Abstract

Earlier approximations for multiple scattering by uncorrelated random distributions of large, low-refracting, absorbing particles are generalized by retaining the internal backscattered flux for the case of negligible coherent reflection. For a plane wave normally incident on a slab-region distribution, we obtain more complete results for the flux into a detecting cone at an arbitrary angle of observation, in terms of one-particle scattering functions, scatterer concentration, and distribution thickness. The approximations cover the full range of concentration; although the results near full packing are in general physically unrealizable, they are shown to be in accord with elementary physical considerations. The special forms for the total transmitted flux and total backscattered flux are similar to forms obtained by the diffusion-equation approach. However, the present development, based on averaging the absolute square of the solution of the wave equation for a large number of particles, provides approximations for the bulk parameters and for the flux in terms of results for an isolated scatterer.

© 1970 Optical Society of America

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References

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  1. V. Twersky, J. Math. Phys. 3, 724 (1962); J. Opt. Soc. Am. 52, 145 (1962). These also cite related developments by others.
    [Crossref]
  2. V. Twersky, Am. Math. Soc. Symp. Appl. Math. 16, 84 (1964); in Electromagnetic Scattering, edited by R. L. Rowell and R. S. Stein (Gordon and Breach, New York, 1968), p. 579; in Turbulence of Fluids and Plasmas, edited by J. Fox (Polytechnic Press, Brooklyn, N. Y., 1969), p. 143.
  3. C. I. Beard, T. H. Kays, and V. Twersky, Appl. Opt. 4, 1299 (1965); IEEE Trans. AP-15, 99 (1967); S. W. Hawley, T. H. Kays, and V. Twersky, IEEE Trans. AP-15, 118 (1967); J. E. Burke, T. H. Kays, J. L. Kulp, and V. Twersky, Appl. Opt. 7, 2392 (1968).
    [Crossref] [PubMed]
  4. S. Q. Duntley, J. Opt. Soc. Am. 32, 61 (1942), who also reviews the earlier work on the problem.
    [Crossref]
  5. V. Twersky, J. Math. Phys. 3, 716 (1962).
    [Crossref]
  6. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  7. A. Guinier, X-Ray Diffraction (Freeman, San Francisco, 1963).
  8. Rayleigh, Proc. Roy. Soc. (London) A-90, 219 (1914).
  9. N. M. Anderson and P. S. Sekelj, Phys. Med. Biol. 12, 173 (1967).
    [Crossref] [PubMed]
  10. E. Loewinger, A. Gordon, A. Weinreb, and J. Gross, J. Appl. Physiol. 19, 1179 (1964).
    [PubMed]

1967 (1)

N. M. Anderson and P. S. Sekelj, Phys. Med. Biol. 12, 173 (1967).
[Crossref] [PubMed]

1965 (1)

1964 (2)

V. Twersky, Am. Math. Soc. Symp. Appl. Math. 16, 84 (1964); in Electromagnetic Scattering, edited by R. L. Rowell and R. S. Stein (Gordon and Breach, New York, 1968), p. 579; in Turbulence of Fluids and Plasmas, edited by J. Fox (Polytechnic Press, Brooklyn, N. Y., 1969), p. 143.

E. Loewinger, A. Gordon, A. Weinreb, and J. Gross, J. Appl. Physiol. 19, 1179 (1964).
[PubMed]

1962 (2)

V. Twersky, J. Math. Phys. 3, 716 (1962).
[Crossref]

V. Twersky, J. Math. Phys. 3, 724 (1962); J. Opt. Soc. Am. 52, 145 (1962). These also cite related developments by others.
[Crossref]

1942 (1)

1914 (1)

Rayleigh, Proc. Roy. Soc. (London) A-90, 219 (1914).

Anderson, N. M.

N. M. Anderson and P. S. Sekelj, Phys. Med. Biol. 12, 173 (1967).
[Crossref] [PubMed]

Beard, C. I.

Duntley, S. Q.

Gordon, A.

E. Loewinger, A. Gordon, A. Weinreb, and J. Gross, J. Appl. Physiol. 19, 1179 (1964).
[PubMed]

Gross, J.

E. Loewinger, A. Gordon, A. Weinreb, and J. Gross, J. Appl. Physiol. 19, 1179 (1964).
[PubMed]

Guinier, A.

A. Guinier, X-Ray Diffraction (Freeman, San Francisco, 1963).

Kays, T. H.

Loewinger, E.

E. Loewinger, A. Gordon, A. Weinreb, and J. Gross, J. Appl. Physiol. 19, 1179 (1964).
[PubMed]

Rayleigh,

Rayleigh, Proc. Roy. Soc. (London) A-90, 219 (1914).

