Abstract

We present a theoretical description of the scattering of a Gaussian beam by a spherical, homogeneous, and isotropic particle. This theory handles particles with arbitrary size and nature having any location relative to the Gaussian beam. The formulation is based on the Bromwich method and closely follows Kerker’s formulation for plane-wave scattering. It provides expressions for the scattered intensities, the phase angle, the cross sections, and the radiation pressure.

© 1988 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
  6. G. Gréhan, G. Gouesbet, “Mie theory calculations: new progress, with emphasis on particle sizing,” Appl. Opt. 18, 3489–3493 (1979).
    [CrossRef] [PubMed]
  7. G. Gréhan, G. Gouesbet, C. Rabasse, “Monotonic relationships between scattered powers and diameters in Lorenz–Mie theory, for simultaneous velocimetry and sizing of single particles,” Appl. Opt. 20, 796–799 (1981).
    [CrossRef] [PubMed]
  8. A. Ungut, G. Gréhan, G. Gouesbet, “Comparisons between geometrical optics and Lorenz–Mie theory,” Appl. Opt. 20, 2911–2918 (1981).
    [CrossRef] [PubMed]
  9. G. Gouesbet, G. Gréhan, B. Maheu, “Single scattering characteristics of volume elements in coal clouds,” Appl. Opt. 22, 2038–2050 (1983).
    [CrossRef] [PubMed]
  10. G. Gréhan, G. Gouesbet, C. Rabasse, “The computer program supermidifor Lorenz–Mie theory and the research of one to one relationships for particle sizing,” in Proceedings of the Symposium on Long Range and Short Range Optical Velocity Measurements, H. J. Pfeifer, ed. (Institut Francoallemand de St. Louis, St. Louis, France, 1980), pp. XVI.1–VI.107.
  11. Y. Kakui, J. Hirono, A. Nishimoto, M. Nango, Y. Kanno, Bull. Electrotech. Lab. Jpn. 46, 640 (1982).
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    [CrossRef]
  13. F. Corbin, G. Gréhan, G. Gouesbet, B. Maheu, “Interaction between a sphere and a Gaussian beam: computations on a microcomputer,” Part. Part. Syst. Character. (to be published).
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  16. A. J. Yule, N. A. Chigier, S. Atakan, A. Ungut, “Particle size and velocity measurement by laser anemometry,” J. Energy 1, 220–228 (1977).
    [CrossRef]
  17. D. Holve, S. A. Self, “Optical particle sizing for in situ measurements,” Part 1, Appl. Opt. 18, 1632–1645 (1979); Part 2, Appl. Opt. 18, 1646–1652 (1979).
    [CrossRef]
  18. G. Gouesbet, G. Gréhan, “Advances in quasi-elastic scattering of light with emphasis on simultaneous measurements of velocities and sizes of particles embedded in flows,” presented at the American Institute of Aeronautics and Astronautics 16th Thermophysics Conference, Palo Alto, Calif., June 23–25, 1981.
  19. D. Allano, G. Gouesbet, G. Gréhan, D. Lisiecki, “Comparative measurements of calibrated droplets using Gabor holography and corrected top-hat laser beam sizing with discussion of simultaneous velocimetry,” in Proceedings of the International Symposium on Applications of Laser Doppler Anemometry to Fluid Mechanics, D. F. G. Durão, ed. (Lisbon University, Lisbon, 1982), pp. 6.1.1–6.1.10.
  20. D. Allano, G. Gouesbet, G. Gréhan, D. Lisiecki, “Droplet sizing using a ‘top-hat’ laser beam technique,” J. Phys. D 17, 43–58 (1984).
    [CrossRef]
  21. G. Gouesbet, G. Gréhan, R. Kleine, “Simultaneous optical measurement of velocity and size of individual particles in flows,” in Proceedings of the Second International Symposium on Applications of Laser Anemometry to Fluid Mechanics, D. F. G. Durão, ed. (Lisbon University, Lisbon, 1984), pp. 18.2.1–18.2.7.
  22. G. Gréhan, G. Gouesbet, R. Kleine, V. Renz, I. Wilhelmi, “Corrected laser beam techniques for simultaneous velocimetry and sizing of particles in flows and applications,” in Proceedings of the Third International Symposium on Application of Laser Anemometry to Fluid Mechanics, D. F. G. Durão, ed. (Lisbon University, Lisbon, 1986), pp. 20.5.1–20.5.57.
  23. G. Gouesbet, P. Gougeon, G. Gréhan, J. N. Letoulouzan, N. Lhuissier, B. Maheu, M. E. Weill, “Laser optical sizing from 100 Å to 1 mm diameter, and from 0 to 10 kg/m3concentration,” presented at the American Institute of Aeronautics and Astronautics 20th Thermophysics Conference, Williamsburg, Va., June 19–21, 1985.
  24. G. Gréhan, G. Gouesbet, “Simultaneous measurements of velocities and sizes of particles in flows using a combined system incorporating a top-hat beam technique,” Appl. Opt. 25, 3527–3538 (1986).
    [CrossRef] [PubMed]
  25. G. Gouesbet, G. Gréhan, eds., Proceedings of the International Symposium on Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988).
    [CrossRef]
  26. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
    [CrossRef]
  27. P. Chylek, V. Ramaswamy, A. Ashkin, J. Dziedzic, “Simultaneous determination of refractive index and size of spherical dielectric particles from light scattering data,” Appl. Opt. 22, 2302–2307 (1983).
    [CrossRef] [PubMed]
  28. G. Roosen, “A theoretical and experimental study of the stable equilibrium positions of spheres levitated by two horizontal laser beams,” Opt. Commun. 21, 189–194 (1977).
    [CrossRef]
  29. G. Roosen, “La lévitation optique de sphère,” Can. J. Phys. 57, 1260–1279 (1979).
    [CrossRef]
  30. G. Gréhan, G. Gouesbet, “Optical lévitation of a single particle to study the theory of the quasi-elastic scattering of light,” Appl. Opt. 19, 2485–2487 (1980).
    [CrossRef] [PubMed]
  31. W. D. Bachalo, M. J. Houser, “An instrument for two-components velocity and particle size measurement,” in Proceedings of the Third International Symposium on Applications of Laser Anemometry to Fluid Mechanics, D. F. G. Durão, ed. (Lisbon University, Lisbon, 1986), pp. 18.3.1–18.3.67.
  32. W. G. Tam, R. Corriveau, “Scattering of electromagnetic beams by spherical objects,” J. Opt. Soc. Am. 68, 763–767 (1978).
    [CrossRef]
  33. N. Morita, T. Tanaka, T. Yamasaki, Y. Nakanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. AP-16, 724–727 (1968).
    [CrossRef]
  34. W. C. Tsai, R. J. Pogorzelski, “Eigenfunction solution of the scattering of beam radiation fields by spherical objects,” J. Opt. Soc. Am. 65, 1457–1463 (1975).
    [CrossRef]
  35. J. S. Kim, S. S. Lee, “Scattering of laser beams and the optical potential well for a homogeneous sphere,” J. Opt. Soc. Am. 73, 303–312 (1983).
    [CrossRef]
  36. G. Gouesbet, G. Gréhan, “The quasi-elastic scattering of light: A lecture with emphasis on particulate diagnosis,” in Proceedings of the NATO Workshop on Soot in Combustion Systems and Its Toxic Properties, J. Lahaye, G. Prado, eds. (Plenum, New York, 1981), pp. 395–412.
  37. G. Gouesbet, “Optical sizing with emphasis on simultaneous measurements of velocities and sizes of particles embedded in flows,” presented at the XVth International Symposium on Heat and Mass Transfer, Dubrovnik, Yugoslavia, September 5–9, 1983 [in R. I. Soloukhin, N. H. Afgan, eds., Measurement Techniques for Heat Mass Transfer (Springer-Verlag, Berlin, 1983), pp. 27–39].
  38. G. Gréhan, “Nouveaux progrès en théorie de Lorenz–Mie. Application la mesure de diamètres de particules dans des écoulements,” thèse de troisième cycle (Rouen University, Rouen, France, 1980).
  39. G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. 13, 97–103 (1982).
    [CrossRef]
  40. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (AcademicNew York, 1969).
  41. G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. 16, 83–93 (1985).
    [CrossRef]
  42. G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. 16, 239–247 (1985).
    [CrossRef]
  43. G. Gréhan, 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]
  44. B. Maheu, G. Gréhan, G. Gouesbet, “Generalized Lorenz–Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23–25 (1987). Note that, although it was not stated explicitly in Refs. 44–46, the approximation iQ≃ 1 was included in the computations in those references. Values obtained without this approximation are published in Ref. 47 and compare well with those in Refs. 44–46.
    [CrossRef] [PubMed]
  45. G. Gréhan, B. Maheu, G. Gouesbet, “Localized approximation to the generalized Lorenz–Mie theory and its application to optical sizing,” presented at the International Congress on Applications of Lasers and Electro-Optics, Arlington, Va., November 10–13, 1986.
  46. B. Maheu, G. Gréhan, G. Gouesbet, “Diffusion de la lumière par une sphère dans le cas d’un faisceau d’extension finie: 1ère partie-théorie de Lorenz–Mie généralisée: les coefficients gn et leur calcul numérique,” J. Aerosol. Sci. 19, 47–53 (1988).
    [CrossRef]
  47. B. Maheu, G. Gréhan, G. Gouesbet, “Laser beam scattering by individual particles: numerical results and applications to optical sizing,” presented at the Symposium on Optical Particle Sizing: Theory and Practice, Mt. St. Aignan, France, May 12–15, 1987 [in G. Gouesbet, G. Gréhan, eds., Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988), pp. 77–88].
    [CrossRef]
  48. C. W. Yeh, S. Colak, P. W. Barber, “Scattering of sharply focused beams by arbitrarily shaped dielectric particles: an exact solution,” Appl. Opt. 21, 4426–4433 (1982).
    [CrossRef] [PubMed]
  49. T. J. Bromwich, “Electromagnetic waves,” Philos. Mag. 38, 143–164 (1919).
  50. P. Poincelot, Précis d’Électromagnétisme Théorique (Masson, Paris, 1963).
  51. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  52. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  53. F. Slimani, G. Gréhan, G. Gouesbet, D. Allano, “Near-field Lorenz–Mie theory and its application to microholography,” Appl. Opt. 23, 4140–4148 (1984).
    [CrossRef] [PubMed]
  54. L. Brillouin, “The scattering cross-sections of spheres for electromagnetic scattering,” J. Appl. Phys. 20, 1110–1125 (1949).
    [CrossRef]
  55. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  56. L. Robin, Fonctions Sphériques de Legendre et Fonctions Sphérdidales (Gauthier-Villars, Paris, 1957–1959), Vols. 1–3.

