Abstract

The terms “independent” and “dependent” scattering are ubiquitous in the phenomenological discipline of light scattering by particulate media. Yet there is a wide range of ad hoc definitions of these terms, many of which are vague and conceptually inconsequential. In this paper we perform a first-principles analysis of these terms based on the rigorous volume-integral-equation formulation of electromagnetic scattering. We argue that scattering by a multi-particle group can be called independent if certain optical observables for the entire group can be expressed in appropriate single-particle observables. Otherwise one deals with the dependent scattering regime. The prime (and perhaps the only) examples of independent scattering are scattering scenarios described by the first-order-scattering approximation and the first-principles radiative transfer theory.

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References

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2018 (5)

B. X. Wang and C. Y. Zhao, “Effect of dependent scattering on light absorption in highly scattering random media,” Int. J. Heat Mass Transfer 125, 1069–1078 (2018).
[Crossref]

B. X. Wang and C. Y. Zhao, “Analysis of dependent scattering mechanism in hard-sphere Yukawa random media,” J. Appl. Phys. 123(22), 223101 (2018).
[Crossref]

M. A. Yurkin and M. I. Mishchenko, “Volume integral equation for electromagnetic scattering: rigorous derivation and analysis for a set of multi-layered particles with piecewise-smooth boundaries in a passive host medium,” Phys. Rev. A 97(4), 043824 (2018).
[Crossref]

M. I. Mishchenko and M. A. Yurkin, “Impressed sources and fields in the volume-integral-equation formulation of electromagnetic scattering by a finite object: a tutorial,” J. Quant. Spectrosc. Radiat. Transf. 214, 158–167 (2018).
[Crossref] [PubMed]

K. Muinonen, J. Markkanen, T. Väisänen, J. Peltoniemi, and A. Penttilä, “Multiple scattering of light in discrete random media using incoherent interactions,” Opt. Lett. 43(4), 683–686 (2018).
[Crossref] [PubMed]

2017 (3)

L. X. Ma, J. Y. Tan, J. M. Zhao, F. Q. Wang, and C. A. Wang, “Multiple and dependent scattering by densely packed discrete spheres: comparison of radiative transfer and Maxwell theory,” J. Quant. Spectrosc. Radiat. Transf. 187, 255–266 (2017).
[Crossref]

L. X. Ma, J. Y. Tan, J. M. Zhao, F. Q. Wang, C. A. Wang, and Y. Y. Wang, “Dependent scattering and absorption by densely packed discrete spherical particles: effects of complex refractive index,” J. Quant. Spectrosc. Radiat. Transf. 196, 94–102 (2017).
[Crossref]

M. I. Mishchenko, “Electromagnetic scattering and emission by a fixed multi-particle object in local thermal equilibrium: General formalism,” J. Quant. Spectrosc. Radiat. Transf. 200, 137–145 (2017).
[Crossref] [PubMed]

2016 (1)

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] [PubMed]

2014 (3)

2013 (5)

A. García-Valenzuela, H. Contreras-Tello, J. A. Olivares, and F. L. S. Cuppo, “Insights into the dependent-scattering contributions to the extinction coefficient in highly scattering suspensions,” J. Opt. Soc. Am. A 30(7), 1328–1334 (2013).
[Crossref] [PubMed]

M. I. Mishchenko, “Measurement of electromagnetic energy flow through a sparse particulate medium: a perspective,” J. Quant. Spectrosc. Radiat. Transf. 123, 122–134 (2013).
[Crossref]

V. P. Tishkovets and E. V. Petrova, “Light scattering by densely packed systems of particles: near-field effects,” Light Scattering Rev. 7, 3–36 (2013).

S. Fitzwater and J. W. Hook, “Response to “Dependent light scattering in white paint films: clarification and application of the theoretical concepts” [Auger, J-C., Stout, B., J. Coat. Technol. Res., DOI 10.1007/s11998-011-9731-9],” J. Coat. Technol. Res. 10(6), 923–927 (2013).
[Crossref]

J.-C. Auger and B. Stout, “Discussion on dependent light scattering phenomenon in white paint films,” J. Coat. Technol. Res. 10(6), 929–931 (2013).
[Crossref]

2012 (1)

J.-C. Auger and B. Stout, “Dependent light scattering in white paint films: clarification and application of the theoretical concepts,” J. Coat. Technol. Res. 9(3), 287–295 (2012).
[Crossref]

2011 (3)

O. Muñoz and J. W. Hovenier, “Laboratory measurements of single light scattering by ensembles of randomly oriented small irregular particles in air. a review,” J. Quant. Spectrosc. Radiat. Transf. 112(11), 1646–1657 (2011).
[Crossref]

V. P. Tishkovets, E. V. Petrova, and M. I. Mishchenko, “Scattering of electromagnetic waves by ensembles of particles and discrete random media,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2095–2127 (2011).
[Crossref]

M. I. Mishchenko, V. P. Tishkovets, L. D. Travis, B. Cairns, J. M. Dlugach, L. Liu, V. K. Rosenbush, and N. N. Kiselev, “Electromagnetic scattering by a morphologically complex object: fundamental concepts and common misconceptions,” J. Quant. Spectrosc. Radiat. Transf. 112(4), 671–692 (2011).
[Crossref]

2008 (1)

M. Francoeur and M. Pinar Mengüç, “Role of fluctuational electrodynamics in near-field radiative heat transfer,” J. Quant. Spectrosc. Radiat. Transf. 109(2), 280–293 (2008).
[Crossref]

2007 (1)

2003 (2)

G. Zaccanti, S. Del Bianco, and F. Martelli, “Measurements of optical properties of high-density media,” Appl. Opt. 42(19), 4023–4030 (2003).
[Crossref] [PubMed]

W. E. Vargas, “Optical properties of pigmented coatings taking into account particle interactions,” J. Quant. Spectrosc. Radiat. Transf. 78(2), 187–195 (2003).
[Crossref]

