Abstract

The fulfillment of the reciprocity by five publicly available scattering programs is investigated for a number of different particles. Reciprocity means that the source and the observation point of a given scattering configuration can be interchanged without changing the result. The programs under consideration are either implementations of T-matrix methods or of the discrete dipole approximation. Similarities and differences concerning their reciprocity behavior are discussed. In particular, it is investigated whether and under which conditions reciprocity tests can be used to evaluate the scattering results obtained by the different programs for the given particles.

© 2012 OSA

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    [CrossRef]
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    [CrossRef]
  48. J. Gasteiger, M. Wiegner, S. Groß, V. Freudenthaler, C. Toledano, M. Tesche, and K. Kandler, “Modelling lidar–relevant optical properties of complex mineral dust aerosols,” Tellus B63, 725–741 (2011).
    [CrossRef]

2011

M. A. Yurkin and A. G. Hoekstra, “The discrete–dipole–approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transfer112, 2234–2247 (2011).
[CrossRef]

D. W. Mackowski and M. I. Mishchenko, “A multiple sphere T-matrix Fortran code for use on parallel computer clusters,” J. Quant. Spectrosc. Radiat. Transfer112, 2182–2192 (2011).
[CrossRef]

J. Gasteiger, M. Wiegner, S. Groß, V. Freudenthaler, C. Toledano, M. Tesche, and K. Kandler, “Modelling lidar–relevant optical properties of complex mineral dust aerosols,” Tellus B63, 725–741 (2011).
[CrossRef]

2010

T. Rother and J. Wauer, “Case study about the accuracy behavior of three different T-matrix methods,” Appl. Opt.49, 5746–5756 (2010).
[CrossRef] [PubMed]

M. A. Yurkin, D. de Kanter, and A. G. Hoekstra, “Accuracy of the discrete dipole approximation for simulation of optical properties of gold nanoparticles,” J. Nanophoton.4, 041585 (2010).
[CrossRef]

M. A. Yurkin, M. Min, and A. G. Hoekstra, “Application of the discrete dipole approximation to very large refractive indices: filtered coupled dipoles revived,” Phys. Rev. E82, 036703 (2010).
[CrossRef]

M. Kahnert, “Electromagnetic scattering by nonspherical particles: recent advances,” J. Quant. Spectrosc. Radiat. Transfer111, 1788–1790 (2010).
[CrossRef]

2009

J. Hellmers and T. Wriedt, “New approaches for a light scattering internet information portal and categorization schemes for light scattering software,” J. Quant. Spectrosc. Radiat. Transfer110, 1511–1517 (2009).
[CrossRef]

2008

T. Wriedt and J. Hellmers, “New scattering information portal for the light–scattering community,” J. Quant. Spectrosc. Radiat. Transfer109, 1536–1542 (2008).
[CrossRef]

S. G. Warren and R. E. Brandt, “Optical constants of ice from the ultraviolet to the microwave: a revised compilation,” J. Geophys. Res.113, D14220 (2008).
[CrossRef]

2007

X. A. Penttila, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. T. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov,“Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transfer106, 417–436 (2007).
[CrossRef]

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer106, 558–589 (2007).
[CrossRef]

2006

2005

M. Kahnert, “Irreducible representations of finite groups in the T-matrix formulation of the electromagnetic scattering problem”, J. Opt. Soc. Am. A22, 1187–1199 (2005).
[CrossRef]

H. Volten, O. Muñoz, J. W. Hovenier, J. F. de Haan, W. Vassen, W. J. van der Zande, and L. B. F. M. Waters, “WWW scattering matrix database for small mineral particles at 441.6 and 632.8 nm,” J. Quant. Spectrosc. Radiat. Transfer90, 191–206 (2005).
[CrossRef]

2004

2003

M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer79–80, 775–824 (2003).
[CrossRef]

1998

T. Rother, “Generalization of the separation of variables method for nonspherical scattering on dielectric objects,” J. Quant. Spectrosc. Radiat. Transfer60, 335–353 (1998).
[CrossRef]

1994

1993

B. T. Draine and J. J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J.405, 685–697 (1993).
[CrossRef]

1991

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London A433, 599–614 (1991).
[CrossRef]

1988

B. T. Draine, “The discrete dipole approximation and its application to interstellar graphite grains,” Astrophys. J.333, 848–872 (1988).
[CrossRef]

1980

A. Mugnai and W. J. Wiscombe, “Scattering of radiation by moderately nonspherical particles,” J. Atmos. Sci.37, 1291–1307 (1980).
[CrossRef]

1971

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D3, 825–839 (1971).
[CrossRef]

Barber, P.

P. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).

Borghese, F.

F. Borghese, P. Denti, and R. Saija, Scattering from Model Nonspherical Particles: Theory and Applications to Environmental Physics(Springer, 2003).

Brandt, R. E.

S. G. Warren and R. E. Brandt, “Optical constants of ice from the ultraviolet to the microwave: a revised compilation,” J. Geophys. Res.113, D14220 (2008).
[CrossRef]

de Haan, J. F.

H. Volten, O. Muñoz, J. W. Hovenier, J. F. de Haan, W. Vassen, W. J. van der Zande, and L. B. F. M. Waters, “WWW scattering matrix database for small mineral particles at 441.6 and 632.8 nm,” J. Quant. Spectrosc. Radiat. Transfer90, 191–206 (2005).
[CrossRef]

de Kanter, D.

M. A. Yurkin, D. de Kanter, and A. G. Hoekstra, “Accuracy of the discrete dipole approximation for simulation of optical properties of gold nanoparticles,” J. Nanophoton.4, 041585 (2010).
[CrossRef]

Denti, P.

F. Borghese, P. Denti, and R. Saija, Scattering from Model Nonspherical Particles: Theory and Applications to Environmental Physics(Springer, 2003).

Doicu, A.

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles. Null-Field Method with Discrete Sources: Theory and Programs (Springer, 2006).

Draine, B. T.

X. A. Penttila, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. T. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov,“Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transfer106, 417–436 (2007).
[CrossRef]

B. T. Draine and P. J. Flatau, “Discrete–dipole approximation for scattering calculations,” J. Opt. Soc. Am. A11, 1491–1499 (1994).
[CrossRef]

B. T. Draine and J. J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J.405, 685–697 (1993).
[CrossRef]

B. T. Draine, “The discrete dipole approximation and its application to interstellar graphite grains,” Astrophys. J.333, 848–872 (1988).
[CrossRef]

Eremin, Y. A.

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles. Null-Field Method with Discrete Sources: Theory and Programs (Springer, 2006).

Ernst, T.

Flatau, P. J.

Freudenthaler, V.

J. Gasteiger, M. Wiegner, S. Groß, V. Freudenthaler, C. Toledano, M. Tesche, and K. Kandler, “Modelling lidar–relevant optical properties of complex mineral dust aerosols,” Tellus B63, 725–741 (2011).
[CrossRef]

Gasteiger, J.

J. Gasteiger, M. Wiegner, S. Groß, V. Freudenthaler, C. Toledano, M. Tesche, and K. Kandler, “Modelling lidar–relevant optical properties of complex mineral dust aerosols,” Tellus B63, 725–741 (2011).
[CrossRef]

Gayet, J.-F.

Goodman, J. J.

B. T. Draine and J. J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J.405, 685–697 (1993).
[CrossRef]

Groß, S.

J. Gasteiger, M. Wiegner, S. Groß, V. Freudenthaler, C. Toledano, M. Tesche, and K. Kandler, “Modelling lidar–relevant optical properties of complex mineral dust aerosols,” Tellus B63, 725–741 (2011).
[CrossRef]

Hellmers, J.

J. Hellmers and T. Wriedt, “New approaches for a light scattering internet information portal and categorization schemes for light scattering software,” J. Quant. Spectrosc. Radiat. Transfer110, 1511–1517 (2009).
[CrossRef]

T. Wriedt and J. Hellmers, “New scattering information portal for the light–scattering community,” J. Quant. Spectrosc. Radiat. Transfer109, 1536–1542 (2008).
[CrossRef]

Hess, M.

Hill, S. C.

P. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).

Hoekstra, A. G.

M. A. Yurkin and A. G. Hoekstra, “The discrete–dipole–approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transfer112, 2234–2247 (2011).
[CrossRef]

M. A. Yurkin, D. de Kanter, and A. G. Hoekstra, “Accuracy of the discrete dipole approximation for simulation of optical properties of gold nanoparticles,” J. Nanophoton.4, 041585 (2010).
[CrossRef]

M. A. Yurkin, M. Min, and A. G. Hoekstra, “Application of the discrete dipole approximation to very large refractive indices: filtered coupled dipoles revived,” Phys. Rev. E82, 036703 (2010).
[CrossRef]

X. A. Penttila, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. T. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov,“Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transfer106, 417–436 (2007).
[CrossRef]

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer106, 558–589 (2007).
[CrossRef]

M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “Convergence of the discrete dipole approximation. II. an extrapolation technique to increase the accuracy,” J. Opt. Soc. Am. A23, 2592–2601 (2006).
[CrossRef]

Hovenier, J. W.