1988 (1)

B. Maheu, G. Gréhan, G. Gouesbet, “Diffusion de la lumière par une sphère dans le cas d’un faisceau d’extension finie: 1ère partie-théorie de Lorenz–Mie généralisée: les coefficients gn et leur calcul numérique,” J. Aerosol. Sci. 19, 47–53 (1988).
[CrossRef]

1987 (1)

1986 (2)

1985 (2)

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

G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. 16, 239–247 (1985).
[CrossRef]

1984 (2)

D. Allano, G. Gouesbet, G. Gréhan, D. Lisiecki, “Droplet sizing using a ‘top-hat’ laser beam technique,” J. Phys. D 17, 43–58 (1984).
[CrossRef]

F. Slimani, G. Gréhan, G. Gouesbet, D. Allano, “Near-field Lorenz–Mie theory and its application to microholography,” Appl. Opt. 23, 4140–4148 (1984).
[CrossRef] [PubMed]

1983 (4)

1982 (3)

Y. Kakui, J. Hirono, A. Nishimoto, M. Nango, Y. Kanno, Bull. Electrotech. Lab. Jpn. 46, 640 (1982).

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. 13, 97–103 (1982).
[CrossRef]

C. W. Yeh, S. Colak, P. W. Barber, “Scattering of sharply focused beams by arbitrarily shaped dielectric particles: an exact solution,” Appl. Opt. 21, 4426–4433 (1982).
[CrossRef] [PubMed]

1981 (2)

1980 (2)

1979 (4)

G. Gréhan, G. Gouesbet, “Mie theory calculations: new progress, with emphasis on particle sizing,” Appl. Opt. 18, 3489–3493 (1979).
[CrossRef] [PubMed]

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

G. Roosen, “La lévitation optique de sphère,” Can. J. Phys. 57, 1260–1279 (1979).
[CrossRef]

D. Holve, S. A. Self, “Optical particle sizing for in situ measurements,” Part 1, Appl. Opt. 18, 1632–1645 (1979); Part 2, Appl. Opt. 18, 1646–1652 (1979).
[CrossRef]

1978 (1)

1977 (2)

G. Roosen, “A theoretical and experimental study of the stable equilibrium positions of spheres levitated by two horizontal laser beams,” Opt. Commun. 21, 189–194 (1977).
[CrossRef]

A. J. Yule, N. A. Chigier, S. Atakan, A. Ungut, “Particle size and velocity measurement by laser anemometry,” J. Energy 1, 220–228 (1977).
[CrossRef]

1976 (1)

1975 (2)

1972 (1)

1970 (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[CrossRef]

1968 (1)

N. Morita, T. Tanaka, T. Yamasaki, Y. Nakanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. AP-16, 724–727 (1968).
[CrossRef]

1949 (1)

L. Brillouin, “The scattering cross-sections of spheres for electromagnetic scattering,” J. Appl. Phys. 20, 1110–1125 (1949).
[CrossRef]

1919 (1)

T. J. Bromwich, “Electromagnetic waves,” Philos. Mag. 38, 143–164 (1919).

1909 (1)

P. Debye, “Der Lichtdruck auf Kugeln von Beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
[CrossRef]

1908 (1)

G. Mie, “Beiträge zur Optik Trüber Medien, speziell Kolloidaler Metallösungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

1890 (1)

L. Lorenz, “Lysbevaegelsen i og uden for en haf plane lysbölger belyst Kulge,” Vidensk. Selk. Skr. 6, 1–62 (1890).

Allano, D.

D. Allano, G. Gouesbet, G. Gréhan, D. Lisiecki, “Droplet sizing using a ‘top-hat’ laser beam technique,” J. Phys. D 17, 43–58 (1984).
[CrossRef]

F. Slimani, G. Gréhan, G. Gouesbet, D. Allano, “Near-field Lorenz–Mie theory and its application to microholography,” Appl. Opt. 23, 4140–4148 (1984).
[CrossRef] [PubMed]

D. Allano, G. Gouesbet, G. Gréhan, D. Lisiecki, “Comparative measurements of calibrated droplets using Gabor holography and corrected top-hat laser beam sizing with discussion of simultaneous velocimetry,” in Proceedings of the International Symposium on Applications of Laser Doppler Anemometry to Fluid Mechanics, D. F. G. Durão, ed. (Lisbon University, Lisbon, 1982), pp. 6.1.1–6.1.10.

Ashkin, A.

Atakan, S.

A. J. Yule, N. A. Chigier, S. Atakan, A. Ungut, “Particle size and velocity measurement by laser anemometry,” J. Energy 1, 220–228 (1977).
[CrossRef]

Bachalo, W. D.

W. D. Bachalo, M. J. Houser, “An instrument for two-components velocity and particle size measurement,” in Proceedings of the Third International Symposium on Applications of Laser Anemometry to Fluid Mechanics, D. F. G. Durão, ed. (Lisbon University, Lisbon, 1986), pp. 18.3.1–18.3.67.

Barber, P. W.

Brillouin, L.

L. Brillouin, “The scattering cross-sections of spheres for electromagnetic scattering,” J. Appl. Phys. 20, 1110–1125 (1949).
[CrossRef]

Bromwich, T. J.

T. J. Bromwich, “Electromagnetic waves,” Philos. Mag. 38, 143–164 (1919).

Chigier, N. A.

A. J. Yule, N. A. Chigier, S. Atakan, A. Ungut, “Particle size and velocity measurement by laser anemometry,” J. Energy 1, 220–228 (1977).
[CrossRef]

Chylek, P.

Colak, S.

Corbin, F.

F. Corbin, G. Gréhan, G. Gouesbet, B. Maheu, “Interaction between a sphere and a Gaussian beam: computations on a microcomputer,” Part. Part. Syst. Character. (to be published).

Corriveau, R.

Davis, L. W.

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

Debye, P.

P. Debye, “Der Lichtdruck auf Kugeln von Beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
[CrossRef]

Dziedzic, J.

Farmer, W. M.

Gouesbet, G.

B. Maheu, G. Gréhan, G. Gouesbet, “Diffusion de la lumière par une sphère dans le cas d’un faisceau d’extension finie: 1ère partie-théorie de Lorenz–Mie généralisée: les coefficients gn et leur calcul numérique,” J. Aerosol. Sci. 19, 47–53 (1988).
[CrossRef]

B. Maheu, G. Gréhan, G. Gouesbet, “Generalized Lorenz–Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23–25 (1987). Note that, although it was not stated explicitly in Refs. 44–46, the approximation iQ≃ 1 was included in the computations in those references. Values obtained without this approximation are published in Ref. 47 and compare well with those in Refs. 44–46.
[CrossRef] [PubMed]

G. Gréhan, G. Gouesbet, “Simultaneous measurements of velocities and sizes of particles in flows using a combined system incorporating a top-hat beam technique,” Appl. Opt. 25, 3527–3538 (1986).
[CrossRef] [PubMed]

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

G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. 16, 239–247 (1985).
[CrossRef]

D. Allano, G. Gouesbet, G. Gréhan, D. Lisiecki, “Droplet sizing using a ‘top-hat’ laser beam technique,” J. Phys. D 17, 43–58 (1984).
[CrossRef]

F. Slimani, G. Gréhan, G. Gouesbet, D. Allano, “Near-field Lorenz–Mie theory and its application to microholography,” Appl. Opt. 23, 4140–4148 (1984).
[CrossRef] [PubMed]

G. Gouesbet, G. Gréhan, B. Maheu, “Single scattering characteristics of volume elements in coal clouds,” Appl. Opt. 22, 2038–2050 (1983).
[CrossRef] [PubMed]

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. 13, 97–103 (1982).
[CrossRef]

A. Ungut, G. Gréhan, G. Gouesbet, “Comparisons between geometrical optics and Lorenz–Mie theory,” Appl. Opt. 20, 2911–2918 (1981).
[CrossRef] [PubMed]

G. Gréhan, G. Gouesbet, C. Rabasse, “Monotonic relationships between scattered powers and diameters in Lorenz–Mie theory, for simultaneous velocimetry and sizing of single particles,” Appl. Opt. 20, 796–799 (1981).
[CrossRef] [PubMed]

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

G. Gréhan, G. Gouesbet, “Mie theory calculations: new progress, with emphasis on particle sizing,” Appl. Opt. 18, 3489–3493 (1979).
[CrossRef] [PubMed]

B. Maheu, G. Gréhan, G. Gouesbet, “Laser beam scattering by individual particles: numerical results and applications to optical sizing,” presented at the Symposium on Optical Particle Sizing: Theory and Practice, Mt. St. Aignan, France, May 12–15, 1987 [in G. Gouesbet, G. Gréhan, eds., Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988), pp. 77–88].
[CrossRef]

G. Gouesbet, “Optical sizing with emphasis on simultaneous measurements of velocities and sizes of particles embedded in flows,” presented at the XVth International Symposium on Heat and Mass Transfer, Dubrovnik, Yugoslavia, September 5–9, 1983 [in R. I. Soloukhin, N. H. Afgan, eds., Measurement Techniques for Heat Mass Transfer (Springer-Verlag, Berlin, 1983), pp. 27–39].

G. Gréhan, B. Maheu, G. Gouesbet, “Localized approximation to the generalized Lorenz–Mie theory and its application to optical sizing,” presented at the International Congress on Applications of Lasers and Electro-Optics, Arlington, Va., November 10–13, 1986.

F. Corbin, G. Gréhan, G. Gouesbet, B. Maheu, “Interaction between a sphere and a Gaussian beam: computations on a microcomputer,” Part. Part. Syst. Character. (to be published).

G. Gouesbet, G. Gréhan, “The quasi-elastic scattering of light: A lecture with emphasis on particulate diagnosis,” in Proceedings of the NATO Workshop on Soot in Combustion Systems and Its Toxic Properties, J. Lahaye, G. Prado, eds. (Plenum, New York, 1981), pp. 395–412.

D. Allano, G. Gouesbet, G. Gréhan, D. Lisiecki, “Comparative measurements of calibrated droplets using Gabor holography and corrected top-hat laser beam sizing with discussion of simultaneous velocimetry,” in Proceedings of the International Symposium on Applications of Laser Doppler Anemometry to Fluid Mechanics, D. F. G. Durão, ed. (Lisbon University, Lisbon, 1982), pp. 6.1.1–6.1.10.

G. Gouesbet, G. Gréhan, “Advances in quasi-elastic scattering of light with emphasis on simultaneous measurements of velocities and sizes of particles embedded in flows,” presented at the American Institute of Aeronautics and Astronautics 16th Thermophysics Conference, Palo Alto, Calif., June 23–25, 1981.

G. Gréhan, G. Gouesbet, C. Rabasse, “The computer program supermidifor Lorenz–Mie theory and the research of one to one relationships for particle sizing,” in Proceedings of the Symposium on Long Range and Short Range Optical Velocity Measurements, H. J. Pfeifer, ed. (Institut Francoallemand de St. Louis, St. Louis, France, 1980), pp. XVI.1–VI.107.

G. Gouesbet, G. Gréhan, R. Kleine, “Simultaneous optical measurement of velocity and size of individual particles in flows,” in Proceedings of the Second International Symposium on Applications of Laser Anemometry to Fluid Mechanics, D. F. G. Durão, ed. (Lisbon University, Lisbon, 1984), pp. 18.2.1–18.2.7.

G. Gréhan, G. Gouesbet, R. Kleine, V. Renz, I. Wilhelmi, “Corrected laser beam techniques for simultaneous velocimetry and sizing of particles in flows and applications,” in Proceedings of the Third International Symposium on Application of Laser Anemometry to Fluid Mechanics, D. F. G. Durão, ed. (Lisbon University, Lisbon, 1986), pp. 20.5.1–20.5.57.

G. Gouesbet, P. Gougeon, G. Gréhan, J. N. Letoulouzan, N. Lhuissier, B. Maheu, M. E. Weill, “Laser optical sizing from 100 Å to 1 mm diameter, and from 0 to 10 kg/m3concentration,” presented at the American Institute of Aeronautics and Astronautics 20th Thermophysics Conference, Williamsburg, Va., June 19–21, 1985.