Sekelj, P. S.

N. M. Anderson and P. S. Sekelj, Phys. Med. Biol. 12, 173 (1967).
[Crossref] [PubMed]

Twersky, V.

C. I. Beard, T. H. Kays, and V. Twersky, Appl. Opt. 4, 1299 (1965); IEEE Trans. AP-15, 99 (1967); S. W. Hawley, T. H. Kays, and V. Twersky, IEEE Trans. AP-15, 118 (1967); J. E. Burke, T. H. Kays, J. L. Kulp, and V. Twersky, Appl. Opt. 7, 2392 (1968).
[Crossref] [PubMed]

V. Twersky, Am. Math. Soc. Symp. Appl. Math. 16, 84 (1964); in Electromagnetic Scattering, edited by R. L. Rowell and R. S. Stein (Gordon and Breach, New York, 1968), p. 579; in Turbulence of Fluids and Plasmas, edited by J. Fox (Polytechnic Press, Brooklyn, N. Y., 1969), p. 143.

V. Twersky, J. Math. Phys. 3, 724 (1962); J. Opt. Soc. Am. 52, 145 (1962). These also cite related developments by others.
[Crossref]

V. Twersky, J. Math. Phys. 3, 716 (1962).
[Crossref]

van de Hulst, H. C.

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

Weinreb, A.

E. Loewinger, A. Gordon, A. Weinreb, and J. Gross, J. Appl. Physiol. 19, 1179 (1964).
[PubMed]

Am. Math. Soc. Symp. Appl. Math. (1)

V. Twersky, Am. Math. Soc. Symp. Appl. Math. 16, 84 (1964); in Electromagnetic Scattering, edited by R. L. Rowell and R. S. Stein (Gordon and Breach, New York, 1968), p. 579; in Turbulence of Fluids and Plasmas, edited by J. Fox (Polytechnic Press, Brooklyn, N. Y., 1969), p. 143.

Appl. Opt. (1)

J. Appl. Physiol. (1)

E. Loewinger, A. Gordon, A. Weinreb, and J. Gross, J. Appl. Physiol. 19, 1179 (1964).
[PubMed]

J. Math. Phys. (2)

V. Twersky, J. Math. Phys. 3, 724 (1962); J. Opt. Soc. Am. 52, 145 (1962). These also cite related developments by others.
[Crossref]

V. Twersky, J. Math. Phys. 3, 716 (1962).
[Crossref]

J. Opt. Soc. Am. (1)

Phys. Med. Biol. (1)

N. M. Anderson and P. S. Sekelj, Phys. Med. Biol. 12, 173 (1967).
[Crossref] [PubMed]

Proc. Roy. Soc. (London) (1)

Rayleigh, Proc. Roy. Soc. (London) A-90, 219 (1914).

Other (2)

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

A. Guinier, X-Ray Diffraction (Freeman, San Francisco, 1963).

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Equations (91)

