Abstract

The discrete-dipole approximation (DDA) is a powerful method for calculating absorption and scattering by targets that have sizes smaller than or comparable to the wavelength of the incident radiation. The DDA can be extended to targets that are singly or doubly periodic. We generalize the scattering amplitude matrix and the 4×4 Mueller matrix to describe scattering by singly and doubly periodic targets and show how these matrices can be calculated using the DDA. The accuracy of DDA calculations using the open-source code DDSCAT is demonstrated by comparison with exact results for infinite cylinders and infinite slabs. A method for using the DDA solution to obtain fields within and near the target is presented, with results shown for infinite slabs.

© 2008 Optical Society of America

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References

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  1. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705-714 (1973).
    [CrossRef]
  2. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848-872 (1988).
    [CrossRef]
  3. B. T. Draine and P. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491-1499 (1994).
    [CrossRef]
  4. B. T. Draine, “The discrete dipole approximation for light scattering by irregular targets,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M.I.Mishchenko, J.W.Hovenier, and L.D.Travis, eds. (Academic, 2000), pp. 131-145.
  5. R. Schmehl, B. M. Nebeker, and E. D. Hirleman, “Discrete-dipole approximation for scattering by features on surfaces by means of a two-dimensional fast Fourier transform technique,” J. Opt. Soc. Am. A 14, 3026-3036 (1997).
    [CrossRef]
  6. M. Paulus and O. J. F. Martin, “Green's tensor technique for scattering in two-dimensional stratified media,” Phys. Rev. E 63, 066615 (2001).
    [CrossRef]
  7. P. Yang and K. N. Liou, “Finite difference time domain method for light scattering by nonspherical and inhomogeneous 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. 173-221.
    [CrossRef]
  8. A. Taflove and S. C. Hagness, Advances in Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2005).
  9. V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure,” J. Mod. Opt. 40, 2281-2291 (1993).
    [CrossRef]
  10. P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, 165404 (2003).
    [CrossRef]
  11. P. C. Chaumet and A. Sentenac, “Numerical simulations of the electromagnetic field scattered by defects in a double-periodic structure,” Phys. Rev. B 72, 205437 (2005).
    [CrossRef]
  12. B. T. Draine and P. Flatau, “User Guide for the Discrete Dipole Approximation Code DDSCAT6.1,” http://arXiv.org/abs/astro-ph/0409262 (2004).
  13. B. T. Draine and J. Goodman, “Beyond Clausius-Mossotti--Wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685-697 (1993).
    [CrossRef]
  14. D. Gutkowicz-Krusin and B. T. Draine, “Propagation of electromagnetic waves on a rectangular lattice of polarizable points,” http://arXiv.org/abs/astro-ph/0403082 (2004).
  15. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
  16. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  17. D. Mackowski, private communication (2007).
  18. Z. N. Utegulov, J. M. Shaw, B. T. Draine, S. A. Kim, and W. L. Johnson, “Surface-plasmon enhancement of Brillouin light scattering from gold-nanodisk arrays on glass,” Proc. SPIE 6641, 66411M (2007).
    [CrossRef]
  19. W. L. Johnson, S. A. Kim, Z. N. Utegulov, and B. T. Draine, “Surface-plasmon fields in two-dimensional arrays of gold nanodisks,” Proc. SPIE7032(2008) (to be published).
  20. http://www.astro.princeton.edu/~draine/DDSCAT.html (2008).
  21. M. A. Botchev, SUBROUTINE ZBCG2, http://www.math.uu.nl/people/vorst/zbcg2.f90 (2001).
  22. J. J. Goodman, B. T. Draine, and P. J. Flatau, “Application of fast-Fourier transform techniques to the discrete dipole approximation,” Opt. Lett. 16, 1198-1200 (1990).
    [CrossRef]
  23. B. T. Draine and P. Flatau, “User Guide for the Discrete Dipole Approximation Code DDSCAT 7.0,” http://arXiv.org/abs/0809.0337 (2008).
  24. http://ddscat.wikidot.com.

