Abstract

Three different methods can be used to numerically compute the gn coefficients in the generalized Lorenz-Mie theory. Two of them are rigorous and involve (i) numerical evaluation of quadratures and (ii) numerical evaluation of finite series. The third way relies on the so-called localized interpretation that we discussed in previous papers. These three methods are discussed and compared.

© 1988 Optical Society of America

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References

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  1. G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz-Mie,” J. Opt. Paris 13, 97 (1982).
    [CrossRef]
  2. G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian Beam by a Mie Scatter Center, Using a Bromwich Formalism,” J. Opt. Paris 16, 83 (1985).
    [CrossRef]
  3. 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. Paris 16, 239 (1985).
    [CrossRef]
  4. G. Gouesbet, B. Maheu, G. Gréhan, “Scattering of a Gaussian Beam by a Sphere Using a Bromwich Formulation: Case of an Arbitrary Location,” Part. Part. Syst. Charact. 5, 1 (1988); G. Gouesbet, G. Gréhan, Eds., Optical Particle Sizing—Theory and Practice (Plenum, New York, 1988), pp. 27–42.
    [CrossRef]
  5. G. Gouesbet, B. Maheu, G. Gréhan, “Light Scattering from a Sphere Arbitrarily Located in a Gaussian Beam, Using a Bromwich Formulation,” J. Opt. Soc. Am. A 5, 1427, (Sept.1988).
    [CrossRef]
  6. B. Maheu, G. Gouesbet, G. Gréhan, “A Concise Presentation of the Generalized Lorenz-Mie Theory for Arbitrary Location of the Scatterer in an Arbitrary Incident Profile,” J. Opt.19, in press, (1988).
    [CrossRef]
  7. G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of Laser Beams by Mie Scatter Centers: Numerical Results Using a Localized Approximation,” Appl. Opt. 25, 3539 (1986).
    [CrossRef] [PubMed]
  8. B. Maheu, G. Gréhan, G. Gouesbet, “Generalized Lorenz-Mie Theory: First Exact Values and Comparisons with the Localized Approximation,” Appl. Opt. 26, 23 (1987).
    [CrossRef] [PubMed]
  9. B. Maheu, G. Grehan, G. Gouesbet, “Laser Beam Scattering by Individual Spherical Particles: Numerical Results and Application to Optical Sizing,” Part. Part. Syst. Charact. 4, 141 (1987); G. Gouesbet, G. Grehan, Eds., Optical Particle Sizing—Theory and Practice (Plenum, New York, 1988), pp. 77–88.
    [CrossRef]
  10. G. Gréhan, B. Maheu, G. Gouesbet, “Localized Approximation to the Generalized Lorenz-Mie and Its Application to Optical Sizing,” in Proceedings, ICALEO’86, Arlington, VA (10–13 Nov. 1986).
  11. B. Maheu, G. Gréhan, G. Gouesbet, “Diffusion de la lumière par une sphère dans le cas d’un faisceau d’extension finie. lère partie: Théorie de Lorenz-Mie Généralisée, les coefficients gn et leur calcul numérique. 2ème partie: Théorie de Lorenz-Mie Généralisée, applications à la granulométrie optique,” J. Aerosol Sci. 19, 47, 55 (1988).
    [CrossRef]
  12. D. C. Ferguson, I. G. Currie, “Theoretical Evaluation of LDA Techniques for Two-Phase Flow Measurements,” presented at Third International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon (7–9 July 1986), paper 18-1. Actually, it must be noted that the paper was not included in the paperback proceedings (due to delay). We are not aware of another publication of this paper and we invite the reader to consider this reference as a private communication.
  13. W. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to Paraxial Optics,” Phys. Rev. A 11, 1365 (1975).
    [CrossRef]
  14. L. W. Davis, “Theory of Electromagnetic Beams,” Phys. Rev. A 19, 1177 (1979).
    [CrossRef]
  15. P. Poincelot, Préis d’Electromagnetisme Théorique (Masson, Paris, 1963).
  16. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  17. G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to Compute the Coefficients gnm in the Generalized Lorenz-Mie Theory Using Finite Series,” J. Opt. (Paris), 19, 35 (1988).
    [CrossRef]
  18. mulisp, mumath, Soft Warehouse, 3615 Harding Ave., Suite 505, Honolulu, HI 96816.
  19. F. Corbin, G. Gréhan, G. Gouesbet, B. Maheu, “Interaction Between a Sphere and a Gaussian Beam: Computations on a Micro Computer,” Part. Part. Syst. Charact. 5, 103 (1988).
    [CrossRef]
  20. L. Robin, Fonctions Sphériques de Legendre et Fonctions Sphéroidales, Tomes 1, 2, and 3 (Gauthier-Villars, Paris, 1957, 1958, 1959).
  21. G. N. Watson, A Treatise of the Theory of Bessel Functions (Cambridge U.P., London, 1962), pp. 524–525.

