Abstract

Light scattering characteristics of spheroidal particles are studied for a wide range of particle parameters and orientations. The method of computation is based on the scattering theory for a homogeneous spheroid developed by us, and the calculation is extended to fairly large spheroidal particles of a size parameter up to 30. Effects of the particle size, shape, index of refraction, and orientation on the scattering efficiency factors and the scattering intensity functions are investigated and interpreted physically. The scattering properties of prolate and oblate spheroids with incidence parallel to the rotation axis constitute the extremes. The prolate spheroids at parallel incidence have steep and high resonance maxima in the scattering efficiency factors and broad and low forwardscattering peaks in the intensity functions; on the other hand, the oblate spheroids at parallel incidence have broad and low resonance maxima and sharp and high forwardscattering peaks. With an increase of the incidence angle, the scattering behavior of prolate spheroids approaches that of oblate spheroids at parallel incidence and vice versa. It is shown that, for oblique incidence, the scattering properties of a long slender prolate spheroid resemble those of an infinitely long circular cylinder. Effects of absorption on the extinction efficiency factors and scattering intensity functions are examined. Some problems in numerical calculation of the spheroidal wave functions and the infinite series solutions are discussed.

© 1979 Optical Society of America

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1978

1977

P. Chýlek, J. Opt. Soc. Am. 67, 175 (1977).
[CrossRef]

T. B. A. Senior, H. Weil, Appl. Opt. 16, 2979 (1977).
[CrossRef] [PubMed]

P. W. Barber, IEEE Trans. Microwave Theory Tech. MTT-25, 373 (1977).
[CrossRef]

N. K. Uzunoglu, A. R. Holt, J. Phys. A: Math. Gen. 10, 413 (1977).
[CrossRef]

V. Khare, H. M. Nussenzveig, Phys. Rev. Lett. 38, 1279 (1977).
[CrossRef]

1976

1975

1974

J. E. Hansen, L. D. Travis, Space Sci. Rev. 16, 527 (1974).
[CrossRef]

1972

1970

D. B. Hodge, J. Math. Phys. 11, 2308 (1970).
[CrossRef]

1968

1966

H. C. Bryant, A. J. Cox, J. Opt. Soc. Am. 56, 1529 (1966).
[CrossRef]

A. C. Lind, J. M. Greenberg, J. Appl. Phys. 37, 3195 (1966).
[CrossRef]

1965

1947

C. J. Bouwkamp, J. Math. Phys. (Cambridge, Mass.) 26, 79 (1947).

Alexopoulos, N. G.

Asano, S.

Baier, R. V.

S. Hanish, C. Shely, R. V. Baier, Naval Research Laboratory Report 6472, Washington D.C. (1966).

S. Hanish, R. V. Baier, W. H. Buckler, B. K. Carmichael, Naval Research Laboratory Report 6502, Washington D.C. (1967).

S. Hanish, R. V. Baier, W. H. Buckler, B. K. Carmichael, Naval Research Laboratory Report 6543, Washington, D.C. (1967).

Barber, P. W.

Bouwkamp, C. J.

C. J. Bouwkamp, J. Math. Phys. (Cambridge, Mass.) 26, 79 (1947).

Bryant, H. C.

Buckler, W. H.

S. Hanish, R. V. Baier, W. H. Buckler, B. K. Carmichael, Naval Research Laboratory Report 6543, Washington, D.C. (1967).

S. Hanish, R. V. Baier, W. H. Buckler, B. K. Carmichael, Naval Research Laboratory Report 6502, Washington D.C. (1967).

Carmichael, B. K.

S. Hanish, R. V. Baier, W. H. Buckler, B. K. Carmichael, Naval Research Laboratory Report 6502, Washington D.C. (1967).

S. Hanish, R. V. Baier, W. H. Buckler, B. K. Carmichael, Naval Research Laboratory Report 6543, Washington, D.C. (1967).

Chang, C.

C. Chang, C. Yeh, USCEE Report 166, Department of Electrical Engineering, U. Southern Calif., Los Angeles (1966).

Chu, C-M.

Chýlek, P.

Cox, A. J.

Evans, B. G.

A. R. Holt, N. K. Uzunoglu, B. G. Evans, IEEE Trans. Antennas Propag. Ap-26, 706 (1978).
[CrossRef]

N. K. Uzunoglu, B. G. Evans, A. R. Holt, Electron. Lett. 12, 312 (1976).
[CrossRef]

Eyges, L.

Fahlen, T. S.

Fikioris, J. G.

Flammer, C.

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

Greenberg, J. M.