2001 (1)

2000 (1)

J.-C. Auger, B. Stout, and J. Lafait, “Dependent light scattering in dense heterogeneous media,” Physica B 279(1-3), 21–24 (2000).
[Crossref]

1997 (1)

J.-C. Simon, “Dependent scattering and radiative transfer in dense inhomogeneous media,” Physica A 241(1-2), 77–81 (1997).
[Crossref]

1996 (1)

Ž. Ivezić and M. P. Mengüç, “An investigation of dependent/independent scattering regimes using a discrete dipole approximation,” Int. J. Heat Mass Transfer 39(4), 811–822 (1996).
[Crossref]

1995 (1)

G. Göbel, J. Kuhn, and J. Fricke, “Dependent scattering effects in latex-sphere suspensions and scattering powders,” Waves Random Media 5(4), 413–426 (1995).
[Crossref]

1992 (1)

M. A. Al-Nimr and V. S. Arpaci, “Radiative properties of interacting particles,” J. Heat Transfer 114(4), 950–957 (1992).
[Crossref]

1991 (2)

1990 (2)

S. Kumar and C. L. Tien, “Dependent absorption and extinction of radiation by small particles,” J. Heat Transfer 112(1), 178–185 (1990).
[Crossref]

Y. Ma, V. K. Varadan, and V. V. Varadan, “Enhanced absorption due to dependent scattering,” J. Heat Transfer 112(2), 402–407 (1990).
[Crossref]

1988 (1)

C. L. Tien, “Thermal radiation in packed and fluidized beds,” J. Heat Transfer 110(4b), 1230–1242 (1988).
[Crossref]

1987 (2)

B. Drolen and C. L. Tien, “Independent and dependent scattering in packed-sphere systems,” J. Thermophys. Heat Transfer 1(1), 63–68 (1987).
[Crossref]

C. L. Tien and B. L. Drolen, “Thermal radiation in particulate media with dependent and independent scattering,” Annu. Rev. Heat Transfer 1(1), 1–32 (1987).
[Crossref]

1986 (2)

J. D. Cartigny, Y. Yamada, and C. L. Tien, “Radiative transfer with dependent scattering by particles: part 1 – Theoretical investigation,” J. Heat Transfer 108(3), 608–613 (1986).
[Crossref]

Y. Yamada, J. D. Cartigny, and C. L. Tien, “Radiative transfer with dependent scattering by particles: part 2 – Experimental investigation,” J. Heat Transfer 108(3), 614–618 (1986).
[Crossref]

1985 (1)

S. Fitzwater and J. W. Hook, “Dependent scattering theory: a new approach to predicting scattering in paints,” J. Coatings Technology 57, 39–47 (1985).

Aernouts, B.

Alexander, D. R.

Al-Nimr, M. A.

M. A. Al-Nimr and V. S. Arpaci, “Radiative properties of interacting particles,” J. Heat Transfer 114(4), 950–957 (1992).
[Crossref]

Arpaci, V. S.

M. A. Al-Nimr and V. S. Arpaci, “Radiative properties of interacting particles,” J. Heat Transfer 114(4), 950–957 (1992).
[Crossref]

Auger, J.-C.

J.-C. Auger and B. Stout, “Discussion on dependent light scattering phenomenon in white paint films,” J. Coat. Technol. Res. 10(6), 929–931 (2013).
[Crossref]

J.-C. Auger and B. Stout, “Dependent light scattering in white paint films: clarification and application of the theoretical concepts,” J. Coat. Technol. Res. 9(3), 287–295 (2012).
[Crossref]

J.-C. Auger, B. Stout, and J. Lafait, “Dependent light scattering in dense heterogeneous media,” Physica B 279(1-3), 21–24 (2000).
[Crossref]

Barton, J. P.

Bi, 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] [PubMed]

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] [PubMed]

M. I. Mishchenko, V. P. Tishkovets, L. D. Travis, B. Cairns, J. M. Dlugach, L. Liu, V. K. Rosenbush, and N. N. Kiselev, “Electromagnetic scattering by a morphologically complex object: fundamental concepts and common misconceptions,” J. Quant. Spectrosc. Radiat. Transf. 112(4), 671–692 (2011).
[Crossref]

M. I. Mishchenko, L. Liu, D. W. Mackowski, B. Cairns, and G. Videen, “Multiple scattering by random particulate media: exact 3D results,” Opt. Express 15(6), 2822–2836 (2007).
[Crossref] [PubMed]

Cartigny, J. D.

J. D. Cartigny, Y. Yamada, and C. L. Tien, “Radiative transfer with dependent scattering by particles: part 1 – Theoretical investigation,” J. Heat Transfer 108(3), 608–613 (1986).
[Crossref]

Y. Yamada, J. D. Cartigny, and C. L. Tien, “Radiative transfer with dependent scattering by particles: part 2 – Experimental investigation,” J. Heat Transfer 108(3), 614–618 (1986).
[Crossref]

Contreras-Tello, H.

Cuppo, F. L. S.

Del Bianco, S.

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] [PubMed]

M. I. Mishchenko, V. P. Tishkovets, L. D. Travis, B. Cairns, J. M. Dlugach, L. Liu, V. K. Rosenbush, and N. N. Kiselev, “Electromagnetic scattering by a morphologically complex object: fundamental concepts and common misconceptions,” J. Quant. Spectrosc. Radiat. Transf. 112(4), 671–692 (2011).
[Crossref]

Dolgos, G.

Drolen, B.

B. Drolen and C. L. Tien, “Independent and dependent scattering in packed-sphere systems,” J. Thermophys. Heat Transfer 1(1), 63–68 (1987).
[Crossref]

Drolen, B. L.

C. L. Tien and B. L. Drolen, “Thermal radiation in particulate media with dependent and independent scattering,” Annu. Rev. Heat Transfer 1(1), 1–32 (1987).
[Crossref]

Fitzwater, S.