Gougeon, P.

G. Gouesbet, P. Gougeon, G. Gréhan, J. N. Letoulouzan, N. Lhuissier, B. Maheu, M. E. Weill, “Laser optical sizing from 100 Å to 1 mm diameter, and from 0 to 10 kg/m3concentration,” presented at the American Institute of Aeronautics and Astronautics 20th Thermophysics Conference, Williamsburg, Va., June 19–21, 1985.

Gréhan, G.

B. Maheu, G. Gréhan, G. Gouesbet, “Diffusion de la lumière par une sphère dans le cas d’un faisceau d’extension finie: 1ère partie-théorie de Lorenz–Mie généralisée: les coefficients gn et leur calcul numérique,” J. Aerosol. Sci. 19, 47–53 (1988).
[CrossRef]

B. Maheu, G. Gréhan, G. Gouesbet, “Generalized Lorenz–Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23–25 (1987). Note that, although it was not stated explicitly in Refs. 44–46, the approximation iQ≃ 1 was included in the computations in those references. Values obtained without this approximation are published in Ref. 47 and compare well with those in Refs. 44–46.
[CrossRef] [PubMed]

G. Gréhan, 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. Gréhan, G. Gouesbet, “Simultaneous measurements of velocities and sizes of particles in flows using a combined system incorporating a top-hat beam technique,” Appl. Opt. 25, 3527–3538 (1986).
[CrossRef] [PubMed]

G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. 16, 239–247 (1985).
[CrossRef]

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

D. Allano, G. Gouesbet, G. Gréhan, D. Lisiecki, “Droplet sizing using a ‘top-hat’ laser beam technique,” J. Phys. D 17, 43–58 (1984).
[CrossRef]

F. Slimani, G. Gréhan, G. Gouesbet, D. Allano, “Near-field Lorenz–Mie theory and its application to microholography,” Appl. Opt. 23, 4140–4148 (1984).
[CrossRef] [PubMed]

G. Gouesbet, G. Gréhan, B. Maheu, “Single scattering characteristics of volume elements in coal clouds,” Appl. Opt. 22, 2038–2050 (1983).
[CrossRef] [PubMed]

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. 13, 97–103 (1982).
[CrossRef]

A. Ungut, G. Gréhan, G. Gouesbet, “Comparisons between geometrical optics and Lorenz–Mie theory,” Appl. Opt. 20, 2911–2918 (1981).
[CrossRef] [PubMed]

G. Gréhan, G. Gouesbet, C. Rabasse, “Monotonic relationships between scattered powers and diameters in Lorenz–Mie theory, for simultaneous velocimetry and sizing of single particles,” Appl. Opt. 20, 796–799 (1981).
[CrossRef] [PubMed]

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

G. Gréhan, G. Gouesbet, “Mie theory calculations: new progress, with emphasis on particle sizing,” Appl. Opt. 18, 3489–3493 (1979).
[CrossRef] [PubMed]

B. Maheu, G. Gréhan, G. Gouesbet, “Laser beam scattering by individual particles: numerical results and applications to optical sizing,” presented at the Symposium on Optical Particle Sizing: Theory and Practice, Mt. St. Aignan, France, May 12–15, 1987 [in G. Gouesbet, G. Gréhan, eds., Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988), pp. 77–88].
[CrossRef]

G. Gréhan, “Nouveaux progrès en théorie de Lorenz–Mie. Application la mesure de diamètres de particules dans des écoulements,” thèse de troisième cycle (Rouen University, Rouen, France, 1980).

G. Gréhan, B. Maheu, G. Gouesbet, “Localized approximation to the generalized Lorenz–Mie theory and its application to optical sizing,” presented at the International Congress on Applications of Lasers and Electro-Optics, Arlington, Va., November 10–13, 1986.

G. Gouesbet, G. Gréhan, “The quasi-elastic scattering of light: A lecture with emphasis on particulate diagnosis,” in Proceedings of the NATO Workshop on Soot in Combustion Systems and Its Toxic Properties, J. Lahaye, G. Prado, eds. (Plenum, New York, 1981), pp. 395–412.

F. Corbin, G. Gréhan, G. Gouesbet, B. Maheu, “Interaction between a sphere and a Gaussian beam: computations on a microcomputer,” Part. Part. Syst. Character. (to be published).

G. Gréhan, G. Gouesbet, R. Kleine, V. Renz, I. Wilhelmi, “Corrected laser beam techniques for simultaneous velocimetry and sizing of particles in flows and applications,” in Proceedings of the Third International Symposium on Application of Laser Anemometry to Fluid Mechanics, D. F. G. Durão, ed. (Lisbon University, Lisbon, 1986), pp. 20.5.1–20.5.57.

D. Allano, G. Gouesbet, G. Gréhan, D. Lisiecki, “Comparative measurements of calibrated droplets using Gabor holography and corrected top-hat laser beam sizing with discussion of simultaneous velocimetry,” in Proceedings of the International Symposium on Applications of Laser Doppler Anemometry to Fluid Mechanics, D. F. G. Durão, ed. (Lisbon University, Lisbon, 1982), pp. 6.1.1–6.1.10.

G. Gouesbet, G. Gréhan, “Advances in quasi-elastic scattering of light with emphasis on simultaneous measurements of velocities and sizes of particles embedded in flows,” presented at the American Institute of Aeronautics and Astronautics 16th Thermophysics Conference, Palo Alto, Calif., June 23–25, 1981.

G. Gouesbet, P. Gougeon, G. Gréhan, J. N. Letoulouzan, N. Lhuissier, B. Maheu, M. E. Weill, “Laser optical sizing from 100 Å to 1 mm diameter, and from 0 to 10 kg/m3concentration,” presented at the American Institute of Aeronautics and Astronautics 20th Thermophysics Conference, Williamsburg, Va., June 19–21, 1985.

G. Gouesbet, G. Gréhan, R. Kleine, “Simultaneous optical measurement of velocity and size of individual particles in flows,” in Proceedings of the Second International Symposium on Applications of Laser Anemometry to Fluid Mechanics, D. F. G. Durão, ed. (Lisbon University, Lisbon, 1984), pp. 18.2.1–18.2.7.

G. Gréhan, G. Gouesbet, C. Rabasse, “The computer program supermidifor Lorenz–Mie theory and the research of one to one relationships for particle sizing,” in Proceedings of the Symposium on Long Range and Short Range Optical Velocity Measurements, H. J. Pfeifer, ed. (Institut Francoallemand de St. Louis, St. Louis, France, 1980), pp. XVI.1–VI.107.

Hirono, J.

Y. Kakui, J. Hirono, A. Nishimoto, M. Nango, Y. Kanno, Bull. Electrotech. Lab. Jpn. 46, 640 (1982).

Holve, D.

D. Holve, S. A. Self, “Optical particle sizing for in situ measurements,” Part 1, Appl. Opt. 18, 1632–1645 (1979); Part 2, Appl. Opt. 18, 1646–1652 (1979).
[CrossRef]

Houser, M. J.

W. D. Bachalo, M. J. Houser, “An instrument for two-components velocity and particle size measurement,” in Proceedings of the Third International Symposium on Applications of Laser Anemometry to Fluid Mechanics, D. F. G. Durão, ed. (Lisbon University, Lisbon, 1986), pp. 18.3.1–18.3.67.

Jones, A. R.

A. R. Jones, “Calculations of the ratio of complex Ricatti–Bessel functions for Mie scattering,” J. Phys. D 16, 149–152 (1983).
[CrossRef]

Kakui, Y.

Y. Kakui, J. Hirono, A. Nishimoto, M. Nango, Y. Kanno, Bull. Electrotech. Lab. Jpn. 46, 640 (1982).

Kanno, Y.

Y. Kakui, J. Hirono, A. Nishimoto, M. Nango, Y. Kanno, Bull. Electrotech. Lab. Jpn. 46, 640 (1982).

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (AcademicNew York, 1969).

Kim, J. S.

Kleine, R.

G. Gouesbet, G. Gréhan, R. Kleine, “Simultaneous optical measurement of velocity and size of individual particles in flows,” in Proceedings of the Second International Symposium on Applications of Laser Anemometry to Fluid Mechanics, D. F. G. Durão, ed. (Lisbon University, Lisbon, 1984), pp. 18.2.1–18.2.7.

G. Gréhan, G. Gouesbet, R. Kleine, V. Renz, I. Wilhelmi, “Corrected laser beam techniques for simultaneous velocimetry and sizing of particles in flows and applications,” in Proceedings of the Third International Symposium on Application of Laser Anemometry to Fluid Mechanics, D. F. G. Durão, ed. (Lisbon University, Lisbon, 1986), pp. 20.5.1–20.5.57.

Lax, M.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Lee, S. S.

Lentz, W. J.

Letoulouzan, J. N.

G. Gouesbet, P. Gougeon, G. Gréhan, J. N. Letoulouzan, N. Lhuissier, B. Maheu, M. E. Weill, “Laser optical sizing from 100 Å to 1 mm diameter, and from 0 to 10 kg/m3concentration,” presented at the American Institute of Aeronautics and Astronautics 20th Thermophysics Conference, Williamsburg, Va., June 19–21, 1985.

Lhuissier, N.

G. Gouesbet, P. Gougeon, G. Gréhan, J. N. Letoulouzan, N. Lhuissier, B. Maheu, M. E. Weill, “Laser optical sizing from 100 Å to 1 mm diameter, and from 0 to 10 kg/m3concentration,” presented at the American Institute of Aeronautics and Astronautics 20th Thermophysics Conference, Williamsburg, Va., June 19–21, 1985.

Lisiecki, D.

D. Allano, G. Gouesbet, G. Gréhan, D. Lisiecki, “Droplet sizing using a ‘top-hat’ laser beam technique,” J. Phys. D 17, 43–58 (1984).
[CrossRef]

D. Allano, G. Gouesbet, G. Gréhan, D. Lisiecki, “Comparative measurements of calibrated droplets using Gabor holography and corrected top-hat laser beam sizing with discussion of simultaneous velocimetry,” in Proceedings of the International Symposium on Applications of Laser Doppler Anemometry to Fluid Mechanics, D. F. G. Durão, ed. (Lisbon University, Lisbon, 1982), pp. 6.1.1–6.1.10.

Lorenz, L.

L. Lorenz, “Lysbevaegelsen i og uden for en haf plane lysbölger belyst Kulge,” Vidensk. Selk. Skr. 6, 1–62 (1890).

L. Lorenz, “Sur la lumière réfléchie et réfractée par une sphère transparente,” in Oeuvres Scientifiques de L. Lorenz, revues et annotées par H. Valentiner (Librairie Lehmann et Stage, Copenhagen, 1898), pp. 405–529.

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Maheu, B.

