Abstract

A steady-state model of third-order sum-frequency generation (TSFG) in liquid droplets is presented. A third-order polarization is generated in a droplet by stimulated Raman scattering (SRS) or by SRS in combination with fields at the incident laser frequency. This third-order polarization radiates inside the sphere as described in the general model of Chew et al. [ Phys. Rev. A 19, 396 ( 1976)]. The frequency of the third-order polarization will only rarely be within a few linewidths of a high-Q output morphology-dependent resonance (MDR). The TSFG intensity is proportional to the spatial and frequency overlap between the generating polarization and the fields of the output modes. For a distribution of particle sizes, the probability of detecting TSFG increases with the density of output MDR’s. Numerical results are presented for the specific case of the generation of the third harmonic of the first-order SRS. Although the comparison with experiment is only qualitative, the theory is consistent with experimental observations that TSFG in droplets is detectable at only a small fraction of the droplet sizes and is 4–6 orders of magnitude weaker than SRS.

© 1993 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
  5. D. H. Leach, W. P. Acker, and R. K. Chang, “The effect of the phase velocity and spatial overlap of spherical resonances on sum-frequency generation in droplets,” Opt. Lett. 15, 894–896 (1990).
    [CrossRef] [PubMed]
  6. D. H. Leach, R. K. Chang, W. P. Acker, and S. C. Hill, “Third-order sum-frequency generation in droplets: experimental results,” J. Opt. Soc. Am. B 10, 34–45 (1993).
    [CrossRef]
  7. J. B. Snow, S.-X. Qian, and R. K. Chang, “Stimulated Raman scattering from individual water and ethanol droplets at morphology-dependent resonances,” Opt. Lett. 10, 37–39 (1985).
    [CrossRef] [PubMed]
  8. 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 (1991).
    [CrossRef]
  9. H.-B. Lin, J. D. Eversole, and A. J. Campillo, “Double-resonance stimulated Raman scattering in micrometer-sized droplets,” J. Opt. Soc. Am. B 7, 2079–2089 (1990).
    [CrossRef]
  10. A. Biswas, H. Latifi, R. L. Armstrong, and R. G. Pinnick, “Double resonance stimulated Raman scattering from optically levitated glycerol droplets,” Phys. Rev. A 40, 7413–7416 (1989).
    [CrossRef] [PubMed]
  11. G. Chen, W. P. Acker, R. K. Chang, and S. C. Hill, “Angular fine structure in the stimulated Raman scattering from single droplets,” Opt. Lett. 16, 117–119 (1991).
    [CrossRef] [PubMed]
  12. H. M. Tzeng, K. F. Wall, M. B. Long, and R. K. Chang, “Laser emission from individual droplets at wavelength corresponding to morphology-dependent resonances,” Opt. Lett. 9, 499–501 (1984).
    [CrossRef] [PubMed]
  13. H.-B. Lin, A. L. Huston, B. L. Justus, and A. J. Campillo, “Some characteristics of a droplet whispering-gallery-mode laser,” Opt. Lett. 11, 614–616 (1986).
    [CrossRef] [PubMed]
  14. 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]
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    [CrossRef]
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  17. H. Chew, M. Sculley, M. Kerker, P. J. McNulty, and D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: results for coherent optical processes,” J. Opt. Soc. Am. 68, 1686–1689 (1978).
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  25. S. D. Ching, H. M. Lai, and K. Young, “Dielectric micro-spheres as optical cavities: Einstein A and B coefficients and level shifts,” J. Opt. Soc. Am. B 4, 2004–2009 (1987).
    [CrossRef]
  26. H. Chew, “Radiation lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
    [CrossRef] [PubMed]
  27. 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]
  28. J.-Z. Zhang, D. H. Leach, and R. K. Chang, “Photon lifetime within a droplet: temporal determination of elastic and stimulated Raman scattering,” Opt. Lett. 13, 270–272 (1988).
    [CrossRef] [PubMed]
  29. S. C. Hill and R. E. Benner, “Morphology-dependent resonances associated with stimulated processes in microspheres,” J. Opt. Soc. Am. B 3, 1509–1514 (1986).
    [CrossRef]
  30. E. E. M. Khaled, S. C. Hill, P. W. Barber, and D. Q. Chowdhury, “Near-resonance excitation of dielectric spheres with plane waves and off-axis Gaussian beams,” Appl. Opt. 31, 1166–1169 (1992).
    [CrossRef] [PubMed]
  31. R. Lang, M. O. Scully, and W. E. Lamb, “Why is the laser line so narrow? A theory of single-quasimode laser operation,” Phys. Rev. A 7, 1788–1797 (1972).
    [CrossRef]
  32. R. Fuchs and K. L. Kliewar, “Optical modes of vibration in an ionic crystal sphere,” J. Opt. Soc. Am. 58, 319–330 (1968).
    [CrossRef]
  33. H. M. Lai, P. T. Leung, K. Young, S. C. Hill, and P. W. Barber, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
    [CrossRef] [PubMed]
  34. G. Chen, R. K. Chang, S. C. Hill, and P. W. Barber, “Frequency splitting of degenerate spherical cavity modes: stimulated Raman spectrum of deformed droplets,” Opt. Lett. 16, 1269–1271 (1991).
    [CrossRef] [PubMed]
  35. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
    [CrossRef]
  36. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
    [CrossRef]
  37. S. C. Hill, C. K. Rushforth, R. E. Benner, and P. R. Conwell, “Sizing dielectric spheres and cylinders by aligning measured and computed resonance locations: algorithm for multiple orders,” Appl. Opt. 24, 2380–2390 (1985).
    [CrossRef] [PubMed]
  38. A. Serpengüzel, G. Chen, R. K. Chang, and W.-F. Hsieh, “Heuristic model for the growth and coupling of nonlinear processes in droplets,” J. Opt. Soc. Am. B 9, 871–883 (1992).
    [CrossRef]
  39. D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Time-dependence of internal intensity of a dielectric sphere on and near resonance,” J. Opt. Soc. Am. A 9, 1364–1373 (1992).
    [CrossRef]

