Abstract

Explicit asymptotic formulas are derived for the positions, widths, and strengths of the morphology-dependent resonances in Mie scattering. These formulas are compared with numerical data and found to be highly accurate, especially for the low-order resonances most relevant to nonlinear processes. They permit the interpretation of experimental data on light scattering from microdroplets without resorting to the full apparatus of the Mie scattering formalism.

© 1992 Optical Society of America

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  1. G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908);M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  2. J. Cooney, A. Gross, “Coherent anti-Stokes Raman scattering by droplets in the Mie size range,” Opt. Lett. 7, 218–220 (1982);J. F. Owen, R. K. Chang, P. W. Barber, “Morphology-dependent resonances in Raman scattering, fluorescence emission, and elastic scattering from microparticles,” Aerosol Sci. Technol 1, 293–302 (1982);S. C. Hill, R. E. Benner, “Morphology-dependent resonances associated with stimulated processes in microspheres,” J. Opt. Soc. Am. B 3, 1509–1514 (1986);J. B. Snow, S.-X. Qian, R. K. Chang, “Stimulated Raman scattering from individual water and ethanol droplets at morphology-dependent resonances,” Opt. Lett. 10, 37–39 (1985);“Nonlinear optics with a micrometer-size droplet,” Opt. News 12(5), 5–7 (1986);S.-X. Qian, J. B. Snow, R. K. Chang, “Coherent Raman mixing and coherent anti-Stokes Raman scattering from individual micrometer-sized droplets,” Opt. Lett. 10, 499–501 (1985);“Nonlinear optical processes in micron-size droplets,” in Proceedings of SEICOLS’85 (Springer-Verlag, Berlin, 1985);S.-X. Qian, R. K. Chang, “Multiorder Stokes emission from micrometer-size droplets,” Phys. Rev. Lett. 56, 926–929 (1986);“Phase-modulation-broadened line shapes from micrometer-size CS2droplets,” Opt. Lett. 11, 371–373 (1986);H.-M. Tzeng, K. F. Wall, M. B. Long, R. K. Chang, “Laser emission from individual droplets at wavelengths corresponding to morphology-dependent resonances,” Opt. Lett. 9, 499–501 (1984);S.-X. Qian, J. B. Snow, H.-M. Tzeng, R. K. Chang, “Lasing droplets: highlighting the liquid-air interface by laser emission,” Science 231, 486–488 (1986);R. K. Chang, S.-X. Qian, J. H. Eickmans, “Stimulated Raman scattering, phase modulation, and coherent anti-Stokes Raman scattering from single micrometer-size liquid droplets,” in Proceedings of the Methods of Laser Spectroscopy Symposium, Y. Prior, A. Ben-Reuven, M. Rosenbluh, eds. (Plenum, New York, 1986);H.-B. Lin, A. L. Huston, B. L. Justus, A. J. Campillo, “Some characteristics of a droplet whispering-gallery mode laser,” Opt. Lett. 11, 614–616 (1986);R. G. Pinnick, A. Biswas, P. Chýlek, R. L. Armstrong, H. Latifi, E. Creegan, V. Srivastava, M. Jarzembski, G. Fernandez, “Stimulated Raman scattering in micrometer-sized droplets: time-resolved measurements,” Opt. Lett. 13, 494–496 (1988);N. M. Belov, V. Yu. Dubrovskii, E. K. Kosyrev, V. A. Motyagin, A. E. Negin, M. A. Iordanskii, V. E. Kostromin, “Nonlinear scattering and self-focusing of laser radiation in an aerosol,” Sov. J. Quantum Electron. 15, 1150–1151 (1985);L. M. Folan, S. Arnold, S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985);M. Kerker, “Electromagnetic scattering from active objects,” Appl. Opt. 17, 3337–3339 (1978);“Resonances in electromagnetic scattering by objects with negative absorption,” Appl. Opt. 18, 1180–1189 (1979);H. Chew, D.-S. Wang, M. Kerker, “Surface enhancement of coherent anti-Stokes Raman scattering by colloidal spheres,” J. Opt. Soc. Am. B 1, 56–66 (1984);H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976);M. Kerker, “An electromagnetic model for surface enhanced Raman scattering on metal colloids,” Ace. Chem. Res. 17, 271–277 (1984).
    [CrossRef] [PubMed]
  3. A. J. Campillo, J. D. Eversole, H.–B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991);D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233–236 (1981);R. G. Hulet, E. S. Hilfer, D. Kleppner, “Inhibited spontaneous emission by a Rydberg atom,” Phys. Rev. Lett. 55, 2137–2140 (1985).
    [CrossRef] [PubMed]
