Abstract

I review the theory of the scattering of a Gaussian laser beam by a dielectric spherical particle and give the details for constructing a computer program to implement the theory. Computational results indicate that if the width of the laser beam is much less than the diameter of the particle and if the axis of the beam is incident near the edge of the particle, the fifth-, sixth-, and ninth-order rainbows should be evident in the far-field scattered intensity. I performed an experiment that yielded tentative evidence for the presence of the sixth-order rainbow.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. D. Bachalo, “Method for measuring the size and velocity of spheres by dual-beam light-scatter interferometry,” Appl. Opt. 19, 363–370 (1980).
    [CrossRef] [PubMed]
  2. W. D. Bachalo, M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).
    [CrossRef]
  3. S. V. Sankar, W. D. Bachalo, “Response characteristics of the phase Doppler particle analyzer for sizing spherical particles larger than the light wavelength,” Appl. Opt. 30, 1487–1496 (1991).
    [CrossRef] [PubMed]
  4. S. V. Sankar, B. J. Weber, D. Y. Kamemoto, W. D. Bachalo, “Sizing fine particles with the phase Doppler interferometric technique,” Appl. Opt. 30, 4914–4920 (1991).
    [CrossRef] [PubMed]
  5. G. Grehan, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of the phase-Doppler system using generalized Lorenz–Mie theory,” presented at the International Conference on Multiphase Flows, University of Tsukuba, Japan, 1991.
  6. S. A. Schaub, D. R. Alexander, “Theoretical analysis of the effects of particle trajectory on the performance of a phase/Doppler particle analyzer,” Appl. Opt. (to be published).
  7. G. Grehan, G. Gouesbet, “Optical levitation of a single particle to study the theory of the quasi-elastic scattering of light,” Appl. Opt. 19, 2485–2487 (1980).
    [CrossRef] [PubMed]
  8. G. Gouesbet, G. Grehan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
    [CrossRef]
  9. G. Gouesbet, B. Maheu, G. Grehan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formalism,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
    [CrossRef]
  10. J. P. Barton, D. R. Alexander, 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]
  11. J. P. Barton, D. R. Alexander, 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]
  12. J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
    [CrossRef]
  13. G. Grehan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
    [CrossRef] [PubMed]
  14. B. Maheu, G. Grehan, G. Gouesbet, “Generalized Lorenz– Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23–25 (1987).
    [CrossRef] [PubMed]
  15. G. Gouesbet, G. Grehan, B. Maheu, “Computations of the gncoefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
    [CrossRef] [PubMed]
  16. G. Gouesbet, G. Grehan, B. Maheu, “Localized interpretation to compute all the coefficients gnmin the generalized Lorenz-Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
    [CrossRef]
  17. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sec. 12.31.
  18. G. Gouesbet, G. Grehan, B. Maheu, “Generalized Lorenz– Mie theory and applications to optical sizing,” in Proceedings of the Second International Congress on Optical Particle Sizing, E. D. Hirleman, ed. (Arizona State U. Press, Tempe, Ariz., 1990) pp. 118–126.
  19. J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
    [CrossRef]
  20. J. D. Walker, “How to create and observe a dozen rainbows in a single drop of water,” Sci. Am. 237 (1), 138–144 (1977).
    [CrossRef]
  21. J. D. Walker, “Mysteries of rainbows, notably their rare supernumerary arcs,” Sci. Am. 242 (6), 174–184 (1980).
    [CrossRef]
  22. T. S. Fahlen, H. C. Bryant, “Optical back scattering from single water droplets,” J. Opt. Soc. Am. 58, 304–310 (1968).
    [CrossRef]
  23. Ref. 17, Sec. 13.11.
  24. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Sec. 7.1.2.
  25. S. I. Rubinow, “Scattering from a penetrable sphere at short wavelengths,” Ann. Phys. (NY.) 14, 305–332 (1961).
    [CrossRef]
  26. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direction reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
    [CrossRef]
  27. J. A. Lock, “Cooperative effects among partial waves in Mie scattering,” J. Opt. Soc. Am. A 5, 2032–2044 (1988).
    [CrossRef]
  28. J. A. Lock, E. A. Hovenac, “Internal caustic structure of illuminated liquid droplets,” J. Opt. Soc. Am. A 8, 1541–1552 (1991).
    [CrossRef]
  29. E. A. Hovenac, J. A. Lock, “Assessing the contributions of surface waves and complex rays to far-field Mie scattering by use of the Debye series,” J. Opt. Soc. Am. A 9, 781–795 (1992).
    [CrossRef]
  30. J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 17.5.
  31. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), Eq. (11.175).
  32. Ref. 31, Sec. 12.5.
  33. Ref. 9, Eq. (66) and Ref. 15, Eq. (19). See also Ref. 31, Eq. (11.174).
  34. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  35. J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
    [CrossRef]
  36. S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficient expression for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989).
    [CrossRef]
  37. J. V. Dave, “Scattering of visible light by large water spheres,” Appl. Opt. 8, 155–164 (1969).
    [CrossRef] [PubMed]
  38. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef] [PubMed]
  39. Ref. 24, Sec. 4.8 and App. A.
  40. B. Maheu, G. Grehan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
    [CrossRef]
  41. V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
    [CrossRef]
  42. H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079, 1193, 1194 (1979).
    [CrossRef]
  43. D. S. Langley, M. J. Morrell, “Rainbow-enhanced forward and backward glory scattering,” Appl. Opt. 30, 3459–3467 (1991).
    [CrossRef] [PubMed]
  44. J. A. Lock, “Theory of the observations made of high-order rainbows from a single water droplet,” Appl. Opt. 26, 5291–5298 (1987).
    [CrossRef] [PubMed]
  45. H. C. van de Hulst, R. T. Wang, “Glare points,” Appl. Opt. 30, 4755–4763 (1991).
    [CrossRef] [PubMed]
  46. J. A. Lock, J. R. Woodruff, “Non-Debye enhancements in the Mie scattering of light from a single water droplet,” Appl. Opt. 28, 523–529 (1989).
    [CrossRef] [PubMed]
  47. G. Grehan, B. Maheu, G. Gouesbet, “Diffusion de la lumiere par une sphere dans le cas d’un faisceau d’estension finie—2. Theorie de Lorenz–Mie generalisee: application a la granulometrie optique,” J. Aerosol Sci. 19, 55–64 (1988).
    [CrossRef]
  48. Ref. 31, p. 665.

