Abstract

The polarization-resolved electric field autocorrelation function for p-order scattering was derived from the order-of-scattering solution of the exact equations for electromagnetic multiple Rayleigh scattering and was calculated for 2 ≤ p ≤ 6 for particles undergoing diffusive motion in an idealized sample cell. It was found that the polarization-channel and the scattering-angle dependence of the p-order autocorrelation function approximately decoupled from the delay-time dependence for p ≳ 3. The polarization-channel and the scattering-angle dependence were analytically calculated, and the delay-time dependence was analytically approximated. The resulting analytical model for the polarization-resolved autocorrelation function for beginning multiple Rayleigh scattering was then tested against experimental autocorrelation data. The data were found to be well fitted by the model.

© 2001 Optical Society of America

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References

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  1. N. A. Clark, J. H. Lunacek, G. B. Benedek, “A study of Brownian motion using light scattering,” Am. J. Phys. 38, 575–585 (1970).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

1998 (3)

P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing light spectroscopies beyond the diffusion limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
[CrossRef]

Y.-L. Xu, “Electromagnetic scattering by an aggregate of spheres: errata,” Appl. Opt. 37, 6494 (1998).
[CrossRef]

V. I. Ovod, “Modeling of multiple-scattering suppression by a one-beam cross-correlation system,” Appl. Opt. 37, 7856–7864 (1998).
[CrossRef]

1997 (3)

1995 (2)

1994 (2)

A. E. Bailey, D. S. Cannell, “Practical method for calculation of multiple light scattering,” Phys. Rev. E 50, 4853–4864 (1994).
[CrossRef]

W. Lou, T. T. Charalampopoulos, “On the electromagnetic scattering and absorption of agglomerated small spherical particles,” J. Phys. D 27, 2258–2270 (1994).
[CrossRef]

1993 (1)

P. Štěpánek, “Static and dynamic properties of multiple light scattering,” J. Chem. Phys. 99, 6384–6393 (1993).
[CrossRef]

1989 (2)

F. C. MacKintosh, S. John, “Diffusing wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[CrossRef]

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

1988 (2)

1987 (1)

1985 (1)

J. K. G. Dhont, “Multiple Rayleigh–Gans–Debye scattering in colloidal systems: dynamic light scattering,” Physica A 129, 374–394 (1985).
[CrossRef]

1979 (1)

A. R. Jones, “Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation,” Proc. R. Soc. London Ser. A 366, 111–127 (1979).
[CrossRef]

1978 (2)

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Multiple scattering from a system of Brownian particles,” Phys. Rev. A 17, 2030–2035 (1978).
[CrossRef]

A. Bøe, O. Lohne, “Dynamical properties of multiply scattered light from independent Brownian particles,” Phys. Rev. A 17, 2023–2029 (1978).
[CrossRef]

1976 (1)

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Depolarized correlation function of light double scattered from a system of Brownian particles,” Phys. Rev. A 14, 1520–1532 (1976).
[CrossRef]

1973 (1)

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

1970 (1)

N. A. Clark, J. H. Lunacek, G. B. Benedek, “A study of Brownian motion using light scattering,” Am. J. Phys. 38, 575–585 (1970).
[CrossRef]

1962 (2)

O. R. Cruzan, “Translation addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

V. Twersky, “On scattering of waves by random distributions. I. Free-space scatterer formalism,” J. Math. Phys. 3, 700–715 (1962).
[CrossRef]

1961 (1)

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

1945 (1)

L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107–119 (1945).
[CrossRef]

Bailey, A. E.

A. E. Bailey, D. S. Cannell, “Practical method for calculation of multiple light scattering,” Phys. Rev. E 50, 4853–4864 (1994).
[CrossRef]

Benedek, G. B.

N. A. Clark, J. H. Lunacek, G. B. Benedek, “A study of Brownian motion using light scattering,” Am. J. Phys. 38, 575–585 (1970).
[CrossRef]

Bøe, A.

A. Bøe, O. Lohne, “Dynamical properties of multiply scattered light from independent Brownian particles,” Phys. Rev. A 17, 2023–2029 (1978).
[CrossRef]

Bohren, C. F.

Cannell, D. S.

