Abstract

The digitized Green’s function (DGF) algorithm and the underlying theory are described. This finite element algorithm models dielectric particles of arbitrary shape and arbitrary optical structure. DGF predictions of differential and total cross sections are compared with predictions of Mie and EBCM algorithms for several homogeneous spheres and spheroids. Results of tests of convergence of the DGF calculation as the number of elements are increased are presented. Computer time and storage requirements as functions of wavelength and particle size, shape, and optical structure are discussed.

© 1988 Optical Society of America

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References

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  1. Lord Rayleigh, “Dispersal of Light by a Dielectric Cylinder,” Philos. Mag. 36, 365 (1918).
    [CrossRef]
  2. J. R. Wait, “Scattering of a Plane Wave from a Circular Dielectric Cylinder of Oblique Incidence,” Can. J. Phys. 33, 189 (1955).
    [CrossRef]
  3. C. Yeh, “The Diffraction of Waves by a Penetrable Ribbon,” J. Math. Phys. 4, 65 (1963); C. Yeh, “Backscattering Cross Section of a Dielectric Elliptical Cylinder,” J. Opt. Soc. Am. 55, 309 (1965).
    [CrossRef]
  4. H. Weil, C. M. Chu, “Scattering and Absorption of Electromagnetic Radiation by Thin Dielectric Disks,” Appl. Opt. 15, 1832 (1976); H. Weil, C. M. Chu, “Scattering and Absorption by Thin Flat Aerosols,” Appl. Opt. 19, 2066 (1980).
    [CrossRef] [PubMed]
  5. S. Asano, G. Yamamoto, “Light Scattering by a Spheroidal Particle,” Appl. Opt. 14, 29 (1975).
    [PubMed]
  6. P. Barber, C. Yeh, “Scattering of Electromagnetic Waves by Arbitrarily Shaped Dielectric Bodies,” Appl. Opt. 14, 2864 (1975).
    [CrossRef] [PubMed]
  7. M. K. Iskander, A. Lakhtakia, C. M. Durney, “Scattering and Absorption by Dielectric Objects: An Improved Extended Boundary Condition Method of Solution,” Proc. IEEE 70, 1316 (1982). M. F. Iskander, A. Lakhtakia, C. M. Durney, “A New Procedure for Improving the Solution Stability and Extending the Frequency Range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317 (1983).
    [CrossRef]
  8. M. F. Iskander, A. Lakhtakia, “Extension of the Iterative EBCM to Calculate Scattering by Low-Loss or Lossless Elongated Dielectric Objects,” Appl. Opt. 23, 948 (1984).
    [CrossRef] [PubMed]
  9. Te-Kao Wu, L. L. Tsai, “Scattering from Arbitrarily Shaped Lossy Dielectric Bodies of Revolution,” Radio Sci. 12, 709 (1977).
    [CrossRef]
  10. A. L. Aden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242 (1951).
    [CrossRef]
  11. D. S. Wang, P. W. Barber, “Scattering by Inhomogeneous Nonspherical Objects,” Appl. Opt. 18, 1190 (1979).
    [CrossRef] [PubMed]
  12. E. M. Purcell, C. R. Pennypacker, “Scattering and Absorption of Light by Nonspherical Dielectric Grains,” Astrophys. J. 186, 705 (1973).
    [CrossRef]
  13. S. B. Singham, G. C. Salzman, “Evaluation of the Scattering Matrix of an Arbitrary Particle Using the Coupled Dipole Approximation,” J. Chem. Phys. 84, 2658 (1986).
    [CrossRef]
  14. M. A. Morgan, K. K. Mei, “Finite-Element Computation of Scattering by Inhomogeneous Penetrable Bodies of Revolution,” IEEE Trans. Antennas Propag. AP-27, 202 (1979).
    [CrossRef]
  15. D. H. Schaubert, D. R. Wilton, A. W. Glisson, “A Tetrahedral Modeling Method for Electromagnetic Scattering by Arbitrarily Shaped Inhomogeneous Dielectric Bodies,” IEEE Trans. Antennas Propag. AP-32, 77 (1984).
    [CrossRef]
  16. R. Holland, L. Simpson, K. S. Kunz, “Finite Difference Analysis of EMP Coupling to Lossy Dielectric Structures,” IEEE Trans. Electromagn. Compat. EMC-22, 203 (1980).
    [CrossRef]
  17. S. G. O’Brien, G. H. Goedecke, “Digitized Green Function Code for Scattering by Irregular Inhomogeneous Particles,” in Snow Symposium V, Vol. 1, (1986), p. 93.
  18. The S-matrix is defined by (E(s)(r)E(s)(r))∼exp(-ikr)ikr(SVHSHVSVHSVV)(EHEV).
  19. The Maxwell Garnett rule yields the average permittivity ∊ in terms of the constituent ∊i and packing fractions ϕi as (26)∊-1∊-2=∑iϕi∊i-1∊i+2.
  20. S. B. Singham, C. F. Bohren, “Light Scattering by an Arbitrary Particle: A Physical Reformulation of the Coupled Dipole Method,” Opt. Lett. 12, 10 (1987).
    [CrossRef] [PubMed]
  21. S. G. O’Brien, G. H. Goedecke, “Scattering of millimeter waves by snow crystals and equivalent homogeneous symmetric particles” and “Propagation of polarized millimeter waves through falling snow,” Appl. Opt. 27, 0000 (1988), same issue.
    [CrossRef]
  22. We do not consider tensor susceptibilities, so we cannot treat the analog of Singham and Salzman’s2 tensor polarizabilities.
  23. A. D. Yaghjian, “Electric Dyadic Green’s Functions in the Source Region,” IEEE Proc. 68, 248 (1980).
    [CrossRef]