A. C. Lind, J. M. Greenberg, J. Appl. Phys. 37, 3195 (1966).
[CrossRef]

A. C. Lind, R. T. Wang, J. M. Greenberg, Appl. Opt. 4, 1555 (1965).
[CrossRef]

Hanish, S.

S. Hanish, C. Shely, R. V. Baier, Naval Research Laboratory Report 6472, Washington D.C. (1966).

S. Hanish, R. V. Baier, W. H. Buckler, B. K. Carmichael, Naval Research Laboratory Report 6543, Washington, D.C. (1967).

S. Hanish, R. V. Baier, W. H. Buckler, B. K. Carmichael, Naval Research Laboratory Report 6502, Washington D.C. (1967).

Hansen, J. E.

J. E. Hansen, L. D. Travis, Space Sci. Rev. 16, 527 (1974).
[CrossRef]

Hodge, D. B.

D. B. Hodge, J. Math. Phys. 11, 2308 (1970).
[CrossRef]

Holt, A. R.

A. R. Holt, N. K. Uzunoglu, B. G. Evans, IEEE Trans. Antennas Propag. Ap-26, 706 (1978).
[CrossRef]

N. K. Uzunoglu, A. R. Holt, J. Phys. A: Math. Gen. 10, 413 (1977).
[CrossRef]

N. K. Uzunoglu, B. G. Evans, A. R. Holt, Electron. Lett. 12, 312 (1976).
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light (Academic, New York, 1969).

Khare, V.

V. Khare, H. M. Nussenzveig, Phys. Rev. Lett. 38, 1279 (1977).
[CrossRef]

Layton, L. L.

M. M. Stucky, L. L. Layton, Applied Mathematics Laboratory, Department of the Navy Report 164, Washington, D.C. (1964).

Lind, A. C.

A. C. Lind, J. M. Greenberg, J. Appl. Phys. 37, 3195 (1966).
[CrossRef]

A. C. Lind, R. T. Wang, J. M. Greenberg, Appl. Opt. 4, 1555 (1965).
[CrossRef]

Liou, K.-N.

MacPhie, R. H.

B. P. Sinha, R. H. MacPhie, J. Math. Phys. 16, 2378 (1975).
[CrossRef]

Nelson, A.

Nussenzveig, H. M.

V. Khare, H. M. Nussenzveig, Phys. Rev. Lett. 38, 1279 (1977).
[CrossRef]

Senior, T. B. A.

Shely, C.

S. Hanish, C. Shely, R. V. Baier, Naval Research Laboratory Report 6472, Washington D.C. (1966).

Shipley, S. T.

Sinha, B. P.

B. P. Sinha, R. H. MacPhie, J. Math. Phys. 16, 2378 (1975).
[CrossRef]

Stucky, M. M.

M. M. Stucky, L. L. Layton, Applied Mathematics Laboratory, Department of the Navy Report 164, Washington, D.C. (1964).

Travis, L. D.

J. E. Hansen, L. D. Travis, Space Sci. Rev. 16, 527 (1974).
[CrossRef]

Uzunoglu, N. K.

A. R. Holt, N. K. Uzunoglu, B. G. Evans, IEEE Trans. Antennas Propag. Ap-26, 706 (1978).
[CrossRef]

N. K. Uzunoglu, N. G. Alexopoulos, J. G. Fikioris, J. Opt. Soc. Am. 68, 194 (1978).
[CrossRef]

N. K. Uzunoglu, A. R. Holt, J. Phys. A: Math. Gen. 10, 413 (1977).
[CrossRef]

N. K. Uzunoglu, B. G. Evans, A. R. Holt, Electron. Lett. 12, 312 (1976).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 172–179 and 342–364.

Wang, D. S.

Wang, R. T.

Weil, H.

Weinman, J. A.

Yamamoto, G.

Yeh, C.

P. W. Barber, C. Yeh, Appl. Opt. 14, 2864 (1975).
[CrossRef] [PubMed]

C. Chang, C. Yeh, USCEE Report 166, Department of Electrical Engineering, U. Southern Calif., Los Angeles (1966).

Appl. Opt.

Electron. Lett.

N. K. Uzunoglu, B. G. Evans, A. R. Holt, Electron. Lett. 12, 312 (1976).
[CrossRef]

IEEE Trans. Antennas Propag.