1988 (5)

G. Gouesbet, B. Maheu, G. Gréhan, “Scattering of a Gaussian Beam by a Sphere Using a Bromwich Formulation: Case of an Arbitrary Location,” Part. Part. Syst. Charact. 5, 1 (1988); G. Gouesbet, G. Gréhan, Eds., Optical Particle Sizing—Theory and Practice (Plenum, New York, 1988), pp. 27–42.
[CrossRef]

B. Maheu, G. Gréhan, G. Gouesbet, “Diffusion de la lumière par une sphère dans le cas d’un faisceau d’extension finie. lère partie: Théorie de Lorenz-Mie Généralisée, les coefficients gn et leur calcul numérique. 2ème partie: Théorie de Lorenz-Mie Généralisée, applications à la granulométrie optique,” J. Aerosol Sci. 19, 47, 55 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to Compute the Coefficients gnm in the Generalized Lorenz-Mie Theory Using Finite Series,” J. Opt. (Paris), 19, 35 (1988).
[CrossRef]

F. Corbin, G. Gréhan, G. Gouesbet, B. Maheu, “Interaction Between a Sphere and a Gaussian Beam: Computations on a Micro Computer,” Part. Part. Syst. Charact. 5, 103 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “Light Scattering from a Sphere Arbitrarily Located in a Gaussian Beam, Using a Bromwich Formulation,” J. Opt. Soc. Am. A 5, 1427, (Sept.1988).
[CrossRef]

1987 (2)

B. Maheu, G. Gréhan, G. Gouesbet, “Generalized Lorenz-Mie Theory: First Exact Values and Comparisons with the Localized Approximation,” Appl. Opt. 26, 23 (1987).
[CrossRef] [PubMed]

B. Maheu, G. Grehan, G. Gouesbet, “Laser Beam Scattering by Individual Spherical Particles: Numerical Results and Application to Optical Sizing,” Part. Part. Syst. Charact. 4, 141 (1987); G. Gouesbet, G. Grehan, Eds., Optical Particle Sizing—Theory and Practice (Plenum, New York, 1988), pp. 77–88.
[CrossRef]

1986 (1)

1985 (2)

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian Beam by a Mie Scatter Center, Using a Bromwich Formalism,” J. Opt. Paris 16, 83 (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. Paris 16, 239 (1985).
[CrossRef]

1982 (1)

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

1979 (1)

L. W. Davis, “Theory of Electromagnetic Beams,” Phys. Rev. A 19, 1177 (1979).
[CrossRef]

1975 (1)

W. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to Paraxial Optics,” Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Corbin, F.

F. Corbin, G. Gréhan, G. Gouesbet, B. Maheu, “Interaction Between a Sphere and a Gaussian Beam: Computations on a Micro Computer,” Part. Part. Syst. Charact. 5, 103 (1988).
[CrossRef]

Currie, I. G.

D. C. Ferguson, I. G. Currie, “Theoretical Evaluation of LDA Techniques for Two-Phase Flow Measurements,” presented at Third International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon (7–9 July 1986), paper 18-1. Actually, it must be noted that the paper was not included in the paperback proceedings (due to delay). We are not aware of another publication of this paper and we invite the reader to consider this reference as a private communication.

Davis, L. W.

L. W. Davis, “Theory of Electromagnetic Beams,” Phys. Rev. A 19, 1177 (1979).
[CrossRef]

Ferguson, D. C.

D. C. Ferguson, I. G. Currie, “Theoretical Evaluation of LDA Techniques for Two-Phase Flow Measurements,” presented at Third International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon (7–9 July 1986), paper 18-1. Actually, it must be noted that the paper was not included in the paperback proceedings (due to delay). We are not aware of another publication of this paper and we invite the reader to consider this reference as a private communication.

Gouesbet, G.

