Abstract

The theory of an arbitrarily oriented, shaped, and located beam scattered by a homogeneous spheroid is developed within the framework of the generalized Lorenz–Mie theory (GLMT). The incident beam is expanded in terms of the spheroidal vector wave functions and described by a set of beam shape coefficients (Gn,TMm,Gn,TEm). Analytical expressions of the far-field scattering and extinction cross sections are derived. As two special cases, plane wave scattering by a spheroid and shaped beam scattered by a sphere can be recovered from the present theory, which is verified both theoretically and numerically. Calculations of the far-field scattering and cross sections are performed to study the shaped beam scattered by a spheroid, which can be prolate or oblate, transparent or absorbing.

© 2006 Optical Society of America

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References

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  1. R. Gupta, T. Mukai, D. B. Vaidya, A. K. Sen, and Y. Okada, 'Interstellar extinction by spheroidal dust grain,' Astron. Astrophys. 441, 555-561 (2005).
    [CrossRef]
  2. K. Naitoh, 'Cyto-fluid dynamic theory of atomization processes,' Oil Gas Sci. Technol. 54, 205-210 (1999).
    [CrossRef]
  3. T. Oguchi, 'Electromagnetic wave propagation and scattering in rain and other hydrometeors,' Proc. IEEE 71, 1029-1078 (1983).
    [CrossRef]
  4. S. Asano and G. Yamamoto, 'Light scattering by a spheroidal particle,' Appl. Opt. 14, 29-49 (1975).
    [PubMed]
  5. J. P. Barton, 'Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,' Appl. Opt. 34, 5542-5551 (1995).
    [CrossRef] [PubMed]
  6. J. P. Barton, 'Internal, near-surface, and scattered electromagnetic fields for a layered spheroid with arbitrary illumination,' Appl. Opt. 40, 3596-3607 (2001).
    [CrossRef]
  7. J. P. Barton, 'Electromagnetic fields for a spheroidal particle with an arbitrary embedded source,' J. Opt. Soc. Am. A 17, 458-464 (2000).
    [CrossRef]
  8. 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]
  9. Y. P. Han and Z. Wu, 'Scattering of a spheroidal particle illuminated by a Gaussian beam,' Appl. Opt. 40, 2501-2509 (2001).
    [CrossRef]
  10. 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]
  11. 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]
  12. 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]
  13. 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]
  14. 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]
  15. K. F. Ren, G. Gouesbet, and G. Gréhan, 'Integral localized approximation in generalized Lorenz-Mie theory,' Appl. Opt. 37, 4218-4225 (1998).
    [CrossRef]
  16. G. Gouesbet, 'Validity of the localized approximation for arbitrary shaped beams in the generalized Lorenz-Mie theory for spheres,' J. Opt. Soc. Am. A 16, 1641-1650 (1999).
    [CrossRef]
  17. C. Flammer, Spheroidal Wave Functions (Stanford U. Press, 1957).
  18. L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K. Y. Tan, 'Computations of spheroidal harmonics with complex arguments: a review with an algorithm,' Phys. Rev. E 58, 6792-6806 (1998).
    [CrossRef]
  19. T. Oguchi, 'Eigenvalues of spheroidal wave functions and their branch points for complex values of propagation constants,' Radio Sci. 5, 1207-1214 (1970).
    [CrossRef]
  20. D. B. Hodge, 'Eigenvalues and eigenfunctions of the spheroidal wave equation,' J. Math. Phys. 11, 2308-2312 (1970).
    [CrossRef]
  21. L. W. Li, M. S. Leong, P. S. Kooi, and T. S. Yeo, 'Spheroidal vector wave eigenfunction expansion of dyadic Green's functions for a dielectric spheroid,' IEEE Trans. Antennas Propag. 49, 645-659 (2001).
    [CrossRef]
  22. L. W. Li, X. K. Kang, and M. S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (Wiley, 2002).
  23. L. W. Li, T. S. Yeo, P. S. Kooi, and M. S. Leong, 'Microwave specific attenuation by oblate spheroidal raindrops: an exact analysis of TCS's in terms of spheroidal wave functions,' Electromagn. Waves 18, 127-150 (1998).
    [CrossRef]
  24. Y. Yong and A. R. Sebak, 'Radiation pattern of aperture coupled prolate hemispheroidal dielectric resonator antenna,' Electromagn. Waves 58, 115-133 (2006).
    [CrossRef]
  25. F. Xu, K. F. Ren, and X. S. Cai, 'Expansion of an arbitrarily oriented, located, and shaped beam in spheroidal coordinates,' J. Opt. Soc. Am. A 24, 000-000 (2007).
  26. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
  27. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  28. L. W. Davis, 'Theory of electromagnetic beams,' Phys. Rev. A 19, 1177-1179 (1979).
    [CrossRef]
  29. K. F. Ren, G. Gréhan, and G. Gouesbet, 'Evaluation of laser-sheet beam shape coefficients in generalized Lorenz-Mie theory by use of a localized approximation,' J. Opt. Soc. Am. A 11, 2072-2079 (1994).
    [CrossRef]
  30. K. F. Ren, G. Gréhan, and G. Gouesbet, 'Electromagnetic field expression of the laser sheet and the order of approximation,' J. Opt. (Paris) 25, 165-176 (1994).
    [CrossRef]

