Abstract

Solution of Maxwell’s equations to the problem of single scattering can be expanded into iterative series in an order-of-scattering form, where the interference between conjugate terms representing reversible sequences of elementary scatterers is constructive at the backscattering direction, resulting in a coherent backscatter enhancement (CBE). The backscattering phase function of randomly oriented particles is amplified by CBE with an amplification factor between 1 and 2 depending on particle habit and refractive index. The angular width of the CBE-induced backscattering peak line for a specific particle habit is inversely proportional to the particle size parameter. The CBE-induced backscattering peak has been identified in the scattering phase function of a wide range of randomly oriented particles, including non-absorptive spheres, spheroids, and hexagonal particles.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (2)

K. Masuda and H. Ishimoto, “Backscatter ratios for nonspherical ice crystals in cirrus clouds calculated by geometrical -optics-integral-equation method,” J. Quant. Spectrosc. Radiat. Transf. 190, 60–68 (2017).
[Crossref]

W. Lin, L. Bi, D. Liu, and K. Zhang, “Use of Debye’s series to determine the optimal edge-effect terms for computing the extinction efficiencies of spheroids,” Opt. Express 25(17), 20298–20312 (2017).
[Crossref] [PubMed]

2016 (3)

M. I. Mishchenko, J. M. Dlugach, M. A. Yurkin, L. Bi, B. Cairns, L. Liu, R. L. Panetta, L. D. Travis, P. Yang, and N. T. Zakharova, “First-principles modeling of electromagnetic scattering by discrete and discretely heterogeneous random media,” Phys. Rep. 632, 1–75 (2016).
[Crossref]

C. Liu and Y. Yin, “Inherent optical properties of pollen particles: a case study for the morning glory pollen,” Opt. Express 24(2), A104–A113 (2016).
[Crossref] [PubMed]

C. T. Collier, E. Hesse, L. Taylor, Z. Ulanowski, A. Penttilä, and T. Nousiainen, “Effects of surface roughness with two scales on light scattering by hexagonal ice crystals large compared to the wavelength: DDA results,” J. Quant. Spectrosc. Radiat. Transf. 182, 225–239 (2016).
[Crossref]

2015 (2)

2014 (3)

L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Radiat. Transf. 138, 17–35 (2014).
[Crossref]

C. Liu, R. L. Panetta, and P. Yang, “The effective equivalence of geometric irregularity and surface roughness in determining particle single-scattering properties,” Opt. Express 22(19), 23620–23627 (2014).
[Crossref] [PubMed]

L. Bi, P. Yang, C. Liu, B. Yi, and S. Hioki, “Optical properties of ice clouds: new modeling capabilities and relevant applications,” Proc. SPIE 9259, 92591A (2014).

2013 (3)

Z. M. Dlugach and M. I. Mishchenko, “Coherent backscattering and opposition effects observed in some atmosphereless bodies of the solar system,” Sol. Syst. Res. 47(6), 454–462 (2013).
[Crossref]

C. Liu, R. L. Panetta, and P. Yang, “The effects of surface roughness on the scattering properties of hexagonal columns with sizes from the Rayleigh to the geometric optics regimes,” J. Quant. Spectrosc. Radiat. Transf. 129, 169–185 (2013).
[Crossref]

A. Borovoi, A. Konoshonkin, and N. Kustova, “Backscattering by hexagonal ice crystals of cirrus clouds,” Opt. Lett. 38(15), 2881–2884 (2013).
[Crossref] [PubMed]

2010 (1)

F. Xu, J. A. Lock, and G. Gouesbet, “Debye series for light scattering by a nonspherical particle,” Phys. Rev. A 81(4), 043824 (2010).
[Crossref]

2009 (1)

A. J. Baran, “A review of the light scattering properties of cirrus,” J. Quant. Spectrosc. Radiat. Transf. 110(14–16), 1239–1260 (2009).
[Crossref]

2005 (1)

2002 (2)