F. Corbin, G. Gréhan, G. Gouesbet, B. Maheu, “Interaction Between a Sphere and a Gaussian Beam: Computations on a Micro Computer,” Part. Part. Syst. Charact. 5, 103 (1988).
[CrossRef]

B. Maheu, G. Gréhan, G. Gouesbet, “Diffusion de la lumière par une sphère dans le cas d’un faisceau d’extension finie. lère partie: Théorie de Lorenz-Mie Généralisée, les coefficients gn et leur calcul numérique. 2ème partie: Théorie de Lorenz-Mie Généralisée, applications à la granulométrie optique,” J. Aerosol Sci. 19, 47, 55 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “Light Scattering from a Sphere Arbitrarily Located in a Gaussian Beam, Using a Bromwich Formulation,” J. Opt. Soc. Am. A 5, 1427, (Sept.1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to Compute the Coefficients gnm in the Generalized Lorenz-Mie Theory Using Finite Series,” J. Opt. (Paris), 19, 35 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “Scattering of a Gaussian Beam by a Sphere Using a Bromwich Formulation: Case of an Arbitrary Location,” Part. Part. Syst. Charact. 5, 1 (1988); G. Gouesbet, G. Gréhan, Eds., Optical Particle Sizing—Theory and Practice (Plenum, New York, 1988), pp. 27–42.
[CrossRef]

B. Maheu, G. Grehan, G. Gouesbet, “Laser Beam Scattering by Individual Spherical Particles: Numerical Results and Application to Optical Sizing,” Part. Part. Syst. Charact. 4, 141 (1987); G. Gouesbet, G. Grehan, Eds., Optical Particle Sizing—Theory and Practice (Plenum, New York, 1988), pp. 77–88.
[CrossRef]

B. Maheu, G. Gréhan, G. Gouesbet, “Generalized Lorenz-Mie Theory: First Exact Values and Comparisons with the Localized Approximation,” Appl. Opt. 26, 23 (1987).
[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 (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. Paris 16, 83 (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. Paris 16, 239 (1985).
[CrossRef]

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

G. Gréhan, B. Maheu, G. Gouesbet, “Localized Approximation to the Generalized Lorenz-Mie and Its Application to Optical Sizing,” in Proceedings, ICALEO’86, Arlington, VA (10–13 Nov. 1986).

B. Maheu, G. Gouesbet, G. Gréhan, “A Concise Presentation of the Generalized Lorenz-Mie Theory for Arbitrary Location of the Scatterer in an Arbitrary Incident Profile,” J. Opt.19, in press, (1988).
[CrossRef]

Grehan, G.

B. Maheu, G. Grehan, G. Gouesbet, “Laser Beam Scattering by Individual Spherical Particles: Numerical Results and Application to Optical Sizing,” Part. Part. Syst. Charact. 4, 141 (1987); G. Gouesbet, G. Grehan, Eds., Optical Particle Sizing—Theory and Practice (Plenum, New York, 1988), pp. 77–88.
[CrossRef]

Gréhan, G.

B. Maheu, G. Gréhan, G. Gouesbet, “Diffusion de la lumière par une sphère dans le cas d’un faisceau d’extension finie. lère partie: Théorie de Lorenz-Mie Généralisée, les coefficients gn et leur calcul numérique. 2ème partie: Théorie de Lorenz-Mie Généralisée, applications à la granulométrie optique,” J. Aerosol Sci. 19, 47, 55 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “Light Scattering from a Sphere Arbitrarily Located in a Gaussian Beam, Using a Bromwich Formulation,” J. Opt. Soc. Am. A 5, 1427, (Sept.1988).
[CrossRef]

F. Corbin, G. Gréhan, G. Gouesbet, B. Maheu, “Interaction Between a Sphere and a Gaussian Beam: Computations on a Micro Computer,” Part. Part. Syst. Charact. 5, 103 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “Scattering of a Gaussian Beam by a Sphere Using a Bromwich Formulation: Case of an Arbitrary Location,” Part. Part. Syst. Charact. 5, 1 (1988); G. Gouesbet, G. Gréhan, Eds., Optical Particle Sizing—Theory and Practice (Plenum, New York, 1988), pp. 27–42.
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to Compute the Coefficients gnm in the Generalized Lorenz-Mie Theory Using Finite Series,” J. Opt. (Paris), 19, 35 (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 (1987).
[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 (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. Paris 16, 239 (1985).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian Beam by a Mie Scatter Center, Using a Bromwich Formalism,” J. Opt. Paris 16, 83 (1985).
[CrossRef]

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

B. Maheu, G. Gouesbet, G. Gréhan, “A Concise Presentation of the Generalized Lorenz-Mie Theory for Arbitrary Location of the Scatterer in an Arbitrary Incident Profile,” J. Opt.19, in press, (1988).
[CrossRef]

G. Gréhan, B. Maheu, G. Gouesbet, “Localized Approximation to the Generalized Lorenz-Mie and Its Application to Optical Sizing,” in Proceedings, ICALEO’86, Arlington, VA (10–13 Nov. 1986).

