Abstract

An exact transform of the scattering coefficients that removes the m-related refractive-index restrictions of the Rayleigh and Thomson approximations is presented. The resulting series is valid for all m and small x. A similar but approximate transform for spheroids also is presented.

© 1993 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 14, pp. 269–292.
  2. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  3. M. Kerker, P. Scheiner, D. D. Cooke, “The range of validity of the Rayleigh and Thomson limits for Lorenz–Mie scattering,” J. Opt. Soc. Am. 68, 135–137 (1978).
    [CrossRef]
  4. A. F. Stevenson, “Electromagnetic scattering by an ellipsoid in the third approximation,” J. Appl. Phys. 24, 1143–1151 (1953).
    [CrossRef]
  5. G. R. Fournier, B. T. N. Evans, “An approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30, 2042–2048 (1991).
    [CrossRef] [PubMed]
  6. S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
    [PubMed]
  7. P. W. Barber, S. C. Hill, A. C. Hill, “Light scattering by size/shape distributions of soil particles and spheroids,” Appl. Opt. 23, 1025–1031 (1984).
    [CrossRef] [PubMed]
  8. T. B. A. Senior, “The scattering of an electromagnetic wave by a spheroid,” Can. J. Phys. 44, 1353–1359 (1966).
    [CrossRef]
  9. T. C. Choy, A. M. Stoneham, “On the microwave loss of granular high Tc superconductors in the 0.1 GHz to 1 THz region,” Proc. R. Soc. London ser. A 434, 555–570 (1991).
    [CrossRef]
  10. B. T. N. Evans, G. R. Fournier, “Simple approximation to extinction efficiency valid over all size parameters,” Appl. Opt. 29, 4666–4670 (1990).
    [CrossRef] [PubMed]

1991 (2)

T. C. Choy, A. M. Stoneham, “On the microwave loss of granular high Tc superconductors in the 0.1 GHz to 1 THz region,” Proc. R. Soc. London ser. A 434, 555–570 (1991).
[CrossRef]

G. R. Fournier, B. T. N. Evans, “An approximation to extinction efficiency for randomly oriented spheroids,” Appl. Opt. 30, 2042–2048 (1991).
[CrossRef] [PubMed]

1990 (1)

1984 (1)

1978 (1)

1975 (1)

1966 (1)

T. B. A. Senior, “The scattering of an electromagnetic wave by a spheroid,” Can. J. Phys. 44, 1353–1359 (1966).
[CrossRef]

1953 (1)

A. F. Stevenson, “Electromagnetic scattering by an ellipsoid in the third approximation,” J. Appl. Phys. 24, 1143–1151 (1953).
[CrossRef]

Asano, S.

Barber, P. W.

Choy, T. C.

T. C. Choy, A. M. Stoneham, “On the microwave loss of granular high Tc superconductors in the 0.1 GHz to 1 THz region,” Proc. R. Soc. London ser. A 434, 555–570 (1991).
[CrossRef]

Cooke, D. D.

Evans, B. T. N.

Fournier, G. R.

Hill, A. C.

Hill, S. C.

Kerker, M.

Scheiner, P.

Senior, T. B. A.

T. B. A. Senior, “The scattering of an electromagnetic wave by a spheroid,” Can. J. Phys. 44, 1353–1359 (1966).
[CrossRef]

Stevenson, A. F.

A. F. Stevenson, “Electromagnetic scattering by an ellipsoid in the third approximation,” J. Appl. Phys. 24, 1143–1151 (1953).
[CrossRef]

Stoneham, A. M.

T. C. Choy, A. M. Stoneham, “On the microwave loss of granular high Tc superconductors in the 0.1 GHz to 1 THz region,” Proc. R. Soc. London ser. A 434, 555–570 (1991).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 14, pp. 269–292.

Yamamoto, G.

Appl. Opt. (4)

Can. J. Phys. (1)

T. B. A. Senior, “The scattering of an electromagnetic wave by a spheroid,” Can. J. Phys. 44, 1353–1359 (1966).
[CrossRef]

J. Appl. Phys. (1)

A. F. Stevenson, “Electromagnetic scattering by an ellipsoid in the third approximation,” J. Appl. Phys. 24, 1143–1151 (1953).
[CrossRef]

J. Opt. Soc. Am. (1)

Proc. R. Soc. London ser. A (1)

T. C. Choy, A. M. Stoneham, “On the microwave loss of granular high Tc superconductors in the 0.1 GHz to 1 THz region,” Proc. R. Soc. London ser. A 434, 555–570 (1991).
[CrossRef]

Other (2)

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 14, pp. 269–292.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

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Figures (14)

Fig. 1
Fig. 1

Ratio of Qsca to the Mie calculation for spheres as a function of size parameter calculated by three series to the fourth order in X. Refractive index m = 500–500i.

