Abstract

A generalized Lorenz–Mie theory for infinite elliptical cylinders is presented. This theory describes the interaction between arbitrary shaped beams and infinitely long cylinders having an elliptical cross section.

© 1999 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).
    [CrossRef]
  2. G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), pp. 339–384.
  3. G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).
    [CrossRef]
  4. J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
    [CrossRef]
  5. 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).
    [CrossRef]
  6. F. Onofri, G. Gréhan, G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113–7124 (1995).
    [CrossRef] [PubMed]
  7. G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions,” J. Opt. (Paris) 26, 225–239 (1995).
    [CrossRef]
  8. G. Gouesbet, “Interaction between an infinite cylinder and an arbitrary shaped beam,” Appl. Opt. 36, 4292–4304 (1997).
    [CrossRef] [PubMed]
  9. K. F. Ren, G. Gréhan, G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz–Mie theory: formulation and numerical results,” J. Opt. Soc. Am. A 14, 3014–3025 (1997).
    [CrossRef]
  10. J. A. Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitely long homogeneous circular cylinder,” J. Opt. Soc. Am. A 14, 640–652 (1997).
    [CrossRef]
  11. J. B. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34, 5542–5551 (1995).
    [CrossRef] [PubMed]
  12. H. Mignon, G. Gréhan, G. Gouesbet, T. H. Hu, C. Tropea, “Measurement of cylindrical particles with phase-Doppler anemometry,” Appl. Opt. 25, 5180–5190 (1996).
    [CrossRef]
  13. N. Gauchet, T. Girasole, K. F. Ren, G. Gréhan, G. Gouesbet, “Application of generalized Lorenz–Mie theory for cylinders to cylindrical characterization by phase-Doppler anemometry,” Opt. Diag. Eng. 2, 1–10 (1997).
  14. X. Han, K. F. Ren, Z. Wu, F. Corbin, G. Gouesbet, G. Gréhan, “Characterization of initial disturbances in liquid jet by rainbow sizing,” Appl. Opt. 37, 8498–8503 (1998).
    [CrossRef]
  15. J. A. Lock, C. L. Adler, B. R. Stone, P. D. Zajak, “Amplification of high-order rainbows of a cylinder with an elliptical cross-section,” Appl. Opt. 37, 1527–1533 (1998).
    [CrossRef]
  16. S. Lange, G. Schweiger, “Structural resonances in the total Raman- and fluorescence-scattering cross section: concentration-profile dependence,” J. Opt. Soc. Am. B 13, 1864–1872 (1996).
    [CrossRef]
  17. G. Gouesbet, L. Mees, G. Gréhan, “Partial-wave description of shaped beams in elliptical-cylinder coordinates,” J. Opt. Soc. Am. A 15, 3028–3038 (1998).
    [CrossRef]
  18. C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys. (N.Y.) 4, 65–71 (1963).
    [CrossRef]
  19. C. Yeh, “Backscattering cross section of a dielectric elliptical cylinder,” J. Opt. Soc. Am. A 55, 309–314 (1965).
    [CrossRef]
  20. P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953).
  21. T. J. Bromwich, “Electromagnetic waves,” Philos. Mag. 38, 143–164 (1919).
    [CrossRef]
  22. F. E. Borgnis, “Elektromagnetische Eigenschwingungen dielektrischer Raüme,” Ann. Phys. (Leipzig) 35, 359–384 (1939).
    [CrossRef]
  23. G. Gouesbet, G. Gréhan, B. Maheu, K. F. Ren, “Electromagnetic scattering of shaped beams (generalized Lorenz–Mie theory),” available from G. Gouesbet, LESP, UMR 6614-CNRS, INSA de Rouen, B.P. 08, 76131 Mont-Saint-Aignan Cedex, France.
  24. R. Campbell, Théorie générale de l’équation de Mathieu (Masson et Cie, Paris, 1955).
  25. N. W. McLachlan, Theory and Application of Mathieu Functions (Clarendon, Oxford, UK, 1951).
  26. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), pp. 723–745.
  27. G. Gouesbet, “Theory of distributions and its application to beam parametrization in light scattering,” Part. Part. Syst. Charact. (to be published).
  28. G. Gouesbet, L. Mees, G. Gréhan, “Partial wave expansions of higher-order Gaussian beams in elliptical cylinder coordinates,” J. Opt. (to be published).
  29. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. 19, 1177–1179 (1979).
    [CrossRef]
  30. J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
    [CrossRef]
  31. 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).
    [CrossRef] [PubMed]
  32. G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “Description of arbitrary beams in elliptical cylinder coordinates, by using a plane wave spectrum approach,” Opt. Commun. 161, 63–78 (1999).
    [CrossRef]
  33. J. A. Lock, G. Gouesbet, “Rigorous justification of the localized approximation to the beam shape coefficients in generalized Lorenz–Mie theory. 1. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
    [CrossRef]
  34. G. Gouesbet, J. A. Lock, “Rigorous justification of the localized approximation to the beam shape coefficients in generalized Lorenz–Mie theory. 2. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
    [CrossRef]
  35. G. Gouesbet, G. Gréhan, K. F. Ren, “Rigorous justification of the cylindrical localized approximation to speed up computations in the generalized Lorenz–Mie theory for cylinders,” J. Opt. Soc. Am. A 15, 511–523 (1998).
    [CrossRef]