H. Volten, O. Muñoz, J. W. Hovenier, J. F. de Haan, W. Vassen, W. J. van der Zande, and L. B. F. M. Waters, “WWW scattering matrix database for small mineral particles at 441.6 and 632.8 nm,” J. Quant. Spectrosc. Radiat. Transfer90, 191–206 (2005).
[CrossRef]

J. W. Hovenier and C. V. M. van der Mee, “Basic relationships for matrices describing scattering by small particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, 2000), pp. 61–85.
[CrossRef]

Kahnert, M.

M. Kahnert, “Electromagnetic scattering by nonspherical particles: recent advances,” J. Quant. Spectrosc. Radiat. Transfer111, 1788–1790 (2010).
[CrossRef]

M. Kahnert, “Irreducible representations of finite groups in the T-matrix formulation of the electromagnetic scattering problem”, J. Opt. Soc. Am. A22, 1187–1199 (2005).
[CrossRef]

M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer79–80, 775–824 (2003).
[CrossRef]

Kandler, K.

J. Gasteiger, M. Wiegner, S. Groß, V. Freudenthaler, C. Toledano, M. Tesche, and K. Kandler, “Modelling lidar–relevant optical properties of complex mineral dust aerosols,” Tellus B63, 725–741 (2011).
[CrossRef]

Kong, J. A.

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

Lacis, A. A.

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

Lumme, K.

X. A. Penttila, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. T. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov,“Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transfer106, 417–436 (2007).
[CrossRef]

Mackowski, D. W.

D. W. Mackowski and M. I. Mishchenko, “A multiple sphere T-matrix Fortran code for use on parallel computer clusters,” J. Quant. Spectrosc. Radiat. Transfer112, 2182–2192 (2011).
[CrossRef]

D. W. Mackowski, “Calculation of total cross sections of multiple sphere clusters,” J. Opt. Soc. Am. A11, 2851–2861 (1994).
[CrossRef]

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London A433, 599–614 (1991).
[CrossRef]

Maltsev, V. P.

Min, M.

M. A. Yurkin, M. Min, and A. G. Hoekstra, “Application of the discrete dipole approximation to very large refractive indices: filtered coupled dipoles revived,” Phys. Rev. E82, 036703 (2010).
[CrossRef]

Mishchenko, M. I.

D. W. Mackowski and M. I. Mishchenko, “A multiple sphere T-matrix Fortran code for use on parallel computer clusters,” J. Quant. Spectrosc. Radiat. Transfer112, 2182–2192 (2011).
[CrossRef]

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

Mugnai, A.

A. Mugnai and W. J. Wiscombe, “Scattering of radiation by moderately nonspherical particles,” J. Atmos. Sci.37, 1291–1307 (1980).
[CrossRef]

Muinonen, K.

X. A. Penttila, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. T. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov,“Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transfer106, 417–436 (2007).
[CrossRef]

Muñoz, O.

H. Volten, O. Muñoz, J. W. Hovenier, J. F. de Haan, W. Vassen, W. J. van der Zande, and L. B. F. M. Waters, “WWW scattering matrix database for small mineral particles at 441.6 and 632.8 nm,” J. Quant. Spectrosc. Radiat. Transfer90, 191–206 (2005).
[CrossRef]

Penttila, X. A.

X. A. Penttila, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. T. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov,“Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transfer106, 417–436 (2007).
[CrossRef]

Rahola, J.

X. A. Penttila, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. T. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov,“Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transfer106, 417–436 (2007).
[CrossRef]

Rother, T.

Saija, R.

F. Borghese, P. Denti, and R. Saija, Scattering from Model Nonspherical Particles: Theory and Applications to Environmental Physics(Springer, 2003).

Schmidt, K.

Shcherbakov, V.

Shin, R. T.

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

Shkuratov, Y.

