Abstract

We establish a localized approximation to evaluate the beam-shape coefficients of a Gaussian beam in elliptical cylinder coordinates. As for the case of spherical coordinates and of circular cylinder coordinates, this approximation provides an efficient way to speed up computations within the framework of a generalized Lorenz–Mie theory for elliptical cylinders.

© 2000 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, Washington, D.C., 1991), Chap. 10, pp. 339–384.
  3. 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]
  4. G. Gouesbet, “Scattering of higher-order Gaussian beams by an infinite cylinder,” J. Opt. 28, 45–65 (1997).
    [CrossRef]
  5. G. Gouesbet, “Interaction between an infinite cylinder and an arbitrary-shaped beam,” Appl. Opt. 36, 4292–4304 (1997).
    [CrossRef] [PubMed]
  6. K. F. Ren, G. Gréhan, G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in GLMT—framework, formulation, and numerical results,” J. Opt. Soc. Am. A 14, 3014–3025 (1997).
    [CrossRef]
  7. H. Mignon, G. Gréhan, G. Gouesbet, T. H. Xu, C. Tropea, “Measurement of cylindrical particles with phase Doppler anemometry,” Appl. Opt. 25, 5180–5190 (1996).
    [CrossRef]
  8. J. P. A. J. van Beeck, M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. 35, 2259–2266 (1996).
    [CrossRef] [PubMed]
  9. 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]
  10. E. Lenglart, G. Gouesbet, “The separability theorem in terms of distributions with discussion of electromagnetic scattering theory,” J. Math. Phys. 37, 4705–4710 (1996).
    [CrossRef]
  11. 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]
  12. G. Gouesbet, L. Mees, G. Gréhan, “Partial wave expansions of higher-order Gaussian beams in elliptical cylindrical coordinates,” J. Opt. A 1, 121–132 (1999).
    [CrossRef]
  13. 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]
  14. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).
  15. 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).
    [CrossRef] [PubMed]
  16. B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
    [CrossRef]
  17. G. Gouesbet, G. Gréhan, B. Maheu, “On the generalized Lorenz–Mie theory: first attempt to design a localized approximation to the computation of the coefficients gnm,” J. Opt. (Paris) 20, 31–43 (1989).
    [CrossRef]
  18. G. Gouesbet, G. Gréhan, B. Maheu, “A localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
    [CrossRef]
  19. K. F. Ren, G. Gréhan, G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 (1994).
    [CrossRef]
  20. 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]
  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).
    [CrossRef]
  22. J. A. Lock, G. Gouesbet, “A 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).
    [CrossRef]
  23. G. Gouesbet, J. A. Lock, “A 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).
    [CrossRef]
  24. G. Gouesbet, G. Gréhan, K. F. Ren, “Rigorous justification of the cylindrical localized approximation to speed up computations in GLMT for cylinders,” J. Opt. Soc. Am. A 15, 511–523 (1998).
    [CrossRef]
  25. G. Gouesbet, K. F. Ren, L. Mees, G. Gréhan, “Cylindrical localized approximation to speed up computations for Gaussian beams in the generalized Lorentz–Mie theory for cylinders with arbitrary location and orientation of the scatterer,” Appl. Opt. 38, 2647–2665 (1999).
    [CrossRef]
  26. J. A. Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitely long homogeneous circular cylinder. I. Parameterization of the incident beam and the far-zone scattered intensity,” J. Opt. Soc. Am. A 14, 640–652 (1997).
    [CrossRef]
  27. C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys. 4, 65–71 (1963).
    [CrossRef]
  28. C. Yeh, “Backscattering cross section of a dielectric elliptical cylinder,” J. Opt. Soc. Am. 55, 309–314 (1965).
    [CrossRef]
  29. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  30. T. J. Bromwich, “Electromagnetic waves,” Philos. Mag. 38, 143–164 (1919).
    [CrossRef]
  31. F. E. Borgnis, “Elektromagnetische Eigenschwingungen dielektrischer Raüme,” Ann. Phys. 35, 359–384 (1939).
    [CrossRef]
  32. P. Poincelot, Précis d’Electromagnétisme Théorique (Masson, Paris, 1965).
  33. G. Gouesbet, G. Gréhan, B. Maheu, K. F. Ren, Electromagnetic Scattering of Shaped Beams (Generalized Lorenz–Mie Theory, textbook manuscript in preparation; available from the authors on request.
  34. R. Campbell, Théorie Générale de l’Équation de Mathieu (Masson, Paris, 1955).
  35. N. W. McLachlan, Theory and Application of Mathieu Functions (Clarendon, Oxford, 1951).
  36. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 20, pp. 723–745.
  37. G. Gouesbet, “Theory of distributions and its application to beam parameterization in light scattering,” Part. Part. Syst. Charact. 16, 147–159 (1999).
    [CrossRef]
  38. G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “Description of arbitrary shaped beams in elliptical cylinder coordinates, by using a plane wave spectrum approach,” Opt. Commun. 161, 63–78 (1999).
    [CrossRef]
  39. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  40. J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
    [CrossRef]
  41. G. Gouesbet, “Higher-order descriptions of Gaussian beams,” J. Opt. 27, 35–50 (1996).
    [CrossRef]