R. Lenke, U. Mack, and G. Maret, “Comparison of the ‘glory’ with coherent backscattering of light in turbid media,” J. Opt. A, Pure Appl. Opt. 4(3), 309–314 (2002).
[Crossref]

R. Lenke, R. Tweer, and G. Maret, “Coherent backscattering of turbid samples containing large Mie spheres,” J. Opt. A, Pure Appl. Opt. 4(3), 293–298 (2002).
[Crossref]

1995 (1)

R. Corey, M. Kissner, and P. Saulnier, “Coherent backscattering of light,” Am. J. Phys. 63(6), 560–564 (1995).
[Crossref]

1991 (1)

1986 (1)

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent Backscattering of Light by Disordered Media: Analysis of the Peak Line Shape,” Phys. Rev. Lett. 56(14), 1471–1474 (1986).
[Crossref] [PubMed]

1985 (1)

P. E. Wolf and G. Maret, “Weak Localization and Coherent Backscattering of Photons in Disordered Media,” Phys. Rev. Lett. 55(24), 2696–2699 (1985).
[Crossref] [PubMed]

1984 (2)

1977 (1)

V. Khare and H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38(22), 1279–1282 (1977).
[Crossref]

1947 (1)

1908 (1)

G. Mie, “Beitrage zur Optik trüber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 330(3), 377–445 (1908).
[Crossref]

Akkermans, E.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent Backscattering of Light by Disordered Media: Analysis of the Peak Line Shape,” Phys. Rev. Lett. 56(14), 1471–1474 (1986).
[Crossref] [PubMed]

Baran, A. J.

A. J. Baran, “A review of the light scattering properties of cirrus,” J. Quant. Spectrosc. Radiat. Transf. 110(14–16), 1239–1260 (2009).
[Crossref]

Bi, L.

W. Lin, L. Bi, D. Liu, and K. Zhang, “Use of Debye’s series to determine the optimal edge-effect terms for computing the extinction efficiencies of spheroids,” Opt. Express 25(17), 20298–20312 (2017).
[Crossref] [PubMed]

M. I. Mishchenko, J. M. Dlugach, M. A. Yurkin, L. Bi, B. Cairns, L. Liu, R. L. Panetta, L. D. Travis, P. Yang, and N. T. Zakharova, “First-principles modeling of electromagnetic scattering by discrete and discretely heterogeneous random media,” Phys. Rep. 632, 1–75 (2016).
[Crossref]

L. Bi, P. Yang, C. Liu, B. Yi, and S. Hioki, “Optical properties of ice clouds: new modeling capabilities and relevant applications,” Proc. SPIE 9259, 92591A (2014).

L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Radiat. Transf. 138, 17–35 (2014).
[Crossref]

Borovoi, A.

Cairns, B.

M. I. Mishchenko, J. M. Dlugach, M. A. Yurkin, L. Bi, B. Cairns, L. Liu, R. L. Panetta, L. D. Travis, P. Yang, and N. T. Zakharova, “First-principles modeling of electromagnetic scattering by discrete and discretely heterogeneous random media,” Phys. Rep. 632, 1–75 (2016).
[Crossref]

Collier, C. T.

C. T. Collier, E. Hesse, L. Taylor, Z. Ulanowski, A. Penttilä, and T. Nousiainen, “Effects of surface roughness with two scales on light scattering by hexagonal ice crystals large compared to the wavelength: DDA results,” J. Quant. Spectrosc. Radiat. Transf. 182, 225–239 (2016).
[Crossref]

Corey, R.

R. Corey, M. Kissner, and P. Saulnier, “Coherent backscattering of light,” Am. J. Phys. 63(6), 560–564 (1995).
[Crossref]

Dlugach, J. M.

M. I. Mishchenko, J. M. Dlugach, M. A. Yurkin, L. Bi, B. Cairns, L. Liu, R. L. Panetta, L. D. Travis, P. Yang, and N. T. Zakharova, “First-principles modeling of electromagnetic scattering by discrete and discretely heterogeneous random media,” Phys. Rep. 632, 1–75 (2016).
[Crossref]

Dlugach, Z. M.