2007 (1)

Z. N. Utegulov, J. M. Shaw, B. T. Draine, S. A. Kim, and W. L. Johnson, “Surface-plasmon enhancement of Brillouin light scattering from gold-nanodisk arrays on glass,” Proc. SPIE 6641, 66411M (2007).
[CrossRef]

2005 (1)

P. C. Chaumet and A. Sentenac, “Numerical simulations of the electromagnetic field scattered by defects in a double-periodic structure,” Phys. Rev. B 72, 205437 (2005).
[CrossRef]

2003 (1)

P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, 165404 (2003).
[CrossRef]

2001 (1)

M. Paulus and O. J. F. Martin, “Green's tensor technique for scattering in two-dimensional stratified media,” Phys. Rev. E 63, 066615 (2001).
[CrossRef]

1997 (1)

1994 (1)

1993 (2)

V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure,” J. Mod. Opt. 40, 2281-2291 (1993).
[CrossRef]

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

1990 (1)

1988 (1)

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

1973 (1)

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

Bohren, C. F.

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

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Botchev, M. A.

M. A. Botchev, SUBROUTINE ZBCG2, http://www.math.uu.nl/people/vorst/zbcg2.f90 (2001).

Bryant, G. W.

P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, 165404 (2003).
[CrossRef]

Chaumet, P. C.

P. C. Chaumet and A. Sentenac, “Numerical simulations of the electromagnetic field scattered by defects in a double-periodic structure,” Phys. Rev. B 72, 205437 (2005).
[CrossRef]

P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, 165404 (2003).
[CrossRef]

Draine, B. T.

Z. N. Utegulov, J. M. Shaw, B. T. Draine, S. A. Kim, and W. L. Johnson, “Surface-plasmon enhancement of Brillouin light scattering from gold-nanodisk arrays on glass,” Proc. SPIE 6641, 66411M (2007).
[CrossRef]

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

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

J. J. Goodman, B. T. Draine, and P. J. Flatau, “Application of fast-Fourier transform techniques to the discrete dipole approximation,” Opt. Lett. 16, 1198-1200 (1990).
[CrossRef]

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

B. T. Draine, “The discrete dipole approximation for light scattering by irregular targets,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M.I.Mishchenko, J.W.Hovenier, and L.D.Travis, eds. (Academic, 2000), pp. 131-145.

B. T. Draine and P. Flatau, “User Guide for the Discrete Dipole Approximation Code DDSCAT6.1,” http://arXiv.org/abs/astro-ph/0409262 (2004).

D. Gutkowicz-Krusin and B. T. Draine, “Propagation of electromagnetic waves on a rectangular lattice of polarizable points,” http://arXiv.org/abs/astro-ph/0403082 (2004).

B. T. Draine and P. Flatau, “User Guide for the Discrete Dipole Approximation Code DDSCAT 7.0,” http://arXiv.org/abs/0809.0337 (2008).

W. L. Johnson, S. A. Kim, Z. N. Utegulov, and B. T. Draine, “Surface-plasmon fields in two-dimensional arrays of gold nanodisks,” Proc. SPIE7032(2008) (to be published).

Flatau, P.

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

B. T. Draine and P. Flatau, “User Guide for the Discrete Dipole Approximation Code DDSCAT6.1,” http://arXiv.org/abs/astro-ph/0409262 (2004).

B. T. Draine and P. Flatau, “User Guide for the Discrete Dipole Approximation Code DDSCAT 7.0,” http://arXiv.org/abs/0809.0337 (2008).

Flatau, P. J.

Goodman, J.

B. T. Draine and 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.

Gutkowicz-Krusin, D.

D. Gutkowicz-Krusin and B. T. Draine, “Propagation of electromagnetic waves on a rectangular lattice of polarizable points,” http://arXiv.org/abs/astro-ph/0403082 (2004).

Hagness, S. C.

