Abstract

A two-dimensional fast Fourier transform technique is proposed for accelerating the computation of scattering characteristics of features on surfaces by using the discrete-dipole approximation. The two-dimensional fast Fourier transform reduces the CPU execution time dependence on the number of dipoles N from O(N2) to O(N log N). The capabilities and flexibility of a discrete-dipole code implementing the technique are demonstrated with scattering results from circuit features on surfaces.

© 1997 Optical Society of America

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References

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  1. P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A 137, 213–214 (1986).
    [CrossRef]
  2. F. L. Assi, “Electromagnetic wave scattering by a sphere on a layered substrate,” M. S. thesis (University of Arizona, Tucson, Ariz., 1990).
  3. I. V. Lindell, A. H. Sihvola, K. O. Muinonen, P. W. Barber, “Scattering by a small object close to an interface. I. Exact-imagetheory formulation,” J. Opt. Soc. Am. A 8, 472–476 (1991).
    [CrossRef]
  4. B. R. Johnson, “Light scattering from a spherical particle on a conducting plane: I. Normal incidence,” J. Opt. Soc. Am. A 9, 1341–1351 (1992).
    [CrossRef]
  5. G. L. Wojcik, D. K. Vaughn, L. K. Galbraith, “Calculation of light scatter from structures on silicon surfaces,” in Lasers in Microlithography, J. S. Batchelder, D. J. Erlich, J. Y. Tsao, eds., Proc. SPIE774, 21–31 (1987).
    [CrossRef]
  6. M. L. Liswith, “Numerical modeling of light scattering by individual submicron spherical particles on optically smooth semiconductor surfaces,” M. S. thesis (Arizona State University, Tempe, Ariz., 1994).
  7. Y. A. Yeremin, N. V. Orlov, A. G. Sveshnikov, “The analysis of complex diffraction problems by the discrete-source method,” Comp. Math. Phys. 35, 731–743 (1995).
  8. E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [CrossRef]
  9. B. T. Draine, “The discrete-dipole approximation and its application to interstellargraphite grains,” Astrophys. J. 333, 848–872 (1988).
    [CrossRef]
  10. B. T. Draine, P. J. Flatau, “The discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
    [CrossRef]
  11. M. A. Taubenblatt, T. K. Tran, “Calculation of light scattering from particles and structure by thecoupled-dipole method,” J. Opt. Soc. Am. A 10, 912–919 (1993).
    [CrossRef]
  12. A. Lakhtakia, “Macroscopic theory of the coupled dipole approximation method,” Opt. Commun. 72, 1–5 (1990).
    [CrossRef]
  13. A. Lakhtakia, “General theory of the Purcell–Pennypacker scattering approachand its extension to bianisotropic scatterers,” Astrophys. J. 394, 494–499 (1992).
    [CrossRef]
  14. J. I. Hage, J. M. Greenberg, “A model for the optical properties of porous grains,” Astrophys. J. 361, 251–259 (1990).
    [CrossRef]
  15. A. G. Hoekstra, P. M. A. Sloot, “Dipolar unit size in coupled-dipole calculations of the scatteringmatrix elements,” Opt. Lett. 18, 1211–1213 (1993).
    [CrossRef] [PubMed]
  16. J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
    [CrossRef]
  17. J. J. Talmonti, R. B. Kay, D. J. Krebs, “Numerical model estimating the capabilities and limitations of thefast Fourier transform technique in absolute interferometry,” Appl. Opt. 35, 2182–2191 (1996).
    [CrossRef]
  18. B. Ritchie, M. D. Feit, “Fast Fourier transform computational method for the propagation ofelectromagnetic pulses through layered dielectric media,” Phys. Rev. E 53, 1976–1981 (1996).
    [CrossRef]
  19. F. Depasse, P. Groussel, “Use of two-dimensional fast Fourier transform in harmonic modulatedthermal diffusion,” Int. J. Heat Mass Transfer 39, 3761–3764 (1996).
    [CrossRef]
  20. J. J. Goodman, B. T. Draine, P. J. Flatau, “Application of fast-Fourier-transform techniques to the discrete-dipoleapproximation,” Opt. Lett. 16, 1198–1200 (1991).
    [CrossRef] [PubMed]
  21. B. T. Draine, J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable pointlattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
    [CrossRef]
  22. R. Schmehl, “The coupled-dipole method for light scattering for light particles on plane surfaces,” M. S. thesis (Arizona State University, Tempe, Ariz., and the Institut für Thermische Strömungsmaschinen, Universität Karlsruhe, Karlsruhe, Germany, 1994).
  23. D. L. Lager, R. J. Lytle, fortran Subroutines for the Numerical Evaluation of Sommerfeld Integrals unter Anterem, (Lawrence Livermore Laboratory, Livermore, Calif., 1975).
  24. M. Petravic, G. Kuo-Petravic, “An ILUCG algorithm which minimizes in the Euclidean norm,” J. Comput. Phys. 32, 263–269 (1979).
    [CrossRef]
  25. N. M. Brenner, “Fast Fourier transform of externally stored data,” IEEE Trans. Audio Electroacoust. AU-17, 128–132 (1969).
    [CrossRef]
  26. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  27. B. M. Nebeker, G. W. Starr, E. D. Hirleman, “Modeling of light scattering from structures with particle contaminants,” in Flatness, Roughness, and Discrete Defect Characterizationfor Computer Disks, Wafers, and Flat Panel Displays, John Stover, eds., Proc. SPIE2862, 139–150 (1996).
    [CrossRef]
  28. H. C. Chen, Theory of Electromagnetic Waves—A Coordinate-Free Approach (McGraw-Hill, New York, 1983).