X. A. Penttila, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. T. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov,“Comparison between discrete dipole implementations and exact techniques,” J. Quant. Spectrosc. Radiat. Transfer106, 417–436 (2007).
[CrossRef]

Tesche, M.

J. Gasteiger, M. Wiegner, S. Groß, V. Freudenthaler, C. Toledano, M. Tesche, and K. Kandler, “Modelling lidar–relevant optical properties of complex mineral dust aerosols,” Tellus B63, 725–741 (2011).
[CrossRef]

Toledano, C.

J. Gasteiger, M. Wiegner, S. Groß, V. Freudenthaler, C. Toledano, M. Tesche, and K. Kandler, “Modelling lidar–relevant optical properties of complex mineral dust aerosols,” Tellus B63, 725–741 (2011).
[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 Press, 2002).

Tsang, L.

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

van de Hulst, H. C.

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

van der Mee, C. V. M.

J. W. Hovenier and C. V. M. van der Mee, “Basic relationships for matrices describing scattering by small particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, 2000), pp. 61–85.
[CrossRef]

van der Zande, W. J.

H. Volten, O. Muñoz, J. W. Hovenier, J. F. de Haan, W. Vassen, W. J. van der Zande, and L. B. F. M. Waters, “WWW scattering matrix database for small mineral particles at 441.6 and 632.8 nm,” J. Quant. Spectrosc. Radiat. Transfer90, 191–206 (2005).
[CrossRef]

Vassen, W.

H. Volten, O. Muñoz, J. W. Hovenier, J. F. de Haan, W. Vassen, W. J. van der Zande, and L. B. F. M. Waters, “WWW scattering matrix database for small mineral particles at 441.6 and 632.8 nm,” J. Quant. Spectrosc. Radiat. Transfer90, 191–206 (2005).
[CrossRef]

Volten, H.

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Figures (7)

Fig. 1
Fig. 1

Configurations of the bisphere with touching components ((a) and (b)) and of the cube ((c) and (d)) where the incident field Einc propagates along the positive z-axis. The y-axis points to the reader. (a) The rotational axis is oriented along the positive z-axis (θp = 0°). The scattered field Esca is observed in positive x-direction (θ = 90°). (b) The rotational axis is oriented along the positive x-axis (θp = 90°). The scattered field Esca is observed in negative x-direction (θ = 270°). (c) The central axis, which stands perpendicularly on the top and bottom faces of the cube while going through its center, is oriented along the positive z-axis (θp = 0°). The scattered field Esca is observed in positive x-direction (θ = 90°) out of a cube edge. (d) The central axis is oriented along the positive x-axis (θp = 90°). The scattered field Esca is observed in negative x-direction (θ = 270°).

Fig. 2
Fig. 2

vv-polarized DSCS of the scatterer cyl_2 in the orientations θp = 0° (black line) and θp = 90° (red line) calculated with mieschka. The corresponding reciprocal scattering angles θ = 90° and θ = 270° are marked by vertical lines.

Fig. 3
Fig. 3

Reciprocity errors evv and ehh according to Eq. (7) for sphd_p_2 in the two reciprocal configurations (θp = 0°, θ = 90°) and (θp = 90°, θ = 270°) using mieschka (a) versus the relative error within the chosen Barber–Hill convergence criterion, and (b) versus ncut. (ncut, lcut, nint) are given by Table 3.

Fig. 4
Fig. 4

Reciprocity errors evv and ehh according to Eq. (7) for bi_sph_2 in the two reciprocal configurations (θp = 0°, θ = 90°) and (θp = 90°, θ = 270°) using mstm versus translation_epsilon.

Fig. 5
Fig. 5

hh-polarized DSCS of the scatterer sphd_o_2 in the orientations θp = 0° and θp = 90° calculated using ADDA with LDR polarizability (black line) and mieschka (red line).

Fig. 6
Fig. 6

Reciprocity errors evv and ehh according to Eq. (7) for sphd_o_1 in the two reciprocal configurations (θp = 0°, θ = 90°) and (θp = 90°, θ = 270°) using ADDA with CLDR polarizability versus discretization level.

Fig. 7
Fig. 7

hh-polarized DSCS of the scatterer (a) sphd_o_1 and (b) sphd_o_2 in the orientation θp = 0° calculated using ADDA at different discretizations and mieschka (green line). (a) ADDA with CLDR polarizability (black line – 32 dipoles in x-direction, red line – 128 dipoles in x-direction, blue line – extrapolated) (b) ADDA with LDR polarizability (black line – 88 dipoles in x-direction, red line – 256 dipoles in x-direction, blue line – extrapolated)

Tables (5)

Tables Icon

Table 1 Scatterer geometries considered in the investigations.