2007 (1)

F. Xu, K. F. Ren, and X. S. Cai, 'Expansion of an arbitrarily oriented, located, and shaped beam in spheroidal coordinates,' J. Opt. Soc. Am. A 24, 000-000 (2007).

2006 (1)

Y. Yong and A. R. Sebak, 'Radiation pattern of aperture coupled prolate hemispheroidal dielectric resonator antenna,' Electromagn. Waves 58, 115-133 (2006).
[CrossRef]

2005 (1)

R. Gupta, T. Mukai, D. B. Vaidya, A. K. Sen, and Y. Okada, 'Interstellar extinction by spheroidal dust grain,' Astron. Astrophys. 441, 555-561 (2005).
[CrossRef]

2003 (1)

2001 (3)

J. P. Barton, 'Internal, near-surface, and scattered electromagnetic fields for a layered spheroid with arbitrary illumination,' Appl. Opt. 40, 3596-3607 (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]

L. W. Li, M. S. Leong, P. S. Kooi, and T. S. Yeo, 'Spheroidal vector wave eigenfunction expansion of dyadic Green's functions for a dielectric spheroid,' IEEE Trans. Antennas Propag. 49, 645-659 (2001).
[CrossRef]

2000 (1)

1999 (2)

1998 (3)

L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K. Y. Tan, 'Computations of spheroidal harmonics with complex arguments: a review with an algorithm,' Phys. Rev. E 58, 6792-6806 (1998).
[CrossRef]

L. W. Li, T. S. Yeo, P. S. Kooi, and M. S. Leong, 'Microwave specific attenuation by oblate spheroidal raindrops: an exact analysis of TCS's in terms of spheroidal wave functions,' Electromagn. Waves 18, 127-150 (1998).
[CrossRef]

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

1996 (1)

1995 (1)

1994 (3)

1988 (3)

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, 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]

1983 (1)

T. Oguchi, 'Electromagnetic wave propagation and scattering in rain and other hydrometeors,' Proc. IEEE 71, 1029-1078 (1983).
[CrossRef]

1979 (1)

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

1975 (1)

1970 (2)

T. Oguchi, 'Eigenvalues of spheroidal wave functions and their branch points for complex values of propagation constants,' Radio Sci. 5, 1207-1214 (1970).
[CrossRef]

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

Asano, S.

Barton, J. P.

J. P. Barton, 'Internal, near-surface, and scattered electromagnetic fields for a layered spheroid with arbitrary illumination,' Appl. Opt. 40, 3596-3607 (2001).
[CrossRef]

J. P. Barton, 'Electromagnetic fields for a spheroidal particle with an arbitrary embedded source,' J. Opt. Soc. Am. A 17, 458-464 (2000).
[CrossRef]

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

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Cai, X. S.

F. Xu, K. F. Ren, and X. S. Cai, 'Expansion of an arbitrarily oriented, located, and shaped beam in spheroidal coordinates,' J. Opt. Soc. Am. A 24, 000-000 (2007).

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).

Gouesbet, G.

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, 'Validity of the localized approximation for arbitrary shaped beams in the generalized Lorenz-Mie theory for spheres,' J. Opt. Soc. Am. A 16, 1641-1650 (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]

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 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, 'Evaluation of laser-sheet beam shape coefficients in generalized Lorenz-Mie theory by use of a localized approximation,' J. Opt. Soc. Am. A 11, 2072-2079 (1994).
[CrossRef]

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

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]

Gréhan, G.

Gupta, R.

R. Gupta, T. Mukai, D. B. Vaidya, A. K. Sen, and Y. Okada, 'Interstellar extinction by spheroidal dust grain,' Astron. Astrophys. 441, 555-561 (2005).
[CrossRef]

Han, Y. P.