1999 (1)

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “Description of arbitrary beams in elliptical cylinder coordinates, by using a plane wave spectrum approach,” Opt. Commun. 161, 63–78 (1999).
[CrossRef]

1998 (4)

1997 (4)

1996 (2)

H. Mignon, G. Gréhan, G. Gouesbet, T. H. Hu, C. Tropea, “Measurement of cylindrical particles with phase-Doppler anemometry,” Appl. Opt. 25, 5180–5190 (1996).
[CrossRef]

S. Lange, G. Schweiger, “Structural resonances in the total Raman- and fluorescence-scattering cross section: concentration-profile dependence,” J. Opt. Soc. Am. B 13, 1864–1872 (1996).
[CrossRef]

1995 (4)

1994 (2)

1993 (1)

1989 (1)

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

1988 (2)

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).
[CrossRef]

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

1982 (1)

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

1979 (1)

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

1965 (1)

C. Yeh, “Backscattering cross section of a dielectric elliptical cylinder,” J. Opt. Soc. Am. A 55, 309–314 (1965).
[CrossRef]

1963 (1)

C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys. (N.Y.) 4, 65–71 (1963).
[CrossRef]

1939 (1)

F. E. Borgnis, “Elektromagnetische Eigenschwingungen dielektrischer Raüme,” Ann. Phys. (Leipzig) 35, 359–384 (1939).
[CrossRef]

1919 (1)

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

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), pp. 723–745.

Adler, C. L.

Alexander, D. R.

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

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

Barton, J. B.

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).
[CrossRef]

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

Borgnis, F. E.

F. E. Borgnis, “Elektromagnetische Eigenschwingungen dielektrischer Raüme,” Ann. Phys. (Leipzig) 35, 359–384 (1939).
[CrossRef]

Bromwich, T. J.

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

Campbell, R.

R. Campbell, Théorie générale de l’équation de Mathieu (Masson et Cie, Paris, 1955).

Corbin, F.

Davis, L. W.

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

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953).

Gauchet, N.

N. Gauchet, T. Girasole, K. F. Ren, G. Gréhan, G. Gouesbet, “Application of generalized Lorenz–Mie theory for cylinders to cylindrical characterization by phase-Doppler anemometry,” Opt. Diag. Eng. 2, 1–10 (1997).

Girasole, T.

N. Gauchet, T. Girasole, K. F. Ren, G. Gréhan, G. Gouesbet, “Application of generalized Lorenz–Mie theory for cylinders to cylindrical characterization by phase-Doppler anemometry,” Opt. Diag. Eng. 2, 1–10 (1997).

Gouesbet, G.

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “Description of arbitrary beams in elliptical cylinder coordinates, by using a plane wave spectrum approach,” Opt. Commun. 161, 63–78 (1999).
[CrossRef]

G. Gouesbet, L. Mees, G. Gréhan, “Partial-wave description of shaped beams in elliptical-cylinder coordinates,” J. Opt. Soc. Am. A 15, 3028–3038 (1998).
[CrossRef]

X. Han, K. F. Ren, Z. Wu, F. Corbin, G. Gouesbet, G. Gréhan, “Characterization of initial disturbances in liquid jet by rainbow sizing,” Appl. Opt. 37, 8498–8503 (1998).
[CrossRef]

G. Gouesbet, G. Gréhan, K. F. Ren, “Rigorous justification of the cylindrical localized approximation to speed up computations in the generalized Lorenz–Mie theory for cylinders,” J. Opt. Soc. Am. A 15, 511–523 (1998).
[CrossRef]

N. Gauchet, T. Girasole, K. F. Ren, G. Gréhan, G. Gouesbet, “Application of generalized Lorenz–Mie theory for cylinders to cylindrical characterization by phase-Doppler anemometry,” Opt. Diag. Eng. 2, 1–10 (1997).

