Abstract

We present what we believe to be the first results of a light-scattering analysis on several Chebyshev particles characterized by higher orders. Chebyshev particles of comparatively lower orders were used in the past to study the effects of nonspherical but concave geometries in remote sensing applications. We will show that, based on the developed methodology, accurate results can also be obtained for particles of higher orders exhibiting a more pronounced surface waviness. The achieved results demonstrate that higher-order Chebyshev particles can be used to estimate the influence of a weak surface roughness on the light-scattering behavior of the underlying smooth scatterer. The effects obtained correspond with the results of other approaches and with the theoretical expectations of a weak surface roughness. In contrast to what is known for regular particles, there can be observed an essential difference between the phase functions of the underlying spherical scatterer and the corresponding higher-order Chebyshev particle if a higher absorptivity of the scattering medium is considered. This paper demonstrates additionally that Chebyshev polynomials can be simply combined with smooth geometries other than spheres.

© 2006 Optical Society of America

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References

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  1. A. Mugnai and W. J. Wiscombe, "Scattering from nonspherical Chebyshev particles. 1: Cross sections, single scattering albedo, asymmetry factor, and backscattered fraction," Appl. Opt. 25, 1235-1244 (1986).
    [CrossRef] [PubMed]
  2. W. J. Wiscombe and A. Mugnai, "Scattering from nonspherical Chebyshev particles. 2: Means of angular scattering patterns," Appl. Opt. 27, 2405-2421 (1988).
    [CrossRef] [PubMed]
  3. A. Mugnai and W. J. Wiscombe, "Scattering from nonspherical Chebyshev particles. 3: Variability in angular scattering patterns," Appl. Opt. 28, 3061-3073 (1989).
    [CrossRef] [PubMed]
  4. M. I. Mishchenko and L. D. Travis, "Light scattering by polydisperse, rotationally symmetric particles: linear polarization," J. Quant. Spectrosc. Radiat. Transf. 51, 759-778 (1994).
    [CrossRef]
  5. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).
  6. J. Wauer, K. Schmidt, T. Rother, T. Ernst, and M. Hess, "Two software tools for plane wave scattering on nonspherical particles in the German Aerospace Center's virtual laboratory," Appl. Opt. 43, 6371-6379 (2004).
    [CrossRef] [PubMed]
  7. T. Ernst, T. Rother, F. Schreier, J. Wauer, and W. Balzer, "DLR's VirtualLab: Scientific Software just a mouse click away," IEEE Comput. Sci. Eng. 5, 70-79 (2003).
    [CrossRef]
  8. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 4: Wave Propagation Through Random Media (Springer, 1989).
  9. R. Schiffer, "Perturbation approach for light scattering by an ensemble of irregular particles of arbitrary material," Appl. Opt. 29, 1536-1550 (1990).
    [CrossRef] [PubMed]
  10. Ch. Li, G. W. Kattawar, and P. Yang, "Effects of surface roughness on light scattering by small particles," J. Quant. Spectrosc. Radiat. Transf. 89, 123-131 (2004).
    [CrossRef]
  11. W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, and G. Videen, "Light scattering by Gaussian particles: a solution with finite-difference time-domain technique," J. Quant. Spectrosc. Radiat. Transf. 79-80, 1083-1090 (2003).
    [CrossRef]
  12. J. F. Gayet, F. Auriol, S. Oshchepkov, F. Schröder, C. Duroure, G. Febvre, J.-F. Fournol, O. Crépel, P. Personne, and D. Daugeron, "In situ measurements of the scattering phase function of stratocumulus, contrails, and cirrus," Geophys. Res. Lett. 25, 971-974 (1998).
    [CrossRef]
  13. W. Sun, N. G. Loeb, G. Videen, and Q. Fu, "Examination of surface roughness on light scattering by long ice columns by use of a two-dimensional finite-difference time-domain algorithm," Appl. Opt. 43, 1957-1964 (2004).
    [CrossRef] [PubMed]
  14. P. W. Barber and R. K. Chang, Optical Effects Associated with Small Particles (World Scientific, 1988).
  15. T. Rother, S. Havemann, and K. Schmidt, "Scattering of plane waves on finite cylinders with noncircular cross sections," in Progress in Electromagnetics Research, J. A. Kong, ed. (EMW Publishing, 1999), Vol. 23, pp. 79-105.
  16. J. A. Lock, "Excitation of morphology-dependent resonances and van de Hulst's localization principle," Opt. Lett. 24, 427-429 (1999).
    [CrossRef]
  17. M. I. Mishchenko and A. A. Lacis, "Morphology-dependent resonances of nearly spherical particles in random orientation," Appl. Opt. 42, 5551-5556 (2003).
    [CrossRef] [PubMed]