Hodge, D. B.

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

Kang, X. K.

L. W. Li, X. K. Kang, and M. S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (Wiley, 2002).

Kooi, P. S.

L. W. Li, M. S. Leong, P. S. Kooi, and T. S. Yeo, 'Spheroidal vector wave eigenfunction expansion of dyadic Green's functions for a dielectric spheroid,' IEEE Trans. Antennas Propag. 49, 645-659 (2001).
[CrossRef]

L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K. Y. Tan, 'Computations of spheroidal harmonics with complex arguments: a review with an algorithm,' Phys. Rev. E 58, 6792-6806 (1998).
[CrossRef]

L. W. Li, T. S. Yeo, P. S. Kooi, and M. S. Leong, 'Microwave specific attenuation by oblate spheroidal raindrops: an exact analysis of TCS's in terms of spheroidal wave functions,' Electromagn. Waves 18, 127-150 (1998).
[CrossRef]

Leong, M. S.

L. W. Li, M. S. Leong, P. S. Kooi, and T. S. Yeo, 'Spheroidal vector wave eigenfunction expansion of dyadic Green's functions for a dielectric spheroid,' IEEE Trans. Antennas Propag. 49, 645-659 (2001).
[CrossRef]

L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K. Y. Tan, 'Computations of spheroidal harmonics with complex arguments: a review with an algorithm,' Phys. Rev. E 58, 6792-6806 (1998).
[CrossRef]

L. W. Li, T. S. Yeo, P. S. Kooi, and M. S. Leong, 'Microwave specific attenuation by oblate spheroidal raindrops: an exact analysis of TCS's in terms of spheroidal wave functions,' Electromagn. Waves 18, 127-150 (1998).
[CrossRef]

L. W. Li, X. K. Kang, and M. S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (Wiley, 2002).

Letellier, C.

Li, L. W.

L. W. Li, M. S. Leong, P. S. Kooi, and T. S. Yeo, 'Spheroidal vector wave eigenfunction expansion of dyadic Green's functions for a dielectric spheroid,' IEEE Trans. Antennas Propag. 49, 645-659 (2001).
[CrossRef]

L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K. Y. Tan, 'Computations of spheroidal harmonics with complex arguments: a review with an algorithm,' Phys. Rev. E 58, 6792-6806 (1998).
[CrossRef]

L. W. Li, T. S. Yeo, P. S. Kooi, and M. S. Leong, 'Microwave specific attenuation by oblate spheroidal raindrops: an exact analysis of TCS's in terms of spheroidal wave functions,' Electromagn. Waves 18, 127-150 (1998).
[CrossRef]

L. W. Li, X. K. Kang, and M. S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (Wiley, 2002).

Lock, J. A.

Maheu, B.

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]

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]

Mukai, T.

R. Gupta, T. Mukai, D. B. Vaidya, A. K. Sen, and Y. Okada, 'Interstellar extinction by spheroidal dust grain,' Astron. Astrophys. 441, 555-561 (2005).
[CrossRef]

Naitoh, K.

K. Naitoh, 'Cyto-fluid dynamic theory of atomization processes,' Oil Gas Sci. Technol. 54, 205-210 (1999).
[CrossRef]

Oguchi, T.

T. Oguchi, 'Electromagnetic wave propagation and scattering in rain and other hydrometeors,' Proc. IEEE 71, 1029-1078 (1983).
[CrossRef]

T. Oguchi, 'Eigenvalues of spheroidal wave functions and their branch points for complex values of propagation constants,' Radio Sci. 5, 1207-1214 (1970).
[CrossRef]

Okada, Y.

R. Gupta, T. Mukai, D. B. Vaidya, A. K. Sen, and Y. Okada, 'Interstellar extinction by spheroidal dust grain,' Astron. Astrophys. 441, 555-561 (2005).
[CrossRef]

Ren, K. F.

Sebak, A. R.

Y. Yong and A. R. Sebak, 'Radiation pattern of aperture coupled prolate hemispheroidal dielectric resonator antenna,' Electromagn. Waves 58, 115-133 (2006).
[CrossRef]

Sen, A. K.

R. Gupta, T. Mukai, D. B. Vaidya, A. K. Sen, and Y. Okada, 'Interstellar extinction by spheroidal dust grain,' Astron. Astrophys. 441, 555-561 (2005).
[CrossRef]

Stratton, J. A.

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

Tan, K. Y.