K. F. Ren, G. Gréhan, G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz–Mie theory: formulation and numerical results,” J. Opt. Soc. Am. A 14, 3014–3025 (1997).
[CrossRef]

G. Gouesbet, “Interaction between an infinite cylinder and an arbitrary shaped beam,” Appl. Opt. 36, 4292–4304 (1997).
[CrossRef] [PubMed]

H. Mignon, G. Gréhan, G. Gouesbet, T. H. Hu, C. Tropea, “Measurement of cylindrical particles with phase-Doppler anemometry,” Appl. Opt. 25, 5180–5190 (1996).
[CrossRef]

F. Onofri, G. Gréhan, G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113–7124 (1995).
[CrossRef] [PubMed]

G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions,” J. Opt. (Paris) 26, 225–239 (1995).
[CrossRef]

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).
[CrossRef] [PubMed]

J. A. Lock, G. Gouesbet, “Rigorous justification of the localized approximation to the beam shape coefficients in generalized Lorenz–Mie theory. 1. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
[CrossRef]

G. Gouesbet, J. A. Lock, “Rigorous justification of the localized approximation to the beam shape coefficients in generalized Lorenz–Mie theory. 2. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
[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, 1427–1443 (1988).
[CrossRef]

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

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), pp. 339–384.

G. Gouesbet, “Theory of distributions and its application to beam parametrization in light scattering,” Part. Part. Syst. Charact. (to be published).

G. Gouesbet, L. Mees, G. Gréhan, “Partial wave expansions of higher-order Gaussian beams in elliptical cylinder coordinates,” J. Opt. (to be published).

Gréhan, G.

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “Description of arbitrary beams in elliptical cylinder coordinates, by using a plane wave spectrum approach,” Opt. Commun. 161, 63–78 (1999).
[CrossRef]

X. Han, K. F. Ren, Z. Wu, F. Corbin, G. Gouesbet, G. Gréhan, “Characterization of initial disturbances in liquid jet by rainbow sizing,” Appl. Opt. 37, 8498–8503 (1998).
[CrossRef]

G. Gouesbet, L. Mees, G. Gréhan, “Partial-wave description of shaped beams in elliptical-cylinder coordinates,” J. Opt. Soc. Am. A 15, 3028–3038 (1998).
[CrossRef]

G. Gouesbet, G. Gréhan, K. F. Ren, “Rigorous justification of the cylindrical localized approximation to speed up computations in the generalized Lorenz–Mie theory for cylinders,” J. Opt. Soc. Am. A 15, 511–523 (1998).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz–Mie theory: formulation and numerical results,” J. Opt. Soc. Am. A 14, 3014–3025 (1997).
[CrossRef]

N. Gauchet, T. Girasole, K. F. Ren, G. Gréhan, G. Gouesbet, “Application of generalized Lorenz–Mie theory for cylinders to cylindrical characterization by phase-Doppler anemometry,” Opt. Diag. Eng. 2, 1–10 (1997).

H. Mignon, G. Gréhan, G. Gouesbet, T. H. Hu, C. Tropea, “Measurement of cylindrical particles with phase-Doppler anemometry,” Appl. Opt. 25, 5180–5190 (1996).
[CrossRef]

F. Onofri, G. Gréhan, G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113–7124 (1995).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

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

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), pp. 339–384.

G. Gouesbet, L. Mees, G. Gréhan, “Partial wave expansions of higher-order Gaussian beams in elliptical cylinder coordinates,” J. Opt. (to be published).

Han, X.

Hu, T. H.

H. Mignon, G. Gréhan, G. Gouesbet, T. H. Hu, C. Tropea, “Measurement of cylindrical particles with phase-Doppler anemometry,” Appl. Opt. 25, 5180–5190 (1996).
[CrossRef]

Lange, S.

Lock, J. A.

Maheu, B.

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).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), pp. 339–384.

McLachlan, N. W.

N. W. McLachlan, Theory and Application of Mathieu Functions (Clarendon, Oxford, UK, 1951).

Mees, L.

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “Description of arbitrary beams in elliptical cylinder coordinates, by using a plane wave spectrum approach,” Opt. Commun. 161, 63–78 (1999).
[CrossRef]

G. Gouesbet, L. Mees, G. Gréhan, “Partial-wave description of shaped beams in elliptical-cylinder coordinates,” J. Opt. Soc. Am. A 15, 3028–3038 (1998).
[CrossRef]

G. Gouesbet, L. Mees, G. Gréhan, “Partial wave expansions of higher-order Gaussian beams in elliptical cylinder coordinates,” J. Opt. (to be published).

Mignon, H.

H. Mignon, G. Gréhan, G. Gouesbet, T. H. Hu, C. Tropea, “Measurement of cylindrical particles with phase-Doppler anemometry,” Appl. Opt. 25, 5180–5190 (1996).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953).

