Abstract

Within the framework of the generalized Lorenz–Mie theory (GLMT), the incident shaped beam of an arbitrary orientation and location is expanded in terms of the spheroidal vector wave functions in given spheroidal coordinates. The beam shape coefficients (BSCs) in spheroidal coordinates are computed by the quadrature method. The classical localization approximation method for BSC evaluation is found to be inapplicable when the Cartesian coordinates of the beam and the particle are not parallel to each other. Once they are parallel, all the symmetry relationships existing for the BSCs in spherical coordinates (spherical BSCs) [J. Opt. Soc. Am. A 11, 1812 (1994) ] still pertain to the BSCs in spheroidal coordinates (spheroidal BSCs). In addition, the spheroidal BSCs computed by our method are verified by comparing them with those evaluated by Asano and Yamamoto for plane wave incidence [Appl. Opt. 14, 29 (1975) ]. Furthermore, formulas are given for field reconstruction from the spheroidal BSCs, and consistency is found between the original incident fields and the reconstructed ones.

© 2006 Optical Society of America

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References

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  1. G. Gouesbet and G. Gréhan, 'Generalized Lorenz-Mie theories, from past to future,' Atomization Sprays 10, 277-333 (2000).
  2. G. Gouesbet, B. Maheu, and 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]
  3. K. F. Ren, G. Gréhan, and G. Gouesbet, 'Laser sheet scattering by spherical particles,' Part. Part. Syst. Charact. 10, 146-151 (1993).
    [CrossRef]
  4. G. Gouesbet, J. A. Lock, and G. Gréhan, 'Partial wave representation of laser beams for use in light scattering calculations,' Appl. Opt. 34, 2133-2143 (1995).
    [CrossRef] [PubMed]
  5. K. F. Ren, G. Gouesbet, and G. Gréhan, 'Integral localized approximation in generalized Lorenz-Mie theory,' Appl. Opt. 37, 4218-4225 (1998).
    [CrossRef]
  6. K. F. Ren, G. Gréhan, and 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]
  7. G. Gouesbet and L. Méès, 'Generalized Lorenz-Mie theory for infinitely long elliptical cylinders,' J. Opt. Soc. Am. A 16, 1333-1341 (1999).
    [CrossRef]
  8. Y. P. Han and Z. Wu, 'Scattering of a spheroidal particle illuminated by a Gaussian beam,' Appl. Opt. 40, 2501-2509 (2001).
    [CrossRef]
  9. Y. P. Han, G. Gréhan, and G. Gouesbet, 'Generalized Lorenz-Mie theory for a spheroidal particle with off-axis Gaussian-beam illumination,' Appl. Opt. 42, 6621-6629 (2003).
    [CrossRef] [PubMed]
  10. J. P. Barton, 'Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,' Appl. Opt. 34, 5542-5551 (1995).
    [CrossRef] [PubMed]
  11. C. Flammer, Spheroidal Wave Functions (Stanford U. Press, 1957).
  12. E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (Chelsea, 1955).
  13. Y. Han and Z. Wu, 'The expansion coefficients of a spheroidal particle illuminated by Gaussian beam,' IEEE Trans. Antennas Propag. 49, 615-620 (2001).
    [CrossRef]
  14. S. Crozier, L. K. Forbes, and M. Brideson, 'Ellipsoidal harmonic (Lamé) MRI shims,' IEEE Trans. Appl. Supercond. 12, 1880-1885 (2002).
    [CrossRef]
  15. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  16. G. Gouesbet and J. A. Lock, 'Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. II. Off-axis beams,' J. Opt. Soc. Am. A 11, 2516-2525 (1994).
    [CrossRef]
  17. B. Maheu, G. Gouesbet, and 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).
    [CrossRef]
  18. J. P. Barton, 'Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,' J. Appl. Phys. 64, 1632-1639 (1988).
    [CrossRef]
  19. G. Gouesbet, C. Letellier, K. F. Ren, and G. Gréhan, 'Discussion of two quadrature methods of evaluating beam-shape coefficients in generalized Lorenz-Mie theory,' Appl. Opt. 35, 1537-1542 (1996).
    [CrossRef] [PubMed]
  20. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).
  21. G. Gouesbet, 'Localized interpretation to calculate all the coefficients gnm in the generalized Lorenz-Mie theory,' J. Opt. Soc. Am. A 7, 998-1007 (1990).
    [CrossRef]
  22. G. Gouesbet, J. A. Lock, and G. Gréhan, 'Partial-wave representations of laser beams for use in light scattering calculations,' Appl. Opt. 34, 2133-2143 (1995).
    [CrossRef] [PubMed]
  23. B. Maheu, G. Gréhan, and G. Gouesbet, 'Ray localization in Gaussian beams,' Opt. Commun. 70, 259-262 (1989).
    [CrossRef]
  24. D. B. Hodge, 'Eigenvalues and eigenfunctions of the spheroidal wave equation,' J. Math. Phys. 11, 2308-2312 (1970).
    [CrossRef]
  25. S. Asano and G. Yamamoto, 'Light scattering by a spheroidal particle,' Appl. Opt. 14, 29-49 (1975).
    [PubMed]
  26. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).
  27. L. W. Davis, 'Theory of electromagnetic beams,' Phys. Rev. A 19, 1177-1179 (1979).
    [CrossRef]
  28. J. P. Barton and D. R. Alexander, 'Fifth order corrected electromagnetic field component for a fundamental Gaussian beam,' J. Appl. Phys. 66, 2800-2802 (1989).
    [CrossRef]
  29. G. Gouesbet, G. Gréhan, and B. Maheu, 'Computations of the coefficients gn in the generalized Lorenz-Mie theory using three different methods,' Appl. Opt. 27, 4874-4883 (1988).
    [CrossRef] [PubMed]
  30. F. Xu, K. Ren, G. Gouesbet, G. Gréhan, and X. Cai, 'Generalized Lorenz-Mie theory for an arbitrarily oriented, located, and shaped beam scattered by a homogeneous spheroid,' J. Opt. Soc. Am. A 24, 119-131 (2007).
    [CrossRef]
  31. K. F. Ren, G. Gréhan, and G. Gouesbet, 'Symmetry relations in generalized Lorenz-Mie theory,' J. Opt. Soc. Am. A 11, 1812-1817 (1994).
    [CrossRef]
  32. K. F. Ren, G. Gréhan, and G. Gouesbet, 'Electromagnetic field expression of a laser sheet and the order of approximation,' J. Opt. (Paris) 25, 165-176 (1994).
    [CrossRef]
  33. A. E. Siegman, Lasers (University Science, 1986).

