Abstract

We consider scattering by arbitrarily shaped particles that satisfy two conditions: (1) that the polarizability of the particle relative to the ambient medium be small compared to 1 and (2) that the phase shift introduced by the particle be less than 2. We solve the integro-differential equation proposed by Shifrin by using the method of successive iterations and then applying a Fourier transform. For the second iteration, results are presented that accurately describe scattering by a broad class of particles. The phase function and other elements of the scattering matrix are shown to be in excellent agreement with Mie theory for spherical scatterers.

© 1976 Optical Society of America

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References

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  1. J. W. Strutt (Lord Rayleigh), Philos. Mag. 12, 81 (1881).
  2. R. Gans, Ann. Phys. 76, 29 (1925).
  3. I. Rocard, Rev. Opt. 9, 97 (1930).
  4. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  5. A popular misconception about the Rayleigh-Gans approximation is that it applies only to small particles. The proper restriction (the second criterion) is that the phase shift introduced by the particle be small. Thus, scattering by large particles (ka > 1) whose refractive index is very close to 1 (|m − 1| ≪ 1) could be described by the Rayleigh-Gans approximation.
  6. K. S. Shifrin, Scattering of Light in a Turbid Medium (Moscow, 1951) (NASA Technical Translation TT F-477, 1968).
  7. In this form, the integral equation is similar to the Lippmann-Schwinger equation used in nuclear scattering [see R. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966)]. However, when solved by the method of successive iterations, the Lippmann-Schwinger equation gives rise to the well-known Born expansion, which is different from the expansion obtained here using Schifrin’s method.
  8. S. Chandrasekhar, Radiative Transfer (Oxford U. P., Oxford, 1950).
  9. A. C. Holland, G. Gagne, Appl. Opt. 9, 1113 (1970).

1970 (1)

1930 (1)

I. Rocard, Rev. Opt. 9, 97 (1930).

1925 (1)

R. Gans, Ann. Phys. 76, 29 (1925).

1881 (1)

J. W. Strutt (Lord Rayleigh), Philos. Mag. 12, 81 (1881).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford U. P., Oxford, 1950).

Gagne, G.

Gans, R.

R. Gans, Ann. Phys. 76, 29 (1925).

Holland, A. C.

Newton, R.

In this form, the integral equation is similar to the Lippmann-Schwinger equation used in nuclear scattering [see R. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966)]. However, when solved by the method of successive iterations, the Lippmann-Schwinger equation gives rise to the well-known Born expansion, which is different from the expansion obtained here using Schifrin’s method.

Rocard, I.

I. Rocard, Rev. Opt. 9, 97 (1930).

Shifrin, K. S.

K. S. Shifrin, Scattering of Light in a Turbid Medium (Moscow, 1951) (NASA Technical Translation TT F-477, 1968).

Strutt, J. W.

J. W. Strutt (Lord Rayleigh), Philos. Mag. 12, 81 (1881).

van de Hulst, H. C.

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

Ann. Phys. (1)

R. Gans, Ann. Phys. 76, 29 (1925).

Appl. Opt. (1)

Philos. Mag. (1)

J. W. Strutt (Lord Rayleigh), Philos. Mag. 12, 81 (1881).

Rev. Opt. (1)

I. Rocard, Rev. Opt. 9, 97 (1930).

Other (5)

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

A popular misconception about the Rayleigh-Gans approximation is that it applies only to small particles. The proper restriction (the second criterion) is that the phase shift introduced by the particle be small. Thus, scattering by large particles (ka > 1) whose refractive index is very close to 1 (|m − 1| ≪ 1) could be described by the Rayleigh-Gans approximation.

K. S. Shifrin, Scattering of Light in a Turbid Medium (Moscow, 1951) (NASA Technical Translation TT F-477, 1968).

In this form, the integral equation is similar to the Lippmann-Schwinger equation used in nuclear scattering [see R. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966)]. However, when solved by the method of successive iterations, the Lippmann-Schwinger equation gives rise to the well-known Born expansion, which is different from the expansion obtained here using Schifrin’s method.

S. Chandrasekhar, Radiative Transfer (Oxford U. P., Oxford, 1950).

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Figures (2)

Fig. 1
Fig. 1

(a)–(d) Plots of S11 vs scattering angle β for various values of the refractive index m and size parameter x.

Fig. 2
Fig. 2

(a)–(d) Plots of S12 vs scattering angle β for various values of the refractive index m and size parameter x.

Equations (38)