Lax, W.

W. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to Paraxial Optics,” Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Louisell, W. H.

W. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to Paraxial Optics,” Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Maheu, B.

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to Compute the Coefficients gnm in the Generalized Lorenz-Mie Theory Using Finite Series,” J. Opt. (Paris), 19, 35 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “Scattering of a Gaussian Beam by a Sphere Using a Bromwich Formulation: Case of an Arbitrary Location,” Part. Part. Syst. Charact. 5, 1 (1988); G. Gouesbet, G. Gréhan, Eds., Optical Particle Sizing—Theory and Practice (Plenum, New York, 1988), pp. 27–42.
[CrossRef]

F. Corbin, G. Gréhan, G. Gouesbet, B. Maheu, “Interaction Between a Sphere and a Gaussian Beam: Computations on a Micro Computer,” Part. Part. Syst. Charact. 5, 103 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “Light Scattering from a Sphere Arbitrarily Located in a Gaussian Beam, Using a Bromwich Formulation,” J. Opt. Soc. Am. A 5, 1427, (Sept.1988).
[CrossRef]

B. Maheu, G. Gréhan, G. Gouesbet, “Diffusion de la lumière par une sphère dans le cas d’un faisceau d’extension finie. lère partie: Théorie de Lorenz-Mie Généralisée, les coefficients gn et leur calcul numérique. 2ème partie: Théorie de Lorenz-Mie Généralisée, applications à la granulométrie optique,” J. Aerosol Sci. 19, 47, 55 (1988).
[CrossRef]

B. Maheu, G. Grehan, G. Gouesbet, “Laser Beam Scattering by Individual Spherical Particles: Numerical Results and Application to Optical Sizing,” Part. Part. Syst. Charact. 4, 141 (1987); G. Gouesbet, G. Grehan, Eds., Optical Particle Sizing—Theory and Practice (Plenum, New York, 1988), pp. 77–88.
[CrossRef]

B. Maheu, G. Gréhan, G. Gouesbet, “Generalized Lorenz-Mie Theory: First Exact Values and Comparisons with the Localized Approximation,” Appl. Opt. 26, 23 (1987).
[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 (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. Paris 16, 83 (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. Paris 16, 239 (1985).
[CrossRef]

G. Gréhan, B. Maheu, G. Gouesbet, “Localized Approximation to the Generalized Lorenz-Mie and Its Application to Optical Sizing,” in Proceedings, ICALEO’86, Arlington, VA (10–13 Nov. 1986).

B. Maheu, G. Gouesbet, G. Gréhan, “A Concise Presentation of the Generalized Lorenz-Mie Theory for Arbitrary Location of the Scatterer in an Arbitrary Incident Profile,” J. Opt.19, in press, (1988).
[CrossRef]

McKnight, W. B.

W. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to Paraxial Optics,” Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Poincelot, P.

P. Poincelot, Préis d’Electromagnetisme Théorique (Masson, Paris, 1963).

Robin, L.

L. Robin, Fonctions Sphériques de Legendre et Fonctions Sphéroidales, Tomes 1, 2, and 3 (Gauthier-Villars, Paris, 1957, 1958, 1959).

van de Hulst, H. C.

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

Watson, G. N.

G. N. Watson, A Treatise of the Theory of Bessel Functions (Cambridge U.P., London, 1962), pp. 524–525.

Appl. Opt. (2)

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. lère partie: Théorie de Lorenz-Mie Généralisée, les coefficients gn et leur calcul numérique. 2ème partie: Théorie de Lorenz-Mie Généralisée, applications à la granulométrie optique,” J. Aerosol Sci. 19, 47, 55 (1988).
[CrossRef]

J. Opt. (Paris) (1)

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to Compute the Coefficients gnm in the Generalized Lorenz-Mie Theory Using Finite Series,” J. Opt. (Paris), 19, 35 (1988).
[CrossRef]

