturbation theory for quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3068 (1994); P. T. Leung, S. Y. Liu, and K. Young, “Completeness and time-independent perturbation of the quasinormal modes of an absorptive and leaky cavity,” Phys. Rev. A 49, 3982 (1994).
[Crossref] [PubMed]

1993 (3)

1992 (2)

1991 (2)

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 (1990).
[Crossref] [PubMed]

1989 (1)

1987 (1)

L. M. Folan, S. Arnold, and D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322 (1987).
[Crossref]

1985 (1)

1984 (2)

1981 (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 (1980).
[Crossref]

1972 (1)

M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Progr. Phys. 35, 315 (1972).
[Crossref]

1960 (1)

R. G. Newton, “Analytic properties of radial wave functions,” J. Math. Phys. 1, 319 (1960).
[Crossref]

1951 (1)

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242 (1951); R. Bhandari, “Scattering coefficients for a multi-layered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1960 (1985).
[Crossref]

1937 (1)

R. E. Langer, “On the connection formulas and solutions of the wave equation,” Phys. Rev. 51, 669 (1937).
[Crossref]

Ackerman, T. P.

Aden, A. L.

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242 (1951); R. Bhandari, “Scattering coefficients for a multi-layered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1960 (1985).
[Crossref]

Armstrong, R. L.

Arnold, S.

L. M. Folan, S. Arnold, and D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322 (1987).
[Crossref]

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 (1992); 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 (1992).
[Crossref]

D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Morphology-dependent resonances in radially inhomogeneous spheres,” J. Opt. Soc. Am. A 8, 1702 (1991).
[Crossref]

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 (1990).
[Crossref] [PubMed]

A similar calculation has been carried out previously. See P. R. Conwell, P. W. Barber, and C. K. Rushforth, “Resonant spectra of dielectric spheres,” J. Opt. Soc. Am. A 1, 62 (1984).
[Crossref]

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 (1980).
[Crossref]

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

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 (1980).
[Crossref]

See, e.g., S. C. Hill and R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber and R. K. Chang, eds. (World Scientific, Singapore, 1988), p. 3.

Berry, M. V.

M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Progr. Phys. 35, 315 (1972).
[Crossref]

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 (1993).
[Crossref]

Campillo, A. J.

A. L. Huston, H.-B. Lin, J. D. Eversole, and A. J. Campillo, Naval Research Laboratory, Washington D.C. 20375 (private communication, 1992).

Chang, R. K.

Ching, E. S. C.

E. S. C. Ching, P. T. Leung, W. M. Suen, and K. Young, “Late-time tail of wave propagation on curved spacetime,” Phys. Rev. Lett. 74, 2414 (1995).
[Crossref] [PubMed]

Chowdhury, D. Q.

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 (1993).
[Crossref]

Conwell, P. R.

Druger, D.

L. M. Folan, S. Arnold, and D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322 (1987).
[Crossref]

Emily, E. S. C.

E. S. C. Emily, P. T. Leung, W. M. Suen, and K. Young, “Quasinormal mode expansion for linearized waves in gravitational systems,” Phys. Rev. Lett. 74, 4588 (1995).
[Crossref]

Eversole, J. D.

A. L. Huston, H.-B. Lin, J. D. Eversole, and A. J. Campillo, Naval Research Laboratory, Washington D.C. 20375 (private communication, 1992).

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Fetter, A. L.

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

Folan, L. M.

L. M. Folan, S. Arnold, and D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322 (1987).
[Crossref]

Goldberger, M. L.

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

Gu, J.

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 (1993).
[Crossref]

Hill, S. C.

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 (1992); 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 (1992).
[Crossref]

D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Morphology-dependent resonances in radially inhomogeneous spheres,” J. Opt. Soc. Am. A 8, 1702 (1991).
[Crossref]

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 (1990).
[Crossref] [PubMed]

See, e.g., S. C. Hill and R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber and R. K. Chang, eds. (World Scientific, Singapore, 1988), p. 3.

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

Huston, A. L.

A. L. Huston, H.-B. Lin, J. D. Eversole, and A. J. Campillo, Naval Research Laboratory, Washington D.C. 20375 (private communication, 1992).