L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K. Y. Tan, 'Computations of spheroidal harmonics with complex arguments: a review with an algorithm,' Phys. Rev. E 58, 6792-6806 (1998).
[CrossRef]

Vaidya, D. B.

R. Gupta, T. Mukai, D. B. Vaidya, A. K. Sen, and Y. Okada, 'Interstellar extinction by spheroidal dust grain,' Astron. Astrophys. 441, 555-561 (2005).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Wu, Z.

Xu, F.

F. Xu, K. F. Ren, and X. S. Cai, 'Expansion of an arbitrarily oriented, located, and shaped beam in spheroidal coordinates,' J. Opt. Soc. Am. A 24, 000-000 (2007).

Yamamoto, G.

Yeo, T. S.

L. W. Li, M. S. Leong, P. S. Kooi, and T. S. Yeo, 'Spheroidal vector wave eigenfunction expansion of dyadic Green's functions for a dielectric spheroid,' IEEE Trans. Antennas Propag. 49, 645-659 (2001).
[CrossRef]

L. W. Li, T. S. Yeo, P. S. Kooi, and M. S. Leong, 'Microwave specific attenuation by oblate spheroidal raindrops: an exact analysis of TCS's in terms of spheroidal wave functions,' Electromagn. Waves 18, 127-150 (1998).
[CrossRef]

L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K. Y. Tan, 'Computations of spheroidal harmonics with complex arguments: a review with an algorithm,' Phys. Rev. E 58, 6792-6806 (1998).
[CrossRef]

Yong, Y.

Y. Yong and A. R. Sebak, 'Radiation pattern of aperture coupled prolate hemispheroidal dielectric resonator antenna,' Electromagn. Waves 58, 115-133 (2006).
[CrossRef]

Appl. Opt. (7)

Astron. Astrophys. (1)

R. Gupta, T. Mukai, D. B. Vaidya, A. K. Sen, and Y. Okada, 'Interstellar extinction by spheroidal dust grain,' Astron. Astrophys. 441, 555-561 (2005).
[CrossRef]

Electromagn. Waves (2)

L. W. Li, T. S. Yeo, P. S. Kooi, and M. S. Leong, 'Microwave specific attenuation by oblate spheroidal raindrops: an exact analysis of TCS's in terms of spheroidal wave functions,' Electromagn. Waves 18, 127-150 (1998).
[CrossRef]

Y. Yong and A. R. Sebak, 'Radiation pattern of aperture coupled prolate hemispheroidal dielectric resonator antenna,' Electromagn. Waves 58, 115-133 (2006).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

L. W. Li, M. S. Leong, P. S. Kooi, and T. S. Yeo, 'Spheroidal vector wave eigenfunction expansion of dyadic Green's functions for a dielectric spheroid,' IEEE Trans. Antennas Propag. 49, 645-659 (2001).
[CrossRef]

J. Appl. Phys. (1)

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. 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)

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

J. Opt. Soc. Am. A (6)

Oil Gas Sci. Technol. (1)

K. Naitoh, 'Cyto-fluid dynamic theory of atomization processes,' Oil Gas Sci. Technol. 54, 205-210 (1999).
[CrossRef]

Phys. Rev. A (1)

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

Phys. Rev. E (1)

L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K. Y. Tan, 'Computations of spheroidal harmonics with complex arguments: a review with an algorithm,' Phys. Rev. E 58, 6792-6806 (1998).
[CrossRef]

Proc. IEEE (1)

T. Oguchi, 'Electromagnetic wave propagation and scattering in rain and other hydrometeors,' Proc. IEEE 71, 1029-1078 (1983).
[CrossRef]

Radio Sci. (1)

T. Oguchi, 'Eigenvalues of spheroidal wave functions and their branch points for complex values of propagation constants,' Radio Sci. 5, 1207-1214 (1970).
[CrossRef]

Other (4)

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

L. W. Li, X. K. Kang, and M. S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (Wiley, 2002).

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

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

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Figures (16)

Fig. 1
Fig. 1

Coordinate systems: O B - u v w is attached to the incident shaped beam and O P - x y z is attached to the spheroid.

Fig. 2
Fig. 2

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 . The beam center O B locates at ( x 0 , y 0 , z 0 ) in O P - x y z .

Fig. 3
Fig. 3

Plane wave scattered by a prolate spheroid of a b = 2.0 and size parameter c I = 1 . The particle is assumed to be a small water droplet suspended in air so that the relative refractive index m ̂ = m ̂ II m ̂ I = 1.33 . The plane wave of wavelength λ 0 = 0.6328 μ m has the incidence angle ϴ bd = 45 ° and polarization angle Φ bd = 90 ° , corresponding to the TE mode. The figure is same as Asano and Yamamoto’s Fig. 6 of Ref. [4]. Note that for ϕ = 0 ° and ϕ = 90 ° , i 2 = 0 .