1988

S. G. O’Brien, G. H. Goedecke, “Scattering of millimeter waves by snow crystals and equivalent homogeneous symmetric particles” and “Propagation of polarized millimeter waves through falling snow,” Appl. Opt. 27, 0000 (1988), same issue.
[CrossRef]

1987

1986

S. B. Singham, G. C. Salzman, “Evaluation of the Scattering Matrix of an Arbitrary Particle Using the Coupled Dipole Approximation,” J. Chem. Phys. 84, 2658 (1986).
[CrossRef]

1984

D. H. Schaubert, D. R. Wilton, A. W. Glisson, “A Tetrahedral Modeling Method for Electromagnetic Scattering by Arbitrarily Shaped Inhomogeneous Dielectric Bodies,” IEEE Trans. Antennas Propag. AP-32, 77 (1984).
[CrossRef]

M. F. Iskander, A. Lakhtakia, “Extension of the Iterative EBCM to Calculate Scattering by Low-Loss or Lossless Elongated Dielectric Objects,” Appl. Opt. 23, 948 (1984).
[CrossRef] [PubMed]

1982

M. K. Iskander, A. Lakhtakia, C. M. Durney, “Scattering and Absorption by Dielectric Objects: An Improved Extended Boundary Condition Method of Solution,” Proc. IEEE 70, 1316 (1982). M. F. Iskander, A. Lakhtakia, C. M. Durney, “A New Procedure for Improving the Solution Stability and Extending the Frequency Range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317 (1983).
[CrossRef]

1980

R. Holland, L. Simpson, K. S. Kunz, “Finite Difference Analysis of EMP Coupling to Lossy Dielectric Structures,” IEEE Trans. Electromagn. Compat. EMC-22, 203 (1980).
[CrossRef]

A. D. Yaghjian, “Electric Dyadic Green’s Functions in the Source Region,” IEEE Proc. 68, 248 (1980).
[CrossRef]

1979

M. A. Morgan, K. K. Mei, “Finite-Element Computation of Scattering by Inhomogeneous Penetrable Bodies of Revolution,” IEEE Trans. Antennas Propag. AP-27, 202 (1979).
[CrossRef]

D. S. Wang, P. W. Barber, “Scattering by Inhomogeneous Nonspherical Objects,” Appl. Opt. 18, 1190 (1979).
[CrossRef] [PubMed]