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E eff ( r ) = E 0 exp ( i k · r ) + α ( k 2 + grad div ) U ( r ) E eff ( r ) G ( r , r ) d 3 r + 4 3 π α U ( r ) E eff ( r ) .
U ( r ) = 1 if r is inside the scatterer , 0 if r is outside the scatterer ,
G ( x , y ) = exp ( i k x - y 4 π x - y .
E eff ( r ) = n = 0 α n E eff ( n ) ( r ) .
E eff ( 0 ) ( r ) = E 0 exp ( i k · r ) ,
E eff ( 1 ) ( r ) = ( k 2 + grad div ) U ( r ) E 0 exp ( i k · r ) · G ( r , r ) d 3 r + 4 3 π U ( r ) E 0 exp ( i k · r ) ,
E eff ( 2 ) ( r ) = ( k 2 + grad div ) U ( r ) E eff ( 1 ) ( r ) · G ( r , r ) d 3 r + 4 3 π U ( r ) E eff ( 1 ) ( r ) .
E scat ( r ) = E eff ( r ) - E eff ( 0 ) ( r ) .
E scat ( 0 ) ( r ) = 0.
E scat ( 1 ) ( r ) = ( k 2 + grad div ) U ( r ) E 0 exp ( i k · r ) · G ( r , r ) d 3 r ,
E scat ( 2 ) ( r ) = ( k 2 + grad div ) U ( r ) E eff ( 1 ) ( r ) · G ( r , r ) d 3 r .
E scat ( 1 ) ( r ) = v k 2 r exp ( i k r ) E 0 f ( 2 a k sin β / 2 ) ,
[ E scat ( 1 ) ( r ) ] e scat ( 1 ) ( p ) = [ k 2 E 0 - p ( p · E 0 ) ] · g ( p ) u ( p - k ) .
g ( p ) = ( 4 π ) / ( p 2 - k 2 ) .
u ( p ) = all space U ( r ) exp ( - i p · r ) d 3 r = scatterer exp ( - i p · r ) d 3 r .
u ( p ) = v f ( a p ) .
e scat ( 1 ) ( p ) = 4 π [ k 2 E 0 - p ( p · E 0 ) ] u ( p - k ) p 2 - k 2 .
E scat ( 1 ) ( r ) = 4 π ( 2 π ) 3 [ k 2 E 0 - p ( p · E 0 ) ] × u ( p - k ) p 2 - k 2 exp ( i p · r ) d 3 p .
r ^ = ( 0 , 0 , 1 ) , k ^ = ( sin β , 0 , cos β ) , E ^ 0 = ( cos β sin ψ , cos ψ , - sin β sin ψ ) , p ^ = ( cos ϕ sin θ , sin ϕ sin θ , cos θ ) .
E scat ( 1 ) ( r ) = 4 π ( 2 π ) 3 i r - 0 2 π p [ k 2 E 0 - p 2 r ^ ( r ^ · E 0 ) ] · exp ( i p r ) u ( p r ^ - k ) p 2 - k 2 d ϕ d p .
E scat ( 1 ) ( r ) = k 2 r exp ( i k r ) E 0 u ( k r ^ - k ) .
u ( p ) = v · [ sin ( h p / 2 ) ( h p / 2 ) ] · [ 2 J 1 ( a p ) a p ] ,
h = cylinder height , a = cylinder radius , p = component of p parallel to the cylinder axis , and p = component of p perpendicular to the cylinder axis .
u ( p ) = v f { a p [ 1 - 2 ( p / p ) 2 ] 1 / 2 } ,
a = semimajor axis of ellipse ( before rotation ) , = eccentricity of ellipse , and p = component of p parallel to the axis of rotation .
u ( p ) = v f ( a p { 1 - 2 [ 1 - ( p / p ) 2 ] } 1 / 2 ) .
e scat ( 2 ) ( p ) = ( k 2 - pp ) g ( p ) ( 1 2 π ) 3 u ( p ) e eff ( 1 ) ( p - p ) d 3 p .
1 r - r exp [ i ( k r - r + k · r ) ]
e eff ( 1 ) ( p ) = [ k 2 E 0 - p ( p · E 0 ) ] g ( p ) u ( p - k ) + 4 3 π E 0 u ( p - k ) .
E scat ( 2 ) ( r ) = 2 π ( 2 π ) 3 ( k 2 - pp ) u ( p ) u ( p - p - k ) ( p 2 - k 2 ) [ ( p - p ) 2 - k 2 ] × exp ( i p · r ) · { 2 3 k 2 E 0 + 1 3 ( p - p ) 2 E 0 - ( p - p ) [ ( p - p ) · E 0 ] } d 3 p d 3 p .
E scat ( 2 ) ( r ) = 2 k 2 ( 2 π ) 2 exp ( i k r ) r u ( q + k r ^ ) u ( q + k ) q 2 - k 2 · [ ( 2 3 k 2 + 1 3 q 2 ) E 0 - ( q · E 0 ) q ] d 3 q .
I scat = E scat 2 = α 2 E scat ( 1 ) 2 + 2 α 3 Re { E scat ( 1 ) · E scat ( 2 ) * } + .
k 2 r 2 [ I scat Q scat U scat V scat ] = [ S 11 S 12 S 13 S 14 S 21 S 22 S 23 S 24 S 31 S 32 S 33 S 34 S 41 S 42 S 43 S 44 ] · [ I 0 Q 0 U 0 V 0 ] .
S 11 = k 2 r 2 [ ( I scat ) / ( I 0 ) ]
S 11 = α 2 k 6 u 2 ( k r ^ - k ) ( 1 + cos 2 β 2 ) + α 3 k 6 π 2 u ( k r ^ - k ) · u ( q + k r ^ ) u ( q + k ) q 2 - k 2 · [ ( 2 k 2 + q 2 ) ( 1 + cos 2 β 6 ) - q 2 2 sin 2 θ ( 1 - cos 2 ϕ sin 2 β ) + q 2 8 sin 2 θ sin 2 β cos ϕ ] d 3 q .
S 12 = α 2 k 6 u 2 ( k r ^ - k ) · ( - sin 2 β 2 ) + α 3 k 6 π 2 u ( k r ^ - k ) · u ( q + k r ^ ) u ( q + k ) q 2 - k 2 · [ ( 2 k 2 + q 2 ) ( - sin 2 β 6 ) - q 2 2 sin 2 θ · ( cos 2 ϕ cos 2 β - sin 2 ϕ ) + q 2 8 sin 2 θ sin 2 β cos ϕ ] d 3 q .
[ S 11 S 12 0 0 S 12 S 11 0 0 0 0 S 33 S 34 0 0 - S 34 S 33 ]
S 11 = ( 4 π 3 ) 2 α 2 x 6 + ( 4 π 3 ) 3 · α 3 x 6 [ 7 2 - 9 16 ( 4 x - 1 x 3 ) · S i ( 4 x ) - 63 128 x 3 sin 4 x - 9 32 x 2 cos 4 x ] .

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