2004 (3)

2003 (3)

M. I. Mishchenko and A. A. Lacis, "Morphology-dependent resonances of nearly spherical particles in random orientation," Appl. Opt. 42, 5551-5556 (2003).
[CrossRef] [PubMed]

W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, and G. Videen, "Light scattering by Gaussian particles: a solution with finite-difference time-domain technique," J. Quant. Spectrosc. Radiat. Transf. 79-80, 1083-1090 (2003).
[CrossRef]

T. Ernst, T. Rother, F. Schreier, J. Wauer, and W. Balzer, "DLR's VirtualLab: Scientific Software just a mouse click away," IEEE Comput. Sci. Eng. 5, 70-79 (2003).
[CrossRef]

1999 (1)

1998 (1)

J. F. Gayet, F. Auriol, S. Oshchepkov, F. Schröder, C. Duroure, G. Febvre, J.-F. Fournol, O. Crépel, P. Personne, and D. Daugeron, "In situ measurements of the scattering phase function of stratocumulus, contrails, and cirrus," Geophys. Res. Lett. 25, 971-974 (1998).
[CrossRef]

1994 (1)

M. I. Mishchenko and L. D. Travis, "Light scattering by polydisperse, rotationally symmetric particles: linear polarization," J. Quant. Spectrosc. Radiat. Transf. 51, 759-778 (1994).
[CrossRef]

1990 (1)

1989 (1)

1988 (1)

1986 (1)

Auriol, F.

J. F. Gayet, F. Auriol, S. Oshchepkov, F. Schröder, C. Duroure, G. Febvre, J.-F. Fournol, O. Crépel, P. Personne, and D. Daugeron, "In situ measurements of the scattering phase function of stratocumulus, contrails, and cirrus," Geophys. Res. Lett. 25, 971-974 (1998).
[CrossRef]

Balzer, W.

T. Ernst, T. Rother, F. Schreier, J. Wauer, and W. Balzer, "DLR's VirtualLab: Scientific Software just a mouse click away," IEEE Comput. Sci. Eng. 5, 70-79 (2003).
[CrossRef]

Barber, P. W.

P. W. Barber and R. K. Chang, Optical Effects Associated with Small Particles (World Scientific, 1988).

Chang, R. K.

P. W. Barber and R. K. Chang, Optical Effects Associated with Small Particles (World Scientific, 1988).

Crépel, O.

J. F. Gayet, F. Auriol, S. Oshchepkov, F. Schröder, C. Duroure, G. Febvre, J.-F. Fournol, O. Crépel, P. Personne, and D. Daugeron, "In situ measurements of the scattering phase function of stratocumulus, contrails, and cirrus," Geophys. Res. Lett. 25, 971-974 (1998).
[CrossRef]

Daugeron, D.

J. F. Gayet, F. Auriol, S. Oshchepkov, F. Schröder, C. Duroure, G. Febvre, J.-F. Fournol, O. Crépel, P. Personne, and D. Daugeron, "In situ measurements of the scattering phase function of stratocumulus, contrails, and cirrus," Geophys. Res. Lett. 25, 971-974 (1998).
[CrossRef]

Duroure, C.

J. F. Gayet, F. Auriol, S. Oshchepkov, F. Schröder, C. Duroure, G. Febvre, J.-F. Fournol, O. Crépel, P. Personne, and D. Daugeron, "In situ measurements of the scattering phase function of stratocumulus, contrails, and cirrus," Geophys. Res. Lett. 25, 971-974 (1998).
[CrossRef]

Ernst, T.

J. Wauer, K. Schmidt, T. Rother, T. Ernst, and M. Hess, "Two software tools for plane wave scattering on nonspherical particles in the German Aerospace Center's virtual laboratory," Appl. Opt. 43, 6371-6379 (2004).
[CrossRef] [PubMed]

T. Ernst, T. Rother, F. Schreier, J. Wauer, and W. Balzer, "DLR's VirtualLab: Scientific Software just a mouse click away," IEEE Comput. Sci. Eng. 5, 70-79 (2003).
[CrossRef]

Febvre, G.