Onofri, F.

Ren, K. F.

Schaub, S. A.

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

Schweiger, G.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), pp. 723–745.

Stone, B. R.

Tropea, C.

H. Mignon, G. Gréhan, G. Gouesbet, T. H. Hu, C. Tropea, “Measurement of cylindrical particles with phase-Doppler anemometry,” Appl. Opt. 25, 5180–5190 (1996).
[CrossRef]

Wu, Z.

Yeh, C.

C. Yeh, “Backscattering cross section of a dielectric elliptical cylinder,” J. Opt. Soc. Am. A 55, 309–314 (1965).
[CrossRef]

C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys. (N.Y.) 4, 65–71 (1963).
[CrossRef]

Zajak, P. D.

Ann. Phys. (Leipzig) (1)

F. E. Borgnis, “Elektromagnetische Eigenschwingungen dielektrischer Raüme,” Ann. Phys. (Leipzig) 35, 359–384 (1939).
[CrossRef]

Appl. Opt. (7)

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).
[CrossRef]

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

J. Math. Phys. (N.Y.) (1)

C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys. (N.Y.) 4, 65–71 (1963).
[CrossRef]

J. Opt. (Paris) (2)

G. Gouesbet, “Interaction between Gaussian beams and infinite cylinders, by using the theory of distributions,” J. Opt. (Paris) 26, 225–239 (1995).
[CrossRef]

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

J. Opt. Soc. Am. A (9)

C. Yeh, “Backscattering cross section of a dielectric elliptical cylinder,” J. Opt. Soc. Am. A 55, 309–314 (1965).
[CrossRef]

G. Gouesbet, L. Mees, G. Gréhan, “Partial-wave description of shaped beams in elliptical-cylinder coordinates,” J. Opt. Soc. Am. A 15, 3028–3038 (1998).
[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, 1427–1443 (1988).
[CrossRef]

J. A. Lock, G. Gouesbet, “Rigorous justification of the localized approximation to the beam shape coefficients in generalized Lorenz–Mie theory. 1. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
[CrossRef]

G. Gouesbet, J. A. Lock, “Rigorous justification of the localized approximation to the beam shape coefficients in generalized Lorenz–Mie theory. 2. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
[CrossRef]

G. Gouesbet, G. Gréhan, K. F. Ren, “Rigorous justification of the cylindrical localized approximation to speed up computations in the generalized Lorenz–Mie theory for cylinders,” J. Opt. Soc. Am. A 15, 511–523 (1998).
[CrossRef]

J. A. Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitely long homogeneous circular cylinder,” J. Opt. Soc. Am. A 14, 640–652 (1997).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz–Mie theory: formulation and numerical results,” J. Opt. Soc. Am. A 14, 3014–3025 (1997).
[CrossRef]

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).
[CrossRef]

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

Opt. Commun. (1)

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “Description of arbitrary beams in elliptical cylinder coordinates, by using a plane wave spectrum approach,” Opt. Commun. 161, 63–78 (1999).
[CrossRef]

Opt. Diag. Eng. (1)

N. Gauchet, T. Girasole, K. F. Ren, G. Gréhan, G. Gouesbet, “Application of generalized Lorenz–Mie theory for cylinders to cylindrical characterization by phase-Doppler anemometry,” Opt. Diag. Eng. 2, 1–10 (1997).

Philos. Mag. (1)

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

Phys. Rev. (1)

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

Other (8)

P. M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953).

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), pp. 339–384.

G. Gouesbet, G. Gréhan, B. Maheu, K. F. Ren, “Electromagnetic scattering of shaped beams (generalized Lorenz–Mie theory),” available from G. Gouesbet, LESP, UMR 6614-CNRS, INSA de Rouen, B.P. 08, 76131 Mont-Saint-Aignan Cedex, France.

R. Campbell, Théorie générale de l’équation de Mathieu (Masson et Cie, Paris, 1955).

N. W. McLachlan, Theory and Application of Mathieu Functions (Clarendon, Oxford, UK, 1951).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), pp. 723–745.

G. Gouesbet, “Theory of distributions and its application to beam parametrization in light scattering,” Part. Part. Syst. Charact. (to be published).

G. Gouesbet, L. Mees, G. Gréhan, “Partial wave expansions of higher-order Gaussian beams in elliptical cylinder coordinates,” J. Opt. (to be published).