S. Fitzwater and J. W. Hook, “Response to “Dependent light scattering in white paint films: clarification and application of the theoretical concepts” [Auger, J-C., Stout, B., J. Coat. Technol. Res., DOI 10.1007/s11998-011-9731-9],” J. Coat. Technol. Res. 10(6), 923–927 (2013).
[Crossref]

S. Fitzwater and J. W. Hook, “Dependent scattering theory: a new approach to predicting scattering in paints,” J. Coatings Technology 57, 39–47 (1985).

Francoeur, M.

M. Francoeur and M. Pinar Mengüç, “Role of fluctuational electrodynamics in near-field radiative heat transfer,” J. Quant. Spectrosc. Radiat. Transf. 109(2), 280–293 (2008).
[Crossref]

Fricke, J.

G. Göbel, J. Kuhn, and J. Fricke, “Dependent scattering effects in latex-sphere suspensions and scattering powders,” Waves Random Media 5(4), 413–426 (1995).
[Crossref]

Fuller, K. A.

García-Valenzuela, A.

Göbel, G.

G. Göbel, J. Kuhn, and J. Fricke, “Dependent scattering effects in latex-sphere suspensions and scattering powders,” Waves Random Media 5(4), 413–426 (1995).
[Crossref]

Greffet, J.-J.

Hespel, L.

Hook, J. W.

S. Fitzwater and J. W. Hook, “Response to “Dependent light scattering in white paint films: clarification and application of the theoretical concepts” [Auger, J-C., Stout, B., J. Coat. Technol. Res., DOI 10.1007/s11998-011-9731-9],” J. Coat. Technol. Res. 10(6), 923–927 (2013).
[Crossref]

S. Fitzwater and J. W. Hook, “Dependent scattering theory: a new approach to predicting scattering in paints,” J. Coatings Technology 57, 39–47 (1985).

Hovenier, J. W.

O. Muñoz and J. W. Hovenier, “Laboratory measurements of single light scattering by ensembles of randomly oriented small irregular particles in air. a review,” J. Quant. Spectrosc. Radiat. Transf. 112(11), 1646–1657 (2011).
[Crossref]

Ivezic, Ž.

Ž. Ivezić and M. P. Mengüç, “An investigation of dependent/independent scattering regimes using a discrete dipole approximation,” Int. J. Heat Mass Transfer 39(4), 811–822 (1996).
[Crossref]

Kiselev, N. N.

M. I. Mishchenko, V. P. Tishkovets, L. D. Travis, B. Cairns, J. M. Dlugach, L. Liu, V. K. Rosenbush, and N. N. Kiselev, “Electromagnetic scattering by a morphologically complex object: fundamental concepts and common misconceptions,” J. Quant. Spectrosc. Radiat. Transf. 112(4), 671–692 (2011).
[Crossref]

Kuhn, J.

G. Göbel, J. Kuhn, and J. Fricke, “Dependent scattering effects in latex-sphere suspensions and scattering powders,” Waves Random Media 5(4), 413–426 (1995).
[Crossref]

Kumar, S.

S. Kumar and C. L. Tien, “Dependent absorption and extinction of radiation by small particles,” J. Heat Transfer 112(1), 178–185 (1990).
[Crossref]

Lafait, J.

J.-C. Auger, B. Stout, and J. Lafait, “Dependent light scattering in dense heterogeneous media,” Physica B 279(1-3), 21–24 (2000).
[Crossref]

Lammertyn, J.

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] [PubMed]

M. I. Mishchenko, V. P. Tishkovets, L. D. Travis, B. Cairns, J. M. Dlugach, L. Liu, V. K. Rosenbush, and N. N. Kiselev, “Electromagnetic scattering by a morphologically complex object: fundamental concepts and common misconceptions,” J. Quant. Spectrosc. Radiat. Transf. 112(4), 671–692 (2011).
[Crossref]

M. I. Mishchenko, L. Liu, D. W. Mackowski, B. Cairns, and G. Videen, “Multiple scattering by random particulate media: exact 3D results,” Opt. Express 15(6), 2822–2836 (2007).
[Crossref] [PubMed]

Ma, L. X.

L. X. Ma, J. Y. Tan, J. M. Zhao, F. Q. Wang, and C. A. Wang, “Multiple and dependent scattering by densely packed discrete spheres: comparison of radiative transfer and Maxwell theory,” J. Quant. Spectrosc. Radiat. Transf. 187, 255–266 (2017).
[Crossref]

L. X. Ma, J. Y. Tan, J. M. Zhao, F. Q. Wang, C. A. Wang, and Y. Y. Wang, “Dependent scattering and absorption by densely packed discrete spherical particles: effects of complex refractive index,” J. Quant. Spectrosc. Radiat. Transf. 196, 94–102 (2017).
[Crossref]

Ma, W.

Ma, Y.

Y. Ma, V. K. Varadan, and V. V. Varadan, “Enhanced absorption due to dependent scattering,” J. Heat Transfer 112(2), 402–407 (1990).
[Crossref]

Mackowski, D. W.

Mainguy, S.

Markkanen, J.

Martelli, F.

Martins, J. V.

Mengüç, M. P.

Ž. Ivezić and M. P. Mengüç, “An investigation of dependent/independent scattering regimes using a discrete dipole approximation,” Int. J. Heat Mass Transfer 39(4), 811–822 (1996).
[Crossref]

Mishchenko, M. I.