1993 (1)

1992 (3)

1991 (4)

G. Chen, W. P. Acker, R. K. Chang, and S. C. Hill, “Angular fine structure in the stimulated Raman scattering from single droplets,” Opt. Lett. 16, 117–119 (1991).
[CrossRef] [PubMed]

G. Chen, R. K. Chang, S. C. Hill, and P. W. Barber, “Frequency splitting of degenerate spherical cavity modes: stimulated Raman spectrum of deformed droplets,” Opt. Lett. 16, 1269–1271 (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 (1991).
[CrossRef]

1990 (4)

1989 (4)

W. P. Acker, D. H. Leach, and R. K. Chang, “Third-order optical sum-frequency generation in micrometer-sized liquid droplets,” Opt. Lett. 14, 402–404 (1989).
[CrossRef] [PubMed]

J.-Z. Zhang and R. K. Chang, “Generation and suppression of stimulated Brillouin scattering in single liquid droplets,” J. Opt. Soc. Am. B 6, 151–153 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

A. Biswas, H. Latifi, R. L. Armstrong, and R. G. Pinnick, “Double resonance stimulated Raman scattering from optically levitated glycerol droplets,” Phys. Rev. A 40, 7413–7416 (1989).
[CrossRef] [PubMed]

1988 (3)

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

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

J.-Z. Zhang, D. H. Leach, and R. K. Chang, “Photon lifetime within a droplet: temporal determination of elastic and stimulated Raman scattering,” Opt. Lett. 13, 270–272 (1988).
[CrossRef] [PubMed]

1987 (1)

1986 (2)

1985 (2)

1984 (2)

1978 (2)

1976 (1)

H. Chew, P. J. McNulty, and M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 19, 396–404 (1976).
[CrossRef]

1972 (1)

R. Lang, M. O. Scully, and W. E. Lamb, “Why is the laser line so narrow? A theory of single-quasimode laser operation,” Phys. Rev. A 7, 1788–1797 (1972).
[CrossRef]

1968 (1)

1965 (1)

P. D. Maker and R. W. Terhune, “Study of the optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. A 137, 801–818 (1965).
[CrossRef]

Acker, W. P.

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Armstrong, R. L.