J. F. Gayet, F. Auriol, S. Oshchepkov, F. Schröder, C. Duroure, G. Febvre, J.-F. Fournol, O. Crépel, P. Personne, and D. Daugeron, "In situ measurements of the scattering phase function of stratocumulus, contrails, and cirrus," Geophys. Res. Lett. 25, 971-974 (1998).
[CrossRef]

Fournol, J.-F.

J. F. Gayet, F. Auriol, S. Oshchepkov, F. Schröder, C. Duroure, G. Febvre, J.-F. Fournol, O. Crépel, P. Personne, and D. Daugeron, "In situ measurements of the scattering phase function of stratocumulus, contrails, and cirrus," Geophys. Res. Lett. 25, 971-974 (1998).
[CrossRef]

Fu, Q.

W. Sun, N. G. Loeb, G. Videen, and Q. Fu, "Examination of surface roughness on light scattering by long ice columns by use of a two-dimensional finite-difference time-domain algorithm," Appl. Opt. 43, 1957-1964 (2004).
[CrossRef] [PubMed]

W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, and G. Videen, "Light scattering by Gaussian particles: a solution with finite-difference time-domain technique," J. Quant. Spectrosc. Radiat. Transf. 79-80, 1083-1090 (2003).
[CrossRef]

Gayet, J. F.

J. F. Gayet, F. Auriol, S. Oshchepkov, F. Schröder, C. Duroure, G. Febvre, J.-F. Fournol, O. Crépel, P. Personne, and D. Daugeron, "In situ measurements of the scattering phase function of stratocumulus, contrails, and cirrus," Geophys. Res. Lett. 25, 971-974 (1998).
[CrossRef]

Havemann, S.

T. Rother, S. Havemann, and K. Schmidt, "Scattering of plane waves on finite cylinders with noncircular cross sections," in Progress in Electromagnetics Research, J. A. Kong, ed. (EMW Publishing, 1999), Vol. 23, pp. 79-105.

Hess, M.

Kattawar, G. W.

Ch. Li, G. W. Kattawar, and P. Yang, "Effects of surface roughness on light scattering by small particles," J. Quant. Spectrosc. Radiat. Transf. 89, 123-131 (2004).
[CrossRef]

Kong, J. A.

T. Rother, S. Havemann, and K. Schmidt, "Scattering of plane waves on finite cylinders with noncircular cross sections," in Progress in Electromagnetics Research, J. A. Kong, ed. (EMW Publishing, 1999), Vol. 23, pp. 79-105.

Kravtsov, Y. A.

M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 4: Wave Propagation Through Random Media (Springer, 1989).

Lacis, A. A.

M. I. Mishchenko and A. A. Lacis, "Morphology-dependent resonances of nearly spherical particles in random orientation," Appl. Opt. 42, 5551-5556 (2003).
[CrossRef] [PubMed]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Li, Ch.

Ch. Li, G. W. Kattawar, and P. Yang, "Effects of surface roughness on light scattering by small particles," J. Quant. Spectrosc. Radiat. Transf. 89, 123-131 (2004).
[CrossRef]

Lock, J. A.

Loeb, N. G.

W. Sun, N. G. Loeb, G. Videen, and Q. Fu, "Examination of surface roughness on light scattering by long ice columns by use of a two-dimensional finite-difference time-domain algorithm," Appl. Opt. 43, 1957-1964 (2004).
[CrossRef] [PubMed]

W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, and G. Videen, "Light scattering by Gaussian particles: a solution with finite-difference time-domain technique," J. Quant. Spectrosc. Radiat. Transf. 79-80, 1083-1090 (2003).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko and A. A. Lacis, "Morphology-dependent resonances of nearly spherical particles in random orientation," Appl. Opt. 42, 5551-5556 (2003).
[CrossRef] [PubMed]

M. I. Mishchenko and L. D. Travis, "Light scattering by polydisperse, rotationally symmetric particles: linear polarization," J. Quant. Spectrosc. Radiat. Transf. 51, 759-778 (1994).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Mugnai, A.

Muinonen, K.