1999 (4)

G. Gouesbet, L. Mees, G. Gréhan, “Partial wave expansions of higher-order Gaussian beams in elliptical cylindrical coordinates,” J. Opt. A 1, 121–132 (1999).
[CrossRef]

G. Gouesbet, K. F. Ren, L. Mees, G. Gréhan, “Cylindrical localized approximation to speed up computations for Gaussian beams in the generalized Lorentz–Mie theory for cylinders with arbitrary location and orientation of the scatterer,” Appl. Opt. 38, 2647–2665 (1999).
[CrossRef]

G. Gouesbet, “Theory of distributions and its application to beam parameterization in light scattering,” Part. Part. Syst. Charact. 16, 147–159 (1999).
[CrossRef]

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

1998 (3)

1997 (4)

1996 (4)

G. Gouesbet, “Higher-order descriptions of Gaussian beams,” J. Opt. 27, 35–50 (1996).
[CrossRef]

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

J. P. A. J. van Beeck, M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. 35, 2259–2266 (1996).
[CrossRef] [PubMed]

E. Lenglart, G. Gouesbet, “The separability theorem in terms of distributions with discussion of electromagnetic scattering theory,” J. Math. Phys. 37, 4705–4710 (1996).
[CrossRef]

1995 (2)

1994 (3)

1993 (1)

1990 (1)

1989 (3)

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

G. Gouesbet, G. Gréhan, B. Maheu, “On the generalized Lorenz–Mie theory: first attempt to design a localized approximation to the computation of the coefficients gnm,” J. Opt. (Paris) 20, 31–43 (1989).
[CrossRef]

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 (1)

1987 (1)

1986 (1)

1979 (1)

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

1965 (1)

1963 (1)

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

1939 (1)

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

1919 (1)

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

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]

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]

Borgnis, F. E.

F. E. Borgnis, “Elektromagnetische Eigenschwingungen dielektrischer Raüme,” Ann. Phys. 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, Paris, 1955).

Corbin, F.

Davis, L. W.

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

Feshbach, H.

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

Gouesbet, G.

G. Gouesbet, “Theory of distributions and its application to beam parameterization in light scattering,” Part. Part. Syst. Charact. 16, 147–159 (1999).
[CrossRef]

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “Description of arbitrary shaped 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 expansions of higher-order Gaussian beams in elliptical cylindrical coordinates,” J. Opt. A 1, 121–132 (1999).
[CrossRef]

G. Gouesbet, K. F. Ren, L. Mees, G. Gréhan, “Cylindrical localized approximation to speed up computations for Gaussian beams in the generalized Lorentz–Mie theory for cylinders with arbitrary location and orientation of the scatterer,” Appl. Opt. 38, 2647–2665 (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 GLMT 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 GLMT—framework, formulation, and numerical results,” J. Opt. Soc. Am. A 14, 3014–3025 (1997).
[CrossRef]

G. Gouesbet, “Scattering of higher-order Gaussian beams by an infinite cylinder,” J. Opt. 28, 45–65 (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. Xu, C. Tropea, “Measurement of cylindrical particles with phase Doppler anemometry,” Appl. Opt. 25, 5180–5190 (1996).
[CrossRef]

E. Lenglart, G. Gouesbet, “The separability theorem in terms of distributions with discussion of electromagnetic scattering theory,” J. Math. Phys. 37, 4705–4710 (1996).
[CrossRef]

G. Gouesbet, “Higher-order descriptions of Gaussian beams,” J. Opt. 27, 35–50 (1996).
[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]

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]

K. F. Ren, G. Gréhan, G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 (1994).
[CrossRef]

J. A. Lock, G. Gouesbet, “A 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).
[CrossRef]

G. Gouesbet, J. A. Lock, “A 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).
[CrossRef]

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

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

G. Gouesbet, G. Gréhan, B. Maheu, “On the generalized Lorenz–Mie theory: first attempt to design a localized approximation to the computation of the coefficients gnm,” J. Opt. (Paris) 20, 31–43 (1989).
[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]

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).
[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, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, Washington, D.C., 1991), Chap. 10, pp. 339–384.

G. Gouesbet, G. Gréhan, B. Maheu, K. F. Ren, Electromagnetic Scattering of Shaped Beams (Generalized Lorenz–Mie Theory, textbook manuscript in preparation; available from the authors on request.

Gréhan, G.

