Abstract

The effects of light absorption and amplification on the transition rates of the electric-dipole emission of atoms or molecules embedded in micrometer-sized dielectric spheres are investigated in the framework of classical electrodynamics. The expression for the form factor of a resonant emission line is derived. Its dependence on the imaginary part of the light’s refractive index is investigated. It is discovered that the active sphere does not act as an optical resonator if light amplification is strong.

© 2002 Optical Society of America

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  4. A. Serpengüzel, J. C. Swindal, R. K. Chang, and W. P. Acker, “Two-dimensional imaging of sprays with fluorescence, lasing, and stimulated Raman scattering,” Appl. Opt. 31, 3543–3551 (1992).
    [CrossRef] [PubMed]
  5. A. J. Campillo, J. D. Eversole, and H.-B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  13. H. B. Lin, A. L. Huston, J. D. Eversole, B. L. Justus, and A. J. Campillo, “Some characteristics of a droplet whispering gallery mode laser,” J. Opt. Soc. Am. B 8, 250–251 (1986).
  14. J. D. Eversole, H.-B. Lin, and A. J. Campillo, “Cavity-mode identification of fluorescence and lasing in dye-doped microdroplets,” Appl. Opt. 31, 1982–1991 (1992).
    [CrossRef] [PubMed]
  15. P. Chýlek, “Partial-wave resonances and the ripple structure in the Mie normalized extinction cross section,” J. Opt. Soc. Am. 66, 285–287 (1976).
    [CrossRef]
  16. R. T. Wang and H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. 30, 106–117 (1991).
    [CrossRef] [PubMed]
  17. A. K. Ray, A. Souyri, E. J. Davis, and T. M. Allen, “Precision of light scattering techniques for measuring optical parameters of microspheres,” Appl. Opt. 30, 3974–3983 (1991).
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  29. W. P. Acker, A. Serpengüzel, R. K. Chang, and S. C. Hill, “Stimulated Raman scattering of fuel droplets. Chemical concentration and size determination,” Appl. Phys. B 51, 9–16 (1990).
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    [CrossRef]
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1999 (2)

S.-S. Yi and O. M. Stafsudd, “Observation of lossless radiative modes of a dielectric sphere,” J. Appl. Phys. 86, 3694–3698 (1999).
[CrossRef]

G. Roll, T. Kaiser, and G. Schweiger, “Eigenmodes of spherical dielectric cavities: coupling of internal and external rays,” J. Opt. Soc. Am. A 16, 882–895 (1999).
[CrossRef]

1996 (1)

1994 (1)

1993 (3)

1992 (6)

V. V. Datsyuk, “Some characteristics of resonant electromagnetic modes in a dielectric sphere,” Appl. Phys. B 54, 184–187 (1992).
[CrossRef]

C. C. Lam, P. T. Leung, and K. Young, “Explicit asymptotic formulas for the positions, widths, and strength of resonances in Mie scattering,” J. Opt. Soc. Am. B 9, 1585–1592 (1992).
[CrossRef]

A. Serpengüzel, J. C. Swindal, R. K. Chang, and W. P. Acker, “Two-dimensional imaging of sprays with fluorescence, lasing, and stimulated Raman scattering,” Appl. Opt. 31, 3543–3551 (1992).
[CrossRef] [PubMed]

A. J. Campillo, J. D. Eversole, and H.-B. Lin, “Cavity quantum electrodynamic enhancement of spontaneous and stimulated emission in microdroplets,” Mod. Phys. Lett. B 6, 447–457 (1992).
[CrossRef]

H.-B. Lin, J. D. Eversole, C. D. Meritt, and A. J. Campillo, “Cavity-modified spontaneous-emission rates in liquid microdroplets,” Phys. Rev. A 45, 6756–6760 (1992).
[CrossRef] [PubMed]

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

1991 (5)

1990 (4)

W. P. Acker, A. Serpengüzel, R. K. Chang, and S. C. Hill, “Stimulated Raman scattering of fuel droplets. Chemical concentration and size determination,” Appl. Phys. B 51, 9–16 (1990).
[CrossRef]

H. Latifi, A. Biswas, R. L. Armstrong, and R. G. Pinnick, “Lasing and stimulated Raman scattering in spherical liquid droplets: time, irradiance, and wavelength dependence,” Appl. Opt. 29, 5387–5392 (1990).
[CrossRef] [PubMed]

H.-B. Lin, J. D. Eversole, and A. J. Campillo, “Identification of morphology dependent resonances in stimulated Raman scattering from microdroplets,” Opt. Commun. 77, 407–410 (1990).
[CrossRef]