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u ( r ˆ ) ~ f ( r ˆ , z ˆ ) e i k r / r , r ˆ = z ˆ cos θ + sin θ ( x ˆ cos φ + y ˆ sin φ ) ,
( 4 π / k ) Im x ˆ · f ( z ˆ , z ˆ ) = σ a + σ s , σ s = π f ( r ˆ , z ˆ ) 2 d Ω ( r ˆ ) σ ( π ) ,
σ ( α ; r ˆ , z ˆ ) = α ; r ˆ f ( R ˆ , z ˆ ) 2 d Ω ( R ˆ ) ,             R ˆ = R ˆ ( Θ , Φ ) .
f ( z ˆ , z ˆ ) = f ( - z ˆ , - z ˆ ) = f x ˆ ,             f ( - z ˆ , z ˆ ) = f ( z ˆ , - z ˆ ) = f x ˆ .
σ ( α ) = σ ( α ; z ˆ , z ˆ ) = σ ( α ; - z ˆ , - z ˆ ) , σ ( α ) = σ ( α ; - z ˆ , z ˆ ) = σ ( α ; z ˆ , - z ˆ ) = σ ( π ) - σ ( π - α ) ,
σ = σ ( π / 2 ) ,             σ = σ ( π / 2 ) = σ ( π ) - σ ( π / 2 ) = σ s - σ .
f = f 0 ( 1 + i δ b ) ,             f 0 = V k δ / 2 π .
b = b ( z ˆ ) = 1 V ( z - z 1 ) d V = 1 V d x d y z 1 z 2 ( z - z 1 ) d z ,
σ s = 2 V δ 2 b 2 V ( Re δ ) 2 b , σ a = 2 V Im δ ( 1 - 2 Im δ b ) 2 V Im δ ,
f ( r ˆ , z ˆ ) 2 = f 0 J 2 ( 1 - sin 2 θ cos 2 φ ) , J = V - 1 exp [ i k ( z ˆ - r ˆ ) · r ] d V ( r ) ,
J exp [ - ( k h θ ) 2 / 10 ] , h 2 = 5 V - 1 ( ζ ˆ · r ) 2 d V ( r ) , ζ ˆ = ( z ˆ - r ˆ ) / z ˆ - r ˆ ,
J ~ - 3 ( cos H ) / H 2 ,             H = 2 k h sin ( θ / 2 ) .
Ψ ( r ) = e i k z + ρ u ( r - r ) · Ψ ( r ) d V ( r ) ,
Ψ ( z ) = e i k z + i 2 π k 0 d e i k z - ζ F ( ± z ˆ , i ˆ ) · Ψ ( ζ ) d ζ ,
Δ = 2 π ρ F / k ,             Δ = 2 π ρ F / k ,
Ψ = A e i K z + A e - i K z = e i k z + i 0 z e i k ( z - ζ ) [ Δ A e i K ζ + Δ A e - i K ζ ] d ζ + i z d e - i k ( z - ζ ) [ Δ A e i K ζ + Δ A e - i K ζ ] d ζ ,
A = 1 + Q 1 - Q 2 e i 2 K d ( 1 + Q ) D , A = - e i 2 K d Q A ,             Q = - ( K - k ) F ( K + k ) F ,
Ψ = [ e i K z - Q e i K ( 2 d - z ) ] ( 1 + Q ) D .
1 = Δ / ( K - k ) - Δ / ( K + k ) = Δ ( 1 + Q ) / ( K - k ) .
Ψ = e i k z + i e i k z 0 d e - i k ζ [ Δ A e i K ζ + Δ A e - i K ζ ] d ζ = T e i k ( z - d ) , T = e i K d ( 1 - Q 2 ) D = e i K d [ 1 - Q 2 ( 1 - e i 2 K d ) D ] ,
Ψ = e i k z + i e - i k z 0 d e i k ζ [ Δ A e i K ζ + Δ A e - i K ζ ] d ζ = e i k z + e - i k z , = Q ( 1 - e i 2 K d ) D = Q [ 1 - e i 2 K d ( 1 - Q 2 ) D ] ,
T = e i K d ( 1 - Q ) ,             = Q ( 1 - e i K d T ) .
η / μ = Δ / ( K - k ) + Δ / ( K + k ) = ( 1 - Q ) / ( 1 + Q ) Z ,
F ( K - k ) e - i k z ψ d V / 2 π ,             K = k η = k ( ) 1 2 ,
ψ exp [ i K z 1 + i K ( z - z 1 ) ] ,
F f 0 [ 1 + i b ( K - K ) ] ,             f 0 = k δ V / 2 π .
F f f 0 J ( - z ˆ , z ˆ ) ~ - f 0 3 cos ( 2 k h ) / ( 2 k h ) 2 ,
K - k = Δ ( 1 + Q ) Δ ( 1 - Δ / 2 k ) Δ = 2 π ρ F / k ,
Q η - 1 3 w / 2 ( 2 k h ) 2 1.
K - k 2 π ρ F / k = w δ [ 1 + i b ( K - K ) ] w δ [ 1 + i b ( 1 - w ) δ ] ,
Re ( K - k ) w Re δ [ 1 - 2 Im δ b ( 1 - w ) ] w Re δ