W. V. Meyer, D. S. Cannell, A. E. Smart, T. W. Taylor, P. Tin, “Multiple-scattering suppression by cross correlation,” Appl. Opt. 36, 7551–7558 (1997).
[CrossRef]

A. E. Bailey, D. S. Cannell, “Practical method for calculation of multiple light scattering,” Phys. Rev. E 50, 4853–4864 (1994).
[CrossRef]

Charalampopoulos, T. T.

W. Lou, T. T. Charalampopoulos, “On the electromagnetic scattering and absorption of agglomerated small spherical particles,” J. Phys. D 27, 2258–2270 (1994).
[CrossRef]

Clark, N. A.

N. A. Clark, J. H. Lunacek, G. B. Benedek, “A study of Brownian motion using light scattering,” Am. J. Phys. 38, 575–585 (1970).
[CrossRef]

Cruzan, O. R.

O. R. Cruzan, “Translation addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

Dhont, J. K. G.

J. K. G. Dhont, “Multiple Rayleigh–Gans–Debye scattering in colloidal systems: dynamic light scattering,” Physica A 129, 374–394 (1985).
[CrossRef]

Durian, D. J.

P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing light spectroscopies beyond the diffusion limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
[CrossRef]

Foldy, L. L.

L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107–119 (1945).
[CrossRef]

Glatter, O.

Griffiths, D. J.

D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice-Hall, Upper Saddle River, N.J., 1999), p. 158.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2, pp. 254–256.

John, S.

F. C. MacKintosh, S. John, “Diffusing wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[CrossRef]

Jones, A. R.

A. R. Jones, “Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation,” Proc. R. Soc. London Ser. A 366, 111–127 (1979).
[CrossRef]

Lemieux, P.-A.

P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing light spectroscopies beyond the diffusion limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
[CrossRef]

Lock, J. A.

Lohne, O.

A. Bøe, O. Lohne, “Dynamical properties of multiply scattered light from independent Brownian particles,” Phys. Rev. A 17, 2023–2029 (1978).
[CrossRef]

Lou, W.

W. Lou, T. T. Charalampopoulos, “On the electromagnetic scattering and absorption of agglomerated small spherical particles,” J. Phys. D 27, 2258–2270 (1994).
[CrossRef]

Lunacek, J. H.

N. A. Clark, J. H. Lunacek, G. B. Benedek, “A study of Brownian motion using light scattering,” Am. J. Phys. 38, 575–585 (1970).
[CrossRef]

MacKintosh, F. C.

F. C. MacKintosh, S. John, “Diffusing wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[CrossRef]

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

Meyer, W. V.

Mockler, R. C.

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Multiple scattering from a system of Brownian particles,” Phys. Rev. A 17, 2030–2035 (1978).
[CrossRef]

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Depolarized correlation function of light double scattered from a system of Brownian particles,” Phys. Rev. A 14, 1520–1532 (1976).
[CrossRef]

O’Sullivan, W. J.

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Multiple scattering from a system of Brownian particles,” Phys. Rev. A 17, 2030–2035 (1978).
[CrossRef]

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Depolarized correlation function of light double scattered from a system of Brownian particles,” Phys. Rev. A 14, 1520–1532 (1976).
[CrossRef]

Ovod, V. I.

Pennypacker, C. R.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Pine, D. J.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

D. A. Weitz, D. J. Pine, “Diffusing-wave spectroscopy,” in Dynamic Light Scattering, the Method and Some Applications, W. Brown, ed. (Clarendon, Oxford, UK, 1993), pp. 652–720.

Purcell, E. M.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Schnablegger, H.

Sengers, J. V.

J. G. Shanks, J. V. Sengers, “Double scattering in critically opalescent fluids,” Phys. Rev. A 38, 885–896 (1988).
[CrossRef] [PubMed]

Shanks, J. G.

J. G. Shanks, J. V. Sengers, “Double scattering in critically opalescent fluids,” Phys. Rev. A 38, 885–896 (1988).
[CrossRef] [PubMed]

Singham, S. B.

Smart, A. E.

Sorensen, C. M.

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Multiple scattering from a system of Brownian particles,” Phys. Rev. A 17, 2030–2035 (1978).
[CrossRef]

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Depolarized correlation function of light double scattered from a system of Brownian particles,” Phys. Rev. A 14, 1520–1532 (1976).
[CrossRef]

Stein, S.