G. Gouesbet, L. Mees, G. Gréhan, “Partial wave expansions of higher-order Gaussian beams in elliptical cylindrical coordinates,” J. Opt. A 1, 121–132 (1999).
[CrossRef]

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

G. Gouesbet, K. F. Ren, L. Mees, G. Gréhan, “Cylindrical localized approximation to speed up computations for Gaussian beams in the generalized Lorentz–Mie theory for cylinders with arbitrary location and orientation of the scatterer,” Appl. Opt. 38, 2647–2665 (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 GLMT 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 GLMT—framework, formulation, and numerical results,” J. Opt. Soc. Am. A 14, 3014–3025 (1997).
[CrossRef]

H. Mignon, G. Gréhan, G. Gouesbet, T. H. Xu, 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]

K. F. Ren, G. Gréhan, G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 (1994).
[CrossRef]

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

G. Gouesbet, G. Gréhan, B. Maheu, “On the generalized Lorenz–Mie theory: first attempt to design a localized approximation to the computation of the coefficients gnm,” J. Opt. (Paris) 20, 31–43 (1989).
[CrossRef]

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

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).
[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, K. F. Ren, Electromagnetic Scattering of Shaped Beams (Generalized Lorenz–Mie Theory, textbook manuscript in preparation; available from the authors on request.

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, Washington, D.C., 1991), Chap. 10, pp. 339–384.

Han, X.

Lenglart, E.

E. Lenglart, G. Gouesbet, “The separability theorem in terms of distributions with discussion of electromagnetic scattering theory,” J. Math. Phys. 37, 4705–4710 (1996).
[CrossRef]

Lock, J. A.

Maheu, B.

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

G. Gouesbet, G. Gréhan, B. Maheu, “On the generalized Lorenz–Mie theory: first attempt to design a localized approximation to the computation of the coefficients gnm,” J. Opt. (Paris) 20, 31–43 (1989).
[CrossRef]

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

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).
[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, K. F. Ren, Electromagnetic Scattering of Shaped Beams (Generalized Lorenz–Mie Theory, textbook manuscript in preparation; available from the authors on request.

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, Washington, D.C., 1991), Chap. 10, pp. 339–384.

McLachlan, N. W.

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

Mees, L.

G. Gouesbet, L. Mees, G. Gréhan, K. F. Ren, “Description of arbitrary shaped 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 expansions of higher-order Gaussian beams in elliptical cylindrical coordinates,” J. Opt. A 1, 121–132 (1999).
[CrossRef]

G. Gouesbet, K. F. Ren, L. Mees, G. Gréhan, “Cylindrical localized approximation to speed up computations for Gaussian beams in the generalized Lorentz–Mie theory for cylinders with arbitrary location and orientation of the scatterer,” Appl. Opt. 38, 2647–2665 (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]

Mignon, H.

H. Mignon, G. Gréhan, G. Gouesbet, T. H. Xu, 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 (McGraw-Hill, New York, 1953).

Onofri, F.

Poincelot, P.

P. Poincelot, Précis d’Electromagnétisme Théorique (Masson, Paris, 1965).

Ren, K. F.

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

G. Gouesbet, K. F. Ren, L. Mees, G. Gréhan, “Cylindrical localized approximation to speed up computations for Gaussian beams in the generalized Lorentz–Mie theory for cylinders with arbitrary location and orientation of the scatterer,” Appl. Opt. 38, 2647–2665 (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, G. Gréhan, K. F. Ren, “Rigorous justification of the cylindrical localized approximation to speed up computations in GLMT 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 GLMT—framework, formulation, and numerical results,” J. Opt. Soc. Am. A 14, 3014–3025 (1997).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 (1994).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, K. F. Ren, Electromagnetic Scattering of Shaped Beams (Generalized Lorenz–Mie Theory, textbook manuscript in preparation; available from the authors on request.

Riethmuller, M. L.

Tropea, C.

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

van Beeck, J. P. A. J.

van de Hulst, H. C.

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

Wu, Z.

Xu, T. H.

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

Yeh, C.

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

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

Ann. Phys. (1)

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

Appl. Opt. (9)

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, K. F. Ren, L. Mees, G. Gréhan, “Cylindrical localized approximation to speed up computations for Gaussian beams in the generalized Lorentz–Mie theory for cylinders with arbitrary location and orientation of the scatterer,” Appl. Opt. 38, 2647–2665 (1999).
[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]

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

J. P. A. J. van Beeck, M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. 35, 2259–2266 (1996).
[CrossRef] [PubMed]

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, “Interaction between an infinite cylinder and an arbitrary-shaped beam,” Appl. Opt. 36, 4292–4304 (1997).
[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, 23–26 (1987).
[CrossRef] [PubMed]

J. Appl. Phys. (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]

J. Math. Phys. (2)

E. Lenglart, G. Gouesbet, “The separability theorem in terms of distributions with discussion of electromagnetic scattering theory,” J. Math. Phys. 37, 4705–4710 (1996).
[CrossRef]

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

J. Opt. (2)

G. Gouesbet, “Scattering of higher-order Gaussian beams by an infinite cylinder,” J. Opt. 28, 45–65 (1997).
[CrossRef]

G. Gouesbet, “Higher-order descriptions of Gaussian beams,” J. Opt. 27, 35–50 (1996).
[CrossRef]

J. Opt. (Paris) (1)

G. Gouesbet, G. Gréhan, B. Maheu, “On the generalized Lorenz–Mie theory: first attempt to design a localized approximation to the computation of the coefficients gnm,” J. Opt. (Paris) 20, 31–43 (1989).
[CrossRef]