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Equations (154)

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x2a2+y2b2=1,a>b.
z=z,x=l cosh μ cos θ,y=l sinh μ sin θ,
ds2=(hz)2 dz2+(hμ)2 dμ2+(hθ)2 dθ2,
hz=1,
hμ=hθ=l(cosh2 µ-cos2 θ)1/2
=lcosh 2µ-cos 2θ21/2.
hz=1,zhμhθ=0.
2Uz2+k2U+2l2(cosh 2µ-cos 2θ)2Uμ2+2Uθ2=0,
Hz,TM=0,
Hμ,TM=iωl(cosh2 μ-cos2 θ)1/2UTMθ,
Hθ,TM=-iωl(cosh2 μ-cos2 θ)1/2UTMμ,
Ez,TM=2UTMz2+k2UTM,
Eμ,TM=1l(cosh2 μ-cos2 θ)1/22UTMzμ,
Eθ,TM=1l(cosh2 μ-cos2 θ)1/22UTMzθ,
Ez,TE=0,
Eμ,TE=-iωαl(cosh2 μ-cos2 θ)1/2UTEθ,
Eθ,TE=iωαl(cosh2 μ-cos2 θ)1/2UTEμ,
Hz,TE=2UTEz2+k2UTE,
Hμ,TE=1l(cosh2 μ-cos2 θ)1/22UTEzμ,
Hθ,TE=1l(cosh2 μ-cos2 θ)1/22UTEzθ,
U=Z(z)M(μ)ϴ(θ).
d2Z(z)dz2+aZ(z)=0.
Z(z)=exp(ikγ z),(kγ)R.
d2M(μ)dμ2-(b-2q2 cosh 2µ)M(μ)=0,b,
d2ϴ(θ)dθ 2+(b-2q2 cos 2θ)ϴ(θ)=0,b,
q=kl21-γ2.
UTMi=E0k2n=0[An,TMi(γ), cehn(μ, q2)×cen(θ, q2)exp(ikγ z)+Bn,TM(γ), sehn(μ, q2)sen(θ, q2)×exp(ikγ z)],
UTEi=H0k2n=0[An,TEi(γ), cehn(μ, q2)×cen(θ, q2)exp(ikγ z)+Bn,TE(γ), sehn(μ, q2)sen(θ, q2)exp(ikγ z)],
UTMi=E0k2n=0[An,TMi(γ)cehn(μ, q2)cen(θ, q2)+Bn,TMi(γ)sehn(μ, q2)sen(θ, q2)]exp(ikγ z)dγ,
UTEi=H0k2n=0[An,TEi(γ)cehn(μ, q2)cen(θ, q2)+Bn,TEi(γ)sehn(μ, q2)sen(θ, q2)]exp(ikγ z)dγ,
Hzi=H0n=0[An,TEi(γ)cehn(μ, q2)cen(θ, q2)+Bn,TEi(γ)sehn(μ, q2)sen(θ, q2)](1-γ 2)×exp(ikγ z)dγ,
Hμi=iH0kDn=0[An,TMi(γ)cehn(μ, q2)cen(θ, q2)+Bn,TMi(γ)sehn(μ, q2)sen(θ, q2)+γAn,TEi(γ)cehn(μ, q2)cen(θ, q2)+γBn,TEi(γ)sehn(μ, q2)sen(θ, q2)]exp(ikγ z)dγ,
Hθi=iH0kDn=0[γAn,TEi(γ)cehn(μ, q2)cen(θ, q2)+γBn,TEi(γ)sehn(μ, q2)sen(θ, q2)-An,TMi(γ)cehn(μ, q2)cen(θ, q2)-Bn,TMi(γ)sehn(μ, q2)sen(θ, q2)]exp(ikγ z)dγ,
Ezi=E0n=0[An,TMi(γ)cehn(μ, q2)cen(θ, q2)+Bn,TMi(γ)sehn(μ, q2)sen(θ, q2)](1-γ 2)×exp(ikγ z)dγ,
Eμi=iE0kDn=0[γAn,TMi(γ)cehn(μ, q2)cen(θ, q2)+γBn,TMi(γ)sehn(μ, q2)sen(θ, q2)-An,TEi(γ)cehn(μ, q2)cen(θ, q2)-Bn,TEi(γ)sehn(μ, q2)sen(θ, q2)]exp(ikγ z)dγ,
Eθi=iE0kDn=0[γAn,TMi(γ)cehn(μ, q2)cen(θ, q2)+γBn,TMi(γ)sehn(μ, q2)sen(θ, q2)+An,TEi(γ)cehn(μ, q2)cen(θ, q2)+Bn,TEi(γ)sehn(μ, q2)sen(θ, q2)]exp(ikγ z)dγ,