B. Maheu, G. Gréhan, G. Gouesbet, “Diffusion de la lumière par une sphère dans le cas d’un faisceau d’extension finie: 1ère partie-théorie de Lorenz–Mie généralisée: les coefficients gn et leur calcul numérique,” J. Aerosol. Sci. 19, 47–53 (1988).
[CrossRef]

B. Maheu, G. Gréhan, G. Gouesbet, “Generalized Lorenz–Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23–25 (1987). Note that, although it was not stated explicitly in Refs. 44–46, the approximation iQ≃ 1 was included in the computations in those references. Values obtained without this approximation are published in Ref. 47 and compare well with those in Refs. 44–46.
[CrossRef] [PubMed]

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

G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. 16, 239–247 (1985).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Single scattering characteristics of volume elements in coal clouds,” Appl. Opt. 22, 2038–2050 (1983).
[CrossRef] [PubMed]

B. Maheu, G. Gréhan, G. Gouesbet, “Laser beam scattering by individual particles: numerical results and applications to optical sizing,” presented at the Symposium on Optical Particle Sizing: Theory and Practice, Mt. St. Aignan, France, May 12–15, 1987 [in G. Gouesbet, G. Gréhan, eds., Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988), pp. 77–88].
[CrossRef]

G. Gréhan, B. Maheu, G. Gouesbet, “Localized approximation to the generalized Lorenz–Mie theory and its application to optical sizing,” presented at the International Congress on Applications of Lasers and Electro-Optics, Arlington, Va., November 10–13, 1986.

F. Corbin, G. Gréhan, G. Gouesbet, B. Maheu, “Interaction between a sphere and a Gaussian beam: computations on a microcomputer,” Part. Part. Syst. Character. (to be published).

G. Gouesbet, P. Gougeon, G. Gréhan, J. N. Letoulouzan, N. Lhuissier, B. Maheu, M. E. Weill, “Laser optical sizing from 100 Å to 1 mm diameter, and from 0 to 10 kg/m3concentration,” presented at the American Institute of Aeronautics and Astronautics 20th Thermophysics Conference, Williamsburg, Va., June 19–21, 1985.

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Mie, G.

G. Mie, “Beiträge zur Optik Trüber Medien, speziell Kolloidaler Metallösungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

Morita, N.

N. Morita, T. Tanaka, T. Yamasaki, Y. Nakanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. AP-16, 724–727 (1968).
[CrossRef]

Nakanishi, Y.

N. Morita, T. Tanaka, T. Yamasaki, Y. Nakanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. AP-16, 724–727 (1968).
[CrossRef]

Nango, M.

Y. Kakui, J. Hirono, A. Nishimoto, M. Nango, Y. Kanno, Bull. Electrotech. Lab. Jpn. 46, 640 (1982).

Nishimoto, A.

Y. Kakui, J. Hirono, A. Nishimoto, M. Nango, Y. Kanno, Bull. Electrotech. Lab. Jpn. 46, 640 (1982).

Pogorzelski, R. J.

Poincelot, P.

P. Poincelot, Précis d’Électromagnétisme Théorique (Masson, Paris, 1963).

Rabasse, C.

G. Gréhan, G. Gouesbet, C. Rabasse, “Monotonic relationships between scattered powers and diameters in Lorenz–Mie theory, for simultaneous velocimetry and sizing of single particles,” Appl. Opt. 20, 796–799 (1981).
[CrossRef] [PubMed]

G. Gréhan, G. Gouesbet, C. Rabasse, “The computer program supermidifor Lorenz–Mie theory and the research of one to one relationships for particle sizing,” in Proceedings of the Symposium on Long Range and Short Range Optical Velocity Measurements, H. J. Pfeifer, ed. (Institut Francoallemand de St. Louis, St. Louis, France, 1980), pp. XVI.1–VI.107.

Ramaswamy, V.

Renz, V.

G. Gréhan, G. Gouesbet, R. Kleine, V. Renz, I. Wilhelmi, “Corrected laser beam techniques for simultaneous velocimetry and sizing of particles in flows and applications,” in Proceedings of the Third International Symposium on Application of Laser Anemometry to Fluid Mechanics, D. F. G. Durão, ed. (Lisbon University, Lisbon, 1986), pp. 20.5.1–20.5.57.

Robin, L.

L. Robin, Fonctions Sphériques de Legendre et Fonctions Sphérdidales (Gauthier-Villars, Paris, 1957–1959), Vols. 1–3.

Roosen, G.

G. Roosen, “La lévitation optique de sphère,” Can. J. Phys. 57, 1260–1279 (1979).
[CrossRef]

G. Roosen, “A theoretical and experimental study of the stable equilibrium positions of spheres levitated by two horizontal laser beams,” Opt. Commun. 21, 189–194 (1977).
[CrossRef]

Self, S. A.

D. Holve, S. A. Self, “Optical particle sizing for in situ measurements,” Part 1, Appl. Opt. 18, 1632–1645 (1979); Part 2, Appl. Opt. 18, 1646–1652 (1979).
[CrossRef]

Slimani, F.

Tam, W. G.

Tanaka, T.

N. Morita, T. Tanaka, T. Yamasaki, Y. Nakanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. AP-16, 724–727 (1968).
[CrossRef]

Tsai, W. C.

Ungut, A.

A. Ungut, G. Gréhan, G. Gouesbet, “Comparisons between geometrical optics and Lorenz–Mie theory,” Appl. Opt. 20, 2911–2918 (1981).
[CrossRef] [PubMed]

A. J. Yule, N. A. Chigier, S. Atakan, A. Ungut, “Particle size and velocity measurement by laser anemometry,” J. Energy 1, 220–228 (1977).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Weill, M. E.

G. Gouesbet, P. Gougeon, G. Gréhan, J. N. Letoulouzan, N. Lhuissier, B. Maheu, M. E. Weill, “Laser optical sizing from 100 Å to 1 mm diameter, and from 0 to 10 kg/m3concentration,” presented at the American Institute of Aeronautics and Astronautics 20th Thermophysics Conference, Williamsburg, Va., June 19–21, 1985.

Wilhelmi, I.

G. Gréhan, G. Gouesbet, R. Kleine, V. Renz, I. Wilhelmi, “Corrected laser beam techniques for simultaneous velocimetry and sizing of particles in flows and applications,” in Proceedings of the Third International Symposium on Application of Laser Anemometry to Fluid Mechanics, D. F. G. Durão, ed. (Lisbon University, Lisbon, 1986), pp. 20.5.1–20.5.57.

Yamasaki, T.

N. Morita, T. Tanaka, T. Yamasaki, Y. Nakanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. AP-16, 724–727 (1968).
[CrossRef]

Yeh, C. W.

Yule, A. J.

A. J. Yule, N. A. Chigier, S. Atakan, A. Ungut, “Particle size and velocity measurement by laser anemometry,” J. Energy 1, 220–228 (1977).
[CrossRef]

Ann. Phys. (2)

G. Mie, “Beiträge zur Optik Trüber Medien, speziell Kolloidaler Metallösungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

P. Debye, “Der Lichtdruck auf Kugeln von Beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
[CrossRef]

Appl. Opt. (14)

W. J. Lentz, “Generating Bessel functions in Mie scattering calculations using continued fractions,” Appl. Opt. 15, 668–671 (1976).
[CrossRef] [PubMed]

G. Gréhan, G. Gouesbet, “Mie theory calculations: new progress, with emphasis on particle sizing,” Appl. Opt. 18, 3489–3493 (1979).
[CrossRef] [PubMed]

G. Gréhan, G. Gouesbet, C. Rabasse, “Monotonic relationships between scattered powers and diameters in Lorenz–Mie theory, for simultaneous velocimetry and sizing of single particles,” Appl. Opt. 20, 796–799 (1981).
[CrossRef] [PubMed]

A. Ungut, G. Gréhan, G. Gouesbet, “Comparisons between geometrical optics and Lorenz–Mie theory,” Appl. Opt. 20, 2911–2918 (1981).
[CrossRef] [PubMed]

G. Gouesbet, G. Gréhan, B. Maheu, “Single scattering characteristics of volume elements in coal clouds,” Appl. Opt. 22, 2038–2050 (1983).
[CrossRef] [PubMed]

W. M. Farmer, “Measurements of particle size, number density and velocity using laser interferometer,” Appl. Opt. 11, 2603–2612 (1972).
[CrossRef] [PubMed]

W. M. Farmer, “Visibility of large spheres observed with a laser velocimeter: a simple model,” Appl. Opt. 19, 3660–3666 (1980).
[CrossRef] [PubMed]

G. Gréhan, G. Gouesbet, “Simultaneous measurements of velocities and sizes of particles in flows using a combined system incorporating a top-hat beam technique,” Appl. Opt. 25, 3527–3538 (1986).
[CrossRef] [PubMed]

P. Chylek, V. Ramaswamy, A. Ashkin, J. Dziedzic, “Simultaneous determination of refractive index and size of spherical dielectric particles from light scattering data,” Appl. Opt. 22, 2302–2307 (1983).
[CrossRef] [PubMed]

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

G. Gréhan, 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]

B. Maheu, G. Gréhan, G. Gouesbet, “Generalized Lorenz–Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23–25 (1987). Note that, although it was not stated explicitly in Refs. 44–46, the approximation iQ≃ 1 was included in the computations in those references. Values obtained without this approximation are published in Ref. 47 and compare well with those in Refs. 44–46.
[CrossRef] [PubMed]

C. W. Yeh, S. Colak, P. W. Barber, “Scattering of sharply focused beams by arbitrarily shaped dielectric particles: an exact solution,” Appl. Opt. 21, 4426–4433 (1982).
[CrossRef] [PubMed]

F. Slimani, G. Gréhan, G. Gouesbet, D. Allano, “Near-field Lorenz–Mie theory and its application to microholography,” Appl. Opt. 23, 4140–4148 (1984).
[CrossRef] [PubMed]

Bull. Electrotech. Lab. Jpn. (1)

Y. Kakui, J. Hirono, A. Nishimoto, M. Nango, Y. Kanno, Bull. Electrotech. Lab. Jpn. 46, 640 (1982).

Can. J. Phys. (1)

G. Roosen, “La lévitation optique de sphère,” Can. J. Phys. 57, 1260–1279 (1979).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

N. Morita, T. Tanaka, T. Yamasaki, Y. Nakanishi, “Scattering of a beam wave by a spherical object,” IEEE Trans. Antennas Propag. AP-16, 724–727 (1968).
[CrossRef]

J. Aerosol. Sci. (1)

B. Maheu, G. Gréhan, G. Gouesbet, “Diffusion de la lumière par une sphère dans le cas d’un faisceau d’extension finie: 1ère partie-théorie de Lorenz–Mie généralisée: les coefficients gn et leur calcul numérique,” J. Aerosol. Sci. 19, 47–53 (1988).
[CrossRef]

J. Appl. Phys. (1)

L. Brillouin, “The scattering cross-sections of spheres for electromagnetic scattering,” J. Appl. Phys. 20, 1110–1125 (1949).
[CrossRef]

J. Energy (1)

A. J. Yule, N. A. Chigier, S. Atakan, A. Ungut, “Particle size and velocity measurement by laser anemometry,” J. Energy 1, 220–228 (1977).
[CrossRef]

J. Opt. (3)

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. 13, 97–103 (1982).
[CrossRef]

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

G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. 16, 239–247 (1985).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Phys. D (2)

D. Allano, G. Gouesbet, G. Gréhan, D. Lisiecki, “Droplet sizing using a ‘top-hat’ laser beam technique,” J. Phys. D 17, 43–58 (1984).
[CrossRef]

A. R. Jones, “Calculations of the ratio of complex Ricatti–Bessel functions for Mie scattering,” J. Phys. D 16, 149–152 (1983).
[CrossRef]

Opt. Commun. (1)

G. Roosen, “A theoretical and experimental study of the stable equilibrium positions of spheres levitated by two horizontal laser beams,” Opt. Commun. 21, 189–194 (1977).
[CrossRef]

Part 1, Appl. Opt. (1)

D. Holve, S. A. Self, “Optical particle sizing for in situ measurements,” Part 1, Appl. Opt. 18, 1632–1645 (1979); Part 2, Appl. Opt. 18, 1646–1652 (1979).
[CrossRef]

Philos. Mag. (1)

T. J. Bromwich, “Electromagnetic waves,” Philos. Mag. 38, 143–164 (1919).

Phys. Rev. A (2)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

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

Phys. Rev. Lett. (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[CrossRef]

Vidensk. Selk. Skr. (1)

L. Lorenz, “Lysbevaegelsen i og uden for en haf plane lysbölger belyst Kulge,” Vidensk. Selk. Skr. 6, 1–62 (1890).

Other (19)

L. Lorenz, “Sur la lumière réfléchie et réfractée par une sphère transparente,” in Oeuvres Scientifiques de L. Lorenz, revues et annotées par H. Valentiner (Librairie Lehmann et Stage, Copenhagen, 1898), pp. 405–529.