  4. A. Ashkin, J. M. Dziedzic, “Observation of optical resonances of dielectric spheres by light scattering,” Appl. Opt. 20, 1803–1814 (1981);P. R. Conwell, C. K. Rushforth, R. E. Benner, S. C. Hill, “Efficient automated algorithm for the sizing of dielectric microspheres using the resonance spectrum,” J. Opt. Soc. Am. A 1, 1181–1187 (1984);P. W. Barber, S. C. Hill, “Effects of particle nonsphericity on light scattering,” in Proceedings of the International Symposium on Optical Sizing: Theory and Practice, G. Gouesbet, G. Grehan, eds. (Plenum, New York, 1988);R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phys. Rev. Lett. 44, 475–478 (1980);H.-M. Tzeng, M. B. Long, R. K. Chang, “Size and shape variations of liquid droplets deduced from morphology-dependent resonances in fluorescence spectra,” in Particle Sizing and Spray Analysis, N. Chigie, G. W. Stewart, eds., Proc. Soc. Photo-Opt. Instrum. Eng.573, 80–83 (1985);H.-M. Tzeng, K. F. Wall, M. B. Long, R. K. Chang, “Evaporation and condensation rates of liquid droplets deduced from structure resonances in the fluorescence spectra,” Opt. Lett. 9, 273–275 (1984).
    [CrossRef] [PubMed]
  5. P. R. Conwell, P. W. Barber, C. K. Rushforth, “Resonant spectra of dielectric spheres,” J. Opt. Soc. Am. A 1, 62–67 (1984);R. Thurn, W. Kiefer, “Structural resonances observed in the Raman spectra of optically levitated liquid droplets,” Appl. Opt. 24, 1515–1519 (1985).
    [CrossRef] [PubMed]
  6. J.-Z. Zhang, D. H. Leach, R. K. Chang, “Photon lifetime within a droplet: temporal determination of elastic and stimulated Raman scattering,” Opt. Lett. 13, 270–272 (1988);S. Arnold, L. M. Folan, “Energy transfer and the photon lifetime within an aerosol particle,” Opt. Lett. 14, 387–389 (1989).
    [CrossRef] [PubMed]
  7. H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988);“Total fluorescent scattering cross sections,” Phys. Rev. A 37, 4107–4110 (1988);“Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987);S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
    [CrossRef] [PubMed]
  8. S. C. Ching, H. M. Lai, K. Young, “Dielectric microspheres as optical cavities: thermal spectrum and density of states,” J. Opt. Soc. Am. B 4, 1995–2003 (1987).
    [CrossRef]
  9. S. C. Ching, H. M. Lai, K. Young, “Dielectric microspheres as optical cavities: Einstein A and B coefficients and level shifts,” J. Opt. Soc. Am. B 4, 2004–2009 (1987);H. M. Lai, P. T. Leung, K. Young, “Electromagnetic decay into a narrow resonance in an optical cavity,” Phys. Rev. A 37, 1597–1606 (1988);P. T. Leung, K. Young, “Theory of enhanced energy transfer in an aerosol particle,” J. Chem. Phys. 89, 2894–2899 (1988).
    [CrossRef] [PubMed]
  10. J. R. Probert-Jones, “Resonance component of backscattering by large dielectric spheres,” J. Opt. Soc. Am. A 1, 822–830 (1984).
    [CrossRef]
  11. P. Chýlek, “Resonance structure of Mie scattering: distance between resonances,” J. Opt. Soc. Am. A 7, 1609–1613 (1990).
    [CrossRef]
  12. J. D. Eversole, H.-B. Lin, A. J. Campillo, “Cavity mode identification of fluorescence and lasing in dye-doped microdroplets,” App. Opt. 31, 1982–1991 (1992).
    [CrossRef]
  13. D. H. Leach, R. K. Chang, W. P. Acker, S. C. Hill, “Third order sum frequency generation in droplets: experimental results,” J. Opt. Soc. Am. B. (to be published);S. C. Hill, D. H. Leach, R. K. Chang, “Third order sum frequency generation in droplets: model with numerical results for third-harmonic generation,” J. Opt. Soc. Am. B (to be published).
  14. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1981);C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Interscience, New York, 1983).
  15. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1972).
  16. When the derivatives are taken, Ai″ also appears, but it can be eliminated by using the Airy equation.
  17. H. M. Lai, C. C. Lam, P. T. Leung, K. Young, “Effect of perturbations on the widths of narrow morphology dependent resonances in Mie scattering,” J. Opt. Soc. Am. B 8, 1962–1973 (1991);P. T. Leung, K. Young, “Time-independent perturbation theory for quasimodes in quantum mechanics,” Phys. Rev. A 44, 3152–3161 (1991).
    [CrossRef] [PubMed]
  18. H. M. Lai, P. T. Leung, K. Young, P. W. Barber, S. C. Hill, “Time-dependent perturbation theory for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
    [CrossRef] [PubMed]
  19. S. Schiller, R. L. Byer, “High-resolution spectroscopy of whispering gallery modes in large dielectric spheres,” Opt. Lett. 16, 1138–1140 (1991).
    [CrossRef] [PubMed]