1992

1991

1990

1989

J. A. Lock, J. R. Woodruff, “Non-Debye enhancements in the Mie scattering of light from a single water droplet,” Appl. Opt. 28, 523–529 (1989).
[CrossRef] [PubMed]

J. P. Barton, D. R. Alexander, 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, S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficient expression for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989).
[CrossRef]

B. Maheu, G. Grehan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
[CrossRef]

1988

J. P. Barton, D. R. Alexander, 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]

G. Gouesbet, B. Maheu, G. Grehan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formalism,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

G. Grehan, B. Maheu, G. Gouesbet, “Diffusion de la lumiere par une sphere dans le cas d’un faisceau d’estension finie—2. Theorie de Lorenz–Mie generalisee: application a la granulometrie optique,” J. Aerosol Sci. 19, 55–64 (1988).
[CrossRef]

J. A. Lock, “Cooperative effects among partial waves in Mie scattering,” J. Opt. Soc. Am. A 5, 2032–2044 (1988).
[CrossRef]

G. Gouesbet, G. Grehan, B. Maheu, “Computations of the gncoefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

1987

1986

1985

G. Gouesbet, G. Grehan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

1984

W. D. Bachalo, M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).
[CrossRef]

1980

1979

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079, 1193, 1194 (1979).
[CrossRef]

1977

J. D. Walker, “How to create and observe a dozen rainbows in a single drop of water,” Sci. Am. 237 (1), 138–144 (1977).
[CrossRef]

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

1976

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

1969

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direction reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

J. V. Dave, “Scattering of visible light by large water spheres,” Appl. Opt. 8, 155–164 (1969).
[CrossRef] [PubMed]

1968

1961

S. I. Rubinow, “Scattering from a penetrable sphere at short wavelengths,” Ann. Phys. (NY.) 14, 305–332 (1961).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, 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, S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficient expression for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, 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]

S. A. Schaub, D. R. Alexander, “Theoretical analysis of the effects of particle trajectory on the performance of a phase/Doppler particle analyzer,” Appl. Opt. (to be published).

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), Eq. (11.175).

Bachalo, W. D.

Barton, J. P.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficient expression for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, 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, 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]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Sec. 7.1.2.

Bryant, H. C.

Christy, R. W.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 17.5.

Dave, J. V.

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Durst, F.

G. Grehan, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of the phase-Doppler system using generalized Lorenz–Mie theory,” presented at the International Conference on Multiphase Flows, University of Tsukuba, Japan, 1991.

Fahlen, T. S.

Gouesbet, G.

G. Gouesbet, G. Grehan, B. Maheu, “Localized interpretation to compute all the coefficients gnmin the generalized Lorenz-Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
[CrossRef]

B. Maheu, G. Grehan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
[CrossRef]

G. Gouesbet, G. Grehan, B. Maheu, “Computations of the gncoefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

G. Grehan, B. Maheu, G. Gouesbet, “Diffusion de la lumiere par une sphere dans le cas d’un faisceau d’estension finie—2. Theorie de Lorenz–Mie generalisee: application a la granulometrie optique,” J. Aerosol Sci. 19, 55–64 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, G. Grehan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formalism,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

B. Maheu, G. Grehan, G. Gouesbet, “Generalized Lorenz– Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23–25 (1987).
[CrossRef] [PubMed]

G. Grehan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

G. Gouesbet, G. Grehan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

G. Grehan, G. Gouesbet, “Optical levitation of a single particle to study the theory of the quasi-elastic scattering of light,” Appl. Opt. 19, 2485–2487 (1980).
[CrossRef] [PubMed]

G. Gouesbet, G. Grehan, B. Maheu, “Generalized Lorenz– Mie theory and applications to optical sizing,” in Proceedings of the Second International Congress on Optical Particle Sizing, E. D. Hirleman, ed. (Arizona State U. Press, Tempe, Ariz., 1990) pp. 118–126.

G. Grehan, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of the phase-Doppler system using generalized Lorenz–Mie theory,” presented at the International Conference on Multiphase Flows, University of Tsukuba, Japan, 1991.

Grehan, G.

