Abstract

A comparison between two quadrature methods of evaluating beam-shape coefficients in generalized Lorenz–Mie theory, in the case of incident Gaussian beams, is carried out. It is shown that, when the electromagnetic description of the Gaussian beams does not perfectly satisfy Maxwell's equations, both quadrature methods are basically flawed. These flaws do not prevent an accurate evaluation of beam-shape coefficients when their nature is correctly identified, because they produce artifacts that can easily be identified and removed.

© 1996 Optical Society of America

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  1. 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, 1427–1443 (1988).
  2. 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. (Paris) 19, 59–67 (1988).
  3. G. Gouesbet, “Generalized Lorenz–Mie theory and applications,” Part. Part. Syst. Charact. 11, 22–34 (1994).
  4. L. Lorenz, “Lysbevaegelsen i og uden for en haf plane lysbolger belyst kulge,” Vidensk. Selk. Skr. 6, 1–62 (1890).
  5. L. Lorenz, “Sur la lumière réfractée par une sphère transparente,” in Oeuvres Scientifiques de L. Lorenz, Revues et Annotées par H. Valentiner (Libraire Lehmann et Stage, Copenhagen, 1898), pp. 405–529.
  6. G. Mie, “Beitrage zur optik truber medien, speziell kolloidaler metallosungen,” Ann. Phys. 25, 377–452 (1908).
  7. G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the coefficients gn in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
  8. 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, 35–48 (1988).
  9. 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).
  10. G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
  11. J. A. Lock, G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in the generalized Lorenz–Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
  12. G. Gouesbet, J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in the generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
  13. G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).
  14. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. 19, 1177–1179 (1979).
  15. G. Gouesbet, J.A. Lock, G. Gréhan, “Do you know what a laser beam is?’’ in Proceedings of the Seventh Workshop on Two-Phase Flows (to be published).
  16. G. Gouesbet, J. A. Lock, G. Gréhan, “Partial-wave representations of laser beams for use in light-scattering calculations,” Appl. Opt. 34, 2133–2143 (1995).
  17. S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficients for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989).
  18. J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
  19. G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1966).
  20. J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle illuminated by a focussed laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
  21. J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A 10, 693–706 (1993).
  22. G. Gouesbet, G. Gréhan, B. Maheu, “Electromagnetic scattering of shaped beams,” is in preparation. A rather polished draft is available on request.
  23. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (New York, Academic, 1980).
  24. B. Maheu, G. Gréhan, G. Gouesbet, “Generalized Lorenz–Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23–26 (1987).
  25. B. Maheu, G. Gréhan, G. Gouesbet, “Diffusion de la lumière par une sphère dans le cas d’un faisceau d’extension finie. lere partie: théorie de Lorenz–Mie généralisée, les coefficients gn et leur calcul numérique,” presented at the 3éme Journées d’ Etudes sur les Aérosols, Paris, 9–10 Décembre, 1986 [published in J. Aerosol Sci.19, 47–53 (1988).]
  26. B. Maheu, G. Gréhan, G. Gouesbet, “Laser beam scattering by individual spherical particles: theoretical progress and applications to optical sizing,” presented at the International Symposium: Optical Particle Sizing: Theory and Practice, Rouen, France, 12–15 May 1987 [published in J. Part. Charact.4, 141–146 (1987).]
  27. B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).

1995 (1)

1994 (3)

1993 (1)

1990 (1)

1989 (3)

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

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

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

1988 (5)

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

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, 1427–1443 (1988).

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. (Paris) 19, 59–67 (1988).

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the coefficients gn in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).

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, 35–48 (1988).

1987 (1)

1986 (1)

1982 (1)

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

1979 (1)

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

1908 (1)

G. Mie, “Beitrage zur optik truber medien, speziell kolloidaler metallosungen,” Ann. Phys. 25, 377–452 (1908).

1890 (1)

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

Alexander, D. R.

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

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

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

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1966).

Barton, J. P.

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

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

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

Davis, L. W.

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

Gouesbet, G.

G. Gouesbet, J. A. Lock, G. Gréhan, “Partial-wave representations of laser beams for use in light-scattering calculations,” Appl. Opt. 34, 2133–2143 (1995).

J. A. Lock, G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in the generalized Lorenz–Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).

G. Gouesbet, J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in the generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).

G. Gouesbet, “Generalized Lorenz–Mie theory and applications,” Part. Part. Syst. Charact. 11, 22–34 (1994).

G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).

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

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the coefficients gn in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).

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, 1427–1443 (1988).

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. (Paris) 19, 59–67 (1988).

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, 35–48 (1988).

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

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).

