Abstract

Several approximate methods for modeling the electromagnetic (em) scattering properties of nonspherical particles are examined and evaluated. Although some of the approaches are applicable to arbitrary shapes we confine our attention here mainly to spheres and cylinders, for which exact solutions are available for comparisons. Evaluations include comparisons of the computed angular phase function, total extinction efficiency, and backscatter efficiency. Approximate methods investigated include the Rayleigh–Gans (RG) approximation, the Wentzel–Kramers–Brillouin or WKB approximation [and the closely related eikonal approximation (EA)], diffraction theory, and the second-order Shifrin iterative technique. Examples using spheres indicate that for weakly absorbing particles of moderate- to large-size parameters with a real refractive index near unity (i.e., the optically soft case), all models work well in representing the phase function over all scattering angles, with the Shifrin approximation showing the best agreement with the exact solutions. For larger refractive indices, however, the Shifrin approximation breaks down, whereas the WKB method continues to perform relatively well for all scattering angles over a wide range of particle sizes, including those appropriate in both the RG (small particle) and the diffraction (large particle) limits. The relationship between the WKB, eikonal, and anomalous diffraction descriptions of particle extinction is discussed briefly. Backscatter is also discussed in the context of the WKB model, and two modifications to improve the description are included: one to add an internal-reflected internal wave and the other to add a multiplicative scaling factor to preserve the correct backscatter result for strong absorption in the geometric optics limit. A major conclusion of the paper is that the WKB method offers a viable alternative to the more widely used RG and diffraction approximations and is a method that offers significant improvement in accuracy with only a slight increase in mathematical complexity.

© 1992 Optical Society of America

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References

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1990 (1)

S. K. Sharma, D. J. Somerford, “The eikonal approximation revisited,” Nuovo Cimento 12, 719–748 (1990).
[CrossRef]

1989 (1)

1988 (2)

1987 (1)

1985 (1)

R. D. Haracz, L. D. Cohen, A. Cohen, “Scattering of linearly polarized light from randomly oriented cylinders and spheroids,” J. Appl. Phys. 58, 3322–3327 (1985).
[CrossRef]

1984 (2)

1983 (2)

L. D. Cohen, R. D. Haracz, A. Cohen, C. Acquista, “Scattering of light from arbitrarily oriented finite cylinders,” Appl. Opt. 22, 742–748 (1983).
[CrossRef] [PubMed]

M. F. Iskander, A. Lakhtakia, C. H. Durney, “A new procedure for improving the solution stability and extending the frequency range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317–324 (1983).
[CrossRef]

1976 (1)

1959 (1)

D. Deirmendjian, “Theory of the solar aureole, part II: applications to atmospheric models,” Ann. Geophys. 15, 218–249. (1959).

1923 (1)

H. Jeffreys, “On certain approximate solutions of linear differential equations of the second order,” Proc. London Math. Soc., 23, 428–436 (1923).
[CrossRef]

1912 (1)

J. W. S. Rayleigh, “On the propagation of waves through a stratified medium, with special reference to the question of reflection,” Proc. R. Soc. London Ser. A 86, 207–223 (1912).
[CrossRef]

Acquista, C.

Bayvel, L. P.

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and its Applications (Applied Science, London, 1981).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles, (Wiley, New York, 1983).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

Chen, T. W.

Cohen, A.

Cohen, L. D.

Deirmendjian, D.

D. Deirmendjian, “Theory of the solar aureole, part II: applications to atmospheric models,” Ann. Geophys. 15, 218–249. (1959).

Durney, C. H.

M. F. Iskander, A. Lakhtakia, C. H. Durney, “A new procedure for improving the solution stability and extending the frequency range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317–324 (1983).
[CrossRef]

Goedecke, G. H.

Haracz, R. D.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles, (Wiley, New York, 1983).

Ikeda, Y.

Y. Ikeda, “Extension of the Rayleigh-Gans theory,” in Electromagnetic Scattering, M. Kerker, ed. (Pergamon, Oxford, 1963).

Iskander, M. F.

M. F. Iskander, A. Lakhtakia, C. H. Durney, “A new procedure for improving the solution stability and extending the frequency range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317–324 (1983).
[CrossRef]

Jeffreys, H.

H. Jeffreys, “On certain approximate solutions of linear differential equations of the second order,” Proc. London Math. Soc., 23, 428–436 (1923).
[CrossRef]

Johnson, B. R.

