Abstract

Formulation of the scattering problem as an integral permits calculation of the scattered field without any restrictions upon particle shape or internal structure (distribution of dielectric constant). However, the internal field must be known. Here, we approximate the internal field for a structured sphere by that of a homogeneous sphere whose dielectric constant is obtained from the volume weighted average of the polarizability. This approximation should be accurate for sufficiently small variations of the dielectric constant. The resulting algorithm has been tested by comparison with the boundary value solution for concentric spheres. For dielectric constants corresponding to aqueous suspensions of biological particles, the results are quite accurate for dimensionless sizes (circumference over wavelength) no greater than 7. The backscatter is particularly sensitive to small changes in morphology, including less symmetrical dispositions of the internal structure than those modeled by concentric spheres. Although the algorithm is also accurate for more highly refractive particles (corresponding to atmospheric aerosols), in this case the scattering is much less sensitive to changes in internal structure, even in the backward directions.

© 1978 Optical Society of America

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References

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  1. D. S. Saxon, Lectures on the Scattering of Light, Department of Meteorology, University of California, Los Angeles, California, 1955. See also, R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966).
  2. A. L. Aden and M. Kerker, J. Appl. Phys. 22, 1242 (1951).
    [Crossref]
  3. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  4. M. Kerker and D. D. Cooke, Appl. Opt. 15, 2105 (1976).
    [Crossref] [PubMed]
  5. P. W. Barber, IEEE Trans. Microwave Theory Tech. MTT-25, 373 (1977).
    [Crossref]
  6. P. Dusel, M. Kerker, and D. D. Cooke, J. Opt. Soc. Am. (to be published).
  7. S. Asano and G. Yamamoto, Appl. Opt. 14, 29 (1975).
    [PubMed]
  8. P. W. Barber and C. Yeh, Appl. Opt. 14, 2864 (1975).
    [Crossref] [PubMed]
  9. We follow the notation of H. Chew, P. J. McNulty, and M. Kerker, Phys. Rev. A 13, 396(1976), Appendix B, with the following modifications: the quantities ω0, k1, ɛ1, k2 there are denoted here by ω, k′, ɛ1, and k, respectively, whereas μ1, μ2, and ɛ2 there are set equal to unity in the present discussion.
    [Crossref]
  10. See, for example, Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun (Natl. Bur. of Stand., U.S. GPO, Washington, D.C.1965).

1977 (1)

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

1976 (2)

M. Kerker and D. D. Cooke, Appl. Opt. 15, 2105 (1976).
[Crossref] [PubMed]

We follow the notation of H. Chew, P. J. McNulty, and M. Kerker, Phys. Rev. A 13, 396(1976), Appendix B, with the following modifications: the quantities ω0, k1, ɛ1, k2 there are denoted here by ω, k′, ɛ1, and k, respectively, whereas μ1, μ2, and ɛ2 there are set equal to unity in the present discussion.
[Crossref]

1975 (2)

1951 (1)

A. L. Aden and M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[Crossref]

Aden, A. L.

A. L. Aden and M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[Crossref]

Asano, S.

Barber, P. W.

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

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

Chew, H.

We follow the notation of H. Chew, P. J. McNulty, and M. Kerker, Phys. Rev. A 13, 396(1976), Appendix B, with the following modifications: the quantities ω0, k1, ɛ1, k2 there are denoted here by ω, k′, ɛ1, and k, respectively, whereas μ1, μ2, and ɛ2 there are set equal to unity in the present discussion.
[Crossref]

Cooke, D. D.

M. Kerker and D. D. Cooke, Appl. Opt. 15, 2105 (1976).
[Crossref] [PubMed]

P. Dusel, M. Kerker, and D. D. Cooke, J. Opt. Soc. Am. (to be published).

Dusel, P.

P. Dusel, M. Kerker, and D. D. Cooke, J. Opt. Soc. Am. (to be published).

Kerker, M.