M. A. Yurkin and M. I. Mishchenko, “Volume integral equation for electromagnetic scattering: rigorous derivation and analysis for a set of multi-layered particles with piecewise-smooth boundaries in a passive host medium,” Phys. Rev. A 97(4), 043824 (2018).
[Crossref]

M. I. Mishchenko and M. A. Yurkin, “Impressed sources and fields in the volume-integral-equation formulation of electromagnetic scattering by a finite object: a tutorial,” J. Quant. Spectrosc. Radiat. Transf. 214, 158–167 (2018).
[Crossref] [PubMed]

M. I. Mishchenko, “Electromagnetic scattering and emission by a fixed multi-particle object in local thermal equilibrium: General formalism,” J. Quant. Spectrosc. Radiat. Transf. 200, 137–145 (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] [PubMed]

M. I. Mishchenko, “Directional radiometry and radiative transfer: the convoluted path from centuries-old phenomenology to physical optics,” J. Quant. Spectrosc. Radiat. Transf. 146, 4–33 (2014).
[Crossref]

M. I. Mishchenko, “Measurement of electromagnetic energy flow through a sparse particulate medium: a perspective,” J. Quant. Spectrosc. Radiat. Transf. 123, 122–134 (2013).
[Crossref]

M. I. Mishchenko, V. P. Tishkovets, L. D. Travis, B. Cairns, J. M. Dlugach, L. Liu, V. K. Rosenbush, and N. N. Kiselev, “Electromagnetic scattering by a morphologically complex object: fundamental concepts and common misconceptions,” J. Quant. Spectrosc. Radiat. Transf. 112(4), 671–692 (2011).
[Crossref]

V. P. Tishkovets, E. V. Petrova, and M. I. Mishchenko, “Scattering of electromagnetic waves by ensembles of particles and discrete random media,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2095–2127 (2011).
[Crossref]

M. I. Mishchenko, L. Liu, D. W. Mackowski, B. Cairns, and G. Videen, “Multiple scattering by random particulate media: exact 3D results,” Opt. Express 15(6), 2822–2836 (2007).
[Crossref] [PubMed]

Muinonen, K.

Muñoz, O.

O. Muñoz and J. W. Hovenier, “Laboratory measurements of single light scattering by ensembles of randomly oriented small irregular particles in air. a review,” J. Quant. Spectrosc. Radiat. Transf. 112(11), 1646–1657 (2011).
[Crossref]

Olivares, J. A.

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] [PubMed]

Peltoniemi, J.

Penttilä, A.

Petrova, E. V.

V. P. Tishkovets and E. V. Petrova, “Light scattering by densely packed systems of particles: near-field effects,” Light Scattering Rev. 7, 3–36 (2013).

V. P. Tishkovets, E. V. Petrova, and M. I. Mishchenko, “Scattering of electromagnetic waves by ensembles of particles and discrete random media,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2095–2127 (2011).
[Crossref]

Pinar Mengüç, M.

M. Francoeur and M. Pinar Mengüç, “Role of fluctuational electrodynamics in near-field radiative heat transfer,” J. Quant. Spectrosc. Radiat. Transf. 109(2), 280–293 (2008).
[Crossref]

Rosenbush, V. K.

M. I. Mishchenko, V. P. Tishkovets, L. D. Travis, B. Cairns, J. M. Dlugach, L. Liu, V. K. Rosenbush, and N. N. Kiselev, “Electromagnetic scattering by a morphologically complex object: fundamental concepts and common misconceptions,” J. Quant. Spectrosc. Radiat. Transf. 112(4), 671–692 (2011).
[Crossref]

Saeys, W.

Schaub, S. A.

Simon, J.-C.

J.-C. Simon, “Dependent scattering and radiative transfer in dense inhomogeneous media,” Physica A 241(1-2), 77–81 (1997).
[Crossref]

Stout, B.

J.-C. Auger and B. Stout, “Discussion on dependent light scattering phenomenon in white paint films,” J. Coat. Technol. Res. 10(6), 929–931 (2013).
[Crossref]

J.-C. Auger and B. Stout, “Dependent light scattering in white paint films: clarification and application of the theoretical concepts,” J. Coat. Technol. Res. 9(3), 287–295 (2012).
[Crossref]

J.-C. Auger, B. Stout, and J. Lafait, “Dependent light scattering in dense heterogeneous media,” Physica B 279(1-3), 21–24 (2000).
[Crossref]

Tan, J. Y.

L. X. Ma, J. Y. Tan, J. M. Zhao, F. Q. Wang, and C. A. Wang, “Multiple and dependent scattering by densely packed discrete spheres: comparison of radiative transfer and Maxwell theory,” J. Quant. Spectrosc. Radiat. Transf. 187, 255–266 (2017).
[Crossref]

L. X. Ma, J. Y. Tan, J. M. Zhao, F. Q. Wang, C. A. Wang, and Y. Y. Wang, “Dependent scattering and absorption by densely packed discrete spherical particles: effects of complex refractive index,” J. Quant. Spectrosc. Radiat. Transf. 196, 94–102 (2017).
[Crossref]

Tien, C. L.

S. Kumar and C. L. Tien, “Dependent absorption and extinction of radiation by small particles,” J. Heat Transfer 112(1), 178–185 (1990).
[Crossref]

C. L. Tien, “Thermal radiation in packed and fluidized beds,” J. Heat Transfer 110(4b), 1230–1242 (1988).
[Crossref]

B. Drolen and C. L. Tien, “Independent and dependent scattering in packed-sphere systems,” J. Thermophys. Heat Transfer 1(1), 63–68 (1987).
[Crossref]

C. L. Tien and B. L. Drolen, “Thermal radiation in particulate media with dependent and independent scattering,” Annu. Rev. Heat Transfer 1(1), 1–32 (1987).
[Crossref]

Y. Yamada, J. D. Cartigny, and C. L. Tien, “Radiative transfer with dependent scattering by particles: part 2 – Experimental investigation,” J. Heat Transfer 108(3), 614–618 (1986).
[Crossref]

J. D. Cartigny, Y. Yamada, and C. L. Tien, “Radiative transfer with dependent scattering by particles: part 1 – Theoretical investigation,” J. Heat Transfer 108(3), 608–613 (1986).
[Crossref]

Tishkovets, V. P.

V. P. Tishkovets and E. V. Petrova, “Light scattering by densely packed systems of particles: near-field effects,” Light Scattering Rev. 7, 3–36 (2013).