G. Gouesbet, G. Grehan, B. Maheu, “Localized interpretation to compute all the coefficients gnmin the generalized Lorenz-Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
[CrossRef]

B. Maheu, G. Grehan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
[CrossRef]

G. Gouesbet, G. Grehan, B. Maheu, “Computations of the gncoefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

G. Gouesbet, B. Maheu, G. Grehan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formalism,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

G. Grehan, B. Maheu, G. Gouesbet, “Diffusion de la lumiere par une sphere dans le cas d’un faisceau d’estension finie—2. Theorie de Lorenz–Mie generalisee: application a la granulometrie optique,” J. Aerosol Sci. 19, 55–64 (1988).
[CrossRef]

B. Maheu, G. Grehan, G. Gouesbet, “Generalized Lorenz– Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23–25 (1987).
[CrossRef] [PubMed]

G. Grehan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

G. Gouesbet, G. Grehan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

G. Grehan, G. Gouesbet, “Optical levitation of a single particle to study the theory of the quasi-elastic scattering of light,” Appl. Opt. 19, 2485–2487 (1980).
[CrossRef] [PubMed]

G. Gouesbet, G. Grehan, B. Maheu, “Generalized Lorenz– Mie theory and applications to optical sizing,” in Proceedings of the Second International Congress on Optical Particle Sizing, E. D. Hirleman, ed. (Arizona State U. Press, Tempe, Ariz., 1990) pp. 118–126.

G. Grehan, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of the phase-Doppler system using generalized Lorenz–Mie theory,” presented at the International Conference on Multiphase Flows, University of Tsukuba, Japan, 1991.

Houser, M. J.

W. D. Bachalo, M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).
[CrossRef]

Hovenac, E. A.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Sec. 7.1.2.

Kamemoto, D. Y.

Khare, V.

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

Langley, D. S.

Lock, J. A.

Maheu, B.

G. Gouesbet, G. Grehan, B. Maheu, “Localized interpretation to compute all the coefficients gnmin the generalized Lorenz-Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
[CrossRef]

B. Maheu, G. Grehan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
[CrossRef]

G. Gouesbet, G. Grehan, B. Maheu, “Computations of the gncoefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

G. Gouesbet, B. Maheu, G. Grehan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formalism,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

G. Grehan, B. Maheu, G. Gouesbet, “Diffusion de la lumiere par une sphere dans le cas d’un faisceau d’estension finie—2. Theorie de Lorenz–Mie generalisee: application a la granulometrie optique,” J. Aerosol Sci. 19, 55–64 (1988).
[CrossRef]

B. Maheu, G. Grehan, G. Gouesbet, “Generalized Lorenz– Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23–25 (1987).
[CrossRef] [PubMed]

G. Grehan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

G. Gouesbet, G. Grehan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

G. Gouesbet, G. Grehan, B. Maheu, “Generalized Lorenz– Mie theory and applications to optical sizing,” in Proceedings of the Second International Congress on Optical Particle Sizing, E. D. Hirleman, ed. (Arizona State U. Press, Tempe, Ariz., 1990) pp. 118–126.

Milford, F. J.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 17.5.

Morrell, M. J.

Naqwi, A.

G. Grehan, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of the phase-Doppler system using generalized Lorenz–Mie theory,” presented at the International Conference on Multiphase Flows, University of Tsukuba, Japan, 1991.

Nussenzveig, H. M.

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079, 1193, 1194 (1979).
[CrossRef]

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direction reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

Reitz, J. R.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 17.5.

Rubinow, S. I.

S. I. Rubinow, “Scattering from a penetrable sphere at short wavelengths,” Ann. Phys. (NY.) 14, 305–332 (1961).
[CrossRef]

Sankar, S. V.

Schaub, S. A.

J. P. Barton, D. R. Alexander, 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]

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficient expression for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, 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]

S. A. Schaub, D. R. Alexander, “Theoretical analysis of the effects of particle trajectory on the performance of a phase/Doppler particle analyzer,” Appl. Opt. (to be published).

van de Hulst, H. C.

H. C. van de Hulst, R. T. Wang, “Glare points,” Appl. Opt. 30, 4755–4763 (1991).
[CrossRef] [PubMed]

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sec. 12.31.

Walker, J. D.

J. D. Walker, “Mysteries of rainbows, notably their rare supernumerary arcs,” Sci. Am. 242 (6), 174–184 (1980).
[CrossRef]

J. D. Walker, “How to create and observe a dozen rainbows in a single drop of water,” Sci. Am. 237 (1), 138–144 (1977).
[CrossRef]

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

Wang, R. T.

Weber, B. J.

Wiscombe, W. J.

Woodruff, J. R.

Am. J. Phys.

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

Ann. Phys. (NY.)

S. I. Rubinow, “Scattering from a penetrable sphere at short wavelengths,” Ann. Phys. (NY.) 14, 305–332 (1961).
[CrossRef]

Appl. Opt.

