Abstract

When the finite-difference time-domain (FDTD) method is applied to light-scattering computations, the far fields can be obtained by means of integrating the near fields either over the volume bounded by the particle’s surface or on a regular surface encompassing the scatterer. For light scattering by a sphere, the accurate near-field components on the FDTD-staggered meshes can be computed from the rigorous Lorenz-Mie theory. We investigate the errors associated with these near- to far-field transform methods for a canonical scattering problem associated with spheres. For a scatterer with a small refractive index, the surface-integral approach is more accurate than its volume counterpart for computation of the phase functions and extinction efficiencies; however, the volume-integral approach is more accurate for computation of other scattering matrix elements, such as P 12, P 33, and P 43, especially for backscattering. If a large refractive index is involved, the results computed from the volume-integration method become less accurate, whereas the surface method still retains the same order of accuracy as in the situation for the small refractive index.

© 2004 Optical Society of America

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  1. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
    [CrossRef]
  2. K. R. Umashankar, A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,” IEEE Trans. Electromagn. Compat. 24, 397–405 (1982).
    [CrossRef]
  3. A. Taflove, S. C. Hagness, Computational Electromagnetics, 2nd ed. (Artech House, Boston, Mass., 2000).
  4. K. S. Kunz, R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton, Fla., 1993).
  5. S. A. Schelkunoff, Electromagnetic Waves (Van Nostrand, New York, 1943).
  6. C. L. Britt, “Solution of electromagnetic scattering problems using time domain techniques,” IEEE Trans. Antennas Propag. 37, 1181–1191 (1989).
    [CrossRef]
  7. P. Yang, K. N. Liou, “Light scattering by hexagonal ice crystals: comparison of finite-difference time domain and geometric optics models,” J. Opt. Soc. Am. A 12, 162–176 (1995).
    [CrossRef]
  8. P. Yang, K. N. Liou, “Finite difference time domain method for light scattering by nonspherical particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, S. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), Chap. 7, pp. 173–221.
    [CrossRef]
  9. P. Yang, K. N. Liou, “Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space,” J. Opt. Soc. Am. A 13, 2072–2085 (1996).
    [CrossRef]
  10. W. Sun, Q. Fu, Z. Chen, “Finite-difference time-domain solution of light scattering by dielectric particles with a perfectly matched layer absorbing boundary condition,” Appl. Opt. 38, 3141–3151 (1999).
    [CrossRef]
  11. S. C. Hill, G. Videen, W. Sun, Q. Fu, “Scattering and internal fields of a microsphere that contains a saturable absorber: finite-difference time-domain simulations,” Appl. Opt. 40, 5487–5494 (2001).
    [CrossRef]
  12. P. J. Flatau, G. L. Stephens, B. T. Draine, “Light scattering by rectangular solids in the discrete-dipole approximation: a new algorithm exploiting the Block-Toeplitz structure,” J. Opt. Soc. Am. A 7, 593–600 (1990).
    [CrossRef]
  13. B. T. Draine, P. J. Flatau, “Discrete-dipole approximation for calculation,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
    [CrossRef]
  14. G. H. Goedecke, S. G. O’Brien, “Scattering by irregular inhomogeneous particles via the digitized Green’s function algorithm,” Appl. Opt. 27, 2431–2437 (1988).
    [CrossRef] [PubMed]
  15. T. Wriedt, U. Comberg, “Comparison of computational scattering method,” J. Quant. Spectrosc. Radiat. Transfer 60, 411–423 (1998).
    [CrossRef]
  16. M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002).
  17. F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 775–824 (2003).
    [CrossRef]
  18. M. I. Mishchenko, L. D. Travis, “Light scattering by polydispersions of randomly oriented spheroids with sizes comparable to wavelengths of observation,” Appl. Opt. 33, 7206–7225 (1994).
    [CrossRef] [PubMed]
  19. T. Wriedt, “Using the T-matrix method for light scattering computations by nonaxisymmetric particles: superellipsoids and realistically shaped particles,” Part. Part. Syst. Charact. 19, 256–268 (2002).
    [CrossRef]
  20. A. J. Baran, P. Yang, S. Havemann, “Calculation of the single-scattering properties of randomly oriented hexagonal ice columns: a comparison of the T-matrix and the finite-difference time-domain methods,” Appl. Opt. 40, 4376–4386 (2001).
    [CrossRef]
  21. G. Mie, “Beigrade zur optik truber medien, speziell kolloidaler metallosungen,” Ann. Phys. (Leipzig) 25, 377–455 (1908).
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  24. P. Yang, B.-C. Gao, W. J. Wiscombe, M. I. Mishchenko, S. E. Platnick, H.-L. Huang, B. A. Baum, Y. X. Hu, D. M. Winker, S.-C. Tsay, S. K. Park, “Inherent and apparent scattering properties of coated or uncoated spheres embedded in an absorbing host medium,” Appl. Opt. 41, 2740–2759 (2002).
    [CrossRef] [PubMed]
  25. W. J. Wiscombe, “Mie scattering calculation,” NCAR Tech. Note TN-140+STR (National Center for Atmospheric Research, Boulder, Colo., 1979).
  26. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [CrossRef] [PubMed]
  27. W. Sun, Q. Fu, “Finite-difference time-domain solution of light scattering by dielectric particles with large complex refractive indices,” Appl. Opt. 39, 5569–5578 (2000).
    [CrossRef]