Jones, A. R.

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and its Applications (Applied Science, London, 1981).
[CrossRef]

Lakhtakia, A.

M. F. Iskander, A. Lakhtakia, C. H. Durney, “A new procedure for improving the solution stability and extending the frequency range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317–324 (1983).
[CrossRef]

O’Brien, S. G.

Rayleigh, J. W. S.

J. W. S. Rayleigh, “On the propagation of waves through a stratified medium, with special reference to the question of reflection,” Proc. R. Soc. London Ser. A 86, 207–223 (1912).
[CrossRef]

Saxon, D. S.

D. S. Saxon, “Lectures on the scattering of light,” UCLA Department of Meterological Science Rep. 9 (University of California at Los Angeles, Los Angeles, Calif., 1955).

D. S. Saxon, in UCLA Conference on Radiation and Remote Probing of the Atmosphere, J. G. Kuriyan, ed. (University of California at Los Angeles, Los Angeles, Calif., 1974), pp. 227–308.

Sharma, S. K.

S. K. Sharma, D. J. Somerford, “The eikonal approximation revisited,” Nuovo Cimento 12, 719–748 (1990).
[CrossRef]

Shifrin, K. S.

K. S. Shifrin, Scattering of Light in a Turbid Medium (Moscow, 1951); NASA Tech. Transl. TT F-477 (National Technical Information Service, Springfield, VA, 1968).

Somerford, D. J.

S. K. Sharma, D. J. Somerford, “The eikonal approximation revisited,” Nuovo Cimento 12, 719–748 (1990).
[CrossRef]

Stephens, G. L.

van de Hulst, H.

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

Wang, R. T.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

Ann. Geophys. (1)

D. Deirmendjian, “Theory of the solar aureole, part II: applications to atmospheric models,” Ann. Geophys. 15, 218–249. (1959).

Appl. Opt. (8)

IEEE Trans. Antennas Propag. (1)

M. F. Iskander, A. Lakhtakia, C. H. Durney, “A new procedure for improving the solution stability and extending the frequency range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317–324 (1983).
[CrossRef]

J. Appl. Phys. (1)

R. D. Haracz, L. D. Cohen, A. Cohen, “Scattering of linearly polarized light from randomly oriented cylinders and spheroids,” J. Appl. Phys. 58, 3322–3327 (1985).
[CrossRef]

Nuovo Cimento (1)

S. K. Sharma, D. J. Somerford, “The eikonal approximation revisited,” Nuovo Cimento 12, 719–748 (1990).
[CrossRef]

Proc. London Math. Soc. (1)

H. Jeffreys, “On certain approximate solutions of linear differential equations of the second order,” Proc. London Math. Soc., 23, 428–436 (1923).
[CrossRef]

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

J. W. S. Rayleigh, “On the propagation of waves through a stratified medium, with special reference to the question of reflection,” Proc. R. Soc. London Ser. A 86, 207–223 (1912).
[CrossRef]

Other (8)

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

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

K. S. Shifrin, Scattering of Light in a Turbid Medium (Moscow, 1951); NASA Tech. Transl. TT F-477 (National Technical Information Service, Springfield, VA, 1968).

D. S. Saxon, “Lectures on the scattering of light,” UCLA Department of Meterological Science Rep. 9 (University of California at Los Angeles, Los Angeles, Calif., 1955).

D. S. Saxon, in UCLA Conference on Radiation and Remote Probing of the Atmosphere, J. G. Kuriyan, ed. (University of California at Los Angeles, Los Angeles, Calif., 1974), pp. 227–308.

Y. Ikeda, “Extension of the Rayleigh-Gans theory,” in Electromagnetic Scattering, M. Kerker, ed. (Pergamon, Oxford, 1963).

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and its Applications (Applied Science, London, 1981).
[CrossRef]

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles, (Wiley, New York, 1983).

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Figures (9)

Fig. 1
Fig. 1

Normalized phase function versus scattering angle for unpolarized light for spheres. Comparison of WKB, RG, and Shifrin approximations with Mie theory: (a) refractive index m = (1.1, 0) and size parameter α = 1.6; (b) m = (1.33, 0), α = 2.0.

