Abstract

We studied the accuracy of volume integral equation simulations of internal fields in small particles illuminated by a monochromatic plane wave as well as the accuracy of the scattered fields. We obtained this accuracy by considering scattering by spheres and comparing the simulated internal and scattered fields with those obtained by Mie theory. The accuracy was measured in several error norms (e.g., mean and root mean square). Furthermore, the distribution of the errors within the particle was obtained. The accuracy was measured as a function of the size parameter and the refractive index of the sphere and as a function of the cube size used in the simulations. The size parameter of the sphere was as large as 10, and three refractive indices were considered. The errors in the internal field are located mostly on the surface of the sphere, and even for fine discretizations they remain relatively large. The errors depend strongly on the refractive index of the particle. If the discretization is kept constant, the errors depend only weakly on the size parameter. We also examined the case of sharp internal field resonances in the sphere. We show that the simulation is able to reproduce the resonances in the internal field, although at a slightly larger refractive index.

© 1998 Optical Society of America

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References

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    [CrossRef]
  29. A. G. Hoekstra, M. D. Grimminck, P. M. A. Sloot, “Large scale simulations of elastic light scattering by a fast discrete dipole approximations,” Int. J. Mod. Phys. C 9, 87–102 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  32. A. G. Hoekstra, P. M. A. Sloot, “Dipolar unit size in coupled dipole calculations of the scattering matrix elements,” Opt. Lett. 18, 1211–1213 (1993).
    [CrossRef] [PubMed]
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    [CrossRef]
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1998

A. G. Hoekstra, M. D. Grimminck, P. M. A. Sloot, “Large scale simulations of elastic light scattering by a fast discrete dipole approximations,” Int. J. Mod. Phys. C 9, 87–102 (1998).
[CrossRef]

1997

A. Doicu, T. Wriedt, “Formulation of the extended boundary condition method for incident Gaussian beams using multiple-multipole expansions,” J. Mod. Opt. 44, 785–801 (1997).
[CrossRef]

1996

J. I. Peltoniemi, “Variational volume integral equation method for electromagnetic scattering by irregular grains,” J. Quant. Spectrosc. Radiat. Transfer 55, 637–647 (1996).
[CrossRef]

J. Rahola, “Solution of dense systems of linear equations in the discrete dipole approximation,” SIAM J. Sci. Stat. Comput. 17, 79–89 (1996).
[CrossRef]

1995

C. Liu, T. Kaiser, S. Lange, G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521–531 (1995).
[CrossRef]

G. Videen, J. Li, P. Chylek, “Resonances and poles of weakly absorbing spheres,” J. Opt. Soc. Am. A 12, 916–921 (1995).
[CrossRef]

1994

B. T. Draine, P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
[CrossRef]

K. Lumme, J. Rahola, “Light scattering by porous dust particles in the discrete-dipole approximation,” Astrophys. J. 425, 653–667 (1994).
[CrossRef]

1993

A. Lakhtakia, G. W. Mulholland, “On two numerical techniques for light scattering by dielectric agglomerated structures,” J. Res. Natl. Inst. Stand. Technol. 98, 699–716 (1993).
[CrossRef]

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

A. G. Hoekstra, P. M. A. Sloot, “Dipolar unit size in coupled dipole calculations of the scattering matrix elements,” Opt. Lett. 18, 1211–1213 (1993).
[CrossRef] [PubMed]

1992

A. Lakhtakia, “Strong and weak forms of the method of moments and the coupled dipole method for scattering of time-harmonic electromagnetic fields,” Int. J. Mod. Phys. C 3, 583–603 (1992).
[CrossRef]

R. W. Freund, “Conjugate gradient type methods for linear systems with complex symmetric coefficient matrices,” SIAM J. Sci. Stat. Comput. 13, 425–488 (1992).
[CrossRef]

1991

1990

J. I. Hage, J. M. Greenberg, “A model for the optical properties of porous grains,” Astrophys. J. 361, 251–259 (1990).
[CrossRef]

A. Lakhtakia, “Macroscopic theory of the coupled dipole approximation method,” Opt. Commun. 79, 1–5 (1990).
[CrossRef]

1989

1988

1985

1984

1979

1974

D. E. Livesay, K. Chen, “Electromagnetic field induced inside arbitrarily shaped biological bodies,” IEEE Trans. Microwave Theory Tech. MTT-22, 1273–1280 (1974).
[CrossRef]

1973

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

1971

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Barber, P. W.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Box, M. A.

Chang, R. K.

Chen, K.