Jackson, J. D.

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

Kerker, M.

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242 (1951); R. Bhandari, “Scattering coefficients for a multi-layered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1960 (1985).
[Crossref]

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 (1991).
[Crossref]

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 (1990).
[Crossref] [PubMed]

Lam, C. C.

Langer, R. E.

R. E. Langer, “On the connection formulas and solutions of the wave equation,” Phys. Rev. 51, 669 (1937).
[Crossref]

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 (1993).
[Crossref]

Leung, P. T.

P. T. Leung and S. T. Ng, “Determination of quasinormal modes in leaky cavities by diagonalization,” J. Phys. A 29, 143 (1996).
[Crossref]

E. S. C. Emily, P. T. Leung, W. M. Suen, and K. Young, “Quasinormal mode expansion for linearized waves in gravitational systems,” Phys. Rev. Lett. 74, 4588 (1995).
[Crossref]

E. S. C. Ching, P. T. Leung, W. M. Suen, and K. Young, “Late-time tail of wave propagation on curved spacetime,” Phys. Rev. Lett. 74, 2414 (1995).
[Crossref] [PubMed]

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057 (1994); P. T. Leung, S. Y. Liu, S. S. Tong, and K. Young, “Time-independent perturbation theory for quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3068 (1994); P. T. Leung, S. Y. Liu, and K. Young, “Completeness and time-independent perturbation of the quasinormal modes of an absorptive and leaky cavity,” Phys. Rev. A 49, 3982 (1994).
[Crossref] [PubMed]

C. C. Lam, P. T. Leung, and K. Young, “Explicit asymptotic formulas for the positions, widths, and strengths of resonances in Mie scattering,” J. Opt. Soc. Am. B 9, 1585 (1992).
[Crossref]

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 (1991).
[Crossref]

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 (1990).
[Crossref] [PubMed]

Lin, H.-B.

A. L. Huston, H.-B. Lin, J. D. Eversole, and A. J. Campillo, Naval Research Laboratory, Washington D.C. 20375 (private communication, 1992).

Liu, S. Y.

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057 (1994); P. T. Leung, S. Y. Liu, S. S. Tong, and K. Young, “Time-independent perturbation theory for quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3068 (1994); P. T. Leung, S. Y. Liu, and K. Young, “Completeness and time-independent perturbation of the quasinormal modes of an absorptive and leaky cavity,” Phys. Rev. A 49, 3982 (1994).
[Crossref] [PubMed]

Long, M. B.

Mazumder, M. M.

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Mount, K. E.

M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Progr. Phys. 35, 315 (1972).
[Crossref]

Newton, R. G.

R. G. Newton, “Analytic properties of radial wave functions,” J. Math. Phys. 1, 319 (1960).
[Crossref]

Ng, S. T.

P. T. Leung and S. T. Ng, “Determination of quasinormal modes in leaky cavities by diagonalization,” J. Phys. A 29, 143 (1996).
[Crossref]

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 (1980).
[Crossref]

Papen, G.

Pinnick, R. 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 (1993).
[Crossref]

Rudin, W.

See, e.g., W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1966).

Ruekgauer, T. E.

Rushforth, C. K.

Sipe, J. E.

Snow, J. B.

Suen, W. M.

E. S. C. Ching, P. T. Leung, W. M. Suen, and K. Young, “Late-time tail of wave propagation on curved spacetime,” Phys. Rev. Lett. 74, 2414 (1995).
[Crossref] [PubMed]

E. S. C. Emily, P. T. Leung, W. M. Suen, and K. Young, “Quasinormal mode expansion for linearized waves in gravitational systems,” Phys. Rev. Lett. 74, 4588 (1995).
[Crossref]

Toon, O. B.

Tzeng, H. M.

Walecka, J. D.

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

Wall, K. F.

Watson, K. M.

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

Xie, J.-G.

Young, K.