2007 (1)

2003 (1)

2002 (1)

S. Crozier, L. K. Forbes, and M. Brideson, 'Ellipsoidal harmonic (Lamé) MRI shims,' IEEE Trans. Appl. Supercond. 12, 1880-1885 (2002).
[CrossRef]

2001 (2)

Y. Han and Z. Wu, 'The expansion coefficients of a spheroidal particle illuminated by Gaussian beam,' IEEE Trans. Antennas Propag. 49, 615-620 (2001).
[CrossRef]

Y. P. Han and Z. Wu, 'Scattering of a spheroidal particle illuminated by a Gaussian beam,' Appl. Opt. 40, 2501-2509 (2001).
[CrossRef]

2000 (1)

G. Gouesbet and G. Gréhan, 'Generalized Lorenz-Mie theories, from past to future,' Atomization Sprays 10, 277-333 (2000).

1999 (1)

1998 (1)

1997 (1)

1996 (1)

1995 (3)

1994 (3)

1993 (1)

K. F. Ren, G. Gréhan, and G. Gouesbet, 'Laser sheet scattering by spherical particles,' Part. Part. Syst. Charact. 10, 146-151 (1993).
[CrossRef]

1990 (1)

1989 (2)

J. P. Barton and D. R. Alexander, 'Fifth order corrected electromagnetic field component for a fundamental Gaussian beam,' J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

B. Maheu, G. Gréhan, and G. Gouesbet, 'Ray localization in Gaussian beams,' Opt. Commun. 70, 259-262 (1989).
[CrossRef]

1988 (4)

B. Maheu, G. Gouesbet, and 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).
[CrossRef]

J. P. Barton, 'Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,' J. Appl. Phys. 64, 1632-1639 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, and B. Maheu, 'Computations of the coefficients gn in the generalized Lorenz-Mie theory using three different methods,' Appl. Opt. 27, 4874-4883 (1988).
[CrossRef] [PubMed]

G. Gouesbet, B. Maheu, and 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]

1979 (1)

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

1975 (1)

1970 (1)

D. B. Hodge, 'Eigenvalues and eigenfunctions of the spheroidal wave equation,' J. Math. Phys. 11, 2308-2312 (1970).
[CrossRef]

Alexander, D. R.

J. P. Barton and D. R. Alexander, 'Fifth order corrected electromagnetic field component for a fundamental Gaussian beam,' J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

Asano, S.

Barton, J. P.