J. Opt. A (1)

G. Gouesbet, L. Mees, G. Gréhan, “Partial wave expansions of higher-order Gaussian beams in elliptical cylindrical coordinates,” J. Opt. A 1, 121–132 (1999).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (10)

J. A. Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitely long homogeneous circular cylinder. I. Parameterization of the incident beam and the far-zone scattered intensity,” J. Opt. Soc. Am. A 14, 640–652 (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. A. Lock, G. Gouesbet, “A 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).
[CrossRef]

G. Gouesbet, J. A. Lock, “A 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).
[CrossRef]

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

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

K. F. Ren, G. Gréhan, G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in GLMT—framework, formulation, and numerical results,” J. Opt. Soc. Am. A 14, 3014–3025 (1997).
[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, L. Mees, G. Gréhan, “Partial wave description of shaped beams in elliptical cylinder coordinates,” J. Opt. Soc. Am. A 15, 3028–3038 (1998).
[CrossRef]

Opt. Commun. (2)

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

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

Part. Part. Syst. Charact. (1)

G. Gouesbet, “Theory of distributions and its application to beam parameterization in light scattering,” Part. Part. Syst. Charact. 16, 147–159 (1999).
[CrossRef]

Philos. Mag. (1)

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

Phys. Rev. A (1)

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

Other (8)

P. Poincelot, Précis d’Electromagnétisme Théorique (Masson, Paris, 1965).

G. Gouesbet, G. Gréhan, B. Maheu, K. F. Ren, Electromagnetic Scattering of Shaped Beams (Generalized Lorenz–Mie Theory, textbook manuscript in preparation; available from the authors on request.

R. Campbell, Théorie Générale de l’Équation de Mathieu (Masson, Paris, 1955).

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

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 20, pp. 723–745.

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

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

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, Washington, D.C., 1991), Chap. 10, pp. 339–384.

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Figures (12)

Fig. 1
Fig. 1

Relationship between Cartesian coordinates (x, y) and (u, w).

Fig. 2
Fig. 2

Real part of A 0,TE(γ) and A0,TE¯(γ) versus γ for s = 10-2, λ = 0.5 µm, β = 0°, and l = 1 µm.

Fig. 3
Fig. 3

Real part of A 0,TE(γ) and A0,TE¯(γ) versus γ for s = 10-3, λ = 0.5 µm, β = 0°, and l = 1 µm.

Fig. 4
Fig. 4

Real part of A 0,TE(γ) and A0,TE¯(γ) versus γ for s = 10-1, λ = 0.5 µm, β = 0°, and l = 1 µm.

Fig. 5
Fig. 5

Real part of A 0,TE(γ) and A0,TE¯(γ) versus γ for s = 10-2, λ = 0.5 µm, β = 0°, and l = 2 µm.

Fig. 6
Fig. 6

Real part of H Z versus μ for s = 10-2, λ = 0.5 µm, β = 0°, l = 2 µm, and θ = π/2, computed directly from the Davis scheme and obtained by summations from ECLAT.

Fig. 7
Fig. 7

Real part of H Z versus μ for s = 10-2, λ = 0.5 µm, β = 0°, l = 3 µm, and θ = π/2, computed directly from the Davis scheme and obtained by summations from ECLAT.

Fig. 8
Fig. 8

Real part of A 0,TE(γ) and A0,TE¯(γ) versus γ for s = 10-2, λ = 0.5 µm, β = 30°, and l = 2 µm.

Fig. 9
Fig. 9

Real part of H Z versus μ for s = 10-2, λ = 0.5 µm, β = 30°, l = 2 µm, and θ = π/2, computed directly from the Davis scheme and obtained by summations from ECLAT.

Fig. 10
Fig. 10

Real part of A 0,TE(γ) and A0,TE¯(γ) versus γ for s = 10-2, λ = 0.5 µm, β = 89.9°, and l = 2 µm.

Fig. 11
Fig. 11

Real part of H Z versus μ for s = 10-2, λ = 0.5 µm, β = 89.9°, l = 2 µm, and θ = π/2, computed directly from the Davis scheme and obtained by summations from ECLAT.

Fig. 12
Fig. 12

Real part of A 30,TE(γ) and A30,TE¯(γ) versus γ for s = 10-2, λ = 0.5 µm, β = 0°, and l = 2 µm.

Equations (182)