J. V. Dave, “Scattering of visible light by large water spheres,” Appl. Opt. 8, 155–164 (1969).
[CrossRef] [PubMed]

W. D. Bachalo, “Method for measuring the size and velocity of spheres by dual-beam light-scatter interferometry,” Appl. Opt. 19, 363–370 (1980).
[CrossRef] [PubMed]

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[CrossRef] [PubMed]

J. A. Lock, “Theory of the observations made of high-order rainbows from a single water droplet,” Appl. Opt. 26, 5291–5298 (1987).
[CrossRef] [PubMed]

J. A. Lock, J. R. Woodruff, “Non-Debye enhancements in the Mie scattering of light from a single water droplet,” Appl. Opt. 28, 523–529 (1989).
[CrossRef] [PubMed]

S. V. Sankar, W. D. Bachalo, “Response characteristics of the phase Doppler particle analyzer for sizing spherical particles larger than the light wavelength,” Appl. Opt. 30, 1487–1496 (1991).
[CrossRef] [PubMed]

D. S. Langley, M. J. Morrell, “Rainbow-enhanced forward and backward glory scattering,” Appl. Opt. 30, 3459–3467 (1991).
[CrossRef] [PubMed]

H. C. van de Hulst, R. T. Wang, “Glare points,” Appl. Opt. 30, 4755–4763 (1991).
[CrossRef] [PubMed]

S. V. Sankar, B. J. Weber, D. Y. Kamemoto, W. D. Bachalo, “Sizing fine particles with the phase Doppler interferometric technique,” Appl. Opt. 30, 4914–4920 (1991).
[CrossRef] [PubMed]

G. Grehan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

G. Gouesbet, G. Grehan, B. Maheu, “Computations of the gncoefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

G. Grehan, G. Gouesbet, “Optical levitation of a single particle to study the theory of the quasi-elastic scattering of light,” Appl. Opt. 19, 2485–2487 (1980).
[CrossRef] [PubMed]

B. Maheu, G. Grehan, G. Gouesbet, “Generalized Lorenz– Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23–25 (1987).
[CrossRef] [PubMed]

Appl. Phys. Lett.

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficient expression for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989).
[CrossRef]

J. Aerosol Sci.

G. Grehan, B. Maheu, G. Gouesbet, “Diffusion de la lumiere par une sphere dans le cas d’un faisceau d’estension finie—2. Theorie de Lorenz–Mie generalisee: application a la granulometrie optique,” J. Aerosol Sci. 19, 55–64 (1988).
[CrossRef]

J. Appl. Phys.

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, 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. P. Barton, D. R. Alexander, 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, S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

J. Math. Phys.

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direction reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

J. Opt. (Paris)

G. Gouesbet, G. Grehan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

J. Opt. Soc. Am.

T. S. Fahlen, H. C. Bryant, “Optical back scattering from single water droplets,” J. Opt. Soc. Am. 58, 304–310 (1968).
[CrossRef]

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079, 1193, 1194 (1979).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

B. Maheu, G. Grehan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
[CrossRef]

Opt. Eng.

W. D. Bachalo, M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).
[CrossRef]

Phys. Rev. A

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Phys. Rev. Lett.

V. Khare, H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

Sci. Am.

J. D. Walker, “How to create and observe a dozen rainbows in a single drop of water,” Sci. Am. 237 (1), 138–144 (1977).
[CrossRef]

J. D. Walker, “Mysteries of rainbows, notably their rare supernumerary arcs,” Sci. Am. 242 (6), 174–184 (1980).
[CrossRef]

Other

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 17.5.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), Eq. (11.175).

Ref. 31, Sec. 12.5.

Ref. 9, Eq. (66) and Ref. 15, Eq. (19). See also Ref. 31, Eq. (11.174).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Sec. 12.31.

G. Gouesbet, G. Grehan, B. Maheu, “Generalized Lorenz– Mie theory and applications to optical sizing,” in Proceedings of the Second International Congress on Optical Particle Sizing, E. D. Hirleman, ed. (Arizona State U. Press, Tempe, Ariz., 1990) pp. 118–126.

G. Grehan, G. Gouesbet, A. Naqwi, F. Durst, “Evaluation of the phase-Doppler system using generalized Lorenz–Mie theory,” presented at the International Conference on Multiphase Flows, University of Tsukuba, Japan, 1991.

S. A. Schaub, D. R. Alexander, “Theoretical analysis of the effects of particle trajectory on the performance of a phase/Doppler particle analyzer,” Appl. Opt. (to be published).

Ref. 24, Sec. 4.8 and App. A.

Ref. 17, Sec. 13.11.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Sec. 7.1.2.

Ref. 31, p. 665.

Cited By

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

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

(a) Intensity of a Gaussian beam as a function of position in the yz plane. The beam is propagating in the positive z direction, and the center of the focal waist is at y = z = 0. The dashed curves represent the nominal width of the beam as a function of z. (b) The planes of constant phase of the Gaussian beam as a function of position in the yz plane. The curvature of the planes of constant phase for z < 0 and z > 0 is due to the convergence and divergence of the beam on each side of the focal plane. The dashed curves represent the nominal width of the beam as a function of z.

Fig. 2
Fig. 2

Scattering geometry. A Gaussian laser beam polarized in the x direction propagating parallel to the z axis and displaced from it by the distance y0 in the yz plane is incident upon a spherical particle. The scattered light is measured at the scattering angle θ to the right of the z direction in the yz plane (ϕ = π/2) by the detector D.

Fig. 3
Fig. 3

Far-field scattered intensity I1(d) = |S1(θ)|2 as a function of scattering angle for a plane wave with λ = 0.5145 μm and polarized in the x direction incident upon a spherical particle with a = 43.3 μm and n = 1.33.

Fig. 4
Fig. 4

Far-field scattered intensity I1(θ,π/2) = |S1(θ,π/2)|2 as a function of scattering angle for a Davis first-order Gaussian laser beam with λ = 0.5145 μm and w = 20 μm and polarized in the x direction incident upon a spherical particle with a = 43.3 μm and n = 1.33. The center of the focal waist of the beam is located at (a) x0 = z0 = 0, y0 = −40 μm; (b) x0 = z0 = 0, y0 = 0 μm; and (c) x0 = z0 = 0, y0 = 40 μm. The center of the particle is at the origin of coordinates.