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]

A. Biswas, H. Latifi, R. L. Armstrong, and R. G. Pinnick, “Double resonance stimulated Raman scattering from optically levitated glycerol droplets,” Phys. Rev. A 40, 7413–7416 (1989).
[CrossRef] [PubMed]

Barber, P. W.

Barton, J. P.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Benner, R. E.

Biswas, A.

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]

A. Biswas, H. Latifi, R. L. Armstrong, and R. G. Pinnick, “Double resonance stimulated Raman scattering from optically levitated glycerol droplets,” Phys. Rev. A 40, 7413–7416 (1989).
[CrossRef] [PubMed]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992).

Campillo, A. J.

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 (1991).
[CrossRef]

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, “Double-resonance stimulated Raman scattering in micrometer-sized droplets,” J. Opt. Soc. Am. B 7, 2079–2089 (1990).
[CrossRef]

H.-B. Lin, A. L. Huston, B. L. Justus, and A. J. Campillo, “Some characteristics of a droplet whispering-gallery-mode laser,” Opt. Lett. 11, 614–616 (1986).
[CrossRef] [PubMed]

Chang, R. K.

D. H. Leach, R. K. Chang, W. P. Acker, and S. C. Hill, “Third-order sum-frequency generation in droplets: experimental results,” J. Opt. Soc. Am. B 10, 34–45 (1993).
[CrossRef]

A. Serpengüzel, G. Chen, R. K. Chang, and W.-F. Hsieh, “Heuristic model for the growth and coupling of nonlinear processes in droplets,” J. Opt. Soc. Am. B 9, 871–883 (1992).
[CrossRef]

G. Chen, R. K. Chang, S. C. Hill, and P. W. Barber, “Frequency splitting of degenerate spherical cavity modes: stimulated Raman spectrum of deformed droplets,” Opt. Lett. 16, 1269–1271 (1991).
[CrossRef] [PubMed]

G. Chen, W. P. Acker, R. K. Chang, and S. C. Hill, “Angular fine structure in the stimulated Raman scattering from single droplets,” Opt. Lett. 16, 117–119 (1991).
[CrossRef] [PubMed]

D. H. Leach, W. P. Acker, and R. K. Chang, “The effect of the phase velocity and spatial overlap of spherical resonances on sum-frequency generation in droplets,” Opt. Lett. 15, 894–896 (1990).
[CrossRef] [PubMed]

W. P. Acker, D. H. Leach, and R. K. Chang, “Third-order optical sum-frequency generation in micrometer-sized liquid droplets,” Opt. Lett. 14, 402–404 (1989).
[CrossRef] [PubMed]

J.-Z. Zhang and R. K. Chang, “Generation and suppression of stimulated Brillouin scattering in single liquid droplets,” J. Opt. Soc. Am. B 6, 151–153 (1989).
[CrossRef]

J.-Z. Zhang, D. H. Leach, and R. K. Chang, “Photon lifetime within a droplet: temporal determination of elastic and stimulated Raman scattering,” Opt. Lett. 13, 270–272 (1988).
[CrossRef] [PubMed]

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

H. M. Tzeng, K. F. Wall, M. B. Long, and R. K. Chang, “Laser emission from individual droplets at wavelength corresponding to morphology-dependent resonances,” Opt. Lett. 9, 499–501 (1984).
[CrossRef] [PubMed]

Chen, G.

Chew, H.

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).

Ching, S. D.

Chowdhury, D. Q.

Conwell, P. R.

Cooke, D. D.

Edmonds, A. R.

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1957).

Eversole, J. D.

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 (1991).
[CrossRef]

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, “Double-resonance stimulated Raman scattering in micrometer-sized droplets,” J. Opt. Soc. Am. B 7, 2079–2089 (1990).
[CrossRef]

Fuchs, R.

Hill, S. C.