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z=z,  x=l cosh μ cos θ,  y=l sinh μ sin θ,
x2a2+y2b2=1,  a>b.
ds2=hz2dz2+hμ2dμ2+hθ2dθ2,
hz=1,
hμ=hθ=lcosh2 μ-cos2 θ1/2=lcosh 2μ-cos 2θ21/2.
hz=1,  zhμhθ=0.
2Uz2+k2U+2l2cosh 2μ-cos 2θ2Uμ2+2Uθ2=0,
Hz=Hz,TE=2UTEz2+k2UTE,
Ez=Ez,TM=2UTMz2+k2UTM.
Zz=expikγz,  kγ,
q2=k2l241-γ2,
UTE=H0k2n=0An,TEγ, cehnμ, q2cenθ, q2expikγz+Bn,TEγ, sehnμ, q2senθ, q2expikγz,
UTE=H0k2n=0-1+1 An,TEγcehnμ, q2cenθ, q2×expikγzdγ+-1+1 Bn,TEγsehnμ, q2×senθ, q2expikγzdγ,
Eu=E0 exp-iW1+s22iW-U2+V2+s4-4W2-4iWU2+V2+3U2+V2+1/2U2+V22+s6-8iW3+12W2U2+V2+18iWU2+6iWV2+3iWU2+V22-6U2V2-4U4-2V4-1/6U2+V23,
Ev=E0UV exp-iW2s4+s612iW-2U2+V2,
Ew=E0U exp-iW2is2+s4-8W-2iU2+V2+s6-24iW2+12WU2+V2+6iV2+6iU2+iU2+V22,
Hu=H0E0 Ev,
Hv=H0E0 Eu+H0 exp-iW-2s4U2-V2+s6-12iWU2-V2+2U4-V4,
Hw=H0E0VU Ew,
U, V, W=ku, kv, kw
s=1/kw0,
Ez=Ev, Hz=Hv,
u=-x sin β+y cos β,
v=z,
w=x cos β+y sin β,
An,TMγ=An,TM4γs4+An,TM6γs6,
An,TEγ=An,TE0γ+An,TE2γs2+An,TE4γs4+An,TE6γs6,
Bn,TMγ=Bn,TM4γs4+Bn,TM6γs6,
Bn,TEγ=Bn,TE0γ+Bn,TE2γs2+Bn,TE4γs4+Bn,TE6γs6,
An,TE0γ=An,TE00δγ,
An,TE00=2 -ippn cenβ, q02,
Bn,TE0γ=Bn,TE00δγ,
Bn,TE00=2 -ipsn senβ, q02,
An,TE2γ=An,TE20δγ+An,TE22δγ,
An,TE22=An,TE00,
An,TE20=-ippn4cenβ, q02+2 d2dβ2 cenβ, q02--ipk2l2q2cenβ, q2pnγ=0,
Bn,TE2γ=Bn,TE20δγ+Bn,TE22δγ,
Bn,TE22=Bn,TE00,
Bn,TE20=-ipsn4 senβ, q02+2 d2dβ2 senβ, q02--ipk2l2q2senβ, q2snγ=0,
An,TE4γ=An,TE40δγ+An,TE42δγ+An,TE44δ4γ,
An,TE44=-ippn cenβ, q02,
An,TE42=-ippn6 cenβ, q02+2 d2dβ2 cenβ, q02--ip3k2l2q2cenβ, q2pnγ=0,
An,TE40=-ippn12 cenβ, q02+10 d2dβ2 cenβ, q02+d4dβ4 cenβ, q02+-ip-3k2l2q2cenβ, q2pnγ=0-k2l2d2dβ2q2cenβ, q2pnγ=0+34 k4l422q2cenβ, q2pnγ=0,
Bn,TE4γ=Bn,TE40δγ+Bn,TE42δγ+Bn,TE44δ4γ,
Bn,TE44=-ipsn senβ, q02,
Bn,TE42=-ipsn6 senβ, q02+2 d2dβ2 senβ, q02--ip3k2l2q2senβ, q2snγ=0,
Bn,TE40=-ipsn12 senβ, q02+10 d2dβ2 senβ, q02+d4dβ4 senβ, q02+-ip-3k2l2q2senβ, q2snγ=0-k2l2d2dβ2q2senβ, q2snγ=0+34 k4l422q2senβ, q2snγ=0,
An,TM4γ=An,TM41δγ,
An,TM41=-4 -ippnddβ cenβ, q02,
Bn,TM4γ=Bn,TM41δγ,
Bn,TM41=-4 -ipsnddβ senβ, q02,
An,TE6γ=An,TE60δγ+An,TE62δ2γ+An,TE64δ4γ+An,TE66δ6γ,
An,TE66=13-ippn cenβ, q02,
An,TE64=-ippn2 cenβ, q02+d2dβ2 cenβ, q02--ip52 k2l2q2cenβ, q2pnγ=0,
An,TE62=-ippn24 cenβ, q02+16 d2dβ2 cenβ, q02+d4dβ4 cenβ, q02+-ip-6k2l2q2cenβ, q2pnγ=0-3k2l2d2dβ2q2cenβ, q2pnγ=0+154 k4l422q2cenβ, q2pnγ=0,