D. E. Livesay, K. Chen, “Electromagnetic field induced inside arbitrarily shaped biological bodies,” IEEE Trans. Microwave Theory Tech. MTT-22, 1273–1280 (1974).
[CrossRef]

Chylek, P.

Conwell, P. R.

Cooke, D. D.

Dobson, C. C.

Doicu, A.

A. Doicu, T. Wriedt, “Formulation of the extended boundary condition method for incident Gaussian beams using multiple-multipole expansions,” J. Mod. Opt. 44, 785–801 (1997).
[CrossRef]

Drain, B. T.

Draine, B. T.

B. T. Draine, P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
[CrossRef]

B. T. Draine, 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]

Dusel, P. W.

Flatau, P. J.

Freund, R. W.

R. W. Freund, “Conjugate gradient type methods for linear systems with complex symmetric coefficient matrices,” SIAM J. Sci. Stat. Comput. 13, 425–488 (1992).
[CrossRef]

Goedecke, G. H.

Goodman, J.

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

Goodman, J. J.

Greenberg, J. M.

J. I. Hage, J. M. Greenberg, “A model for the optical properties of porous grains,” Astrophys. J. 361, 251–259 (1990).
[CrossRef]

Grimminck, M. D.

A. G. Hoekstra, M. D. Grimminck, P. M. A. Sloot, “Large scale simulations of elastic light scattering by a fast discrete dipole approximations,” Int. J. Mod. Phys. C 9, 87–102 (1998).
[CrossRef]

A. G. Hoekstra, M. D. Grimminck, P. M. A. Sloot, “Simulating light scattering from micron-sized particles: a parallel fast discrete dipole approximation,” in Proceedings of High Performance Computing and Networking Europe 1996, H. Lidell, A. Colbrook, B. Hertzberger, P. Sloot, eds., Vol. 1067 of Lecture Notes in Computer Science (Springer-Verlag, Berlin, 1996), pp. 269–275.
[CrossRef]

Hafner, C.

C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech, Norwood, Mass., 1990).

Hage, J. I.

J. I. Hage, J. M. Greenberg, “A model for the optical properties of porous grains,” Astrophys. J. 361, 251–259 (1990).
[CrossRef]

J. I. Hage, “The optics of porous particles and the nature of comets,” Ph.D. dissertation (University of Leiden, Leiden, The Netherlands, 1991).

Hoekstra, A. G.

A. G. Hoekstra, M. D. Grimminck, P. M. A. Sloot, “Large scale simulations of elastic light scattering by a fast discrete dipole approximations,” Int. J. Mod. Phys. C 9, 87–102 (1998).
[CrossRef]

A. G. Hoekstra, P. M. A. Sloot, “Dipolar unit size in coupled dipole calculations of the scattering matrix elements,” Opt. Lett. 18, 1211–1213 (1993).
[CrossRef] [PubMed]

A. G. Hoekstra, M. D. Grimminck, P. M. A. Sloot, “Simulating light scattering from micron-sized particles: a parallel fast discrete dipole approximation,” in Proceedings of High Performance Computing and Networking Europe 1996, H. Lidell, A. Colbrook, B. Hertzberger, P. Sloot, eds., Vol. 1067 of Lecture Notes in Computer Science (Springer-Verlag, Berlin, 1996), pp. 269–275.
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Hunter, B. A.

Kaiser, T.

C. Liu, T. Kaiser, S. Lange, G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521–531 (1995).
[CrossRef]

Kerker, M.

Kiefer, W.

Lakhtakia, A.

A. Lakhtakia, G. W. Mulholland, “On two numerical techniques for light scattering by dielectric agglomerated structures,” J. Res. Natl. Inst. Stand. Technol. 98, 699–716 (1993).
[CrossRef]

A. Lakhtakia, “Strong and weak forms of the method of moments and the coupled dipole method for scattering of time-harmonic electromagnetic fields,” Int. J. Mod. Phys. C 3, 583–603 (1992).
[CrossRef]

A. Lakhtakia, “Macroscopic theory of the coupled dipole approximation method,” Opt. Commun. 79, 1–5 (1990).
[CrossRef]

Lange, S.

C. Liu, T. Kaiser, S. Lange, G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521–531 (1995).
[CrossRef]

Lewis, J. W. I.

Li, J.

Liu, C.

C. Liu, T. Kaiser, S. Lange, G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521–531 (1995).
[CrossRef]

Livesay, D. E.