W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, and G. Videen, "Light scattering by Gaussian particles: a solution with finite-difference time-domain technique," J. Quant. Spectrosc. Radiat. Transf. 79-80, 1083-1090 (2003).
[CrossRef]

Nousiainen, T.

W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, and G. Videen, "Light scattering by Gaussian particles: a solution with finite-difference time-domain technique," J. Quant. Spectrosc. Radiat. Transf. 79-80, 1083-1090 (2003).
[CrossRef]

Oshchepkov, S.

J. F. Gayet, F. Auriol, S. Oshchepkov, F. Schröder, C. Duroure, G. Febvre, J.-F. Fournol, O. Crépel, P. Personne, and D. Daugeron, "In situ measurements of the scattering phase function of stratocumulus, contrails, and cirrus," Geophys. Res. Lett. 25, 971-974 (1998).
[CrossRef]

Personne, P.

J. F. Gayet, F. Auriol, S. Oshchepkov, F. Schröder, C. Duroure, G. Febvre, J.-F. Fournol, O. Crépel, P. Personne, and D. Daugeron, "In situ measurements of the scattering phase function of stratocumulus, contrails, and cirrus," Geophys. Res. Lett. 25, 971-974 (1998).
[CrossRef]

Rother, T.

J. Wauer, K. Schmidt, T. Rother, T. Ernst, and M. Hess, "Two software tools for plane wave scattering on nonspherical particles in the German Aerospace Center's virtual laboratory," Appl. Opt. 43, 6371-6379 (2004).
[CrossRef] [PubMed]

T. Ernst, T. Rother, F. Schreier, J. Wauer, and W. Balzer, "DLR's VirtualLab: Scientific Software just a mouse click away," IEEE Comput. Sci. Eng. 5, 70-79 (2003).
[CrossRef]

T. Rother, S. Havemann, and K. Schmidt, "Scattering of plane waves on finite cylinders with noncircular cross sections," in Progress in Electromagnetics Research, J. A. Kong, ed. (EMW Publishing, 1999), Vol. 23, pp. 79-105.

Rytov, M.

M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 4: Wave Propagation Through Random Media (Springer, 1989).

Schiffer, R.

Schmidt, K.

J. Wauer, K. Schmidt, T. Rother, T. Ernst, and M. Hess, "Two software tools for plane wave scattering on nonspherical particles in the German Aerospace Center's virtual laboratory," Appl. Opt. 43, 6371-6379 (2004).
[CrossRef] [PubMed]

T. Rother, S. Havemann, and K. Schmidt, "Scattering of plane waves on finite cylinders with noncircular cross sections," in Progress in Electromagnetics Research, J. A. Kong, ed. (EMW Publishing, 1999), Vol. 23, pp. 79-105.

Schreier, F.

T. Ernst, T. Rother, F. Schreier, J. Wauer, and W. Balzer, "DLR's VirtualLab: Scientific Software just a mouse click away," IEEE Comput. Sci. Eng. 5, 70-79 (2003).
[CrossRef]

Schröder, F.

J. F. Gayet, F. Auriol, S. Oshchepkov, F. Schröder, C. Duroure, G. Febvre, J.-F. Fournol, O. Crépel, P. Personne, and D. Daugeron, "In situ measurements of the scattering phase function of stratocumulus, contrails, and cirrus," Geophys. Res. Lett. 25, 971-974 (1998).
[CrossRef]

Sun, W.

W. Sun, N. G. Loeb, G. Videen, and Q. Fu, "Examination of surface roughness on light scattering by long ice columns by use of a two-dimensional finite-difference time-domain algorithm," Appl. Opt. 43, 1957-1964 (2004).
[CrossRef] [PubMed]

W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, and G. Videen, "Light scattering by Gaussian particles: a solution with finite-difference time-domain technique," J. Quant. Spectrosc. Radiat. Transf. 79-80, 1083-1090 (2003).
[CrossRef]

Tatarskii, V. I.

M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 4: Wave Propagation Through Random Media (Springer, 1989).

Travis, L. D.

M. I. Mishchenko and L. D. Travis, "Light scattering by polydisperse, rotationally symmetric particles: linear polarization," J. Quant. Spectrosc. Radiat. Transf. 51, 759-778 (1994).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Videen, G.

