Abstract

We introduce analytical models for scattering potentials of particles that have ellipsoid-, cylinder-, and parallelepiped-like shapes and adjustable edge sharpness with the help of the three-dimensional versions of the multi-Gaussian functions. The far fields produced upon scattering from such potentials are examined in detail and are shown to qualitatively and quantitatively depend on the scatterer’s symmetry type as well as its orientation and edge sharpness.

© 2014 Optical Society of America

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  1. G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 330, 377–445 (1908).
    [CrossRef]
  2. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  3. T. Shirai and T. Asakura, “Multiple light scattering from spatially random media under the second-order Born approximation,” Opt. Commun. 123, 234–249 (1996).
    [CrossRef]
  4. D. Cabaret, S. Rossano, and C. Brouder, “Mie scattering of a partially coherent beam,” Opt. Commun. 150, 239–250 (1998).
    [CrossRef]
  5. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  6. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).
  7. S. K. Sharma and D. J. Somerford, Light Scattering by Optically Soft Particles (Praxis, 2006).
  8. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  9. J. Jannson, T. Jannson, and E. Wolf, “Spatial coherence discrimination on scattering,” Opt. Lett. 13, 1060–1062 (1988).
    [CrossRef]
  10. E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6, 1142–1149 (1989).
    [CrossRef]
  11. A. Dogariu and E. Wolf, “Spectral changes produced by static scattering on a system of particles,” Opt. Lett. 23, 1340–1342 (1998).
    [CrossRef]
  12. O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E 75, 056609 (2007).
    [CrossRef]
  13. S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A 78, 063815 (2008).
    [CrossRef]
  14. S. Sahin and O. Korotkova, “Effect of the pair-structure factor of a particulate medium on scalar wave scattering in the first Born approximation,” Opt. Lett. 34, 1762–1764 (2009).
    [CrossRef]
  15. X. Du and D. Zhao, “Scattering of light by Gaussian-correlated quasi-homogeneous anisotropic media,” Opt. Lett. 35, 384–386 (2010).
    [CrossRef]
  16. Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82, 033836 (2010).
    [CrossRef]
  17. X. Du and D. Zhao, “Scattering of light by a system of anisotropic particles,” Opt. Lett. 35, 1518–1520 (2010).
    [CrossRef]
  18. T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
    [CrossRef]
  19. Z. Tong and O. Korotkova, “Pair-structure matrix of random collections of particles: implications for light scattering,” Opt. Commun. 284, 5598–5600 (2011).
    [CrossRef]
  20. Z. Tong and O. Korotkova, “Momentum of light scattered from collections of particles,” Phys. Rev. A 84, 043835 (2011).
    [CrossRef]
  21. S. Sukhov, D. Haefner, J. Bae, D. Ma, D. R. Carter, and A. Dogariu, “Effect of spatial coherence on scattering from optically inhomogeneous media,” J. Opt. Soc. Am. A 29, 85–88 (2012).
    [CrossRef]
  22. Z. Mei and O. Korotkova, “Random light scattering by collections of ellipsoids,” Opt. Express 20, 29296–29307 (2012).
    [CrossRef]
  23. T. D. Visser and E. Wolf, “Potential scattering with field discontinuities at the boundaries,” Phys. Rev. E 59, 2355–2360 (1999).
    [CrossRef]
  24. S. Sahin, O. Korotkova, and G. Gbur, “Scattering of light from particles with semisoft boundaries,” Opt. Lett. 36, 3957–3959 (2011).
    [CrossRef]
  25. Y. Li, H. Lee, and E. Wolf, “Effect of edge rounding and sloping of sidewalls on the readout signal of the information pits,” Opt. Eng. 42, 2707–2720 (2003).
    [CrossRef]
  26. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  27. Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. 27, 1007–1009 (2002).
    [CrossRef]
  28. Y. Li, “New expressions for flat-topped beams,” Opt. Commun. 206, 225–234 (2002).
    [CrossRef]
  29. Y. Li, “Flat-topped beam with non-circular cross-sections,” J. Mod. Opt. 50, 1957–1966 (2003).
    [CrossRef]
  30. Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A 6, 390–395 (2004).
    [CrossRef]
  31. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37, 2970–2972 (2012).
    [CrossRef]
  32. O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29, 2159–2164 (2012).
    [CrossRef]

2012 (4)

2011 (3)

S. Sahin, O. Korotkova, and G. Gbur, “Scattering of light from particles with semisoft boundaries,” Opt. Lett. 36, 3957–3959 (2011).
[CrossRef]

Z. Tong and O. Korotkova, “Pair-structure matrix of random collections of particles: implications for light scattering,” Opt. Commun. 284, 5598–5600 (2011).
[CrossRef]