D. E. Livesay, K. Chen, “Electromagnetic field induced inside arbitrarily shaped biological bodies,” IEEE Trans. Microwave Theory Tech. MTT-22, 1273–1280 (1974).
[CrossRef]

Long, M. B.

Lumme, K.

K. Lumme, J. Rahola, “Light scattering by porous dust particles in the discrete-dipole approximation,” Astrophys. J. 425, 653–667 (1994).
[CrossRef]

Maier, B.

Mishchenko, M. I.

Mulholland, G. W.

A. Lakhtakia, G. W. Mulholland, “On two numerical techniques for light scattering by dielectric agglomerated structures,” J. Res. Natl. Inst. Stand. Technol. 98, 699–716 (1993).
[CrossRef]

O’Brien, S. G.

Peltoniemi, J. I.

J. I. Peltoniemi, “Variational volume integral equation method for electromagnetic scattering by irregular grains,” J. Quant. Spectrosc. Radiat. Transfer 55, 637–647 (1996).
[CrossRef]

Pendleton, J. D.

Pennypacker, C. R.

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

Purcell, E. M.

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

Qian, S. X.

Rahola, J.

J. Rahola, “Solution of dense systems of linear equations in the discrete dipole approximation,” SIAM J. Sci. Stat. Comput. 17, 79–89 (1996).
[CrossRef]

K. Lumme, J. Rahola, “Light scattering by porous dust particles in the discrete-dipole approximation,” Astrophys. J. 425, 653–667 (1994).
[CrossRef]

J. Rahola, “Efficient solution of dense systems of linear equations in electromagnetic scattering calculations,” Ph.D. dissertation (Helsinki University of Technology, Espoo, Finland, 1996).

Rushforth, C. K.

Schweiger, G.

C. Liu, T. Kaiser, S. Lange, G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521–531 (1995).
[CrossRef]

Sloot, P. M. A.

A. G. Hoekstra, M. D. Grimminck, P. M. A. Sloot, “Large scale simulations of elastic light scattering by a fast discrete dipole approximations,” Int. J. Mod. Phys. C 9, 87–102 (1998).
[CrossRef]

A. G. Hoekstra, P. M. A. Sloot, “Dipolar unit size in coupled dipole calculations of the scattering matrix elements,” Opt. Lett. 18, 1211–1213 (1993).
[CrossRef] [PubMed]

A. G. Hoekstra, M. D. Grimminck, P. M. A. Sloot, “Simulating light scattering from micron-sized particles: a parallel fast discrete dipole approximation,” in Proceedings of High Performance Computing and Networking Europe 1996, H. Lidell, A. Colbrook, B. Hertzberger, P. Sloot, eds., Vol. 1067 of Lecture Notes in Computer Science (Springer-Verlag, Berlin, 1996), pp. 269–275.
[CrossRef]

Snow, J. B.

Thurn, R.

Tzeng, H. M.

Videen, G.

Wall, K. F.

Waterman, P. C.

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Wriedt, T.

A. Doicu, T. Wriedt, “Formulation of the extended boundary condition method for incident Gaussian beams using multiple-multipole expansions,” J. Mod. Opt. 44, 785–801 (1997).
[CrossRef]

Appl. Opt.

Astrophys. J.

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

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

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

K. Lumme, J. Rahola, “Light scattering by porous dust particles in the discrete-dipole approximation,” Astrophys. J. 425, 653–667 (1994).
[CrossRef]

J. I. Hage, J. M. Greenberg, “A model for the optical properties of porous grains,” Astrophys. J. 361, 251–259 (1990).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

D. E. Livesay, K. Chen, “Electromagnetic field induced inside arbitrarily shaped biological bodies,” IEEE Trans. Microwave Theory Tech. MTT-22, 1273–1280 (1974).
[CrossRef]

Int. J. Mod. Phys. C

A. G. Hoekstra, M. D. Grimminck, P. M. A. Sloot, “Large scale simulations of elastic light scattering by a fast discrete dipole approximations,” Int. J. Mod. Phys. C 9, 87–102 (1998).
[CrossRef]

A. Lakhtakia, “Strong and weak forms of the method of moments and the coupled dipole method for scattering of time-harmonic electromagnetic fields,” Int. J. Mod. Phys. C 3, 583–603 (1992).
[CrossRef]

J. Mod. Opt.

A. Doicu, T. Wriedt, “Formulation of the extended boundary condition method for incident Gaussian beams using multiple-multipole expansions,” J. Mod. Opt. 44, 785–801 (1997).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

J. I. Peltoniemi, “Variational volume integral equation method for electromagnetic scattering by irregular grains,” J. Quant. Spectrosc. Radiat. Transfer 55, 637–647 (1996).
[CrossRef]

J. Res. Natl. Inst. Stand. Technol.

A. Lakhtakia, G. W. Mulholland, “On two numerical techniques for light scattering by dielectric agglomerated structures,” J. Res. Natl. Inst. Stand. Technol. 98, 699–716 (1993).
[CrossRef]

Opt. Commun.