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

Štepánek, P.

P. Štěpánek, “Static and dynamic properties of multiple light scattering,” J. Chem. Phys. 99, 6384–6393 (1993).
[CrossRef]

Taylor, T. W.

Tin, P.

Twersky, V.

V. Twersky, “On scattering of waves by random distributions. I. Free-space scatterer formalism,” J. Math. Phys. 3, 700–715 (1962).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), p. 144.

Vera, M. U.

P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing light spectroscopies beyond the diffusion limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
[CrossRef]

Weitz, D. A.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

D. A. Weitz, D. J. Pine, “Diffusing-wave spectroscopy,” in Dynamic Light Scattering, the Method and Some Applications, W. Brown, ed. (Clarendon, Oxford, UK, 1993), pp. 652–720.

Xu, Y.-L.

Zhu, J. X.

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

Am. J. Phys. (1)

N. A. Clark, J. H. Lunacek, G. B. Benedek, “A study of Brownian motion using light scattering,” Am. J. Phys. 38, 575–585 (1970).
[CrossRef]

Appl. Opt. (7)

Astrophys. J. (1)

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

J. Chem. Phys. (1)

P. Štěpánek, “Static and dynamic properties of multiple light scattering,” J. Chem. Phys. 99, 6384–6393 (1993).
[CrossRef]

J. Math. Phys. (1)

V. Twersky, “On scattering of waves by random distributions. I. Free-space scatterer formalism,” J. Math. Phys. 3, 700–715 (1962).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. D (1)

W. Lou, T. T. Charalampopoulos, “On the electromagnetic scattering and absorption of agglomerated small spherical particles,” J. Phys. D 27, 2258–2270 (1994).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107–119 (1945).
[CrossRef]

Phys. Rev. A (4)

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Depolarized correlation function of light double scattered from a system of Brownian particles,” Phys. Rev. A 14, 1520–1532 (1976).
[CrossRef]

C. M. Sorensen, R. C. Mockler, W. J. O’Sullivan, “Multiple scattering from a system of Brownian particles,” Phys. Rev. A 17, 2030–2035 (1978).
[CrossRef]

A. Bøe, O. Lohne, “Dynamical properties of multiply scattered light from independent Brownian particles,” Phys. Rev. A 17, 2023–2029 (1978).
[CrossRef]

J. G. Shanks, J. V. Sengers, “Double scattering in critically opalescent fluids,” Phys. Rev. A 38, 885–896 (1988).
[CrossRef] [PubMed]

Phys. Rev. B (2)

F. C. MacKintosh, S. John, “Diffusing wave spectroscopy and multiple scattering of light in correlated random media,” Phys. Rev. B 40, 2383–2406 (1989).
[CrossRef]

F. C. MacKintosh, J. X. Zhu, D. J. Pine, D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342–9345 (1989).
[CrossRef]

Phys. Rev. E (2)

P.-A. Lemieux, M. U. Vera, D. J. Durian, “Diffusing light spectroscopies beyond the diffusion limit: the role of ballistic transport and anisotropic scattering,” Phys. Rev. E 57, 4498–4515 (1998).
[CrossRef]

A. E. Bailey, D. S. Cannell, “Practical method for calculation of multiple light scattering,” Phys. Rev. E 50, 4853–4864 (1994).
[CrossRef]

Physica A (1)

J. K. G. Dhont, “Multiple Rayleigh–Gans–Debye scattering in colloidal systems: dynamic light scattering,” Physica A 129, 374–394 (1985).
[CrossRef]

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

A. R. Jones, “Electromagnetic wave scattering by assemblies of particles in the Rayleigh approximation,” Proc. R. Soc. London Ser. A 366, 111–127 (1979).
[CrossRef]

Q. Appl. Math. (2)

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

O. R. Cruzan, “Translation addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

Other (4)

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), p. 144.

D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice-Hall, Upper Saddle River, N.J., 1999), p. 158.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2, pp. 254–256.

D. A. Weitz, D. J. Pine, “Diffusing-wave spectroscopy,” in Dynamic Light Scattering, the Method and Some Applications, W. Brown, ed. (Clarendon, Oxford, UK, 1993), pp. 652–720.