Fig. 4
Fig. 4

Same as Fig. 3 but for the TM mode. This figure is same as Asano and Yamamoto’s Fig. 8 of Ref. [4]. Note that for ϕ = 0 ° , i 1 = 0 .

Fig. 5
Fig. 5

Laser sheet scattering by a prolate spheroid of refractive index m ̂ = 1.33 , axis ratio a b = 1.0001 , semiminor axis b = 0.5 μ m . The parameters for the beam are ω 0 x = 1.0 μ m , ω 0 y = 1.5 μ m , λ 0 = 0.6328 μ m , ϴ bd = Φ bd = 0 ° . Its center O B has coordinates x 0 = y 0 = 0.5 μ m and z 0 = 0.0 μ m in the particle system O P - x y z . The particle is assumed to be suspended in air. In the GLMT calculation for the sphere, its radius is given as 0.5 μ m .

Fig. 6
Fig. 6

Laser sheet scattering by a prolate water spheroid of refractive index m ̂ = 1.33 , axis ratio a b = 1.2 , semiminor axis b = 0.5 μ m . The parameters for the beam are ω 0 x = 1.0 μ m , ω 0 y = 1.5 μ m , λ 0 = 0.6328 μ m , ϴ bd = Φ bd = 0 ° . Its center O B has the coordinates x 0 = y 0 = 0.5 μ m and z 0 = 0.0 μ m in the particle system O P - x y z . The particle is assumed to be suspended in air.

Fig. 7
Fig. 7

Same as Fig. 6, but for an oblate spheroid of a b = 1.2 .

Fig. 8
Fig. 8

Same as Fig. 6 but for ϴ bd = 45 ° .

Fig. 9
Fig. 9

Same as Fig. 6 but for ϴ bd = 90 ° .

Fig. 10
Fig. 10

Same as Fig. 9 but m ̂ = 1.33 + 0.1 I .

Fig. 11
Fig. 11

Extinction cross section C ext versus axis ratio a b of the spheroid. The beam with parameters ϴ bd = 0 ° , Φ bd = 0 ° , x 0 = y 0 = z 0 = 0 μ m , ω 0 x = 1.0 μ m , ω 0 y = 1.5 μ m , and λ 0 = 0.6328 μ m illuminates the prolate spheroid of semiminor axis length b = 0.5 μ m . The particle is assumed suspended in air. Five curves are plotted for particles of refractive indices m ̂ = 1.33 , 1.33 + 0.005 i , 1.33 + 0.01 i , 1.33 + 0.05 i , and 1.33 + 0.1 i .

Fig. 12
Fig. 12

Scattering cross section C sca versus axis ratio a b of the spheroid. Parameters of the incident beam and the prolate spheroid are the same as those in Fig. 11.

Fig. 13
Fig. 13

Extinction cross section C ext versus axis ratio a b of the spheroid. Parameters of the incident beam and the prolate spheroid are the same as those in Fig. 11 except ϴ bd = 45 ° .

Fig. 14
Fig. 14

Scattering cross section C sca versus axis ratio a b of the spheroid. Parameters of the incident beam and the prolate spheroid are the same as those in Fig. 13.

Fig. 15
Fig. 15

Scattering cross section C sca versus size parameter c I of a spheroidal droplet. The beam with parameters Φ bd = 0 ° , x 0 = y 0 = z 0 = 0 μ m , ω 0 x = 1.0 μ m , ω 0 y = 1.5 μ m , and λ 0 = 0.6328 μ m illuminates the prolate spheroid of axis ratio a b = 2.0 and refractive index m ̂ = 1.33 . The particle is assumed suspended in air. Five curves are plotted for incidence angles of ϴ bd = 0 ° , 22.5°, 45°, 67.5° and 90°.

Fig. 16
Fig. 16

Scattering cross section C sca versus size parameter c I of a spheroidal water droplet. The beam with parameters ϴ bd = 45 ° , x 0 = y 0 = z 0 = 0 μ m , ω 0 x = 1.0 μ m , ω 0 y = 1.5 μ m , and λ 0 = 0.6328 μ m illuminates the prolate spheroid of axis ratio a b = 2.0 and refractive index m ̂ = 1.33 . The particle is assumed suspended in air. Three curves are plotted for polarization angles Φ bd = 0 ° , 45°, and 90°.

