Abstract

A relation is developed between point-group symmetries of light-scattering particles and symmetry relations for the electromagnetic scattering solution in the T-matrix formulation. A systematic derivation of a representation of symmetry operations is presented in the vector space on which the T matrix operates. From this the set of symmetry relations of the T matrix is obtained for various point groups. As examples several symmetry groups relevant to modeling atmospheric particles are treated, such as the K group of spherical symmetry, the Cv group of axial symmetry, and the Dh group of dihedral axial symmetry. The Dh symmetry relations for the T matrix in spheroidal coordinates (denoted by script font) are also derived. Previously known symmetry relations of the T matrix can be verified, and new relations are found for DNh symmetry, i.e., for the important case of particles with dihedral symmetry and an N-fold axis of rotation.

© 1999 Optical Society of America

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References

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  1. M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8, 871–882 (1991).
    [CrossRef]
  2. F. M. Schulz, K. Stamnes, J. J. Stamnes, “Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875–7896 (1998).
    [CrossRef]
  3. C.-R. Hu, G. W. Kattawar, M. E. Parkin, P. Herb, “Symmetry theorems on the forward and backward scattering Mueller matrices for light scattering from a nonspherical dielectric scatterer,” Appl. Opt. 26, 4159–4173 (1987).
    [CrossRef] [PubMed]
  4. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  5. P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1970).
    [CrossRef]
  6. P. C. Waterman, “Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979).
    [CrossRef]
  7. P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
    [CrossRef]
  8. P. Barber, “Differential scattering of electromagnetic waves by homogeneous isotropic dielectric bodies,” Ph.D. thesis (University of California, Los Angeles, Los Angeles, Calif., 1973).
  9. P. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975).
    [CrossRef] [PubMed]
  10. L. Tsang, J. A. Kong, R. T. Shin, “Radiative transfer theory for active remote sensing of a layer of nonspherical particles,” Radio Sci. 19, 629–642 (1984).
    [CrossRef]
  11. V. V. Varadan, V. K. Varadan, “Multiple scattering of electromagnetic waves by randomly distributed and oriented dielectric scatterers,” Phys. Rev. D 21, 388–394 (1980).
    [CrossRef]
  12. N. G. Khlebtsov, “Orientational averaging of light-scattering observables in the T-matrix approach,” Appl. Opt. 31, 5359–5365 (1992).
    [CrossRef] [PubMed]
  13. D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
  14. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1995).
  15. M. I. Mishchenko, “Light scattering by size–shape distributions of randomly oriented axially symmetric particles of a size comparable to a wavelength,” Appl. Opt. 32, 4652–4665 (1993).
    [CrossRef] [PubMed]
  16. M. I. Mishchenko, L. D. Travis, “Light scattering by polydispersions of randomly oriented spheroids with sizes comparable to wavelengths of observation,” Appl. Opt. 33, 7206–7225 (1994).
    [CrossRef] [PubMed]
  17. M. I. Mishchenko, L. D. Travis, “Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight,” J. Geophys. Res. 102, 16989–17013 (1997).
    [CrossRef]
  18. M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
    [CrossRef] [PubMed]
  19. S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
    [CrossRef] [PubMed]
  20. S. Asano, M. Sato, “Light scattering by randomly oriented spheroidal particles,” Appl. Opt. 19, 962–974 (1980).
    [CrossRef] [PubMed]
  21. V. G. Farafonov, “Scattering of electromagnetic waves by a perfectly conducting spheroid,” Radio Eng. Electron. Phys. 29, 39–45 (1984).
  22. N. V. Voshchinnikov, V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
    [CrossRef]
  23. N. V. Voshchinnikov, “Electromagnetic scattering by homogeneous and coated spheroids: calculations using the separation of variable method,” J. Quant. Spectrosc. Radiat. Transf. 55, 627–636 (1996).
    [CrossRef]
  24. P. N. Francis, “Some aircraft observation of the scattering properties of ice crystals,” J. Atmos. Sci. 52, 1142–1154 (1995).
    [CrossRef]
  25. H. Laitinen, K. Lumme, “T-matrix method for general star-shaped particles: first results,” J. Quant. Spectrosc. Radiat. Transf. 60, 325–334 (1998).
    [CrossRef]
  26. M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
    [CrossRef]
  27. F. M. Schulz, K. Stamnes, J. J. Stamnes, “Shape-dependence of the optical properties in size–shape distributions of randomly oriented prolate spheroids, including highly elongated shapes,” J. Geophys. Res. (to be published).
  28. F. M. Schulz, K. Stamnes, J. J. Stamnes, “Modeling of the radiative transfer properties of media containing particles of moderately and highly elongated shape,” Geophys. Res. Lett. 25, 4481–4484 (1998).
    [CrossRef]
  29. C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).
  30. C. J. Bouwkamp, “On spheroidal wave functions of order zero,” J. Math. Phys. 26, 79–93 (1947).
  31. M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transf. 55, 535–575 (1996).
    [CrossRef]