An,TE60=-ippn48 cenβ, q02+1603d2dβ2 cenβ, q02+263d4dβ4 cenβ, q02+13d6dβ6 cenβ, q02+-ip12k2l2q2cenβ, q2pnγ=0-8k2l2d2dβ2q2cenβ, q2pnγ=0-12 k2l2d4dβ4q2cenβ, q2pnγ=0+32 k4l422q2cenβ, q2pnγ=0+34 k4l4d2dβ222q2cenβ, q2pnγ=0-58 k6l633q2cenβ, q2pnγ=0,
Bn,TE6γ=Bn,TE60δγ+Bn,TE62δγ+Bn,TE64δ4γ+Bn,TE66δ6γ,
Bn,TE66=13-ipsn senβ, q02,
Bn,TE64=-ipsn2 senβ, q02+d2dβ2 senβ, q02--ip52 k2l2q2senβ, q2snγ=0,
Bn,TE62=-ipsn24 senβ, q02+16 d2dβ2 senβ, q02+d4dβ4 senβ, q02+-ip-6k2l2q2senβ, q2snγ=0-3k2l2d2dβ2q2senβ, q2snγ=0+154 k4l422q2senβ, q2snγ=0,
Bn,TE60=-ipsn48 senβ, q02+1603d2dβ2 senβ, q02+263d4dβ4 senβ, q02+13d6dβ6 senβ, q02+-ip-12k2l2q2senβ, q2snγ=0-8k2l2d2dβ2q2senβ, q2snγ=0-12 k2l2d4dβ4q2senβ, q2snγ=0+32 k4l422q2senβ, q2snγ=0+34 k4l4d2dβ222q2senβ, q2snγ=0-58 k6l633q2senβ, q2snγ=0,
An,TM6=An,TM61δγ+An,TM63δ3γ,
An,TM63=-4 -ippnddβ cenβ, q02,
An,TM61=-ippn-40 ddβ cenβ, q02-4 d3dβ3 cenβ, q02+-ip6k2l2ddβq2cenβ, q2pnγ=0,
Bn,TM6γ=Bn,TM61δγ+Bn,TM63δ3γ,
Bn,TM63=-4 -ipsnddβ senβ, q02,
Bn,TM61=-ipsn-40 ddβ senβ, q02-4 d3dβ3 senβ, q02+-ip6k2l2ddβq2senβ, q2snγ=0,
q02=q2γ=0=k2l24.
Fˆfkr, θ=fn+12, π2,
Gˆgkr, φ=g-n, π2,
Eu=E0ψ0 exp-ikw,
Ev=0,
Ew=-2QuL Eu,
Hu=0,
Hv=H0ψ0 exp-ikw,
Hw=-2QvL Hv,
ψ0=iQ exp-iQ u2+v2w02,
Q=1i+2w/L,
L=kw02.
Ez=0,
Hz=H0ψ0 exp-iklcosh μ cos θ cos β+sinh μ sin θ sin β,
ψ0=iQ exp-iQs2k2l2cosh μ cos θ sin β-sinh μ sin θ cos β2+k2z2,
Q=i+2s2klcosh μ cos θ cos β+sinh μ sin θ sin β-1.
An,TEγ=12π31-γ202π02π-+ Hz+cenμ, q2cen×θ, q2exp-iγZdμdθdZ,
Bn,TEγ=12π31-γ202π02π-+ Hz-senμ, q2sen×θ, q2exp-iγZdμdθdZ,
Z=kz,
Hz+=12H0Hzz, iμ, θ+Hzz, -iμ, θ=12H0Hzz, iμ, θ+Hzz, iμ, -θ,
Hz-=12iH0Hzz, iμ, θ-Hzz, -iμ, θ=12iH0Hzz, iμ, θ-Hzz, iμ, -θ,
An,TMγ=Bn,TMγ=0,
Hz+=1/2ε1ψ0z, iμ, θ+ε1˜ψ0˜z, iμ, θ,
Hz-=-i/2ε1ψ0z, iμ, θ-ε1˜ψ0˜z, iμ, θ,
ε1=exp-iklcos μ cos θ cos β+i sin μ sin θ sin β,
ψ0z, iμ, θ=iQiμ, θ×exp-iQiμ, θs2k2l2ξ2+Z2,
Qiμ, θ=1i+2s2klη,
2ε1β2=iklηε1-k2l2ξ2ε1,
η=cos μ cos θ cos β+i sin μ sin θ sin β,
ξ2=ηβ2.
Ãθ=A-θ.
Hz+=1/2ε11+s2a-Z2+1/2ε1˜1+s2ã-Z2,
Hz-=-i/2ε11+s2a-Z2+i/2ε1˜1+s2ã-Z2,
a=2iklη-k2l2ξ2.
An,TEγ=12π31-γ202π02π-+Hzz, iμ, θH0 cen×μ, q2cenθ, q2exp-iγZdμdθdZ,
Bn,TEγ=-i2π31-γ202π02π-+Hzz, iμ, θH0 sen×μ, q2senθ, q2exp-iγZdμdθdZ,
Hzz, iμ, θ=H0ψ0z, iμ, θε1,
Hzz, iμ, θ=H0ε11+s2a-Z2.
An,TM¯γ=Bn,TM¯γ=0,
Hzz, iμ, θ=ε1Hz0z, iμ, θ;
ε1¯=1π-ippn cenβ, q2;
μμ0,  θθ0
An,TE¯γ=ε1¯1-γ2-+Hz0z, iμ0, θ0H0 exp-iγZdZ.
ε1=2 n=0-ippn cenμ, q02cenθ, q02cenβ, q02+i -ipsn senμ, q02senθ, q02senβ, q02.