E. S. C. Emily, P. T. Leung, W. M. Suen, and K. Young, “Quasinormal mode expansion for linearized waves in gravitational systems,” Phys. Rev. Lett. 74, 4588 (1995).
[Crossref]

E. S. C. Ching, P. T. Leung, W. M. Suen, and K. Young, “Late-time tail of wave propagation on curved spacetime,” Phys. Rev. Lett. 74, 2414 (1995).
[Crossref] [PubMed]

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057 (1994); P. T. Leung, S. Y. Liu, S. S. Tong, and K. Young, “Time-independent perturbation theory for quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3068 (1994); P. T. Leung, S. Y. Liu, and K. Young, “Completeness and time-independent perturbation of the quasinormal modes of an absorptive and leaky cavity,” Phys. Rev. A 49, 3982 (1994).
[Crossref] [PubMed]

C. C. Lam, P. T. Leung, and K. Young, “Explicit asymptotic formulas for the positions, widths, and strengths of resonances in Mie scattering,” J. Opt. Soc. Am. B 9, 1585 (1992).
[Crossref]

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 (1991).
[Crossref]

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 (1990).
[Crossref] [PubMed]

Zhang, J.-Z.

Appl. Opt. (1)

Chem. Phys. Lett. (1)

L. M. Folan, S. Arnold, and D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322 (1987).
[Crossref]

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 (1993).
[Crossref]

J. Appl. Phys. (1)

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242 (1951); R. Bhandari, “Scattering coefficients for a multi-layered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1960 (1985).
[Crossref]

J. Math. Phys. (1)

R. G. Newton, “Analytic properties of radial wave functions,” J. Math. Phys. 1, 319 (1960).
[Crossref]

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

J. Opt. Soc. Am. B (4)

J. Phys. A (1)

P. T. Leung and S. T. Ng, “Determination of quasinormal modes in leaky cavities by diagonalization,” J. Phys. A 29, 143 (1996).
[Crossref]

Opt. Lett. (3)

Phys. Rev. (1)

R. E. Langer, “On the connection formulas and solutions of the wave equation,” Phys. Rev. 51, 669 (1937).
[Crossref]

Phys. Rev. A (2)

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 (1990).
[Crossref] [PubMed]

P. T. Leung, S. Y. Liu, and K. Young, “Completeness and orthogonality of quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3057 (1994); P. T. Leung, S. Y. Liu, S. S. Tong, and K. Young, “Time-independent perturbation theory for quasinormal modes in leaky optical cavities,” Phys. Rev. A 49, 3068 (1994); P. T. Leung, S. Y. Liu, and K. Young, “Completeness and time-independent perturbation of the quasinormal modes of an absorptive and leaky cavity,” Phys. Rev. A 49, 3982 (1994).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

E. S. C. Emily, P. T. Leung, W. M. Suen, and K. Young, “Quasinormal mode expansion for linearized waves in gravitational systems,” Phys. Rev. Lett. 74, 4588 (1995).
[Crossref]

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 (1980).
[Crossref]

E. S. C. Ching, P. T. Leung, W. M. Suen, and K. Young, “Late-time tail of wave propagation on curved spacetime,” Phys. Rev. Lett. 74, 2414 (1995).
[Crossref] [PubMed]

Rep. Progr. Phys. (1)

M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Progr. Phys. 35, 315 (1972).
[Crossref]

Other (9)

See, e.g., W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1966).

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

See, e.g., S. C. Hill and R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber and R. K. Chang, eds. (World Scientific, Singapore, 1988), p. 3.

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

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

For problems defined on a half-line x ∈ [0, ∞), the origin is counted as a discontinuity so that only one more discontinuity in the dielectric constant distribution is required for establishment of the completeness relation.

A. L. Huston, H.-B. Lin, J. D. Eversole, and A. J. Campillo, Naval Research Laboratory, Washington D.C. 20375 (private communication, 1992).

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

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Locations of the TE (open circles) and the TM (filled circles) QNM’s (l = 10) of a uniform dielectric sphere with n = 1.33 in the complex xωa plane. The dashed line shows the asymptotic value of ωI predicted by Eq. (3.30).

Fig. 2
Fig. 2

Radial function φ(r) of QNM’s (l = 10) of a uniform dielectric sphere with n = 1.33: (a) TE mode with ωa = (8.4232, −4.1649), (b) TM mode with ωa = (7.7316, −4.4029). The solid and the dashed curves, respectively, show the real and the imaginary parts of φ(r).