1998

F. M. Schulz, K. Stamnes, J. J. Stamnes, “Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875–7896 (1998).
[CrossRef]

H. Laitinen, K. Lumme, “T-matrix method for general star-shaped particles: first results,” J. Quant. Spectrosc. Radiat. Transf. 60, 325–334 (1998).
[CrossRef]

F. M. Schulz, K. Stamnes, J. J. Stamnes, “Modeling of the radiative transfer properties of media containing particles of moderately and highly elongated shape,” Geophys. Res. Lett. 25, 4481–4484 (1998).
[CrossRef]

1997

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight,” J. Geophys. Res. 102, 16989–17013 (1997).
[CrossRef]

1996

M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
[CrossRef] [PubMed]

N. V. Voshchinnikov, “Electromagnetic scattering by homogeneous and coated spheroids: calculations using the separation of variable method,” J. Quant. Spectrosc. Radiat. Transf. 55, 627–636 (1996).
[CrossRef]

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transf. 55, 535–575 (1996).
[CrossRef]

1995

P. N. Francis, “Some aircraft observation of the scattering properties of ice crystals,” J. Atmos. Sci. 52, 1142–1154 (1995).
[CrossRef]

1994

1993

1992

1991

1987

1984

L. Tsang, J. A. Kong, R. T. Shin, “Radiative transfer theory for active remote sensing of a layer of nonspherical particles,” Radio Sci. 19, 629–642 (1984).
[CrossRef]

V. G. Farafonov, “Scattering of electromagnetic waves by a perfectly conducting spheroid,” Radio Eng. Electron. Phys. 29, 39–45 (1984).

1980

S. Asano, M. Sato, “Light scattering by randomly oriented spheroidal particles,” Appl. Opt. 19, 962–974 (1980).
[CrossRef] [PubMed]

V. V. Varadan, V. K. Varadan, “Multiple scattering of electromagnetic waves by randomly distributed and oriented dielectric scatterers,” Phys. Rev. D 21, 388–394 (1980).
[CrossRef]

1979

P. C. Waterman, “Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979).
[CrossRef]

1975

1970

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1970).
[CrossRef]

1965

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[CrossRef]

1947

C. J. Bouwkamp, “On spheroidal wave functions of order zero,” J. Math. Phys. 26, 79–93 (1947).

Asano, S.

Barber, P.

P. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975).
[CrossRef] [PubMed]

P. Barber, “Differential scattering of electromagnetic waves by homogeneous isotropic dielectric bodies,” Ph.D. thesis (University of California, Los Angeles, Los Angeles, Calif., 1973).

Bouwkamp, C. J.

C. J. Bouwkamp, “On spheroidal wave functions of order zero,” J. Math. Phys. 26, 79–93 (1947).

Farafonov, V. G.

N. V. Voshchinnikov, V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
[CrossRef]

V. G. Farafonov, “Scattering of electromagnetic waves by a perfectly conducting spheroid,” Radio Eng. Electron. Phys. 29, 39–45 (1984).

Flammer, C.

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

Francis, P. N.

P. N. Francis, “Some aircraft observation of the scattering properties of ice crystals,” J. Atmos. Sci. 52, 1142–1154 (1995).
[CrossRef]

Herb, P.

Hu, C.-R.

Kahn, R. A.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
[CrossRef]

Kattawar, G. W.

Khersonskii, V. K.

D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).

Khlebtsov, N. G.

Kong, J. A.

L. Tsang, J. A. Kong, R. T. Shin, “Radiative transfer theory for active remote sensing of a layer of nonspherical particles,” Radio Sci. 19, 629–642 (1984).
[CrossRef]

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1995).

Laitinen, H.