1996 (3)

J. J. Talmonti, R. B. Kay, D. J. Krebs, “Numerical model estimating the capabilities and limitations of thefast Fourier transform technique in absolute interferometry,” Appl. Opt. 35, 2182–2191 (1996).
[CrossRef]

B. Ritchie, M. D. Feit, “Fast Fourier transform computational method for the propagation ofelectromagnetic pulses through layered dielectric media,” Phys. Rev. E 53, 1976–1981 (1996).
[CrossRef]

F. Depasse, P. Groussel, “Use of two-dimensional fast Fourier transform in harmonic modulatedthermal diffusion,” Int. J. Heat Mass Transfer 39, 3761–3764 (1996).
[CrossRef]

1995 (1)

Y. A. Yeremin, N. V. Orlov, A. G. Sveshnikov, “The analysis of complex diffraction problems by the discrete-source method,” Comp. Math. Phys. 35, 731–743 (1995).

1994 (1)

1993 (3)

1992 (2)

A. Lakhtakia, “General theory of the Purcell–Pennypacker scattering approachand its extension to bianisotropic scatterers,” Astrophys. J. 394, 494–499 (1992).
[CrossRef]

B. R. Johnson, “Light scattering from a spherical particle on a conducting plane: I. Normal incidence,” J. Opt. Soc. Am. A 9, 1341–1351 (1992).
[CrossRef]

1991 (2)

1990 (2)

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. 72, 1–5 (1990).
[CrossRef]

1988 (1)

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

1986 (1)

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A 137, 213–214 (1986).
[CrossRef]

1979 (1)

M. Petravic, G. Kuo-Petravic, “An ILUCG algorithm which minimizes in the Euclidean norm,” J. Comput. Phys. 32, 263–269 (1979).
[CrossRef]

1973 (1)

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

1969 (1)

N. M. Brenner, “Fast Fourier transform of externally stored data,” IEEE Trans. Audio Electroacoust. AU-17, 128–132 (1969).
[CrossRef]

1965 (1)

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Assi, F. L.

F. L. Assi, “Electromagnetic wave scattering by a sphere on a layered substrate,” M. S. thesis (University of Arizona, Tucson, Ariz., 1990).

Barber, P. W.

Bobbert, P. A.

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A 137, 213–214 (1986).
[CrossRef]

Bohren, C. F.

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

Brenner, N. M.

N. M. Brenner, “Fast Fourier transform of externally stored data,” IEEE Trans. Audio Electroacoust. AU-17, 128–132 (1969).
[CrossRef]

Chen, H. C.