Fig. 2
Fig. 2

Normalized phase function versus scattering angle for unpolarized light with refractive index m = (1.0, 0.07) and size parameter α = 20. Comparison of WKB, RG, and diffraction approximations with rigorous theory: (a) sphere, (b) cylinder.

Fig. 3
Fig. 3

Normalized phase function versus scattering angle for unpolarized light with refractive index m = (10, 2) and size parameter α = 20. Comparison of WKB, RG, and diffraction approximations with rigorous theory: (a) sphere, (b) cylinder.

Fig. 4
Fig. 4

WKB or ADA extinction efficiency versus phase parameter as calculated for a circular cylinder perpendicular to incident radiation, an ensemble of uniform randomly oriented circular cylinders, and a sphere of the same radius as the cylinders.

Fig. 5
Fig. 5

Backscatter efficiency versus phase parameter for a sphere without absorption. Comparison of single-wave and two-wave WKB approximations with rigorous theory: (a) m = (1.33, 0); (b) m = (3.0, 0).

Fig. 6
Fig. 6

Backscatter efficiency versus phase parameter for a sphere with absorption. Comparison of unsealed and scaled single-wave and two-wave WKB approximations with rigorous theory: (a) m = (1.33, 0.1); (b) m = (3.0, 1.0).

Fig. 7
Fig. 7

Phase function versus scattering angle for unpolarized light incident on spheres. Comparison of single-wave and two-wave WKB approximations with rigorous theory: (a) m = (1.2, 0), α = 1.0; (b) m = (1.2, 0.1), α = 1.0.

Fig. 8
Fig. 8

Phase function versus scattering angle for unpolarized light incident on spheres. Comparison of single-wave and two-wave WKB approximations with rigorous theory: (a) m = (1.33, 0), α = 2.0; (b) m = (1.33, 0.1), α = 2.0.

Fig. 9
Fig. 9

Phase function versus scattering angle for unpolarized light incident on spheres. Comparison of single-wave and two-wave WKB approximations with rigorous theory: (a) m = (3.0, 0), a = 7.5; (b) m = (3.0, 1.0), α = 7.5.

Equations (88)