A. Taflove and S. C. Hagness, Advances in Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2005).

Hirleman, E. D.

Huffman, D. R.

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

Johnson, W. L.

Z. N. Utegulov, J. M. Shaw, B. T. Draine, S. A. Kim, and W. L. Johnson, “Surface-plasmon enhancement of Brillouin light scattering from gold-nanodisk arrays on glass,” Proc. SPIE 6641, 66411M (2007).
[CrossRef]

W. L. Johnson, S. A. Kim, Z. N. Utegulov, and B. T. Draine, “Surface-plasmon fields in two-dimensional arrays of gold nanodisks,” Proc. SPIE7032(2008) (to be published).

Kim, S. A.

Z. N. Utegulov, J. M. Shaw, B. T. Draine, S. A. Kim, and W. L. Johnson, “Surface-plasmon enhancement of Brillouin light scattering from gold-nanodisk arrays on glass,” Proc. SPIE 6641, 66411M (2007).
[CrossRef]

W. L. Johnson, S. A. Kim, Z. N. Utegulov, and B. T. Draine, “Surface-plasmon fields in two-dimensional arrays of gold nanodisks,” Proc. SPIE7032(2008) (to be published).

Liou, K. N.

P. Yang and K. N. Liou, “Finite difference time domain method for light scattering by nonspherical and inhomogeneous 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. 173-221.
[CrossRef]

Mackowski, D.

D. Mackowski, private communication (2007).

Markel, V. A.

V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure,” J. Mod. Opt. 40, 2281-2291 (1993).
[CrossRef]

Martin, O. J. F.

M. Paulus and O. J. F. Martin, “Green's tensor technique for scattering in two-dimensional stratified media,” Phys. Rev. E 63, 066615 (2001).
[CrossRef]

Nebeker, B. M.

Paulus, M.

M. Paulus and O. J. F. Martin, “Green's tensor technique for scattering in two-dimensional stratified media,” Phys. Rev. E 63, 066615 (2001).
[CrossRef]

Pennypacker, C. R.

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

Rahmani, A.

P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, 165404 (2003).
[CrossRef]

Schmehl, R.

Sentenac, A.

P. C. Chaumet and A. Sentenac, “Numerical simulations of the electromagnetic field scattered by defects in a double-periodic structure,” Phys. Rev. B 72, 205437 (2005).
[CrossRef]

Shaw, J. M.

Z. N. Utegulov, J. M. Shaw, B. T. Draine, S. A. Kim, and W. L. Johnson, “Surface-plasmon enhancement of Brillouin light scattering from gold-nanodisk arrays on glass,” Proc. SPIE 6641, 66411M (2007).
[CrossRef]

Taflove, A.

A. Taflove and S. C. Hagness, Advances in Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2005).

Utegulov, Z. N.

Z. N. Utegulov, J. M. Shaw, B. T. Draine, S. A. Kim, and W. L. Johnson, “Surface-plasmon enhancement of Brillouin light scattering from gold-nanodisk arrays on glass,” Proc. SPIE 6641, 66411M (2007).
[CrossRef]

W. L. Johnson, S. A. Kim, Z. N. Utegulov, and B. T. Draine, “Surface-plasmon fields in two-dimensional arrays of gold nanodisks,” Proc. SPIE7032(2008) (to be published).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Yang, P.

P. Yang and K. N. Liou, “Finite difference time domain method for light scattering by nonspherical and inhomogeneous 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. 173-221.
[CrossRef]

Astrophys. J. (3)

E. M. Purcell and 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 interstellar graphite grains,” Astrophys. J. 333, 848-872 (1988).
[CrossRef]

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

J. Mod. Opt. (1)

V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure,” J. Mod. Opt. 40, 2281-2291 (1993).
[CrossRef]

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

Opt. Lett. (1)

Phys. Rev. B (2)

P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, 165404 (2003).
[CrossRef]