H. Laitinen, K. Lumme, “T-matrix method for general star-shaped particles: first results,” J. Quant. Spectrosc. Radiat. Transf. 60, 325–334 (1998).
[CrossRef]

Lumme, K.

H. Laitinen, K. Lumme, “T-matrix method for general star-shaped particles: first results,” J. Quant. Spectrosc. Radiat. Transf. 60, 325–334 (1998).
[CrossRef]

Macke, A.

Mackowski, D. W.

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transf. 55, 535–575 (1996).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight,” J. Geophys. Res. 102, 16989–17013 (1997).
[CrossRef]

M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
[CrossRef] [PubMed]

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transf. 55, 535–575 (1996).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “Light scattering by polydispersions of randomly oriented spheroids with sizes comparable to wavelengths of observation,” Appl. Opt. 33, 7206–7225 (1994).
[CrossRef] [PubMed]

M. I. Mishchenko, “Light scattering by size–shape distributions of randomly oriented axially symmetric particles of a size comparable to a wavelength,” Appl. Opt. 32, 4652–4665 (1993).
[CrossRef] [PubMed]

M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8, 871–882 (1991).
[CrossRef]

Moskalev, A. N.

D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).

Parkin, M. E.

Sato, M.

Schulz, F. M.

F. M. Schulz, K. Stamnes, J. J. Stamnes, “Modeling of the radiative transfer properties of media containing particles of moderately and highly elongated shape,” Geophys. Res. Lett. 25, 4481–4484 (1998).
[CrossRef]

F. M. Schulz, K. Stamnes, J. J. Stamnes, “Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875–7896 (1998).
[CrossRef]

F. M. Schulz, K. Stamnes, J. J. Stamnes, “Shape-dependence of the optical properties in size–shape distributions of randomly oriented prolate spheroids, including highly elongated shapes,” J. Geophys. Res. (to be published).

Shin, R. T.

L. Tsang, J. A. Kong, R. T. Shin, “Radiative transfer theory for active remote sensing of a layer of nonspherical particles,” Radio Sci. 19, 629–642 (1984).
[CrossRef]

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1995).

Stamnes, J. J.

F. M. Schulz, K. Stamnes, J. J. Stamnes, “Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875–7896 (1998).
[CrossRef]

F. M. Schulz, K. Stamnes, J. J. Stamnes, “Modeling of the radiative transfer properties of media containing particles of moderately and highly elongated shape,” Geophys. Res. Lett. 25, 4481–4484 (1998).
[CrossRef]

F. M. Schulz, K. Stamnes, J. J. Stamnes, “Shape-dependence of the optical properties in size–shape distributions of randomly oriented prolate spheroids, including highly elongated shapes,” J. Geophys. Res. (to be published).

Stamnes, K.

F. M. Schulz, K. Stamnes, J. J. Stamnes, “Modeling of the radiative transfer properties of media containing particles of moderately and highly elongated shape,” Geophys. Res. Lett. 25, 4481–4484 (1998).
[CrossRef]

F. M. Schulz, K. Stamnes, J. J. Stamnes, “Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875–7896 (1998).
[CrossRef]

F. M. Schulz, K. Stamnes, J. J. Stamnes, “Shape-dependence of the optical properties in size–shape distributions of randomly oriented prolate spheroids, including highly elongated shapes,” J. Geophys. Res. (to be published).

Travis, L. D.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight,” J. Geophys. Res. 102, 16989–17013 (1997).
[CrossRef]

M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
[CrossRef] [PubMed]

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transf. 55, 535–575 (1996).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “Light scattering by polydispersions of randomly oriented spheroids with sizes comparable to wavelengths of observation,” Appl. Opt. 33, 7206–7225 (1994).
[CrossRef] [PubMed]

Tsang, L.

L. Tsang, J. A. Kong, R. T. Shin, “Radiative transfer theory for active remote sensing of a layer of nonspherical particles,” Radio Sci. 19, 629–642 (1984).
[CrossRef]

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1995).

van de Hulst, H. C.

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

Varadan, V. K.

V. V. Varadan, V. K. Varadan, “Multiple scattering of electromagnetic waves by randomly distributed and oriented dielectric scatterers,” Phys. Rev. D 21, 388–394 (1980).
[CrossRef]

Varadan, V. V.

V. V. Varadan, V. K. Varadan, “Multiple scattering of electromagnetic waves by randomly distributed and oriented dielectric scatterers,” Phys. Rev. D 21, 388–394 (1980).
[CrossRef]

Varshalovich, D. A.