J. Opt. Paris (3)

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

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian Beam by a Mie Scatter Center, Using a Bromwich Formalism,” J. Opt. Paris 16, 83 (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. Paris 16, 239 (1985).
[CrossRef]

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

Part. Part. Syst. Charact. (3)

G. Gouesbet, B. Maheu, G. Gréhan, “Scattering of a Gaussian Beam by a Sphere Using a Bromwich Formulation: Case of an Arbitrary Location,” Part. Part. Syst. Charact. 5, 1 (1988); G. Gouesbet, G. Gréhan, Eds., Optical Particle Sizing—Theory and Practice (Plenum, New York, 1988), pp. 27–42.
[CrossRef]

F. Corbin, G. Gréhan, G. Gouesbet, B. Maheu, “Interaction Between a Sphere and a Gaussian Beam: Computations on a Micro Computer,” Part. Part. Syst. Charact. 5, 103 (1988).
[CrossRef]

B. Maheu, G. Grehan, G. Gouesbet, “Laser Beam Scattering by Individual Spherical Particles: Numerical Results and Application to Optical Sizing,” Part. Part. Syst. Charact. 4, 141 (1987); G. Gouesbet, G. Grehan, Eds., Optical Particle Sizing—Theory and Practice (Plenum, New York, 1988), pp. 77–88.
[CrossRef]

Phys. Rev. A (2)

W. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to Paraxial Optics,” Phys. Rev. A 11, 1365 (1975).
[CrossRef]

L. W. Davis, “Theory of Electromagnetic Beams,” Phys. Rev. A 19, 1177 (1979).
[CrossRef]

Other (8)

P. Poincelot, Préis d’Electromagnetisme Théorique (Masson, Paris, 1963).

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

G. Gréhan, B. Maheu, G. Gouesbet, “Localized Approximation to the Generalized Lorenz-Mie and Its Application to Optical Sizing,” in Proceedings, ICALEO’86, Arlington, VA (10–13 Nov. 1986).

D. C. Ferguson, I. G. Currie, “Theoretical Evaluation of LDA Techniques for Two-Phase Flow Measurements,” presented at Third International Symposium on Applications of Laser Anemometry to Fluid Mechanics, Lisbon (7–9 July 1986), paper 18-1. Actually, it must be noted that the paper was not included in the paperback proceedings (due to delay). We are not aware of another publication of this paper and we invite the reader to consider this reference as a private communication.

L. Robin, Fonctions Sphériques de Legendre et Fonctions Sphéroidales, Tomes 1, 2, and 3 (Gauthier-Villars, Paris, 1957, 1958, 1959).

G. N. Watson, A Treatise of the Theory of Bessel Functions (Cambridge U.P., London, 1962), pp. 524–525.

mulisp, mumath, Soft Warehouse, 3615 Harding Ave., Suite 505, Honolulu, HI 96816.

B. Maheu, G. Gouesbet, G. Gréhan, “A Concise Presentation of the Generalized Lorenz-Mie Theory for Arbitrary Location of the Scatterer in an Arbitrary Incident Profile,” J. Opt.19, in press, (1988).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Geometry of the problem.

Tables (8)

Tables Icon

Table I Comparison of the gn Coefficients for a Beam Waist Center Location

Tables Icon

Table II Comparison of the gn Coefficients for a Beam Waist Center Location

Tables Icon

Table III Comparison of the gn Coefficients for a Beam Waist Center Location

Tables Icon

Table IV Comparison of the Incident Electric Field Computed Either with gn Coefficients (Right-Hand Side) or Without (Left-Hand Side)

Tables Icon

Table V Study of the n Domain Where the Finite Series and the Localized Interpretation are Identical

Tables Icon

Table VI Comparison of the gn Coefficients for a Location Along the Beam Axis

Tables Icon

Table VII Comparison of the gn Coefficients for a Location Along the Beam Axis

Tables Icon

Table VIII Comparison of the gn Coefficients for a Location Along the Beam Axis

Equations (67)