1992 (1)

J. D. Eversole, H.-B. Lin, A. J. Campillo, “Cavity mode identification of fluorescence and lasing in dye-doped microdroplets,” App. Opt. 31, 1982–1991 (1992).
[CrossRef]

1991 (3)

H. M. Lai, C. C. Lam, P. T. Leung, K. Young, “Effect of perturbations on the widths of narrow morphology dependent resonances in Mie scattering,” J. Opt. Soc. Am. B 8, 1962–1973 (1991);P. T. Leung, K. Young, “Time-independent perturbation theory for quasimodes in quantum mechanics,” Phys. Rev. A 44, 3152–3161 (1991).
[CrossRef] [PubMed]

S. Schiller, R. L. Byer, “High-resolution spectroscopy of whispering gallery modes in large dielectric spheres,” Opt. Lett. 16, 1138–1140 (1991).
[CrossRef] [PubMed]

A. J. Campillo, J. D. Eversole, H.–B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991);D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233–236 (1981);R. G. Hulet, E. S. Hilfer, D. Kleppner, “Inhibited spontaneous emission by a Rydberg atom,” Phys. Rev. Lett. 55, 2137–2140 (1985).
[CrossRef] [PubMed]

1990 (2)

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

P. Chýlek, “Resonance structure of Mie scattering: distance between resonances,” J. Opt. Soc. Am. A 7, 1609–1613 (1990).
[CrossRef]

1988 (2)

J.-Z. Zhang, D. H. Leach, R. K. Chang, “Photon lifetime within a droplet: temporal determination of elastic and stimulated Raman scattering,” Opt. Lett. 13, 270–272 (1988);S. Arnold, L. M. Folan, “Energy transfer and the photon lifetime within an aerosol particle,” Opt. Lett. 14, 387–389 (1989).
[CrossRef] [PubMed]

H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988);“Total fluorescent scattering cross sections,” Phys. Rev. A 37, 4107–4110 (1988);“Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987);S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
[CrossRef] [PubMed]

1987 (2)

1984 (2)

1982 (1)

J. Cooney, A. Gross, “Coherent anti-Stokes Raman scattering by droplets in the Mie size range,” Opt. Lett. 7, 218–220 (1982);J. F. Owen, R. K. Chang, P. W. Barber, “Morphology-dependent resonances in Raman scattering, fluorescence emission, and elastic scattering from microparticles,” Aerosol Sci. Technol 1, 293–302 (1982);S. C. Hill, R. E. Benner, “Morphology-dependent resonances associated with stimulated processes in microspheres,” J. Opt. Soc. Am. B 3, 1509–1514 (1986);J. B. Snow, S.-X. Qian, R. K. Chang, “Stimulated Raman scattering from individual water and ethanol droplets at morphology-dependent resonances,” Opt. Lett. 10, 37–39 (1985);“Nonlinear optics with a micrometer-size droplet,” Opt. News 12(5), 5–7 (1986);S.-X. Qian, J. B. Snow, R. K. Chang, “Coherent Raman mixing and coherent anti-Stokes Raman scattering from individual micrometer-sized droplets,” Opt. Lett. 10, 499–501 (1985);“Nonlinear optical processes in micron-size droplets,” in Proceedings of SEICOLS’85 (Springer-Verlag, Berlin, 1985);S.-X. Qian, R. K. Chang, “Multiorder Stokes emission from micrometer-size droplets,” Phys. Rev. Lett. 56, 926–929 (1986);“Phase-modulation-broadened line shapes from micrometer-size CS2droplets,” Opt. Lett. 11, 371–373 (1986);H.-M. Tzeng, K. F. Wall, M. B. Long, R. K. Chang, “Laser emission from individual droplets at wavelengths corresponding to morphology-dependent resonances,” Opt. Lett. 9, 499–501 (1984);S.-X. Qian, J. B. Snow, H.-M. Tzeng, R. K. Chang, “Lasing droplets: highlighting the liquid-air interface by laser emission,” Science 231, 486–488 (1986);R. K. Chang, S.-X. Qian, J. H. Eickmans, “Stimulated Raman scattering, phase modulation, and coherent anti-Stokes Raman scattering from single micrometer-size liquid droplets,” in Proceedings of the Methods of Laser Spectroscopy Symposium, Y. Prior, A. Ben-Reuven, M. Rosenbluh, eds. (Plenum, New York, 1986);H.-B. Lin, A. L. Huston, B. L. Justus, A. J. Campillo, “Some characteristics of a droplet whispering-gallery mode laser,” Opt. Lett. 11, 614–616 (1986);R. G. Pinnick, A. Biswas, P. Chýlek, R. L. Armstrong, H. Latifi, E. Creegan, V. Srivastava, M. Jarzembski, G. Fernandez, “Stimulated Raman scattering in micrometer-sized droplets: time-resolved measurements,” Opt. Lett. 13, 494–496 (1988);N. M. Belov, V. Yu. Dubrovskii, E. K. Kosyrev, V. A. Motyagin, A. E. Negin, M. A. Iordanskii, V. E. Kostromin, “Nonlinear scattering and self-focusing of laser radiation in an aerosol,” Sov. J. Quantum Electron. 15, 1150–1151 (1985);L. M. Folan, S. Arnold, S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985);M. Kerker, “Electromagnetic scattering from active objects,” Appl. Opt. 17, 3337–3339 (1978);“Resonances in electromagnetic scattering by objects with negative absorption,” Appl. Opt. 18, 1180–1189 (1979);H. Chew, D.-S. Wang, M. Kerker, “Surface enhancement of coherent anti-Stokes Raman scattering by colloidal spheres,” J. Opt. Soc. Am. B 1, 56–66 (1984);H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976);M. Kerker, “An electromagnetic model for surface enhanced Raman scattering on metal colloids,” Ace. Chem. Res. 17, 271–277 (1984).
[CrossRef] [PubMed]

1981 (1)

1908 (1)

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908);M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Acker, W. P.

D. H. Leach, R. K. Chang, W. P. Acker, S. C. Hill, “Third order sum frequency generation in droplets: experimental results,” J. Opt. Soc. Am. B. (to be published);S. C. Hill, D. H. Leach, R. K. Chang, “Third order sum frequency generation in droplets: model with numerical results for third-harmonic generation,” J. Opt. Soc. Am. B (to be published).

Ashkin, A.

Barber, P. W.