H. C. Chen, Theory of Electromagnetic Waves—A Coordinate-Free Approach (McGraw-Hill, New York, 1983).

Cooley, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Depasse, F.

F. Depasse, P. Groussel, “Use of two-dimensional fast Fourier transform in harmonic modulatedthermal diffusion,” Int. J. Heat Mass Transfer 39, 3761–3764 (1996).
[CrossRef]

Draine, B. T.

B. T. Draine, P. J. Flatau, “The 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 pointlattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[CrossRef]

J. J. Goodman, B. T. Draine, P. J. Flatau, “Application of fast-Fourier-transform techniques to the discrete-dipoleapproximation,” Opt. Lett. 16, 1198–1200 (1991).
[CrossRef] [PubMed]

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

Feit, M. D.

B. Ritchie, M. D. Feit, “Fast Fourier transform computational method for the propagation ofelectromagnetic pulses through layered dielectric media,” Phys. Rev. E 53, 1976–1981 (1996).
[CrossRef]

Flatau, P. J.

Galbraith, L. K.

G. L. Wojcik, D. K. Vaughn, L. K. Galbraith, “Calculation of light scatter from structures on silicon surfaces,” in Lasers in Microlithography, J. S. Batchelder, D. J. Erlich, J. Y. Tsao, eds., Proc. SPIE774, 21–31 (1987).
[CrossRef]

Goodman, J.

B. T. Draine, J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable pointlattice 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]

Groussel, P.

F. Depasse, P. Groussel, “Use of two-dimensional fast Fourier transform in harmonic modulatedthermal diffusion,” Int. J. Heat Mass Transfer 39, 3761–3764 (1996).
[CrossRef]

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]

Hirleman, E. D.

B. M. Nebeker, G. W. Starr, E. D. Hirleman, “Modeling of light scattering from structures with particle contaminants,” in Flatness, Roughness, and Discrete Defect Characterizationfor Computer Disks, Wafers, and Flat Panel Displays, John Stover, eds., Proc. SPIE2862, 139–150 (1996).
[CrossRef]

Hoekstra, A. G.

Huffman, D. R.

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

Johnson, B. R.

Kay, R. B.

Krebs, D. J.

Kuo-Petravic, G.

M. Petravic, G. Kuo-Petravic, “An ILUCG algorithm which minimizes in the Euclidean norm,” J. Comput. Phys. 32, 263–269 (1979).
[CrossRef]

Lager, D. L.

D. L. Lager, R. J. Lytle, fortran Subroutines for the Numerical Evaluation of Sommerfeld Integrals unter Anterem, (Lawrence Livermore Laboratory, Livermore, Calif., 1975).

Lakhtakia, A.

A. Lakhtakia, “General theory of the Purcell–Pennypacker scattering approachand its extension to bianisotropic scatterers,” Astrophys. J. 394, 494–499 (1992).
[CrossRef]

A. Lakhtakia, “Macroscopic theory of the coupled dipole approximation method,” Opt. Commun. 72, 1–5 (1990).
[CrossRef]

Lindell, I. V.

Liswith, M. L.

M. L. Liswith, “Numerical modeling of light scattering by individual submicron spherical particles on optically smooth semiconductor surfaces,” M. S. thesis (Arizona State University, Tempe, Ariz., 1994).

Lytle, R. J.

D. L. Lager, R. J. Lytle, fortran Subroutines for the Numerical Evaluation of Sommerfeld Integrals unter Anterem, (Lawrence Livermore Laboratory, Livermore, Calif., 1975).

Muinonen, K. O.

Nebeker, B. M.

B. M. Nebeker, G. W. Starr, E. D. Hirleman, “Modeling of light scattering from structures with particle contaminants,” in Flatness, Roughness, and Discrete Defect Characterizationfor Computer Disks, Wafers, and Flat Panel Displays, John Stover, eds., Proc. SPIE2862, 139–150 (1996).
[CrossRef]

Orlov, N. V.

Y. A. Yeremin, N. V. Orlov, A. G. Sveshnikov, “The analysis of complex diffraction problems by the discrete-source method,” Comp. Math. Phys. 35, 731–743 (1995).