Fig. 2
Fig. 2

Qext for spheres as calculated by Mie theory, the new series [Eq. (15)] and the Rayleigh approximation. Refractive index m = 500–500i.

Fig. 3
Fig. 3

Qext for spheres as calculated by Mie theory and the new series [Eq. (15)]. Refractive index m = 100–10−6i.

Fig. 4
Fig. 4

Qext for water spheres at 3.3 GHz. Refractive index m = 8.743–0.6409i.

Fig. 5
Fig. 5

Qext for water spheres at 9.4 GHz. Refractive index m = 8.075–1.824i.

Fig. 6
Fig. 6

Qext for water spheroids, r = 2, at 3.3 GHz. Refractive index m = 8.743–0.6409i.

Fig. 7
Fig. 7

Qext for water spheroids, r = 2, at 9.4 GHz. Refractive index m = 8.075–1.824i.

Fig. 8
Fig. 8

Qext for water spheroids, r = 0.5, at 3.3 GHz. Refractive index m = 8.743–0.6409i.

Fig. 9
Fig. 9

Qext for water spheroids, r = 0.5, at 9.4 GHz. Refractive index m = 8.075–1.824i.

Fig. 10
Fig. 10

Qext for amorphous carbon, r = 1.5, at 94 GHz. Refractive index m = 50–50i.

Fig. 11
Fig. 11

Qext for copper oblate spheroids, r = 0.333, in the infrared. Refractive index m = 35–35i.

Fig. 12
Fig. 12

Qext for spheres of YBa2Cu3O7−δ at 77 °K, r = 1, at 10 GHz. Refractive index m = 1.2–17683i.

Fig. 13
Fig. 13

Qext for spheroids of YBa2Cu3O7−δ at 77 °K, r = 10, at 10 GHz. Refractive index m = 1.2–17683i.

Fig. 14
Fig. 14

Qext for spheroids of YBa2Cu3O7−δ at 77 °K, r = 0.1, at 10 GHz. Refractive index m = 1.2–17683i.

Equations (41)