Fig. 3
Fig. 3

Log–log plot of the difference (absolute value) between the partial sum of the N terms of S in Eq. (3.33) and the step function θ(rr) for the l = 10 QNM’s of a uniform dielectric sphere (n = 1.33) against N. (a) r = 0.4, r = 0.5; (b) r = 0.6, r = 0.5. In (a) and (b) the smooth lines (continuous and dotted lines) show the upper limit of the difference. The continuous and the dotted lines denote, respectively, results of TE and TM modes.

Fig. 4
Fig. 4

Exact positions (open circles) and approximate positions (pluses) of the l = 10 QNM’s in a uniform dielectric sphere with dielectric constant n2(1 + α). Here n = 1.33, and α = −0.1. The approximate eigenfrequencies are obtained from the second-order perturbation theory. (a) TE QNM’s, (b) TM QNM’s.

Fig. 5
Fig. 5

Error in QNM’s frequencies Δ ω ω j = 0 k μ j ω ( j ) is plotted against the mode order J, which is assigned in the order of increasing ωR. QNM’s with angular momentum l = 10, in a coated sphere with dielectric constant n2(1 + α). Here n = 1.33, and α = −0.1. The squares and the open and filled circles, respectively, show results obtained from the zeroth-(k = 0), the first- (k = 1), and the second-order (k = 2) expansions. (a) TE modes, log10(|Δω|a) versus J; (b) TM modes, log10(|Δω|a) versus J.

Fig. 6
Fig. 6

Exact positions (open circles) and approximate positions (pluses) of the QNM’s (l = 10) in a coated dielectric sphere described by Eq. (4.33) with b/a = 0.9, n = 1.33, and α = +0.1. The approximate positions are obtained from second-order perturbation theory. (a) TE modes, (b) TM modes.

Fig. 7
Fig. 7

Error in the QNM’s frequencies Δ ω ω j = 0 k μ j ω ( j ) is plotted against the mode order J, which is assigned in the order of increasing ωR. The QNM’s (l = 10) are those of a uniform dielectric sphere with n = 1.33, α = −0.1, and b/a = 0.9 are considered. The squares and the open and filled circles, respectively, show results obtained from the zeroth- (k = 0), the first- (k = 1), and the second-order (k = 2) expansions. (a) TE modes, log10(|Δω|a) versus J; (b) TM modes, log10(|Δω|a) versus J.

Equations (83)

Equations on this page are rendered with MathJax. Learn more.