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

G. Gouesbet, J.A. Lock, G. Gréhan, “Do you know what a laser beam is?’’ in Proceedings of the Seventh Workshop on Two-Phase Flows (to be published).

B. Maheu, G. Gréhan, G. Gouesbet, “Diffusion de la lumière par une sphère dans le cas d’un faisceau d’extension finie. lere partie: théorie de Lorenz–Mie généralisée, les coefficients gn et leur calcul numérique,” presented at the 3éme Journées d’ Etudes sur les Aérosols, Paris, 9–10 Décembre, 1986 [published in J. Aerosol Sci.19, 47–53 (1988).]

G. Gouesbet, G. Gréhan, B. Maheu, “Electromagnetic scattering of shaped beams,” is in preparation. A rather polished draft is available on request.

B. Maheu, G. Gréhan, G. Gouesbet, “Laser beam scattering by individual spherical particles: theoretical progress and applications to optical sizing,” presented at the International Symposium: Optical Particle Sizing: Theory and Practice, Rouen, France, 12–15 May 1987 [published in J. Part. Charact.4, 141–146 (1987).]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (New York, Academic, 1980).

Gréhan, G.

G. Gouesbet, J. A. Lock, G. Gréhan, “Partial-wave representations of laser beams for use in light-scattering calculations,” Appl. Opt. 34, 2133–2143 (1995).

G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).

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

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the coefficients gn in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).

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, 35–48 (1988).

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. (Paris) 19, 59–67 (1988).

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, 1427–1443 (1988).

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

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).

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

G. Gouesbet, G. Gréhan, B. Maheu, “Electromagnetic scattering of shaped beams,” is in preparation. A rather polished draft is available on request.

G. Gouesbet, J.A. Lock, G. Gréhan, “Do you know what a laser beam is?’’ in Proceedings of the Seventh Workshop on Two-Phase Flows (to be published).

B. Maheu, G. Gréhan, G. Gouesbet, “Diffusion de la lumière par une sphère dans le cas d’un faisceau d’extension finie. lere partie: théorie de Lorenz–Mie généralisée, les coefficients gn et leur calcul numérique,” presented at the 3éme Journées d’ Etudes sur les Aérosols, Paris, 9–10 Décembre, 1986 [published in J. Aerosol Sci.19, 47–53 (1988).]

B. Maheu, G. Gréhan, G. Gouesbet, “Laser beam scattering by individual spherical particles: theoretical progress and applications to optical sizing,” presented at the International Symposium: Optical Particle Sizing: Theory and Practice, Rouen, France, 12–15 May 1987 [published in J. Part. Charact.4, 141–146 (1987).]

Lock, J. A.

Lock, J.A.

G. Gouesbet, J.A. Lock, G. Gréhan, “Do you know what a laser beam is?’’ in Proceedings of the Seventh Workshop on Two-Phase Flows (to be published).

Lorenz, L.

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

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

Maheu, B.

G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).

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

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the coefficients gn in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).

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. (Paris) 19, 59–67 (1988).

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, 1427–1443 (1988).

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, 35–48 (1988).

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

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).

G. Gouesbet, G. Gréhan, B. Maheu, “Electromagnetic scattering of shaped beams,” is in preparation. A rather polished draft is available on request.

B. Maheu, G. Gréhan, G. Gouesbet, “Laser beam scattering by individual spherical particles: theoretical progress and applications to optical sizing,” presented at the International Symposium: Optical Particle Sizing: Theory and Practice, Rouen, France, 12–15 May 1987 [published in J. Part. Charact.4, 141–146 (1987).]

B. Maheu, G. Gréhan, G. Gouesbet, “Diffusion de la lumière par une sphère dans le cas d’un faisceau d’extension finie. lere partie: théorie de Lorenz–Mie généralisée, les coefficients gn et leur calcul numérique,” presented at the 3éme Journées d’ Etudes sur les Aérosols, Paris, 9–10 Décembre, 1986 [published in J. Aerosol Sci.19, 47–53 (1988).]

Mie, G.

G. Mie, “Beitrage zur optik truber medien, speziell kolloidaler metallosungen,” Ann. Phys. 25, 377–452 (1908).

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (New York, Academic, 1980).

Schaub, S. A.

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

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

Ann. Phys. (1)

G. Mie, “Beitrage zur optik truber medien, speziell kolloidaler metallosungen,” Ann. Phys. 25, 377–452 (1908).

Appl. Opt. (4)

Appl. Phys. Lett. (1)

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

J. Appl. Phys. (2)

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

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

J. Opt. (Paris) (3)

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, 35–48 (1988).

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

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. (Paris) 19, 59–67 (1988).