We follow the notation of H. Chew, P. J. McNulty, and M. Kerker, Phys. Rev. A 13, 396(1976), Appendix B, with the following modifications: the quantities ω0, k1, ɛ1, k2 there are denoted here by ω, k′, ɛ1, and k, respectively, whereas μ1, μ2, and ɛ2 there are set equal to unity in the present discussion.
[Crossref]

M. Kerker and D. D. Cooke, Appl. Opt. 15, 2105 (1976).
[Crossref] [PubMed]

A. L. Aden and M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[Crossref]

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

P. Dusel, M. Kerker, and D. D. Cooke, J. Opt. Soc. Am. (to be published).

McNulty, P. J.

We follow the notation of H. Chew, P. J. McNulty, and M. Kerker, Phys. Rev. A 13, 396(1976), Appendix B, with the following modifications: the quantities ω0, k1, ɛ1, k2 there are denoted here by ω, k′, ɛ1, and k, respectively, whereas μ1, μ2, and ɛ2 there are set equal to unity in the present discussion.
[Crossref]

Saxon, D. S.

D. S. Saxon, Lectures on the Scattering of Light, Department of Meteorology, University of California, Los Angeles, California, 1955. See also, R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966).

Yamamoto, G.

Yeh, C.

Appl. Opt. (3)

IEEE Trans. Microwave Theory Tech. (1)

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

J. Appl. Phys. (1)

A. L. Aden and M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[Crossref]

Phys. Rev. A (1)

We follow the notation of H. Chew, P. J. McNulty, and M. Kerker, Phys. Rev. A 13, 396(1976), Appendix B, with the following modifications: the quantities ω0, k1, ɛ1, k2 there are denoted here by ω, k′, ɛ1, and k, respectively, whereas μ1, μ2, and ɛ2 there are set equal to unity in the present discussion.
[Crossref]

Other (4)

See, for example, Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun (Natl. Bur. of Stand., U.S. GPO, Washington, D.C.1965).

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

P. Dusel, M. Kerker, and D. D. Cooke, J. Opt. Soc. Am. (to be published).

D. S. Saxon, Lectures on the Scattering of Light, Department of Meteorology, University of California, Los Angeles, California, 1955. See also, R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966).

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Figures (15)

FIG. 1
FIG. 1

Concentric sphere model used to test the algorithm; ɛ1 and ɛ2 are relative dielectric constants, a and b are radii. Dimensionless sizes are α = 2πa/λ and β = 2πb/λ.

FIG. 2
FIG. 2

Rayleigh ratio R2 vs. scattering angle: β = 3.0, α = 1.5, ɛ1 = 1.21, ɛ2 = 1.04, 1.12. ——AK, …FA, - - -LMA, –·–·–·–· RD.

FIG. 3
FIG. 3

Rayleigh ratio R2 vs. scattering angle; β = 3.0, α = 1.5, ɛ1 = 1.04, ɛ2 = 1.06, 1.12. ——AK, - - -LMA, –·–·–·–· RD. FA is indistinguishable from AK.

FIG. 4
FIG. 4

Rayleigh ratio R2 vs. core size α at scattering angles θ = 60°, 120°, and 180°: β = 3.0, ɛ1 = 1.21, ɛ2 = 1.12. ——AK, …FA, - - -LMA. FA and AK are indistinguishable at 60°.

FIG. 5
FIG. 5

Rayleigh ratio R2 vs. scattering angle for β = 5, α = 2.5, ɛ1 = 1.21, ɛ2 = 1.17. ——AK, … FA, - - -LMA, –··–··–··LM.

FIG. 6
FIG. 6

Rayleigh ratio R2 vs. core size α at θ = 180° and 120°. β = 5.0, ɛ1 = 1.21, ɛ2 = 1.17. ——AK, …FA, - - -LMA.