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]

Petravic, M.

M. Petravic, G. Kuo-Petravic, “An ILUCG algorithm which minimizes in the Euclidean norm,” J. Comput. Phys. 32, 263–269 (1979).
[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]

Ritchie, B.

B. Ritchie, M. D. Feit, “Fast Fourier transform computational method for the propagation ofelectromagnetic pulses through layered dielectric media,” Phys. Rev. E 53, 1976–1981 (1996).
[CrossRef]

Schmehl, R.

R. Schmehl, “The coupled-dipole method for light scattering for light particles on plane surfaces,” M. S. thesis (Arizona State University, Tempe, Ariz., and the Institut für Thermische Strömungsmaschinen, Universität Karlsruhe, Karlsruhe, Germany, 1994).

Sihvola, A. H.

Sloot, P. M. A.

Starr, G. W.

B. M. Nebeker, G. W. Starr, E. D. Hirleman, “Modeling of light scattering from structures with particle contaminants,” in Flatness, Roughness, and Discrete Defect Characterizationfor Computer Disks, Wafers, and Flat Panel Displays, John Stover, eds., Proc. SPIE2862, 139–150 (1996).
[CrossRef]

Sveshnikov, A. G.

Y. A. Yeremin, N. V. Orlov, A. G. Sveshnikov, “The analysis of complex diffraction problems by the discrete-source method,” Comp. Math. Phys. 35, 731–743 (1995).

Talmonti, J. J.

Taubenblatt, M. A.

Tran, T. K.

Tukey, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Vaughn, D. K.

G. L. Wojcik, D. K. Vaughn, L. K. Galbraith, “Calculation of light scatter from structures on silicon surfaces,” in Lasers in Microlithography, J. S. Batchelder, D. J. Erlich, J. Y. Tsao, eds., Proc. SPIE774, 21–31 (1987).
[CrossRef]

Vlieger, J.

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A 137, 213–214 (1986).
[CrossRef]

Wojcik, G. L.

G. L. Wojcik, D. K. Vaughn, L. K. Galbraith, “Calculation of light scatter from structures on silicon surfaces,” in Lasers in Microlithography, J. S. Batchelder, D. J. Erlich, J. Y. Tsao, eds., Proc. SPIE774, 21–31 (1987).
[CrossRef]

Yeremin, Y. A.

Y. A. Yeremin, N. V. Orlov, A. G. Sveshnikov, “The analysis of complex diffraction problems by the discrete-source method,” Comp. Math. Phys. 35, 731–743 (1995).

Appl. Opt. (1)

Astrophys. J. (5)

A. Lakhtakia, “General theory of the Purcell–Pennypacker scattering approachand its extension to bianisotropic scatterers,” Astrophys. J. 394, 494–499 (1992).
[CrossRef]

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

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

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

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

Comp. Math. Phys. (1)

Y. A. Yeremin, N. V. Orlov, A. G. Sveshnikov, “The analysis of complex diffraction problems by the discrete-source method,” Comp. Math. Phys. 35, 731–743 (1995).

IEEE Trans. Audio Electroacoust. (1)

N. M. Brenner, “Fast Fourier transform of externally stored data,” IEEE Trans. Audio Electroacoust. AU-17, 128–132 (1969).
[CrossRef]

Int. J. Heat Mass Transfer (1)

F. Depasse, P. Groussel, “Use of two-dimensional fast Fourier transform in harmonic modulatedthermal diffusion,” Int. J. Heat Mass Transfer 39, 3761–3764 (1996).
[CrossRef]

J. Comput. Phys. (1)

M. Petravic, G. Kuo-Petravic, “An ILUCG algorithm which minimizes in the Euclidean norm,” J. Comput. Phys. 32, 263–269 (1979).
[CrossRef]

J. Opt. Soc. Am. A (4)

Math. Comput. (1)

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Opt. Commun. (1)

A. Lakhtakia, “Macroscopic theory of the coupled dipole approximation method,” Opt. Commun. 72, 1–5 (1990).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. E (1)

B. Ritchie, M. D. Feit, “Fast Fourier transform computational method for the propagation ofelectromagnetic pulses through layered dielectric media,” Phys. Rev. E 53, 1976–1981 (1996).
[CrossRef]

Physica A (1)

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A 137, 213–214 (1986).
[CrossRef]

Other (8)

F. L. Assi, “Electromagnetic wave scattering by a sphere on a layered substrate,” M. S. thesis (University of Arizona, Tucson, Ariz., 1990).