D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).

Voshchinnikov, N. V.

N. V. Voshchinnikov, “Electromagnetic scattering by homogeneous and coated spheroids: calculations using the separation of variable method,” J. Quant. Spectrosc. Radiat. Transf. 55, 627–636 (1996).
[CrossRef]

N. V. Voshchinnikov, V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
[CrossRef]

Waterman, P. C.

P. C. Waterman, “Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979).
[CrossRef]

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1970).
[CrossRef]

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[CrossRef]

West, R. A.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
[CrossRef]

Yamamoto, G.

Yeh, C.

Appl. Opt.

F. M. Schulz, K. Stamnes, J. J. Stamnes, “Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875–7896 (1998).
[CrossRef]

C.-R. Hu, G. W. Kattawar, M. E. Parkin, P. Herb, “Symmetry theorems on the forward and backward scattering Mueller matrices for light scattering from a nonspherical dielectric scatterer,” Appl. Opt. 26, 4159–4173 (1987).
[CrossRef] [PubMed]

P. Barber, C. Yeh, “Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies,” Appl. Opt. 14, 2864–2872 (1975).
[CrossRef] [PubMed]

N. G. Khlebtsov, “Orientational averaging of light-scattering observables in the T-matrix approach,” Appl. Opt. 31, 5359–5365 (1992).
[CrossRef] [PubMed]

M. I. Mishchenko, “Light scattering by size–shape distributions of randomly oriented axially symmetric particles of a size comparable to a wavelength,” Appl. Opt. 32, 4652–4665 (1993).
[CrossRef] [PubMed]

M. I. Mishchenko, L. D. Travis, “Light scattering by polydispersions of randomly oriented spheroids with sizes comparable to wavelengths of observation,” Appl. Opt. 33, 7206–7225 (1994).
[CrossRef] [PubMed]

M. I. Mishchenko, L. D. Travis, A. Macke, “Scattering of light by polydisperse, randomly oriented, finite circular cylinders,” Appl. Opt. 35, 4927–4940 (1996).
[CrossRef] [PubMed]

S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
[CrossRef] [PubMed]

S. Asano, M. Sato, “Light scattering by randomly oriented spheroidal particles,” Appl. Opt. 19, 962–974 (1980).
[CrossRef] [PubMed]

Astrophys. Space Sci.

N. V. Voshchinnikov, V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
[CrossRef]

Geophys. Res. Lett.

F. M. Schulz, K. Stamnes, J. J. Stamnes, “Modeling of the radiative transfer properties of media containing particles of moderately and highly elongated shape,” Geophys. Res. Lett. 25, 4481–4484 (1998).
[CrossRef]

J. Appl. Phys.

P. C. Waterman, “Matrix methods in potential theory and electromagnetic scattering,” J. Appl. Phys. 50, 4550–4566 (1979).
[CrossRef]

J. Atmos. Sci.

P. N. Francis, “Some aircraft observation of the scattering properties of ice crystals,” J. Atmos. Sci. 52, 1142–1154 (1995).
[CrossRef]

J. Geophys. Res.

M. I. Mishchenko, L. D. Travis, R. A. Kahn, R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997).
[CrossRef]

M. I. Mishchenko, L. D. Travis, “Satellite retrieval of aerosol properties over the ocean using polarization as well as intensity of reflected sunlight,” J. Geophys. Res. 102, 16989–17013 (1997).
[CrossRef]

J. Math. Phys.

C. J. Bouwkamp, “On spheroidal wave functions of order zero,” J. Math. Phys. 26, 79–93 (1947).

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transf.

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transf. 55, 535–575 (1996).
[CrossRef]

H. Laitinen, K. Lumme, “T-matrix method for general star-shaped particles: first results,” J. Quant. Spectrosc. Radiat. Transf. 60, 325–334 (1998).
[CrossRef]

N. V. Voshchinnikov, “Electromagnetic scattering by homogeneous and coated spheroids: calculations using the separation of variable method,” J. Quant. Spectrosc. Radiat. Transf. 55, 627–636 (1996).
[CrossRef]

Phys. Rev. D

V. V. Varadan, V. K. Varadan, “Multiple scattering of electromagnetic waves by randomly distributed and oriented dielectric scatterers,” Phys. Rev. D 21, 388–394 (1980).
[CrossRef]

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1970).
[CrossRef]

Proc. IEEE

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[CrossRef]

Radio Eng. Electron. Phys.