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

P. R. Conwell, P. W. Barber, C. K. Rushforth, “Resonant spectra of dielectric spheres,” J. Opt. Soc. Am. A 1, 62–67 (1984);R. Thurn, W. Kiefer, “Structural resonances observed in the Raman spectra of optically levitated liquid droplets,” Appl. Opt. 24, 1515–1519 (1985).
[CrossRef] [PubMed]

Byer, R. L.

Campillo, A. J.

J. D. Eversole, H.-B. Lin, A. J. Campillo, “Cavity mode identification of fluorescence and lasing in dye-doped microdroplets,” App. Opt. 31, 1982–1991 (1992).
[CrossRef]

A. J. Campillo, J. D. Eversole, H.–B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991);D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233–236 (1981);R. G. Hulet, E. S. Hilfer, D. Kleppner, “Inhibited spontaneous emission by a Rydberg atom,” Phys. Rev. Lett. 55, 2137–2140 (1985).
[CrossRef] [PubMed]

Chang, R. K.

J.-Z. Zhang, D. H. Leach, R. K. Chang, “Photon lifetime within a droplet: temporal determination of elastic and stimulated Raman scattering,” Opt. Lett. 13, 270–272 (1988);S. Arnold, L. M. Folan, “Energy transfer and the photon lifetime within an aerosol particle,” Opt. Lett. 14, 387–389 (1989).
[CrossRef] [PubMed]

D. H. Leach, R. K. Chang, W. P. Acker, S. C. Hill, “Third order sum frequency generation in droplets: experimental results,” J. Opt. Soc. Am. B. (to be published);S. C. Hill, D. H. Leach, R. K. Chang, “Third order sum frequency generation in droplets: model with numerical results for third-harmonic generation,” J. Opt. Soc. Am. B (to be published).

Chew, H.

H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988);“Total fluorescent scattering cross sections,” Phys. Rev. A 37, 4107–4110 (1988);“Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987);S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
[CrossRef] [PubMed]

Ching, S. C.

Chýlek, P.

Conwell, P. R.

Cooney, J.

J. Cooney, A. Gross, “Coherent anti-Stokes Raman scattering by droplets in the Mie size range,” Opt. Lett. 7, 218–220 (1982);J. F. Owen, R. K. Chang, P. W. Barber, “Morphology-dependent resonances in Raman scattering, fluorescence emission, and elastic scattering from microparticles,” Aerosol Sci. Technol 1, 293–302 (1982);S. C. Hill, R. E. Benner, “Morphology-dependent resonances associated with stimulated processes in microspheres,” J. Opt. Soc. Am. B 3, 1509–1514 (1986);J. B. Snow, S.-X. Qian, R. K. Chang, “Stimulated Raman scattering from individual water and ethanol droplets at morphology-dependent resonances,” Opt. Lett. 10, 37–39 (1985);“Nonlinear optics with a micrometer-size droplet,” Opt. News 12(5), 5–7 (1986);S.-X. Qian, J. B. Snow, R. K. Chang, “Coherent Raman mixing and coherent anti-Stokes Raman scattering from individual micrometer-sized droplets,” Opt. Lett. 10, 499–501 (1985);“Nonlinear optical processes in micron-size droplets,” in Proceedings of SEICOLS’85 (Springer-Verlag, Berlin, 1985);S.-X. Qian, R. K. Chang, “Multiorder Stokes emission from micrometer-size droplets,” Phys. Rev. Lett. 56, 926–929 (1986);“Phase-modulation-broadened line shapes from micrometer-size CS2droplets,” Opt. Lett. 11, 371–373 (1986);H.-M. Tzeng, K. F. Wall, M. B. Long, R. K. Chang, “Laser emission from individual droplets at wavelengths corresponding to morphology-dependent resonances,” Opt. Lett. 9, 499–501 (1984);S.-X. Qian, J. B. Snow, H.-M. Tzeng, R. K. Chang, “Lasing droplets: highlighting the liquid-air interface by laser emission,” Science 231, 486–488 (1986);R. K. Chang, S.-X. Qian, J. H. Eickmans, “Stimulated Raman scattering, phase modulation, and coherent anti-Stokes Raman scattering from single micrometer-size liquid droplets,” in Proceedings of the Methods of Laser Spectroscopy Symposium, Y. Prior, A. Ben-Reuven, M. Rosenbluh, eds. (Plenum, New York, 1986);H.-B. Lin, A. L. Huston, B. L. Justus, A. J. Campillo, “Some characteristics of a droplet whispering-gallery mode laser,” Opt. Lett. 11, 614–616 (1986);R. G. Pinnick, A. Biswas, P. Chýlek, R. L. Armstrong, H. Latifi, E. Creegan, V. Srivastava, M. Jarzembski, G. Fernandez, “Stimulated Raman scattering in micrometer-sized droplets: time-resolved measurements,” Opt. Lett. 13, 494–496 (1988);N. M. Belov, V. Yu. Dubrovskii, E. K. Kosyrev, V. A. Motyagin, A. E. Negin, M. A. Iordanskii, V. E. Kostromin, “Nonlinear scattering and self-focusing of laser radiation in an aerosol,” Sov. J. Quantum Electron. 15, 1150–1151 (1985);L. M. Folan, S. Arnold, S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985);M. Kerker, “Electromagnetic scattering from active objects,” Appl. Opt. 17, 3337–3339 (1978);“Resonances in electromagnetic scattering by objects with negative absorption,” Appl. Opt. 18, 1180–1189 (1979);H. Chew, D.-S. Wang, M. Kerker, “Surface enhancement of coherent anti-Stokes Raman scattering by colloidal spheres,” J. Opt. Soc. Am. B 1, 56–66 (1984);H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976);M. Kerker, “An electromagnetic model for surface enhanced Raman scattering on metal colloids,” Ace. Chem. Res. 17, 271–277 (1984).
[CrossRef] [PubMed]

Dziedzic, J. M.