P. C. Chaumet and A. Sentenac, “Numerical simulations of the electromagnetic field scattered by defects in a double-periodic structure,” Phys. Rev. B 72, 205437 (2005).
[CrossRef]

Phys. Rev. E (1)

M. Paulus and O. J. F. Martin, “Green's tensor technique for scattering in two-dimensional stratified media,” Phys. Rev. E 63, 066615 (2001).
[CrossRef]

Proc. SPIE (1)

Z. N. Utegulov, J. M. Shaw, B. T. Draine, S. A. Kim, and W. L. Johnson, “Surface-plasmon enhancement of Brillouin light scattering from gold-nanodisk arrays on glass,” Proc. SPIE 6641, 66411M (2007).
[CrossRef]

Other (13)

W. L. Johnson, S. A. Kim, Z. N. Utegulov, and B. T. Draine, “Surface-plasmon fields in two-dimensional arrays of gold nanodisks,” Proc. SPIE7032(2008) (to be published).

http://www.astro.princeton.edu/~draine/DDSCAT.html (2008).

M. A. Botchev, SUBROUTINE ZBCG2, http://www.math.uu.nl/people/vorst/zbcg2.f90 (2001).

B. T. Draine and P. Flatau, “User Guide for the Discrete Dipole Approximation Code DDSCAT 7.0,” http://arXiv.org/abs/0809.0337 (2008).

http://ddscat.wikidot.com.

P. Yang and K. N. Liou, “Finite difference time domain method for light scattering by nonspherical and inhomogeneous 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. 173-221.
[CrossRef]

A. Taflove and S. C. Hagness, Advances in Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2005).

B. T. Draine, “The discrete dipole approximation for light scattering by irregular targets,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M.I.Mishchenko, J.W.Hovenier, and L.D.Travis, eds. (Academic, 2000), pp. 131-145.

B. T. Draine and P. Flatau, “User Guide for the Discrete Dipole Approximation Code DDSCAT6.1,” http://arXiv.org/abs/astro-ph/0409262 (2004).

D. Gutkowicz-Krusin and B. T. Draine, “Propagation of electromagnetic waves on a rectangular lattice of polarizable points,” http://arXiv.org/abs/astro-ph/0403082 (2004).

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

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

D. Mackowski, private communication (2007).

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Figures (7)

Fig. 1
Fig. 1

(a) Target consisting of a 1-D array of TUCs and (b) showing how an infinite cylinder can be constructed from disklike TUCs (lower). The M = 0 scattering cone with α s = α 0 is illustrated.

Fig. 2
Fig. 2

(a) Target consisting of a 2-D array of TUCs and (b) showing how an infinite slab is created from TUCs consisting of a single “line” of dipoles.

Fig. 3
Fig. 3

Scattering by an infinite cylinder with diameter D and m = 1.33 + 0.01 i , for radiation with x = π D λ = 50 and incidence angle α 0 = 60 ° . (a) S 11 ( 1 d ) : solid curve, exact solution; broken curves, DDA results for D d = 256 , 360, and 512 ( N = 51,676 , 102,036, 206,300 dipoles per TUC). (b) Fractional error in S 11 ( 1 d ) ( DDA ) . (c) S 21 ( 1 d ) . (d) Error in S 21 ( 1 d ) .

Fig. 4
Fig. 4

Scattering by an infinite cylinder with diameter D and m = 1.33 + 0.01 i , for radiation with π D λ = 50 and incidence angle α 0 = 60 ° . (a) Exact solution (solid curve) and DDA results for D d = 512 and various values of the interaction cutoff parameter γ. (b) Fractional error in S 11 ( 1 d ) . (c), (d) Same as (a), (b), but expanding the region 0 < ζ < 20 ° . For this case, results computed with γ = 0.002 and 0.001 are nearly indistinguishable.