V. P. Tishkovets, E. V. Petrova, and M. I. Mishchenko, “Scattering of electromagnetic waves by ensembles of particles and discrete random media,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2095–2127 (2011).
[Crossref]

M. I. Mishchenko, V. P. Tishkovets, L. D. Travis, B. Cairns, J. M. Dlugach, L. Liu, V. K. Rosenbush, and N. N. Kiselev, “Electromagnetic scattering by a morphologically complex object: fundamental concepts and common misconceptions,” J. Quant. Spectrosc. Radiat. Transf. 112(4), 671–692 (2011).
[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] [PubMed]

M. I. Mishchenko, V. P. Tishkovets, L. D. Travis, B. Cairns, J. M. Dlugach, L. Liu, V. K. Rosenbush, and N. N. Kiselev, “Electromagnetic scattering by a morphologically complex object: fundamental concepts and common misconceptions,” J. Quant. Spectrosc. Radiat. Transf. 112(4), 671–692 (2011).
[Crossref]

Väisänen, T.

Van Beers, R.

Varadan, V. K.

Y. Ma, V. K. Varadan, and V. V. Varadan, “Enhanced absorption due to dependent scattering,” J. Heat Transfer 112(2), 402–407 (1990).
[Crossref]

Varadan, V. V.

Y. Ma, V. K. Varadan, and V. V. Varadan, “Enhanced absorption due to dependent scattering,” J. Heat Transfer 112(2), 402–407 (1990).
[Crossref]

Vargas, W. E.

W. E. Vargas, “Optical properties of pigmented coatings taking into account particle interactions,” J. Quant. Spectrosc. Radiat. Transf. 78(2), 187–195 (2003).
[Crossref]

Videen, G.

Wang, B. X.

B. X. Wang and C. Y. Zhao, “Analysis of dependent scattering mechanism in hard-sphere Yukawa random media,” J. Appl. Phys. 123(22), 223101 (2018).
[Crossref]

B. X. Wang and C. Y. Zhao, “Effect of dependent scattering on light absorption in highly scattering random media,” Int. J. Heat Mass Transfer 125, 1069–1078 (2018).
[Crossref]

Wang, C. A.

L. X. Ma, J. Y. Tan, J. M. Zhao, F. Q. Wang, C. A. Wang, and Y. Y. Wang, “Dependent scattering and absorption by densely packed discrete spherical particles: effects of complex refractive index,” J. Quant. Spectrosc. Radiat. Transf. 196, 94–102 (2017).
[Crossref]

L. X. Ma, J. Y. Tan, J. M. Zhao, F. Q. Wang, and C. A. Wang, “Multiple and dependent scattering by densely packed discrete spheres: comparison of radiative transfer and Maxwell theory,” J. Quant. Spectrosc. Radiat. Transf. 187, 255–266 (2017).
[Crossref]

Wang, F. Q.

L. X. Ma, J. Y. Tan, J. M. Zhao, F. Q. Wang, C. A. Wang, and Y. Y. Wang, “Dependent scattering and absorption by densely packed discrete spherical particles: effects of complex refractive index,” J. Quant. Spectrosc. Radiat. Transf. 196, 94–102 (2017).
[Crossref]

L. X. Ma, J. Y. Tan, J. M. Zhao, F. Q. Wang, and C. A. Wang, “Multiple and dependent scattering by densely packed discrete spheres: comparison of radiative transfer and Maxwell theory,” J. Quant. Spectrosc. Radiat. Transf. 187, 255–266 (2017).
[Crossref]

Wang, Y. Y.

L. X. Ma, J. Y. Tan, J. M. Zhao, F. Q. Wang, C. A. Wang, and Y. Y. Wang, “Dependent scattering and absorption by densely packed discrete spherical particles: effects of complex refractive index,” J. Quant. Spectrosc. Radiat. Transf. 196, 94–102 (2017).
[Crossref]

Watté, R.

Yamada, Y.

Y. Yamada, J. D. Cartigny, and C. L. Tien, “Radiative transfer with dependent scattering by particles: part 2 – Experimental investigation,” J. Heat Transfer 108(3), 614–618 (1986).
[Crossref]

J. D. Cartigny, Y. Yamada, and C. L. Tien, “Radiative transfer with dependent scattering by particles: part 1 – Theoretical investigation,” J. Heat Transfer 108(3), 608–613 (1986).
[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] [PubMed]

Yurkin, M. A.

M. I. Mishchenko and M. A. Yurkin, “Impressed sources and fields in the volume-integral-equation formulation of electromagnetic scattering by a finite object: a tutorial,” J. Quant. Spectrosc. Radiat. Transf. 214, 158–167 (2018).
[Crossref] [PubMed]

M. A. Yurkin and M. I. Mishchenko, “Volume integral equation for electromagnetic scattering: rigorous derivation and analysis for a set of multi-layered particles with piecewise-smooth boundaries in a passive host medium,” Phys. Rev. A 97(4), 043824 (2018).
[Crossref]

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] [PubMed]

Zaccanti, G.

Zakharova, N. T.

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] [PubMed]

Zhao, C. Y.

B. X. Wang and C. Y. Zhao, “Analysis of dependent scattering mechanism in hard-sphere Yukawa random media,” J. Appl. Phys. 123(22), 223101 (2018).
[Crossref]

B. X. Wang and C. Y. Zhao, “Effect of dependent scattering on light absorption in highly scattering random media,” Int. J. Heat Mass Transfer 125, 1069–1078 (2018).
[Crossref]

Zhao, J. M.