Eversole, J. D.

J. D. Eversole, H.-B. Lin, A. J. Campillo, “Cavity mode identification of fluorescence and lasing in dye-doped microdroplets,” App. Opt. 31, 1982–1991 (1992).
[CrossRef]

A. J. Campillo, J. D. Eversole, H.–B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991);D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233–236 (1981);R. G. Hulet, E. S. Hilfer, D. Kleppner, “Inhibited spontaneous emission by a Rydberg atom,” Phys. Rev. Lett. 55, 2137–2140 (1985).
[CrossRef] [PubMed]

Gross, A.

J. Cooney, A. Gross, “Coherent anti-Stokes Raman scattering by droplets in the Mie size range,” Opt. Lett. 7, 218–220 (1982);J. F. Owen, R. K. Chang, P. W. Barber, “Morphology-dependent resonances in Raman scattering, fluorescence emission, and elastic scattering from microparticles,” Aerosol Sci. Technol 1, 293–302 (1982);S. C. Hill, R. E. Benner, “Morphology-dependent resonances associated with stimulated processes in microspheres,” J. Opt. Soc. Am. B 3, 1509–1514 (1986);J. B. Snow, S.-X. Qian, R. K. Chang, “Stimulated Raman scattering from individual water and ethanol droplets at morphology-dependent resonances,” Opt. Lett. 10, 37–39 (1985);“Nonlinear optics with a micrometer-size droplet,” Opt. News 12(5), 5–7 (1986);S.-X. Qian, J. B. Snow, R. K. Chang, “Coherent Raman mixing and coherent anti-Stokes Raman scattering from individual micrometer-sized droplets,” Opt. Lett. 10, 499–501 (1985);“Nonlinear optical processes in micron-size droplets,” in Proceedings of SEICOLS’85 (Springer-Verlag, Berlin, 1985);S.-X. Qian, R. K. Chang, “Multiorder Stokes emission from micrometer-size droplets,” Phys. Rev. Lett. 56, 926–929 (1986);“Phase-modulation-broadened line shapes from micrometer-size CS2droplets,” Opt. Lett. 11, 371–373 (1986);H.-M. Tzeng, K. F. Wall, M. B. Long, R. K. Chang, “Laser emission from individual droplets at wavelengths corresponding to morphology-dependent resonances,” Opt. Lett. 9, 499–501 (1984);S.-X. Qian, J. B. Snow, H.-M. Tzeng, R. K. Chang, “Lasing droplets: highlighting the liquid-air interface by laser emission,” Science 231, 486–488 (1986);R. K. Chang, S.-X. Qian, J. H. Eickmans, “Stimulated Raman scattering, phase modulation, and coherent anti-Stokes Raman scattering from single micrometer-size liquid droplets,” in Proceedings of the Methods of Laser Spectroscopy Symposium, Y. Prior, A. Ben-Reuven, M. Rosenbluh, eds. (Plenum, New York, 1986);H.-B. Lin, A. L. Huston, B. L. Justus, A. J. Campillo, “Some characteristics of a droplet whispering-gallery mode laser,” Opt. Lett. 11, 614–616 (1986);R. G. Pinnick, A. Biswas, P. Chýlek, R. L. Armstrong, H. Latifi, E. Creegan, V. Srivastava, M. Jarzembski, G. Fernandez, “Stimulated Raman scattering in micrometer-sized droplets: time-resolved measurements,” Opt. Lett. 13, 494–496 (1988);N. M. Belov, V. Yu. Dubrovskii, E. K. Kosyrev, V. A. Motyagin, A. E. Negin, M. A. Iordanskii, V. E. Kostromin, “Nonlinear scattering and self-focusing of laser radiation in an aerosol,” Sov. J. Quantum Electron. 15, 1150–1151 (1985);L. M. Folan, S. Arnold, S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985);M. Kerker, “Electromagnetic scattering from active objects,” Appl. Opt. 17, 3337–3339 (1978);“Resonances in electromagnetic scattering by objects with negative absorption,” Appl. Opt. 18, 1180–1189 (1979);H. Chew, D.-S. Wang, M. Kerker, “Surface enhancement of coherent anti-Stokes Raman scattering by colloidal spheres,” J. Opt. Soc. Am. B 1, 56–66 (1984);H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976);M. Kerker, “An electromagnetic model for surface enhanced Raman scattering on metal colloids,” Ace. Chem. Res. 17, 271–277 (1984).
[CrossRef] [PubMed]

Hill, S. C.

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

D. H. Leach, R. K. Chang, W. P. Acker, S. C. Hill, “Third order sum frequency generation in droplets: experimental results,” J. Opt. Soc. Am. B. (to be published);S. C. Hill, D. H. Leach, R. K. Chang, “Third order sum frequency generation in droplets: model with numerical results for third-harmonic generation,” J. Opt. Soc. Am. B (to be published).