μ0¯=iμ0,
Hz0z, iμ0, θ0H0=Hz0z, μ0¯, θ0H0=1+s2a-iμ0¯, θ0-Z2.
-+ exp-iγZdZ=2πδγ
-+ Z2 exp-iγZdZ=-2πδγ
1H0-+ Hz0z, μ0¯, θ0exp-iγZdZ=2π1+s2a-iμ0¯, θ0δγ+s2δγ,
An,TE¯γ=2-ip1+a-iμ0¯, θ0s21-γ2cenβ, q2pn δγ+s21-γ2cenβ, q2pn δγ.
δγ1-γ2cenβ, q2pn, φγ=δγ, cenβ, q2pnφγ1-γ2=cenβ, q2φγpn1-γ2γ=0=cenβ, q02pn φ0,
δγ1-γ2cenβ, q2pn=cenβ, q02pn δγ.
δγ1-γ2cenβ, q2pn, φγ=2γ2cenβ, q2pnφγ1-γ2γ=0=2 cenβ, q02pn φ0-k2l22q2cenβ, q2pnγ=0φ0+cenβ, q02pn φ0,
δγ1-γ2cenβ, q2pn=2 cenβ, q02pn δγ-k2l22q2cenβ, q2pnγ=0δγ+cenβ, q02pnδγ.
An,TE¯γ=-ip2 cenβ, q02pn δγ+4s2cenβ, q02pn δγ+2a-iμ0¯, θ0s2cenβ, q02pn δγAn,TE¯γ=-ip2 cenβ, q02pn δγ+4s2cenβ, q02pn δγ+2a-iμ0¯, θ0s2cenβ, q02pn δγ-k2l2s2q2cenβ, q2pnγ=0δγ+2s2cenβ, q02pn δγ,
An,TE¯γ=An,TE00¯δγ+s2An,TE20¯δγ+An,TE22¯δγ,
An,TE00¯=2 -ippn cenβ, q02=An,TE22¯,
An,TE20¯=-ippn4 cenβ, q02+2a-iμ0¯, θ0cenβ, q02--ipk2l2q2cenβ, q2pnγ=0.
Cn=a-iμ0¯, θ0cenβ, q02-d2dβ2 cenβ, q02=0.
ε1¯=1π-ipsn senβ, q2;
Bn,TE¯γ=ε1¯1-γ2-+Hz0z, iμ0, θ0H0 exp-iγZdZ.
Sn=a-iμ0¯, θ0senβ, q02-d2dβ2 senβ, q02=0.
θ=β+φ-π.
θ0=β-π/2,
a-iμ0¯, β-π2=2iY-Xsin β cos β-X cos2 β+Y sin2 β2,
X=kl sinhμ0¯,
Y=kl coshμ0¯.
2ikl sin β cos β exp-μ0¯-k2l2 cos4 β exp-2μ0¯-k2l2 cosh2μ0¯+2k2l2 cos2 β coshμ0¯exp-μ0¯cenβ, q02=d2dβ2 cenβ, q02,
a-iμ0¯, -π2=-2iX sin β-X2 cos2 β.
cos2 β cenβ, q02X2+2i sin β cenβ, q02X+d2dβ2 cenβ, q02=0,
X=-i sin βcos2 β±icos2 β×sin2 β+cos2 βcenβ, q02d2dβ2 cenβ, q021/2,
X=±i1cenβ, q02d2dβ2 cenβ, q021/2.
a-iμ0¯, 0=2iY cos β-Y2 sin2 β,
sin2 β cenβ, q02Y2-2i cos β cenβ, q02Y+d2dβ2 cenβ, q02=0,
Y=i cos βsin2 β+isin2 β×cos2 β+sin2 βcenβ, q02d2dβ2 cenβ, q021/2,
An,TE¯γ=1π1-γ2-ippn cenβ, q2Q0×exp-Q0s2X2 cos2 β,
Q0=iQμ0¯, -π2=11+2is2X sin β
=-+ exp-Q0s2Z2exp-iγZdZ.
-+ exp-αZ2exp-iγZdZ=παexp-γ24α
An,TE¯γ=1sπ1-γ2-ippn cenβ, q2×11+2is2X sin β1/2 exp-s2X2 cos2 β1+2is2X sin β×exp-γ21+2is2X sin β4s2.
Bn,TE¯γ=1sπ1-γ2-ipsn senβ, q2×11+2is2X sin β1/2 exp-s2X2 cos2 β1+2is2X sin β×exp-γ21+2is2X sin β4s2.
An,TEs46¯γ=ε1¯1-γ2-+H40+H42Z2+H44Z4s4+H60+H62Z2+H64Z4+H66Z6s6exp-iγZdZ,
H40=-4X2 sin2 β+4i sin β cos2βX3+1/2X4 cos4 β,
H42=4iX sin β+X2 cos2 β,
H44=1/2,
H60=8iX3 sin3 β+12X4 sin2 β cos2 β-3iX5 sin β cos4 β-1/6X6 cos6 β,
H62=12X2 sin2 β-6iX3 sin β cos2 β-1/2X4 cos4 β,
H64=-3iX sin β-1/2X2 cos2 β,
H66=-1/6.
-+ Z2n exp-iγZdZ=2π-1nδ2nγ,