L. X. Ma, J. Y. Tan, J. M. Zhao, F. Q. Wang, C. A. Wang, and Y. Y. Wang, “Dependent scattering and absorption by densely packed discrete spherical particles: effects of complex refractive index,” J. Quant. Spectrosc. Radiat. Transf. 196, 94–102 (2017).
[Crossref]

L. X. Ma, J. Y. Tan, J. M. Zhao, F. Q. Wang, and C. A. Wang, “Multiple and dependent scattering by densely packed discrete spheres: comparison of radiative transfer and Maxwell theory,” J. Quant. Spectrosc. Radiat. Transf. 187, 255–266 (2017).
[Crossref]

Annu. Rev. Heat Transfer (1)

C. L. Tien and B. L. Drolen, “Thermal radiation in particulate media with dependent and independent scattering,” Annu. Rev. Heat Transfer 1(1), 1–32 (1987).
[Crossref]

Appl. Opt. (3)

Int. J. Heat Mass Transfer (2)

B. X. Wang and C. Y. Zhao, “Effect of dependent scattering on light absorption in highly scattering random media,” Int. J. Heat Mass Transfer 125, 1069–1078 (2018).
[Crossref]

Ž. Ivezić and M. P. Mengüç, “An investigation of dependent/independent scattering regimes using a discrete dipole approximation,” Int. J. Heat Mass Transfer 39(4), 811–822 (1996).
[Crossref]

J. Appl. Phys. (1)

B. X. Wang and C. Y. Zhao, “Analysis of dependent scattering mechanism in hard-sphere Yukawa random media,” J. Appl. Phys. 123(22), 223101 (2018).
[Crossref]

J. Coat. Technol. Res. (3)

S. Fitzwater and J. W. Hook, “Response to “Dependent light scattering in white paint films: clarification and application of the theoretical concepts” [Auger, J-C., Stout, B., J. Coat. Technol. Res., DOI 10.1007/s11998-011-9731-9],” J. Coat. Technol. Res. 10(6), 923–927 (2013).
[Crossref]

J.-C. Auger and B. Stout, “Discussion on dependent light scattering phenomenon in white paint films,” J. Coat. Technol. Res. 10(6), 929–931 (2013).
[Crossref]

J.-C. Auger and B. Stout, “Dependent light scattering in white paint films: clarification and application of the theoretical concepts,” J. Coat. Technol. Res. 9(3), 287–295 (2012).
[Crossref]

J. Coatings Technology (1)

S. Fitzwater and J. W. Hook, “Dependent scattering theory: a new approach to predicting scattering in paints,” J. Coatings Technology 57, 39–47 (1985).

J. Heat Transfer (6)

J. D. Cartigny, Y. Yamada, and C. L. Tien, “Radiative transfer with dependent scattering by particles: part 1 – Theoretical investigation,” J. Heat Transfer 108(3), 608–613 (1986).
[Crossref]

Y. Yamada, J. D. Cartigny, and C. L. Tien, “Radiative transfer with dependent scattering by particles: part 2 – Experimental investigation,” J. Heat Transfer 108(3), 614–618 (1986).
[Crossref]

C. L. Tien, “Thermal radiation in packed and fluidized beds,” J. Heat Transfer 110(4b), 1230–1242 (1988).
[Crossref]

S. Kumar and C. L. Tien, “Dependent absorption and extinction of radiation by small particles,” J. Heat Transfer 112(1), 178–185 (1990).
[Crossref]

Y. Ma, V. K. Varadan, and V. V. Varadan, “Enhanced absorption due to dependent scattering,” J. Heat Transfer 112(2), 402–407 (1990).
[Crossref]

M. A. Al-Nimr and V. S. Arpaci, “Radiative properties of interacting particles,” J. Heat Transfer 114(4), 950–957 (1992).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Quant. Spectrosc. Radiat. Transf. (11)

W. E. Vargas, “Optical properties of pigmented coatings taking into account particle interactions,” J. Quant. Spectrosc. Radiat. Transf. 78(2), 187–195 (2003).
[Crossref]

V. P. Tishkovets, E. V. Petrova, and M. I. Mishchenko, “Scattering of electromagnetic waves by ensembles of particles and discrete random media,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2095–2127 (2011).
[Crossref]

L. X. Ma, J. Y. Tan, J. M. Zhao, F. Q. Wang, and C. A. Wang, “Multiple and dependent scattering by densely packed discrete spheres: comparison of radiative transfer and Maxwell theory,” J. Quant. Spectrosc. Radiat. Transf. 187, 255–266 (2017).
[Crossref]

L. X. Ma, J. Y. Tan, J. M. Zhao, F. Q. Wang, C. A. Wang, and Y. Y. Wang, “Dependent scattering and absorption by densely packed discrete spherical particles: effects of complex refractive index,” J. Quant. Spectrosc. Radiat. Transf. 196, 94–102 (2017).
[Crossref]

M. I. Mishchenko and M. A. Yurkin, “Impressed sources and fields in the volume-integral-equation formulation of electromagnetic scattering by a finite object: a tutorial,” J. Quant. Spectrosc. Radiat. Transf. 214, 158–167 (2018).
[Crossref] [PubMed]

M. I. Mishchenko, V. P. Tishkovets, L. D. Travis, B. Cairns, J. M. Dlugach, L. Liu, V. K. Rosenbush, and N. N. Kiselev, “Electromagnetic scattering by a morphologically complex object: fundamental concepts and common misconceptions,” J. Quant. Spectrosc. Radiat. Transf. 112(4), 671–692 (2011).
[Crossref]

M. I. Mishchenko, “Measurement of electromagnetic energy flow through a sparse particulate medium: a perspective,” J. Quant. Spectrosc. Radiat. Transf. 123, 122–134 (2013).
[Crossref]

O. Muñoz and J. W. Hovenier, “Laboratory measurements of single light scattering by ensembles of randomly oriented small irregular particles in air. a review,” J. Quant. Spectrosc. Radiat. Transf. 112(11), 1646–1657 (2011).
[Crossref]

M. I. Mishchenko, “Directional radiometry and radiative transfer: the convoluted path from centuries-old phenomenology to physical optics,” J. Quant. Spectrosc. Radiat. Transf. 146, 4–33 (2014).
[Crossref]

M. Francoeur and M. Pinar Mengüç, “Role of fluctuational electrodynamics in near-field radiative heat transfer,” J. Quant. Spectrosc. Radiat. Transf. 109(2), 280–293 (2008).
[Crossref]

M. I. Mishchenko, “Electromagnetic scattering and emission by a fixed multi-particle object in local thermal equilibrium: General formalism,” J. Quant. Spectrosc. Radiat. Transf. 200, 137–145 (2017).
[Crossref] [PubMed]

J. Thermophys. Heat Transfer (1)

B. Drolen and C. L. Tien, “Independent and dependent scattering in packed-sphere systems,” J. Thermophys. Heat Transfer 1(1), 63–68 (1987).
[Crossref]

Light Scattering Rev. (1)

V. P. Tishkovets and E. V. Petrova, “Light scattering by densely packed systems of particles: near-field effects,” Light Scattering Rev. 7, 3–36 (2013).