Lai, H. M.

Lam, C. C.

Leach, D. H.

J.-Z. Zhang, D. H. Leach, R. K. Chang, “Photon lifetime within a droplet: temporal determination of elastic and stimulated Raman scattering,” Opt. Lett. 13, 270–272 (1988);S. Arnold, L. M. Folan, “Energy transfer and the photon lifetime within an aerosol particle,” Opt. Lett. 14, 387–389 (1989).
[CrossRef] [PubMed]

D. H. Leach, R. K. Chang, W. P. Acker, S. C. Hill, “Third order sum frequency generation in droplets: experimental results,” J. Opt. Soc. Am. B. (to be published);S. C. Hill, D. H. Leach, R. K. Chang, “Third order sum frequency generation in droplets: model with numerical results for third-harmonic generation,” J. Opt. Soc. Am. B (to be published).

Leung, P. T.

Lin, H.-B.

J. D. Eversole, H.-B. Lin, A. J. Campillo, “Cavity mode identification of fluorescence and lasing in dye-doped microdroplets,” App. Opt. 31, 1982–1991 (1992).
[CrossRef]

Lin, H.–B.

A. J. Campillo, J. D. Eversole, H.–B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991);D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233–236 (1981);R. G. Hulet, E. S. Hilfer, D. Kleppner, “Inhibited spontaneous emission by a Rydberg atom,” Phys. Rev. Lett. 55, 2137–2140 (1985).
[CrossRef] [PubMed]

Mie, G.

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908);M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Probert-Jones, J. R.

Rushforth, C. K.

Schiller, S.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1981);C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Interscience, New York, 1983).

Young, K.

Zhang, J.-Z.

Ann. Phys. (Leipzig) (1)

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908);M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

App. Opt. (1)

J. D. Eversole, H.-B. Lin, A. J. Campillo, “Cavity mode identification of fluorescence and lasing in dye-doped microdroplets,” App. Opt. 31, 1982–1991 (1992).
[CrossRef]

Appl. Opt. (1)

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

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

Opt. Lett. (3)

J. Cooney, A. Gross, “Coherent anti-Stokes Raman scattering by droplets in the Mie size range,” Opt. Lett. 7, 218–220 (1982);J. F. Owen, R. K. Chang, P. W. Barber, “Morphology-dependent resonances in Raman scattering, fluorescence emission, and elastic scattering from microparticles,” Aerosol Sci. Technol 1, 293–302 (1982);S. C. Hill, R. E. Benner, “Morphology-dependent resonances associated with stimulated processes in microspheres,” J. Opt. Soc. Am. B 3, 1509–1514 (1986);J. B. Snow, S.-X. Qian, R. K. Chang, “Stimulated Raman scattering from individual water and ethanol droplets at morphology-dependent resonances,” Opt. Lett. 10, 37–39 (1985);“Nonlinear optics with a micrometer-size droplet,” Opt. News 12(5), 5–7 (1986);S.-X. Qian, J. B. Snow, R. K. Chang, “Coherent Raman mixing and coherent anti-Stokes Raman scattering from individual micrometer-sized droplets,” Opt. Lett. 10, 499–501 (1985);“Nonlinear optical processes in micron-size droplets,” in Proceedings of SEICOLS’85 (Springer-Verlag, Berlin, 1985);S.-X. Qian, R. K. Chang, “Multiorder Stokes emission from micrometer-size droplets,” Phys. Rev. Lett. 56, 926–929 (1986);“Phase-modulation-broadened line shapes from micrometer-size CS2droplets,” Opt. Lett. 11, 371–373 (1986);H.-M. Tzeng, K. F. Wall, M. B. Long, R. K. Chang, “Laser emission from individual droplets at wavelengths corresponding to morphology-dependent resonances,” Opt. Lett. 9, 499–501 (1984);S.-X. Qian, J. B. Snow, H.-M. Tzeng, R. K. Chang, “Lasing droplets: highlighting the liquid-air interface by laser emission,” Science 231, 486–488 (1986);R. K. Chang, S.-X. Qian, J. H. Eickmans, “Stimulated Raman scattering, phase modulation, and coherent anti-Stokes Raman scattering from single micrometer-size liquid droplets,” in Proceedings of the Methods of Laser Spectroscopy Symposium, Y. Prior, A. Ben-Reuven, M. Rosenbluh, eds. (Plenum, New York, 1986);H.-B. Lin, A. L. Huston, B. L. Justus, A. J. Campillo, “Some characteristics of a droplet whispering-gallery mode laser,” Opt. Lett. 11, 614–616 (1986);R. G. Pinnick, A. Biswas, P. Chýlek, R. L. Armstrong, H. Latifi, E. Creegan, V. Srivastava, M. Jarzembski, G. Fernandez, “Stimulated Raman scattering in micrometer-sized droplets: time-resolved measurements,” Opt. Lett. 13, 494–496 (1988);N. M. Belov, V. Yu. Dubrovskii, E. K. Kosyrev, V. A. Motyagin, A. E. Negin, M. A. Iordanskii, V. E. Kostromin, “Nonlinear scattering and self-focusing of laser radiation in an aerosol,” Sov. J. Quantum Electron. 15, 1150–1151 (1985);L. M. Folan, S. Arnold, S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985);M. Kerker, “Electromagnetic scattering from active objects,” Appl. Opt. 17, 3337–3339 (1978);“Resonances in electromagnetic scattering by objects with negative absorption,” Appl. Opt. 18, 1180–1189 (1979);H. Chew, D.-S. Wang, M. Kerker, “Surface enhancement of coherent anti-Stokes Raman scattering by colloidal spheres,” J. Opt. Soc. Am. B 1, 56–66 (1984);H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976);M. Kerker, “An electromagnetic model for surface enhanced Raman scattering on metal colloids,” Ace. Chem. Res. 17, 271–277 (1984).
[CrossRef] [PubMed]