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

O P O G = ( x 0 , y 0 , z 0 ) .
s = 1 / ( k w 0 ) = w 0 / l ,
E r = E 0 Ψ 0 [ cos φ sin θ ( 1 - 2 Q l ɛ L r cos θ ) + 2 Q l ɛ L x 0 cos θ ] × exp ( - i k r cos θ ) exp ( i k 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 ) ] ,
Q = 1 i + 2 l ( z - z 0 ) .
ɛ L = 1 , at order L 0 , at order L - .
U TM i = E 0 n = 1 m = - n + n C n p w g n , TM m r Ψ n 1 ( k r ) P n m ( cos θ ) exp ( i m φ ) ,
C n p w = 1 k i n - 1 ( - 1 ) n 2 n + 1 n ( n + 1 ) .
x 0 = y 0 = 0 ,
g n , TM m = 0 , m 1 g n , TM 1 = g n , TM - 1 = 1 2 g n } ,
E r = E 0 cos φ sin θ ( 1 - 2 Q l ɛ L r cos θ ) i Q exp ( - i Q r 2 sin 2 θ w 0 2 ) × exp ( - i k r cos θ ) exp ( i k z 0 ) ,
U TM i = E 0 cos φ n = 1 C n p w g n r Ψ n 1 ( k r ) P n 1 ( cos θ ) .
E r = 2 U TM i r 2 + k 2 U TM i .
[ d 2 d r 2 + k 2 ] [ r Ψ n 1 ( k r ) ] = n ( n + 1 ) r Ψ n 1 ( k r ) ,
E r = E 0 k r cos φ n = 1 i n - 1 ( - 1 ) n ( 2 n + 1 ) g n Ψ n 1 ( k r ) P n 1 ( cos θ ) .
k r sin θ ( 1 - 2 Q l ɛ L r cos θ ) i Q exp ( - i Q r 2 sin 2 θ w 0 2 ) × exp ( - i k r cos θ ) exp ( i k z 0 ) = n = 1 i n - 1 ( - 1 ) n ( 2 n + 1 ) g n Ψ n 1 ( k r ) P n 1 ( cos θ ) .
0 Ψ n 1 ( k r ) Ψ m 1 ( k r ) d ( k r ) = π 2 ( 2 n + 1 ) δ n m ,
0 π P n 1 ( cos θ ) P l 1 ( cos θ ) sin θ d θ = 2 n ( n + 1 ) 2 n + 1 δ n l .
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 ) ,
F = i Q exp ( - i Q r 2 sin 2 θ w 0 2 ) sin θ ( 1 - 2 Q l ɛ L r cos θ ) exp ( i k z 0 ) × exp ( - i k r cos θ ) .
n ! ! = 2 ( n + 1 ) / 2 π Γ ( n 2 + 1 ) ,
k r Z 0 exp ( - Z 0 r 2 w 0 2 ) exp ( i k z 0 ) = p = 1 2 ( 4 p - 1 ) π Γ ( p + ½ ) Γ ( p ) × g 2 p - 1 Ψ 2 p - 1 1 ( k r ) ,
Z 0 = i Q ( θ = π / 2 ) = 1 1 + 2 i z 0 l .
k r 2 exp ( i k z 0 ) exp ( - Z 0 r 2 w 0 2 ) [ ( k Z 0 + k Z 0 - 2 l ɛ L Z 0 2 ) - k Z 0 Z 0 r 2 w 0 2 ] = p = 1 ( 4 p + 2 ) ( 4 p + 1 ) π Γ ( p + ½ ) Γ ( p ) g 2 p Ψ 2 p 1 ( k r ) ,
Z 0 = 2 k l [ i - 2 z 0 L ] 2 = 1 k r Q θ | θ = π / 2 .
Ψ n 1 ( k r ) = π 2 k r J n + 1 / 2 ( k r ) ,
x 1 / 2 g ( x ) = p = 1 2 ( 4 p - 1 ) Γ ( p + ½ ) Γ ( p ) g 2 p - 1 J 2 p - 1 / 2 ( x ) ,
g ( x ) = x Z 0 exp ( - Z 0 s 2 x 2 ) exp ( i k z 0 ) .
x 1 / 2 g ( x ) = n = 0 O 2 ( 2 n + 1 ) Γ ( n 2 + 1 ) Γ ( n 2 + 1 2 ) g n J n + 1 / 2 ( x ) ,
n = 0 O
c n = 0 , n even c n = 2 ( 2 n + 1 ) Γ ( n 2 + 1 ) Γ ( n 2 + 1 2 ) g n , n odd } .