ωα=kH0/E0E0/H0
D=l(cosh2 μ-cos2 θ)1/2.
mehn(μ, q2)=cehn(μ, q2)±ifehn(μ, q2),
nehn(μ, q2)=sehn(μ, q2)±igehn(μ, q2),
UTMs=E0k2n=0[An,TMs(γ)mehn(μ, q2)cen(θ, q2)+Bn,TMs(γ)nehn(μ, q2)sen(θ, q2)]exp(ikγ z)dγ,
UTEs=H0k2n=0[An,TEs(γ)mehn(μ, q2)cen(θ, q2)+Bn,TEs(γ)nehn(μ, q2)sen(θ, q2)]exp(ikγ z)dγ,
Hzs=H0n=0[An,TEs(γ)mehn(μ, q2)cen(θ, q2)+Bn,TEs(γ)nehn(μ, q2)sen(θ, q2)](1-γ 2)×exp(ikγ z)dγ,
Hμs=iH0kDn=0[An,TMs(γ)mehn(μ, q2)cen(θ, q2)+Bn,TMs(γ)nehn(μ, q2)sen(θ, q2)+γAn,TEs(γ)mehn(μ, q2)cen(θ, q2)+γBn,TEs(γ)nehn(μ, q2)sen(θ, q2)]exp(ikγ z)dγ,
Hθs=iH0kDn=0[γAn,TEs(γ)mehn(μ, q2)cen(θ, q2)+γBn,TEs(γ)nehn(μ, q2)sen(θ, q2)-An,TMs(γ)mehn(μ, q2)cen(θ, q2)-Bn,TMs(γ)nehn(μ, q2)sen(θ, q2)]exp(ikγ z)dγ,
Ezs=E0n=0[An,TMs(γ)mehn(μ, q2)cen(θ, q2)+Bn,TMs(γ)nehn(μ, q2)sen(θ, q2)](1-γ 2)×exp(ikγ z)dγ,
Eμs=iE0kDn=0[γAn,TMs(γ)mehn(μ, q2)cen(θ, q2)+γBn,TMs(γ)nehn(μ, q2)sen(θ, q2)-An,TEs(γ)mehn(μ, q2)cen(θ, q2)-Bn,TEs(γ)nehn(μ, q2)sen(θ, q2)]exp(ikγ z)dγ,
Eθs=iE0kDn=0[γAn,TMs(γ)mehn(μ, q2)cen(θ, q2)+γBn,TMs(γ)nehn(μ, q2)sen(θ, q2)+An,TEs(γ)mehn(μ, q2)cen(θ, q2)+Bn,TEs(γ)nehn(μ, q2)sen(θ, q2)]exp(ikγ z)dγ.
kc=Mk,
UTMc=E0k2n=0[An,TMc(γ)cehn(μ, qc2)cen(θ, qc2)+Bn,TMc(γ)sehn(μ, qc2)sen(θ, qc2)]exp(ikγ z)dγ,
UTEc=H0k2n=0[An,TEc(γ)cehn(μ, qc2)cen(θ, qc2)+Bn,TEc(γ)sehn(μ, qc2)sen(θ, qc2)]exp(ikγ z)dγ,
qc=kl2M2-γ 2.
c=M2,
Hzc=H0n=0[An,TEc(γ)cehn(μ, qc2)cen(θ, qc2)+Bn,TEc(γ)sehn(μ, qc2)sen(θ, qc2)](M2-γ 2)×exp(ikγ z)dγ,
Hμc=iH0kDn=0[M2An,TMc(γ)cehn(μ, qc2)cen(θ, qc2)+M2Bn,TMc(γ)sehn(μ, qc2)sen(θ, qc2)+γAn,TEc(γ)cehn(μ, qc2)cen(θ, qc2)+γBn,TEc(γ)sehn(μ, qc2)sen(θ, qc2)]×exp(ikγ z)dγ,
Hθc=iH0kDn=0[γAn,TEc(γ)cehn(μ, qc2)cen(θ, qc2)+γBn,TEc(γ)sehn(μ, qc2)sen(θ, qc2)-M2An,TMc(γ)cehn(μ, qc2)cen(θ, qc2)-M2Bn,TMc(γ)sehn(μ, qc2)sen(θ, qc2)]×exp(ikγ z)dγ,
Ezc=E0n=0[An,TMc(γ)cehn(μ, qc2)cen(θ, qc2)+Bn,TMc(γ)sehn(μ, qc2)sen(θ, qc2)](M2-γ 2)×exp(ikγ z)dγ,
Eμc=iE0kDn=0[γAn,TMc(γ)cehn(μ, qc2)cen(θ, qc2)+γBn,TMc(γ)sehn(μ, qc2)sen(θ, qc2)-An,TEc(γ)cehn(μ, qc2)cen(θ, qc2)-Bn,TEc(γ)sehn(μ, qc2)sen(θ, qc2)]×exp(ikγ z)dγ,