Z. M. Dlugach and M. I. Mishchenko, “Coherent backscattering and opposition effects observed in some atmosphereless bodies of the solar system,” Sol. Syst. Res. 47(6), 454–462 (2013).
[Crossref]

Gouesbet, G.

F. Xu, J. A. Lock, and G. Gouesbet, “Debye series for light scattering by a nonspherical particle,” Phys. Rev. A 81(4), 043824 (2010).
[Crossref]

Hesse, E.

C. T. Collier, E. Hesse, L. Taylor, Z. Ulanowski, A. Penttilä, and T. Nousiainen, “Effects of surface roughness with two scales on light scattering by hexagonal ice crystals large compared to the wavelength: DDA results,” J. Quant. Spectrosc. Radiat. Transf. 182, 225–239 (2016).
[Crossref]

Hioki, S.

L. Bi, P. Yang, C. Liu, B. Yi, and S. Hioki, “Optical properties of ice clouds: new modeling capabilities and relevant applications,” Proc. SPIE 9259, 92591A (2014).

Ishimaru, A.

Ishimoto, H.

K. Masuda and H. Ishimoto, “Backscatter ratios for nonspherical ice crystals in cirrus clouds calculated by geometrical -optics-integral-equation method,” J. Quant. Spectrosc. Radiat. Transf. 190, 60–68 (2017).
[Crossref]

Khare, V.

V. Khare and H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38(22), 1279–1282 (1977).
[Crossref]

Kissner, M.

R. Corey, M. Kissner, and P. Saulnier, “Coherent backscattering of light,” Am. J. Phys. 63(6), 560–564 (1995).
[Crossref]

Konoshonkin, A.

Kuga, Y.

Kustova, N.

Laven, P.

Lenke, R.

R. Lenke, U. Mack, and G. Maret, “Comparison of the ‘glory’ with coherent backscattering of light in turbid media,” J. Opt. A, Pure Appl. Opt. 4(3), 309–314 (2002).
[Crossref]

R. Lenke, R. Tweer, and G. Maret, “Coherent backscattering of turbid samples containing large Mie spheres,” J. Opt. A, Pure Appl. Opt. 4(3), 293–298 (2002).
[Crossref]

Lin, W.

Liu, C.

C. Liu and Y. Yin, “Inherent optical properties of pollen particles: a case study for the morning glory pollen,” Opt. Express 24(2), A104–A113 (2016).
[Crossref] [PubMed]

L. Bi, P. Yang, C. Liu, B. Yi, and S. Hioki, “Optical properties of ice clouds: new modeling capabilities and relevant applications,” Proc. SPIE 9259, 92591A (2014).

C. Liu, R. L. Panetta, and P. Yang, “The effective equivalence of geometric irregularity and surface roughness in determining particle single-scattering properties,” Opt. Express 22(19), 23620–23627 (2014).
[Crossref] [PubMed]

C. Liu, R. L. Panetta, and P. Yang, “The effects of surface roughness on the scattering properties of hexagonal columns with sizes from the Rayleigh to the geometric optics regimes,” J. Quant. Spectrosc. Radiat. Transf. 129, 169–185 (2013).
[Crossref]

Liu, D.

Liu, L.

M. I. Mishchenko, J. M. Dlugach, M. A. Yurkin, L. Bi, B. Cairns, L. Liu, R. L. Panetta, L. D. Travis, P. Yang, and N. T. Zakharova, “First-principles modeling of electromagnetic scattering by discrete and discretely heterogeneous random media,” Phys. Rep. 632, 1–75 (2016).
[Crossref]

Lock, J. A.

F. Xu, J. A. Lock, and G. Gouesbet, “Debye series for light scattering by a nonspherical particle,” Phys. Rev. A 81(4), 043824 (2010).
[Crossref]

Mack, U.

R. Lenke, U. Mack, and G. Maret, “Comparison of the ‘glory’ with coherent backscattering of light in turbid media,” J. Opt. A, Pure Appl. Opt. 4(3), 309–314 (2002).
[Crossref]

Maret, G.