g ( x ) = n = 0 O Z 0 exp ( i k z 0 ) ( - Z 0 s 2 ) n - 1 2 ( n - 1 2 ) ! x n ,
b n = 0 , n even b n = Z 0 exp ( i k z 0 ) ( - Z 0 s 2 ) n - 1 2 ( n - 1 2 ) ! , n odd } .
g 2 p + 1 = Z 0 exp ( i k z 0 ) j = 0 p p ! Γ ( p + j + / 2 3 ) j ! ( p - j ) ! Γ ( p + / 2 3 ) ( - 4 Z 0 s 2 ) j .
g 2 p + 2 = 1 k exp ( i k z 0 ) j = 0 p p ! j ! ( p - j ) ! Γ ( p + j + / 2 5 ) Γ ( p + / 2 5 ) × [ A - j B Z 0 ] ( - 4 Z 0 s 2 ) j ,
A = k Z 0 + k Z 0 - 2 l ɛ L Z 0 2 ,
B = - k Z 0 Z 0 .
g 2 p + 1 = j = 0 p p ! j ! ( p - j ) ! Γ ( p + j + / 2 3 ) Γ ( p + / 2 3 ) ( - 4 s 2 ) j ,
g 2 p + 2 = j = 0 p p ! j ! ( p - j ) ! Γ ( p + j + / 2 5 ) Γ ( p + / 2 5 ) ( - 4 s 2 ) j .
g n 1 a = exp { - [ ( n + ½ ) λ 2 π w 0 ] 2 } ,
E x / E 0 = H y / H 0 = i Q exp [ - i Q r 2 sin 2 θ w 0 2 ] exp ( - i k z ) ,
Q = 1 i + 2 r cos θ l .
E x E 0 = H y H 0 exp ( - r 2 sin 2 θ w 0 2 ) exp ( - i k z ) .
ρ n = ( n + ½ ) λ 2 π .
E x n E 0 = H y n H 0 exp ( - ρ n 2 w 0 2 ) exp ( - i k z ) .
E x E 0 = H y H 0 = exp ( - i k z ) ,
g n l a = exp ( - ρ n 2 w 0 2 ) ,
g n 1 a = i Q ¯ exp [ - i Q ¯ ( ρ n w 0 ) 2 ] ,
h ¯ ( z , r sin θ ) = h ( 0 , ρ n ) .
E r = { E 0 exp ( - i k z ) cos φ sin θ } i Q exp [ - i Q r 2 sin 2 θ w 0 2 ] .
g n , TM 1 a = i Q exp [ - i Q r 2 sin 2 θ w 0 2 ] ¯ .
H r = { H 0 exp ( - i k z ) sin φ sin θ } i Q exp [ - i Q r 2 sin 2 θ w 0 2 ] .
g n , TE 1 a = g n , TM 1 a .
g n , TM 1 a = g n , TE 1 a = g n 1 a = i Q ¯ exp [ - i Q ¯ ( ρ n w 0 ) 2 ] ,
( E x H y ) = ( E 0 H 0 ) i Q exp ( - i Q r 2 sin 2 θ w 0 2 ) exp ( i k z 0 ) exp ( - i k z ) ,
( E r H r ) = { ( E 0 cos φ H 0 sin φ ) exp ( - i k z ) sin θ } × i Q exp [ - i Q r 2 sin 2 θ w 0 2 ] exp ( i k z 0 ) .
g n , TM 1 a = g n , TE 1 a = g n 1 a = i Q exp [ - i Q r 2 sin 2 θ w 0 2 ] exp ( i k z 0 ) ¯ ,
g n 1 a = i Q ¯ exp [ - i Q ¯ ( ρ n w 0 ) 2 ] exp ( i k z 0 ) ,
Q ¯ = Q ( z = 0 ) = 1 i - 2 z 0 l .
P n 1 ( 0 ) = ( - 1 ) ( n + 1 ) / 2 n ! ! 2 ( n - 1 ) / 2 ( n - 1 2 ) ! , n odd = 0 , n even } ,
d P n 1 ( cos θ ) d cos θ = 0 , n odd = ( - 1 ) n / 2 ( n + 1 ) ! ! 2 ( n - 2 ) / 2 ( n - 2 2 ) ! , n even } if cos θ = 0 ,
n ! ! = 1.3 n ( - 1 ) ! ! = 1 } .
x 1 / 2 g ( x ) = n = 0 c n J n + 1 / 2 ( x ) ,
g ( x ) = n = 0 b n x n .
c n = ( n + ½ ) m = 0 n / 2 2 ( n - 2 m + 1 / 2 ) Γ ( ½ + n - m ) m ! b n - 2 m .

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