F. Corbin, G. Gréhan, G. Gouesbet, B. Maheu, “Interaction between a sphere and a Gaussian beam: computations on a microcomputer,” Part. Part. Syst. Character. (to be published).

G. Gouesbet, G. Gréhan, “Advances in quasi-elastic scattering of light with emphasis on simultaneous measurements of velocities and sizes of particles embedded in flows,” presented at the American Institute of Aeronautics and Astronautics 16th Thermophysics Conference, Palo Alto, Calif., June 23–25, 1981.

D. Allano, G. Gouesbet, G. Gréhan, D. Lisiecki, “Comparative measurements of calibrated droplets using Gabor holography and corrected top-hat laser beam sizing with discussion of simultaneous velocimetry,” in Proceedings of the International Symposium on Applications of Laser Doppler Anemometry to Fluid Mechanics, D. F. G. Durão, ed. (Lisbon University, Lisbon, 1982), pp. 6.1.1–6.1.10.

G. Gréhan, G. Gouesbet, C. Rabasse, “The computer program supermidifor Lorenz–Mie theory and the research of one to one relationships for particle sizing,” in Proceedings of the Symposium on Long Range and Short Range Optical Velocity Measurements, H. J. Pfeifer, ed. (Institut Francoallemand de St. Louis, St. Louis, France, 1980), pp. XVI.1–VI.107.

W. D. Bachalo, M. J. Houser, “An instrument for two-components velocity and particle size measurement,” in Proceedings of the Third International Symposium on Applications of Laser Anemometry to Fluid Mechanics, D. F. G. Durão, ed. (Lisbon University, Lisbon, 1986), pp. 18.3.1–18.3.67.

G. Gouesbet, G. Gréhan, “The quasi-elastic scattering of light: A lecture with emphasis on particulate diagnosis,” in Proceedings of the NATO Workshop on Soot in Combustion Systems and Its Toxic Properties, J. Lahaye, G. Prado, eds. (Plenum, New York, 1981), pp. 395–412.

G. Gouesbet, “Optical sizing with emphasis on simultaneous measurements of velocities and sizes of particles embedded in flows,” presented at the XVth International Symposium on Heat and Mass Transfer, Dubrovnik, Yugoslavia, September 5–9, 1983 [in R. I. Soloukhin, N. H. Afgan, eds., Measurement Techniques for Heat Mass Transfer (Springer-Verlag, Berlin, 1983), pp. 27–39].

G. Gréhan, “Nouveaux progrès en théorie de Lorenz–Mie. Application la mesure de diamètres de particules dans des écoulements,” thèse de troisième cycle (Rouen University, Rouen, France, 1980).

G. Gouesbet, G. Gréhan, eds., Proceedings of the International Symposium on Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988).
[CrossRef]

G. Gouesbet, G. Gréhan, R. Kleine, “Simultaneous optical measurement of velocity and size of individual particles in flows,” in Proceedings of the Second International Symposium on Applications of Laser Anemometry to Fluid Mechanics, D. F. G. Durão, ed. (Lisbon University, Lisbon, 1984), pp. 18.2.1–18.2.7.

G. Gréhan, G. Gouesbet, R. Kleine, V. Renz, I. Wilhelmi, “Corrected laser beam techniques for simultaneous velocimetry and sizing of particles in flows and applications,” in Proceedings of the Third International Symposium on Application of Laser Anemometry to Fluid Mechanics, D. F. G. Durão, ed. (Lisbon University, Lisbon, 1986), pp. 20.5.1–20.5.57.

G. Gouesbet, P. Gougeon, G. Gréhan, J. N. Letoulouzan, N. Lhuissier, B. Maheu, M. E. Weill, “Laser optical sizing from 100 Å to 1 mm diameter, and from 0 to 10 kg/m3concentration,” presented at the American Institute of Aeronautics and Astronautics 20th Thermophysics Conference, Williamsburg, Va., June 19–21, 1985.

P. Poincelot, Précis d’Électromagnétisme Théorique (Masson, Paris, 1963).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

L. Robin, Fonctions Sphériques de Legendre et Fonctions Sphérdidales (Gauthier-Villars, Paris, 1957–1959), Vols. 1–3.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (AcademicNew York, 1969).

B. Maheu, G. Gréhan, G. Gouesbet, “Laser beam scattering by individual particles: numerical results and applications to optical sizing,” presented at the Symposium on Optical Particle Sizing: Theory and Practice, Mt. St. Aignan, France, May 12–15, 1987 [in G. Gouesbet, G. Gréhan, eds., Optical Particle Sizing: Theory and Practice (Plenum, New York, 1988), pp. 77–88].
[CrossRef]

G. Gréhan, B. Maheu, G. Gouesbet, “Localized approximation to the generalized Lorenz–Mie theory and its application to optical sizing,” presented at the International Congress on Applications of Lasers and Electro-Optics, Arlington, Va., November 10–13, 1986.

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Figures (1)

Fig. 1
Fig. 1

Coordinate systems for the beam description (beam center OG) and for the scattered fields (scatterer center OP; scattered field computed at point P).

Equations (234)