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Figures (6)

Fig. 1
Fig. 1

Plane wave with the wave vector k inc incident upon a collection of spherical particles: Light scattered at the angle θ in the xz plane of the x, y, z laboratory coordinate system has the wave vector k scatt. The position of particle ℓ with respect to particle j is also shown, as is the V and the H polarization direction of the incoming and the outgoing electric field.

Fig. 2
Fig. 2

Normalized p = 2 autocorrelation function plotted as a function of the scaled delay time T for scattering angles of (a) 0°, (b) 45°, (c) 90°, (d) 135°, (e) 180° for VV, VH, HV, and HH scattering and for multiple s-wave scattering of scalar waves, denoted I. The dashed curves represent the function exp(-2T).

Fig. 3
Fig. 3

Normalized p = 3 autocorrelation function plotted as a function of the scaled delay time T for scattering angles of (a) 0°, (b) 45°, (c) 90°, (d) 135°, (e) 180° for VV, VH, HV, and HH scattering and for multiple s-wave scattering of scalar waves. The dashed curves represent the function exp(-3T).

Fig. 4
Fig. 4

Normalized p = 4 autocorrelation function plotted as a function of the scaled delay time T for scattering angles of (a) 0°, (b) 45°, (c) 90°, (d) 135°, (e) 180° for VV, VH, HV, and HH scattering (open circles) and for multiple s-wave scattering of scalar waves (solid curve). The dashed curves represent the function exp(-4T).

Fig. 5
Fig. 5

Autocorrelation function for multiple s-wave scattering of scalar waves plotted as a function of the scaled delay time T for (a) p = 2, (b) p = 3, (c) p = 4, (d) p = 5, (e) p = 6 and for θ = 0°, 45°, 90°, 135°, 180°. The approximation to Eq. (62) is denoted A, and the dashed curve represents the function exp(-pT).

Fig. 6
Fig. 6

Intensity autocorrelation function plotted as a function of the delay time τ for suspensions of a = 53.5 nm PSL spheres in water with volume fractions f of (a) 1.7 × 10-5, (b) 2.0 × 10-4, (c) 1.0 × 10-3, (d) 2.0 × 10-3. The data are represented by the open circles, and the fit to Eqs. (63) and (64) by the solid curves. In (b)–(d) values of R ave = 3.11, 1.78, 1.23 mm were used, respectively.

Tables (1)

Tables Icon

Table 1 Parameters of the Fit of the Multiple-Scattering Model of Eq. (63) to the Experimental Data for Suspensions of PSL Spheres in Water as Shown in Figs. 6(b)6(d),a

Equations (88)

Equations on this page are rendered with MathJax. Learn more.