D. H. Leach, R. K. Chang, W. P. Acker, and S. C. Hill, “Third-order sum-frequency generation in droplets: experimental results,” J. Opt. Soc. Am. B 10, 34–45 (1993).
[CrossRef]

D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Time-dependence of internal intensity of a dielectric sphere on and near resonance,” J. Opt. Soc. Am. A 9, 1364–1373 (1992).
[CrossRef]

E. E. M. Khaled, S. C. Hill, P. W. Barber, and D. Q. Chowdhury, “Near-resonance excitation of dielectric spheres with plane waves and off-axis Gaussian beams,” Appl. Opt. 31, 1166–1169 (1992).
[CrossRef] [PubMed]

G. Chen, R. K. Chang, S. C. Hill, and P. W. Barber, “Frequency splitting of degenerate spherical cavity modes: stimulated Raman spectrum of deformed droplets,” Opt. Lett. 16, 1269–1271 (1991).
[CrossRef] [PubMed]

G. Chen, W. P. Acker, R. K. Chang, and S. C. Hill, “Angular fine structure in the stimulated Raman scattering from single droplets,” Opt. Lett. 16, 117–119 (1991).
[CrossRef] [PubMed]

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

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

S. C. Hill, C. K. Rushforth, R. E. Benner, and P. R. Conwell, “Sizing dielectric spheres and cylinders by aligning measured and computed resonance locations: algorithm for multiple orders,” Appl. Opt. 24, 2380–2390 (1985).
[CrossRef] [PubMed]

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

Hsieh, W.-F.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Huston, A. L.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Chap. 16.

Justus, B. L.

Kerker, M.

Khaled, E. E. M.

Kliewar, K. L.

Lai, H. M.

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

S. D. Ching, H. M. Lai, and K. Young, “Dielectric micro-spheres as optical cavities: Einstein A and B coefficients and level shifts,” J. Opt. Soc. Am. B 4, 2004–2009 (1987).
[CrossRef]

Lamb, W. E.

R. Lang, M. O. Scully, and W. E. Lamb, “Why is the laser line so narrow? A theory of single-quasimode laser operation,” Phys. Rev. A 7, 1788–1797 (1972).
[CrossRef]

Lang, R.

R. Lang, M. O. Scully, and W. E. Lamb, “Why is the laser line so narrow? A theory of single-quasimode laser operation,” Phys. Rev. A 7, 1788–1797 (1972).
[CrossRef]

Latifi, H.

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]

A. Biswas, H. Latifi, R. L. Armstrong, and R. G. Pinnick, “Double resonance stimulated Raman scattering from optically levitated glycerol droplets,” Phys. Rev. A 40, 7413–7416 (1989).
[CrossRef] [PubMed]

Leach, D. H.

Leung, P. T.

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

Lin, H.-B.

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 (1991).
[CrossRef]

H.-B. Lin, J. D. Eversole, and A. J. Campillo, “Double-resonance stimulated Raman scattering in micrometer-sized droplets,” J. Opt. Soc. Am. B 7, 2079–2089 (1990).
[CrossRef]

H.-B. Lin, A. L. Huston, B. L. Justus, and A. J. Campillo, “Some characteristics of a droplet whispering-gallery-mode laser,” Opt. Lett. 11, 614–616 (1986).
[CrossRef] [PubMed]

Long, M. B.

Maker, P. D.

P. D. Maker and R. W. Terhune, “Study of the optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. A 137, 801–818 (1965).
[CrossRef]

McNulty, P. J.

Pinnick, R. G.

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]

A. Biswas, H. Latifi, R. L. Armstrong, and R. G. Pinnick, “Double resonance stimulated Raman scattering from optically levitated glycerol droplets,” Phys. Rev. A 40, 7413–7416 (1989).
[CrossRef] [PubMed]

Qian, S.-X.

Rushforth, C. K.

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Sculley, M.

Scully, M. O.

R. Lang, M. O. Scully, and W. E. Lamb, “Why is the laser line so narrow? A theory of single-quasimode laser operation,” Phys. Rev. A 7, 1788–1797 (1972).
[CrossRef]

Serpengüzel, A.

Snow, J. B.

Terhune, R. W.

P. D. Maker and R. W. Terhune, “Study of the optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. A 137, 801–818 (1965).
[CrossRef]

Tzeng, H. M.

Wall, K. F.

Wang, D.-S.

Yariv, A.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), p. 455.

Young, K.

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

S. D. Ching, H. M. Lai, and K. Young, “Dielectric micro-spheres as optical cavities: Einstein A and B coefficients and level shifts,” J. Opt. Soc. Am. B 4, 2004–2009 (1987).
[CrossRef]

Zhang, J.-Z.