Fig. 5
Fig. 5

Light scattered by an infinite cylinder with m = 2 + i for radiation with x = 2 π R λ = 25 and incidence angle α 0 = 60 ° . (a) S 11 ( 1 d ) : solid curve, exact solution; broken curves, DDA results for D d = 128 , 180, and 256 ( N = 12,972 , 25,600, 51,676 dipoles per TUC). (b) Fractional error in S 11 ( 1 d ) .

Fig. 6
Fig. 6

Transmission and reflection coefficients for radiation of wavelength λ incident at angle θ i = ( π 2 α 0 ) = 40 ° relative to the normal on a slab with thickness h, incident E and ⊥ to the scattering plane, as a function of Re ( m ) h λ . (a) Nonabsorbing slab with m = 1.5 . (b) Absorbing slab with m = 1.5 + 0.02 i . Solid curves exact solution; symbols, results calculated with the DDA using dipole spacing d = h 10 , h 20 , and h 40 .

Fig. 7
Fig. 7

E 2 E 0 2 within and near the dielectric slab of Fig. 6b for slab thickness h = 0.2 λ , incidence angle α i = 40 ° , and incident polarizations ∥ and ⊥ to the scattering plane. Results were calculated using Eq. (76) with the slab represented by N x = 10 and N x = 20 dipole layers (i.e., dipole spacing d = 0.1 h and 0.05 h ). The circles along track 1 are at points where dipoles are located.

Tables (1)

Tables Icon

Table 1 CPU Time to Calculate Scattering by m = 1.33 + 0.01 i Infinite Cylinders on Single-Core 2.4 GHz AMD Opteron Model 250

Equations (88)