[ ( x ) 2 t 2 2 x 2 ] φ ( x , t ) = 0 ,
[ ( x ) 2 t 2 2 x 2 ] G ( x , x ; t ) = δ ( x x ) δ ( t ) ,
× ( × E ) + ( r ) 2 E t 2 = J t ,
× ( × E ) + ( r ) 2 E t 2 = 0 ,
× ( × e ) + ( r ) ω 2 e = 0 .
· ( E ) = 0 ,
e ( r ) = l m ϕ l m ( r ) X l m + 1 ( r ) × [ ψ l m ( r ) X l m ] ,
d 2 Φ l m d r 2 + [ ( r ) ω 2 l ( l + 1 ) r 2 ] Φ l m = 0 ,
d d r 1 ( r ) d Ψ l m d r + [ ω 2 l ( l + 1 ) ( r ) r 2 ] Ψ l m = 0 ,
d d r ρ ( r ) d φ ( r ) d r + ρ ( r ) [ ( r ) ω 2 l ( l + 1 ) r 2 ] φ ( r ) = 0 .
[ ρ ( r ) ( r ) 2 t 2 r ρ ( r ) r + ρ ( r ) l ( l + 1 ) r 2 ] G ( r , r ; t ) = δ ( t ) δ ( r r ) ,
G ˜ ( r , r ; ω ) = 0 d t G ( r , r ; t ) exp ( i ω t )
D ˜ ( ω ) G ˜ [ ρ ( r ) ( r ) ω 2 d d r ρ ( r ) d d r + ρ ( r ) l ( l + 1 ) r 2 ] × G ˜ ( r , r ; ω ) = δ ( r r )
W ( ω , r ) = g d f d r f d g d r .
G ˜ ( r , r ; ω ) = f ( ω , r ) g ( ω , r ) / W ¯ ( ω ) , r < r , = f ( ω , r ) g ( ω , r ) / W ¯ ( ω ) , r < r ,
lim r [ exp ( i n 0 ω r ) g ( ω , r ) ] = 1 ,
f ( ω , r = 0 ) = 0 , lim r 0 r ( l + 1 ) f ( ω , r ) = 1 ,
R j = f ( ω j , r ) g ( ω j , r ) / [ W ¯ ( ω = ω j ) / ω ] ,
( ω 2 ω j 2 ) 0 X d r ρ ( r ) ( r ) f ( ω j , r ) g ( ω , r ) = ρ ( r ) [ g ( ω , r ) f ( ω j , r ) g ( ω , r ) f ( ω j , r ) ] | 0 X ,
ρ ( X ) { i ( ω j ω ) ( X ) 1 / 2 f ( ω j , X ) g ( ω , X ) [ g ( ω , 0 ) f ( ω j , 0 ) g ( ω , 0 ) f ( ω j , 0 ) ] }
lim X 0 X d x ρ ( r ) ( r ) f ( ω j , r ) g ( ω j , r ) + i 2 ω j ρ ( X ) ( X ) 1 / 2 f ( ω j , X ) g ( ω j , X ) = 1 2 ω j W ¯ ( ω = ω j ) ω .
f j | f j lim X 0 X d x ρ ( r ) ( r ) f ( ω j , r ) f ( ω j , r ) + i 2 ω j ρ ( X ) ( X ) 1 / 2 f ( ω j , X ) f ( ω j , X )
R j = f j ( r ) f j ( r ) 2 ω j f j | f j .
G ( r , r ; t ) = i 2 j f j ( r ) f j ( r ) exp ( i ω j t ) ω j f j | f j .
G ˙ ( r , r ; t = 0 + ) = δ ( r r ) ρ ( r ) ( r ) = lim t 0 + j f j ( r ) f j ( r ) exp ( i ω j t ) 2 f j | f j .
j ρ ( r ) ( r ) f j ( r ) f j ( r ) 2 f j | f j = δ ( r r ) ,
φ ¯ ( r ) φ ( r ) / ρ ( r ) .
d 2 φ ¯ ( r ) d r 2 + [ ( r ) ω 2 + ( d ρ / d r 2 ρ ) 2 d 2 ρ / d r 2 2 ρ l ( l + 1 ) r 2 ] × φ ¯ ( r ) = 0 ,
φ ¯ ( r ) [ k ( r ) ] 1 / 2 exp [ ± i S ( r ) ] ,
k ( r ) = [ ( r ) ω 2 + ( d p / d r 2 ρ ) 2 d 2 p / d r 2 2 ρ ( l + 1 / 2 ) 2 r 2 ] 1 / 2 .
f ( ω , r ) 1 [ ρ ( r ) k ( r ) ] 1 / 2 sin [ π 4 + r 0 r k ( r ) d r ] .