J. P. Barton, 'Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,' Appl. Opt. 34, 5542-5551 (1995).
[CrossRef] [PubMed]

J. P. Barton and D. R. Alexander, 'Fifth order corrected electromagnetic field component for a fundamental Gaussian beam,' J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

J. P. Barton, 'Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,' J. Appl. Phys. 64, 1632-1639 (1988).
[CrossRef]

Brideson, M.

S. Crozier, L. K. Forbes, and M. Brideson, 'Ellipsoidal harmonic (Lamé) MRI shims,' IEEE Trans. Appl. Supercond. 12, 1880-1885 (2002).
[CrossRef]

Cai, X.

Crozier, S.

S. Crozier, L. K. Forbes, and M. Brideson, 'Ellipsoidal harmonic (Lamé) MRI shims,' IEEE Trans. Appl. Supercond. 12, 1880-1885 (2002).
[CrossRef]

Davis, L. W.

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

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, 1957).

Forbes, L. K.

S. Crozier, L. K. Forbes, and M. Brideson, 'Ellipsoidal harmonic (Lamé) MRI shims,' IEEE Trans. Appl. Supercond. 12, 1880-1885 (2002).
[CrossRef]

Gouesbet, G.

F. Xu, K. Ren, G. Gouesbet, G. Gréhan, and X. Cai, 'Generalized Lorenz-Mie theory for an arbitrarily oriented, located, and shaped beam scattered by a homogeneous spheroid,' J. Opt. Soc. Am. A 24, 119-131 (2007).
[CrossRef]

Y. P. Han, G. Gréhan, and G. Gouesbet, 'Generalized Lorenz-Mie theory for a spheroidal particle with off-axis Gaussian-beam illumination,' Appl. Opt. 42, 6621-6629 (2003).
[CrossRef] [PubMed]

G. Gouesbet and G. Gréhan, 'Generalized Lorenz-Mie theories, from past to future,' Atomization Sprays 10, 277-333 (2000).

G. Gouesbet and L. Méès, 'Generalized Lorenz-Mie theory for infinitely long elliptical cylinders,' J. Opt. Soc. Am. A 16, 1333-1341 (1999).
[CrossRef]

K. F. Ren, G. Gouesbet, and G. Gréhan, 'Integral localized approximation in generalized Lorenz-Mie theory,' Appl. Opt. 37, 4218-4225 (1998).
[CrossRef]

K. F. Ren, G. Gréhan, and 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, C. Letellier, K. F. Ren, and G. Gréhan, 'Discussion of two quadrature methods of evaluating beam-shape coefficients in generalized Lorenz-Mie theory,' Appl. Opt. 35, 1537-1542 (1996).
[CrossRef] [PubMed]

G. Gouesbet, J. A. Lock, and G. Gréhan, 'Partial wave representation of laser beams for use in light scattering calculations,' Appl. Opt. 34, 2133-2143 (1995).
[CrossRef] [PubMed]

G. Gouesbet, J. A. Lock, and 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, and G. Gouesbet, 'Symmetry relations in generalized Lorenz-Mie theory,' J. Opt. Soc. Am. A 11, 1812-1817 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, and G. Gouesbet, 'Electromagnetic field expression of a laser sheet and the order of approximation,' J. Opt. (Paris) 25, 165-176 (1994).
[CrossRef]

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

K. F. Ren, G. Gréhan, and G. Gouesbet, 'Laser sheet scattering by spherical particles,' Part. Part. Syst. Charact. 10, 146-151 (1993).
[CrossRef]

G. Gouesbet, 'Localized interpretation to calculate all the coefficients gnm in the generalized Lorenz-Mie theory,' J. Opt. Soc. Am. A 7, 998-1007 (1990).
[CrossRef]

B. Maheu, G. Gréhan, and G. Gouesbet, 'Ray localization in Gaussian beams,' Opt. Commun. 70, 259-262 (1989).
[CrossRef]

G. Gouesbet, G. Gréhan, and B. Maheu, 'Computations of the coefficients gn in the generalized Lorenz-Mie theory using three different methods,' Appl. Opt. 27, 4874-4883 (1988).
[CrossRef] [PubMed]

G. Gouesbet, B. Maheu, and 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. Gouesbet, and 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).
[CrossRef]

Gréhan, G.