1977

Te-Kao Wu, L. L. Tsai, “Scattering from Arbitrarily Shaped Lossy Dielectric Bodies of Revolution,” Radio Sci. 12, 709 (1977).
[CrossRef]

1976

1975

1973

E. M. Purcell, C. R. Pennypacker, “Scattering and Absorption of Light by Nonspherical Dielectric Grains,” Astrophys. J. 186, 705 (1973).
[CrossRef]

1963

C. Yeh, “The Diffraction of Waves by a Penetrable Ribbon,” J. Math. Phys. 4, 65 (1963); C. Yeh, “Backscattering Cross Section of a Dielectric Elliptical Cylinder,” J. Opt. Soc. Am. 55, 309 (1965).
[CrossRef]

1955

J. R. Wait, “Scattering of a Plane Wave from a Circular Dielectric Cylinder of Oblique Incidence,” Can. J. Phys. 33, 189 (1955).
[CrossRef]

1951

A. L. Aden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

1918

Lord Rayleigh, “Dispersal of Light by a Dielectric Cylinder,” Philos. Mag. 36, 365 (1918).
[CrossRef]

Aden, A. L.

A. L. Aden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

Asano, S.

Barber, P.

Barber, P. W.

Bohren, C. F.

Chu, C. M.

Durney, C. M.

M. K. Iskander, A. Lakhtakia, C. M. Durney, “Scattering and Absorption by Dielectric Objects: An Improved Extended Boundary Condition Method of Solution,” Proc. IEEE 70, 1316 (1982). M. F. Iskander, A. Lakhtakia, C. M. Durney, “A New Procedure for Improving the Solution Stability and Extending the Frequency Range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317 (1983).
[CrossRef]

Glisson, A. W.

D. H. Schaubert, D. R. Wilton, A. W. Glisson, “A Tetrahedral Modeling Method for Electromagnetic Scattering by Arbitrarily Shaped Inhomogeneous Dielectric Bodies,” IEEE Trans. Antennas Propag. AP-32, 77 (1984).
[CrossRef]

Goedecke, G. H.

S. G. O’Brien, G. H. Goedecke, “Scattering of millimeter waves by snow crystals and equivalent homogeneous symmetric particles” and “Propagation of polarized millimeter waves through falling snow,” Appl. Opt. 27, 0000 (1988), same issue.
[CrossRef]

S. G. O’Brien, G. H. Goedecke, “Digitized Green Function Code for Scattering by Irregular Inhomogeneous Particles,” in Snow Symposium V, Vol. 1, (1986), p. 93.

Holland, R.

R. Holland, L. Simpson, K. S. Kunz, “Finite Difference Analysis of EMP Coupling to Lossy Dielectric Structures,” IEEE Trans. Electromagn. Compat. EMC-22, 203 (1980).
[CrossRef]

Iskander, M. F.

Iskander, M. K.

M. K. Iskander, A. Lakhtakia, C. M. Durney, “Scattering and Absorption by Dielectric Objects: An Improved Extended Boundary Condition Method of Solution,” Proc. IEEE 70, 1316 (1982). M. F. Iskander, A. Lakhtakia, C. M. Durney, “A New Procedure for Improving the Solution Stability and Extending the Frequency Range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317 (1983).
[CrossRef]

Kerker, M.

A. L. Aden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

Kunz, K. S.

R. Holland, L. Simpson, K. S. Kunz, “Finite Difference Analysis of EMP Coupling to Lossy Dielectric Structures,” IEEE Trans. Electromagn. Compat. EMC-22, 203 (1980).
[CrossRef]

Lakhtakia, A.

M. F. Iskander, A. Lakhtakia, “Extension of the Iterative EBCM to Calculate Scattering by Low-Loss or Lossless Elongated Dielectric Objects,” Appl. Opt. 23, 948 (1984).
[CrossRef] [PubMed]

M. K. Iskander, A. Lakhtakia, C. M. Durney, “Scattering and Absorption by Dielectric Objects: An Improved Extended Boundary Condition Method of Solution,” Proc. IEEE 70, 1316 (1982). M. F. Iskander, A. Lakhtakia, C. M. Durney, “A New Procedure for Improving the Solution Stability and Extending the Frequency Range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317 (1983).
[CrossRef]

Mei, K. K.