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

E inc ( r , t ) = E 0 exp ( i k 0 r i ω t ) ,
r j m n = r j 00 + m L u + n L v ,
A TUC L u × L v = L u L v sin θ u v ,
P j m n ( t ) = P j 00 ( t ) exp [ i ( m k 0 L u + n k 0 L v ) ] .
A ̃ j , k = m = n = n max n max ( 1 δ j k δ m 0 δ n 0 ) A j , k m n exp [ i ( m k 0 L u + n k 0 L v ) ] ,
A ̃ j , k m , n A j , k m n exp [ i ( m k 0 L u + n k 0 L v ) ( γ k 0 r j , k m n ) 4 ] ,
P j 00 = α j [ E inc ( r j ) k j A ̃ j , k P k 00 ] .
E j m n = k 0 2 exp ( i k 0 r r j m n ) r r j m n [ 1 ( r r j m n ) ( r r j m n ) r r j m n 2 ] P j m n ,
r r j m n = [ r 2 2 r r j m n + r j m n 2 ] 1 2 r { 1 r r j m n r 2 + 1 2 r 2 [ r j m n 2 ( r r j m n r ) 2 ] + } .
1 r r j m n [ 1 ( r r j m n ) ( r r j m n ) r r j m n 2 ] P j m n 1 r [ 1 k ̂ s k ̂ s ] P j m n .
E ( r ) = k 0 3 k 0 r exp ( i k 0 r ) [ 1 k ̂ s k ̂ s ] j P j 00 m , n exp ( i ψ j m n ) ,
ψ j m n m k 0 L u + n k 0 L v k s r j m n + k 0 2 r [ r j m n 2 ( k ̂ s r j m n ) 2 ]
k s r j 00 + m ( k 0 k s ) L u + n ( k 0 k s ) L v + 1 2 k 0 r [ m 2 ( k 0 2 k s u 0 ) L u 2 + n 2 ( k 0 2 k s v 2 ) L v 2 + 2 m n ( k 0 2 L u L v k s u k s v L u L v ) ] + O ( m L r ) ,
E s { k 0 2 r exp ( i k 0 r ) [ 1 k ̂ s k ̂ s ] j P j 00 exp ( i k s r j 00 ) } G ( r , k s ) ,
G ( r , k s ) m , n exp ( i Φ m n ) ,
Φ m n m ( k 0 k s ) L u + n ( k 0 k s ) L v + 1 2 k 0 r [ m 2 ( k 0 2 k s u 2 ) L u 2 + n 2 ( k 0 2 k s v 2 ) L v 2 + 2 m n ( k 0 2 L u L v k s u k s v L u L v ) ] .
F TUC ( k ̂ s ) k 0 3 [ 1 k ̂ s k ̂ s ] j = 1 N P j 00 exp ( i ω t i k s r j 00 ) ,
E s = exp ( i k s r i ω t ) k 0 r F TUC ( k ̂ s ) G ( r , k s ) .
E s = exp ( i k 0 r i ω t ) k 0 r F TUC ( k ̂ s ) .
I s = c E s 2 8 π = c 8 π k 0 2 r 2 F TUC 2 ,
d C sca d Ω = 1 k 0 2 F TUC 2 E 0 2 .
k s y = k 0 y + M 2 π L y M = 0 , ± 1 , ± 2 , ,
( k 0 y k 0 ) L y 2 π M ( k 0 y + k 0 ) L y 2 π .
k 0 y = k 0 cos α 0 ,
k s y = k 0 cos α s .
k s = k s y y ̂ + ( k 0 2 k s y 2 ) 1 2 sin α s sin α 0 [ ( k ̂ 0 y ̂ cos α 0 ) cos ζ + y ̂ × k ̂ 0 sin ζ ] ,
G = m = exp ( i Φ m 0 ) = m = exp [ 2 π i M m + i m 2 2 k 0 r ( k 0 2 k s y 2 ) L y 2 ]
lim ϵ 0 + d m exp [ i ( 1 + i ϵ ) 2 k 0 r m 2 ( k 0 2 k s y 2 ) L y 2 ]
= ( 2 π i k 0 r ) 1 2 ( k 0 2 k s y 2 ) 1 2 L y = ( 2 π i k 0 r ) 1 2 k 0 L y sin α s ,
E s = ( 2 π i k 0 r ) 1 2 exp ( i k 0 r i ω t ) k 0 L y sin α s F TUC ( k ̂ s )