FIG. 7
FIG. 7

Rayleigh ratio R2 vs. scattering angle for β = 7, α = 1.75, ɛ1 = 1.21, ɛ2 = 1.12. ——AK, …FA, - - -LMA, –··–··–··LM.

FIG. 8
FIG. 8

Rayleigh ratio R2 vs. core size α at scattering angles 180° and 120° for β = 7, ɛ1 = 1.21, ɛ2 = 1.17. ——AK, …FA.

FIG. 9
FIG. 9

Rayleigh ratio R2 vs. scattering angle for β = 3, α = 1.5, ɛ1 = 2.25 and ɛ2 = 2.50 and 2.13. ——AK, …FA.

FIG. 10
FIG. 10

Rayleigh ratio vs. core size α at scattering angle θ = 180° for β = 3, α = 1.5, ɛ1 = 2.25 and ɛ2 = 2.13 and 2.04. ——AK, …FA.

FIG. 11
FIG. 11

Rayleigh ratio R2 vs. core size α at scattering angle θ = 180° for β = 8, ɛ1 = 2.25, ɛ2 = 2.19. ——AK, …FA, - - -LMA.

FIG. 12
FIG. 12

Rayleigh ratio R2 vs. scattering angle θ for β = 3, α = 0.75, ɛ1 = 1.21, ɛ2 = 1.04. Core at center, 0; core displaced half radial distance in direction of incident beam, 0.5; displaced radial distance, 1; displaced two radii, 2; displaced three radii (touching outer surface) 3.

FIG. 13
FIG. 13

Same as Fig. 12 for curve 0. Curves 1, 2, 3 each correspond to a pair of similar cores displaced plus and minus 1, 2, and 3 core radial distances along the diameter parallel to the incident beam.

FIG. 14
FIG. 14

Curve 1 represents the Rayleigh ratio R2 vs. scattering angle θ for β = 3, α = 0.75, ɛ1 = 1.21, ɛ2 = 1.04. Other curves are for equivolume prolate spheroids with indicated axial ratios (ratio of figure axis to one of two equal axes).

FIG. 15
FIG. 15

Rayleigh ratio R2 vs. oblate ellipsoid axial ratio at scattering angles 120° and 180° when axial ratio is 1, β = 3, α = 0.75, ɛ1 = 1.21, ɛ2 = 1.04. Larger axial ratios are for oblate equivolume spheroids.

Equations (37)