M. A. Morgan, K. K. Mei, “Finite-Element Computation of Scattering by Inhomogeneous Penetrable Bodies of Revolution,” IEEE Trans. Antennas Propag. AP-27, 202 (1979).
[CrossRef]

Morgan, M. A.

M. A. Morgan, K. K. Mei, “Finite-Element Computation of Scattering by Inhomogeneous Penetrable Bodies of Revolution,” IEEE Trans. Antennas Propag. AP-27, 202 (1979).
[CrossRef]

O’Brien, S. G.

S. G. O’Brien, G. H. Goedecke, “Scattering of millimeter waves by snow crystals and equivalent homogeneous symmetric particles” and “Propagation of polarized millimeter waves through falling snow,” Appl. Opt. 27, 0000 (1988), same issue.
[CrossRef]

S. G. O’Brien, G. H. Goedecke, “Digitized Green Function Code for Scattering by Irregular Inhomogeneous Particles,” in Snow Symposium V, Vol. 1, (1986), p. 93.

Pennypacker, C. R.

E. M. Purcell, C. R. Pennypacker, “Scattering and Absorption of Light by Nonspherical Dielectric Grains,” Astrophys. J. 186, 705 (1973).
[CrossRef]

Purcell, E. M.

E. M. Purcell, C. R. Pennypacker, “Scattering and Absorption of Light by Nonspherical Dielectric Grains,” Astrophys. J. 186, 705 (1973).
[CrossRef]

Rayleigh, Lord

Lord Rayleigh, “Dispersal of Light by a Dielectric Cylinder,” Philos. Mag. 36, 365 (1918).
[CrossRef]

Salzman, G. C.

S. B. Singham, G. C. Salzman, “Evaluation of the Scattering Matrix of an Arbitrary Particle Using the Coupled Dipole Approximation,” J. Chem. Phys. 84, 2658 (1986).
[CrossRef]

Schaubert, D. H.

D. H. Schaubert, D. R. Wilton, A. W. Glisson, “A Tetrahedral Modeling Method for Electromagnetic Scattering by Arbitrarily Shaped Inhomogeneous Dielectric Bodies,” IEEE Trans. Antennas Propag. AP-32, 77 (1984).
[CrossRef]

Simpson, L.

R. Holland, L. Simpson, K. S. Kunz, “Finite Difference Analysis of EMP Coupling to Lossy Dielectric Structures,” IEEE Trans. Electromagn. Compat. EMC-22, 203 (1980).
[CrossRef]

Singham, S. B.

S. B. Singham, C. F. Bohren, “Light Scattering by an Arbitrary Particle: A Physical Reformulation of the Coupled Dipole Method,” Opt. Lett. 12, 10 (1987).
[CrossRef] [PubMed]

S. B. Singham, G. C. Salzman, “Evaluation of the Scattering Matrix of an Arbitrary Particle Using the Coupled Dipole Approximation,” J. Chem. Phys. 84, 2658 (1986).
[CrossRef]

Tsai, L. L.

Te-Kao Wu, L. L. Tsai, “Scattering from Arbitrarily Shaped Lossy Dielectric Bodies of Revolution,” Radio Sci. 12, 709 (1977).
[CrossRef]

Wait, J. R.

J. R. Wait, “Scattering of a Plane Wave from a Circular Dielectric Cylinder of Oblique Incidence,” Can. J. Phys. 33, 189 (1955).
[CrossRef]

Wang, D. S.

Weil, H.

Wilton, D. R.

D. H. Schaubert, D. R. Wilton, A. W. Glisson, “A Tetrahedral Modeling Method for Electromagnetic Scattering by Arbitrarily Shaped Inhomogeneous Dielectric Bodies,” IEEE Trans. Antennas Propag. AP-32, 77 (1984).
[CrossRef]

Wu, Te-Kao

Te-Kao Wu, L. L. Tsai, “Scattering from Arbitrarily Shaped Lossy Dielectric Bodies of Revolution,” Radio Sci. 12, 709 (1977).
[CrossRef]

Yaghjian, A. D.