Fig. 5
Fig. 5

Dominant ray trajectories for plane-wave incidence are specular reflection (p = 0), transmission (p = 1), transmission after one internal reflection (p = 2), and transmission after two internal reflections (p = 3).

Fig. 6
Fig. 6

Dominant ray trajectories for Gaussian beam incidence for wa are (a) p = 1,2 for y0 < 0; (b) p = 0,1 for y0 ≈ 0; and (c)p = 0,3 for y0 > 0.

Fig. 7
Fig. 7

Debye-series contributions to I1(θ) = |S1(θ)|2 for plane-wave incidence with the polarization in the x direction, λ = 0.5145 μm, a = 43.3 μm, and n = 1.33. (a) The sum of the contributions that are due to diffraction plus p = 0,1,2,3 almost exactly fit the full Mie intensity of Fig. 3. (b) The contributions that are due to the 4 ≤ p ≤ 12 rainbows are weak compared with the contributions of p ≤ 3. The dashed curve represents the sum of 0 ≤ p ≤ 3 from Fig. 7(a).

Fig. 8
Fig. 8

Debye-series contributions to I1(θ,ir/2) = |S1(θ,π/2)|2 for a Davis first-order Gaussian beam incident with the polarization in the x direction, λ = 0.5145 μm, w = 20 μm, a = 43.3 /on, and n = 1.33. (a) For y0 = −40.0 μm, the sum of the contributions that are due to diffraction plus p = 0,1,2,6,10 almost exactly fit the full wave-theory intensity of Fig. 4(a). (b) For y0 = 0 μm, the sum of the contributions that are due to diffraction plus p = 0,1,2 almost exactly fit the full wave-theory intensity of Fig. 4(b). (c) For y0 = 40.0 μm, the sum of the contributions that are due to diffraction plus p = 0,3,7,11 almost exactly fit the full wave-theory intensity of Fig. 4(c).

Fig. 9
Fig. 9

(a) Experimental intensity spectrum for 30° ≤ 0 ≤ 150° (left to right) for a plane wave that is polarized in the x direction and with λ = 0.5145 μm incident upon a spherical water droplet with a ≈ 23.9 μm. The reflection–transmission interference is on the left-hand side of the photograph, and the second- and first-order rainbows along with their first few supernumeraries are on the right-hand side, (b) I1(θ) = |S1(θ)|2 for plane -wave incidence with the same parameters as in the experiment whose results are shown in (a).

Fig. 10
Fig. 10

(a) Experimental intensity spectra for 30° ≤ θ ≤ 150° (left to right) for a focused Gaussian laser beam polarized in the x direction and with λ = 0.5145 μm and w ≈ 20 μm incident upon a spherical water droplet with a = 23.9 μm and y0 ≈ -22 μm. The transmitted light is on the left-hand side of the photograph, and the first-order rainbow is on the right-hand side. (b) The experimental intensity for a ≈ 43.3 μm and y0 ≈ 40 μm. The specularly reflected light is on the left-hand side of the photograph, and the second-order rainbow is on the right-hand side. The single wide dim fringe to the right of the main second-order rainbow fringe is tentatively identified as the main fringe of the sixth-order rainbow. Compare this photograph with Fig. 4(c). (c) The experimental intensity for a ≈ 23.9 μm and y0 ≈ 7 μm. The scattered light is much dimmer, and both the first- and second-order rainbows are absent, (d) I1(θ,π/2) = |S1(θ,π/2)|2 with the same parameters as in the experiment whose result is shown in (a), (e) I1(θ,π/2) = |S1(θ,π/2)|2 with the same parameters as in the experiment whose result is shown in (c).

Equations (77)