Opt. Express (3)

Opt. Lett. (1)

Phys. Rep. (1)

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] [PubMed]

Phys. Rev. A (1)

M. A. Yurkin and M. I. Mishchenko, “Volume integral equation for electromagnetic scattering: rigorous derivation and analysis for a set of multi-layered particles with piecewise-smooth boundaries in a passive host medium,” Phys. Rev. A 97(4), 043824 (2018).
[Crossref]

Physica A (1)

J.-C. Simon, “Dependent scattering and radiative transfer in dense inhomogeneous media,” Physica A 241(1-2), 77–81 (1997).
[Crossref]

Physica B (1)

J.-C. Auger, B. Stout, and J. Lafait, “Dependent light scattering in dense heterogeneous media,” Physica B 279(1-3), 21–24 (2000).
[Crossref]

Waves Random Media (1)

G. Göbel, J. Kuhn, and J. Fricke, “Dependent scattering effects in latex-sphere suspensions and scattering powders,” Waves Random Media 5(4), 413–426 (1995).
[Crossref]

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L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, “Concepts, terms, notation,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, 2000), pp. 3–27.

A. P. Ivanov, V. A. Loiko, and V. P. Dik, Propagation of Light in Densely Packed Disperse Media (Nauka i Tekhnika, 1988) (in Russian).

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

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (Wiley, 2001).

J. R. Howell, R. Siegel, and M. P. Mengüç, Thermal Radiation Heat Transfer (CRC Press, 2011).

L. A. Dombrovsky and D. Baillis, Thermal Radiation in Disperse Systems: An Engineering Approach (Begell House, 2010).

M. F. Modest, Radiative Heat Transfer (Academic, 2013).

G. Kristensson, Scattering of Electromagnetic Waves by Obstacles (Scitech Publishing, 2016).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge, 2002), https://www.giss.nasa.gov/staff/mmishchenko/books.html .

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge, 2006), https://www.giss.nasa.gov/staff/mmishchenko/books.html .

J. C. Dainty, Laser Speckle and Related Phenomena (Springer, 1975).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts & Company, 2007).

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

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

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Figures (7)

Fig. 1
Fig. 1 Electromagnetic scattering by an arbitrary fixed finite object.
Fig. 2
Fig. 2 Far-field scattering by a single particle.
Fig. 3
Fig. 3 Notation used in Eq. (17).
Fig. 4
Fig. 4 Origin of the speckle pattern.
Fig. 5
Fig. 5 The net polarized signal recorded by a WCR depends on the line of sight.
Fig. 6
Fig. 6 A random group of N particles populating the volume V is observed from a large distance.
Fig. 7
Fig. 7 Sparse discrete random medium. The sizes of the particles and the WCR are exaggerated for demonstration purposes.

Equations (39)