F. Xu, K. Ren, G. Gouesbet, G. Gréhan, and X. Cai, 'Generalized Lorenz-Mie theory for an arbitrarily oriented, located, and shaped beam scattered by a homogeneous spheroid,' J. Opt. Soc. Am. A 24, 119-131 (2007).
[CrossRef]

Y. P. Han, G. Gréhan, and G. Gouesbet, 'Generalized Lorenz-Mie theory for a spheroidal particle with off-axis Gaussian-beam illumination,' Appl. Opt. 42, 6621-6629 (2003).
[CrossRef] [PubMed]

G. Gouesbet and G. Gréhan, 'Generalized Lorenz-Mie theories, from past to future,' Atomization Sprays 10, 277-333 (2000).

K. F. Ren, G. Gouesbet, and G. Gréhan, 'Integral localized approximation in generalized Lorenz-Mie theory,' Appl. Opt. 37, 4218-4225 (1998).
[CrossRef]

K. F. Ren, G. Gréhan, and 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, C. Letellier, K. F. Ren, and G. Gréhan, 'Discussion of two quadrature methods of evaluating beam-shape coefficients in generalized Lorenz-Mie theory,' Appl. Opt. 35, 1537-1542 (1996).
[CrossRef] [PubMed]

G. Gouesbet, J. A. Lock, and G. Gréhan, 'Partial wave representation of laser beams for use in light scattering calculations,' Appl. Opt. 34, 2133-2143 (1995).
[CrossRef] [PubMed]

G. Gouesbet, J. A. Lock, and 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, and G. Gouesbet, 'Electromagnetic field expression of a laser sheet and the order of approximation,' J. Opt. (Paris) 25, 165-176 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, and G. Gouesbet, 'Symmetry relations in generalized Lorenz-Mie theory,' J. Opt. Soc. Am. A 11, 1812-1817 (1994).
[CrossRef]

K. F. Ren, G. Gréhan, and G. Gouesbet, 'Laser sheet scattering by spherical particles,' Part. Part. Syst. Charact. 10, 146-151 (1993).
[CrossRef]

B. Maheu, G. Gréhan, and G. Gouesbet, 'Ray localization in Gaussian beams,' Opt. Commun. 70, 259-262 (1989).
[CrossRef]

G. Gouesbet, B. Maheu, and 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, and B. Maheu, 'Computations of the coefficients gn in the generalized Lorenz-Mie theory using three different methods,' Appl. Opt. 27, 4874-4883 (1988).
[CrossRef] [PubMed]

B. Maheu, G. Gouesbet, and 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).
[CrossRef]

Han, Y.

Y. Han and Z. Wu, 'The expansion coefficients of a spheroidal particle illuminated by Gaussian beam,' IEEE Trans. Antennas Propag. 49, 615-620 (2001).
[CrossRef]

Han, Y. P.

Hobson, E. W.

E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (Chelsea, 1955).

Hodge, D. B.

D. B. Hodge, 'Eigenvalues and eigenfunctions of the spheroidal wave equation,' J. Math. Phys. 11, 2308-2312 (1970).
[CrossRef]

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

Letellier, C.

Lock, J. A.

Maheu, B.

B. Maheu, G. Gréhan, and G. Gouesbet, 'Ray localization in Gaussian beams,' Opt. Commun. 70, 259-262 (1989).
[CrossRef]

G. Gouesbet, G. Gréhan, and B. Maheu, 'Computations of the coefficients gn in the generalized Lorenz-Mie theory using three different methods,' Appl. Opt. 27, 4874-4883 (1988).
[CrossRef] [PubMed]

G. Gouesbet, B. Maheu, and 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. Gouesbet, and 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).
[CrossRef]

Méès, L.

Ren, K.

Ren, K. F.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

Wu, Z.

Y. Han and Z. Wu, 'The expansion coefficients of a spheroidal particle illuminated by Gaussian beam,' IEEE Trans. Antennas Propag. 49, 615-620 (2001).
[CrossRef]

Y. P. Han and Z. Wu, 'Scattering of a spheroidal particle illuminated by a Gaussian beam,' Appl. Opt. 40, 2501-2509 (2001).
[CrossRef]

Xu, F.

Yamamoto, G.

Appl. Opt. (9)

S. Asano and G. Yamamoto, 'Light scattering by a spheroidal particle,' Appl. Opt. 14, 29-49 (1975).
[PubMed]

G. Gouesbet, J. A. Lock, and G. Gréhan, 'Partial-wave representations of laser beams for use in light scattering calculations,' Appl. Opt. 34, 2133-2143 (1995).
[CrossRef] [PubMed]

G. Gouesbet, J. A. Lock, and G. Gréhan, 'Partial wave representation of laser beams for use in light scattering calculations,' Appl. Opt. 34, 2133-2143 (1995).
[CrossRef] [PubMed]

J. P. Barton, 'Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,' Appl. Opt. 34, 5542-5551 (1995).
[CrossRef] [PubMed]