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

k = 2 π λ = ω c
Ψ inc ( r , t ) = ψ inc ( r ) exp ( i ω t ) ,
2 ψ inc + k 2 ψ inc = 0 .
ψ inc TE = l m i l 2 l + 1 2 l ( l + 1 ) B l m j l ( k r ) P l | m | [ cos ( θ ) ] exp ( i m φ ) , ψ inc TM = l m i l 2 l + 1 2 l ( l + 1 ) A l m j l ( k r ) P l | m | [ cos ( θ ) ] exp ( i m φ ) ,
A l m = ( i ) l 1 2 π k r j l ( k r ) ( l | m | ) ! ( l + | m | ) ! 0 π sin ( θ ) d θ 0 2 π d ϕ P l | m | [ cos ( θ ) ] × exp ( i m φ ) E inc rad ( r , θ , ϕ ) , B l m = ( i ) l 1 2 π k r j l ( k r ) ( l | m | ) ! ( l + | m | ) ! 0 π sin ( θ ) d θ 0 2 π d ϕ P l | m | [ cos ( θ ) ] × exp ( i m φ ) c B inc rad ( r , θ , ϕ ) ,
A l m = ( i ) l 1 π 2 ( 2 l + 1 ) ( l | m | ) ! ( l + | m | ) ! 0 k r d ( k r ) 0 π sin ( θ ) d θ × 0 2 π d ϕ j l ( k r ) P l | m | [ cos ( θ ) ] exp ( i m ϕ ) E inc rad ( r , θ , ϕ ) , B l m = ( i ) l 1 π 2 ( 2 l + 1 ) ( l | m | ) ! ( l + | m | ) ! 0 k r d ( k r ) 0 π sin ( θ ) d θ × 0 2 π d ϕ j l ( k r ) P l | m | [ cos ( θ ) ] exp ( i m ϕ ) c B inc rad ( r , θ , ϕ ) .
0 d ( k r ) j l ( k r ) j l ( k r ) = π 2 ( 2 l + 1 ) δ l l .
ψ scatterd TE = l = 0 m = 1 l i l 2 l + 1 l ( l + 1 ) ( β l m ) h l ( 1 ) ( k r ) P l | m | [ cos ( θ ) ] × exp ( i m φ ) , ψ scatterd TM = l = 0 m = 1 l i l 2 l + 1 l ( l + 1 ) ( α l m ) h l ( 1 ) ( k r ) P l | m | [ cos ( θ ) ] × exp ( i m φ )
ψ inside TE = l = 0 m = 1 l i l 2 l + 1 l ( l + 1 ) ( n δ l m ) j l ( nkr ) P l | m | [ cos ( θ ) ] × exp ( i m φ ) , ψ inside TM = l = 0 m = 1 l i l 2 l + 1 l ( l + 1 ) ( n γ l m ) j l ( nkr ) P l | m | [ cos ( θ ) ] × exp ( i m φ )
lim r E scattered ( r , θ , ϕ ) = i exp ( ikr ) k r [ S 2 ( θ , ϕ ) û θ S 1 ( θ , ϕ ) û ϕ ] , lim r B scattered ( r , θ , ϕ ) = i exp ( ikr ) ckr [ S 1 ( θ , ϕ ) û θ + S 2 ( θ , ϕ ) û ϕ ] ,
lim r I scattered ( r , θ , ϕ ) = 1 2 μ 0 c 1 k 2 r 2 [ | S 1 ( θ , ϕ ) | 2 + | S 2 ( θ , ϕ ) | 2 ] ,
S 1 ( θ , ϕ ) = l = 0 m = l l 2 l + 1 2 l ( l + 1 ) [ i m α l m π l | m | ( θ ) + β l m τ l | m | ( θ ) ] × exp ( i m φ ) , S 2 ( θ , ϕ ) = l = 0 m = l l 2 l + 1 2 l ( l + 1 ) [ i m β l m π l | m | ( θ ) + α l m τ l | m | ( θ ) ] × exp ( i m φ )
π l | m | ( θ ) = 1 sin θ P l | m | [ cos ( θ ) ] , τ l | m | ( θ ) = d d θ P l | m | [ cos ( θ ) ] ,
α l m = A l m a l , β l m = B l m b l ,
E inc rad ( r , θ , ϕ ) = exp [ ikr cos ( θ ) ] sin ( θ ) cos ( ϕ ) , c B inc rad ( r , θ , ϕ ) = exp [ ikr cos ( θ ) ] sin ( θ ) sin ( ϕ ) .
A l m = { 1 if m = ± 1 0 if otherwise ,
B l m = { i if m = ± 1 0 if otherwise ,
S 1 ( θ , ϕ ) = l = 1 2 l + 1 l ( l + 1 ) [ a l π l 1 ( θ ) + b l τ l 1 ( θ ) ] sin ( ϕ ) , S 2 ( θ , ϕ ) = l = 1 2 l + 1 l ( l + 1 ) [ a l τ l 1 ( θ ) + b l π l 1 ( θ ) ] cos ( ϕ ) .
E inc = exp ( ikz ) 1 + ( 2 izs ) / w exp [ ( x 2 + y 2 ) / w 2 1 + ( 2 izs ) / w ] × [ û x 2 ixs w ( 1 + 2 izs w ) 1 û z ] ,
E inc rad = E inc x sin ( θ ) cos ( ϕ ) + E inc z cos ( θ ) ,
c B inc = exp ( ikz ) 1 + ( 2 izs ) / w exp [ ( x 2 + y 2 ) / w 2 1 + ( 2 izs ) / w ] × [ û y 2 iys w ( 1 + 2 izs w ) 1 û z ] ,
c B inc rad = c B inc y sin ( θ ) sin ( ϕ ) + c B inc z cos ( θ ) .