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E(r)= E inc (r)+ E sca (r),   r 3 ,
E sca (r)= V INT d 3 r G (r, r ) V INT d 3 r T ( r , r ) E inc ( r ) .
V INT i=1 N   V INT i .
T (r, r )=U(r)δ(r r ) I +U(r) V INT d 3 r G (r, r ) T ( r ,  r ) ,  r, r V INT ,
U(r){ 0,   r 3 \ V INT , ω 2 ε 2 (r) μ 0 k 1 2 ,   r V INT
B ^ E 3 d 3 r B (r, r )E( r ) ,( B ^ C ^ )E B ^ ( C ^ E ),
E= E inc + E sca = E inc + G ^ T ^ E inc ,
T ^ = U ^ + U ^ G ^ T ^ ,
U (r, r )= i=1 N U i (r, r ),    r 3 ,
U i (r, r ){ 0 ,r 3 \ V INT i , U(r)δ(r r ) I ,    r V INT i .
T ^ i = U ^ i + U ^ i G ^ T ^ i ,i=1,...,N
T ^ = i=1 N T ^ i +     i=1 j(i)=1 N T ^ i G ^ T ^ j +     i=1 j(i)=1 l(j)=1 N T ^ i G ^ T ^ j G ^ T ^ l +.
E= E inc + i=1 N G ^ T ^ i E inc +     i=1 j(i)=1 N G ^ T ^ i G ^ T ^ j E inc +     i=1 j(i)=1 l(j)=1 N G ^ T ^ i G ^ T ^ j G ^ T ^ l E inc +.
E inc (r)=exp(i k 1 n ^ inc r) E 0 inc , E 0 inc n ^ inc =0.
E sca (r) = r g(r) A ( r ^ , n ^ inc ) E 0 inc , E sca ( r ^ ) r ^ =0,
A ( r ^ , n ^ inc )= 1 4π ( I r ^ r ^ ) V INT d 3 r exp(i k 1 r ^ r ) × V INT d 3 r T ( r , r )( I n ^ inc n ^ inc ) exp(i k 1 n ^ inc r ),
E(r)= E inc (r)+ i=1 N g( r i ) A i ( r ^ i , n ^ inc ) E inc ( R i ) + i=1 N j(i)=1 N g( r i ) A i ( r ^ i , R ^ ij )g( R ij ) A j ( R ^ ij , n ^ inc ) E inc ( R j ) + i=1 N j(i)=1 N l(j)=1 N g( r i ) A i ( r ^ i , R ^ ij )g( R ij ) A j ( R ^ ij , R ^ jl ) g( R jl ) A l ( R ^ jl , n ^ inc ) E inc ( R l )+.
E sca (r)= i=1 N g( r i ) A i ( r ^ i , n ^ inc ) E inc ( R i ) .
ρ sca (r)= i=1 N ρ i sca (r)+ i=1 N j(i)=1 N ρ ij sca (r) ,
ρ i sca (r)= 1 r i 2 A i ( r ^ i , n ^ inc ) ρ inc [ A i ( r ^ i , n ^ inc ) ] T ,
ρ ij sca (r)= exp(i Δ ij ) r i r j A i ( r ^ i , n ^ inc ) ρ inc [ A j ( r ^ j , n ^ inc ) ] T ,
ρ sca (r) t = ρ sca (r) R = i=1 N ρ i sca (r) R .
E sca (r r ^ ) = r g(r)S( r ^ , n ^ inc ) E 0 inc ,
S( r ^ , n ^ inc )=[ θ ^ A ( r ^ , n ^ inc ) θ ^ inc θ ^ A ( r ^ , n ^ inc ) φ ^ inc φ ^ A ( r ^ , n ^ inc ) θ ^ inc φ ^ A ( r ^ , n ^ inc ) φ ^ inc ].
K( n ^ inc )= 2π k 1 [ Im( S 11 + S 22 ) Im( S 11 S 22 ) Im( S 12 + S 21 ) Re( S 21 S 12 ) Im( S 11 S 22 ) Im( S 11 + S 22 ) Im( S 21 S 12 ) Re( S 12 + S 21 ) Im( S 12 + S 21 ) Im( S 21 S 12 ) Im( S 11 + S 22 ) Re( S 22 S 11 ) Re( S 21 S 12 ) Re( S 12 + S 21 ) Re( S 22 S 11 ) Im( S 11 + S 22 ) ],
Z= [ 1 2 (| S 11 | 2 +| S 12 | 2 +| S 21 | 2 +| S 22 | 2 ) 1 2 (| S 11 | 2 | S 12 | 2 +| S 21 | 2 | S 22 | 2 ) 1 2 (| S 11 | 2 +| S 12 | 2 | S 21 | 2 | S 22 | 2 ) 1 2 (| S 11 | 2 | S 12 | 2 | S 21 | 2 +| S 22 | 2 ) Re( S 11 S 21 + S 22 S 12 ) Re( S 11 S 21 S 22 S 12 ) Im( S 21 S 11 + S 22 S 12 ) Im( S 21 S 11 S 22 S 12 ) Re( S 11 S 12 + S 22 S 21 ) Im( S 11 S 12 S 22 S 21 ) Re( S 11 S 12 S 22 S 21 ) Im( S 11 S 12 + S 22 S 21 ) Re( S 11 S 22 + S 12 S 21 ) Im( S 11 S 22 + S 21 S 12 ) Im( S 22 S 11 S 12 S 21 ) Re( S 22 S 11 S 12 S 21 ) ].
Signal1= S ol I sca ( r ^ )= S ol r 2 Z( r ^ , n ^ inc ) I inc ,
Signal2= S ol I inc K( n ^ inc ) I inc + S ol r 2 Z( n ^ inc , n ^ inc ) I inc .
Signal1 t = Signal1 R,ξ = S ol r 2 i=1 N Z i ( r ^ , n ^ inc ) ξ I inc ,
Signal2 t = Signal2 R,ξ = S ol I inc i=1 N K i ( n ^ inc ) ξ I inc + S ol r 2 i=1 N Z i ( n ^ inc , n ^ inc ) ξ I inc ,
W Σ t = W Σ R,ξ =Re Σ d 2 r S(r) R,ξ r ^ ,
W Σ R,ξ = i=1 N [ Κ 11 i ( n ^ inc ) ξ I inc + Κ 12 i ( n ^ inc ) ξ Q inc + Κ 13 i ( n ^ inc ) ξ U inc + Κ 14 i ( n ^ inc ) ξ V inc ] i=1 N 4π d r ^ [ Z 11 i ( r ^ , n ^ inc ) ξ I inc + Z 12 i ( r ^ , n ^ inc ) ξ Q inc + Z 13 i ( r ^ , n ^ inc ) ξ U inc + Z 14 i ( r ^ , n ^ inc ) ξ V inc ].
Z( q ^ , q ^ ) ξ = 1 N i=1 N Z i ( q ^ , q ^ ) ξ , Κ( q ^ ) ξ = 1 N i=1 N Κ i ( q ^ ) ξ .
I ˜ (r, q ^ )=δ( n ^ inc q ^ ) I c (r)+ I ˜  d (r, q ^ ),
n ^ inc I c (r)= n 0 K( n ^ inc ) ξ I c (r), I c (r)| r S ill = I inc ,
q ^ I ˜  d (r, q ^ )= n 0 K( q ^ ) ξ I ˜  d (r, q ^ )+ n 0 4π d q ^ Z( q ^ , q ^ ) ξ I ˜  d (r, q ^ ) + n 0 Z( q ^ , n ^ inc ) ξ I c (r), I ˜  d (r, q ^ )| rS =0.
Signal(r, q ^ ) R,ξ = S ol Δ Ω q ^ d q ^ I ˜ (r, q ^ ){ S ol I c (r)+ S ol ΔΩ I ˜  d (r, n ^ inc )  if   q ^ = n ^ inc , S ol ΔΩ I ˜  d (r, q ^ )  if   q ^ n ^ inc ,
W Σ t = W Σ R,ξ =Re Σ d 2 r S(r) R,ξ n ^ (r) ,
Re S(r) R,ξ = 4π d q ^ q ^ I ˜ (r, q ^ ),