G. Schweiger, “Raman scattering on single aerosol particles and on flowing aerosols: a review,” J. Aerosol Sci. 21, 483–509 (1990).
[CrossRef]

1988 (1)

H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
[CrossRef] [PubMed]

1987 (3)

1986 (4)

S.-X. Qian, J. B. Snow, H.-M. Tzeng, and R. K. Chang, “Lasing droplets: highlighting the liquid–air interface by laser emission,” Science 231, 486–488 (1986).
[CrossRef] [PubMed]

W.-F. Hsieh, H.-M. Tzeng, and R. K. Chang, “High intensity laser interactions with micrometer size dye droplets,” Acad. Sin. 16, 1–8 (1986).

S. C. Hill and R. E. Benner, “Morphology-dependent resonances associated with stimulated processes in microdroplets,” J. Opt. Soc. Am. B 3, 1509–1514 (1986).
[CrossRef]

H. B. Lin, A. L. Huston, J. D. Eversole, B. L. Justus, and A. J. Campillo, “Some characteristics of a droplet whispering gallery mode laser,” J. Opt. Soc. Am. B 8, 250–251 (1986).

1984 (1)

1978 (1)

1976 (2)

H. Chew, P. J. McNulty, and M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. 13, 396–404 (1976).
[CrossRef]

P. Chýlek, “Partial-wave resonances and the ripple structure in the Mie normalized extinction cross section,” J. Opt. Soc. Am. 66, 285–287 (1976).
[CrossRef]

1974 (1)

V. S. Pekar, “On the theory of spontaneous and stimulated emission of electromagnetic waves in media and resonators inhomogeneous in one direction,” Zh. Eksp. Teor. Fiz. 67, 471–480 (1974).

1961 (1)

C. G. B. Garrett, W. Kaiser, and W. L. Bond, “Stimulated emission into optical whispering modes of spheres,” Phys. Rev. 124, 1807–1809 (1961).
[CrossRef]

Acker, W. P.

A. Serpengüzel, J. C. Swindal, R. K. Chang, and W. P. Acker, “Two-dimensional imaging of sprays with fluorescence, lasing, and stimulated Raman scattering,” Appl. Opt. 31, 3543–3551 (1992).
[CrossRef] [PubMed]

W. P. Acker, A. Serpengüzel, R. K. Chang, and S. C. Hill, “Stimulated Raman scattering of fuel droplets. Chemical concentration and size determination,” Appl. Phys. B 51, 9–16 (1990).
[CrossRef]

Alexopoulos, N. G.

Allen, T. M.

Armstrong, R. L.

Benner, R. E.

Biswas, A.

Bond, W. L.

C. G. B. Garrett, W. Kaiser, and W. L. Bond, “Stimulated emission into optical whispering modes of spheres,” Phys. Rev. 124, 1807–1809 (1961).
[CrossRef]

Campillo, A. J.

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

A. J. Campillo, J. D. Eversole, and H.-B. Lin, “Cavity quantum electrodynamic enhancement of spontaneous and stimulated emission in microdroplets,” Mod. Phys. Lett. B 6, 447–457 (1992).
[CrossRef]

H.-B. Lin, J. D. Eversole, C. D. Meritt, and A. J. Campillo, “Cavity-modified spontaneous-emission rates in liquid microdroplets,” Phys. Rev. A 45, 6756–6760 (1992).
[CrossRef] [PubMed]

A. J. Campillo, J. D. Eversole, and H.-B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[CrossRef] [PubMed]

P. Chýlek, H.-B. Lin, J. D. Eversole, and A. J. Campillo, “Absorption effects on microdroplet resonant emission structure,” Opt. Lett. 16, 1723–1725 (1991).
[CrossRef] [PubMed]

H.-B. Lin, J. D. Eversole, and A. J. Campillo, “Identification of morphology dependent resonances in stimulated Raman scattering from microdroplets,” Opt. Commun. 77, 407–410 (1990).
[CrossRef]

H. B. Lin, A. L. Huston, J. D. Eversole, B. L. Justus, and A. J. Campillo, “Some characteristics of a droplet whispering gallery mode laser,” J. Opt. Soc. Am. B 8, 250–251 (1986).

Chan, C. K.

Chang, R. K.

Chew, H.

H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
[CrossRef] [PubMed]

H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987).
[CrossRef]

H. Chew, P. J. McNulty, and M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. 13, 396–404 (1976).
[CrossRef]

Ching, S. C.

Chýlek, P.