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2 U r 2 + k 2 U + 1 r 2 sin θ θ ( sin θ U θ ) + 1 r 2 sin 2 θ 2 U φ 2 = 0 ,
k = ω ( μ ) 1 / 2 = M ω c ,
U = r Ψ n 1 ( k r ) P n m ( cos θ ) { sin cos } ( m φ )
Ψ n 1 ( k r ) = ( π 2 k r ) 1 / 2 J n + ( 1 / 2 ) ( k r ) ,
P n m ( cos θ ) = ( 1 ) m ( sin θ ) m d m P n ( cos θ ) [ d ( cos θ ) ] m ,
Ψ n ( k r ) = k r Ψ n 1 ( k r ) ,
ξ n ( k r ) = Ψ n ( k r ) + i ( 1 ) n ( π k r 2 ) 1 / 2 J n ( 1 / 2 ) ( k r ) ,
U = ξ n ( k r ) P n m ( cos θ ) { sin cos } ( m φ )
H r , TM = E r , TE = 0.
E r , TM = 2 U TM r 2 + k 2 U TM ,
E θ , TM = 1 r 2 U TM r θ ,
E φ , TM = 1 r sin θ 2 U TM r φ ,
H θ , TM = i ω r sin θ U TM φ ,
H φ , TM = i ω r U TM θ ,
E θ , TE = i ω μ r sin θ U TE φ ,
E φ , TE = i ω μ r U TE θ ,
H r , TE = 2 U TE r 2 + k 2 U TE ,
H θ , TE = 1 r 2 U TE r θ ,
H φ , TE = 1 r sin θ 2 U TE r φ .
s = w 0 / l = 1 / ( k w 0 ) ,
E υ = H u = 0 ,
E u = E 0 Ψ 0 exp ( i k w ) ,
E w = 2 Q u l E u ,
H υ = H 0 Ψ 0 exp ( i k w ) ,
H w = 2 Q υ l H υ ,
E 0 / H 0 = ( μ / ) 1 / 2 .
Ψ 0 = i Q exp ( i Q h + 2 ) ,
h + 2 = u + 2 + υ + 2 ,
Q = 1 i + 2 w + .
u + = u / w 0 , υ + = υ / w 0 ,
w + = w / l .
E y = H x = 0 ,
E x = E 0 Ψ 0 exp [ i k ( z z 0 ) ] ,
E z = 2 Q l ( x x 0 ) E x ,
H y = H 0 Ψ 0 exp [ i k ( z z 0 ) ] ,
H z = 2 Q l ( y y 0 ) H y ,
h + 2 = 1 w 0 2 [ ( x x 0 ) 2 + ( y y 0 ) 2 ] ,
Q = 1 i + 2 ( ζ ζ 0 ) ,
ζ = z l , ζ 0 = z 0 l .
E r = E 0 Ψ 0 [ cos φ sin θ ( 1 2 Q l r cos θ ) + 2 Q l x 0 cos θ ] exp ( K ) ,
E θ = E 0 Ψ 0 [ cos φ ( cos θ + 2 Q l r sin 2 θ ) 2 Q l x 0 sin θ ] exp ( K ) ,
E φ = E 0 Ψ 0 sin φ exp ( K ) ,
H r = H 0 Ψ 0 [ sin φ sin θ ( 1 2 Q l r cos θ ) + 2 Q l y 0 cos θ ] exp ( K ) ,
H θ = H 0 Ψ 0 [ sin φ ( cos θ + 2 Q l r sin 2 θ ) 2 Q l y 0 sin θ ] exp ( K ) ,
H φ = H 0 Ψ 0 cos φ exp ( K ) ,
K = i k ( r cos θ z 0 ) .
Ψ 0 = Ψ 0 0 Ψ 0 φ ,
Ψ 0 0 = i Q exp ( i Q r 2 sin 2 θ w 0 2 ) exp ( i Q x 0 2 + y 0 2 w 0 2 ) ,
Ψ 0 φ = exp [ 2 i Q w 0 2 r sin θ ( x 0 cos φ + y 0 sin φ ) ] ,
Ψ 0 φ = j p Ψ j p exp [ i φ ( j 2 p ) ] ,
j p = j = 0 p = 0 j ,
Ψ j p = ( i Q r sin θ w 0 2 ) j ( x 0 i y 0 ) j p ( x 0 + i y 0 ) p ( j p ) ! ( p ) ! .
Ψ 00 = 1.
E r = E 0 F 2 [ j p Ψ j p exp ( i j + φ ) + j p Ψ j p exp ( i j φ ) ] + E 0 x 0 G j p Ψ j p exp ( i j 0 φ ) ,
H r = H 0 F 2 i [ j p Ψ j p exp ( i j + φ ) j p Ψ j p exp ( i j φ ) ] + H 0 y 0 G j p Ψ j p exp ( i j 0 φ ) ,
F = Ψ 0 0 sin θ ( 1 2 Q l r cos θ ) exp ( K ) ,
G = Ψ 0 0 2 Q l cos θ exp ( K ) ,
j + = j + 1 2 p = j 0 + 1 ,
j = j 1 2 p = j 0 1.
E r = E r , TM = 2 U TM i r 2 + k 2 U TM i ,
U TM i = E 0 n = 1 m = n + n C n pw g n , TM m r Ψ n 1 ( k r ) P n | m | ( cos θ ) exp ( i m φ ) ,
C n pw = 1 k i n 1 ( 1 ) n 2 n + 1 n ( n + 1 ) .
( d 2 d r 2 + k 2 ) [ r Ψ n 1 ( k r ) ] = n ( n + 1 ) r Ψ n 1 ( k r ) .
0 2 π exp [ i ( m m ) φ ] d φ = 2 π δ m m ,
0 π P n m ( cos θ ) P l m ( cos θ ) sin θ d θ = 2 2 n + 1 ( n + m ) ! ( n m ) ! δ n l .
0 Ψ n 1 ( k r ) Ψ m 1 ( k r ) d ( k r ) = π 2 ( 2 n + 1 ) δ n m .
g n , TM m = 1 C n pw ( 2 n + 1 ) 2 π n ( n + 1 ) ( n | m | ) ! ( n + | m | ) ! × 0 π 0 [ F 2 ( j + = m j p Ψ j p + j = m j p Ψ j p ) + x 0 G j 0 = m j p Ψ j p ] × r Ψ n 1 ( k r ) P n | m | ( cos θ ) sin θ d θ d ( k r ) .
U TM i = E 0 k n = 1 m = n + n C n pw g n , TM m Ψ n ( k r ) P n | m | ( cos θ ) exp ( i m φ ) .
U TE i = H 0 k n = 1 m = n + n C n pw g n , TE m Ψ n ( k r ) P n | m | ( cos θ ) exp ( i m φ ) ,
g n , TE m = 1 C n pw ( 2 n + 1 ) 2 π n ( n + 1 ) ( n | m | ) ! ( n + | m | ) ! × 0 π 0 [ F 2 ( i j + = m j p Ψ j p + i j = m j p Ψ j p ) ] + y 0 G j 0 = m j p Ψ j p ] r Ψ n 1 ( k r ) P n | m | ( cos θ ) sin θ d θ d ( k r ) .
U TM s = E 0 k n = 1 m = n + n C n pw A n m ξ n ( k r ) P n | m | ( cos θ ) exp ( i m φ ) ,
U TE s = H 0 k n = 1 m = n + n C n pw B n m ξ n ( k r ) P n | m | ( cos θ ) exp ( i m φ ) ,
U TM sp = k E 0 k sp 2 n = 1 m = n + n C n pw E n m Ψ n ( k sp r ) P n | m | ( cos θ ) exp ( i m φ ) ,
U TE sp = k H 0 k sp 2 n = 1 m = n + n C n pw D n m Ψ n ( k sp r ) P n | m | ( cos θ ) exp ( i m φ ) .
V θ , X i + V θ , X s = V θ , X sp ,
M [ g n , TM m Ψ n ( α ) A n m ξ n ( α ) ] = E n m Ψ n ( β ) ,
M 2 [ g n , TE m Ψ n ( α ) B n m ξ n ( α ) ] = D n m Ψ n ( β ) ,
[ g n , TM m Ψ n ( α ) A n m ξ n ( α ) ] = E n m Ψ n ( β ) ,
M [ g n , TE m Ψ n ( α ) B n m ξ n ( α ) ] = D n m Ψ n ( β ) ,
M = k sp k = ( sp ) 1 / 2 .
A n m = a n g n , TM m ,
B n m = b n g n , TE m ,
a n = Ψ n ( α ) Ψ n ( β ) M Ψ n ( α ) Ψ n ( β ) ξ n ( α ) Ψ n ( β ) M ξ n ( α ) Ψ n ( β ) ,
b n = M Ψ n ( α ) Ψ n ( β ) Ψ n ( α ) Ψ n ( β ) M ξ n ( α ) Ψ n ( β ) ξ n ( α ) Ψ n ( β ) .
E r = k E 0 n = 1 m = n + n C n pw a n g n , TM m [ ξ n ( k r ) + ξ n ( k r ) ] × P n | m | ( cos θ ) exp ( i m φ ) ,
E θ = E 0 r n = 1 m = n + n C n pw [ a n g n , TM m ξ n ( k r ) τ n | m | ( cos θ ) + m b n g n , TE m ξ n ( k r ) Π n | m | ( cos θ ) ] exp ( i m φ ) ,
E φ = i E 0 r n = 1 m = n + n C n pw [ m a n g n , TM m ξ n ( k r ) Π n | m | ( cos θ ) + b n g n , TE m ξ n ( k r ) τ n | m | ( cos θ ) ] exp ( i m φ ) ,
H r = k H 0 n = 1 m = n + n C n pw b n g n , TE m [ ξ n ( k r ) + ξ n ( k r ) ] × P n | m | ( cos θ ) exp ( i m φ ) ,
H θ = H 0 r n = 1 m = n + n C n pw [ m a n g n , TM m ξ n ( k r ) Π n | m | ( cos θ ) b n g n , TE m ξ n ( k r ) τ n | m | ( cos θ ) ] exp ( i m φ ) ,
H φ = i H 0 r n = 1 m = n + n C n pw [ a n g n , TM m ξ n ( k r ) τ n | m | ( cos θ ) m b n g n , TE m ξ n ( k r ) Π n | m | ( cos θ ) ] exp ( i m φ ) ,
k = ω μ H 0 E 0 = ω E 0 H 0 .
τ n k ( cos θ ) = d d θ P n k ( cos θ ) ,
Π n k ( cos θ ) = P n k ( cos θ ) sin θ .
ξ n ( k r ) i n + 1 exp ( i k r ) .
ξ n ( k r ) + ξ n ( k r ) = 0 ,
E r = H r = 0.
E θ = i E 0 k r exp ( i k r ) n = 1 m = n + n 2 n + 1 n ( n + 1 ) [ a n g n , TM m τ n | m | ( cos θ ) + i m b n g n , TE m Π n | m | ( cos θ ) ] exp ( i m φ ) ,
E φ = E 0 k r exp ( i k r ) n = 1 m = n + n 2 n + 1 n ( n + 1 ) [ m a n g n , TM m × Π n | m | ( cos θ ) + i b n g n , TE m τ n | m | ( cos θ ) ] exp ( i m φ ) ,
H φ = H 0 E 0 E θ ,
H θ = H 0 E 0 E φ .
S + = 1 2 Re ( E θ H φ * E φ H θ * ) ,
1 2 ( μ ) 1 / 2 E 0 2 = 1.
( I θ + I φ + ) = λ 2 4 π 2 r 2 ( | S 2 | 2 | S 1 | 2 ) ,
E θ = i E 0 k r exp ( i k r ) S 2 ,
E φ = E 0 k r exp ( i k r ) S 1 .
E θ = i S 2 = A θ exp ( i φ 2 ) ,
E φ = S 1 = A φ exp ( i φ 1 ) ,
tan δ = tan ( φ 2 φ 1 ) = Re ( S 1 ) Re ( S 2 ) + Im ( S 1 ) Im ( S 2 ) Im ( S 1 ) Re ( S 2 ) Re ( S 1 ) Im ( S 2 ) .
E j t = E j i + E j s ,
H j t = H j i + H j s .
S + = 1 2 Re ( E θ t H φ t * E φ t H θ t * ) .
C abs = + ( S ) S + d S ,
T i = 0 π 0 2 π 1 2 Re ( E θ i H φ i * E φ i H θ i * ) r 2 sin θ d θ d φ ,
T s = 0 π 0 2 π 1 2 Re ( E θ s H φ s * E φ s H θ s * ) r 2 sin θ d θ d φ ,
T is = 0 π 0 2 π 1 2 Re ( E θ i H φ s * + E θ s H φ i * E φ i H θ s * E φ s H θ i * ) × r 2 sin θ d θ d φ .
T is = + ( S ) S + d S T i T s = C abs C sca = C ext .
E θ i = E 0 r n = 1 m = n + n C n pw ( g n , TM m Ψ n τ n | m | + m g n , TE m Ψ n Π n | m | ) × exp ( i m φ ) ,
E φ i = E 0 r n = 1 m = n + n C n pw ( i m g n , TM m Ψ n Π n | m | + i g n , TE m Ψ n τ n | m | ) × exp ( i m φ ) ,
H θ i = H 0 r n = 1 m = n + n C n pw ( m g n , TM m Ψ n Π n | m | + g n , TE m Ψ n τ n | m | ) × exp ( i m φ ) ,