G. Gouesbet, C. Letellier, K. F. Ren, and G. Gréhan, 'Discussion of two quadrature methods of evaluating beam-shape coefficients in generalized Lorenz-Mie theory,' Appl. Opt. 35, 1537-1542 (1996).
[CrossRef] [PubMed]

K. F. Ren, G. Gouesbet, and G. Gréhan, 'Integral localized approximation in generalized Lorenz-Mie theory,' Appl. Opt. 37, 4218-4225 (1998).
[CrossRef]

G. Gouesbet, G. Gréhan, and B. Maheu, 'Computations of the coefficients gn in the generalized Lorenz-Mie theory using three different methods,' Appl. Opt. 27, 4874-4883 (1988).
[CrossRef] [PubMed]

Y. P. Han and Z. Wu, 'Scattering of a spheroidal particle illuminated by a Gaussian beam,' Appl. Opt. 40, 2501-2509 (2001).
[CrossRef]

Y. P. Han, G. Gréhan, and G. Gouesbet, 'Generalized Lorenz-Mie theory for a spheroidal particle with off-axis Gaussian-beam illumination,' Appl. Opt. 42, 6621-6629 (2003).
[CrossRef] [PubMed]

Atomization Sprays (1)

G. Gouesbet and G. Gréhan, 'Generalized Lorenz-Mie theories, from past to future,' Atomization Sprays 10, 277-333 (2000).

IEEE Trans. Antennas Propag. (1)

Y. Han and Z. Wu, 'The expansion coefficients of a spheroidal particle illuminated by Gaussian beam,' IEEE Trans. Antennas Propag. 49, 615-620 (2001).
[CrossRef]

IEEE Trans. Appl. Supercond. (1)

S. Crozier, L. K. Forbes, and M. Brideson, 'Ellipsoidal harmonic (Lamé) MRI shims,' IEEE Trans. Appl. Supercond. 12, 1880-1885 (2002).
[CrossRef]

J. Appl. Phys. (2)

J. P. Barton, '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. P. Barton and D. R. Alexander, 'Fifth order corrected electromagnetic field component for a fundamental Gaussian beam,' J. Appl. Phys. 66, 2800-2802 (1989).
[CrossRef]

J. Math. Phys. (1)

D. B. Hodge, 'Eigenvalues and eigenfunctions of the spheroidal wave equation,' J. Math. Phys. 11, 2308-2312 (1970).
[CrossRef]

J. Opt. (Paris) (2)

B. Maheu, G. Gouesbet, and 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).
[CrossRef]

K. F. Ren, G. Gréhan, and G. Gouesbet, 'Electromagnetic field expression of a laser sheet and the order of approximation,' J. Opt. (Paris) 25, 165-176 (1994).
[CrossRef]

J. Opt. Soc. Am. A (7)

Opt. Commun. (1)

B. Maheu, G. Gréhan, and G. Gouesbet, 'Ray localization in Gaussian beams,' Opt. Commun. 70, 259-262 (1989).
[CrossRef]

Part. Part. Syst. Charact. (1)

K. F. Ren, G. Gréhan, and G. Gouesbet, 'Laser sheet scattering by spherical particles,' Part. Part. Syst. Charact. 10, 146-151 (1993).
[CrossRef]

Phys. Rev. A (1)

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

Other (6)

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, 1957).

E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (Chelsea, 1955).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

A. E. Siegman, Lasers (University Science, 1986).

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Figures (4)

Fig. 1
Fig. 1

Geometry of Cartesian coordinates of the beam and spheroid. O P - x y z is the Cartesian coordinates of the particle and O P - u v w is parallelly translated from the beam coordinates O B - u v w with location O B ( x 0 , y 0 , z 0 ) in O P - x y z .

Fig. 2
Fig. 2

Reconstruction of E x ( x , 0 , 0 ) from the spheroidal BSCs calculated by the localization approximation and quadrature method, respectively, for the incidence of a highly focused Gaussian beam with ω 0 = λ 0 = 0.6328 μ m . Location of the beam center is x 0 = y 0 = z 0 = 0 μ m and the angle set is ( α = β = γ = 0 ° ) .

Fig. 3
Fig. 3

Reconstruction of E x ( x , 0 , 0 ) from the spheroidal BSCs calculated by the localization approximation and quadrature method, respectively. The beam has waist radii of ω 0 u = 1 μ m , ω 0 v = 1.5 μ m , wavelength λ 0 = 0.6328 μ m , and the center location x 0 = y 0 = z 0 = 0 μ m . The angle set of the beam is α = 45 ° , β = 45 ° , and γ = 0 ° . The locations of the waists along the u and v axes, w u and w v , are set as 0 and 5 μ m , respectively.