Datsyuk, V. V.

V. V. Datsyuk, “Some characteristics of resonant electromagnetic modes in a dielectric sphere,” Appl. Phys. B 54, 184–187 (1992).
[CrossRef]

Davis, E. J.

Eversole, J. D.

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

A. J. Campillo, J. D. Eversole, and H.-B. Lin, “Cavity quantum electrodynamic enhancement of spontaneous and stimulated emission in microdroplets,” Mod. Phys. Lett. B 6, 447–457 (1992).
[CrossRef]

H.-B. Lin, J. D. Eversole, C. D. Meritt, and A. J. Campillo, “Cavity-modified spontaneous-emission rates in liquid microdroplets,” Phys. Rev. A 45, 6756–6760 (1992).
[CrossRef] [PubMed]

A. J. Campillo, J. D. Eversole, and H.-B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[CrossRef] [PubMed]

P. Chýlek, H.-B. Lin, J. D. Eversole, and A. J. Campillo, “Absorption effects on microdroplet resonant emission structure,” Opt. Lett. 16, 1723–1725 (1991).
[CrossRef] [PubMed]

H.-B. Lin, J. D. Eversole, and A. J. Campillo, “Identification of morphology dependent resonances in stimulated Raman scattering from microdroplets,” Opt. Commun. 77, 407–410 (1990).
[CrossRef]

H. B. Lin, A. L. Huston, J. D. Eversole, B. L. Justus, and A. J. Campillo, “Some characteristics of a droplet whispering gallery mode laser,” J. Opt. Soc. Am. B 8, 250–251 (1986).

Flagan, R. C.

Garrett, C. G. B.

C. G. B. Garrett, W. Kaiser, and W. L. Bond, “Stimulated emission into optical whispering modes of spheres,” Phys. Rev. 124, 1807–1809 (1961).
[CrossRef]

Hill, S. C.

W. P. Acker, A. Serpengüzel, R. K. Chang, and S. C. Hill, “Stimulated Raman scattering of fuel droplets. Chemical concentration and size determination,” Appl. Phys. B 51, 9–16 (1990).
[CrossRef]

S. C. Hill and R. E. Benner, “Morphology-dependent resonances associated with stimulated processes in microdroplets,” J. Opt. Soc. Am. B 3, 1509–1514 (1986).
[CrossRef]

Hsieh, W.-F.

W.-F. Hsieh, H.-M. Tzeng, and R. K. Chang, “High intensity laser interactions with micrometer size dye droplets,” Acad. Sin. 16, 1–8 (1986).

Huston, A. L.

Johnson, B. R.

Justus, B. L.

Kaiser, T.

Kaiser, W.

C. G. B. Garrett, W. Kaiser, and W. L. Bond, “Stimulated emission into optical whispering modes of spheres,” Phys. Rev. 124, 1807–1809 (1961).
[CrossRef]

Kerker, M.

H. Chew, P. J. McNulty, and M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. 13, 396–404 (1976).
[CrossRef]

Kwok, A. S.

Lai, H. M.

Lam, C. C.

Lange, S.

Latifi, H.

Leung, P. T.

Lin, H. B.

Lin, H.-B.

H.-B. Lin, J. D. Eversole, C. D. Meritt, and A. J. Campillo, “Cavity-modified spontaneous-emission rates in liquid microdroplets,” Phys. Rev. A 45, 6756–6760 (1992).
[CrossRef] [PubMed]

A. J. Campillo, J. D. Eversole, and H.-B. Lin, “Cavity quantum electrodynamic enhancement of spontaneous and stimulated emission in microdroplets,” Mod. Phys. Lett. B 6, 447–457 (1992).
[CrossRef]

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

P. Chýlek, H.-B. Lin, J. D. Eversole, and A. J. Campillo, “Absorption effects on microdroplet resonant emission structure,” Opt. Lett. 16, 1723–1725 (1991).
[CrossRef] [PubMed]

A. J. Campillo, J. D. Eversole, and H.-B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[CrossRef] [PubMed]

H.-B. Lin, J. D. Eversole, and A. J. Campillo, “Identification of morphology dependent resonances in stimulated Raman scattering from microdroplets,” Opt. Commun. 77, 407–410 (1990).
[CrossRef]

Long, M. B.

McNulty, P. J.

H. Chew, P. J. McNulty, and M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. 13, 396–404 (1976).
[CrossRef]

Meritt, C. D.