J. Opt. Soc. Am. A (4)

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

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, 1427–1443 (1988).

Opt. Commun. (1)

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

Part. Part. Syst. Charact. (1)

G. Gouesbet, “Generalized Lorenz–Mie theory and applications,” Part. Part. Syst. Charact. 11, 22–34 (1994).

Phys. Rev. (1)

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

Vidensk. Selk. Skr. (1)

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

Other (7)

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

G. Gouesbet, J.A. Lock, G. Gréhan, “Do you know what a laser beam is?’’ in Proceedings of the Seventh Workshop on Two-Phase Flows (to be published).

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1966).

G. Gouesbet, G. Gréhan, B. Maheu, “Electromagnetic scattering of shaped beams,” is in preparation. A rather polished draft is available on request.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (New York, Academic, 1980).

B. Maheu, G. Gréhan, G. Gouesbet, “Diffusion de la lumière par une sphère dans le cas d’un faisceau d’extension finie. lere partie: théorie de Lorenz–Mie généralisée, les coefficients gn et leur calcul numérique,” presented at the 3éme Journées d’ Etudes sur les Aérosols, Paris, 9–10 Décembre, 1986 [published in J. Aerosol Sci.19, 47–53 (1988).]

B. Maheu, G. Gréhan, G. Gouesbet, “Laser beam scattering by individual spherical particles: theoretical progress and applications to optical sizing,” presented at the International Symposium: Optical Particle Sizing: Theory and Practice, Rouen, France, 12–15 May 1987 [published in J. Part. Charact.4, 141–146 (1987).]

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Figures (1)

Fig. 1.
Fig. 1.

Geometry under study.

Equations (47)