Z. Tong and O. Korotkova, “Momentum of light scattered from collections of particles,” Phys. Rev. A 84, 043835 (2011).
[CrossRef]

2010 (4)

X. Du and D. Zhao, “Scattering of light by Gaussian-correlated quasi-homogeneous anisotropic media,” Opt. Lett. 35, 384–386 (2010).
[CrossRef]

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82, 033836 (2010).
[CrossRef]

X. Du and D. Zhao, “Scattering of light by a system of anisotropic particles,” Opt. Lett. 35, 1518–1520 (2010).
[CrossRef]

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[CrossRef]

2009 (1)

2008 (1)

S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A 78, 063815 (2008).
[CrossRef]

2007 (1)

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E 75, 056609 (2007).
[CrossRef]

2004 (1)

Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A 6, 390–395 (2004).
[CrossRef]

2003 (2)

Y. Li, “Flat-topped beam with non-circular cross-sections,” J. Mod. Opt. 50, 1957–1966 (2003).
[CrossRef]

Y. Li, H. Lee, and E. Wolf, “Effect of edge rounding and sloping of sidewalls on the readout signal of the information pits,” Opt. Eng. 42, 2707–2720 (2003).
[CrossRef]

2002 (2)

Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. 27, 1007–1009 (2002).
[CrossRef]

Y. Li, “New expressions for flat-topped beams,” Opt. Commun. 206, 225–234 (2002).
[CrossRef]

1999 (1)

T. D. Visser and E. Wolf, “Potential scattering with field discontinuities at the boundaries,” Phys. Rev. E 59, 2355–2360 (1999).
[CrossRef]

1998 (2)

A. Dogariu and E. Wolf, “Spectral changes produced by static scattering on a system of particles,” Opt. Lett. 23, 1340–1342 (1998).
[CrossRef]

D. Cabaret, S. Rossano, and C. Brouder, “Mie scattering of a partially coherent beam,” Opt. Commun. 150, 239–250 (1998).
[CrossRef]

1996 (1)

T. Shirai and T. Asakura, “Multiple light scattering from spatially random media under the second-order Born approximation,” Opt. Commun. 123, 234–249 (1996).
[CrossRef]

1994 (1)

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

1989 (1)

1988 (1)

1908 (1)

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

Asakura, T.

T. Shirai and T. Asakura, “Multiple light scattering from spatially random media under the second-order Born approximation,” Opt. Commun. 123, 234–249 (1996).
[CrossRef]

Bae, J.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Brouder, C.

D. Cabaret, S. Rossano, and C. Brouder, “Mie scattering of a partially coherent beam,” Opt. Commun. 150, 239–250 (1998).
[CrossRef]

Cabaret, D.

D. Cabaret, S. Rossano, and C. Brouder, “Mie scattering of a partially coherent beam,” Opt. Commun. 150, 239–250 (1998).
[CrossRef]

Cai, Y.

Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A 6, 390–395 (2004).
[CrossRef]

Carter, D. R.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Dogariu, A.

Du, X.

Fischer, D. G.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[CrossRef]

Foley, J. T.

Gbur, G.

Gori, F.

Haefner, D.

Jannson, J.

Jannson, T.

Korotkova, O.

Z. Mei and O. Korotkova, “Random light scattering by collections of ellipsoids,” Opt. Express 20, 29296–29307 (2012).
[CrossRef]

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37, 2970–2972 (2012).
[CrossRef]

O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29, 2159–2164 (2012).
[CrossRef]

S. Sahin, O. Korotkova, and G. Gbur, “Scattering of light from particles with semisoft boundaries,” Opt. Lett. 36, 3957–3959 (2011).
[CrossRef]

Z. Tong and O. Korotkova, “Pair-structure matrix of random collections of particles: implications for light scattering,” Opt. Commun. 284, 5598–5600 (2011).
[CrossRef]

Z. Tong and O. Korotkova, “Momentum of light scattered from collections of particles,” Phys. Rev. A 84, 043835 (2011).
[CrossRef]

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82, 033836 (2010).
[CrossRef]

S. Sahin and O. Korotkova, “Effect of the pair-structure factor of a particulate medium on scalar wave scattering in the first Born approximation,” Opt. Lett. 34, 1762–1764 (2009).
[CrossRef]

S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A 78, 063815 (2008).
[CrossRef]

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E 75, 056609 (2007).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Lee, H.

Y. Li, H. Lee, and E. Wolf, “Effect of edge rounding and sloping of sidewalls on the readout signal of the information pits,” Opt. Eng. 42, 2707–2720 (2003).
[CrossRef]

Li, Y.