J.-Z. Zhang, D. H. Leach, R. K. Chang, “Photon lifetime within a droplet: temporal determination of elastic and stimulated Raman scattering,” Opt. Lett. 13, 270–272 (1988);S. Arnold, L. M. Folan, “Energy transfer and the photon lifetime within an aerosol particle,” Opt. Lett. 14, 387–389 (1989).
[CrossRef] [PubMed]

S. Schiller, R. L. Byer, “High-resolution spectroscopy of whispering gallery modes in large dielectric spheres,” Opt. Lett. 16, 1138–1140 (1991).
[CrossRef] [PubMed]

Phys. Rev. A (2)

H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988);“Total fluorescent scattering cross sections,” Phys. Rev. A 37, 4107–4110 (1988);“Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987);S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
[CrossRef] [PubMed]

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

Phys. Rev. Lett. (1)

A. J. Campillo, J. D. Eversole, H.–B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991);D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233–236 (1981);R. G. Hulet, E. S. Hilfer, D. Kleppner, “Inhibited spontaneous emission by a Rydberg atom,” Phys. Rev. Lett. 55, 2137–2140 (1985).
[CrossRef] [PubMed]

Other (4)

D. H. Leach, R. K. Chang, W. P. Acker, S. C. Hill, “Third order sum frequency generation in droplets: experimental results,” J. Opt. Soc. Am. B. (to be published);S. C. Hill, D. H. Leach, R. K. Chang, “Third order sum frequency generation in droplets: model with numerical results for third-harmonic generation,” J. Opt. Soc. Am. B (to be published).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1981);C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Interscience, New York, 1983).

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1972).

When the derivatives are taken, Ai″ also appears, but it can be eliminated by using the Airy equation.

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Figures (4)

Fig. 1
Fig. 1

nxν for TE resonances are plotted against ν1/3 for (a) n = 1.33, i = 1,2,3; (b) n = 1.33, i = 8,9,10, and (c) n = 1.50, i = 8,9,10.

Fig. 2
Fig. 2

Approximate locations of TE resonances obtained from our result, Eq. (1.1) (squares) and that of Probert-Jones, Eq. (2.12) (circles), are normalized to the exact values xMie and plotted against mode order i for n = 1.33 and l = 130.

Fig. 3
Fig. 3

Approximate widths Γ of TE resonances (normalized to exact values ΓMie) are evaluated from Eq. (1.3) by substituting approximate values of x given by Eq. (1.1) (crosses) and exact values of x (squares), and plotted against l for n = 1.33 and i = 1.

Fig. 4
Fig. 4

Approximate widths Γ of TE resonances (normalized to exact values ΓMie) are evaluated from Eq. (1.3) (squares) and Eq. (3.5) (circles). (a) Γ/ΓMie versus l for i = 1; and (b) Γ/ΓMie versus i for l = 130. The refractive index is n = 1.33 in both figures. Note the difference in vertical scale between Figs. 3 and 4.

Tables (3)

Tables Icon

Table 1 Roots αi of Ai(− z)

Tables Icon

Table 2 Position x = xl,i for a Mode with Mode Number l and Mode Order i, and Δx = Δxl,i = xl,+1,ixl,ia

Tables Icon

Table 3 Analysis of Our Asymptotic Expansion and Chýlek’s Result in Terms of the General Expression

Equations (55)