H φ i = H 0 r n = 1 m = n + n C n pw ( i g n , TM m Ψ n τ n | m | + i m g n , TE m Ψ n Π n | m | ) × exp ( i m φ ) ,
T i = 2 π Re { i p = + n = | p | 0 m = | p | 0 C n pw C n p w * × [ I 1 ( g n , TM p g m , TM p * Ψ n Ψ m g n , TE p g m , TE p * Ψ m Ψ n ) + p I 2 ( g n , TE p g m , TM p * Ψ n Ψ m g n , TM p g m , TE p * Ψ m Ψ n ) ] } ,
I 1 = 0 π ( τ n | p | τ m | p | + p 2 Π n | p | Π m | p | ) sin θ d θ ,
I 1 = 2 n ( n + 1 ) 2 n + 1 ( n + | p | ) ! ( n | p | ) ! δ n m
I 2 = 0 π ( Π n | p | τ m | p | + Π m | p | τ n | p | ) sin θ d θ = 0 if p 0.
T i = Im [ p = + n = | p | 0 A n p | C n pw | 2 ( | g n , TM p | 2 | g n , TE p | 2 Ψ n Ψ n ) ] = 0.
C sca = 0 π 0 2 π ( I θ + + I φ + ) r 2 sin θ d θ d φ ,
C sca = λ 2 π n = 1 m = n + n 2 n + 1 n ( n + 1 ) ( n + | m | ) ! ( n | m | ) ! × ( | a n | 2 | g n , TM m | 2 + | b n | 2 | g n , TE m | 2 ) .
C ext = 0 π 0 2 π 1 2 Re ( E φ i H θ s * + E φ s H θ i * E θ i H φ s * E θ s H φ i * ) × r 2 sin θ d θ d φ .
C ext = 2 π Re ( p = + n = | p | 0 m = | p | 0 C n pw C m p w * × { I 1 [ ( i ) m exp ( i k r ) ( a m * g n , TM p g m , TM p * Ψ n i b m * g n , TE p g m , TE p * Ψ n ) + i n exp ( i k r ) ( i a n g n , TM p g m , TM p * Ψ m + b n g n , TE p g m , TE p * Ψ m ) ] + p I 2 [ ( i ) m exp ( i k r ) ( a m * g n , TE p g m , TM p * Ψ n i b m * g n , TM p g m , TE p * Ψ n ) i n exp ( i k r ) ( i a n Ψ m g n , TM p g m , TE p * + b n Ψ m g n , TE p g m , TM p * ) ] } ) .
C ext = 4 π k 2 Re { n = 1 m = n + n 2 n + 1 n ( n + 1 ) ( n + | m | ) ! ( n | m | ) ! [ ( i ) n exp ( i k r ) ( a n * Ψ n | g n , TM m | 2 i b n * Ψ n | g n , TE m | 2 ) + i n exp ( i k r ) ( i a n Ψ n | g n , TM m | 2 + b n Ψ n | g n , TE m | 2 ) ] } .
Ψ n ( k r ) 1 2 [ ( i ) n + 1 exp ( i k r ) + i n + 1 exp ( i k r ) ] .
C ext = λ 2 π Re [ n = 1 m = n + n 2 n + 1 n ( n + 1 ) ( n + | m | ) ! ( n | m | ) ! × ( a n | g n , TM m | 2 + b n | g n , TE m | 2 ) ] .
C pr , z = c F z + = ( cos θ ¯ ) C ext ( cos θ ¯ ) C sca ,
( cos θ ¯ ) C sca = 0 π 0 2 π ( I θ + + I φ + ) r 2 sin θ cos θ d θ d φ .
( cos θ ¯ ) C sca = 2 π k 2 p = + n = | p | 0 m = | p | 0 2 n + 1 n ( n + 1 ) 2 m + 1 m ( m + 1 ) × [ I 3 ( a n a m * g n , TM p g m , TM p * + b n b m * g n , TE p g m , TE p * ) + i p I 4 ( b n a m * g n , TE p g m , TM p * a n b m * g n , TM p g m , TE p * ) ] ,
I 3 = 0 π ( τ n | p | τ m | p | + p 2 Π n | p | Π m | p | ) cos θ sin θ d θ = 2 ( n 1 ) ( n + 1 ) ( n + | p | ) ! ( 2 n 1 ) ( 2 n + 1 ) ( n 1 | p | ) ! δ m , n 1 + 2 ( m 1 ) ( m + 1 ) ( m + | p | ) ! ( 2 m 1 ) ( 2 m + 1 ) ( m 1 | p | ) ! δ n , m 1 ,
I 4 = 0 π ( τ n | p | Π m | p | + τ m | p | Π n | p | ) cos θ sin θ d θ = 2 ( n + | p | ) ! 2 n + 1 ( n | p | ) ! δ n m .
( cos θ ¯ ) C sca = 2 λ 2 π n = 1 p = n + n p 2 n + 1 n 2 ( n + 1 ) 2 ( n + | p | ) ! ( n | p | ) ! × Re ( i a n b n * g n , TM p g n , TE p * ) 1 ( n + 1 ) 2 × ( n + 1 + | p | ) ! ( n | p | ) ! Re ( a n a n + 1 * g n , TM p g n + 1 , TM p * + b n b n + 1 * g n , TE p g n + 1 , TE p * ) .
( cos θ ¯ ) C ext = 0 π 0 2 π 1 2 Re ( E φ i H θ s * + E φ s H θ i * E θ i H φ s * E θ s H φ i * ) × r 2 sin θ cos θ d θ d φ .
( cos θ ¯ ) C ext = 2 π Re { p = + n = | p | 0 2 n ( n + 2 ) ( 2 n + 1 ) ( 2 n + 3 ) × ( n + 1 + | p | ) ! ( n | p | ) ! C n pw C n + 1 p w * [ ( i ) n + 1 exp ( i k r ) ( Ψ n i Ψ n ) × ( a n + 1 * g n , TM p g n + 1 , TM p * + b n + 1 * g n , TE p g n + 1 , TE p * ) + i n exp ( i k r ) × ( Ψ n + 1 + i Ψ n + 1 ) ( a n g n , TM p g n + 1 , TM p * + b n g n , TE p g n + 1 , TE p * ) ] + p = + n = | p | 0 2 p 2 n + 1 ( n + | p | ) ! ( n | p | ) ! | C n pw | 2 i n exp ( i k r ) × ( Ψ n i Ψ n ) ( a n g n , TM p g n , TE p * b n g n , TE p g n , TM p * ) } ;
( cos θ ¯ ) C ext = λ 2 π n = 1 p = n + n { 1 ( n + 1 ) 2 ( n + 1 + | p | ) ! ( n | p | ) ! × Re [ ( a n + a n + 1 * ) g n , TM p g n + 1 , TM p * + ( b n + b n + 1 * ) g n , TE p g n + 1 , TE p * ] p 2 n + 1 n 2 ( n + 1 ) 2 ( n + | p | ) ! ( n | p | ) ! Re [ i ( a n + b n * ) ] g n , TM p g n , TE p * ] } .
C pr , z = λ 2 π n = 1 p = n + n { 1 ( n + 1 ) 2 ( n + 1 + | p | ) ! ( n | p | ) ! × Re [ ( a n + a n + 1 * 2 a n a n + 1 * ) g n , TM p g n + 1 , TM p * + ( b n + b n + 1 * 2 b n b n + 1 * ) g n , TE p g n + 1 , TE p * ] + p 2 n + 1 n 2 ( n + 1 ) 2 ( n + | p | ) ! ( n | p | ) ! × Re [ i ( 2 a n b n * a n b n * ) g n , TM p g n , TE p * ) ] } .
C pr , x = c F x + = ( sin θ cos φ ¯ ) C ext ( sin θ cos φ ¯ ) C sca .
( sin θ cos φ ¯ ) C sca = 0 π 0 2 π ( I θ + + I φ + ) r 2 sin 2 θ cos φ d θ d φ .
0 2 π cos φ exp ( i k φ ) exp ( i k φ ) d φ = π ( δ k , k + 1 + δ k , k + 1 ) ,
( sin θ cos φ ¯ ) C sca = λ 2 4 π p = + n = | p | 0 m = | p + 1 | 0 2 n + 1 n ( n + 1 ) × 2 m + 1 m ( m + 1 ) [ I 5 Re ( U n m p ) + I 6 Re ( V n m p ) ] ,
U n m p = a n a m * g n , TM p g m , TM p + 1 * + b n b m * g n , TE p g m , TE p + 1 * ,
V n m p = i b n a m * g n , TE p g m , TM p + 1 * i a n b m * g n , TM p g m , TE p + 1 * ,
I 5 = 0 π ( τ n | p | τ m | p + 1 | + p ( p + 1 ) Π n | p | Π m | p + 1 | ) sin 2 θ d θ = { 2 ( 2 n + 1 ) ( 2 m + 1 ) ( m + p + 1 ) ! ( m p 1 ) ! [ ( n 1 ) ( n + 1 ) δ n , m + 1 ( m 1 ) ( m + 1 ) δ m , n + 1 ] , p 0 2 ( 2 n + 1 ) ( 2 m + 1 ) ( n p ) ! ( n + p ) ! [ ( m 1 ) ( m + 1 ) δ m , n + 1 ( n 1 ) ( n + 1 ) δ n , m + 1 ] , p < 0 ,
I 6 = 0 π ( p Π n | p | τ m | p + 1 | + ( p + 1 ) Π m | p + 1 | τ n | p | ) sin 2 θ d θ = { 2 2 n + 1 ( n + p + 1 ) ! ( n p 1 ) ! δ n m , p 0 2 2 n + 1 ( n p ) ! ( n + p ) ! δ n m , p < 0 .
( sin θ cos φ ¯ ) C sca = λ 2 π { p = 0 n = | p | 0 m = | p + 1 | 0 ( m + p + 1 ) ! ( m p 1 ) ! × [ Re ( U n m p ) ( 1 n 2 δ n , m + 1 1 m 2 δ m , n + 1 ) + Re ( V n m p ) 2 n + 1 n 2 ( n + 1 ) 2 δ n m ] + p = 1 n = | p | 0 m = | p + 1 | 0 ( n p ) ! ( n + p ) ! × [ Re ( U n m p ) ( 1 m 2 δ m , n + 1 1 n 2 δ n , m + 1 ) Re ( V n m p ) 2 n + 1 n 2 ( n + 1 ) 2 δ n m ] } ,
( sin θ cos φ ¯ ) C sca = λ 2 π p = 1 n = p m = p 1 0 ( n + p ) ! ( n p ) ! × [ Re ( U m n p 1 + U n m p ) ( 1 m 2 δ m , n + 1 1 n 2 δ n , m + 1 ) + 2 n + 1 n 2 ( n + 1 ) 2 δ n m Re ( V m n p 1 + V n m p ) ] .
( sin θ cos φ ¯ ) C ext = 0 π 0 2 π 1 2 Re ( E φ i H θ s * + E φ s H θ i * E θ i H φ s * E θ s H φ i * ) r 2 sin 2 θ cos φ d θ d φ .
( sin θ cos φ ¯ ) C ext = λ 2 4 π Re p = + n = | p | 0 m = | p + 1 | 0 2 n + 1 n ( n + 1 ) × 2 m + 1 m ( m + 1 ) { I 5 [ i n exp ( i k r ) ( Ψ n + i Ψ n ) ( a m g m , TM p + 1 g n , TM p * + b m g m , TE p + 1 g n , TE p * ) + i m exp ( i k r ) ( Ψ m + i Ψ m ) ( a n g n , TM p g m , TM p + 1 * + b n g n , TE p g m , TE p + 1 * ) ] + I 6 [ i m 1 exp ( i k r ) ( Ψ m + i Ψ m ) × ( a n g n , TM p g m , TE p + 1 * b n g n , TE p g m , TM p + 1 * ) + i n 1 exp ( i k r ) × ( Ψ n + i Ψ n ) ( a m g m , TM p + 1 g n , TE p * b m g m , TE p + 1 g n , TM p * ) ] } .
S n m p = ( a n + a m * ) g n , TM p g m , TM p + 1 * + ( b n + b m * ) g n , TE p g m , TE p + 1 * ,
T n m p = i ( a n + b m * ) g n , TM p g m , TE p + 1 * + i ( b n + a m * ) g n , TE p g m , TM p + 1 * .