G. L. Wojcik, D. K. Vaughn, L. K. Galbraith, “Calculation of light scatter from structures on silicon surfaces,” in Lasers in Microlithography, J. S. Batchelder, D. J. Erlich, J. Y. Tsao, eds., Proc. SPIE774, 21–31 (1987).
[CrossRef]

M. L. Liswith, “Numerical modeling of light scattering by individual submicron spherical particles on optically smooth semiconductor surfaces,” M. S. thesis (Arizona State University, Tempe, Ariz., 1994).

R. Schmehl, “The coupled-dipole method for light scattering for light particles on plane surfaces,” M. S. thesis (Arizona State University, Tempe, Ariz., and the Institut für Thermische Strömungsmaschinen, Universität Karlsruhe, Karlsruhe, Germany, 1994).

D. L. Lager, R. J. Lytle, fortran Subroutines for the Numerical Evaluation of Sommerfeld Integrals unter Anterem, (Lawrence Livermore Laboratory, Livermore, Calif., 1975).

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

B. M. Nebeker, G. W. Starr, E. D. Hirleman, “Modeling of light scattering from structures with particle contaminants,” in Flatness, Roughness, and Discrete Defect Characterizationfor Computer Disks, Wafers, and Flat Panel Displays, John Stover, eds., Proc. SPIE2862, 139–150 (1996).
[CrossRef]

H. C. Chen, Theory of Electromagnetic Waves—A Coordinate-Free Approach (McGraw-Hill, New York, 1983).

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Figures (12)

Fig. 1
Fig. 1

Dipole configuration for a spherical particle on a surface.

Fig. 2
Fig. 2

Comparison of required CPU time per iteration for ddsurf with and without the 2-D FFT. Computations were performed on an IBM 370 RS6000 workstation.

Fig. 3
Fig. 3

Prediction of scattering in the x z plane as a function of the angle measured from the surface normal by ddsurf compared with predictions by Taubenblatt and Tran 11 and Wojcik et al.5

Fig. 4
Fig. 4

Arizona State University scatterometer used for measuring light scatter from samples.

Fig. 5
Fig. 5

Configuration of the ring/wedge photodetector used to measure light scatter.

Fig. 6
Fig. 6

Plane of detection for the ring/wedge photodetector with respect to the sample die.

Fig. 7
Fig. 7

Predicted intensity distribution of light scattered from a 0.482-μm PSL sphere on a Si surface.

Fig. 8
Fig. 8

Numerical predictions and experimental calculations of the differential scattering cross sections for (a) the ring region and (b) the wedge region of the ring/wedge photodetector for a 0.482-μm PSL sphere on a Si surface.

Fig. 9
Fig. 9

Dipole arrangement for cornered-feature–contaminant configuration.

Fig. 10
Fig. 10

Predicted intensity distribution of light scattered from a SiO 2 cornered-feature–PSL sphere contaminant configuration on a Si surface.

Fig. 11
Fig. 11

Numerical predictions and experimental calculations of the differential scattering cross sections for (a) the ring region and (b) the wedge region of the ring/wedge photodetector for a SiO 2 cornered-feature–PSL sphere contaminant configuration on a Si surface.

Fig. 12
Fig. 12

Geometry of two dipoles and the terms referred to in the Green's function in Eq. (A3).