Eθc=iE0kDn=0[γAn,TMc(γ)cehn(μ, qc2)cen(θ, qc2)+γBn,TMc(γ)sehn(μ, qc2)sen(θ, qc2)+An,TEc(γ)cehn(μ, qc2)cen(θ, qc2)+Bn,TEc(γ)sehn(μ, qc2)sen(θ, qc2)]×exp(ikγ z)dγ.
μ=μ0=const.
Ezc-Ezs=Ezi,
Eθc-Eθs=Eθi,
Hzc-Hzs=Hzi,
Hθc-Hθs=Hθi,
δ(γ-γ)=12π-+ exp[i(γ-γ)z]dz,
f(γ)δ(γ-γ)dγ=f(γ),
n=0[(M2-γ 2)An,TMc(γ)cehn(μ0, qc2)cen(θ, qc2)+(M2-γ 2)Bn,TMc(γ)sehn(μ0, qc2)sen(θ, qc2)
-(1-γ 2)An,TMs(γ)mehn(μ0, q2)cen(θ, q2)-(1-γ 2)Bn,TMs(γ)nehn(μ0, q2)sen(θ, q2)]
=n=0(1-γ 2)[An,TMi(γ)cehn(μ0, q2)cen(θ, q2)+Bn,TMi(γ)sehn(μ0, q2)sen(θ, q2)],
n=0[γAn,TMc(γ)cehn(μ0, qc2)cen(θ, qc2)+γBn,TMc(γ)sehn(μ0, qc2)sen(θ, qc2)
+An,TEc(γ)cehn(μ0, qc2)cen(θ, qc2)+Bn,TEc(γ)sehn(μ0, qc2)sen(θ, qc2)
-γAn,TMs(γ)mehn(μ0, q2)cen(θ, q2)-γBn,TMs(γ)nehn(μ0, q2)sen(θ, q2)
-An,TEs(γ)mehn(μ0, q2)cen(θ, q2)-Bn,TEs(γ)nehn(μ0, q2)sen(θ, q2)]
=n=0[γAn,TMi(γ)cehn(μ0, q2)cen(θ, q2)+γBn,TMi(γ)sehn(μ0, q2)sen(θ, q2)+An,TEi(γ)cehn(μ0, q2)cen(θ, q2)+Bn,TEi(γ)sehn(μ0, q2)sen(θ, q2)],
n=0[(M2-γ 2)An,TEc(γ)cehn(μ0, qc2)cen(θ, qc2)+(M2-γ 2)Bn,TEc(γ)sehn(μ0, qc2)sen(θ, qc2)
-(1-γ 2)An,TEs(γ)mehn(μ0, q2)cen(θ, q2)-(1-γ 2)Bn,TEs(γ)nehn(μ0, q2)sen(θ, q2)]
=n=0(1-γ 2)[An,TEi(γ)cehn(μ0, q2)cen(θ, q2)+Bn,TEi(γ)sehn(μ0, q2)sen(θ, q2)],
n=0[γAn,TEc(γ)cehn(μ0, qc2)cen(θ, qc2)+γBn,TEc(γ)sehn(μ0, qc2)sen(θ, qc2)
-M2An,TMc(γ)cehn(μ0, qc2)cen(θ, qc2)-M2Bn,TMc(γ)sehn(μ0, qc2)sen(θ, qc2)
-γAn,TEs(γ)mehn(μ0, q2)cen(θ, q2)-γBn,TEs(γ)nehn(μ0, q2)sen(θ, q2)
+An,TMs(γ)mehn(μ0, q2)cen(θ, q2)+Bn,TMs(γ)nehn(μ0, q2)sen(θ, q2)]
=n=0[γAn,TEi(γ)cehn(μ0, q2)cen(θ, q2)+γBn,TEi(γ)sehn(μ0, q2)sen(θ, q2)-An,TMi(γ)cehn(μ0, q2)cen(θ, q2)-Bn,TMi(γ)sehn(μ0, q2)sen(θ, q2)].
cen(θ, qc2)=pαnpcep(θ, q2),
sen(θ, qc2)=pβnpsep(θ, q2),
I={0, 2, 4,},neven{1, 3, 5,},nodd,
02πcen(x, q2)cem(x, q2)dx=πδ nm,
02πsen(x, q2)sem(x, q2)dx=πδ nm,
02πcen(x, q2)sem(x, q2)dx=0.
amAm,TMs+n=0anm An,TMc=Im1,
bmBm,TMs+n=0bnm Bn,TMc=Im2,
amAm,TEs+n=0anmAn,TEc=Im3,
bmBm,TEs+n=0bnmBn,TEc=Im4,