Equations (117)

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{ 2 E + k 2 E = 0 2 H + k 2 H = 0 } ,
2 ψ + k 2 ψ = 0 .
ψ m n = z n ( r ) P n m ( cos θ ) exp ( i m ϕ ) ,
{ m m n = × ( r ψ m n ) n m n = 1 k × m m n } .
E ( i ) = m = + n = m , n 1 + c n , p w i n + 1 ( i g n , T E m m m n ( i ) ( r , θ , ϕ ) + g n , T M m n m n ( i ) ( r , θ , ϕ ) ) ,
H ( i ) = i k I ω μ 0 m = + n = m , n 1 + c n , p w i n + 1 ( g n , T M m m m n ( i ) ( r , θ , ϕ ) + i g n , T E m n m n ( i ) ( r , θ , ϕ ) ) ,
c n , p w = ( 2 n + 1 ) n ( n + 1 ) ,
g n , T M 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 , T E 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 ) .
ψ m n = S m n ( c , η ) R m n ( c , ξ ) exp ( i m ϕ ) ,
d d ξ [ ( ξ 2 1 ) d R m n ( c , ξ ) d ξ ] [ λ m n ( c ) c 2 ξ 2 m 2 ξ 2 1 ] R m n ( c , ξ ) = 0 ,
d d η [ ( 1 η 2 ) d S m n ( c , η ) d η ] + [ λ m n ( c ) c 2 η 2 m 2 1 η 2 ] S m n ( c , η ) = 0 .
{ M m n = × ( r Ψ m n ) N m n = 1 k × M m n } .
E ( i ) = m = n = m , n 0 i n + 1 [ i G n , T E m M m n ( i ) ( c I ; ξ , η , ϕ ) + G n , T M 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 [ G n , T M m M m n ( i ) ( c I ; ξ , η , ϕ ) + i G n , T E m N m n ( i ) ( c I ; ξ , η , ϕ ) ] ,
E ( s ) = m = n = m , n 0 i n + 1 [ B n m M m n ( s ) ( c I ; ξ , η , ϕ ) + A n m N m n ( s ) ( c I ; ξ , η , ϕ ) ] ,
H ( s ) = i k I ω μ 0 m = n = m , n 0 i n + 1 [ A n m M m n ( s ) ( c I ; ξ , η , ϕ ) + B n m N m n ( s ) ( c I ; ξ , η , ϕ ) ] .
E ( t ) = m = n = m , n 0 i n + 1 [ D n m M m n ( t ) ( c II ; ξ , η , ϕ ) + C n m N m n ( t ) ( c II ; ξ , η , ϕ ) ] ,
H ( t ) = i k II ω μ 0 m = n = m , n 0 i n + 1 [ D n m M m n ( t ) ( c II ; ξ , η , ϕ ) + C n m N m n ( t ) ( c II ; ξ , η , ϕ ) ] .
E η ( i ) + E η ( s ) = E η ( t ) ,
E ϕ ( i ) + E ϕ ( s ) = E ϕ ( t ) ,
H η ( i ) + H η ( s ) = H η ( t ) ,
H ϕ ( i ) + H ϕ ( s ) = H ϕ ( t ) .
( x x 0 y y 0 z z 0 ) = A ( u v w ) ,
A = [ cos ϴ bd cos Φ bd cos ϴ bd sin Φ bd sin ϴ bd sin Φ bd cos Φ bd 0 sin ϴ bd cos Φ bd sin ϴ bd sin Φ bd cos ϴ bd ] .
{ E x = cos ϴ bd cos Φ bd E u cos ϴ bd sin Φ bd E v + sin ϴ bd E w E y = sin Φ bd E u + cos Φ bd E v E z = sin ϴ bd cos Φ bd E u + sin ϴ bd sin Φ bd E v + cos ϴ bd E w } .
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 .
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 , T E m = 1 N m n ( c I ) r = 0 , 1 g r + m , T E m 2 ( r + 2 m ) ! ( r + m ) ( r + m + 1 ) r ! d r m n ( c I ) ,
G n , T M m = 1 N m n ( c I ) r = 0 , 1 g r + m , T M m 2 ( r + 2 m ) ! ( r + m ) ( r + m + 1 ) r ! d r m n ( c I ) .
S m n ( c , η ) = r = 0 , 1 d r m n ( c ) P m + r m ( cos θ ) ,