Escattered by jr=MjEincident on jrj,
Etotalr=Ebeamr+j=1JEscattered by jr,
Eincident on jrj=Ebeamrj+=1jJEscattered by lrj.
kscatt=ksin θ cos ϕux+sin θ sin ϕuy+cos θuz.
kinc=kuz.
a1=-2i/3n2-1/n2+2ka3.
Escattered by jr, θ, ϕ=i expikr/kr×S2θ, ϕuθ-S1θ, ϕuϕ, Bscattered by jr, θ, ϕ=i expikr/ckr×S1θ, ϕuθ+S2θ, ϕuϕ,
S1θ, ϕ=-P+ cos ϕ+P- sin ϕ, S2θ, ϕ=P+ cos θ sin ϕ-P0 sin θ+P- cos θ cos ϕ,
P+=j=1Jexp-ikscatt·rjP+j, P0=j=1Jexp-ikscatt·rjP0j, P-=j=1Jexp-ikscatt·rjP-j.
P+jP0jP-j=3/2a1E0 expikinc·rjb+b0b--ia1=1jJ h01krjP+P0P-+3i/2a1=1jJ h21krjQjP+P0P-.
Qj=1/rj2×rj2/3-yj2-yjzj-xjyj-yjzjrj2/3-zj2-xjzj-xjyj-xjzjrj2/3-xj2,
xj=rj sin θj cos ϕj, yj=rj sin θj sin ϕj, zj=rj cos θj.
b+=0,  b0=0,  b-=1,
b+=1,  b0=0,  b-=0.
EVVtotalr=i expikr/krP+uϕ, EVHtotalr=i expikr/kr-P0 sin θ+P- cos θuθ, EHVtotalr=i expikr/krP+uϕ, EHHtotalr=i expikr/kr-P0 sin θ+P- cos θuθ,
kRaveka3  1.
P+jP0jP-j=3/2a1E0 expikinc·rjb+b0b-+3i/2a1=1jJexpkrj/krjTjP+P0P-,
Tj=1/rj2xj2+zj2-yjzj-xjyj-yjzjxj2+yj2-xjzj-xjyj-xjzjyj2+zj2.
3fkRaveka3n2-1/n2+221.5.
P+P0P-1=3/2a1E0j=1Jexpikinc-kscatt·rjb+b0b-,
P+P0P-p=3a1/2pi/kp-1E0×CpexpiΦrnmrj-1×TjTnmb+b0b-,
Φ=expikinc-knm·rnexpikj-kscatt·rj.
Eαβtotalpr=E0 expikr/rn2-1k2a3/n2+2p×CpexpiΦWαβpθ, Ωnm,, Ωj/rnmrj,
ψtotalr=i expikr/krj=1J ψj exp-ikscatt·rj,
ψj=ψ0a0 expikinc·rj+ia0=1jJ ψ expikrj/krj.
a0=-i/3n2-1ka3,
ψptotalr=ψ0 expikr/ria0/kp×CpexpiΦ/rnmrj.
PjΔ=exp-Δ2/4Dτ/4πDτ3/2,
gαβt, τ, θ=|Eαβ*t+τ, θ×Bαβt, θ|/2μ0c,
gαβτ, θ=JIVVeachp=1 Qp-1gαβpτ, θ,
gαβ1τ, θ=Wαβ1θ2 exp-|kinc-kscatt|2Dτ
WVV1=1,  WVH1=WHV1=0,  WHH1=cos2 θ,
gαβpτ, θ=4πp-1  dΩjdΩnm×Wαβpθ, Ωnm,, Ωj2×exp-|kinc-knm|2Dτ×exp-|kj-kscatt|2Dτ
Q=3fkRaveka3n2-1/n2+22.
gIτ, θ=JIeachp=1Qp-1gIpτ, θ,
gI1τ, θ=exp-|kinc-kscatt|2Dτ
gIpτ, θ=4πp-1  dΩjdΩnm×exp-|kinc-knm|2Dτ×exp-|kj-kscatt|2Dτ
Q=fkRaveka3n2-12/3.
gIp0, θ=1.
Iαβpθgαβp0, θ
IVVpθ=1/15p-1Ap, IVHpθ=IHVpθ=1/15p-1Bp, IHHpθ=1/15p-1Ap-Bpcos2 θ+Bp,
A1=1,  B1=0
Ap=8Ap-1+2Bp-1, Bp=Ap-1+9Bp-1.
IVV2=8/15, IVV4=1.0988/153, IVV6=1.3598/155, IVV8=1.8528/157, IVV10=2.6858/159
IVH2=0.125IVV2, IVH4=0.390IVV4, IVH6=0.623IVV6, IVH8=0.788IVV8, IVH10=0.888IVV10
IHH2=0.125+0.875 cos2 θIVV2, IHH4=0.390+0.610 cos2 θIVV4, IHH6=0.623+0.377 cos2 θIVV6, IHH8=0.788+0.212 cos2 θIVV8, IHH10=0.888+0.112 cos2 θIVV10