A. D. Yaghjian, “Electric Dyadic Green’s Functions in the Source Region,” IEEE Proc. 68, 248 (1980).
[CrossRef]

Yamamoto, G.

Yeh, C.

P. Barber, C. Yeh, “Scattering of Electromagnetic Waves by Arbitrarily Shaped Dielectric Bodies,” Appl. Opt. 14, 2864 (1975).
[CrossRef] [PubMed]

C. Yeh, “The Diffraction of Waves by a Penetrable Ribbon,” J. Math. Phys. 4, 65 (1963); C. Yeh, “Backscattering Cross Section of a Dielectric Elliptical Cylinder,” J. Opt. Soc. Am. 55, 309 (1965).
[CrossRef]

Appl. Opt.

Astrophys. J.

E. M. Purcell, C. R. Pennypacker, “Scattering and Absorption of Light by Nonspherical Dielectric Grains,” Astrophys. J. 186, 705 (1973).
[CrossRef]

Can. J. Phys.

J. R. Wait, “Scattering of a Plane Wave from a Circular Dielectric Cylinder of Oblique Incidence,” Can. J. Phys. 33, 189 (1955).
[CrossRef]

IEEE Proc

A. D. Yaghjian, “Electric Dyadic Green’s Functions in the Source Region,” IEEE Proc. 68, 248 (1980).
[CrossRef]

IEEE Trans. Antennas Propag.

M. A. Morgan, K. K. Mei, “Finite-Element Computation of Scattering by Inhomogeneous Penetrable Bodies of Revolution,” IEEE Trans. Antennas Propag. AP-27, 202 (1979).
[CrossRef]

D. H. Schaubert, D. R. Wilton, A. W. Glisson, “A Tetrahedral Modeling Method for Electromagnetic Scattering by Arbitrarily Shaped Inhomogeneous Dielectric Bodies,” IEEE Trans. Antennas Propag. AP-32, 77 (1984).
[CrossRef]

IEEE Trans. Electromagn. Compat.

R. Holland, L. Simpson, K. S. Kunz, “Finite Difference Analysis of EMP Coupling to Lossy Dielectric Structures,” IEEE Trans. Electromagn. Compat. EMC-22, 203 (1980).
[CrossRef]

J. Appl. Phys.

A. L. Aden, M. Kerker, “Scattering of Electromagnetic Waves from Two Concentric Spheres,” J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

J. Chem. Phys.

S. B. Singham, G. C. Salzman, “Evaluation of the Scattering Matrix of an Arbitrary Particle Using the Coupled Dipole Approximation,” J. Chem. Phys. 84, 2658 (1986).
[CrossRef]

J. Math. Phys.

C. Yeh, “The Diffraction of Waves by a Penetrable Ribbon,” J. Math. Phys. 4, 65 (1963); C. Yeh, “Backscattering Cross Section of a Dielectric Elliptical Cylinder,” J. Opt. Soc. Am. 55, 309 (1965).
[CrossRef]

Opt. Lett.

Philos. Mag.

Lord Rayleigh, “Dispersal of Light by a Dielectric Cylinder,” Philos. Mag. 36, 365 (1918).
[CrossRef]

Proc. IEEE

M. K. Iskander, A. Lakhtakia, C. M. Durney, “Scattering and Absorption by Dielectric Objects: An Improved Extended Boundary Condition Method of Solution,” Proc. IEEE 70, 1316 (1982). M. F. Iskander, A. Lakhtakia, C. M. Durney, “A New Procedure for Improving the Solution Stability and Extending the Frequency Range of the EBCM,” IEEE Trans. Antennas Propag. AP-31, 317 (1983).
[CrossRef]

Radio Sci.

Te-Kao Wu, L. L. Tsai, “Scattering from Arbitrarily Shaped Lossy Dielectric Bodies of Revolution,” Radio Sci. 12, 709 (1977).
[CrossRef]

Other

We do not consider tensor susceptibilities, so we cannot treat the analog of Singham and Salzman’s2 tensor polarizabilities.