( sin θ cos φ ¯ ) C ext = λ 2 2 π { p = 0 n = | p | 0 m = | p + 1 | 0 ( m + p + 1 ) ! ( m p 1 ) ! × [ Re ( S n m p ) ( 1 n 2 δ n , m + 1 1 m 2 δ m , n + 1 ) + Re ( T n m p ) 2 m + 1 m 2 ( m + 1 ) 2 δ n m ] + p = 1 n = | p | 0 m = | p + 1 | 0 ( n p ) ! ( n + p ) ! × [ Re ( S n m p ) ( 1 m 2 δ m , n + 1 1 n 2 δ n , m + 1 ) Re ( T n m p ) 2 n + 1 n 2 ( n + 1 ) 2 δ n m ] } ,
( sin θ cos φ ¯ ) C ext = λ 2 2 π p = 1 n = p m = p 1 0 ( n + p ) ! ( n p ) ! × [ Re ( S m n p 1 + S n m p ) ( 1 m 2 δ m , n + 1 1 n 2 δ n , m + 1 ) + Re ( T m n p 1 T n m p ) 2 n + 1 n 2 ( n + 1 ) 2 δ n m ] .
C pr , x = c F x + = λ 2 2 π p = 1 n = p m = p 1 0 ( n + p ) ! ( n p ) ! × [ Re ( S m n p 1 + S n m p 2 U m n p 1 2 U n m p ) × ( 1 m 2 δ m , n + 1 1 n 2 δ n , m + 1 ) + 2 n + 1 n 2 ( n + 1 ) 2 δ n m × Re ( T m n p 1 T n m p 2 V m n p 1 + 2 V n m p ) ] .
C pr , y = c F y + = ( sin θ sin φ ¯ ) C ext ( sin θ sin φ ¯ ) C sca .
( sin θ sin φ ¯ ) C sca = 0 π 0 2 π ( I θ + + I φ + ) r 2 sin 2 θ sin φ d θ d φ .
0 2 π sin φ exp ( i k φ ) exp ( i k φ ) d φ = i π ( δ k , k + 1 δ k , k + 1 ) .
C pr , y = c F y + = λ 2 2 π p = 1 n = p m = p 1 0 ( n + p ) ! ( n p ) ! × [ Im ( S m n p 1 + S n m p 2 U m n p 1 2 U n m p ) × ( 1 m 2 δ m , n + 1 1 n 2 δ n , m + 1 ) + 2 n + 1 n 2 ( n + 1 ) 2 δ n m × Im ( T m n p 1 T n m p 2 V m n p 1 + 2 V n m p ) ] .
S 1 = n = 1 2 n + 1 n ( n + 1 ) g n [ a n Π n ( cos θ ) + b n τ n ( cos θ ) ] ,
S 2 = n = 1 2 n + 1 n ( n + 1 ) g n [ a n τ n ( cos θ ) + b n Π n ( cos θ ) ] ,
g n = k ( 2 n + 1 ) i n 1 ( 1 ) n π n ( n + 1 ) 0 π 0 F r Ψ n 1 ( k r ) P n 1 ( cos θ ) × sin θ d θ d ( k r ) .
S 2 = ( cos φ ) S 2 + S 2 ,
S 1 = i ( sin φ ) S 1 + S 1 ,
S 2 = n = 1 m = n | m | 1 + n 2 n + 1 n ( n + 1 ) [ a n g n , TM m τ n | m | ( cos θ ) + i m b n g n , TE m Π n | m | ( cos θ ) ] exp ( i m φ ) + n = 1 2 n + 1 n ( n + 1 ) × { a n τ n ( cos θ ) [ ( cos φ ) G n , TM + + i ( sin φ ) G n , TM ] + b n Π n ( cos θ ) [ i ( cos φ ) G n , TE ( sin φ ) G n , TE + ] } ,
S 1 = n = 1 m = n | m | 1 + n 2 n + 1 n ( n + 1 ) [ ma n g n , TM m Π n | m | ( cos θ ) + i b n g n , TE m τ n | m | ( cos θ ) ] exp ( i m φ ) + n = 1 2 n + 1 n ( n + 1 ) × { a n Π n ( cos θ ) [ ( cos φ ) G n , TM + i ( sin φ ) G n , TM + ] + b n τ n ( cos θ ) [ i ( cos φ ) G n , TE + ( sin φ ) G n , TE ] } ,
G n , TM + = g n , TM 1 + g n , TM 1 g n ,
G n , TE + = g n , TE 1 + g n , TE 1 ,
G n , TM = g n , TM 1 g n , TM 1 ,
G n , TE = g n , TE 1 g n , TE 1 + i g n .
( I θ + I φ + ) = ( I θ + L I φ + L ) + ( I θ + C I φ + C ) + ( I θ + S I φ + S ) ,
( I θ + L I φ + L ) = λ 2 4 π 2 r 2 ( i 2 cos 2 φ i 1 sin 2 φ ) ,
( I θ + C I φ + C ) = 2 λ 2 4 π 2 r 2 [ cos φ Re ( S 2 S 2 * ) sin φ Re ( i S 1 S 1 * ) ] ,
( I θ + S I φ + S ) = λ 2 4 π 2 r 2 ( | S 2 | 2 | S 1 | 2 ) .
tan δ 0 = Re ( S 1 ) Im ( S 2 ) Re ( S 2 ) Im ( S 1 ) Re ( S 1 ) Re ( S 2 ) + Im ( S 1 ) Im ( S 2 ) ,
tan δ = tan ( δ 0 + δ 1 ) + tan δ 2 ,
Ψ j p ( x 0 = y 0 = 0 ) = δ j 0 .
g n , TM 1 = g n , TM 1 = 1 2 g n ,
g n , TE 1 = g n , TE 1 = i 2 g n ,
G n , TM + = G n , TE + = G n , TM = G n , TE = 0.
S 2 = S 1 = 0.
[ I θ + ( x 0 = y 0 = 0 ) I φ + ( x 0 = y 0 = 0 ) ] = λ 2 4 π 2 r 2 [ i 2 ( x 0 = y 0 = 0 ) cos 2 φ i 1 ( x 0 = y 0 = 0 ) sin 2 φ ] .
δ 1 = δ 2 = 0 ,
tan δ = tan δ 0 .
C sca = C sca L = λ 2 2 π n = 1 ( 2 n + 1 ) | g n | 2 [ | a n | 2 + | b n | 2 ] .
C ext = C ext L = λ 2 2 π Re n = 1 ( 2 n + 1 ) | g n | 2 ( a n + b n ) .
F x + = F x L + = F y + = F y L + = 0
C pr , z = ( cos θ ¯ ) C ext L ( cos θ ¯ ) C sca L ,
C pr , z = λ 2 2 π n = 1 2 n + 1 n ( n + 1 ) | g n | 2 Re ( a n + b n 2 a n b n * ) + n ( n + 2 ) n + 1 Re [ g n g n + 1 * ( a n + b n + a n + 1 * + b n + 1 * 2 a n a n + 1 * 2 b n b n + 1 * ) ] .
g n = 2 n + 1 π n ( n + 1 ) 1 ( 1 ) n i n 1 0 π 0 i Q exp ( i Q r 2 sin 2 θ w 0 2 ) × exp ( i k z 0 ) exp ( i k r cos θ ) ( 1 2 Q l r cos θ ) × Ψ n ( k r ) P n 1 ( cos θ ) sin 2 θ d θ d ( k r ) .
g n = 1 i n 1 ( 1 ) n π 2 n + 1 n ( n + 1 ) 0 π 0 ( sin 2 θ ) f exp ( i k r cos θ ) × Ψ n ( k r ) P n 1 ( cos θ ) d θ d ( k r ) ,
f = i Q exp ( i Q r 2 sin 2 θ w 0 2 ) ( 1 2 Q l r cos θ ) ,
f = i Q exp ( i Q r 2 sin 2 θ w 0 2 ) ,
Ψ 00 = 1 ,
g n , TM 1 = g n , TM 1 = 1 2 ,
g n , TE 1 = g n , TE 1 = i 2 .
I 1 = 0 π ( τ n k τ m k + k 2 Π n k Π m k ) sin θ d θ ,
I 2 = 0 π ( τ n k Π m k + Π n k τ m k ) sin θ d θ .
I 2 = 0 π ( P m k d P n k + P n k d P m k ) .
I 2 = P m k ( 1 ) P n k ( 1 ) P m k ( 1 ) P n k ( 1 ) .
P n k ( ± 1 ) = 0 ,
I 2 = 0.
I 1 = I 1 1 + I 1 2 = 0 π ( d P n k d θ d P m k d θ ) sin θ d θ + 0 π k 2 P n k sin θ P m k sin θ sin θ d θ .
d d ( cos θ ) sin 2 θ d P n k ( cos θ ) d ( cos θ ) + [ n ( n + 1 ) k 2 sin 2 θ ] P n k ( cos θ ) = 0 ,
I 1 1 = m ( m + 1 ) 0 π P n k ( cos θ ) P m k ( cos θ ) sin θ d θ k 2 0 π P n k ( cos θ ) sin θ P m k ( cos θ ) sin θ sin θ d θ ,
I 1 = m ( m + 1 ) 0 π P n k ( cos θ ) P m k ( cos θ ) sin θ d θ .
0 π P n k ( cos θ ) P m k ( cos θ ) sin θ d θ = 2 2 n + 1 ( n + k ) ! ( n k ) ! δ n m .
I 1 = 2 m ( m + 1 ) ( m + k ) ! ( 2 m + 1 ) ( m k ) ! δ n m .
I 3 = 0 π ( τ n k τ m k + k 2 Π n k Π m k ) cos θ sin θ d θ ,
I 4 = 0 π ( τ n k Π n k + τ m k Π n k ) cos θ sin θ d θ .
I 4 = 0 π P n k d P m k d ( cos θ ) cos θ sin θ d θ 0 π P m k d P n k d ( cos θ ) cos θ sin θ d θ .
I 3 = 0 π k 2 P n k P m k sin θ cos θ d θ + 0 π d P n k d θ d P m k d θ cos θ sin θ d θ = I 3 1 + I 3 2 .
1 sin θ d d θ sin θ d P m k d θ + m ( m + 1 ) P m k = k 2 P m k sin 2 θ ,
I 3 = m ( m + 1 ) 0 π P n k P m k sin θ cos θ d θ 0 π P n k [ sin 2 θ d P m k d ( cos θ ) ] sin θ d θ .
sin 2 θ d P m k d ( cos θ ) = k ( cos θ ) P m k ( sin θ ) P m k + 1
( sin θ ) P m k + 1 = ( m k ) ( cos θ ) P m k ( m + k ) P m 1 k .
I 3 = m ( m + 2 ) 0 π P n k P m k cos θ sin θ d θ ( m + k ) × 0 π P n k P m 1 k sin θ d θ .
A = n + k 2 + n + m 2 2 m + 1 ( m + k ) ! ( m k ) ! δ m , n 1 + m + k 2 + n + m 2 2 n + 1 ( n + k ) ! ( n k ) ! δ n , m 1
I 3 = 2 ( n 1 ) ( n + 1 ) ( 2 n 1 ) ( 2 n + 1 ) ( n + k ) ! ( n 1 k ) ! δ m , n 1 + 2 ( m 1 ) ( m + 1 ) ( 2 m 1 ) ( 2 m + 1 ) ( m + k ) ! ( m 1 k ) ! δ n , m 1 .
I 5 = 0 π ( τ n | p | τ m | p + 1 | + p ( p + 1 ) Π n | p | Π m | p + 1 | ) sin 2 θ d θ ,
I 6 = 0 π ( p τ m | p + 1 | Π n | p | + ( p + 1 ) τ n | p | Π m | p + 1 | ) sin 2 θ d θ ,
I 6 ( p > 0 ) = 0 π [ p Π n p τ m p + 1 + ( p + 1 ) τ n p Π m p + 1 ] sin 2 θ d θ .
I 6 ( p > 0 ) = 0 π P n p [ sin θ d P m p + 1 d θ + ( p + 1 ) ( cos θ ) P m p + 1 ] d θ .
( 2 l + 1 ) sin θ d P l m d θ = l ( l m + 1 ) P l + 1 m ( l + 1 ) ( l + m ) P l 1 m ,
( 2 l + 1 ) ( cos θ ) P l m = ( l + m ) P l 1 m + ( l m + 1 ) P l + 1 m ,
I 6 ( p > 0 ) = ( p m ) ( m + p + 1 ) 2 m + 1 0 π P n p ( P m + 1 p + 1 P m 1 p + 1 ) d θ .
( 2 l + 1 ) ( sin θ ) P l m = P l 1 m + 1 P l + 1 m + 1 ,
I 6 ( p > 0 ) = ( m p ) ( m + p + 1 ) 0 π P n p P m p sin θ d θ .
I 5 ( p > 0 ) = 0 π [ τ n p τ m p + 1 + p ( p + 1 ) Π n p Π m p + 1 ] sin 2 θ d θ .
I 5 ( p > 0 ) = 0 π P n p { p ( p + 1 ) P m p + 1 + m ( m p ) ( m + 1 ) ( m + 2 ) ( 2 m + 1 ) ( 2 m + 3 ) × ( P m p + 1 P m + 2 p + 1 ) + m p ( m p ) ( 2 m + 1 ) ( 2 m + 3 ) × [ ( m + 1 ) P m p + 1 + ( m + 2 ) P m + 2 p + 1 ] + m ( m 1 ) ( m + 1 ) ( m + p + 1 ) ( 2 m 1 ) ( 2 m + 1 ) ( P m p + 1 P m 2 p + 1 ) p ( m + 1 ) ( m + p + 1 ) ( 2 m 1 ) ( 2 m + 1 ) × [ m P m p + 1 + ( m 1 ) P m 2 p + 1 ] } d θ .

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