R. Lenke, U. Mack, and G. Maret, “Comparison of the ‘glory’ with coherent backscattering of light in turbid media,” J. Opt. A, Pure Appl. Opt. 4(3), 309–314 (2002).
[Crossref]

R. Lenke, R. Tweer, and G. Maret, “Coherent backscattering of turbid samples containing large Mie spheres,” J. Opt. A, Pure Appl. Opt. 4(3), 293–298 (2002).
[Crossref]

P. E. Wolf and G. Maret, “Weak Localization and Coherent Backscattering of Photons in Disordered Media,” Phys. Rev. Lett. 55(24), 2696–2699 (1985).
[Crossref] [PubMed]

Masuda, K.

K. Masuda and H. Ishimoto, “Backscatter ratios for nonspherical ice crystals in cirrus clouds calculated by geometrical -optics-integral-equation method,” J. Quant. Spectrosc. Radiat. Transf. 190, 60–68 (2017).
[Crossref]

Maynard, R.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent Backscattering of Light by Disordered Media: Analysis of the Peak Line Shape,” Phys. Rev. Lett. 56(14), 1471–1474 (1986).
[Crossref] [PubMed]

Mie, G.

G. Mie, “Beitrage zur Optik trüber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 330(3), 377–445 (1908).
[Crossref]

Mishchenko, M. I.

M. I. Mishchenko, J. M. Dlugach, M. A. Yurkin, L. Bi, B. Cairns, L. Liu, R. L. Panetta, L. D. Travis, P. Yang, and N. T. Zakharova, “First-principles modeling of electromagnetic scattering by discrete and discretely heterogeneous random media,” Phys. Rep. 632, 1–75 (2016).
[Crossref]

Z. M. Dlugach and M. I. Mishchenko, “Coherent backscattering and opposition effects observed in some atmosphereless bodies of the solar system,” Sol. Syst. Res. 47(6), 454–462 (2013).
[Crossref]

M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8(6), 871–882 (1991).
[Crossref]

Nousiainen, T.

C. T. Collier, E. Hesse, L. Taylor, Z. Ulanowski, A. Penttilä, and T. Nousiainen, “Effects of surface roughness with two scales on light scattering by hexagonal ice crystals large compared to the wavelength: DDA results,” J. Quant. Spectrosc. Radiat. Transf. 182, 225–239 (2016).
[Crossref]

Nussenzveig, H. M.

V. Khare and H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38(22), 1279–1282 (1977).
[Crossref]

Panetta, R. L.

M. I. Mishchenko, J. M. Dlugach, M. A. Yurkin, L. Bi, B. Cairns, L. Liu, R. L. Panetta, L. D. Travis, P. Yang, and N. T. Zakharova, “First-principles modeling of electromagnetic scattering by discrete and discretely heterogeneous random media,” Phys. Rep. 632, 1–75 (2016).
[Crossref]

C. Liu, R. L. Panetta, and P. Yang, “The effective equivalence of geometric irregularity and surface roughness in determining particle single-scattering properties,” Opt. Express 22(19), 23620–23627 (2014).
[Crossref] [PubMed]

C. Liu, R. L. Panetta, and P. Yang, “The effects of surface roughness on the scattering properties of hexagonal columns with sizes from the Rayleigh to the geometric optics regimes,” J. Quant. Spectrosc. Radiat. Transf. 129, 169–185 (2013).
[Crossref]

Penttilä, A.

C. T. Collier, E. Hesse, L. Taylor, Z. Ulanowski, A. Penttilä, and T. Nousiainen, “Effects of surface roughness with two scales on light scattering by hexagonal ice crystals large compared to the wavelength: DDA results,” J. Quant. Spectrosc. Radiat. Transf. 182, 225–239 (2016).
[Crossref]

Saulnier, P.

R. Corey, M. Kissner, and P. Saulnier, “Coherent backscattering of light,” Am. J. Phys. 63(6), 560–564 (1995).
[Crossref]

Taylor, L.