Y. Li, H. Lee, and E. Wolf, “Effect of edge rounding and sloping of sidewalls on the readout signal of the information pits,” Opt. Eng. 42, 2707–2720 (2003).
[CrossRef]

Y. Li, “Flat-topped beam with non-circular cross-sections,” J. Mod. Opt. 50, 1957–1966 (2003).
[CrossRef]

Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. 27, 1007–1009 (2002).
[CrossRef]

Y. Li, “New expressions for flat-topped beams,” Opt. Commun. 206, 225–234 (2002).
[CrossRef]

Lin, Q.

Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A 6, 390–395 (2004).
[CrossRef]

Ma, D.

Mei, Z.

Mie, G.

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Rossano, S.

D. Cabaret, S. Rossano, and C. Brouder, “Mie scattering of a partially coherent beam,” Opt. Commun. 150, 239–250 (1998).
[CrossRef]

Sahin, S.

Sharma, S. K.

S. K. Sharma and D. J. Somerford, Light Scattering by Optically Soft Particles (Praxis, 2006).

Shchepakina, E.

Shirai, T.

T. Shirai and T. Asakura, “Multiple light scattering from spatially random media under the second-order Born approximation,” Opt. Commun. 123, 234–249 (1996).
[CrossRef]

Somerford, D. J.

S. K. Sharma and D. J. Somerford, Light Scattering by Optically Soft Particles (Praxis, 2006).

Sukhov, S.

Tong, Z.

Z. Tong and O. Korotkova, “Pair-structure matrix of random collections of particles: implications for light scattering,” Opt. Commun. 284, 5598–5600 (2011).
[CrossRef]

Z. Tong and O. Korotkova, “Momentum of light scattered from collections of particles,” Phys. Rev. A 84, 043835 (2011).
[CrossRef]

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82, 033836 (2010).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

van Dijk, T.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[CrossRef]

Visser, T. D.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[CrossRef]

T. D. Visser and E. Wolf, “Potential scattering with field discontinuities at the boundaries,” Phys. Rev. E 59, 2355–2360 (1999).
[CrossRef]

Wolf, E.

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[CrossRef]

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E 75, 056609 (2007).
[CrossRef]

Y. Li, H. Lee, and E. Wolf, “Effect of edge rounding and sloping of sidewalls on the readout signal of the information pits,” Opt. Eng. 42, 2707–2720 (2003).
[CrossRef]

T. D. Visser and E. Wolf, “Potential scattering with field discontinuities at the boundaries,” Phys. Rev. E 59, 2355–2360 (1999).
[CrossRef]

A. Dogariu and E. Wolf, “Spectral changes produced by static scattering on a system of particles,” Opt. Lett. 23, 1340–1342 (1998).
[CrossRef]

E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6, 1142–1149 (1989).
[CrossRef]

J. Jannson, T. Jannson, and E. Wolf, “Spatial coherence discrimination on scattering,” Opt. Lett. 13, 1060–1062 (1988).
[CrossRef]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Zhao, D.

Ann. Phys. (1)

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

J. Mod. Opt. (1)

Y. Li, “Flat-topped beam with non-circular cross-sections,” J. Mod. Opt. 50, 1957–1966 (2003).
[CrossRef]

J. Opt. A (1)

Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A 6, 390–395 (2004).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (5)

T. Shirai and T. Asakura, “Multiple light scattering from spatially random media under the second-order Born approximation,” Opt. Commun. 123, 234–249 (1996).
[CrossRef]

D. Cabaret, S. Rossano, and C. Brouder, “Mie scattering of a partially coherent beam,” Opt. Commun. 150, 239–250 (1998).
[CrossRef]

Z. Tong and O. Korotkova, “Pair-structure matrix of random collections of particles: implications for light scattering,” Opt. Commun. 284, 5598–5600 (2011).
[CrossRef]

Y. Li, “New expressions for flat-topped beams,” Opt. Commun. 206, 225–234 (2002).
[CrossRef]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

Opt. Eng. (1)

Y. Li, H. Lee, and E. Wolf, “Effect of edge rounding and sloping of sidewalls on the readout signal of the information pits,” Opt. Eng. 42, 2707–2720 (2003).
[CrossRef]

Opt. Express (1)

Opt. Lett. (8)

Phys. Rev. A (3)

Z. Tong and O. Korotkova, “Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media,” Phys. Rev. A 82, 033836 (2010).
[CrossRef]

S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A 78, 063815 (2008).
[CrossRef]

Z. Tong and O. Korotkova, “Momentum of light scattered from collections of particles,” Phys. Rev. A 84, 043835 (2011).
[CrossRef]

Phys. Rev. E (2)

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E 75, 056609 (2007).
[CrossRef]

T. D. Visser and E. Wolf, “Potential scattering with field discontinuities at the boundaries,” Phys. Rev. E 59, 2355–2360 (1999).
[CrossRef]

Phys. Rev. Lett. (1)

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[CrossRef]

Other (5)

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

S. K. Sharma and D. J. Somerford, Light Scattering by Optically Soft Particles (Praxis, 2006).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

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Figures (8)

Fig. 1.
Fig. 1.