gVV1τ, θ=exp-2T sin2θ/2, gVH1τ, θ=gHV1τ, θ=0, gHH1τ, θ=cos2 θ exp-2T sin2 θ/2,
T=2k2Dτ.
gVV2τ, θ=exp-2T×3/8F0τ, θ+1/4F2τ, θ+3/8F4τ, θ, gVH2τ, θ=gHV2τ, θ=exp-2T×1/8cos2θ/2F0τ, θ+-3/4cos2θ/2+1/2F2τ, θ+5/8cos2θ/2-1/2F4τ, θ, gHH2τ, θ=exp-2T×19/8cos4θ/2-3 cos2θ/2+1F0τ, θ+-7/4cos4θ/2+4 cos2θ/2-2F2τ, θ+3/8cos4θ/2-cos2θ/2+1F4τ, θ,
F0τ, θ=1/2-11du exp2Tu cosθ/2=sinhs/s=even n sn/n!n+1,
F2τ, θ=1/2-11du u2 exp2Tu cosθ/2=sinhs/s-2 coshs/s2+2 sinhs/s3=even n sn/n!n+3,
F4τ, θ=1/2-11du u4 exp2Tu cosθ/2=sinhs/s-4 coshs/s2+12 sinhs/s3-24 coshs/s4+24 sinhs/s5=even n sn/n!n+5,
s=2T cosθ/2.
gαβpτ, θ=IαβpθgαβNpτ, θ
gαβNp0, θ=1
gI1τ, θ=exp-2T sin2θ/2.
02πdϕj expγ cos ϕj=2πI0γ,
02πsin θjdθj expB1+cos θcos θj×I0B sin θ sin θj=2 sinh2B cosθ/2/2B cosθ/2
gI2τ, θ=exp-2T×sinh2T cosθ/2/2T cosθ/2,
gI2τ, θ=exp-2T1+2/3T2-2/3T2 sin2θ/2+.
gI3τ, θ=exp-3T1+3/6T2+1/9T3-2/9T3 sin2θ/2+, gI4τ, θ=exp-4T1+4/6T2+32/135T4-2/27T4 sin2θ/2+, gI5τ, θ=exp-5T1+5/6T2+.
gapproxpτ=exp-pTsinhp1/2T/p1/2T=exp-pT1+p/6T2+p2/120T4+.
gαβτ, θJIeachgαβ1τ, θ+Qgαβ2τ, θ+p=3 Qp-1Iαβpθgapproxpτ,
GNτ, θ=gVV2τ, θ+gVH2τ, θgVV20, θ+gVH20, θ.
WVV2=xj2+zj2/rj2,
WVH2=B cosθ-A sinθ,
A=-yjzj/rj2,
B=-xjyj/rj2.
WHV2=-xjyj/rj2,
WHH2=D cosθ-C sinθ,
C=-xjzj/rj2,
D=yj2+zj2/rj2.
WVV3=xj2+zj2xm2+zm2+yjzjymzm+xjyjxmymrj2rm2.
A=[-yjzjxm2+zm2-xj2+yj2ymzm+xjzjxmymrlj2rml2,
B=-xjyjxm2+zm2+xjzjymzm-yj2+zj2xmymrj2rm2.
WHV3=-xj2+zj2xmym+yjzjxmzm-xjyjym2+zm2rj2rm2,
C=yjzjxmym-xj2+yj2xmzm-xjzjym2+zm2rj2rm2,
D=xjyjxmym+xjzjxmzj+yj2+zj2ym2+zm2rj2rm2.
|Eαβ*t+τ, θ×Bαβt, θ/2μ0c|p=E02/2μ0c×1/r21/k2pn2-1ka3/n2+22p×Cpexpikinc-knm·Δn×expikj-kscatt·Δj×Wαβp2/rnm2rj2.
gαβτ, θp= d3rj  d3rnJ/Vp d3Δj× d3ΔnPΔjPΔn×|Eαβ*t+τ, θ×Bαβt, θ/2μ0c|p,
 d3 Δi expiKi·ΔiPΔi=exp-Ki2Dτ
 d3rj=V.
(gαβ(τ, θ))(p)=(E02/2μ0c)(1/r2)(1/k2p)×V(J/V)p[(n2-1)(ka)3/(n2+2)]2p× d3rjd3rnm×exp(-i|kinc-knm|2Dτ)×exp(-i|kj-kscatt|2Dτ)×[Wαβ(p)]2/(rnm2rj2).
 d3ruv=0Rave ruv2druv  dΩuv,
0Rave ruv2 druv/ruv2=Rave,
(gαβ(τ, θ))(p)=(E02/2μ0c)(1/r2)(1/k2p)×(J/V)p[(n2-1)(ka)3/(n2+2)]2p×V(Rave)p-1  dΩjdΩnm×exp(-|kinc-knm|2Dτ)×exp(-|kj-kscatt|2Dτ)×[Wαβ(p)(θ, Ωnm,, Ωj)]2.
IVVeach=(E02/2μ0c)(1/r2)[(n2-1)(ka)3/(n2+2)]2
f=4πa3/3J/V

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