A. Lakhtakia, “Macroscopic theory of the coupled dipole approximation method,” Opt. Commun. 79, 1–5 (1990).
[CrossRef]

C. Liu, T. Kaiser, S. Lange, G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521–531 (1995).
[CrossRef]

Opt. Lett.

Phys. Rev. D

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

SIAM J. Sci. Stat. Comput.

J. Rahola, “Solution of dense systems of linear equations in the discrete dipole approximation,” SIAM J. Sci. Stat. Comput. 17, 79–89 (1996).
[CrossRef]

R. W. Freund, “Conjugate gradient type methods for linear systems with complex symmetric coefficient matrices,” SIAM J. Sci. Stat. Comput. 13, 425–488 (1992).
[CrossRef]

Other

J. I. Hage, “The optics of porous particles and the nature of comets,” Ph.D. dissertation (University of Leiden, Leiden, The Netherlands, 1991).

A. G. Hoekstra, M. D. Grimminck, P. M. A. Sloot, “Simulating light scattering from micron-sized particles: a parallel fast discrete dipole approximation,” in Proceedings of High Performance Computing and Networking Europe 1996, H. Lidell, A. Colbrook, B. Hertzberger, P. Sloot, eds., Vol. 1067 of Lecture Notes in Computer Science (Springer-Verlag, Berlin, 1996), pp. 269–275.
[CrossRef]

C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech, Norwood, Mass., 1990).

L. Shafai, ed., thematic issues on computational electromagnetics, Comput. Phys. Commun.68(1–3) (1991).

J. W. Hovenier, ed., special issue on light scattering by non-spherical particles, J. Quant. Spectrosc. Radiat. Transfer55(5) (1996).

K. Lumme, J. W. Hovenier, K. Muinonen, J. Rahola, H. Laitinen, eds., Proceedings of the Workshop on Light Scattering by Non-Spherical Particles (University of Helsinki, Helsinki, 1997).

J. Rahola, “Efficient solution of dense systems of linear equations in electromagnetic scattering calculations,” Ph.D. dissertation (Helsinki University of Technology, Espoo, Finland, 1996).

WWW pages containing other figures and additional internal field visualizations: http://www.wins.uva.nl/∼alfons/int/int-f.html .

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

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Figures (17)

Fig. 1
Fig. 1

Energy density for the internal field obtained by Mie calculation. Left, sphere 1 (x = 9, m = 1.05, plotted in the y = 0.21 plane); middle, sphere 2 (x = 9, m = 1.33 + 0.01i, plotted in the y = 0.157 plane); right, sphere 3 (x = 5, m = 2.5 + 1.4i, plotted in the y = 0.09 plane). All scales in the plots are linear.

Fig. 2
Fig. 2

Same as in Fig. 1 but for the energy density for the difference field defined in Eq. (2).

Fig. 3
Fig. 3

Amplitudes of top, the internal Mie field and bottom, difference field as functions of radial distance in the sphere. Left, sphere 1; middle, sphere 2; right, sphere 3. (See Table 1 for properties of the spheres.)

Fig. 4
Fig. 4

Scattering matrix elements as a function of scattering angle for sphere 1; pol is defined as -S 12/S 11; S33 (S 33) and S34 (S 34) are normalized by S11 (S 11). Solid curves, Mie results; dashed curves, VIEF results.

Fig. 5
Fig. 5

Same as Fig. 4 but for sphere 2.

Fig. 6
Fig. 6

Same as Fig. 4 but for sphere 3.

Fig. 7
Fig. 7

Top, absolute and bottom, relative errors in VIEF simulations of the internal field for a sphere with x = 5 and m = 1.33 + 0.01i as a function of cpwl. Upper solid curves, maximum error; lower solid curves, minimum error. Filled circles, mean errors; bars, standard deviations. Dashed curves, the rms. Left, all the data; right, enlargements of data at left.

Fig. 8
Fig. 8

Relative errors left, in the cross sections (solid curve, C ext; dotted curve, C sca; dashed curve, C abs) and right, in the S 11 element of the scattering matrix (meaning of the curves as in Fig. 7) as a function of cpwl for a sphere with x = 5 and m = 1.33 + 0.01i.