C. T. Collier, E. Hesse, L. Taylor, Z. Ulanowski, A. Penttilä, and T. Nousiainen, “Effects of surface roughness with two scales on light scattering by hexagonal ice crystals large compared to the wavelength: DDA results,” J. Quant. Spectrosc. Radiat. Transf. 182, 225–239 (2016).
[Crossref]

Travis, L. D.

M. I. Mishchenko, J. M. Dlugach, M. A. Yurkin, L. Bi, B. Cairns, L. Liu, R. L. Panetta, L. D. Travis, P. Yang, and N. T. Zakharova, “First-principles modeling of electromagnetic scattering by discrete and discretely heterogeneous random media,” Phys. Rep. 632, 1–75 (2016).
[Crossref]

Tsang, L.

Tweer, R.

R. Lenke, R. Tweer, and G. Maret, “Coherent backscattering of turbid samples containing large Mie spheres,” J. Opt. A, Pure Appl. Opt. 4(3), 293–298 (2002).
[Crossref]

Ulanowski, Z.

C. T. Collier, E. Hesse, L. Taylor, Z. Ulanowski, A. Penttilä, and T. Nousiainen, “Effects of surface roughness with two scales on light scattering by hexagonal ice crystals large compared to the wavelength: DDA results,” J. Quant. Spectrosc. Radiat. Transf. 182, 225–239 (2016).
[Crossref]

Van De Hulst, H. C.

Wolf, P. E.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent Backscattering of Light by Disordered Media: Analysis of the Peak Line Shape,” Phys. Rev. Lett. 56(14), 1471–1474 (1986).
[Crossref] [PubMed]

P. E. Wolf and G. Maret, “Weak Localization and Coherent Backscattering of Photons in Disordered Media,” Phys. Rev. Lett. 55(24), 2696–2699 (1985).
[Crossref] [PubMed]

Xu, F.

F. Xu, J. A. Lock, and G. Gouesbet, “Debye series for light scattering by a nonspherical particle,” Phys. Rev. A 81(4), 043824 (2010).
[Crossref]

Yang, P.

M. I. Mishchenko, J. M. Dlugach, M. A. Yurkin, L. Bi, B. Cairns, L. Liu, R. L. Panetta, L. D. Travis, P. Yang, and N. T. Zakharova, “First-principles modeling of electromagnetic scattering by discrete and discretely heterogeneous random media,” Phys. Rep. 632, 1–75 (2016).
[Crossref]

C. Zhou and P. Yang, “Backscattering peak of ice cloud particles,” Opt. Express 23(9), 11995–12003 (2015).
[Crossref] [PubMed]

L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Radiat. Transf. 138, 17–35 (2014).
[Crossref]

C. Liu, R. L. Panetta, and P. Yang, “The effective equivalence of geometric irregularity and surface roughness in determining particle single-scattering properties,” Opt. Express 22(19), 23620–23627 (2014).
[Crossref] [PubMed]

L. Bi, P. Yang, C. Liu, B. Yi, and S. Hioki, “Optical properties of ice clouds: new modeling capabilities and relevant applications,” Proc. SPIE 9259, 92591A (2014).

C. Liu, R. L. Panetta, and P. Yang, “The effects of surface roughness on the scattering properties of hexagonal columns with sizes from the Rayleigh to the geometric optics regimes,” J. Quant. Spectrosc. Radiat. Transf. 129, 169–185 (2013).
[Crossref]

Yi, B.

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M. I. Mishchenko, J. M. Dlugach, M. A. Yurkin, L. Bi, B. Cairns, L. Liu, R. L. Panetta, L. D. Travis, P. Yang, and N. T. Zakharova, “First-principles modeling of electromagnetic scattering by discrete and discretely heterogeneous random media,” Phys. Rep. 632, 1–75 (2016).
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M. I. Mishchenko, J. M. Dlugach, M. A. Yurkin, L. Bi, B. Cairns, L. Liu, R. L. Panetta, L. D. Travis, P. Yang, and N. T. Zakharova, “First-principles modeling of electromagnetic scattering by discrete and discretely heterogeneous random media,” Phys. Rep. 632, 1–75 (2016).
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M. I. Mishchenko, V. K. Rosenbush, N. N. Kiselev, D. F. Lupishko, V. P. Tishkovets, V. G. Kaydash, I. N. Belskaya, Y. S. Efimov, and N. M. Shakhovskoy, “Polarimetric Remote Sensing of Solar System Objects,” (Akademperiodyka, 2010).