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Q sca = 2 x 2 n = 1 ( 2 n + 1 ) ( a n 2 + b n 2 ) ,
Q ext = 2 x 2 n = 1 ( 2 n + 1 ) [ Re ( a n + b n ) ] ,
a n = μ ψ n ( μ x ) ψ n ( x ) - ψ n ( μ x ) ψ n ( x ) μ ψ n ( μ x ) ζ n ( x ) - ψ n ( μ x ) ζ n ( x ) ,
b n = ψ n ( μ x ) ψ n ( x ) - μ ψ n ( μ x ) ψ n ( x ) ψ n ( μ x ) ζ n ( x ) - μ ψ n ( μ x ) ζ n ( x ) ,
a 1 = 2 3 i ( - 1 + 2 ) x 3 + 1 5 i [ 2 - 6 + 4 + 2 μ ( + 2 ) 2 ] x 5 + 4 9 ( - 1 + 2 ) 2 x 6 + 1 175 i [ 4 μ 2 + 3 ( 9 μ 2 + 35 μ - 25 ) - 2 ( 70 μ + 150 ) + 200 ( 2 - 1 ) ( + 2 ) 3 ] x 7 + ,
a 2 = 1 15 i ( - 1 2 + 3 ) x 5 + ,             b n = a n ,             μ .
a 1 = 2 3 i ( m 2 - 1 m 2 + 2 ) x 3 + 2 5 i [ ( m 2 - 1 ) ( m 2 - 2 ) ( m 2 + 2 ) 2 ] x 5 + 4 9 ( m 2 - 1 m 2 + 2 ) 2 x 6 + 1 175 × i [ ( m 2 - 1 ) ( m 6 + 20 m 4 - 200 m 2 + 200 ) ( m 2 + 2 ) 3 ] x 7 + ,
b 1 = / 45 1 i ( m 2 - 1 ) x 5 + .
a 1 = i x 3 + i x 5 + / 9 4 x 6 - / 7 1 i x 7 + ,
b 1 = - i x 3 + i x 5 + .
a n = ψ n ( x ) - [ ψ n ( μ x ) μ ψ n ( μ x ) ] ψ n ( x ) ζ n ( x ) - [ ψ n ( μ x ) μ ψ n ( μ x ) ] ζ n ( x ) .
lim μ 0 a n = ψ n ( x ) - ( x n + 1 ) ψ n ( x ) ζ n ( x ) - ( x n + 1 ) ζ n ( x ) .
μ ( n + 1 ) x ψ n ( μ x ) ψ n ( μ x ) = E n ,
μ μ ( n + 1 ) x ψ n ( μ x ) ψ n ( μ x ) = U n .
a 1 = 2 3 i ( E 1 - 1 E 1 + 2 ) x 3 + 1 5 i [ E 1 2 - 6 E 1 + 4 ( E 1 + 2 ) 2 ] x 5 + 4 9 ( E 1 - 1 E 1 + 2 ) 2 x 6 - 1 7 i ( E 1 3 + 6 E 1 2 - 16 E 1 + 8 ( E 1 + 2 ) 3 ) x 7 + ,
a 2 = 1 15 i ( E 2 - 1 2 E 2 + 3 ) x 5 + ,
b n = a n ,             E n U n .
E n = - ( n + 1 ) / n ,
U n = - ( n + 1 ) / n ,
E 1 ( z ) = 2 F ( 1 + F ) ,             E 2 ( z ) = - 3 ( 1 + 3 F ) 2 + 6 F - F z 2 ,
F ( z ) = z cot ( z ) - 1 z 2             z = μ x = ( n - i k ) x , U n = E n ,             μ .
Q ray = Q sca + Q abs ,
Q sca = 8 3 b 4 r 2 p [ sin 2 θ 2 ( η 1 2 + η 1 2 ) + ( 1 + cos 2 θ ) 2 ( η 2 2 + η 2 2 ) ] ,
Q abs = 4 b r p Re { i [ sin 2 θ 2 ( η 1 + η 1 ) + ( 1 + cos 2 θ ) 2 ( η 2 + η 2 ) ] } ,
p = cos 2 θ + r 2 sin 2 θ ,             a = 2 π α / λ ,             b = 2 π β / λ .
η 1 = 1 3 ( L 1 + 1 - 1 ) ,             η 1 = 1 3 ( L 1 + 1 μ - 1 ) ,
η 2 = 1 3 ( L 2 + 1 - 1 ) ,             η 2 = 1 3 ( L 2 + 1 μ - 1 ) ,
L 1 = ( 1 - g 2 ) g 2 [ - 1 = 1 2 g ln ( 1 + g 1 - g ) ] ,
L 2 = 1 - L 1 2 ,
g 2 = 1 - 1 r 2 .
L 1 = 1 + f 2 f 2 ( 1 - tan - 1 f f ) ,
L 2 = 1 - L 1 2 ,
f 2 = 1 r 2 - 1.
z ¯ = { μ b [ 1 + ν ( 1 - 1 / r 2 ) ] prolates μ b ( r 2 ) ν oblates ,             for η 1 ,
z ¯ = { μ b [ 1 + ν 1 / 3 ( 1 - 1 / r ) ] prolates μ b ( r ) ν 1 / 3 oblates ,             for η 2 ,
Q ¯ s c a = 16 9 ( b 4 r 2 A ¯ ) [ η ˜ 1 2 + η ˜ 1 2 + 2 ( η ˜ 2 2 + η ˜ 2 2 ) ] ,
Q ¯ a b s = 8 3 ( b r A ¯ ) Re { i [ η ˜ 1 + η ˜ 1 + 2 ( η ˜ 2 + η ˜ 2 ) ] } ,
A ¯ = 1 + r 2 r 2 - 1 sin - 1 ( r 2 - 1 r ) ,
A ¯ = 1 + r 2 1 - r 2 ln ( 1 + 1 - r 2 r ) ,
η ˜ 1 = 1 3 ( L 1 + 1 E 1 - 1 ) ,             η ˜ 1 = 1 3 ( L 1 + 1 U 1 - 1 ) ,
η ˜ 2 = 1 3 ( L 2 + 1 E 1 - 1 ) ,             η ˜ 2 = 1 3 ( L 2 + 1 U 1 - 1 ) .

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