Appl. Opt. (3)

J. Appl. Phys. (2)

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

J. Opt. Soc. Am. (3)

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

J. Opt. Soc. Am. B (7)

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 (1991).
[CrossRef]

Opt. Lett. (8)

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Figures (9)

Fig. 1
Fig. 1

Radial variations jn13 and jn3 (two upper curves in each panel) and the product jn1jn3, which is the integrand of Kj (lower curve). The jn1, and jn3 are the spherical Bessel functions, where n1 is the mode number of the generating wave and n3 is the mode number of the output wave. The third-order polarization that generates the TSFG is proportional to jn13. The fields of the output MDR are proportional to jn3. (a) n1 = 163, l1 = 1, n3 = 508, and l3 = 1, where l1 and l3 are the mode orders of the generating and output waves. (b) n1 = 163, l1 = 1, n3 = 473, l3 = 5. (c) n1 = 143, l1 = 4, n3 = 411, l3 = 16. The refractive indices at the two frequencies are m(ω1) = 1.32 and m(ω3) = 1.335.

Fig. 2
Fig. 2

Radial-overlap integral Kj2, i.e., the square of the integral of jn13jn3r2, integrated from r = 0 to r = a, for TE MDR’s with 125 < x1 < 135 and |x3 − 3x1| < 0.05, plotted as a function of n3/n1, the ratio of the mode numbers. The refractive indices are m(ω1) = 1.32 and m(ω3) = 1.335.

Fig. 3
Fig. 3

Radial-overlap integral Kj2, i.e., the square of the integral of jn13jn1r2, integrated from r = 0 to r = a, for TE MDR’s with 125 < x1 < 135 and |x3 − 3x1| < 0.05, plotted as a function of l3/l1, the ratio of the mode orders.

Fig. 4
Fig. 4

Relative polarization (θ component) of a TE mode as a function of zenith angle θ. The mode number n1 is 150 for each case. The azimuthal mode number m1 is 150 (dashed curve) or 140 (solid curve). The polarization is essentially zero for those θ not shown.

Fig. 5
Fig. 5

Example integrands of the in-phase part of the angular-overlap integral, KY, plotted as a function of the zenith angle θ. The mode number of the generating wave in each case is n1 = 150. For the solid curve the azimuthal mode number of the generating wave, m1, is 144, the azimuthal mode number of the output wave, m3, is 432, and the mode number of the output wave, n3, is 454. The long-dashed curve, which has one large peak at θ = 90°, has m1 = 150, m3 = 450, and n3 = 450. The short-dashed curve has m1 = 140, m3 = 420, and n3 = 450.

Fig. 6
Fig. 6

Angular-overlap integral |KY|2 plotted as a function of m3/n3, which is ≈sin(θp), where θp is the angle of the peak intensity of the spherical harmonic. The mode number of the generating wave, n1, is 150 for each of the curves. The mode number of the output wave, n3, varies from 430 to 456 as marked. The curves are obtained by varying m3 = 3m1 for each of the fixed Values of n3. values of the integral are computed only for integer values of m1 = m3/3 but are connected by curves for clarity.

Fig. 7
Fig. 7

Angular-overlap integral |KY|2 plotted as a function of m3/n3 ≅ sin(θp), where θp is the angle of the peak intensity of the spherical harmonic. The mode number of the generating wave is 150, and the azimuthal mode number of the input is m1 = m3/3 = 150. The points are obtained by varying n3, the mode number of the output wave, from 420 to 462.

Fig. 8
Fig. 8

Integrand of the frequency-overlap integral, KL, plotted as a function of (xxT)/Δx1, where x is the size parameter (i.e., a normalized frequency, which is the ratio of the circumference to the wavelength), xT is the center frequency of the nonlinear polarization, and Δx1 is the linewidth of the MDR supporting the wave that generates the nonlinear polarization. Results are shown for three different values of the detuning, D = xTx3, where x3 is the resonance frequency of the output MDR. The detunings D are shown in terms of the number of linewidths of the input MDR, which are 0, 30, and 122 linewidths. The quality factors of the input and the output MDR’s are Q1 = Q3 = 5 × 106.