exp [ ( x 2 + y 2 ) / w 2 1 + ( 2 izs ) / w ]
x x x 0 , y y y 0 , z z z 0
E inc rad = exp [ ikr cos ( θ ) ] f ( k r , θ ) sin ( θ ) cos ( ϕ ) , c B inc rad = exp [ ikr cos ( θ ) ] f ( k r , θ ) sin ( θ ) sin ( ϕ ) .
S 1 ( θ , ϕ ) = l = 1 2 l + 1 l ( l + 1 ) I l [ a l π l 1 ( θ ) + b l τ l 1 ( θ ) ] sin ( ϕ ) , S 2 ( θ , ϕ ) = l = 1 2 l + 1 l ( l + 1 ) I l [ a l τ l 1 ( θ ) + b l π l 1 ( θ ) ] cos ( ϕ ) .
I l = ( i ) l 1 2 k r j l ( k r ) 1 l ( l + 1 ) × 0 π sin 2 ( θ ) d θ f ( k r , θ ) exp [ ikr cos ( θ ) ] P l 1 [ cos ( θ ) ] .
P l 1 l [ cos ( θ ) ] = 0 . P l l [ cos ( θ ) ] = 1 × 3 × 5 ( 2 l 1 ) sin l ( θ ) ( 2 l 1 ) ! ! sin l ( θ )
P l + 1 m [ cos ( θ ) ] = 2 l + 1 l + 1 m cos ( θ ) P l m [ cos ( θ ) ] l + m l + 1 m P l 1 m [ cos ( θ ) ] .
P 0 [ cos ( θ ) ] = 1 , P 1 [ cos ( θ ) ] = cos ( θ )
P l + 1 [ cos ( θ ) ] = 2 l + 1 l + 1 cos ( θ ) P l [ cos ( θ ) ] l l + 1 P l 1 [ cos ( θ ) ] .
Δ θ = π / ( 10 k a )
l max = k a + 4.05 ( k a ) 1 / 3 + 2 ,
Δ ϕ = 2 π 100 + 20 m
j 0 ( k a ) = sin ( k a ) k a , j 1 ( k a ) = sin ( k a ) ( k a ) 2 cos ( k a ) k a ,
j l + 1 ( k a ) = 2 l + 1 k a j l ( k a ) j l 1 ( k a ) .
A l m = A l m , B l m = B l m
A l m = { A l m for m = even A l m for m = odd , B l m = { B l m for m = even B l m for m = odd .
π l 1 l ( θ ) = 0 , π l l ( θ ) = ( 2 l 1 ) ! ! sin l 1 ( θ )
π l + 1 | m | ( θ ) = 2 l + 1 l + 1 | m | cos ( θ ) π l | m | ( θ ) l + | m | l + 1 | m | π l 1 | m | ( θ ) .
τ l | m | ( θ ) = l cos ( θ ) π l | m | ( θ ) ( l + | m | ) π l 1 | m | ( θ ) .
P 0 [ cos ( θ ) ] = 0 , P l [ cos ( θ ) ] = l P l 1 [ cos ( θ ) ] + cos ( θ ) P l 1 [ cos ( θ ) ] .
τ l 0 ( θ ) = sin ( θ ) P l [ cos ( θ ) ] .
P l | m | [ cos ( θ ) ] ( l + | m | ) ! ( l | m | ) ! ( l + 1 2 ) | m | J | m | [ ( l + 1 2 ) θ ]
τ l | m | ( θ ) | m | π l | m | ( θ ) 1 2 ( l + | m | ) ! ( l | m | ) ! ( θ / 2 ) | m | 1 ( | m | 1 ) !
P l | m | [ cos ( θ ) ] ( 2 π ) 1 / 2 [ sin ( θ ) ] 1 / 2 l | m | 1 / 2 × cos [ ( l + 1 2 ) θ + | m | π 2 π 4 ] ,
π l | m | ( θ ) ( 2 π ) 1 / 2 [ sin ( θ ) ] 3 / 2 l | m | 1 / 2 × cos [ ( l + 1 2 ) θ + | m | π 2 π 4 ] , π l | m | ( θ ) ( 2 π ) 1 / 2 sin ( θ ) 1 / 2 l | m | + 1 / 2 × sin [ ( l + 1 2 ) θ + | m | π 2 π 4 ] l π l | m | ( θ ) .
S 1 ( θ , ϕ ) = l = 1 l max 2 l + 1 2 l ( l + 1 ) B l 0 b l τ l 0 ( θ ) + m = 1 l = m l max 2 l + 1 2 l ( l + 1 ) i m a l π l m ( θ ) × [ A l m + exp ( i m ϕ ) + A l m exp ( i m ϕ ) ] + m = 1 l = m l max 2 l + 1 2 l ( l + 1 ) b l τ l m ( θ ) × [ B l m + exp ( i m ϕ ) + B l m exp ( i m ϕ ) ] S 2 ( θ , ϕ ) = l = 1 l max 2 l + l 2 l ( l + 1 ) A l 0 a l τ l 0 ( θ ) + m = 1 l = m l max 2 l + 1 2 l ( l + 1 ) a l τ l m ( θ ) × [ A l m + exp ( i m ϕ ) + A l m exp ( i m ϕ ) ] + m = 1 l = m l max 2 l + 1 2 l ( l + 1 ) i m b l π l m ( θ ) × [ B l m + exp ( i m ϕ ) B l m exp ( i m ϕ ) ] ,
A l m + A l m for m 1 , A l m A l m for m 1
A l m B l m exp [ ( l + 1 / 2 ) 2 / k 2 w 2 ] ( a w ) | m | 1 s | m | 1 ( | m | 1 ) ! .
I l exp [ ( l + 1 / 2 ) 2 / k 2 w 2 1 ( 2 i z 0 s ) / w ] 1 ( 2 i z 0 s ) / w .