anm=(M2-γ 2)cehn(μ0, qc2)αnm,
bnm=(M2-γ 2)sehn(μ0, qc2)βnm,
am=-(1-γ 2)mehm(μ0, q2),
bm=-(1-γ 2)nehm(μ0, q2),
Im1=Am,TMi cehm(μ0, q2)(1-γ 2),
Im2=Bm,TMi sehm(μ0, q2)(1-γ 2),
Im3=Am,TEi cehm(μ0, q2)(1-γ 2),
Im4=Bm,TEi sehm(μ0, q2)(1-γ 2).
ce2n(x, q2)=r=0A2r2n(q2)cos(2rx),
ce2n+1(x, q2)=r=0A2r+12n+1(q2)cos[(2r+1)x],
se2n+1(x, q2)=r=0B2r+12n+1 sin[(2r+1)x],
se2n+2(x, q2)=r=0B2r+22n+2 sin[(2r+2)x].
02πcen(θ, q2)cem(θ, q2)dθ=0,
02πsen(θ, q2)sem(θ, q2)dθ=0,
Lnm=1π02πsen(θ, q2)cem(θ, q2)dθ=r=0(2r+2)B2r+22N+2A2r+2m,n=2N+2r=0(2r+1)B2r+12N+1A2r+1m,n=2N+1,
Snm=1π02πcen(θ, q2)sem(θ, q2)dθ=-s=0(2s+2)B2s+22M+2A2s+2nm=2M+2-s=0(2s+1)B2s+12M+1A2s+1nm=2M+1,
02πcen(θ, qc2)cem(θ, q2)dθ=0,
02πsen(θ, qc2)sem(θ, q2)dθ=0,
02πsen(θ, qc2)cem(θ, q2)dθ=πpβnpLpm=πγnm,
02πcen(θ, qc2)sem(θ, q2)dθ=πpαnpSpm=πnm.
Lnm=-Smn.
cm Am,TEs
+n=0[cnmBn,TMc+dnm An,TEc+enmBn,TMs]=Im5,
dmBm,TEs
+n=0[fnm An,TMc+gnmBn,TEc+hnm An,TMs]=Im6,
-cm Am,TMs
+n=0[cnmBn,TEc-M2dnm An,TMc+enmBn,TEs]=Im7,
-dmBm,TMs
+n=0[fnm An,TEc-M2gnmBn,TMc+hnm An,TEs]=Im8,
cnm=γ sehn(μ0, qc2)γ nm,
dnm=cehn(μ0, qc2)αnm,
enm=-γ nehn(μ0, q2)Lnm,
fnm=γ cehn(μ0, qc2)nm,
gnm=sehn(μ0, qc2)βnm,
hnm=-γ mehn(μ0, q2)Snm,
cm=-mehm(μ0, q2),
dm=-nehm(μ0, q2),
Im5=Am,TEi cehm(μ0, q2)+n=0γ Bn,TMisehn(μ0, q2)Lnm,
Im6=Bm,TEi sehm(μ0, q2)+n=0γ An,TMi cehn(μ0, q2)Snm,
Im7=-Am,TMi cehm(μ0, q2)+n=0γ Bn,TEi sehn(μ0, q2)Lnm,
Im8=-Bm,TMi sehm(μ0, q2)+n=0γ An,TEi cehn(μ0, q2)Snm.
enmam=-bnhmn.
n=0(Bn,TMc Anm+An,TEc Bnm)=Jm1,
n=0(Bn,TMc Cnm+An,TEc Dnm)=Jm4,
n=0(An,TMc Dnm+Bn,TEc Enm)=Jm2,
n=0(An,TMc Fnm+Bn,TEc Anm)=Jm3,
Anm=amcnm+p=0bnphmp,
Bnm=amdnm-cmanm,
Cnm=dmbnm-M2bm gnm,
Dnm=bm fnm+p=0empanp,
Enm=bm gnm-dmbnm,
Fnm=cmanm-M2amdnm,
Jm1=am Im5-cm Im3+n=0hmn In2,
Jm2=bm Im6-dm Im4+n=0emn In1,
Jm3=am Im7+cm Im1+n=0hmn In4,
Jm4=bm Im8+dm Im2+n=0emn In3.
n=0[Bn,TMc(Anm+Cnm)+An,TEc(Bnm+Dnm)]
=Jm1+Jm4,
n=0[An,TMc(Dnm+Fnm)+Bn,TEc(Enm+Anm)]
=Jm2+Jm3.
Ai jVj=Wi,
Vk=Aki-1Wi,

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