Fig. 9
Fig. 9

Frequency-overlap integral, KL, plotted as a function of the detuning, x3xT. The upper axis gives the absolute detuning. The lower axis gives the detuning in terms of the number of linewidths, Δx1, of the input SRS MDR. The input MDR has x1 = 131. The output MDR has x3 = 393. The different curves are for different values of the output Q3’s. (a) Q1 = 5 × 106, and Q3 ranges from 5 × 107 to 5 × 103. (b) Q1 = 108, and Q3 ranges from 108 to 104.

Tables (1)

Tables Icon

Table 1 Polarization and Output MDR Frequencies

Equations (55)

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P NLS ( r , ω ) = χ ( 3 ) E a ( r , ω a ) E b ( r , ω b ) E c ( r , ω c ) ,
P i NLS ( r , ω ) = D j k l χ i j k l ( 3 ) E j ( r , ω a ) E k ( r , ω b ) E l ( r , ω c ) ,
χ i j k l = χ 1122 δ i j δ k l + χ 1212 δ i k δ j l + χ 1221 δ i l δ j k .
P θ NLS = χ 1111 ( 3 ) E a θ E b θ E c θ + χ 1122 ( 3 ) E a θ E b θ E c θ + χ 1212 ( 3 ) E a ϕ E b θ E c ϕ × χ 1221 ( 3 ) E a ϕ E b ϕ E c θ ,
χ i j k l ( 3 ) = χ 1122 ( 3 ) ( δ i j δ k j + δ i k δ j l + δ i l δ j k ) .
P θ NLS = 3 χ 1122 ( 3 ) ( E a θ 3 + E a θ E a ϕ 2 ) ,
P ϕ NLS = 3 χ 1122 ( 3 ) ( E a ϕ 3 + E a ϕ E a θ 2 ) .
B ( r , ω ) = B ( r , r , ω ) d 3 r = exp ( i k r ) k r n , m ( - i ) n + 1 [ f n Y n n m ( θ , ϕ ) a E ( n , m ) d 3 r + g n r ^ × Y n n m ( θ , ϕ ) a M ( n , m ) d 3 r ] ,
Y n n m ( θ , ϕ ) = - i / [ n ( n + 1 ) ] 1 / 2 r × Y n m ( θ , ϕ ) ,
Y n m ( θ , ϕ ) = [ ( 2 n + 1 ) ( n - m ) ! 4 π ( n + m ) ] 1 / 2 P n m ( cos θ ) exp ( i m ϕ ) ,
Y n n m ( θ , ϕ ) = - [ ( 2 n + 1 ) ( n - m ) ! 4 π n ( n + 1 ) ( n + m ) ! ] 1 / 2 × ( i ϕ ^ P n m θ + θ ^ m P n m sin θ ) exp ( i m ϕ ) .
f n = i / k t a m ( ω ) 2 j n ( k t a ) [ k a h n ( 1 ) ( k a ) ] - h n ( 1 ) ( k a ) [ k t a j n ( k t a ) ] ,
g n = i / k t a j n ( k t a ) [ k a h n ( 1 ) ( k a ) ] - h n ( 1 ) ( k a ) [ k t a j n ( k t a ) ] ,
a E ( n , m ) d 3 r = 4 π k 2 p ( r , ω ) · { × [ j n ( k t r ) Y n n m * ( θ , ϕ ) ] } d 3 r ,
a M ( n , m ) d 3 r = 4 π i k 3 p ( r , ω ) · j n ( k t r ) Y n n m * ( θ , ϕ ) d 3 r .
j n ( k r ) Y n n m ( θ , ϕ ) + Y n n m * ( θ , ϕ ) 2 = - [ ( 2 n + 1 ) ( n - m ) ! 4 π n ( n + 1 ) ( n + m ) ! ] 1 / 2 M o m n 1 ,
j n ( k r ) Y n n m ( θ , ϕ ) - Y n n m * ( θ , ϕ ) 2 = - i [ ( 2 n + 1 ) ( n - m ) ! 4 π n ( n + 1 ) ( n + m ) ! ] 1 / 2 M e m n 1 ,