H.-B. Lin, J. D. Eversole, C. D. Meritt, and A. J. Campillo, “Cavity-modified spontaneous-emission rates in liquid microdroplets,” Phys. Rev. A 45, 6756–6760 (1992).
[CrossRef] [PubMed]

Pekar, V. S.

V. S. Pekar, “On the theory of spontaneous and stimulated emission of electromagnetic waves in media and resonators inhomogeneous in one direction,” Zh. Eksp. Teor. Fiz. 67, 471–480 (1974).

Pinnick, R. G.

Qian, S.-X.

S.-X. Qian, J. B. Snow, H.-M. Tzeng, and R. K. Chang, “Lasing droplets: highlighting the liquid–air interface by laser emission,” Science 231, 486–488 (1986).
[CrossRef] [PubMed]

Ray, A. K.

Roll, G.

Schweiger, G.

Seinfeld, J. H.

Serpengüzel, A.

A. Serpengüzel, J. C. Swindal, R. K. Chang, and W. P. Acker, “Two-dimensional imaging of sprays with fluorescence, lasing, and stimulated Raman scattering,” Appl. Opt. 31, 3543–3551 (1992).
[CrossRef] [PubMed]

W. P. Acker, A. Serpengüzel, R. K. Chang, and S. C. Hill, “Stimulated Raman scattering of fuel droplets. Chemical concentration and size determination,” Appl. Phys. B 51, 9–16 (1990).
[CrossRef]

Snow, J. B.

S.-X. Qian, J. B. Snow, H.-M. Tzeng, and R. K. Chang, “Lasing droplets: highlighting the liquid–air interface by laser emission,” Science 231, 486–488 (1986).
[CrossRef] [PubMed]

Souyri, A.

Stafsudd, O. M.

S.-S. Yi and O. M. Stafsudd, “Observation of lossless radiative modes of a dielectric sphere,” J. Appl. Phys. 86, 3694–3698 (1999).
[CrossRef]

Swindal, J. C.

Tzeng, H.-M.

S.-X. Qian, J. B. Snow, H.-M. Tzeng, and R. K. Chang, “Lasing droplets: highlighting the liquid–air interface by laser emission,” Science 231, 486–488 (1986).
[CrossRef] [PubMed]

W.-F. Hsieh, H.-M. Tzeng, and R. K. Chang, “High intensity laser interactions with micrometer size dye droplets,” Acad. Sin. 16, 1–8 (1986).

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).
[CrossRef] [PubMed]

Uzunoglu, N. K.

van de Hulst, H. C.

Wall, K. F.

Wang, R. T.

Yi, S.-S.

S.-S. Yi and O. M. Stafsudd, “Observation of lossless radiative modes of a dielectric sphere,” J. Appl. Phys. 86, 3694–3698 (1999).
[CrossRef]

Young, K.

Acad. Sin. (1)

W.-F. Hsieh, H.-M. Tzeng, and R. K. Chang, “High intensity laser interactions with micrometer size dye droplets,” Acad. Sin. 16, 1–8 (1986).

Appl. Opt. (8)

A. Serpengüzel, J. C. Swindal, R. K. Chang, and W. P. Acker, “Two-dimensional imaging of sprays with fluorescence, lasing, and stimulated Raman scattering,” Appl. Opt. 31, 3543–3551 (1992).
[CrossRef] [PubMed]

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

R. T. Wang and H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. 30, 106–117 (1991).
[CrossRef] [PubMed]

A. K. Ray, A. Souyri, E. J. Davis, and T. M. Allen, “Precision of light scattering techniques for measuring optical parameters of microspheres,” Appl. Opt. 30, 3974–3983 (1991).
[CrossRef] [PubMed]

C. K. Chan, R. C. Flagan, and J. H. Seinfeld, “Resonance structures in elastic and Raman scattering from microspheres,” Appl. Opt. 30, 459–467 (1991).
[CrossRef] [PubMed]

T. Kaiser, G. Roll, and G. Schweiger, “Investigation of coated droplets in an optical trap: Raman scattering, elastic-scattering and evaporation characteristics,” Appl. Opt. 35, 5918–5924 (1996).
[CrossRef] [PubMed]

H. Latifi, A. Biswas, R. L. Armstrong, and R. G. Pinnick, “Lasing and stimulated Raman scattering in spherical liquid droplets: time, irradiance, and wavelength dependence,” Appl. Opt. 29, 5387–5392 (1990).
[CrossRef] [PubMed]

N. G. Alexopoulos and N. K. Uzunoglu, “Electromagnetic scattering from active objects: invisible scatterers,” Appl. Opt. 17, 235–239 (1978).
[CrossRef] [PubMed]