A. R. Holt, N. K. Uzunoglu, B. G. Evans, IEEE Trans. Antennas Propag. Ap-26, 706 (1978).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

P. W. Barber, IEEE Trans. Microwave Theory Tech. MTT-25, 373 (1977).
[CrossRef]

J. Appl. Phys.

A. C. Lind, J. M. Greenberg, J. Appl. Phys. 37, 3195 (1966).
[CrossRef]

J. Math. Phys.

D. B. Hodge, J. Math. Phys. 11, 2308 (1970).
[CrossRef]

B. P. Sinha, R. H. MacPhie, J. Math. Phys. 16, 2378 (1975).
[CrossRef]

J. Math. Phys. (Cambridge, Mass.)

C. J. Bouwkamp, J. Math. Phys. (Cambridge, Mass.) 26, 79 (1947).

J. Opt. Soc. Am.

J. Phys. A: Math. Gen.

N. K. Uzunoglu, A. R. Holt, J. Phys. A: Math. Gen. 10, 413 (1977).
[CrossRef]

Phys. Rev. Lett.

V. Khare, H. M. Nussenzveig, Phys. Rev. Lett. 38, 1279 (1977).
[CrossRef]

Space Sci. Rev.

J. E. Hansen, L. D. Travis, Space Sci. Rev. 16, 527 (1974).
[CrossRef]

Other

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 172–179 and 342–364.

M. Kerker, The Scattering of Light (Academic, New York, 1969).

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

M. M. Stucky, L. L. Layton, Applied Mathematics Laboratory, Department of the Navy Report 164, Washington, D.C. (1964).

C. Chang, C. Yeh, USCEE Report 166, Department of Electrical Engineering, U. Southern Calif., Los Angeles (1966).

S. Hanish, C. Shely, R. V. Baier, Naval Research Laboratory Report 6472, Washington D.C. (1966).

S. Hanish, R. V. Baier, W. H. Buckler, B. K. Carmichael, Naval Research Laboratory Report 6502, Washington D.C. (1967).

S. Hanish, R. V. Baier, W. H. Buckler, B. K. Carmichael, Naval Research Laboratory Report 6543, Washington, D.C. (1967).

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Figures (17)

Fig. 1
Fig. 1

Scattering geometry. The spherical coordinate system is adopted to represent the scattered field in the far-field zone. The origin of the coordinate system and the z axis are the center and the axis of revolution of the spheroid, respectively. The angle of incidence ζ is the angle in the plane of incidence (the xz plane) between the direction of incidence and the z axis.

Fig. 2
Fig. 2

(a) Value of the mth term of the infinite summation series, Eq. (3), for the extinction cross sections with oblique incidence of the TE (solid lines) and TM (broken lines) mode polarization waves for the prolate spheroid with m ˜ = 1.50, a/b = 5, and α = 20. (b) Value of the mth term of the infinite summation series, Eq. (3), for oblate spheroids with m ˜ = 1.33, a/b = 1.5, and α = 5 (broken lines) and α = 20 (solid lines) for oblique incidence of the TE mode polarization wave.

Fig. 3
Fig. 3

Scattering efficiency factors Qsca at ζ = 0° as a function of the size parameter 2πa/λ for the prolate spheroids with m ˜ = 1.50 for several values of the shape parameter a/b.

Fig. 4
Fig. 4

Scattering cross sections of prolate spheroids normalized by the area m ˜ of a sphere of the same volume as a function of the size parameter of the sphere 2πrv/λ for prolate spheroids with m ˜ = 1.50 and a/b = 1.5 2,3, and 5. The curve for spheres is shown by the dotted line.

Fig. 5
Fig. 5

Scattering efficiency factors Qsca at ζ = 0° as a function of the size parameter 2πa/λ for oblate spheroids with m ˜ = 1.33 and a/b = 1.1, 1.5, 2, 3, and 5.

Fig. 6
Fig. 6

Scattering cross sections of oblate spheroids normalized by the area π r v 2 of a sphere of the same volume as a function of the size parameter of the sphere 2πrv/λ for oblate spheroids with m ˜ = 1.33 and a/b = 1.5, 2, 3, and 5. The curve for spheres is shown by the dotted line.

Fig. 7
Fig. 7

Scattering efficiency factors Qsca as a function of the size parameter 2πa/λ for prolate spheroids with m ˜ = 1.50 and a/b = 5 for incidence of the TE (solid lines) and TM (broken lines) mode polarization waves at incidence angles ζ = 0°, 45°, and 90°.

Fig. 8
Fig. 8

Scattering efficiency factors Qsca as a function of the size parameter 2πa/λ for oblate spheroids with m ˜ = 1.33 and a/b = 3 for incidence of the TE (solid lines) and TM (broken lines) mode polarization waves at incidence angles ζ = 0°, 45°, and 90°.

Fig. 9
Fig. 9

Intensity functions of forwardscattering i(0°) and backscattering i(180°) at parallel incidence as a function of the size parameter 2πa/λ for the slightly nonspherical prolate spheroids with m ˜ = 1.50 and a/b = 1.1 and 1.5. The intensity functions for spheres are shown by dotted lines.