d 2 P ¯ sca d L d ζ = E 0 2 8 π c d 2 C sca d L d ζ ,
d 2 C sca d L d ζ = 8 π E 0 2 1 c d 2 P ¯ sca d L d ζ = 8 π E 0 2 c E 2 c 8 π R sin α s
= 2 π k 0 3 L y 2 F TUC 2 E 0 2 .
( k s k 0 ) L u = 2 π M , M = 0 , ± 1 , ± 2 , ,
( k s k 0 ) L v = 2 π N , N = 0 , ± 1 , ± 2 , .
u 2 π x ̂ × L v x ̂ ( L u × L v ) , v 2 π x ̂ × L u x ̂ ( L v × L u ) .
k s k 0 + M u + N v .
k s x 2 = k 0 2 k 0 + M u + N v 2 > 0 .
sin α 0 k 0 x k 0 ,
sin α s k s x k 0 .
G = m , n exp ( i Φ m n ) = m , n exp { i 2 k 0 r [ ( k 0 2 k s u 2 ) L u 2 m 2 + ( k 0 2 k s v 2 ) L v 2 n 2 + 2 ( k 0 2 L u : L v k s u k s v L u L v ) m n ] } lim ϵ 0 + d m d n exp { i ( 1 + i ϵ ) 2 k 0 r [ ( k 0 2 k s u 2 ) L u 2 m 2 + ( k 0 2 k s v 2 ) L v 2 n 2 + 2 ( k 0 2 L u L v k s u k s v L u L v ) m n ] } = lim ϵ 0 + 1 A TUC d y d z exp { i ( 1 + i ϵ ) 2 k 0 r [ k 0 2 ( y 2 + z 2 ) ( k s y y + k s z z ) 2 ] } = 2 π i r k 0 A TUC sin α s .
E s = 2 π i exp ( i k s r i ω t ) k 0 2 A TUC sin α s F TUC ( k ̂ s ) .
E = exp ( i k 0 r i ω t ) [ E 0 + 2 π i F TUC ( k ̂ s = k ̂ 0 ) k 0 2 A TUC sin α s ] .
d C sca ( M , N ) d A = E 2 sin α s E 0 2 sin α 0
= 4 π 2 k 0 4 A TUC 2 sin α 0 sin α s F TUC ( k ̂ s ) 2 E 0 2 ,
T ( 0 , 0 ) = 1 E 0 2 E 0 + 2 π i F TUC ( k ̂ s = k ̂ 0 ) k 0 2 A TUC sin α 0 2 .
[ E s e ̂ s E s e ̂ s ] = i exp ( i k s r i ω t ) k 0 r [ S 2 ( 0 d ) S 3 ( 0 d ) S 4 ( 0 d ) S 1 ( 0 d ) ] [ E 0 e ̂ i E 0 e ̂ i ] ,
e ̂ i = e ̂ s k ̂ s × k ̂ 0 k ̂ s × k ̂ 0 = k ̂ s × k ̂ 0 1 ( k ̂ s k ̂ 0 ) 2 = ϕ ̂ s ,
e ̂ i k ̂ 0 × e ̂ i = k ̂ s ( k ̂ s k ̂ 0 ) k ̂ 0 1 ( k ̂ s k ̂ 0 ) 2 ,
e ̂ s k ̂ s × e ̂ s = k ̂ 0 + ( k ̂ s k ̂ 0 ) k ̂ s 1 ( k ̂ s k ̂ 0 ) 2 = θ ̂ s
[ E s e ̂ s E s e ̂ s ] = i exp ( i k s r i ω t ) ( k 0 R ) 1 2 [ S 2 ( 1 d ) S 3 ( 1 d ) S 4 ( 1 d ) S 1 ( 1 d ) ] [ E 0 e ̂ i E 0 e ̂ i ]
[ E s e ̂ s E s e ̂ s ] = i exp ( i k s r i ω t ) [ S 2 ( 2 d ) S 3 ( 2 d ) S 4 ( 2 d ) S 1 ( 2 d ) ] [ E 0 e ̂ i E 0 e ̂ i ]
e ̂ i = e ̂ s k 0 × k s k 0 × k s ,
[ E e ̂ s E e ̂ s ] = i exp ( i k 0 r i ω t ) [ ( S 2 ( 2 d ) i ) 0 0 ( S 1 ( 2 d ) i ) ] [ E 0 e ̂ i E 0 e ̂ i ] .
S 1 ( ν d ) = C ν e ̂ s F TUC ( k ̂ s , E 0 = e ̂ i ) ,
S 2 ( ν d ) = C ν e ̂ s F TUC ( k ̂ s , E 0 = e ̂ i ) ,
S 3 ( ν d ) = C ν e ̂ s F TUC ( k ̂ s , E 0 = e ̂ i ) ,
S 4 ( ν d ) = C ν e ̂ s F TUC ( k ̂ s , E 0 = e ̂ i ) ,
C 0 = i ,
C 1 = ( 2 π i sin α s ) 1 2 i k 0 L y ,
C 2 = 2 π k 0 2 A TUC sin α s ,
I sca , α 1 ( k 0 r ) 2 β = 1 4 S α β ( 0 d ) I inc , β ,