f ( ω , r ) 1 [ ω ρ ( r ) ] 1 / 2 ( r ) 1 / 4 sin [ θ ( ω , r ) + ω 0 r ( r ) 1 / 2 d r ] ,
g ( ω , r ) 1 [ ω ρ ( r ) ] 1 / 2 ( r ) 1 / 4 exp [ i ω I ( a , r ) ] ,
R = ρ | a ρ | a + ρ | a + ρ | a + .
g ( ω , r ) { exp [ i ω I ( r , a ) ] + R exp [ i ω I ( r , a ) ] } [ ω ρ ( r ) ] 1 / 2 ( r ) 1 / 4 ( 1 + R )
G ˜ ( r , r ; ω ) sin [ θ + ω I ( 0 , r ) ] { exp [ i ω I ( r , a ) ] + R exp [ i ω I ( r , a ) ] } ρ ( r ) 1 / 2 ( r ) 1 / 4 ρ ( r ) 1 / 2 ( r ) 1 / 4 ω ( exp { i [ θ + ω I ( 0 , a ) ] } + R exp { i [ θ + ω I ( 0 , a ) ] } ) .
G ˜ ( r , r ; ω ) exp ( i ω t ) exp [ i ω ( t + r r ) ] ω , r r .
φ ( ω , r ) = r j l ( n ω r )
ρ ( a ) r j l ( n ω r ) d [ r j l ( n ω r ) ] d r | r = a = 1 r h l ( 1 ) ( ω r ) d [ r h l ( 1 ) ( ω r ) ] d r | r = a ,
ω I 1 2 n a ln n + 1 n 1 ,
f j | f j = N l ( a 3 / 2 ) j l 2 ( n x j ) ,
N l = n 2 1 ,
N l = ( 1 1 n 2 ) { [ j l ( n x j ) j l ( n x j ) + 1 n x j ] 2 + l 2 + l x j 2 }
S = j [ 0 r d r 1 ρ ( r 1 ) ( r 1 ) f j ( r 1 ) ] f j ( r ) 2 f j | f j = 0 r d r 1 δ ( r 1 r ) = θ ( r r ) .
e 1 j l m ( r ) f 1 j ( r ) r X l m ,
e 2 j l m ( r ) 1 ω 2 j ( r ) × [ f 2 j ( r ) r X l m ] .
e ν j l m | e ν j l m lim X r X d 3 r ( r ) [ e ν j l m ( r ) ] · e ν j l m ( r ) + i 2 ω ν j ( X ) 1 / 2 d 3 r δ ( r X ) × [ e ν j l m ( r ) ] · e ν j l m ( r ) ,
e ν j l m | e ν j l m = f ν j | f ν j ,
e ( r ) = ν j l m α ν j l m e ν j l m ( r ) .
f i | f j lim X 0 X d x ρ ( x ) ( r ) f ( ω i , r ) f ( ω j , r ) + i ω i + ω j ρ ( X ) ( X ) 1 / 2 f ( ω i , X ) f ( ω j , X ) ,
( r ) = ( r ) ( 0 ) + δ ( r )
[ ρ ( r ) ( 0 ) ( r ) ( 0 ) ω 2 d d r ρ ( r ) ( 0 ) d d r + ρ ( r ) ( 0 ) l ( l + 1 ) r 2 ] × D ˜ ( r , r ; ω ) = δ ( r r ) ,
D ˜ ( r , r ; ω ) = 1 2 j f j ( 0 ) ( r ) f j ( 0 ) ( r ) ω j ( 0 ) ( ω ω j ( 0 ) ) f j ( 0 ) | f j ( 0 ) ,
[ ρ ( r ) ( 0 ) ( r ) ( 0 ) ω 2 d d r ρ ( r ) ( 0 ) d d r + ρ ( r ) ( 0 ) l ( l + 1 ) r 2 ] × G ˜ ( r , r ; ω ) = δ ( r r ) + Δ G ˜ ( r , r ; ω ) ,
Δ G ˜ ( r , r ; ω ) [ ( δ ρ ( 0 ) + ρ ( 0 ) δ + δ ρ δ ) ω 2 + d d r δ ρ d d r δ ρ l ( l + 1 ) r 2 ] G ˜ ( r , r ; ω ) .
G ˜ = D ˜ + D ˜ Δ D ˜ + D ˜ Δ D ˜ Δ D ˜ + .
[ D ˜ Δ D ˜ ] ( r , r ; ω ) = 0 a d r 1 D ˜ ( r , r 1 ; ω ) Δ D ˜ ( r 1 , r ; ω ) .
G ˜ ( r , r ; ω ) = j k f j ( 0 ) ( r ) G j k f k ( 0 ) ( r ) .