Fig. 4
Fig. 4

Scattering profiles calculated from two cases of ( α = 0 ° , β = 0 ° , and γ = 0 ° ) and ( α = 45 ° , β = 0 ° , γ = 0 ° ) in the scattering plane of the azimuthal angle ϕ = 0 ° . A droplet of refractive index 1.333 and radius 2.5 μ m is illuminated by a laser sheet of waist radius ω 0 u = 1.0 μ m , ω 0 v = 1.5 μ m , wavelength λ 0 = 0.6328 μ m , and its center location x 0 = y 0 = z 0 = 0 μ m in Cartesian coordinates of the particle.

Tables (2)

Tables Icon

Table 1 Beam Shape Coefficients in Spheroidal Coordinates: Oblique Incidence of a Plane Wave a

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Table 2 Beam Shape Coefficients in Spheroidal Coordinates: Parallel Incidence of a Gaussian Beam a

Equations (73)

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E ( i ) = n = 1 + m = n + n c n , p w i n + 1 [ i g n , TE m m m n ( i ) + g n , TM m n m n ( i ) ] ,
H ( i ) = i k I ω μ 0 n = 1 + m = n + n c n , p w i n + 1 [ g n , TM m m m n ( i ) + i g n , TE m n m n ( i ) ] ,
c n , p w = ( 2 n + 1 ) n ( n + 1 ) .
g n , TM m = ( 2 n + 1 ) ( i ) n + 1 2 π 2 ( n m ) ! ( n + m ) ! 0 k I r z n ( k I r ) 0 2 π exp ( i m ϕ ) 0 π E r ( i ) ( r , θ , ϕ ) E 0 P n m ( cos θ ) sin θ d θ d ϕ d ( k I r ) ,
g n , TE m = ( 2 n + 1 ) ( i ) n + 1 2 π 2 ( n m ) ! ( n + m ) ! 0 k I r z n ( k I r ) 0 2 π exp ( i m ϕ ) 0 π H r ( i ) ( r , θ , ϕ ) H 0 P n m ( cos θ ) sin θ d θ d ϕ d ( k I r ) .
g n , TM m = ( i ) n + 1 4 π ( n m ) ! ( n + m ) ! k I r z n ( k I r ) 0 π 0 2 π E r ( i ) ( r , θ , ϕ ) E 0 P n m ( cos θ ) sin θ exp ( i m ϕ ) d θ d ϕ ,
g n , TE m = ( i ) n + 1 4 π ( n m ) ! ( n + m ) ! k I r z n ( k I r ) 0 π 0 2 π H r ( i ) ( r , θ , ϕ ) H 0 P n m ( cos θ ) sin θ exp ( i m ϕ ) d θ d ϕ .
E r ( i ) = sin θ cos ϕ E x + sin θ sin ϕ E y + cos θ E z ,
H r ( i ) = sin θ cos ϕ H x + sin θ sin ϕ H y + cos θ H z .
E ( i ) = m = n = m , n 0 i n + 1 [ i G n , TE m M m n ( i ) ( c I ; ξ , η , ϕ ) + G n , TM m N m n ( i ) ( c I ; ξ , η , ϕ ) ] ,
H ( i ) = i k I ω μ 0 m = n = m , n 0 i n + 1 [ i G n , TM m M m n ( i ) ( c I ; ξ , η , ϕ ) + G n , TE m N m n ( i ) ( c I ; ξ , η , ϕ ) ] ,
m m n ( i ) ( r , θ , ϕ ) = l = m , m + 1 2 ( n + m ) ! ( 2 n + 1 ) ( n m ) ! i l n N m l d n m m l M m l ( i ) ( c ; ξ , η , ϕ ) ,
n m n ( i ) ( r , θ , ϕ ) = l = m , m + 1 2 ( n + m ) ! ( 2 n + 1 ) ( n m ) ! i l n N m l d n m m l N m l ( i ) ( c ; ξ , η , ϕ ) ,