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E ( r ) = E 0 exp ( i k · r ) + [ k 2 + ( · ) ] V ( m 2 1 ) E ( r ) G ( r , r ) d 3 r ,
E s ( r ) = E 0 A ( k , r ) exp ( i k r ) r ,
A ( k , r ) = k 2 [ 1 r ˆ ( r ˆ · ) ] I ,
I = ( m 2 1 ) 4 π V E ( r ) E 0 exp ( i k r ˆ · r ˆ ) d 3 r .
| m 1 | 1 ,
k d | m 1 | 1 .
I RG = ( m 2 1 ) 4 π V exp ( i k s · r ) d 3 r ,
R = u ( k s ) V = 1 V V exp ( i k s · r ) d 3 r .
[ 1 r ˆ ( r ˆ · ) ] e ˆ i = cos ( ϕ ν ) cos θ e ˆ θ sin ( ϕ ν ) e ˆ ϕ .
σ = | A | 2 ,
σ RG = k 4 V 2 32 π 2 | ( m 2 1 ) R RG | 2 ( 1 + cos 2 θ ) ,
= k 4 2 γ 2 ( m 2 + 2 3 ) 2 u 2 ( k s ) ( 1 + cos 2 θ ) ,
σ SH ( 1 ) = k 4 2 γ 2 u 2 ( k s ) ( 1 + cos 2 θ ) .
I WKB = ( m 2 1 ) 4 π V exp { i [ k s · r + χ ( r ) ] } d 3 r ,
χ ( r ) = ( m 1 ) k · ( r r 1 )
σ WKB = k 4 | I WKB | 2 ( 1 + cos 2 θ ) 2 .
σ T WKB = 4 π k Im [ A ( k ˆ , k ˆ ) ] · e ˆ i .
σ T WKB = 4 π k Im A ( k ˆ , k ˆ ) .
A ( k ˆ , k ˆ ) = k 2 ( m 2 1 ) 4 π V exp [ i χ ( r ) ] d 3 r .
V exp ( i χ ) d 3 r = P d P z exp ( i χ ) d z ,
z e i χ d z = 0 l exp [ i ( m 1 ) k z ] d z = e i ψ 1 i ( m 1 ) k ,
A ( k ˆ ) , k ˆ ) = i k ( m + 1 ) 4 π P [ exp ( i ψ ) 1 ] d P .
E s = S ( θ ) exp ( i k r ) i k r ,
σ T ADA = 4 π k 2 Re S ( 0 ) .
S ( 0 ) = k 2 2 π P { 1 exp [ i ψ ( ξ, η ) ] } d P .
[ 2 m + 1 A ] * = i k S ( 0 ) ,
Im [ 2 m + 1 A ( k ˆ , k ˆ ) ] = 1 k Re S ( 0 ) .
E = f ( r ) exp [ i k ζ ( r ) ] ,
| ζ | 2 = m 2 .
E ( r ) = E 0 exp ( i k · r i χ e ) ,
χ e = 1 2 k z V e d z .
V e = k 2 ( m 2 1 ) ,
χ e = ( m 2 1 ) 2 k · ( r r 1 ) ,
σ D ( θ ) = [ 2 J 1 ( α sin θ ) α sin θ ] 2 ( 1 + cos 2 θ ) 2 ,
R RG ( θ ) = 1 V 0 2 π d ϕ 0 π sin θ d θ 0 A r 2 d r exp ( i k s r cos θ ) = 3 j 1 ( k s a ) k s a = 3 ( k s a ) 3 ( sin k s a k s a cos k s a ) ,
r ˆ · r = z cos θ + ρ sin θ cos ( ϕ ϕ ) .
I WKB = ( m 2 1 ) 4 π 0 a ρ exp [ i k ( m 1 ) z 1 ] d ρ × 0 2 π exp [ i k ρ sin θ cos ( ϕ ϕ ) ] d ϕ z 1 z 1 exp [ i k ( m cos θ ) z ] d z .
I WKB = a 2 ( m 2 1 ) k ( m cos θ ) 0 1 d x x J 0 ( α x sin θ ) × sin [ α ( m cos θ ) 1 x 2 ] exp [ i α ( m 1 ) 1 x 2 ] .
E s = n = 1 γ n E s , n ,
| E s | 2 = γ 2 | E s , 1 | 2 + 2 γ 3 | E s , 1 · E s , 2 * | + O ( γ 4 ) + .
σ SH = σ SH ( 1 ) + γ 3 k 4 16 π 4 u ( k s ) × u ( q + k r ˆ ) u ( q + k ) q 2 k 2 f ( q , k , θ ) d 3 q + O ( γ 4 ) ,
f ( q , k , θ ) = ( 2 k 2 + q 2 ) 1 + cos 2 θ 6 q 2 sin 2 θ 2 × ( 1 cos 2 ϕ sin 2 θ ) + q 2 8 sin 2 θ sin 2 θ cos ϕ .
P ( θ ) = π 2 4 α 2 [ ( 1 + cos θ ) π α sin ( sin θ ) α sin θ ] 2 ,
R RG = 1 V L L exp ( i k s cos ψ z ) d z 0 a ρ d ρ × 0 2 π d ϕ exp [ i k s sin ψ ρ cos ( ϕ ϕ ) ] ] = 2 J 1 ( z d sin ψ ) z d sin ψ sin ( z n cos ψ ) z n cos ψ ,
z d = k s a = 2 k a sin ( θ / 2 ) , z n = k s L = 2 k L sin ( θ / 2 ) .