V. G. Farafonov, “Scattering of electromagnetic waves by a perfectly conducting spheroid,” Radio Eng. Electron. Phys. 29, 39–45 (1984).

Radio Sci.

L. Tsang, J. A. Kong, R. T. Shin, “Radiative transfer theory for active remote sensing of a layer of nonspherical particles,” Radio Sci. 19, 629–642 (1984).
[CrossRef]

Other

P. Barber, “Differential scattering of electromagnetic waves by homogeneous isotropic dielectric bodies,” Ph.D. thesis (University of California, Los Angeles, Los Angeles, Calif., 1973).

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

D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1995).

F. M. Schulz, K. Stamnes, J. J. Stamnes, “Shape-dependence of the optical properties in size–shape distributions of randomly oriented prolate spheroids, including highly elongated shapes,” J. Geophys. Res. (to be published).

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

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Figures (6)

Fig. 1
Fig. 1

Transformation of a vector function A(r) under a reflection in the xy plane.

Fig. 2
Fig. 2

Decomposition of the reflection operation σyz into C4-1σxzC4, demonstrated on a square object.

Fig. 3
Fig. 3

Decomposition of the reflection operation σv into C8-1σxzC8, demonstrated on a square object.

Fig. 4
Fig. 4

Decomposition of the rotation–reflection operation S2 into σhC2. Note that C2 denotes in this case a rotation about the vertical axis. Note also that neither C2 nor σh is a symmetry operation of the object shown in the figure.

Fig. 5
Fig. 5

Triagonal column with symmetry rotation axes C3 and C2 and symmetry reflection planes σv and σh.

Fig. 6
Fig. 6

Number of independent, nonzero T-matrix elements versus T-matrix truncation index n¯ for a general particle (squares) and for particles with D3h (circles), D4h (up-pointing triangles), D6h (down-pointing triangles), Dh (diamonds), and K (crosses) symmetries.

Tables (1)

Tables Icon

Table 1 Group Multiplication Table of the Dihedral Point Group D3h

Equations (109)

Equations on this page are rendered with MathJax. Learn more.