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M. I. Mishchenko, Electromagnetic Scattering by Particles and Particle Groups: An Introduction (Cambridge University, 2014).

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Figures (3)

Fig. 1
Fig. 1 Illustration of conjugate reversible sequences of elementary scatterers.
Fig. 2
Fig. 2 Scattering properties of roughened hexagonal particles (a-c), smooth hexagonal particles (d-f), spheres (g-i) and spheroids (j-l). The refractive index of the hexagons is 1.31 + 0.0i (ice crystals), while spheres and spheroids are calculated with refractive index of 1.33 + 0.0i (water droplets). Left column denotes the scattering phase function. Middle column shows the phase function near the backscattering direction normalized by the backscattering phase function. The right column denotes the angular width (half width at half height) of backscattering peak as a function of particle size parameter. The blue lines are for size parameter of 50, and the red lines are for size parameter of 100. Considering that the backscattering properties of spheres oscillate dramatically as a function of particle size due to lack of ensemble average (gray line), the phase functions for these spheres are averaged within each particle size interval to reduce noises. Note that the scattering angle θ = 180°-α.
Fig. 3
Fig. 3 Parallel and perpendicular components of light scattered by roughened hexagonal particles (a-b), smooth hexagonal particles (c-d), and spheroids (e-f), when the incident wave is linearly polarized with a Stokes vector of I0[1 1 0 0]T. Pij denotes element of the phase matrix.

Equations (42)