2 Im K β + γ ;             β 2 w δ 2 b ( 1 - w ) ρ ( 1 - w ) σ s γ 2 w Im δ [ 1 - 2 Im δ b ( 1 - w ) ] 2 w Im δ ρ σ a
Ψ = e i K z ,             T = e i K d ,             = 0 ,
Ψ ( z ) = e i k z + i Δ 0 z e i k ( z - ζ ) Ψ ( ζ ) d ζ .
Ψ ( r ) 2 = Ψ ( r ) 2 + ρ v ( r - r ) 2 · Ψ ( r ) 2 d V ( r )
v ( r - r ) = u ( r - r ) + ρ u ( r - r ) · v ( r - r ) d V ( r ) .
Ψ 2 = C = e i K z 2 = e - L z ,             L 2 Im K ,
v ( r ) 2 ~ G ( r ˆ , z ˆ ) exp [ i K ( r ˆ ) r ] / r 2 = G / r 2 exp [ - L ( r ˆ ) z sec θ ] ,
Ψ ( r ) 2 = e - L z + ρ ± d Ω G ( R ˆ , i ˆ ) 2 sec Θ × 0 d exp [ - L ( z - z ) sec Θ ] Ψ ( r ) 2 d z ,
Ψ 2 = A e - M z + A e M z = e - L z + 0 z e - L ( z - ζ ) [ S A e - M ζ + S A e M ζ ] d ζ + z d e L ( z - ζ ) [ S A e - M ζ + S A e M ζ ] d ζ ,
A = 1 + P 1 - P 2 e - 2 M d ( 1 + P ) D ,             A = - e - 2 M d P A , P = ( L - M ) S ( L + M ) S ,
Ψ 2 = [ e - M z - P e - M ( 2 d - z ) ] ( 1 + P ) D .
Re Ψ * Ψ / i k · z ˆ = ( A e - M z - A e M z ) ( 1 - P ) / ( 1 + P ) = [ e - M z + P e - M ( 2 d - z ) ] ( 1 + P ) D .
1 = S / ( L - M ) + S / ( L + M ) = S ( 1 + P ) / ( L - M ) .
Ξ = ( 1 - P ) / ( 1 + P ) = S / ( L - M ) - S / ( L + M ) ,
S ( α ; r ˆ , z ˆ ) ρ ( 1 - w ) σ ( α ; r ˆ , z ˆ ) ,
T = C + I ( z ˆ ) = e - L d + 0 d e - L ( d - z ) × [ S A e - M z + S A e M z ] d z .
I ( α ; z ˆ ) = q ( α ) e - M d [ 1 - e - ( L - M ) d ] D - p ( α ) P 2 e - M d [ 1 - e - ( L + M ) d ] D , q ( α ) S ( α ) / S σ ( α ) / σ , p ( α ) S ( α ) / S σ ( α ) / σ .
T t = C + I t = e - M d ( 1 - P 2 ) D = e - M d [ 1 - P 2 ( 1 - e - 2 M d ) D ] .
I ( - z ˆ ) = 0 d e - L z [ S A e - M z + S A e M z ] d z ,
I ( α ; - z ˆ ) = p ( α ) P [ 1 - e - ( L + M ) d ] D - q ( α ) P e - 2 M d [ 1 - e - ( L - M ) d ] D R ( α ) .
R t = P [ 1 - e - 2 M d ] D = P [ 1 - e - 2 M d ( 1 - P 2 ) D ] .
T t = e - M d ( 1 - P R t ) ,             R t = P ( 1 - e - M d T t ) .
T t + R t = ( e - M d + P ) / ( 1 + P e - M d ) = e - M d + P ( 1 - e - 2 M d ) / ( 1 + P e - M d ) ,
1 - T t - R t = ( 1 - P ) ( 1 - e - M d ) / ( 1 + P e - M d ) .
M = L - S - S + S 2 M / ( L + M ) γ [ 1 + 2 S / ( 2 γ + β ) ] .
P = S ( L - M ) / S ( L + M ) σ β / σ ( 2 γ + β ) .
T 0 ( α ) = T 0 [ q ] = e - L d + q ( α ) ( e - M d - e - L d ) = C + I 0 ,
R ( α ) = R [ p ] = p ( α ) P q ( α ) β / ( 2 γ + β ) , q ( α ) σ ( α ) / σ .
T ( α ) = T 0 [ q ] - R [ p ] P e - M d { 1 - e - M d T 0 [ q / p ] } D ,
R ( α ) = R [ p ] { 1 - e - M d T 0 [ q / p ] } D R 0 ( α ) D ,
T = T 0 [ q ] - P e - M d R ,
I ( r ˆ ) = S ( α ; r ˆ , z ˆ ) L - M cos θ [ e - M d - exp ( - L d sec θ ) ] A - S ( α ; r ˆ , - z ˆ ) L + M cos θ P [ e - M d - exp ( - 2 M d - L d sec θ ) ] A ,