Appl. Phys. B (2)

V. V. Datsyuk, “Some characteristics of resonant electromagnetic modes in a dielectric sphere,” Appl. Phys. B 54, 184–187 (1992).
[CrossRef]

W. P. Acker, A. Serpengüzel, R. K. Chang, and S. C. Hill, “Stimulated Raman scattering of fuel droplets. Chemical concentration and size determination,” Appl. Phys. B 51, 9–16 (1990).
[CrossRef]

J. Aerosol Sci. (1)

G. Schweiger, “Raman scattering on single aerosol particles and on flowing aerosols: a review,” J. Aerosol Sci. 21, 483–509 (1990).
[CrossRef]

J. Appl. Phys. (1)

S.-S. Yi and O. M. Stafsudd, “Observation of lossless radiative modes of a dielectric sphere,” J. Appl. Phys. 86, 3694–3698 (1999).
[CrossRef]

J. Chem. Phys. (1)

H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (6)

Mod. Phys. Lett. B (1)

A. J. Campillo, J. D. Eversole, and H.-B. Lin, “Cavity quantum electrodynamic enhancement of spontaneous and stimulated emission in microdroplets,” Mod. Phys. Lett. B 6, 447–457 (1992).
[CrossRef]

Opt. Commun. (1)

H.-B. Lin, J. D. Eversole, and A. J. Campillo, “Identification of morphology dependent resonances in stimulated Raman scattering from microdroplets,” Opt. Commun. 77, 407–410 (1990).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. (2)

H. Chew, P. J. McNulty, and M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. 13, 396–404 (1976).
[CrossRef]

C. G. B. Garrett, W. Kaiser, and W. L. Bond, “Stimulated emission into optical whispering modes of spheres,” Phys. Rev. 124, 1807–1809 (1961).
[CrossRef]

Phys. Rev. A (2)

H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
[CrossRef] [PubMed]

H.-B. Lin, J. D. Eversole, C. D. Meritt, and A. J. Campillo, “Cavity-modified spontaneous-emission rates in liquid microdroplets,” Phys. Rev. A 45, 6756–6760 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

A. J. Campillo, J. D. Eversole, and H.-B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[CrossRef] [PubMed]

Science (1)

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[CrossRef] [PubMed]

Zh. Eksp. Teor. Fiz. (1)

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Other (3)

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Figures (3)

Fig. 1
Fig. 1

Wavelength dependence of the average rate R/Rv for (a) n=1.33, (b) n=1.33+ι10-4, and (c) n=1.33-ι10-4. The large peaks in (a), from left to right, are due to the first-order WGMs TE58, TM58, TE57, and TM57, respectively. The small maxima, from the left to right, are caused by the second-order TM53, TE52, and TM52 modes.

Fig. 2
Fig. 2

Average rate R/Rv versus |ni| at (a) λ=0.90184311 µm, where the TE58 resonance is dominant, and (b) λ=0.91000211 µm, where the TM58 resonance is dominant. Solid and dashed curves, light amplification and absorption, respectively.

Fig. 3
Fig. 3

Inelastic emission spectra from R6G in ethanol droplets at input-laser intensities of (a) 0.01, (b) 0.1, and (c) 0.8 GW/cm2.

Tables (1)

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Table 1 Suppression of the Resonances with Increasing Input Laser Intensity

Equations (16)

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Q-1=Q0-1+Qa-1,
II0QQ0=QaQ0+Qa.
R(ξ)Rv=12(xξ)2 l=1(2l+1)|ψl(zξ)|2|D0l|2+(l+1)|ψl-1(zξ)|2+l|ψl+1(zξ)|2|nD1l|2,
Dpl(x, n)n1-2pψl(nx)ζl(x)-ψl(nx)ξl(x),
RRv=3 01R(ξ)Rvξ2dξ.
R/Δω1,
ψl(nx)=πβ Ai(tq+y),
ζl(x)=-ι/sinh ηeT1+ι2e-2T,
|npDpl(x, n)|-2=2xρ0x ddxψl(nx)-2F(x),
xρ0=1nr l+12-tqβ-nr1-2p 1sinh η,
F(x)=1Qr 4x-xρ0x2+1Q2-1,
Qr=(½)xρ0nr2p sinh ηe2T.
F(xρ0±(½)wρ)=(½)F(xρ0),
wρ(ni)=|wρ(0)+2xρ0(ni/nr)|.
II0|Q|Qr=QaQr+Qa.
R/Δω=(R/Δω)|ni=0×|1-(N/N1)|-3.

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