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× E ( r , t ) = i ω μ ( r ) H ( r , t ) ,
× H ( r , t ) = i ω ε ( r ) E ( r , t ) ,
× × E ( r , t ) k 1 2 E ( r , t ) = 0 , [ r V e x t ] , and
× × E ( r , t ) k 2 ( r ) 2 E ( r , t ) = 0 , [ r V int ] ,
× × E ( r , t ) k 1 2 E ( r , t ) = j ( r , t ) ,
j ( r , t ) = k 1 2 ( m ˜ 2 ( r ) 1 ) E ( r , t ) ,
m ˜ ( r ) = 1 , [ r V e x t ] ,
m ˜ ( r ) = m 2 ( r ) / m , 1 [ r V int ] .
× × E i n c ( r , t ) k 1 2 E i n c ( r , t ) = 0 , [ r V int V e x t ] ,
× × G ( r , r ' ) k 1 2 G ( r , r ' ) = I δ ( r r ' ) ,
× × [ G ( r , r ' ) j ( r ' , t ) ] k 1 2 [ G ( r , r ' ) j ( r ' , t ) ] = I j ( r ' , t ) δ ( r r ' ) ,
E s c a ( r , t ) = V int d r ' G ( r , r ' ) j ( r ' , t ) ,
E ( r , t ) = E i n c ( r , t ) + V int d r ' G ( r , r ' ) E ( r ' , t ) k 1 2 ( m ˜ 2 ( r ' ) 1 ) .
G ( r , r ' ) = ( I + 1 k 1 2 ) G ( r , r ' ) ,
G ( r , r ' ) = exp ( i k 1 | r r ' | ) 4 π | r r ' | .
E s c a ( r s , t ) = V int d r ' G ( r s , r ' ) E ( r ' , t ) k 1 2 ( m ˜ 2 ( r ' ) 1 ) ,
E s c a ( r s , t ) = V int d r 1 k 1 2 ( m ˜ 2 ( r 1 ) 1 ) G ( r s , r 1 ) [ E i n c ( r 1 , t ) + V int d r 2 k 1 2 ( m ˜ 2 ( r 2 ) 1 ) G ( r 1 , r 2 ) E i n c ( r 2 , t ) ] ,
E s c a ( r s , t ) = V int d r 1 k 1 2 ( m ˜ 2 ( r 1 ) 1 ) G ( r s , r 1 ) E i n c ( r 1 , t ) + n = 2 V int ... V int d r 1 ... d r n k 1 2 n i = 1 n ( m ˜ 2 ( r i ) 1 ) G ( r s , r 1 ) G ( r 1 , r 2 ) ... G ( r n 1 , r n ) E i n c ( r n , t ) ,
E s c a ( r s , t ) = E n c ( r s , t ) + E c 1 ( r s , t ) + E c 2 ( r s , t ) ,
E n c ( r s , t ) = V int d r 1 k 1 2 ( m ˜ 2 ( r 1 ) 1 ) G ( r s , r 1 ) E i n c ( r 1 , t ) + n = 2 V int ... V int , r 1 = r n d r 1 ... d r n k 1 2 n i = 1 n ( m ˜ 2 ( r i ) 1 ) G ( r s , r 1 ) ... G ( r n 1 , r n ) E i n c ( r n , t ) ,
E c 1 ( r s , t ) = n = 2 V int ... V int , | r n | > | r 1 | d r 1 ... d r n k 1 2 n i = 1 n ( m ˜ 2 ( r i ) 1 ) G ( r s , r 1 ) ... G ( r n 1 , r n ) E i n c ( r n , t ) ,
E c 2 ( r s , t ) = n = 2 V int ... V int , | r n | < | r 1 | d r 1 ... d r n k 1 2 n i = 1 n ( m ˜ 2 ( r i ) 1 ) G ( r s , r 1 ) ... G ( r n 1 , r n ) E i n c ( r n , t ) = n = 2 V int ... V int , | r n | > | r 1 | d r n ... d r 1 k 1 2 n i = n 1 ( m ˜ 2 ( r i ) 1 ) G ( r s , r n ) ... G ( r 2 , r 1 ) E i n c ( r 1 , t ) ,
G ( r i , r j ) = { ( 3 k 1 2 r i , j 2 3 i k 1 r i , j 1 ) r ^ i , j r ^ i , j + ( 1 + i k 1 r i , j 1 k 1 2 r i , j 2 ) I } G ( r i , r j )
r i , j = r j , i = | r i r j | ,
r ^ i , j = r ^ j , i = r i r j | r i r j | ,
I = r ^ s r ^ s + r ^ r ^ + r ^ r ^ ,
G ( r s , r i ) = ( r ^ r ^ + r ^ r ^ ) G ( r s , r i ) ,
E i n c ( r , t ) = r ^ 0 E 0 exp ( i k i r i ω t ) ,
r ^ 0 = r ^ r ^ , [ b a c k s c a t t e r ]