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A i = ( A u , 0 , 0 ) ,
A u = Ψ ( u , υ , w ) exp ( i k w ) ,
ξ = u w 0 ,  η = υ w 0 ,  ζ = w l = s w w 0
s = w 0 l = 1 k w 0 .
Δ A u + k 2 A u = 0 ,
( 2 ξ 2 + 2 η 2 ) Ψ 2 i Ψ ζ + s 2 2 Ψ ζ 2 = 0.
Ψ = Ψ 0 + s 2 Ψ 2 + s 4 Ψ 4 + ....
Ψ 0 = i Q  exp [ i Q ( ξ 2 + η 2 ) ] ,
Q = 1 i + 2 ζ .
( 2 ξ 2 + 2 η 2 2 i ζ ) Ψ 2 n + 2 = 2 ζ 2 Ψ 2 n , n 0.
E i = i c k ( grad div  A j ) i i w A i ,
H i = ( curl  A j ) i μ ,
E u = E 0 [ Ψ 0 + s 2 ( Ψ 2 + 2 Ψ 0 ξ 2 ) + ... ] exp ( i k w ) ,
E υ = E 0 ( s 2 2 Ψ 0 ξ η + s 4 2 Ψ 2 ξ η + ... ) exp ( i k w )
E w = E 0 [ i s Ψ 0 ξ i s 3 ( Ψ 2 ξ + i 2 Ψ 0 ξ ζ ) + ... ] exp ( i k w ) ,
H u = 0 ,
H υ = H 0 [ Ψ 0 + s 2 ( Ψ 2 + i Ψ 0 ζ ) + ... ] exp ( i k w ) ,
H w = H 0 ( i s Ψ 0 η i s 3 Ψ 2 η ) exp ( i k w ) .
( E r k E 0 H r k H 0 ) = exp ( i R  cos  θ ) f k ( R , θ ) sin θ ( cos φ sin φ ) ,
f k = D 0 exp ( s 2 R 2 D 0 sin 2   θ ) h k ( R , θ ) ,
D 0 = ( 1 2 i ζ ) 1 = i Q ,
h 1 = D 0 ,
h 3 = D 0 ( 1 + 3 s 4 D 0 2 R 2 sin 2 θ s 6 D 0 3 R 4 sin 4 θ ) ,
h 5 = D 0 ( 1 + 3 s 4 D 0 2 R 2 sin 2 θ s 6 D 0 3 R 4 sin 4 θ    + 10 s 8 D 0 4 R 4 sin 4 θ 5 s 10 D 0 5 R 6 sin 6 θ    + 1 2 s 12 D 0 6 R 8 sin 8 θ ) .
g n F 2 = ( 1 ) n 2 n + 1 π i n 1 n ( n + 1 ) 0 R d R 0 π sin 2 θ    × exp ( i R cosθ ) f ( R , θ ) R n 1 ( cosθ ) Ψ n 1 ( R ) dθ,
P n 1 ( cosθ ) = sinθ d P n ( cosθ ) d cosθ ,
g n F 1 = ( 1 ) n 2 i n 1 n ( n + 1 ) R Ψ n 1 ( R ) 0 π sin 2 θ    × exp ( i R cosθ ) f ( R , θ ) P n 1 ( cosθ ) dθ,
0 Ψ n 1 ( R ) Ψ n 1 ( R ) d R = π 2 ( 2 n + 1 ) ,
g n F 2 = 2 ( 2 n + 1 ) π 0 [ Ψ n 1 ( R ) ] 2 g n F 1 d R .
g n F 1 , 1 = 1 n 1 s 2 + NCT 1 ,
g n F 1 , 3 = 1 n 1 s 2 + 1 2 n 2 s 4 1 6 n 3 s 6 + NCT 3 ,
g n F 1 , 5 = 1 n 1 s 2 + 1 2 n 2 s 4 1 6 n 3 s 6 + 1 24 n 4 s 8    1 120 n 5 s 10 + NCT 5 ,
n l = ( n l ) ( n l + 1 ) ... ( n 1 ) ( n + 2 ) ( n + 3 )     ... ( n + l + 1 ) ,
R ( n + 1 2 ) [ ( n 1 ) ( n + 2 ) ] 1 2 ,
NCT 1 = 1 2 s 4 ( n 1 ) ( n + 2 ) ( n 3 ) ( n + 4 ) + R G n G n ( 12 s 4 )    1 6 s 6 ( n 1 ) ( n + 2 ) ( n 4 + 2 n 3 41 n 2 42 n + 72 ) + R G n G n [ 24 ( n 1 ) ( n + 2 ) s 6 ]    + 1 24 ( n 1 ) ( n + 2 ) ( n 6 + 3 n 5 95 n 4 195 n 3    + 694 n 2 + 792 n + 8640 ) s 8 240 R 2 s 8    + 20 ( n 4 + 2 n 3 13 n 2    14 n 36 ) R G n G n s 8 + O ( s 10 ) ,
G n ( R ) = Ψ n 1 ( R ) R
NCT 3 = 240 R 2 s 8 + 120 R ( n 2 + n + 8 ) G n G n s 8    + 1 24 ( n 1 ) ( n + 2 ) [ ( n 1 ) 3 ( n + 2 ) 3    32 ( n 1 ) 2 ( n + 2 ) 2    + 52 ( n 1 ) ( n + 2 ) 9840 ] s 8 + O ( s 10 ) ,
NCT 5 = s 12 [ 840 R ( 8 R 2 n 4 2 n 3 180 49 n 2    48 n ) G n G n + 1680 ( 3 n 3 + 3 n + 22 ) R 2    + 1 720 ( n 1 ) ( n + 2 ) ( n 10 + 5 n 9 100 n 8    430 n 7 + 3773 n 6 + 12 , 845 n 5 92 , 450 n 4    206 , 820 n 3 4 , 447 , 224 n 2 4 , 341 , 600 n    47 , 174 , 400 ) ] + O ( s 14 ) .
K F 2 = K F 1 ,
F ( R ) = R G n G n .
I 1 = 0 [ Ψ n 1 ( R ) ] 2 R G n G n d R = 0 [ Ψ n 1 ( R ) ] 2 R ( Ψ n 1 R ) ( R Ψ n 1 ) d R = 0 R Ψ n 1 Ψ n 1 d R J 0 ( Ψ n 1 ) 2 d R . π 2 ( 2 n + 1 )
( Ψ n 1 ) = 1 2 n + 1 [ n Ψ n 1 1 ( n + 1 ) Ψ n + 1 1 ] ,
J = π n 2 ( 2 n + 1 ) 0 J n + ( 1 / 2 ) J n ( 1 / 2 ) d R 1 2    π ( n + 1 ) 2 ( 2 n + 1 ) 0 J n + ( 1 / 2 ) J n + ( 3 / 2 ) d R 1 2 = π 4 ( 2 n + 1 ) .
I 1 = 3 π 4 ( 2 n + 1 ) .
I 2 = 0 ( Ψ n 1 ) 2 R 2 d R .
I 2 ( n = 1 ) = 0 ( Ψ n 1 ) 2 R 2 d R = 0 ( sin R R 2 cos R R ) 2 R 2 d R ,
g n F 2 , 1 = g n F 1 , 1 + s 4 [ 1 2 ( n 1 ) ( n + 2 ) ( n 3 ) ( n + 4 ) 18 ]    + s 6 [ 1 6 ( n 1 ) ( n + 2 ) ( n 4 + 2 n 3 41 n 2    42 n + 72 ) + 36 ( n 1 ) ( n + 2 ) ] + DT ,

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