I sca , α 1 k 0 R β = 1 4 S α β ( 1 d ) I inc , β ,
I sca , α β = 1 4 S α β ( 2 d ) I inc , β .
S 11 ( ν d ) = 1 2 ( S 1 ( ν d ) 2 + S 2 ( ν d ) 2 + S 3 ( ν d ) 2 + S 4 ( ν d ) 2 ) ,
S 21 ( ν d ) = 1 2 ( S 2 ( ν d ) 2 S 1 ( ν d ) 2 S 4 ( ν d ) 2 + S 3 ( ν d ) 2 ) ,
S 14 ( ν d ) = Im ( S 2 ( ν d ) S 3 ( ν d ) * S 1 ( ν d ) S 4 ( ν d ) * ) .
S 11 ( 2 d ) ( k s = k 0 ) = 1 2 ( S 1 ( 2 d ) i 2 + S 2 ( 2 d ) i 2 + S 3 ( 2 d ) 2 + S 4 ( 2 d ) 2 ) .
R α β ( M , N ) = sin α s sin α 0 S α β ( 2 d ) for k s x k 0 x < 0 ,
T α β ( M , N ) = sin α s sin α 0 S α β ( 2 d ) for k s x k 0 x > 0 .
P abs Area E 0 2 c sin α 0 8 π = 1 M , N β = 1 4 [ R 1 β ( M , N ) + T 1 β ( M , N ) ] I inc , β I inc , 1 ,
R = S 11 ( 1 d ) ( k s x = k 0 x ) + S 12 ( 1 d ) ( k s x = k 0 x ) ,
R = S 11 ( 1 d ) ( k s x = k 0 x ) S 12 ( 1 d ) ( k s x = k 0 x ) ,
T = S 11 ( 1 d ) ( k s x = k 0 x ) + S 12 ( 1 d ) ( k s x = k 0 x ) ,
T = S 11 ( 1 d ) ( k s x = k 0 x ) S 12 ( 1 d ) ( k s x = k 0 x ) .
E ( r , t ) = e i ω t j m , n exp ( i k 0 R j m n ) R j m n 3 ϕ ( R j m n ) { k 0 2 R j m n × ( P j m n × R j m n ) + ( 1 i k 0 R j m n ) R j m n 2 [ 3 R j m n ( R j m n P j m n ) R j m n 2 P j m n ] } + E 0 exp ( i k 0 r i ω t ) ,
B ( r , t ) = e i ω t j m , n k 2 exp ( i k 0 R j m n ) R j m n 2 ϕ ( R j m n ) ( R j m n × P j m n ) ( 1 1 i k 0 R j m n ) + k ̂ 0 × E 0 exp ( i k 0 r i ω t ) ,
R j m n r r j m n ,
ϕ ( R ) exp [ γ ( k 0 R ) 4 ] × { 1 for R d ( R d ) 4 for R < d } .
d C x d L = 1 L y Q x π a eff 2
d C x d A = Q x π a eff 2 L u L v sin θ u v .
[ E s e ̂ s ( c k ) E s e ̂ s ( c k ) ] = i exp ( i k s r i ω t ) ( 2 i π k 0 R sin α ) 1 2 [ T 1 T 3 T 3 T 2 ] [ E 0 e ̂ i ( c k ) E 0 e ̂ i ( c k ) ] ,
θ = arccos [ 1 ( 1 cos ζ ) sin 2 α ] .
S = ( 2 i π sin α ) 1 2 A T B 1 ,
S [ S 2 ( 1 d ) S 3 ( 1 d ) S 4 ( 1 d ) S 1 ( 1 d ) ] , T [ T 1 T 3 T 3 T 2 ] ,
A [ e ̂ s e ̂ s ( c k ) e ̂ s e ̂ s ( c k ) e ̂ s e ̂ s ( c k ) e ̂ s e ̂ s ( c k ) ] = 1 sin θ [ cot α ( 1 cos θ ) sin α sin ζ sin α sin ζ cot α ( 1 cos θ ) ] ,
B [ e ̂ i e ̂ i ( c k ) e ̂ i e ̂ i ( c k ) e ̂ i e ̂ i ( c k ) e ̂ i e ̂ i ( c k ) ] = 1 sin θ [ cot α ( 1 cos θ ) sin α sin ζ sin α sin ζ cot α ( 1 cos θ ) ] ,
B 1 = sin θ cot 2 α ( 1 cos θ ) 2 + sin 2 α sin 2 ζ [ cot α ( 1 cos θ ) sin α sin ζ sin α sin ζ cot α ( 1 cos θ ) ] .

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