W. Sun, N. G. Loeb, G. Videen, and Q. Fu, "Examination of surface roughness on light scattering by long ice columns by use of a two-dimensional finite-difference time-domain algorithm," Appl. Opt. 43, 1957-1964 (2004).
[CrossRef] [PubMed]

W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, and G. Videen, "Light scattering by Gaussian particles: a solution with finite-difference time-domain technique," J. Quant. Spectrosc. Radiat. Transf. 79-80, 1083-1090 (2003).
[CrossRef]

Wauer, J.

J. Wauer, K. Schmidt, T. Rother, T. Ernst, and M. Hess, "Two software tools for plane wave scattering on nonspherical particles in the German Aerospace Center's virtual laboratory," Appl. Opt. 43, 6371-6379 (2004).
[CrossRef] [PubMed]

T. Ernst, T. Rother, F. Schreier, J. Wauer, and W. Balzer, "DLR's VirtualLab: Scientific Software just a mouse click away," IEEE Comput. Sci. Eng. 5, 70-79 (2003).
[CrossRef]

Wiscombe, W. J.

Yang, P.

Ch. Li, G. W. Kattawar, and P. Yang, "Effects of surface roughness on light scattering by small particles," J. Quant. Spectrosc. Radiat. Transf. 89, 123-131 (2004).
[CrossRef]

Appl. Opt. (7)

Geophys. Res. Lett. (1)

J. F. Gayet, F. Auriol, S. Oshchepkov, F. Schröder, C. Duroure, G. Febvre, J.-F. Fournol, O. Crépel, P. Personne, and D. Daugeron, "In situ measurements of the scattering phase function of stratocumulus, contrails, and cirrus," Geophys. Res. Lett. 25, 971-974 (1998).
[CrossRef]

IEEE Comput. Sci. Eng. (1)

T. Ernst, T. Rother, F. Schreier, J. Wauer, and W. Balzer, "DLR's VirtualLab: Scientific Software just a mouse click away," IEEE Comput. Sci. Eng. 5, 70-79 (2003).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf. (3)

Ch. Li, G. W. Kattawar, and P. Yang, "Effects of surface roughness on light scattering by small particles," J. Quant. Spectrosc. Radiat. Transf. 89, 123-131 (2004).
[CrossRef]

W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, and G. Videen, "Light scattering by Gaussian particles: a solution with finite-difference time-domain technique," J. Quant. Spectrosc. Radiat. Transf. 79-80, 1083-1090 (2003).
[CrossRef]

M. I. Mishchenko and L. D. Travis, "Light scattering by polydisperse, rotationally symmetric particles: linear polarization," J. Quant. Spectrosc. Radiat. Transf. 51, 759-778 (1994).
[CrossRef]

Opt. Lett. (1)

Other (4)

P. W. Barber and R. K. Chang, Optical Effects Associated with Small Particles (World Scientific, 1988).

T. Rother, S. Havemann, and K. Schmidt, "Scattering of plane waves on finite cylinders with noncircular cross sections," in Progress in Electromagnetics Research, J. A. Kong, ed. (EMW Publishing, 1999), Vol. 23, pp. 79-105.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge U. Press, 2002).

M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 4: Wave Propagation Through Random Media (Springer, 1989).

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Figures (15)

Fig. 1
Fig. 1

Shapes of the conventional Chebyshev particle (dotted curves) for different orders of the polynomial. The radius of the underlying sphere (solid curve) is r 0 = 3.5 μ m .

Fig. 2
Fig. 2

Shape of the combination of an underlying spheroidal geometry (solid curve) with a Chebyshev polynomial of order n o = 4 5 (dotted curve). The aspect ratio of the spheroidal boundary is 1 .105 .

Fig. 3
Fig. 3

Normalized phase functions (random orientation) of the conventional Chebyshev particles according to Fig. 1 at a size parameter of 27.5, a refractive index of n r = 1.31 , and for increasing orders: (a) orders 5 and 25, (b) orders 35 and 45.

Fig. 4
Fig. 4

Normalized phase function (random orientation) of the conventional Chebyshev particle of order 45 (dotted curve) compared to the corresponding phase function of the underlying sphere (solid curve) at a size parameter of 27.5 and a refractive index of n r = 1.31.

Fig. 5
Fig. 5

Phase matrix elements, 33∕11 (solid curve) and 44∕11 (dotted curve) of the conventional Chebyshev particle of order 5 at a size parameter of 27.5 and a refractive index of n r = 1.31 in random orientation.