Hψ=Eψ
p=T·a.
Ei=n=1m=-nn[an,m(1)Mn,m(1)(kr)+an,m(2)Nn,m(1)(kr)],
Es=n=1m=-nn[pn,m(1)Mn,m(3)(kr)+pn,m(2)Nn,m(3)(kr)],
ψn,m(j)=γn,mPn(m)(cos θ)zn(j)(kr)exp(imϕ),j=1,, 4,
γn,m=4πn(n+1)2n+1(n-m)!(n+m)!1/2.
Mn,m(j)=×[r·ψn,m(j)],
Nn,m(j)=k-1·×Mn,m(j),
Mn,m(3)(kr)=γn,mhn(1)(kr)×exp(imϕ)θˆPn(m)(u)imsin θ-ϕˆdPn(m)(u)dθ,
Nn,m(3)(kr)=γn,m exp(imϕ)rˆn(n+1)krhn(1)(kr)Pn(m)(u)+1kr[krhn(1)(kr)]×θˆdPn(m)(u)dθ+ϕˆPn(m)(u)imsin θ.
pn,m(k)=n=1m=-nnk=12Tn,m,n,m(k, k)an,m(k).
Ar(r, θ, ϕ)Aθ(r, θ, ϕ)Aϕ(r, θ, ϕ)σhAr(r, π-θ, ϕ)-Aθ(r, π-θ, ϕ)Aϕ(r, π-θ, ϕ).
Mn,m(j)σh-(-1)n+mMn,m(j),
Nn,m(j)σh(-1)n+mNn,m(j).
Eiσhn=1m=-nn(-1)n+m[-an,m(1)Mn,m(1)(kr)+an,m(2)Nn,m(1)(kr)],
an,m(k)σh(-1)n+m+kan,m(k).
(Σh)n,m,n,mk,k=(-1)n+m+kδn,nδm,mδk,k.
TσhΣh·T·Σh-1
Tn,m,n,m(k, k)σh(-1)n+n(-1)m+m(-1)k+kTn,m,n,m(k, k).
T=Σh·T·Σh-1
[T, Σh]=0,
Tn,m,n,m(k, k)=0
unlessn+n+m+m+k+kiseven.
Mn,m(j)i-(-1)nMn,m(j),
Nn,m(j)i(-1)nNn,m(j).
Tn,m,n,m(k, k)=0unlessn+n+k+kiseven.
U(α, β, γ)=exp(-iαJz)exp(-iβJy)exp(-iγJz),
UMn,m(j)=U×[a·ψn,m(j)]=×[a·Uψn,m(j)]
|n, m=Yn,m(θ, ϕ)=2n+14π(n-m)!(n+m)!1/2Pn(m)(cos θ)exp(imϕ).
Dm,m(n)(α, β, γ)=n, m|U(α, β, γ)|n, m=exp(-imα)dm,m(n)(β)exp(-imγ),
dm,m(n)(β)=n, m|exp(-iJyβ)|n, m.
U(α, β, γ)|n, m=mDm,m(n)(α, β, γ)|n, m,
Tn,m,n,m(k, k)(α, β, γ)
=m1=-nnm2=-nnDm,m1(n)(α, β, γ)Tn,m1,n,m2(k, k)(0, 0, 0)
×(D-1)m2,m(n)(α, β, γ).
dm,m(n)(0)=δm,m
Tn,m,n,m(k, k)(2π/N, 0, 0)=exp[-i2π(m-m)/N]Tn,m,n,m(k, k)(0, 0, 0).
l=1:m-m=0, 6, 12, 18,,
l=2:m-m=0, 3, 6, 9,,
l=3:m-m=0, 2, 4, 6,,
l=4:m-m=0, 3, 6, 9,,
l=5:m-m=0, 6, 12, 18,,
Tn,m,n,m(k, k)=0unless(m-m)(mod 2)=0or(m-m)(mod 3)=0,
Tn,m,n,m(k, k)=0unless(m-m)(mod 3)=0.
Tn,m,n,m(k, k)=0
unless(m-m)(mod p)=0forsomepM.
Tn,m,n,m(k, k)(α, 0, 0)
=exp[-i(m-m)α]Tn,m,n,m(k, k)(0, 0, 0)
=Tn,m,n,m(k, k)(0, 0, 0)α,
Tn,m,n,m(k, k)=δm,mTn,m,n,m(k, k).
Tn,m,n,m(k, k)(0, π, 0)=m1=-nnm2=-nndm,m1(n)(π)Tn,m1,n,m2(k, k)×(0, 0, 0)(d-1)m2,m(n)(π).
Tn,m,n,m(k, k)(0, π, 0)=(-1)n+n+m+mTn,-m,n,-m(k, k)(0, 0, 0).
Tn,-m,n,-m(k, k)=(-1)n+n+m+mTn,m,n,m(k, k).
Mn,m(j)σxz-(-1)mMn,-m(j),
Nn,m(j)σxz(-1)mNn,-m(j).
Tn,-m,n,-m(k, k)=(-1)m+m(-1)k+kTn,m,n,m(k, k).
Mn,m(j)σyz-Mn,-m(j),
Nn,m(j)σyzNn,-m(j).
Tn,-m,n,-m(k, k)=(-1)k+kTn,m,n,m(k, k).
Tn,m,n,m(k, k)C4-1σxzC4 exp[i(-m+m)(-π/2)](-1)m+m+k+k exp[i(m-m)π/2]Tn,-m,n,-m(k, k)=(-1)k+kTn,-m,n,-m(k, k),