Fig. 10
Fig. 10

Intensity functions of forwardscattering i(ζ,0°) and backscattering i(180° − ζ, 180°) for unpolarized incident light at ζ = 0°, 45°, and 90° as a function of the size parameter 2πa/λ for prolate spheroids with m ˜ = 1.50 and a/b = 5.

Fig. 11
Fig. 11

Intensity functions of forwardscattering i(ζ,0°) and backscattering i(180° − ζ, 180°) for unpolarized incident light at ζ = 0°, 45°, and 90° as a function of the size parameter 2πa/λ for oblate spheroids with m ˜ = 1.33 and a/b = 3.

Fig. 12
Fig. 12

Angular distribution of the intensity functions i(θ,ϕ) for unpolarized incident light as a function of the zenith angle θ on the incidence plane (ϕ = 0° − 180°) for the prolate spheroid with m ˜ = 1.50, a/b = 5, and α = 10 at incidence angles θ = 0°, 45°, and 90°.

Fig. 13
Fig. 13

Angular distribution of the intensity functions i(θ,ϕ) for unpolarized incident light as a function of the zenith angle θ on the incidence plane (ϕ = 0° − 180°) for the oblate spheroid with m ˜ = 1.33, a/b = 3, and α = 10 at incidence angles ζ = 0°, 45°, and 90°.

Fig. 14
Fig. 14

Angular distribution of the intensity functions i(θ,ϕ) for unpolarized incidence light for the prolate spheroid with m ˜ = 1.50, a/b = 5, and α = 20 at ζ = 45°. The. figure shows, as a function of the zenith angle θ, the distribution patterns in three scattering planes through the z axis: one parallel to the incidence plane (ϕ = 0° and 180°), one inclining from it by an angle 45° (ϕ = 45° and 225°), and one normal to it (ϕ = 90° and 270°).

Fig. 15
Fig. 15

Efficiency factors for extinction Qext, scattering Qsca, and absorption Qabs at ζ = 0° as a function of the size parameter 2πa/λ for absorbing prolate spheroids with m ˜ = 1.50 + 0.1i and a/b = 2. The extinction efficiency factors for nonabsorbing prolate spheroids with m ˜ = 1.50 + 0.0i are also shown.

Fig. 16
Fig. 16

Angular distribution of the intensity function i1(θ) at parallel incidence of the TE mode polarization wave as a function of the zenith angle θ for absorbing ( m ˜ = 1.50 + 0.1i) and nonabsorbing ( m ˜ = 1.50 + 0.0i) prolate spheroids with a/b = 2 and α = 9.

Fig. 17
Fig. 17

Intensity functions for forwardscattering i(0°) and backscattering i(180°) at parallel incidence as a function of the size parameter 2πa/λ for absorbing ( m ˜ = 1.50 + 0.0i) and nonabsorbing ( m ˜ = 1.50 + 0.0i) prolate spheroids with a/b = 2.

Equations (11)

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α = 2 π a / λ ,
r = 0 , 1 ʹ ( 2 m + r ) ! r ! d r m n ( c )
R m n ( 2 ) ( - i c ; i ξ ) = 1 i [ R m n ( 3 ) ( - i c ; i ξ ) - R m n ( 1 ) ( - i c ; i ξ ) ] ,
C 1 , ext 2 ( m ) = - Re n = m N [ α 1 , m n 2 σ m n ( ζ ) + β 1 , m n 2 χ m n ( ζ ) ]
cos Θ = cos ζ · cos θ + sin ζ · sin θ · cos ϕ ,
G ( ζ ) = π b ( a 2 sin 2 ζ + b 2 cos 2 ζ ) 1 / 2 ,
G ( ζ ) = π a ( b 2 sin 2 ζ + a 2 cos 2 ζ ) 1 / 2
i ( ζ , ) = ½ [ i 11 ( ζ , ) + i 22 ( ζ , ) ,
i ( 180° - ζ , 180° ) = ½ [ i 11 ( 180° - ζ , 180° ) + i 22 ( 180° - ζ , 180° ) ] .
i 1 ( ) = i 2 ( ) = | n = 1 [ r = 0 , 1 ( r + 1 ) ( r + 2 ) 2 d r ln ] ( α ln + β ln ) | 2 ,
i 1 ( 180° ) = i 2 ( 180° ) = | n = 1 [ r = 0 , 1 ( r + 1 ) ( r + 2 ) 2 d r ln ] ( - 1 ) n ( α ln - β ln ) | 2 ,

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