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n x l , i = ν + 2 1 / 3 α i ν 1 / 3 P ( n 2 1 ) 1 / 2 + ( 3 10 2 2 / 3 ) α i 2 ν 1 / 3 2 1 / 3 P ( n 2 2 P 2 / 3 ) ( n 2 1 ) 3 / 2 α i ν 2 / 3 + O ( ν 1 ) ,
P = { n for TE modes 1 / n for TM modes
Γ l , i = 2 [ N x 2 n l ( x ) 2 ] 1 ,
N = { n 2 1 for TE modes ( n 2 1 ) [ μ 2 + ( μ 2 / n 2 1 ) ] for TM modes
μ = ν / x ,
n j l ( n x ) / j i ( n x ) = n l ( x ) / n l ( x )
n J ν ( n x ) / J ν ( n x ) = Y ν ( x ) / Y ν ( x ) ,
ν = n k a sin θ = n x sin θ .
n x = ν + t ν 1 / 3 .
J ν ( n x ) 2 1 / 3 ν 1 / 3 Ai ( 2 1 / 3 t ) [ 1 + j = 1 f j ( t ) / ν 2 j / 3 ] + 2 2 / 3 ν A i ( 2 1 / 3 t ) j = 0 g j ( t ) / ν 2 j / 3 ,
Y ν ( x ) exp [ ν ( β tanh β ) ] ( π ν tanh β / 2 ) 1 / 2 [ 1 + j = 1 ( ) j u j ( coth β ) ν j ] ,
cosh β = μ .
n J ν ( n x ) / J ν ( n x ) n ( 2 / ν ) 1 / 3 A i ( 2 1 / 3 t ) / Ai ( 2 1 / 3 t )
Y ν ( x ) / Y ν ( x ) | sinh β | = ( μ 2 1 ) 1 / 2 .
Ai ( 2 1 / 3 t ) = O ( ν 1 / 3 ) 0 ,
2 1 / 3 t = α i + O ( ν 1 / 3 ) .
n x l , i = ν + 2 1 / 3 α i ν 1 / 3 + O ( 1 ) ,
d / a 2 1 / 3 α i ν 2 / 3 .
2 1 / 3 t = α i j = 1 c j ν j / 3 ,
Ai ( 2 1 / 3 t ) = Ai ( α i + Δ ) = Δ Ai ( α i ) + ( Δ 2 / 2 ) Ai ( α i ) + ,
cosh β = ν x = n 1 + 2 1 / 3 ν 2 / 3 ( α i Δ ) ,
tan ϕ = n P ρ [ ( 2 u ν ) 1 / 2 1 4 u ] ,
ϕ = ν ( ρ tan 1 ρ ) ρ u + u 2 2 ν ρ π 4 ,
ρ 2 = n 2 1 , u = ν x .
n Δ x l , i = 1 + 2 1 / 3 3 α i ν 2 / 3 2 2 / 3 10 α i 2 ν 4 / 3 + [ 2 2 3 3 P ( n 2 2 P 2 / 3 ) ( n 2 1 ) 2 / 3 2 1 / 3 9 ] α i ν 5 / 3 + O ( ν 2 ) .
Δ x = x l tan 1 [ ( n x / l ) 2 1 ] 1 / 2 [ ( n x / l ) 2 1 ] 1 / 2 for | x l | 1 / 2 ,
Δ x = tan 1 [ ( n 2 1 ) 1 / 2 ] ( n 2 1 ) 1 / 2 for x / l 1 .
n x = ν + a 1 ν 1 / 3 + a 0 ν 0 + a 1 ν 1 / 3 + ,
M ( z ) h l ( 1 ) ( z ) / h l ( 1 ) ( z ) n j l ( n z ) / j l ( n z ) ,
M ( x 0 ) = i ( n 2 1 ) y 0 .
h l ( 1 ) ( x ) / h l ( 1 ) ( x ) n l ( x ) n l ( x ) [ 1 + i x 2 n l ( x ) n l ( x ) ] .
Im M ( x 0 ) 1 x 0 2 n l ( x 0 ) 2 .
Γ 2 = n P ( n 2 1 ) [ ( 2 u 0 ν ) 1 / 2 1 4 u 0 ] exp [ 4 u 0 3 ( 2 u 0 ν ) 1 / 2 ] ,
e ( r ) = { β j l ( n k r ) L Y l m ( θ , ϕ ) r < a [ γ ( 1 ) h l ( 1 ) ( k r ) + γ ( 2 ) h l ( 2 ) ( k r ) ] L Y l m ( θ , ϕ ) r > a .
k = Λ π d k .
r Λ n 2 ( r ) | e | 2 d V = 1 ,
| γ ( 1 ) | 2 = | γ ( 2 ) | 2 = k 2 2 l ( l + 1 ) Λ .
ρ E C ( k ) = r a n 2 | e | 2 d V .
k ρ E C = Λ π ρ E C ( k ) d k 1 ,
modes ( fraction of each mode in C ) = total number of modes in C .
ρ E C ( k ) = 2 n 2 k 2 Λ 0 a r 2 j l ( n k r ) 2 d r x 4 | W [ j l ( n x ) , h l ( 1 ) ( x ) ] | 2 ,
W [ f ( x ) , g ( x ) ] = f ( x ) g ( x ) f ( x ) g ( x ) ,
M ( x ) ( n 2 1 ) [ x ( x 0 + i y 0 ) ] = ( n 2 1 ) [ ( x x 0 ) + i Γ / 2 ] ,
| W [ j l ( n x ) , h l ( 1 ) ( x ) ] | 2 = | j l ( n x ) h l ( 1 ) ( x ) | 2 | M ( x ) | 2 = | j l ( n x ) h l ( 1 ) ( x ) | 2 ( n 2 1 ) 2 [ ( x x 0 ) 2 + Γ 2 / 4 ] .
0 a r 2 j l ( n k r ) 2 d r a 3 2 j l ( n x ) 2 ( n 2 1 ) / n 2 .
| h l ( 1 ) ( x ) | 2 | n l ( x ) | 2 .
ρ E C ( k ) α Λ Γ / 2 ( x x 0 ) 2 + ( Γ / 2 ) 2 ,
γ ( 2 ) = i l [ π ( 2 l + 1 ) 2 l ( l + 1 ) ] 1 / 2 E 0 .
E 2 res E 0 2 = 3 4 ( 2 l + 1 ) x 0 2 Γ / 2 ( x x 0 ) 2 + ( Γ / 2 ) 2 .
c l n l ( x 0 ) j l ( x 0 ) Γ / 2 ( x x 0 ) + i Γ / 2 ,
d l 1 n n l ( x 0 ) j l ( x 0 ) Γ / 2 ( x x 0 ) + i Γ / 2 .
p ( n 2 1 ) 1 / 2
3 10 2 2 / 3 α i 2
1 2
3 20 c 2 a

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