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E sc ( r ) = - [ exp ( i k r ) k 2 / 4 π r ] × v exp ( - i k r ˆ · r ) [ ɛ ( r ) - 1 ] × [ r ˆ × ( r ˆ × E ( r ) ) ] d v ,
ɛ ¯ - 1 ɛ ¯ + 2 = 1 v v ɛ ( r ) - 1 ɛ ( r ) + 2 d v .
E inc ( r ) = ( ɛ ˆ 1 ± i ɛ ˆ 2 ) e i k z = l , m ( i k α E ( l , m ) × [ j l ( k r ) Y l l m ( r ˆ ) ] + α M ( l , m ) j l ( k r ) Y l l m ( r ) ) ,
α M ( l , m ) = i l [ 4 π ( 2 l + 1 ) ] 1 / 2 δ m ± 1 , α E ( l , m ) = i α M ( l , m ) .
E in ( r ) = l , m [ ( i c ɛ ω ) γ E ( l , m ) × [ j l ( k r ) Y l l m ( r ˆ ) ] + γ M ( l , m ) j l ( k r ) Y l l m ( r ˆ ) ] ,
γ E ( l , m ) = ( i ɛ / k a ) α E ( l , m ) ɛ j l ( k a ) [ k a h l ( 1 ) ( k a ) ] - h l ( 1 ) ( k a ) [ k a j l ( k a ) ]
γ M ( l , m ) = - ( i / k a ) α M ( l , m ) h l ( 1 ) ( k a ) [ k a j l ( k a ) ] - j l ( k a ) [ k a h l ( 1 ) ( k a ) ] .
k 2 ( ɛ - 1 ) e i k r 4 π r e - i k · r [ E in ( r ) - r ˆ · E in ( r ˆ ) r ˆ ] d 3 r .
e - i k · r = 4 π l , m i - l j l ( k r ) Y l m * ( r ˆ ) Y l m ( r ˆ ) ,
× [ j l ( k r ) Y l l m ( r ˆ ) ] = i k ( 2 l + 1 ) 1 / 2 [ - l j l + 1 ( k r ) Y l , l + 1 , m ( r ˆ ) + ( l + 1 ) 1 / 2 j l - 1 ( k r ) Y l , l - 1 , m ( r ˆ ) ] ,
0 a j l ( k r ) j l ( k r ) r 2 d r I l = a [ k a j l + 1 ( k a ) j l ( k a ) - k a j l ( k a ) j l + 1 ( k a ) ] / ( k 2 - k 2 )
e - i k · r E in ( r ) d 3 r = 4 π l , m i - l { - i c ɛ ω γ E ( l , m ) k [ ( l 2 l + 1 ) 1 / 2 I l + 1 Y l , l + 1 , m ( r ˆ ) + ( l + 1 2 l + 1 ) 1 / 2 I l - 1 Y l , l - 1 , m ( r ˆ ) ] + γ m ( l , m ) I l Y l l m ( r ˆ ) } 4 π l , m i - l [ A l m Y l , l + 1 , m ( r ˆ ) + B l m Y l , l - 1 , m ( r ˆ ) + γ m ( l , m ) I l Y l l m ( r ˆ ) ] .
e - i k · r [ E in ( r ) - r ˆ · E in ( r ) r ˆ ] d 3 r = 4 π l , m i - l { A l m Y l , l + 1 , m ( r ˆ ) + B l m Y l , l - 1 , m ( r ˆ ) - [ A l m Y l , l + 1 , m ( r ˆ ) · r ˆ + B l m Y l , l - 1 , m ( r ˆ ) · r ] r ˆ + γ m ( l , m ) I l Y l l m ( r ) }
Y l , l + 1 , m ( r ˆ ) · r ˆ = - [ ( l + 1 ) / ( 2 l + 1 ) ] 1 / 2 Y l m ( r ˆ ) ,
Y l , l - 1 , m ( r ˆ ) · r ˆ = [ l / ( 2 l + 1 ) ] 1 / 2 Y l m ( r ˆ ) ,
A l m Y l , l + 1 , m ( r ˆ ) + B l m Y l , l - 1 , m ( r ˆ ) - [ A l m Y l , l + 1 , m ( r ˆ ) · r ˆ + B l m Y l , l - 1 , m ( r ˆ ) · r ˆ ] r ˆ = 1 2 l + 1 [ l A l m + ( l + 1 ) 1 / 2 B l m ] × [ l Y l , l + 1 , m ( r ˆ ) + ( l + 1 ) 1 / 2 Y l , l - 1 , m ( r ˆ ) ] .