P ( ω ) = c 8 π k 2 n , m [ | f n a E ( n , m ) d 3 r | 2 + | g n a M ( n , m ) d 3 r | 2 ] .
g n ( x , x i , Δ x i ) G i Δ x i 1 / 2 ( x - x 1 ) + i ( x i / 2 Q i ) ,
L i ( x , x i , Δ x i ) = Δ x i ( x - x i ) 2 + ( Δ x i / 2 ) 2 .
δ x = x e 6 [ 3 m 2 n ( n + 1 ) - 1 ] ,
E s ( r , ω s ) = A M s g n s , m s ( ω s ) j n s ( k s t r ) × [ Y n s n s m s ( θ , ϕ ) + Y n s n s m s * ( θ , ϕ ) ] / 2
E s ( r , ω s ) = A E s f n s , m s ( ω s ) × j n s ( k s t r ) × [ Y n s n s m s ( θ , ϕ ) + Y n s n s m s * ( θ , ϕ ) ] / 2
P n s m s SRS ( ω s ) = c A M s 2 g n s , m s ( ω s ) 2 16 π k s 2 ,
P n s , m s T = 0 P n s , m s ( x s ) d x s = c A M s 2 G n s 4 8 k s 2 .
P n 3 , m 3 ( ω ) = c 8 π k 2 | g n 3 , m 3 ( ω ) a M ( n 3 , m 3 ) d 3 r | 2 = 2 π c k 4 g n 3 , m 3 ( ω ) 2 × | { χ ( 3 ) [ A M a g n a , m a ( ω a ) j n a ( k a t r ) Y n a n a m a + Y n a n a m a * 2 ] × [ A M b g n b , m b ( ω b ) j n b ( k b t r ) Y n b n b m b + Y n b n b m b * 2 ] × [ A M c g n c , m c ( ω c ) j n c ( k c t r ) Y n c n c m c + Y n c n c m c 2 ] · j n 3 ( k t r ) Y n 3 n 3 m 3 * d 3 r | 2 .
P n 3 , m 3 T = 0 P n 3 , m 3 ( ω ) d ω .
P n 3 , m 3 T = 2 π c k 4 A M a 2 A M b 2 A M c 2 × | 0 a j n a ( k a t r ) j n b ( k b t r ) j n c ( k c t r ) j n 3 ( k t r ) r 2 d r | 2 × | [ χ ( 3 ) ( Y n a n a m a + Y n a n a m a * 2 ) ( Y n b n b m b + Y n b n b m b * 2 ) × ( Y n c n c m c + Y n c n c m c * 2 ) ] · Y n 3 n 3 m 3 * d Ω | 2 × 0 | ω b = 0 ω ω a = ω - ω b ω g n a , m a ( ω a ) g n b , m b ( ω b ) × g n c , m c ( ω c = ω - ω a - ω b ) d ω a d ω b | 2 g n 3 , m 3 ( ω ) 2 d ω .
P n 3 , m 3 T = 2 π c k 4 A M a 2 A M b 2 A M c 2 K j 2 χ ( 3 ) K Y 2 K L ,
K j 2 = | 0 a j n a ( k a t r ) j n b ( k b t r ) j n c ( k c t r ) j n 3 ( k t r ) r 2 d r | 2 ,
χ ( 3 ) K Y 2 = | [ χ ( 3 ) ( Y n a n a m a + Y n a n a m a * 2 ) × ( Y n b n b m b + Y n b n b m b * 2 ) ( Y n c n c m c + Y n c n c m c * 2 ) ] · Y n 3 n 3 m 3 * d Ω | 2 .
0 | ω b = 0 ω ω a = 0 ω - ω b g n a , m a ( ω a ) g n b , m b ( ω b ) × g n c , m c ( ω c = ω - ω a - ω b ) d ω a d ω b | 2 g n 3 , m 3 ( ω ) 2 d ω .
R = 16 π k 4 A M a 2 A M b 2 A M c 2 K j 2 χ ( 3 ) K Y 2 K L k a 2 G n a 4 A M a 2 + k b 2 G n b 4 A M b 2 + k c 2 G n c 4 A M c 2 .
0 2 π i sin ( m a ϕ ) sin ( m b ϕ ) sin ( m c ϕ ) exp ( - j m 3 ϕ ) d ϕ = π / 4             for             m 3 = ± ( m a + m b + m c ) ,