δ4γ1-γ2cenβ, q2pn=24 cenβ, q02pn δγ-6k2l2q2cenβ, q2pnγ=0δγ+34 k4l422q2cenβ, q2pnγ=0δγ+12 cenβ, q02pn δγ-3k2l2q2cenβ, q2pnγ=0δγ+cenβ, q02pn δ4γ,
δ6γ1-γ2cenβ, q2pn=720 cenβ, q02pn δγ-180k2l2q2cenβ, q2pnγ=0δγ+452 k4l422q2cenβ, q2pnγ=0δγ-158 k6l633q2cenβ, q2pnγ=0 δγ+360 cenβ, q02pn δγ-90k2l2q2cenβ, q2pnγ=0δγ+454 k4l422q2cenβ, q2pnγ=0δγ+30 cenβ, q02pn δ4γ-152 k2l2q2cenβ, q2pnγ=0δ4γ+cenβ, q02pn δ6γ.
An,TE44¯=-ippn cenβ, q02,
An,TE42¯=-ippn12 cenβ, q02+2 d2dβ2 cenβ, q02-4iX sin β cenβ, q02--ip3k2l2q2cenβ, q2pnγ=0,
An,TE40¯=-ippn24 cenβ, q02+4 d2dβ2 cenβ, q02-8iX sin β cenβ, q02-8X2 sin2 β cenβ, q02+8i sin β cos2 βX3cenβ, q02+X4 cos4 β cenβ, q02+-ip-6k2l2q2cenβ, q2pnγ=0+4iX sin βk2l2q2cenβ, q2pnγ=0+X2 cos2 βk2l2q2cenβ, q2pnγ=0+34 k4l422q2cenβ, q2pnγ=0,
An,TE66¯=13-ippn cenβ, q02,
An,TE64¯=-ippn10 cenβ, q02-4iX sin β cenβ, q02+d2dβ2 cenβ, q02-52 k2l2-ipq2cenβ, q2pnγ=0,
An,TE62¯=-ippn120 cenβ, q02-48iX sin β cenβ, q02+12 d2dβ2 cenβ, q02-24X2 sin2 β cenβ, q02+12iX3 sin β cos2 β cenβ, q02+X4 cos4 β cenβ, q02+-ip-30k2l2q2cenβ, q2pnγ=0An,TE62¯=-ippn120 cenβ, q02-48iX sin β cenβ, q02+12 d2dβ2 cenβ, q02-24X2 sin2 β cenβ, q02+12iX3 sin β cos2 β cenβ, q02+X4 cos4 β cenβ, q02+-ip-30k2l2q2cenβ, q2pnγ=0+18iX sin βk2l2q2cenβ, q2pnγ=0+3X2 cos2 βk2l2q2cenβ, q2pnγ=0+154 k4l422q2cenβ, q2pnγ=0,
An,TE60¯=-ippn240 cenβ, q02-96iX sin β cenβ, q02+24 d2dβ2 cenβ, q02-48X2 sin2 β cenβ, q02+16iX3 sin3 β cenβ, q02+24iX3 sin β cos2 β cenβ, q02+2X4 cos4 β cenβ, q02+24X4 sin2 β cos2 β cenβ, q02-6iX5 sin β cos4 β cenβ, q02-13 X6 cos6 β cenβ, q02+-ip-60k2l2q2cenβ, q2pnγ=0+36iX sin βk2l2q2cenβ, q2pnγ=0+6X2 cos2 βk2l2q2cenβ, q2pnγ=0+12X2 sin2 βk2l2q2cenβ, q2pnγ=0-6iX3 sin β cos2 βk2l2q2cenβ, q2pnγ=0-12 X4 cos4 βk2l2q2cenβ, q2pnγ=0+152 k4l422q2cenβ, q2pnγ=0-92 iX sin βk4l422q2cenβ, q2pnγ=0-34 X2 cos2 βk4l422q2cenβ, q2pnγ=0-58 k6l633q2cenβ, q2pnγ=0.
An,TE44¯=An,TE44,
An,TE66¯=An,TE66.
cenβ, q02Ann cos nβ,
X-n.
An,TE422 -ippnd2dβ2 cos nβ=-2n2-ippn cos nβAn,TE42¯,
X-iε+iε2-n2.
An,TE40¯-ippn X4 cos4 β cos nβ,
An,TE40-ippnd4dβ4 cos nβ=-ippn n4 cos nβ.
X2 cos2 β cos nβ-d2dβ2 cos nβ=n2 cos nβ,
X4 cos4 β cos nβn4 cos nβ
An,TE40¯An,TE40.
An,TE6j¯An,TE6j,  j=0, 2, 4.
a-iμ0¯, -π2=-2iX sin β-X2 cos2 β-X2 cos2 β,
a-iμ0¯, +π2=-2iX sin β-X2 cos2 β-X2 cos2 β.
a-iμ0¯, 0=2iY cos β-Y2 sin2 β-Y2 sin2 β,
a-iμ0¯, -π=-2iY cos β-Y2 sin2 β-Y2 sin2 β.

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