Tables (1)

Tables Icon

Table 1 Number of Dipoles Needed and CPU Time per Iteration for a Spherical Particle on a Surface, with FFT and non-FFT Routines

Equations (55)

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

P i = α i E tot , i .
E tot , i = E inc , i + E direct , i + E reflected , i .
( α i ) - 1 P i - E direct , i - E reflected , i = E inc , i .
E direct , i = k 0 2 0   j i G ij P j .
E reflected , i = j = 1 N S ij + k 2 2 0   k 1 2 - k 2 2 k 1 2 + k 2 2   G ij I P ¯ j .
G ij I = - G ij I R ,
( B + A + R ) P = E inc .
B = diag ( α 1 x - 1 ,   α 1 y - 1 ,   α 1 z - 1 ,   ,   α Nx - 1 ,   α Ny - 1 ,   α Nz - 1 , ) ,
A = A 11 A 1 N A N 1 A NN , R = R 11 R 1 N R N 1 R NN ,
P = ( P 1 P N ) T , E inc = ( E inc , 1 E inc , N ) T .
E direct , i = j z = 1 N j y = 1 2 N y j x = 1 2 N x A ij P j ,
A i - j 0 for i = j A ij for i j ,
E direct , i = j z = 1 N z j y = 1 2 N y j x = 1 2 N x A i - j P j .
E ^ direct , ( n x , n y , i z ) = j z = 1 N z A ^ ( n x , n y , i z - j z ) P ^ ( n x , n y , j z ) ,
E reflected , i = j z = 1 N z j y = 1 2 N y j x = 1 2 N x R ij P j .
R i - j R ij for all i ,   j ,
E reflected , i = j z = 1 N z j y = 1 2 N y j x = 1 2 N x R i - j P j .
E ^ reflected , ( n x , n y , i z ) = j z = 1 N z R ^ ( n x , n y , i z + j z ) P ^ ( n x , n y , j z ) .
Y ^ ( n x , n y , i z ) = j z = 1 N z ( A ^ ( n x , n y , i z - j z ) + R ^ ( n x , n y , i z + j z ) ) P ^ ( n x , n y , j z ) ,
Y ^ ( n x , n y , i z ) * = j z = 1 N z ( A ^ ( n x , n y , i z - j z ) + R ^ ( n x , n y , i z + j z ) ) * P ^ ( n x , n y , j z ) .
E sca ( r ) = k 0 2   exp ( ik 0 r ) 4 π r   j = 1 N { exp ( - ik sca r j ) [ ( P j e ^ 1 ) e ^ 1 + ( P 1 e ^ 2 ) e ^ 2 ] + exp ( - ik I , sca r j ) × [ R TM ( P j e ^ 1 ) e ^ 1 + R TE ( P j e ^ 2 ) e ^ 2 ] } ,
I sca = E sca E sca - .
d C sca d Ω = lim Ω 0 C sca Ω I sca A I inc ( A / r 2 ) = r 2 I sca I inc ,
α i = α i ( 0 ) 1 + ( α ( 0 ) / d 3 ) [ ( b 1 + m 2 b 2 + m 2 b 3 S ) ( k 0 d ) 2 - ( 2 / 3 ) i ( k 0 d ) 3 ] ,
α j ( 0 ) = 3 0   m j 2 - 1 m j 2 + 2   Δ V j ,
b 1 = - 1.8915316 ,
b 2 = 0.1648469 ,
b 3 = - 1.7700004 ,
S j ( a j e j ( 0 ) ) 2 ,
G ( R ) = ( I - R ˆ R ˆ ) g ( R ) + i kR   ( I - 3 R ˆ R ˆ ) g ( R ) - 1 k 2 R 2   ( I - 3 R ˆ R ˆ ) g ( R ) ,
R = r - r ,
R = | r - r | ,
R ˆ = R / R ,
g ( R ) = ( 4 π R ) - 1   exp ( ikR ) .
A ii = 0 0 0 0 0 0 0 0 0 .