0 2 π cos ( m a ϕ ) cos ( m b ϕ ) cos ( m c ϕ ) exp ( - j m 3 ϕ ) d ϕ = π / 4             for             m 3 = ± ( m a + m b + m c ) ,
0 2 π ( cos m a ) sin ( m b ϕ ) sin ( m c ϕ ) exp ( - j m 3 ϕ ) d ϕ = - π / 4             for             m 3 = ± ( m a + m b + m c ) ,
0 2 π i cos ( m a ϕ ) cos ( m b ϕ ) sin ( m c ϕ ) exp ( - j m 3 ϕ ) d ϕ = ± π / 4             for             m 3 = ± ( m a + m b + m c ) .
V a V b V c V 3 χ 1111 ( 3 ) m a m b m c m 3 cos ( m a ϕ ) cos ( m b ϕ ) cos ( m c ϕ ) × P n a m a sin θ P n b m b sin θ P n c m c sin θ P n 3 m 3 sin θ exp ( - j m 3 ϕ ) d Ω ,
V i = [ ( 2 n i + 1 ) ( n i - m i ) ! 4 π n i ( n i + 1 ) ( n i + m i ) ! ] 1 / 2 .
χ ( 3 ) K Y = V a V b V c V 3 π 4 χ 1111 ( 3 ) m a m b m c m 3 0 π { P n b m a sin θ P n b m b sin θ P n 3 m 3 sin θ } × ( sin θ ) d θ .
P θ NLS = 3 χ 1122 ( 3 ) [ m a 3 cos 3 ( m a ϕ ) ( P n a m a sin θ ) 3 + m a cos ( m a ϕ ) sin 2 ( m a ϕ ) P n a m a sin θ ( P n a m a θ ) 2 ] ,
P ϕ NLS = 3 χ 1122 ( 3 ) [ sin 3 ( m a ϕ ) ( P n a m a θ ) 3 + m a 2 cos 2 ( m a ϕ ) sin ( m a ϕ ) P n a m a θ ( P n a m a sin θ ) 2 ] .
K Y = V a 3 V 3 ( π / 4 ) ( T 1 + i T 2 ) ,
K Y 2 = [ V a 3 V 3 ( π / 4 ) ] 2 ( T 1 2 + T 2 2 ) ,
T 1 = 0 π [ m a 3 m 3 ( P n a m a sin θ ) 3 P n 3 m 3 sin θ + m a 2 P n a m a sin θ ( P n a m a θ ) 2 P n 3 m 3 sin θ ] ( sin θ ) d θ ,
T 2 = 0 π [ m a 2 P n a m a θ ( P n a m a sin θ ) 2 P n 3 m 3 θ + ( P n a m a θ ) 3 P n 3 m 3 θ ] ( sin θ ) d θ .
ω b = 0 ω a = 0 g n a , m a ( ω a ) g n b , m b ( ω b ) g n c , m c ( ω c = ω - ω a - ω b ) × d ω a d ω b
x b = 0 x a = 0 0 G a G b G c Δ x a Δ x b Δ x c d x a d x b [ x a - x 1 a + i ( Δ x a / 2 ) ] [ x b - x 1 b + i ( Δ x b / 2 ) ] [ x - x a - x b - x 1 c + i ( Δ x c / 2 ) ] .
G a G b G c ( Δ x a Δ x b Δ x c ) 1 / 2 x - x 1 a - x 1 b - x 1 c + i ( Δ x a + Δ x b + Δ x c ) / 2 ,
K L = G a 2 G b 2 G c 2 G 3 2 - Δ x a Δ x b Δ x c ( x - x T ) 2 + ( Δ x T / 2 ) 2 × Δ 3 ( x - x 3 ) 2 + ( Δ x 3 / 2 ) 2 d x .
K L = G a 2 G b 2 G c 2 G 3 2 × 8 π Δ x a Δ x b Δ x c ( Δ x T + Δ x 3 ) 4 Δ x T D 2 + Δ x T 3 + 2 Δ x 3 Δ x T 2 + Δ x 3 2 Δ x T ,
F = 2 d E N Δ x 3 = 2 N d E x 3 / Q 3 .
N 81 x 1 / Q 1 Δ x 3 27.
F = 2 d E N Δ x 3 = 2 N d E x 3 / Q 3 = 0.085 ,
0.966 n 1 = 145 m 1 n 1 = 150.

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