S. G. O’Brien, G. H. Goedecke, “Digitized Green Function Code for Scattering by Irregular Inhomogeneous Particles,” in Snow Symposium V, Vol. 1, (1986), p. 93.

The S-matrix is defined by (E(s)(r)E(s)(r))∼exp(-ikr)ikr(SVHSHVSVHSVV)(EHEV).

The Maxwell Garnett rule yields the average permittivity ∊ in terms of the constituent ∊i and packing fractions ϕi as (26)∊-1∊-2=∑iϕi∊i-1∊i+2.

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Figures (5)

Fig. 1
Fig. 1

Comparison of EBCM and DGF for a homogeneous sphere. Wavelength = 10 mm, diameter = 2.25 mm, refractive index = 1.33–i0.1.

Fig. 2
Fig. 2

Comparison of the EBCM and DGF for a homogeneous prolate spheroid. Wavelength = 10 mm, major axis = 3.11 mm, minor axis = 1.03 mm, refractive index = 1.2–i0.05.

Fig. 3
Fig. 3

Comparison of the EBCM and DGF for a homogeneous oblate spheroid. Wavelength = 10 mm, major axis = 3.75 mm, minor axis = 0.75 mm, refractive index = 1.33–i0.01.

Fig. 4
Fig. 4

Variation of differential scattering cross section of the cylindrical rod for large even values of NM. Wavelength = 10 mm, length = 4 mm, diameter = 354 mm, refractive index = 1.785–i0.00324.

Fig. 5
Fig. 5

Convergence of total extinction cross section for the cylindrical rod. Wavelength = 10 mm, length = 4 mm, diameter = 0.354 mm, refractive index = 1.785–i0.00324.

Tables (2)

Tables Icon

Table I Memory Requirements for DGF Scattering Matrix (Kilobytes, Single Precision)

Tables Icon

Table II DGF Execution Times for Cylindrical Rod on MicroVAX Computer System

Equations (57)