Tables Icon

Table 2 Reciprocity errors ehh and evv, defined by Eq. (7), for the reciprocal configurations (θp = 0°, θ = 90°) and (θp = 90°, θ = 270°) using mieschka, ADDA with LDR polarizability, and DDSCATwith GKDLDR (CLDR) polarizability.

Tables Icon

Table 3 Convergence parameters (ncut, lcut, nint) determined automatically in the standard mode of mieschka for sphd_p_2 in the orientation θp = 90° at different input values of the relative error within the chosen Barber–Hill convergence criterion. Note that the relative integration error has been fixed to 0.1%.

Tables Icon

Table 4 Reciprocity errors ehh and evv, defined by Eq. (7), for the reciprocal configurations (θp = 0°, θ = 90°) and (θp = 90°, θ = 270°) using scsmfo1b and mstm for the bispheres.

Tables Icon

Table 5 Polarized DSCS and reciprocity errors ehh and evv, defined by Eq. (7), for the reciprocal configurations c1 = (θp = 0°, θ = 90°) and c2 = (θp = 90°, θ = 270°) using ADDA with RRC polarizability for cube 1 at different program parameters.

Equations (20)

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

E inc = e 0 exp ( i k 0 z )
( E sca v E sca h ) = exp ( i k 0 r ) r ( F vv ( θ , ϕ ) F vh ( θ , ϕ ) F hv ( θ , ϕ ) F hh ( θ , ϕ ) ) ( E inc v E inc h )
d σ α β d Ω ( θ , ϕ ) = k 0 2 | F α β ( θ , ϕ ) | 2
d σ α β d Ω ( n inc , n sca ) = d σ β α d Ω ( n sca , n inc ) .
d σ α α d Ω ( θ p = 0 ° , θ = θ 1 ) = d σ α α d Ω ( θ p = 180 ° θ 1 , θ = 360 ° θ 1 )
d σ α α d Ω ( θ p = 0 ° , θ = θ 1 ) = d σ α α d Ω ( θ p = θ 1 180 ° , θ = θ 1 )
e α α ( c 1 , c 2 ) = | d σ α α ( c 1 ) / d Ω d σ α α ( c 2 ) / d Ω | d σ α α ( c 1 ) / d Ω 100 %
R ( γ ) = r 0 ( 1 + ε cos ( n γ ) ) , γ [ 0 ° , 180 ° ] .
E sca ( k 0 r ) = τ = 1 2 n = 1 l = n n f τ n l Ψ τ n l ( k 0 r )
E int ( k s r ) = τ = 1 2 n = 1 l = n n p τ n l Rg Ψ τ n l ( k s r )
E inc ( k 0 r ) = τ = 1 2 n = 1 l = n n a τ n l Rg Ψ τ n l ( k 0 r )
f = 𝒯 a .
α ¯ = V d χ { I ¯ + [ ( 4 π / 3 ) I ¯ M ¯ ] χ } 1 = α CM ( I ¯ M ¯ α CM / V d ) 1 ,
α CM = V d 3 4 π ε 1 ε + 2
M ¯ LDR = I ¯ [ ( b 1 LDR + b 2 LDR m 2 + b 3 LDR m 2 S ) ( k 0 d ) 2 + ( 2 / 3 ) i ( k 0 d ) 3 ] ,
b 1 LDR 1.8915316 , b 2 LDR 0.1648469 , b 3 LDR 1.7700004 , S = μ ( a μ e μ 0 ) 2 ,
M μ ν CLDR = δ μ ν [ ( b 1 LDR + b 2 LDR m 2 + b 3 LDR m 2 a μ 2 ) ( k 0 d ) 2 + ( 2 / 3 ) i ( k 0 d ) 3 ]
M ¯ RRC = I ¯ ( 2 / 3 ) i ( k 0 d ) 3 .
α i 1 P i j i G ¯ i j P j = E i inc ,
G ¯ i j = G ¯ ( r i , r j ) = exp ( i k 0 R ) R [ k 0 2 ( I ¯ R ^ R ^ R 2 ) 1 i k 0 R R 2 ( I ¯ 3 R ^ R ^ R 2 ) ] ,

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