- ( i c k / ɛ ω ) γ E ( l , m ) ( 2 l + 1 ) - 1 / 2 [ l I l + 1 + ( l + 1 ) I l - 1 ] .
e - i k · r [ E in ( r ) - r ˆ · E in ( r ˆ ) r ˆ ] d 3 r = 4 π l , m i - l ( - γ E ( l , m ) l I l + 1 + ( l - 1 ) I l - 1 ( 2 l + 1 ) n r ˆ × Y l l m ( r ˆ ) + γ m ( l , m ) I l Y l l m ( r ˆ ) ) .
I l = [ a / k 2 ( ɛ - 1 ) ] { j l ( k a ) [ k a j l ( k a ) ] - j l ( k a ) [ k a j l ( k a ) ] } ,
I l - 1 = [ a / k 2 ( ɛ - 1 ) ] { k a j l ( k a ) j l + 1 ( k a ) - k a j l ( k a ) j l + 1 ( k a ) + ( 2 l + 1 ) ( n - 1 / n ) j l ( k a ) j l ( k a ) } ,
l I l + 1 + ( l + 1 ) I l - 1 = - [ a ( 2 l + 1 ) / n k 2 ( ɛ - 1 ) ] { j l ( k a ) [ k a j l ( k a ) ] - n 2 j l ( k a ) [ k a j l ( k a ) ] } ,
e - i k · r [ E in ( r ) - r ˆ · E in ( r ) r ˆ ] d 3 r = 4 π a ( ɛ - 1 ) k 2 l , m i - l ( ( i ɛ / k a ) α E ( l , m ) [ [ j l ( k a ) [ k a j l ( k a ) ] - n 2 j l ( k a ) [ k a j l ( k a ) ] ] / n ɛ j l ( k a ) [ k a h l ( 1 ) ( k a ) ] - h l ( 1 ) ( k a ) [ k a j l ( k a ) ] r ˆ × Y l l m ( r ˆ ) + ( - i / k a ) α m ( l , m ) [ j l ( k a ) [ k a j l ( k a ) ] - j l ( k a ) [ k a j l ( k a ) ] ] h l ( 1 ) ( k a ) [ k a j l ( k a ) ] - j l ( k a ) [ k a h l ( 1 ) ( k a ) ] Y l l m ( r ˆ ) )
= 4 π i ( ɛ - 1 ) k 3 l , m i - 1 [ β E ( l , m ) r ˆ × Y l l m ( r ˆ ) - β M ( l , m ) Y l l m ( r ˆ ) ] ,
k 2 ( ɛ - 1 ) e i k r 4 π r e - i k · r [ E in ( r ) - r ˆ · E in ( r ) r ˆ ] d 3 r = i e i k r k r l , m i - l { β E ( l , m ) r ˆ × Y l l m ( r ˆ ) - β M ( l , m ) Y l l m ( r ˆ ) } ,
E sc ( r ) = - exp ( i k r ) r k 2 4 π 0 a r 2 d r 0 π sin θ d θ 0 2 π d ϕ exp ( - i k r ˆ · r ˆ ) [ ɛ ( r ) - 1 ] [ E ( r , θ , ϕ ) - r ˆ · E ( r , θ , ϕ ) r ˆ ] .
E r = cos ϕ γ r ,
E θ = cos ϕ γ θ ,
E ϕ = sin ϕ γ ϕ ,
A ( r , θ , ϕ ) = v exp ( - i k r ˆ · r ) [ ɛ ( r ) - 1 ] E ( r , θ , ϕ ) d v .
E x ( r , θ , ϕ ) = cos 2 ϕ ( sin θ γ r + cos θ γ θ ) - sin 2 ϕ γ ϕ ,
E y ( r , θ , ϕ ) = sin ϕ cos ϕ ( sin θ γ r + cos θ γ θ + γ ϕ ) ,
E z ( r , θ , ϕ ) = cos ϕ ( cos θ γ r - sin θ γ θ ) ,
r ˆ · r = r [ cos θ cos θ + sin θ sin θ cos ( ϕ - ϕ ) ] ,
A x = π 0 a r 2 d r 0 π sin θ d θ × exp ( - i k r cos θ cos θ ) [ ɛ ( r ) - 1 ] × { [ J 0 ( μ ) - J 2 ( μ ) ] [ sin θ γ r + cos θ γ θ ] - [ J 0 ( μ ) + J 2 ( μ ) ] γ ϕ }
A y = 0 ,
A z = - 2 π i 0 a r 2 d r 0 π sin θ d θ × exp ( - i k 0 r cos θ cos θ ) [ ɛ ( r ) - 1 ] J 1 ( μ ) ] × ( cos θ γ r - sin θ γ θ ) ,
μ = k r sin θ sin θ .