(A) Scattering potential (7) with λ=0.633μm, kσ=5; L=1 (solid curve), L=4 (dashed curve), L=10 (dotted curve), and L=40 (dash-dotted curve). (B) Intensity of the plane wave calculated from Eq. (9) scattered from potentials in (A).

Fig. 2.
Fig. 2.

Illustration of the semi-hard-edge ellipsoid, σx=σ, σy=1.5σ, and σz=2σ.

Fig. 3.
Fig. 3.

Far-zone intensity of the plane wave scattered from the semi-hard-edge ellipsoids of Fig. 2 with σx=σ=5/k, σy=1.5σ, and σz=2σ, given on the logarithmic scale.

Fig. 4.
Fig. 4.

Illustration of the semi-hard-edge cylinder, σx=σ, σy=1.5σ, and σz=2σ.

Fig. 5.
Fig. 5.

Far-zone intensity of the plane wave scattered from the semi-hard-edge cylinders of Fig. 4 oriented along the z axis, with σx=σ=5/k, σy=1.5σ, and σz=2σ, given on the logarithmic scale.

Fig. 6.
Fig. 6.

Dependence of the scattered intensity distribution on the orientation of the cylinder (C) of Fig. 4 given on the logarithmic scale: (A) orientation along the x axis, σx=σ=5/k, σy=1.5σ, and σz=2σ; (B) orientation along the y axis, σy=σ=5/k, σz=1.5σ, and σx=2σ.

Fig. 7.
Fig. 7.

Illustration of the semi-hard-edge cube, σx=σ, σy=1.5σ, and σz=2σ.

Fig. 8.
Fig. 8.

Far-zone intensity of the plane wave scattered from the semi-hard-edge cubes of Fig. 7 with σx=σ=5/k, σy=1.5σ, and σz=2σ, given on the logarithmic scale.

Equations (15)

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

U(i)(r,ω)=a(i)(ω)exp[iks0·r]
S(s)(rs,ω)=1r2S(i)(ω)|F˜[k(ss0),ω]|2,
F˜(K)=DF(r;ω)exp[iK·r]d3r,
F(r,ω)={k24π[n2(r,ω)1],rD,0,otherwise.
F(G)(r;ω)=exp[x2+y2+z22σ2],
F(H)(r;ω)={1,rb,0,otherwise.
F(MG)(r;ω)=1Cll=1L(1)l1(Ll)exp[lx2+y2+z22σ2],
Cl=l=1L(1)l1(Ll)
SMG(s)(rs;ω)=S(i)(ω)(2π)3σ3r2Cl×(l=1L(1)l1l3/2(Ll)exp{k22lσ2[(sxs0x)2+(sys0y)2+(szs0z)2]})2.
FE(r;ω)=1Cll=1L(1)l1(Ll)exp[l(x22σx2+y22σy2+z22σz2)],
SE(s)(rs;ω)=S(i)(ω)(2π)3σxσyσzr2Cl×(l=1L(1)l1l3/2(Ll)exp{k22l[σx2(sxs0x)2+σy2(sys0y)2+σz2(szs0z)2]})2.
FC(r;ω)=1Cll=1L(1)l1(Ll)exp[l(x22σx2+y22σy2)]×1Cmm=1M(1)m1(Mm)exp[mz22σz2],
SC(s)(rs;ω)=S(i)(ω)(2π)3σxσyσzr2ClCn{l=1L(1)l1l(Ll)×exp{k22l[σx2(sxs0x)2+σy2(sys0y)2]}×m=1M(1)m1m(Mm)exp[k2σz22m(szs0z)2]}2.
FE(r;ω)=1Cnn=1N(1)n1(Nn)exp[nx22σx2]×1Cpp=1P(1)p1(Pp)exp[py22σy2]×1Cmm=1M(1)m1(Mm)exp[mz22σz2],
SP(s)(rs;ω)=S(i)(ω)(2π)3σxσyσzr2CnCpCm×{n=1N(1)n1n(Nn)exp[k2σx22n(sxs0x)2]×p=1P(1)p1p(Pp)exp[k2σy22p(sys0y)2]×m=1M(1)m1m(Mm)exp[k2σz22m(szs0z)2]}2.

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