Tn,m,n,m(k, k)C8-1σxzC8im-m(-1)m+m+k+kTn,-m,n,-m(k, k).
Tn,m,n,m(k, k)SNexp[-i2π(m-m)/N](-1)n+n×(-1)m+m(-1)k+kTn,m,n,m(k, k).
exp{2πi[(n+n+k+k+m+m)/
2+(m-m)l/N]}=1,
n+n+k+k+m+m2+(m-m)lN
Tn,m,n,m(k, k)(0, 0, 0)=Tn,m,n,m(k, k)(α, β, γ)α,β,γ.
Tn,m,n,m(k, k)(0, 0, 0)=m1=-nndm,m1(n)(β)dm,m1(n)(β)×Tn,m1,n,m1(k, k)(0, 0, 0)β.
(2n+1)Tn,m,n,m(k, k)=δn,nm1=-nnTn,m1,n,m1(k, k).
Tn,m,n,m(k, k)=δk,kδn,nδm,mTn,
Tn,m,n,m(k, k)=δm,mTn,m,n,m(k, k).
Tn,-m,n,-m(k, k)=(-1)k+kTn,m,n,m(k, k),
Tn,m,n,m(k, k)=0unlessn+n+k+kiseven.
Tn,m,n,m(k, k)=0unlessm-m=0, 3, 6, 9,,
Tn,m,n,m(k, k)=0unlessn+n+m+m+k+kiseven,
Tn,-m,n,-m(k, k)=(-1)k+kTn,m,n,m(k, k).
T=QTQ-1,
T=RTR-1,
T=STS-1.
S=RQ,
Tn,m,n,m(k, k)=0unless(m-m)(mod N)=0.
Tn,m,n,m(k, k)=0unlessm-m=0, 4, 8, 12,,
Tn,m,n,m(k, k)=0unlessn+n+k+kiseven,
Tn,-m,n,-m(k, k)=(-1)k+kTn,m,n,m(k, k).
Tn,m,n,m(k, k)=0unlessm-m=0, 6, 12, 18,,
Tn,m,n,m(k, k)=0unlessn+n+k+kiseven,
Tn,-m,n,-m(k, k)=(-1)k+kTn,m,n,m(k, k),
Tn,m,n,m(k, k)=0unlessm-m=0, N, 2N, 3N,,
Tn,m,n,m(k, k)=0unlessn+n+m+m+k+kiseven,
Tn,-m,n,-m(k, k)=(-1)k+kTn,m,n,m(k, k).
Vn,m(j)=×[a·φn,m(j)],
Wn,m(j)=k-1·×Vn,m(j),
φn,m(j)(c;η, ξ, ϕ)=Sn,m(c;η)Rn,m(j)(c;ξ)exp(imϕ).
Sn,m(c;η)=r=0,1dr(m, n)(c)P|m|+r(m)(η),
-11dη Sn,m*(c;η)Sn,m(c;η)=Nm,nδn,n,
Nm,n=r=0,1(|m|+m+r)!(|m|-m+r)!22(|m|+r)+1×dr(m, n)*(c)dr(m, n)(c)
fr(m, n)(c)=1Nm,n22(|m|+r)+1(|m|+m+r)!(|m|-m+r)!1/2dr(m, n)(c)ifrandn-mhavethesameparityand|m|n,0otherwise
r=0,1fr(m, n)*(c)fr(m, n)(c)=δn,n.
Aη(η, ξ, ϕ)Aξ(η, ξ, ϕ)Aϕ(η, ξ, ϕ)σh-Aη(π-η, ξ, ϕ)Aξ(π-η, ξ, ϕ)Aϕ(π-η, ξ, ϕ).
Tn,m,n,m(k, k)=0unlessn+n+m+m+k+kiseven.
Tn,-m,n,-m(k, k)=(-1)m+m(-1)k+kTn,m,n,m(k, k).
Tn,m,n,m(k, k)=0unlessn+n+k+kiseven.
Tn,m,n,m(k, k)(α, β, γ)=n1,m1n2,m2(D-1)m2,m(n, n2)(α, β, γ)×Tn1,m1,n2,m2(k, k)(0, 0, 0)×Dm,m1(n, n1)(α, β, γ),
Dm,m(n, n)(α, β, γ)=q=|m|,|m|+1fq-|m|(m, n)*(c)×fq-|m|(m, n)(c)Dm,m(q)(α, β, γ),
(D-1)m2,m(n, n2)(α, β, γ)=Dm2,m(n, n2)(-γ, -β, -α).
Tn,m,n,m(k, k)(α, 0, 0)=n1,n2q=|m|,|m|+1 q=|m|,|m|+1×fq-|m|(m, n)*fq-|m|(m, n1)fq-|m|(m, n)fq-|m|(m, n2)*×exp[-i(m-m)α]×Tn1,m,n2,m(k, k)(0, 0, 0),
Tn,m,n,m(k, k)(α, 0, 0)=exp[-i(m-m)α]×Tn,m,n,m(k, k)(0, 0, 0)α,
Tn,m,n,m(k, k)=δm,mTn,m,n,m(k, k),
Tn,-m,n,-m(k, k)=(-1)k+kTn,m,n,m(k, k),
Tn,m,n,m(k, k)=0unlessn+n+k+kiseven.

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