E c 1 , ( r s , t ) = n = 2 V int ... V int , | r n | > | r 1 | d r 1 ... d r n k 1 2 n i = 1 n ( m ˜ 2 ( r i ) 1 ) ( r ^ r ^ ) i = 1 n 1 { ( 3 k 1 2 r i , i + 1 2 3 i k 1 r i , i + 1 1 ) r ^ i , i + 1 r ^ i , i + 1 + ( 1 + i k 1 r i , i + 1 1 k 1 2 r i , i + 1 2 ) I } r ^ 0 E 0 exp ( i k i r n i ω t ) G ( r s , r 1 ) G ( r 1 , r 2 ) ... G ( r n 1 , r n ) ,
E c 2 , ( r s , t ) = n = 2 V int ... V int , | r n | > | r 1 | d r 1 ... d r n k 1 2 n i = 1 n ( m ˜ 2 ( r i ) 1 ) ( r ^ r ^ ) i = n 1 1 { ( 3 k 1 2 r i + 1 , i 2 3 i k 1 r i + 1 , i 1 ) r ^ i + 1 , i r ^ i + 1 , i + ( 1 + i k 1 r i + 1 , i 1 k 1 2 r i + 1 , i 2 ) I } r ^ 0 E 0 exp ( i k i r 1 i ω t ) G ( r s , r n ) G ( r n , r n 1 ) ... G ( r 2 , r 1 ) .
( a b ) c = a ( b c ) ,
( r ^ r ^ ) ( r ^ i 1 , i 1 + 1 r ^ i 1 , i 1 + 1 ) ( r ^ i 2 , i 2 + 1 r ^ i 2 , i 2 + 1 ) ... ( r ^ i n , i n + 1 r ^ i n , i n + 1 ) r ^ = ( r ^ r ^ ) ( r ^ i n + 1 , i n r ^ i n + 1 , i n ) ... ( r ^ i 2 , i 2 + 1 r ^ i 2 , i 2 + 1 ) ( r ^ i 1 + 1 , i 1 r ^ i 1 + 1 , i 1 ) r ^ , = r ^ ( r ^ r ^ i 1 , i 1 + 1 ) ( r ^ i 1 , i 1 + 1 r ^ i 2 , i 2 + 1 ) ... ( r ^ i n , i n + 1 r ^ ) .
( r ^ r ^ ) i = 1 n 1 { ( 3 k 1 2 r i , i + 1 2 3 i k 1 r i , i + 1 1 ) r ^ i , i + 1 r ^ i , i + 1 + ( 1 + i k r i , i + 1 1 k 2 r i , i + 1 2 ) I } r ^ 0 = ( r ^ r ^ ) i = n 1 1 { ( 3 k 1 2 r i + 1 , i 2 3 i k 1 r i + 1 , i 1 ) r ^ i + 1 , i r ^ i + 1 , i + ( 1 + i k r i + 1 , i 1 k 2 r i + 1 , i 2 ) I } r ^ 0 ,
E c 1 , ( r s , t ) + E c 2 , ( r s , t ) = n = 2 V int ... V int , | r n | > | r 1 | d r 1 ... d r n k 1 2 n i = 1 n ( m ˜ 2 ( r i ) 1 ) ( r ^ r ^ ) i = 1 n 1 { ( 3 k 1 2 r i , i + 1 2 3 i k 1 r i , i + 1 1 ) r ^ i , i + 1 r ^ i , i + 1 + ( 1 + i k r i , i + 1 1 k 2 r i , i + 1 2 ) I } r ^ 0 G ( r 1 , r 2 ) ... G ( r n 1 , r n ) E 0 [ exp ( i k i r n i ω t ) G ( r s , r 1 ) + exp ( i k i r 1 i ω t ) G ( r s , r n ) ] .
1 4 π | r s r n | 1 4 π | r s r 1 | 1 4 π | r s | ,
exp ( i k i r n i ω t ) G ( r s , r 1 ) + exp ( i k i r 1 i ω t ) G ( r s , r n ) = exp ( i k 1 | r s r 1 | + i k i r n i ω t ) 4 π | r s r 1 | + exp ( i k 1 | r s r n | + i k i r 1 i ω t ) 4 π | r s r n | = exp ( i k i r n i ω t ) G ( r s , r 1 ) [ 1 + exp ( i ( k i + k s ) ( r 1 r n ) ) ] ,
E c 1 , ( r s , t ) + E c 2 , ( r s , t ) = E c 1 , ( r s , t ) [ 1 + exp ( i ( k i + k s ) ( r 1 r n ) ) ] ,
ζ = 1 + cos [ ( k i + k s ) ( r 1 r n ) ] .
| E c 1 , ( r s , t ) + E c 2 , ( r s , t ) | 2 = | 2 E c 1 , ( r s , t ) | 2 = 2 ( | E c 1 , ( r s , t ) | 2 + | E c 2 , ( r s , t ) | 2 ) , [ b a c k s c a t t e r ]
| E c 1 , ( r s , t ) + E c 2 , ( r s , t ) | 2 < 2 ( | E c 1 , ( r s , t ) | 2 + | E c 2 , ( r s , t ) | 2 ) , [ o f f b a c k s c a t t e r ] .
max [ ( k i + k s ) ( r 1 r n ) ] = α k D = 2 π D λ α , [ α 0 ] ,

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