G n , TE m = 1 N m n r = 0 , 1 g r + m , TE m 2 ( r + 2 m ) ! ( r + m ) ( r + m + 1 ) r ! d r m l ( c I ) ,
G n , TM m = 1 N m n r = 0 , 1 g r + m , TM m 2 ( r + 2 m ) ! ( r + m ) ( r + m + 1 ) r ! d r m l ( c I ) ,
N m n = r = 0 , 1 2 ( r + 2 m ) ! ( 2 r + 2 m + 1 ) r ! [ d r m n ( c I ) ] 2 .
( x x 0 y y 0 z z 0 ) = A ( u v w ) ,
A = [ cos β cos τ ( sin γ cos β sin τ + cos γ cos α sin β ) cos γ sin α sin β cos γ cos α cos β cos τ + cos γ sin α cos β sin τ sin γ cos β sin τ cos γ sin α sin β sin γ cos β cos τ cos γ cos α ] ,
τ = α + arcsin ( tan γ tan β ) .
( E x E y E z ) = A ( E u E v E w ) .
A = [ cos α cos β cos α sin β sin α sin β cos β 0 sin α cos β sin α sin β cos α ] .
{ E x = cos α cos β E u cos α sin β E v + sin α E w E y = sin β E u + cos β E v E z = sin α cos β E u + sin α sin β E v + cos α E w } .
( u v w ) = A T ( r sin θ cos ϕ x 0 r sin θ sin ϕ y 0 r cos θ z 0 ) .
E ξ = m = n = m , n 0 i n + 1 [ i G n , TE m M m n , ξ ( i ) + G n , TE m N m n , ξ ( i ) ] ,
E η = m = n = m , n 0 i n + 1 [ i G n , TE m M m n , η ( i ) + G n , TM m N m n , η ( i ) ] ,
E ϕ = m = n = m , n 0 i n + 1 [ i G n , TE m M m n , ϕ ( i ) + G n , TM m N m n , ϕ ( i ) ] ,
H ξ = i k I ω μ 0 m = n = m , n 0 i n + 1 [ G n , TM m M m n , ξ ( i ) + i G n , TE m N m n , ξ ( i ) ] ,
H η = i k I ω μ 0 m = n = m , n 0 i n + 1 [ G n , TM m M m n , η ( i ) + i G n , TE m N m n , η ( i ) ] ,
H ϕ = i k I ω μ 0 m = n = m , n 0 i n + 1 [ G n , TM m M m n , ϕ ( i ) + i G n , TE m N m n , ϕ ( i ) ] ,
( E x E y E z ) = [ ξ ( 1 η 2 ξ 2 η 2 ) 1 2 cos ϕ η ( ξ 2 1 ξ 2 η 2 ) 1 2 cos ϕ sin ϕ ξ ( 1 η 2 ξ 2 η 2 ) 1 2 sin ϕ η ( ξ 2 1 ξ 2 η 2 ) 1 2 sin ϕ cos ϕ η ( ξ 2 1 ξ 2 η 2 ) 1 2 ξ ( 1 η 2 ξ 2 η 2 ) 1 2 0 ] { E ξ E η E ϕ } .
η = 2 z x 2 + y 2 + z 2 + f 2 + 2 f z + x 2 + y 2 + z 2 + f 2 2 f z ,
ξ = x 2 + y 2 + z 2 + f 2 + 2 f z + x 2 + y 2 + z 2 + f 2 2 f z 2 f ,
η = 2 z y 2 + z 2 + x 2 f 2 + y 4 + 2 y 2 z 2 + 2 y 2 x 2 2 y 2 f 2 + z 4 + 2 x 2 z 2 + 2 f 2 z 2 + x 4 2 f 2 x 2 + f 4 ,
ξ = y 2 + z 2 + x 2 f 2 + y 4 + 2 y 2 z 2 + 2 y 2 x 2 2 y 2 f 2 + z 4 + 2 x 2 z 2 + 2 f 2 z 2 + x 4 2 f 2 x 2 + f 4 2 f .
ϕ = arctan ( y x ) , x > 0 , y > 0 ,
ϕ = 2 π + arctan ( y x ) , x > 0 , y < 0 ,
ϕ = π + arctan ( y x ) , x < 0 , y > 0 ,
ϕ = π + arctan ( y x ) , x < 0 , y < 0 .
x = f ( 1 η 2 ) 1 2 ( ξ 2 1 ) 1 2 cos ϕ ,
y = f ( 1 η 2 ) 1 2 ( ξ 2 1 ) 1 2 sin ϕ ,
z = f η ξ .
{ G n , TM m = G n , TM m = g m n 2 , m 1 G n , TM m = g m n , m = 0 } ,