N m n ( c ) = r = 0 , 1 2 ( r + 2 m ) ! ( 2 r + 2 m + 1 ) r ! [ d r m n ( c ) ] 2 .
M m n , η ( s ) = ( i ) n e i k I r k I r m S m n ( c I , cos θ ) sin θ exp ( i m ϕ ) ,
M m n , ϕ ( s ) = ( i ) n + 1 e i k I r k I r d S m n ( c I , cos θ ) d θ exp ( i m ϕ ) ,
N m n , η ( s ) = ( i ) n e i k I r k I r d S m n ( c I , cos θ ) d θ exp ( i m ϕ ) ,
N m n , ϕ ( s ) = ( i ) n + 1 e i k I r k I r m S m n ( c I , cos θ ) sin θ exp ( i m ϕ ) .
E η ( s ) = i k I r e i k I r S 1 ,
E ϕ ( s ) = 1 k I r e i k I r S 2 ,
H η ( s ) = k I ω μ 0 E ϕ ( s ) ,
H ϕ ( s ) = k I ω μ 0 E η ( s ) ,
S 1 = m = n = m , n 0 [ A n m d S m n ( c I , cos θ ) d θ + B n m m S m n ( c I , cos θ ) sin θ ] exp ( i m ϕ ) ,
S 2 = m = n = m , n 0 [ A n m m S m n ( c I , cos θ ) sin θ + B n m d S m n ( c I , cos θ ) d θ ] exp ( i m ϕ ) .
C sca = Re { Σ ( E ( s ) × H ( s ) * ) n d S } ,
C ext = Re { Σ ( E ( i ) × H ( s ) * + E ( s ) × H ( i ) * ) n d S } ,
C sca = 0 π 0 2 π 1 2 Re ( E η ( s ) H ϕ ( s ) * E ϕ ( s ) H η ( s ) * ) r 2 sin θ d θ d ϕ ,
C ext = 0 π 0 2 π 1 2 Re ( E ϕ ( i ) H η ( s ) + E ϕ ( s ) H η ( i ) * E η ( i ) H ϕ ( s ) E η ( s ) H ϕ ( i ) * ) r 2 sin θ d θ d ϕ .
M m n , η ( i ) = 1 2 [ ( i ) n e i k I r k I r i n e i k I r k I r ] m S m n ( c I , cos θ ) sin θ exp ( i m ϕ ) ,
M m n , ϕ ( i ) = 1 2 [ ( i ) n + 1 e i k I r k I r + i n + 1 e i k I r k I r ] d S m n ( c I , cos θ ) d θ exp ( i m ϕ ) ,
N m n , η ( i ) = 1 2 [ ( i ) n e i k I r k I r i n e i k I r k I r ] d S m n ( c I , cos θ ) d θ exp ( i m ϕ ) ,
N m n , ϕ ( i ) = 1 2 [ ( i ) n + 1 e i k I r k I r + i n + 1 e i k I r k I r ] m S m n ( c I , cos θ ) sin θ exp ( i m ϕ ) ,
0 2 π exp [ i ( m m ) ϕ ] d ϕ = 2 π δ m m ,
0 π ( τ n m τ n m + m 2 π n m π n m ) sin θ d θ = 2 n ( n + 1 ) ( n + m ) ! ( 2 n + 1 ) ( n m ) ! δ n n ,
0 π ( τ n m π n m + τ n m π n m ) sin θ d θ = 0 ,
τ n m = d P n m ( cos θ ) d θ ,
π n m = P n m ( cos θ ) sin θ ,
C sca = λ 2 π Re [ p = + n = p 0 + n = p 0 + Π n n p ( A n p A n p , * + B n p B n p , * ) ] ,
C ext = λ 2 π Re [ p = + n = p 0 + n = p 0 + Π n n p ( A n p G n , T M p , * + B n p G n , T E p , * ) ] ,
Π n n p = { 0 , n n odd r = 0 , 1 ( r + p ) ( r + p + 1 ) 2 r + 2 p + 1 ( r + 2 p ) ! r ! d r p n d r p n , n n even } .
M m n = M e m n + i M o m n ,
N m n = N e m n + i N o m n .
{ M e m n = M e ( m ) n M o m n = M o ( m ) n } ,
{ N e m n = N e ( m ) n N o m n = N o ( m ) n } .
E ( i ) = m = 0 n = m , n 0 i n + 1 ( 2 δ 0 m ) 2 [ i ( G n , T E m + G n , T E m ) M e m n ( i ) ( G n , T E m G n , T E m ) M o m n ( i ) + ( G n , T M m + G n , T M m ) N e m n ( i ) + ( G n , T M m G n , T M m ) i N o m n ( i ) ] ,