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

E ( r ) [ 1 + 4 π χ ( r ) / 3 ] = E in ( r ) + d 3 r χ ( r ) G ( R ) · E ( r ) ,
E ( r ) E in ( r ) + ( i k r ) - 1 exp ( - i k r ) F ( r ^ ) ,
F ( r ^ ) = ( - r ^ r ^ ) · i k 3 d 3 r exp ( i k r ^ · r ) χ ( r ) E ( r ) ,
E in ( r ) = E 0 exp ( - i k · r ) , E 0 = 1.
σ s = k - 2 d Ω F ( r ^ ) 2 ;
σ a = - 4 π k d 3 r E ( r ) 2 Im [ χ ( r ) ] ;
σ e = 4 π k - 2 Re [ E 0 * · F ( k ^ ) ] ,
σ e = σ s + σ a .
d σ s / d Ω = k - 2 F ( r ^ ) 2 .
F ( r ^ ) = θ ^ F H ( r ^ ) + ϕ ^ F V ( r ^ ) ,
F V ( r ^ ) = ϕ ^ · F ( r ^ ) = - sin ϕ F 1 ( r ^ ) + cos ϕ F 2 ( r ^ ) .
E 0 = i ^ cos β + j ^ sin β .
S H H ( θ ) F H , β = 0 ( θ ) ,             S V H ( θ ) F V , β = 0 ( θ ) .
S H V ( θ ) F H , β = π / 2 ( θ ) ,             S V V ( θ ) F V , β = π / 2 ( θ ) .
σ a b ( θ ) = S a b ( θ ) 2 ,
σ H H = k - 2 d Ω F H ( r ^ ) 2 ,             σ V H = k - 2 d Ω F V ( r ^ ) 2 , σ H = σ H H + σ V H .
r α i = ( - D i / 2 ) + ( n α i - ½ ) d ,             n α i = ( 1 , N i ) ,
E α i ( 1 + 4 π χ α / 3 ) = E α i in + d 3 β = α G i j ( r α β ) χ β E β j + S α i ,
S α i = ( 4 π x α / 3 ) Γ E α i ,
Γ = ( 3 / 4 π ) 2 / 3 ( k d ) 2 - i ( k d ) 3 / 2 π ,
Λ α E α i = E α i in + d 3 β α G i j ( r α β ) χ β E β j
Λ α = 1 + ( 4 π χ α / 3 ) ( 1 - Γ ) .
E α i = I α i β j E β j in .
F i ( r ^ ) = ( δ i j - r ^ i r ^ j ) i k 3 d 3 α exp ( i k r ^ · r α ) χ α E α j
· E = - 4 π · P · B = 0 , × E = - i k B × B = i k ( E + 4 π P ) .
( 2 + k 2 ) E = - 4 π ( k 2 + ) · P ,
E ( r ) = E in ( r ) + d 3 r G 0 ( R ) ( k 2 + ) · P ( r ) ,
G 0 ( R ) = R - 1 exp ( - i k R ) ,             R = r - r ,
( k 2 + ) · d 3 r G 0 ( R ) P ( r ) .
G 0 ( R ) = G 0 ( R ) [ - k 2 R ^ R ^ - ( - 3 R ^ R ^ ) ( R - 2 + i k R - 1 ) ] - ( 4 π / 3 ) δ ( R ) ,
E ( r ) [ 1 + 4 π χ ( r ) / 3 ] = E in ( r ) + d 3 r G ( R ) · P ( r ) ,
G i j ( R ) = G 0 ( R ) [ k 2 ( δ i j - R ^ i R ^ j ) - ( δ i j - 3 R ^ i R ^ j ) ( R - 2 + i k R - 1 ) ] .
P ( r ) = χ ( r ) E ( r ) ,
χ ( r ) = ( 4 π ) - 1 [ ( r ) - 1 ] , m ( r ) = n ( r ) - i κ ( r ) = [ ( r ) ] 1 / 2 ,
S α i = χ α d 3 y G i j ( y ) E α j = χ α P i j E α j ,
P i j = cell d 3 y exp ( - i k y ) [ k 2 ( δ i j - y ^ i y ^ j ) y - 1 - ( δ i j - 3 y ^ i y ^ j ) ( y - 3 + i k y - 2 ) ] ,
d Ω y y ^ i y ^ j = d Ω y δ i j ,
P i j = ( 2 k 2 / 3 ) δ i j d 3 y y - 1 exp ( - i k y ) .
exp ( - i k y ) = 1 - i k y
P i j ( 2 k 2 / 3 ) δ i j ( d 3 y y - 1 - i k d 3 ) .
P i j = ( 4 π / 3 ) δ i j Γ ,
Γ = ( 3 / 4 π ) 2 / 3 ( k d ) 2 - i ( k d ) 3 / 2 π ,
k - 3 G 22 g = exp ( - i k d ) [ 1 / k d - 1 / ( k d ) 3 - i / ( k d ) 2 ] ,
E ( r ) = j ^ [ T ( r ) / Λ ] exp ( - i k · r ) ,
Λ = 1 + ( 4 π / 3 ) χ ( 1 - Γ )
T = ( 1 - f g ) - 1 ,
f ( k d ) 3 χ / Λ .
F ( r ^ ) = j ^ · ( - r ^ r ^ ) ( i f T ) β exp [ i k r β · ( r ^ - k ^ ) ] .
F ( k ^ ) = j ^ 2 i f T ,
σ e = ( 4 π / k 2 ) i ( f T - f * T * ) = i ( 4 π / k 2 ) T 2 [ ( f - f * ) + f 2 ( g - g * ) ] .
σ s = k - 2 d Ω F ( r ^ ) 2
σ s = [ ( 16 π / 3 k 2 ) + i ( 4 π / k 2 ) ( g - g * ) ] f 2 T 2 .
σ a = - ( 8 π / k 2 ) ( k d ) 3 Im ( χ ) T 2 / Λ 2 .
σ e = σ s + σ a
σ s + σ a - σ e = ( 16 π / 3 k 2 ) f 2 T 2 .
(E(s)(r)E(s)(r))exp(-ikr)ikr(SVHSHVSVHSVV)(EHEV).
-1-2=iϕii-1i+2.

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