Fig. 9
Fig. 9

rms for the relative errors in the internal field as a function of c′ = cpwl/Re(m) for x = 3 (solid curves), x = 5 (dotted curves), and x = 9 (dashed curves).

Fig. 10
Fig. 10

rms for the scattering matrix elements as a function of c′ = cpwl/Re(m). Upper left, relative errors of S 11. Absolute errors are shown for polarization (upper right, S 33/S 11 (lower left), and S 34/S 11 (lower right). These plots do not distinguish among the parameters in the simulation (i.e., size and refractive index of the sphere).

Fig. 11
Fig. 11

Relative errors in the internal fields as a function of the size parameter. The refractive index was m = 1.33 + 0.01i, and cpwl/Re(m) = 15. Upper solid curve, maximum error; lower solid curve, minimum error. Filled circles, mean errors; bars, the standard deviation. Dashed curve, the rms.

Fig. 12
Fig. 12

Extinction coefficient as a function of refractive index for Mie calculations (solid curve) and VIEF simulations (filled circles).

Fig. 13
Fig. 13

Energy density for the internal field obtained by Mie calculation, VIEF simulation, and the difference field for m = 2.2. The energy density is plotted in the y = 0.08727 plane. The data are not normalized; all scales in the plots are linear.

Fig. 14
Fig. 14

Energy density for the internal field obtained by Mie calculation and VIEF simulation for m = 2.3009279. The energy density is plotted in the y = 0.08727 plane. The data are not normalized; all scales in the plots are linear.

Fig. 15
Fig. 15

Same as Fig. 14 but for m = 2.3534695. The energy density is now plotted in the x = 0.08727 plane.

Fig. 16
Fig. 16

Same as Fig. 14 but for m = 2.32.

Fig. 17
Fig. 17

Same as Fig. 14 but for m = 2.36. The energy density is now plotted in the x = 0.08727 plane.

Tables (5)

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Table 1 Overview of Size Parameter and Refractive Index of the Three Spheres Studied in This Sectiona

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Table 2 Errors in the VIEF Simulations of the Internal Fields for the Spheres Defined in Table 1, a

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Table 3 Scattering Cross Sections for Spheres 1–3 for Mie Calculations and VIEF Simulations and Relative Errorsa

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Table 4 Statistical Errors over the Scattering Angles (RMS and Max) for the VIEF Simulations of the Scattering Matrix for the Three Spheres from Table 1a

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Table 5 Typical Mean RMS Errors in the Range 1 ≤ x ≤ 10

Equations (14)

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E r = E inc r + k 3 V m r 2 - 1 G r ,   r · E r d 3 r ,
G r ,   r = 1 + k 2 g | r - r | ,
g r = exp ikr 4 π kr .
1 - m r i 2 - 1 k 3 M - 1 3 E r i = E inc r i + k 3 V 1 4 π j = 1 j i N m r j 2 - 1 T ij · E r j ,
M = 2 3 k 3 1 - ikd 3 / 4 π 1 / 3 exp ikd 3 / 4 π 1 / 3 - 1 ,
T ij = exp i ρ ij ρ ij 3 ρ ij 2 + i ρ ij - 1 1 + exp i ρ ij ρ ij 3 - ρ ij 2 - 3 i ρ ij + 3 r ˆ ij r ˆ ij , ρ ij = k | r i - r j | , r ˆ ij = r i - r j / | r i - r j | .
a n = m ψ n mx ψ n x - ψ n x ψ n mx m ψ n mx ξ n x - ξ n x ψ n mx , b n = ψ n mx ψ n x - m ψ n x ψ n mx ψ n mx ξ n x - m ξ n x ψ n mx ,
c n = m ψ n x ξ n x - m ξ n x ψ n x ψ n mx ξ n x - m ξ n x ψ n mx , d n = m ψ n x ξ n x - m ξ n x ψ n mx m ψ n mx ξ n x - ξ n x ψ n mx .
ψ n x = xj n x ,     ξ n x = xh n x ,
a n x ,   m = A n x ,   m A n x ,   m - iC n x ,   m , b n x ,   m = B n x ,   m B n x ,   m - iD n x ,   m , c n x ,   m = im B n x ,   m - iD n x ,   m , d n x ,   m = im A n x ,   m - iC n x ,   m .
E diff i = E Mie i - E VIEF i .
s j i = E j i * E j i ,
ε abs i = | E diff i |
ε rel i = | E diff i | / | E Mie i | .

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