H ( i ) = i k I ω μ 0 m = 0 n = m , n 0 i n + 1 ( 2 δ 0 m ) 2 [ ( G n , T M m + G n , T M m ) M e m n ( i ) + ( G n , T M m G n , T M m ) i M o m n ( i ) + i ( G n , T E m + G n , T E m ) N e m n ( i ) ( G n , T E m G n , T E m ) N o m n ( i ) ] .
{ G n , T M m = G n , T M m = g m n 2 , m 1 G n , T M m = g m n , m = 0 } ,
{ G n , T E m = G n , T E m = i f m n 2 , m 1 G n , T E m = i f m n = 0 , m = 0 } ,
E ( i ) = m = 0 n = m , n 0 i n + 1 [ ( G n , T E m G n , T E m ) M o m n ( i ) + ( 2 δ 0 m ) 2 ( G n , T M m + G n , T M m ) N e m n ( i ) ] ,
H ( i ) = i k I ω μ 0 m = 0 n = m , n 0 i n + 1 [ ( 2 δ 0 m ) 2 ( G n , T M m + G n , T M m ) M e m n ( i ) ( G n , T E m G n , T E m ) N o m n ( i ) ] .
E ( i ) = m = 0 + n = m , n 0 + i n ( f m n M o m n ( i ) i g m n N e m n ( i ) ) ,
H ( i ) = k I ω μ 0 m = 0 + n = m , n 0 + i n ( g m n M e m n ( i ) + i f m n N o m n ( i ) ) ,
E ( i ) = ( i cos ζ k sin ζ ) exp [ i k I ( x sin ζ + z cos ζ ) ] ,
H ( i ) = k I ω μ 0 j exp [ i k I ( x sin ζ + z cos ζ ) ] .
{ G n , T M m = G n , T M m = i g m n 2 , m 1 G n , T M m = i g m n , m = 0 } ,
{ G n , T E m = G n , T E m = f m n 2 , m 1 G n , T E m = f m n = 0 , m = 0 } .
E ( i ) = m = 0 n = m , n 0 i n [ g m n M e m n ( i ) + f m n i N o m n ( i ) ] ,
H ( i ) = k I ω μ 0 m = 0 n = m , n 0 i n [ f m n M o m n r ( i ) i g m n N e m n r ( i ) ] .
E ( i ) = j exp [ i k I ( x sin ζ + z cos ζ ) ] ,
H ( i ) = k I ω μ 0 ( i cos ζ k sin ζ ) exp [ i k I ( x sin ζ + z cos ζ ) ] .
d n m m l ( 0 ) = { 0 n l 1 n = l } .
m m n ( i ) ( r , θ , ϕ ) = 2 ( n + m ) ! ( 2 n + 1 ) ( n m ) ! 1 N m n M m n ( i ) ( 0 ; η , ξ , ϕ ) ,
n m n ( i ) ( r , θ , ϕ ) = 2 ( n + m ) ! ( 2 n + 1 ) ( n m ) ! 1 N m n N m n ( i ) ( 0 ; η , ξ , ϕ ) .
G n , T E m = 1 N m n 2 ( n + m ) ! n ( n + 1 ) ( n m ) ! ,
G n , T M m = 1 N m n 2 ( n + m ) ! n ( n + 1 ) ( n m ) ! ,
N m l = 2 ( n + m ) ! ( 2 n + 1 ) ( n m ) ! .
G n , T E m = F p l g n , T E m ,
G n , T M m = F p l g n , T M m ,
F p l = 2 n + 1 n ( n + 1 ) .
m m n ( i ) ( r , θ , ϕ ) = M m n ( i ) ( 0 ; η , ξ , ϕ ) ,
n m n ( i ) ( r , θ , ϕ ) = N m n ( i ) ( 0 ; η , ξ , ϕ ) .
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 + M 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 ) { 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 ( 1 ξ 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 u = E 0 ψ 0 s h exp ( i k w ) ,
E v = 0 ,
E w = 2 Q u u l u E u ,
H u = 0 ,
H v = H 0 ψ 0 s h exp ( i k w ) ,
H w = 2 Q v v l v H v ,
ψ 0 s h = i Q u Q v exp [ i ( Q v u 2 w 0 u 2 + Q v v 2 w 0 v 2 ) ] ,
Q u = 1 i + 2 w l u ,
Q v = 1 i + 2 w l v ,
l u = k w 0 u 2 ,
l v = k w 0 v 2 ,

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