R RG , d = 2 J 1 ( z d sin ψ ) z d sin ψ ,
R RG , n = sin ( z n cos ψ ) z n cos ψ .
χ = k ( m 1 ) ( z + L ) sec β .
I WKB = e i k ( m 1 ) L sec β V exp [ i ( k s · r ) ] d 3 r ,
k s = k s + k ( m 1 ) sec β e ˆ z .
I WKB , d = ( m 2 1 ) V 2 π exp [ i k ( m 1 ) L sec β ] ] × J 1 ( z d sin ψ ) z d sin ψ sin [ z n cos ψ + h ( m 1 ) L sec β ] [ z n cos ψ + k ( m 1 ) L sec β ]
χ = k ( m 1 ) ( z z 1 ) , z 1 = a 2 y 2 ,
r ˆ · r = x sin θ cos ϕ + y sin θ sin ϕ + z cos θ,
I WKB = L L d x exp [ i k x sin θ cos ϕ ] × a a d y exp { i k [ y sin θ sin ϕ + ( m 1 ) z 1 ] } × z 1 z 1 d z exp [ i k ( m cos θ ) z ] .
I WKB = ( m 2 1 ) 2 π 2 Θ z sin Θ x Θ x 1 1 d x × sin [ Θ z 1 x 2 ] exp { i α [ ( m 1 ) 1 x 2 x sin θ sin ϕ ] } ,
Θ x = k L sin θ cos ϕ , Θ z = α ( m cos θ ) .
I WKB , n = ( m 2 1 ) V 2 π 2 Θ z 1 1 d x × cos ( α x sin Θ ) sin [ Θ z 1 x 2 ] exp [ i α ( m 1 ) 1 x 2 ] .
k ˆ = e ˆ x cos β + e ˆ z sin β,
I WKB = ( m 2 1 ) V 2 π 2 Θ z ' sin Θ x Θ x 1 1 d x × sin [ Θ x 1 x 2 ] exp { i α [ ( m 1 ) csc β 1 x 2 x sin θ sin ϕ ] } ,
Θ x = k L ( sin θ cos ϕ cos β ) , Θ z = α [ sin β + ( m 1 ) csc β cos θ ] .
σ T = 4 π k Im [ I WKB ( θ = 0 ) ] .
I WKB ( θ = 0 ) = 2 ( m 1 ) V π 2 ρ s 1 1 × sin [ ρ s 2 1 x 2 ] exp [ i ρ s 2 1 x 2 ] d x ,
ρ s = 2 α ( m 1 ) csc β
σ T = 8 ( m 1 ) V k πρ s 1 1 sin 2 [ ρ s 2 1 x 2 ] d x
= 4 ( m 1 ) V k ρ s H 1 ( ρ s ) ,
Q WKB = π H 1 ( ρ s ) .
n ( β ) d β = 2 π n 0 sin β d β,
Q WKB = 0 π/ 2 n ( β ) Q ( β ) sin β d β 0 π/ 2 n ( β ) sin β d β = 4 0 π/ 2 H 1 ( ρ s ) sin 2 β d β ,
σ back = | A ( k , k ) | 2 = | A ( θ = π ) | 2 ,
A ( π ) = 2 k a 2 ( m 1 m + 1 ) I 1 ,
I 1 = 0 1 x e i ρ x sin [ α ( m + 1 ) x ] d x = 1 2 { i ρ 1 2 ( i + ρ 1 ) e i ρ 1 ρ 1 2 i ρ 2 2 + ( i ρ 2 ) e i ρ 2 ρ 2 2 } .
ρ 2 = 2 α m .
ρ 2 = 2 α .
E = exp [ i ( m 1 ) k · r 1 ] [ exp ( i m k · r ) + R exp ( i ρ 2 ) exp ( i m k · r ) ] ,
R = ( 1 m m + 1 )
A ( π ) = 2 k a 2 [ I 1 exp ( i ρ 1 ) I 2 ] ,
I 2 = 0 1 x e x p ( i ρ x ) sin ρ x d x = i 2 [ ( i ρ 3 1 ) e i ρ 3 ρ 3 2 + 1 ρ 3 2 + 1 2 ] ,
ρ 3 = 2 α ( m 1 ) .
Q back = 4 π α 2 | R | 2 | I 1 + exp ( i ρ 1 ) I 2 | 2 .
Q back,geo = | R | 2 .
lim m i α 1 Q back = | R | 2 4 π .
C = 1 + [ 1 exp ( m i α ) ] ( 2 π 1 ) .
P ( θ ) = 4 π | A | 2 4 π | A | 2 d Ω .
A ( k , r ) = [ 1 r ˆ ( r ˆ · ) ] H ,
H = 2 k a 2 ( m 1 ) [ H 1 + exp ( i ρ 1 ) R H 2 ] .
H 1 ( θ ) = 1 ( m cos θ ) 0 1 d x x J 0 ( α x sin θ ) × sin [ α ( m cos θ ) 1 x 2 ] exp [ i α ( m 1 ) 1 x 2 ] .
H 2 ( θ ) = H 1 ( π θ ) .
P ( θ ) = 2 ( 1 + cos 2 θ ) | H 1 + exp ( i ρ 1 ) R H 2 | 2 0 π ( 1 + cos 2 θ ) | H 1 + exp ( i ρ 1 ) R H 2 | 2 sin θ d θ .

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