A ij = C ij β ij + γ ij r ^ ij , x 2 γ ij r ^ ij , x r ^ ij , y γ ij r ^ ij , x r ^ ij , z γ ij r ^ ij , y r ^ ij , x β ij + γ ij r ^ j , y 2 γ ij r ^ ij , y r ^ ij , z γ ij r ^ ij , z r ^ ij , x γ ij r ^ ij , z r ^ ij , y β ij + γ ij r ^ ij , z 2 ,
r ^ ij , x = r ij , x r ij , r ^ ij , y = r ij , y r ij , r ^ ij , z = r ij , z r ij ,
C ij = - k 0 2 4 π 0   exp ( ik 0 r ij ) ( r ij ) - 1 ,
β ij = [ 1 - ( k 0 r ij ) - 2 + i ( k 0 r ij ) - 1 ] ,
γ ij = - [ 1 - 3 ( k 0 r ij ) - 2 + 3 ( k 0 r ij ) - 1 i ] ,
r ij = [ ( x i - x r ) 2 + ( y i - y j ) 2 + ( z i - z j ) 2 ] 1 / 2 .
E x ( x ) = C x ρ 2 I ρ H - y ρ 2 I ϕ H - k 1 2 - k 2 2 k 1 2 + k 2 2 × 2 x 2 + k 2 2 4 π g I P x ,
E y ( x ) = C xy ρ 2   ( I ρ H + I ϕ H ) - k 1 2 - k 2 2 k 1 2 + k 2 2 × 2 x y   4 π g I P x ,
E z ( x ) = - C x ρ   I ρ V + k 1 2 - k 2 2 k 1 2 + k 2 2   2 x z   4 π g I P x ;
E x ( y ) = C xy ρ 2   ( I ρ H + I ϕ H ) - k 1 2 - k 2 2 k 1 2 + k 2 2   2 x y   4 π g I P y ,
E y ( y ) = C y ρ 2 I ρ H - x ρ 2 I ϕ H - k 1 2 - k 2 2 k 1 2 + k 2 2 × 2 y 2 + k 2 2 4 π g I P y ,
E z ( y ) = - C y ρ   I ρ V + k 1 2 - k 2 2 k 1 2 + k 2 2   2 y z   4 π g I P y ;
E x ( z ) = C x ρ   I ρ V + k 1 2 - k 2 2 k 1 2 + k 2 2   2 x z   4 π g I P z ,
E y ( z ) = C y ρ   I ρ V + k 1 2 - k 2 2 k 1 2 + k 2 2   2 y z   4 π g I P z ,
E z ( z ) = C I z V + k 1 2 - k 2 2 k 1 2 + k 2 2 × 2 z 2 + k 2 2 4 π g I P z ,
R ij = - ( 4 π 0 ) - 1 r ^ I , ij , x 2 I ρ H - r ^ I , ij , y 2 I ϕ H r ^ I , ij , x r ^ I , ij , y [ I ρ H + I ϕ H ] r ^ I , ij , x I ρ V r ^ I , ij , x r ^ I , ij , y [ I ρ H + I ϕ H ] r ^ I , ij , y 2 I ρ H - r ^ I , ij , x 2 I ϕ H r ^ I , ij , y I ρ V - r ^ I , ij , x I ρ V - r ^ I , ij , y I ρ V I z V - k 1 2 - k 2 2 k 1 2 + k 2 2   exp ( ik 0 r I , ij ) 4 π 0 r I , ij   - ( β I , ij + γ I , ij r ^ I , ij , x 2 ) - ( γ I , ij r ^ I , ij , x r ^ I , ij , y ) γ I , ij r ^ I , ij , x r ^ I , ij , z - ( γ I , ij r ^ I , ij , y r ^ I , ij , x ) - ( β I , ij + γ I , ij r ^ I , ij , y 2 ) γ I , ij r ^ I , ij , y r ^ I , ij , z - ( γ I , ij r ^ I , ij , z r ^ I , ij , x ) - ( γ I , ij r ^ I , ij , z r ^ I , ij , y ) β I , ij + γ I , ij r ^ I , ij , z 2 ,
r ^ I , ij , x = r I , ij , x r I , ij , r ^ I , ij , y = r I , ij , y r I , ij , r ^ I , ij , z = r I , ij , z r I , ij ,
β I , ij = [ 1 - ( k 0 r I , ij ) - 2 + i ( k 0 r I , ij ) - 1 ] ,
γ I , ij = - [ 1 - 3 ( k 0 r I , ij ) - 2 + 3 ( k 0 r I , ij ) - 1 i ] ,
r I , ij = | r i - I ¯ R r j | = [ ( x i - x j ) 2 + ( y i - y j ) 2 + ( z i + z j ) 2 ] 1 / 2 .

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