L ( r ˆ ) = L ( - r ˆ ) = 2 Im K ( r ˆ ) γ + β b ( r ˆ ) / b ρ σ a + ρ ( 1 - w ) σ s ( r ˆ ) ,
I ( - r ˆ ) = S ( α ; - r ˆ , z ˆ ) L + M cos θ [ 1 - exp ( - M d - L d sec θ ) ] A - S ( α ; - r ˆ , - z ˆ ) L - M cos θ P [ e - 2 M d - exp ( - M d - L d sec θ ) ] A ,
q ( α ; r ˆ ) = σ ( α ; r ˆ , z ˆ ) / σ , p ( α ; - r ˆ ) = σ ( α ; - r ˆ , z ˆ ) / σ , q ˜ ( α ; r ˆ ) = q ( α ; r ˆ ) L ( z ˆ ) - M L ( r ˆ ) - M cos θ , p ˜ ( α ; - r ˆ ) = p ( α ; - r ˆ ) L ( z ˆ ) + M L ( r ˆ ) + M cos θ ,
I ( r ˆ ) = q ˜ e - M d { 1 - exp [ - ( L sec θ - M ) d ] } D - p ˜ P 2 e - M d { 1 - exp [ - ( L sec θ + M ) d ] } D ,
I ( - r ˆ ) = p ˜ P { 1 - exp [ - ( L sec θ + M ) d ] } D - q ˜ P e - 2 M d { 1 - exp [ - ( L sec θ - M ) d ] } D .
I 0 ( r ˆ ) = q ˜ ( α ; r ˆ ) [ e - M d - exp ( - L d sec θ ) ] ,
I ( r ˆ ) = I 0 ( r ˆ ) - p ˜ P 2 e - M d { 1 - e - M d T 0 [ L sec θ ; q ˜ / p ˜ ] } D ,
T 0 [ L ( r ˆ ) sec θ ; q ˜ / p ˜ ] exp ( - L d sec θ ) + ( q ˜ / p ˜ ) [ e - M d - exp ( - L d sec θ ) ] .
I ( - r ) = p ˜ ( α ; - r ˆ ) P = q ( α ; - r ˆ ) [ L ( z ˆ ) - M ] / [ L ( r ˆ ) + M cos θ ] ,             q ( α ; - r ˆ ) = σ ( α ; - r ˆ , z ˆ ) / σ ,
I ( - r ˆ ) = I ( - r ˆ ) { 1 - e - M d T 0 [ L ( r ˆ ) sec θ ; q ˜ / p ˜ ] } D I 0 ( - r ˆ ) D ,
I ( r ˆ ) = I 0 ( r ˆ ) - P e - M d I ( - r ˆ ) .
T 0 ( α ) e - γ d [ e - β d + q ( α ) ( 1 - e - β d ) ] ,             0 < q < 1 ,
I 0 ( r ˆ ) q ˜ ( α ; r ˆ ) { e - γ d - exp [ - ( γ + β ) d sec θ ] } ,             β = β ( r ˆ ) ,
R ( α ) q ( α ) β / ( 2 γ + β ) ,             0 < q < q ,
I ( - r ˆ ) q ( α ; - r ˆ ) β ( z ˆ ) / [ β ( r ˆ ) + γ ( 1 + cos θ ) ] , q ( α ; - r ˆ ) = σ ( α ; - r ˆ , z ˆ ) / σ ,
Ψ ( z ) 2 = e - L z + S 0 z e - L ( z - ζ ) Ψ ( ζ ) 2 d ζ = e - M z ,
R 0 R - R e - 2 γ d [ e - β d + ( q / p ) ( 1 - e - β d ) ] = R { 1 - e - γ d T 0 [ q / p ] } ,             q / p > 1 ,
I 0 ( - r ˆ ) = I - I e - γ d { exp ( - L d sec θ ) + ( q ˜ / p ˜ ) [ e - γ d - exp ( - L d sec θ ) ] } ,             L = γ + β ( r ˆ ) ,
Q 2 ( η - 1 ) 3 w / 2 ( 2 k a ) 2 2 1.
Q 2 / P 3 ( η - 1 ) w / 4 k a 2 × [ 2 σ a + σ s ( 1 - w ) ] / σ s ( 1 - w ) ,
η b = [ η 0 + w ( η 1 - η 0 ) ] + i η 1 - η 0 2 w ( 1 - w ) 2 π b / λ = K λ / 2 π ,             η 1 = η η 0 ,
η ˜ = w 0 η 0 + w 1 η 1 ,
η ˜ 2 = w 0 η 0 2 + w 1 η 1 2 ,
η ˜ - η ˜ 2 = η ˜ 2 - η ˜ 2 = ( w 0 η 0 2 + w 1 η 1 2 ) - w 0 η 0 + w 1 η 1 2 = w ( 1 - w ) η 1 - η 0 2 .
η b = η ˜ + i η ˜ - η ˜ 2 2 π b / λ .
η 1 - η ˜ 2 = η 1 2 - 2 Re η η 1 * + η ˜ 2 = ( 1 - w ) η 1 - η 0 2
G 2 f ( η 1 - η ˜ ) 2 ( 1 - w ) f ( η 1 - η 0 ) 2 .
G 2 d Ω ( 1 - w ) σ