{ G n , TE m = G n , TE m = i f m n 2 , m 1 G n , TE m = i f m n = 0 , m = 0 } .
{ G n , TM m = G n , TM m = i g m n 2 , m 1 G n , TM m = i g m n , m = 0 } ,
{ G n , TE m = G n , TE m = f m n 2 , m 1 G n , TE m = f m n = 0 , m = 0 } .
G n , TE + 1 = G n , TE 1
G n , TM + 1 = G n , TM 1
m m n = m m n , θ e θ + m m n , ϕ e ϕ ,
n m n = n m n , r e r + n m n , θ e θ + n m n , ϕ e ϕ ,
m m n , θ = m sin θ z n ( k r ) P m n ( cos θ ) exp ( i m ϕ ) ,
m m n , ϕ = z n ( k r ) P m n ( cos θ ) exp ( i m ϕ ) ,
n m n , r = n ( n + 1 ) z n ( k r ) k r P m n ( cos θ ) exp ( i m ϕ ) ,
n m n , θ = 1 k r [ r z n ( k r ) ] r P m n ( cos θ ) exp ( i m ϕ ) ,
n m n , ϕ = i m sin θ [ r z n ( k r ) ] r P m n ( cos θ ) exp ( i m ϕ ) ,
M m n = M m n , ξ a ξ + M m n , η a η + M m n , ϕ a ϕ ,
N m n = N m n , ξ r ( j ) a ξ + N m n , η a η + N m n , ϕ a ϕ ,
M m n , ξ = ± i m η ( ξ 2 η 2 ) ( ξ 2 1 ) R m n ( j ) ( c , ξ ) S m n ( c , η ) exp ( i m ϕ ) ,
M m n , η = i m ξ ( ξ 2 η 2 ) ( 1 η 2 ) R m n ( j ) ( c , ξ ) S m n ( c , η ) exp ( i m ϕ ) ,
M m n , ϕ = ( ξ 2 1 ) ( 1 η 2 ) ( ξ 2 η 2 ) [ ξ R m n ( j ) ( c , ξ ) S m n ( c , η ) η R m n , ( j ) ( c , ξ ) S m n ( c , η ) ] exp ( i m ϕ ) ,
N m n , ξ = ( ξ 2 1 ) 1 2 c ( ξ 2 η 2 ) 3 2 { ξ [ λ m n c 2 η 2 ± m 2 ξ 2 1 ] S m n ( c , η ) R m n ( j ) ( c , ξ ) 2 ξ η ( 1 η 2 ) ξ 2 η 2 S m n ( c , η ) R m n ( j ) ( c , ξ ) ± η ( 1 η 2 ) S m n ( c , η ) R m n , ( j ) ( c , ξ ) ± [ ξ 2 ( 1 3 η 2 ) ± η 2 ( η 2 + 1 ) ξ 2 η ̇ 2 ] S m n ( c , η ) R m n , ( j ) ( c , ξ ) } exp ( i m ϕ ) ,
N m n , η = 1 η 2 c ( ξ 2 η 2 ) 3 2 [ ξ ( ξ 2 1 ) R m n , ( j ) ( c , ξ ) S m n ( c , η ) + ξ 2 ( ξ 2 1 ) + η 2 ( 1 3 ξ 2 ) ( ξ 2 η 2 ) R m n ( j ) ( c , ξ ) S m n ( c , η ) η ( λ m n ξ 2 η 2 m 2 1 η 2 ) R m n ( j ) ( c , ξ ) S m n ( c , η ) ± 2 ξ η ( ξ 2 1 ) ( ξ 2 η 2 ) R m n , ( j ) ( c , ξ ) S m n ( c , η ) ] exp ( i m ϕ ) ,
N m n , ϕ = i m ( ξ 2 1 ) ( 1 η 2 ) c ( ξ 2 η 2 ) [ ± η ( ξ 2 1 ) S m n ( c , η ) R m n ( j ) ( c , ξ ) + ξ 1 η 2 R m n , ( j ) ( c , ξ ) S m n ( c , η ) + ( ξ 2 η 2 ) ( ξ 2 1 ) ( 1 η 2 ) S m n ( c , η ) R m n ( j ) ( c , ξ ) ] exp ( i m ϕ ) ,
E v = 0 ,
E u = E 0 ψ 0 sh exp ( i k w ) ,
E w = 2 Q u u l u E u ,
H u = 0 ,
H v = H 0 ψ 0 sh exp ( i k w ) ,
H w = 2 Q v v l v H v ,
ψ 0 sh = i Q u Q v exp [ i ( Q v u 2 ω 0 u 2 + Q v v 2 ω 0 v 2 ) ] ,
Q u = 1 i + 2 ( w w u ) l u ,
Q v = 1 i + 2 ( w w v ) l v ,
l u = k ω 0 u 2 ,
l v = k ω 0 v 2 ,

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