A l m K l m 2 exp { [ k 2 x 0 2 + k 2 y 0 2 + ( l + 1 / 2 ) 2 ] / k 2 w 2 1 ( 2 i z 0 s ) / w } 1 ( 2 i z 0 s ) / w × j = 0 p = 0 j ( Ψ j p δ j 2 p + 1 , m + Ψ j p δ j 2 p 1 , m ) exp ( i k z 0 ) , B l m K l m 2 i exp { [ k 2 x 0 2 + k 2 y 0 2 + ( l + 1 / 2 ) 2 ] / k 2 w 2 1 ( 2 i z 0 s ) / w } 1 ( 2 i z 0 s ) / w × j = 0 p = 0 j ( Ψ j p δ j 2 p + 1 , m Ψ j p δ j 2 p 1 , m ) exp ( i k z 0 ) ,
K l m = { 2 l ( l + 1 ) 2 l + 1 ( 2 i ) for m = 0 ( 2 2 l + 1 ) | m | 1 ( 2 i ) ( 1 ) | m | / 2 for m = even and | m | 2 ( 2 2 l + 1 ) | m | 1 ( 2 ) ( 1 ) | m | / 2 1 / 2 for m = odd ,
Ψ j p = ( x 0 i y 0 ) j p ( x 0 i y 0 ) p ( j p ) ! p ! [ ( l + 1 / 2 ) s / w 1 ( 2 i z 0 s ) / w ] j .
a l b l } = 1 2 [ 1 R l 22 p = 1 T l 21 ( R l 11 ) p 1 T l 12 ] ,
sin ( θ ) P l 1 [ cos ( θ ) ] = l ( l + 1 ) 2 l + 1 { P l 1 [ cos ( θ ) ] P l + 1 [ cos ( θ ) ] } ,
0 π sin ( θ ) d θ P l [ cos ( θ ) ] exp [ ikr cos ( θ ) ] = 2 i l j l ( k r ) ,
0 π sin 2 ( θ ) d θ P l 1 [ cos ( θ ) ] exp [ ikr cos ( θ ) ] = 2 i l 1 l ( l + 1 ) × j l ( k r ) / k r .
J l = 0 π sin 2 ( θ ) d θ P l 1 [ cos ( θ ) ] f ( k r , θ ) exp [ ikr cos ( θ ) ] .
f ( k r , θ ) = 1 D exp [ k 2 r 2 sin 2 ( θ ) / k 2 w 2 D ] × [ 1 2 iskr cos ( θ ) / k w D ] exp ( i k z 0 ) ,
D = 1 + 2 i [ k r cos ( θ ) k z 0 ] s k w .
J l 0 π sin 2 ( θ ) d θ [ 2 l π sin ( θ ) ] 1 / 2 sin [ ( l + 1 2 ) θ π 4 ] × f ( k r , θ ) exp [ ikr cos ( θ ) ] .
α = θ π 2
J l = π / 2 π / 2 cos 2 ( α ) d α [ 2 l π cos ( α ) ] 1 / 2 sin [ ( l + 1 2 ) α + l π 2 ] × f ( k r , α + π 2 ) exp [ ikr sin ( α ) ] = i l 2 i π / 2 π / 2 cos 2 ( α ) d α [ 2 l π cos ( α ) ] 1 / 2 f ( k r , α + π 2 ) × exp [ i ( l + 1 2 ) α ikr sin ( α ) ] ( i ) l 2 i × π / 2 π / 2 cos 2 ( α ) d α [ 2 l π cos ( α ) ] 1 / 2 f ( k r , α + π 2 ) × exp [ i ( l + 1 2 ) α ikr sin ( α ) ] .
k r = l + 1 2 , θ = π 2
k r cos ( θ ) 0 , k r sin ( θ ) l + 1 2
J l f ( l + 1 2 , π 2 ) 0 π sin 2 ( θ ) d θ P l 1 [ cos ( θ ) ] exp [ ikr cos ( θ ) ] = f ( l + 1 2 , π 2 ) 2 i l 1 l ( l + 1 ) j l ( k r ) / k r .
I l f ( l + 1 2 , π 2 ) ,
f ( k r , θ ) = exp [ k 2 r 2 sin 2 ( θ ) / k 2 w 2 | D | 2 ] ,
E inc rad ( r , θ , ϕ ) = [ F cos ( ϕ ) + G x 0 ) ] × exp { 2 r sin ( θ ) [ x 0 cos ( ϕ ) + y 0 sin ( ϕ ) ] / w 2 D } = F 2 j = 0 p = 0 j Ψ j p exp [ i ( j 2 p + 1 ) ϕ ] + F 2 j = 0 p = 0 j Ψ j p exp [ i ( j 2 p 1 ) ϕ ] + G x 0 j = 0 p = 0 j Ψ j p exp [ i ( j 2 p ) ϕ ] ,
F = ψ 00 exp [ ikr cos ( θ ) ] sin ( θ ) [ 1 2 isr cos ( θ ) / w D ] ,
G = ψ 00 exp [ ikr cos ( θ ) ] 2 i s cos ( θ ) / w D ,
ψ 00 = 1 D exp { [ r 2 sin 2 ( θ ) + x 0 2 + y 0 2 ] / w 2 D } exp ( i k z 0 ) ,
Ψ j p = ( x 0 i y 0 ) j p ( x 0 i y 0 ) p ( j p ) ! p ! [ r sin ( θ ) / w 2 D ] j .
A l m = ( i ) l 1 k r j l ( k r ) ( l | m | ) ! ( l + | m | ) ! 0 π sin ( θ ) d θ P l m [ cos ( θ ) ] × ( F 2 j = 0 p = 0 j Ψ j p δ j 2 p + 1 , m + F 2 j = 0 p = 0 j Ψ j p δ j 2 p 1 , m + G x 0 j = 0 p = 0 j Ψ j p δ j 2 p , m ) .
0 π sin 2 ( θ ) d θ P l m [ cos ( θ ) ] f m ( k r , θ ) exp [ ikr cos ( θ ) ] ,
0 π sin 2 ( θ ) d θ P l m [ cos ( θ ) ] exp [ ikr cos ( θ ) ] ,

Metrics