G = D + D Δ D + D Δ D Δ D + .
D j k ( ω ) = 1 2 1 ω j ( 0 ) ( ω ω j ( 0 ) ) δ j k ,
Δ j k ( ω ) = 0 a d r f j ( 0 ) ( r ) Δ f k ( 0 ) ( r ) .
G ˜ ( r , r ; ω ) = 1 2 j f j ( r ) f j ( r ) ω j ( ω ω j ) f j | f j + G ˜ A ( r , r ; ω ) .
W j k p ( ω ) Δ j k + l p Δ j l D l l Δ l k + l p q p Δ j l D l l Δ l q D q q Δ q k + .
G j k = D j k + D j j W j k p D k k + D j j W j p p D p p n = 0 [ W p p p D p p ] n × W p k p D k k .
G ˜ ( r , r ; ω ) = j k f j ( 0 ) ( r ) f k ( 0 ) ( r ) [ D j k + D j j W j k p D k k + D j j W j p p W p k p D k k 2 ω p ( 0 ) ( ω ω p ( 0 ) ) W p p p ] .
F ( ω ) 2 ω p ( 0 ) ( ω ω p ( 0 ) ) W ( ω ) p p p .
f p ( r ) f p ( r ) f p | f p = lim ω ω p [ 2 ω p ( ω ω p ) G ˜ ( r , r ; ω ) ] ,
f p ( r ) = [ f p ( 0 ) ( r ) + j p W ( ω p ) p j p D j j ( ω p ) f j ( 0 ) ( r ) ] ÷ { 1 + 1 2 ω p [ W p ω ] ω = ω p } 1 / 2 ,
μ V ν ( r ) ( r ) ( r ) ( 0 ) , ν = 1 ; = [ ( r ) ( 0 ) ] 2 [ 1 / ( r ) ( 0 ) 1 / ( r ) ] , ν = 2 ;
V ν j k = r a d 3 r V ν ( r ) [ e ν j l m ( 0 ) ( r ) ] · e ν k l m ( 0 ) ( r ) e ν j l m ( 0 ) | e ν j l m ( 0 ) 1 / 2 e ν k l m ( 0 ) | e ν k l m ( 0 ) 1 / 2 ,
Δ j k ( ω ) μ ω 2 V 1 j k ,
Δ j k ( ω ) μ ω j ( 0 ) ω k ( 0 ) V 2 j k
ω p = ω p ( 0 ) + μ ω p ( 1 ) + μ 2 ω p ( 2 ) + .
ω p ( 1 ) = ω p ( 0 ) 2 V 1 p p ,
ω p ( 2 ) = ω p ( 0 ) 4 { 2 V 1 p p 2 + m p V 1 p m [ ω p ( 0 ) ] 2 ω m ( 0 ) ( ω p ( 0 ) ω m ( 0 ) ) V 1 m p } ,
ω p ( 3 ) = ω p ( 0 ) 8 { 5 V 1 p p 3 + V 1 p p m p × V 1 p m [ ω p ( 0 ) ] 2 ( 6 ω m ( 0 ) 5 ω p ( 0 ) ) ω m ( 0 ) ( ω p ( 0 ) ω m ( 0 ) ) 2 V 1 m p m p i p V 1 p m [ ω p ( 0 ) ] 2 ω m ( 0 ) ( ω p ( 0 ) ω m ( 0 ) ) × V 1 m i [ ω p ( 0 ) ] 2 ω i ( 0 ) ( ω p ( 0 ) ω i ( 0 ) ) V 1 i p } .
ω p ( 1 ) = ω p ( 0 ) V 2 p p 2 ,
ω p ( 2 ) = ω p ( 0 ) 4 m p V 2 p m ω m ( 0 ) ( ω p ( 0 ) ω m ( 0 ) ) V 2 m p ,
ω p ( 3 ) = ω p ( 0 ) 8 m p i p V 2 p m ω m ( 0 ) ( ω p ( 0 ) ω m ( 0 ) ) × V 2 m i ω i ( 0 ) ( ω p ( 0 ) ω i ( 0 ) ) V 2 i p + ω p ( 0 ) 8 V 2 p p m p V 2 p m ω p ( 0 ) ω m ( 0 ) ( ω m ( 0 ) ω p ( 0 ) ) 2 V 2 m p .
( r ) = n 2 ( 1 + α ) , 0 r a ; = 1 , r > a .
μ V ν ( r ) = α n 2 , ν = 1 , = α n 2 ( 1 + α ) , ν = 2 ,
( r ) = n 2 , 0 r b , = n 2 ( 1 + α ) , b < r a , = 1 , r > a .
μ V ν ( r ) = α n 2 , ν = 1 , = α n 2 ( 1 + α ) , ν = 2

Metrics