Fig. 6
Fig. 6

Phase matrix elements, 33∕11 (solid curve) and 44∕11 (dotted curve) of the conventional Chebyshev particle of order 45 at a size parameter of 27.5 and a refractive index of n r = 1.31 in random orientation.

Fig. 7
Fig. 7

Normalized phase function (random orientation) of the conventional Chebyshev particle of order 45 (dotted curve) compared with the corresponding phase function of the underlying sphere (solid curve) at a size parameter of 27.5 and a refractive index of n r = 1.4717 + 0.389 i .

Fig. 8
Fig. 8

Normalized phase function (random orientation) of the conventional Chebyshev particle of order 45, a refractive index of n r = 1.4717 + 0.389 i , and a deformation parameter of 0.05 (dotted curves) and of the corresponding sphere (solid curves) at size parameters of (a) 15, (b) 24, and (c) 27.5.

Fig. 9
Fig. 9

Morphology-dependent resonances in (a) the backscattering and (b) the extinction efficiency of the conventional Chebyshev particle of order 45. Computations have been performed for different deformation parameters with an incident plane wave along the axis of symmetry.

Fig. 10
Fig. 10

Morphology-dependent resonances in (a) the backscattering and (b) extinction efficiency of the conventional Chebyshev particle of order 45. Computations have been performed for different deformation parameters with a perpendicular incident plane wave.

Fig. 11
Fig. 11

Normalized phase functions of the circular and Chebyshev cylinder. Calculations have been performed at a refractive index of n r = 1.311 and a size parameter of 24.

Fig. 12
Fig. 12

Normalized phase functions of the circular and Chebyshev cylinder. Calculations have been performed at a refractive index of n r = 1.4717 + 0.389 i and a size parameter of 24.

Fig. 13
Fig. 13

Normalized phase functions of the circular and Chebyshev cylinder. Calculations have been performed at a refractive index of n r = 1.4717 + 0.389 i and a size parameter of 28.

Fig. 14
Fig. 14

Normalized phase function (random orientation) of the combination of an underlying spheroidal geometry with a Chebyshev polynomial of order 45 according to Fig. 2 (dotted curve) compared to the corresponding phase function of the underlying spheroid (solid curve). Computations have been performed at a size parameter of 27.5 and a refractive index of n r = 1.31 + 0 i .

Fig. 15
Fig. 15

Phase matrix elements, 33∕11 (solid curve) and 44∕11 (dotted curve) of the combination of an underlying spheroidal geometry with a Chebyshev polynomial of order 45 according to Fig. 2 in random orientation. Computations have been performed at a size parameter of 27.5 and a refractive index of n r = 1.31 .

Tables (1)

Tables Icon

Table 1 Table of Convergence Parameters

Equations (12)

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

R ( θ ) = r 0 [ 1 + ϵ cos ( n o θ ) ] ,
R ( θ ) = r spd ( θ ) [ 1 + ϵ cos ( n o θ ) ] ,
r spd ( θ ) = a [ cos 2 θ + ( a b ) 2 sin 2 θ ] .
E s ( k o r ) = τ = 1 2 n = 1 l = n n f τ n l Ψ τ n l ( k o r ) ,
E int ( k s r ) = τ = 1 2 n = 1 l = n n p τ n l R g Ψ τ n l ( k s r ) ,
E inc ( k o r ) = τ = 1 2 n = 1 l = n n a τ n l R g Ψ τ n l ( k o r ) .
f = T · a
T = Q 1 · R g Q ,
Q τ n l ; τ nl = S d S { 1 μ s [ × Rg Ψ τ n l ( k s r ) ] × Ψ τ n l ( k o r ) + 1 μ o  Rg Ψ τ n l ( k s r ) × [ × Ψ τ n l ( k o r ) ] } ,
Rg Q τ n l ; τ n l = S d S { 1 μ s [ × Rg Ψ τ n l ( k s r ) ] × Rg Ψ τ n l ( k o r ) + 1 μ o  Rg Ψ τ n l ( k s r ) × [ × Rg Ψ τ n l ( k o r ) ] } .
τ = 1 2 n = 1 l = n n l = n = | l | τ = 1 2
l = n = | l | l = l cut l cut n = | l | n cut ,   l cut n cut .

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