2003 (1)

F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 775–824 (2003).
[CrossRef]

2002 (2)

2001 (2)

2000 (1)

1999 (1)

1998 (1)

T. Wriedt, U. Comberg, “Comparison of computational scattering method,” J. Quant. Spectrosc. Radiat. Transfer 60, 411–423 (1998).
[CrossRef]

1996 (1)

1995 (1)

1994 (2)

1990 (1)

1989 (1)

C. L. Britt, “Solution of electromagnetic scattering problems using time domain techniques,” IEEE Trans. Antennas Propag. 37, 1181–1191 (1989).
[CrossRef]

1988 (1)

1982 (1)

K. R. Umashankar, A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,” IEEE Trans. Electromagn. Compat. 24, 397–405 (1982).
[CrossRef]

1980 (1)

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

1908 (1)

G. Mie, “Beigrade zur optik truber medien, speziell kolloidaler metallosungen,” Ann. Phys. (Leipzig) 25, 377–455 (1908).

Baran, A. J.

Baum, B. A.

Bohren, C. F.

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

Britt, C. L.

C. L. Britt, “Solution of electromagnetic scattering problems using time domain techniques,” IEEE Trans. Antennas Propag. 37, 1181–1191 (1989).
[CrossRef]

Chen, Z.

Comberg, U.

T. Wriedt, U. Comberg, “Comparison of computational scattering method,” J. Quant. Spectrosc. Radiat. Transfer 60, 411–423 (1998).
[CrossRef]

Draine, B. T.

Flatau, P. J.

Fu, Q.

Gao, B.-C.

Goedecke, G. H.

Hagness, S. C.

A. Taflove, S. C. Hagness, Computational Electromagnetics, 2nd ed. (Artech House, Boston, Mass., 2000).

Havemann, S.

Hill, S. C.

Hu, Y. X.

Huang, H.-L.

Huffman, D. R.

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

Kahnert, F. M.

F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 775–824 (2003).
[CrossRef]

Kunz, K. S.

K. S. Kunz, R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton, Fla., 1993).

Lacis, A. A.

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

Liou, K. N.

P. Yang, K. N. Liou, “Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space,” J. Opt. Soc. Am. A 13, 2072–2085 (1996).
[CrossRef]

P. Yang, K. N. Liou, “Light scattering by hexagonal ice crystals: comparison of finite-difference time domain and geometric optics models,” J. Opt. Soc. Am. A 12, 162–176 (1995).
[CrossRef]

P. Yang, K. N. Liou, “Finite difference time domain method for light scattering by nonspherical particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, S. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), Chap. 7, pp. 173–221.
[CrossRef]

Luebbers, R. J.

K. S. Kunz, R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton, Fla., 1993).

Mie, G.

G. Mie, “Beigrade zur optik truber medien, speziell kolloidaler metallosungen,” Ann. Phys. (Leipzig) 25, 377–455 (1908).

Mishchenko, M. I.

O’Brien, S. G.

Park, S. K.

Platnick, S. E.

Schelkunoff, S. A.

S. A. Schelkunoff, Electromagnetic Waves (Van Nostrand, New York, 1943).

Stephens, G. L.

Sun, W.

Taflove, A.

K. R. Umashankar, A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,” IEEE Trans. Electromagn. Compat. 24, 397–405 (1982).
[CrossRef]

A. Taflove, S. C. Hagness, Computational Electromagnetics, 2nd ed. (Artech House, Boston, Mass., 2000).

Travis, L. D.

M. I. Mishchenko, L. D. Travis, “Light scattering by polydispersions of randomly oriented spheroids with sizes comparable to wavelengths of observation,” Appl. Opt. 33, 7206–7225 (1994).
[CrossRef] [PubMed]

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

Tsay, S.-C.

Umashankar, K. R.

K. R. Umashankar, A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,” IEEE Trans. Electromagn. Compat. 24, 397–405 (1982).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Videen, G.

Winker, D. M.

Wiscombe, W. J.

Wriedt, T.

T. Wriedt, “Using the T-matrix method for light scattering computations by nonaxisymmetric particles: superellipsoids and realistically shaped particles,” Part. Part. Syst. Charact. 19, 256–268 (2002).
[CrossRef]

T. Wriedt, U. Comberg, “Comparison of computational scattering method,” J. Quant. Spectrosc. Radiat. Transfer 60, 411–423 (1998).
[CrossRef]

Yang, P.

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

Ann. Phys. (Leipzig) (1)

G. Mie, “Beigrade zur optik truber medien, speziell kolloidaler metallosungen,” Ann. Phys. (Leipzig) 25, 377–455 (1908).

Appl. Opt. (8)

M. I. Mishchenko, L. D. Travis, “Light scattering by polydispersions of randomly oriented spheroids with sizes comparable to wavelengths of observation,” Appl. Opt. 33, 7206–7225 (1994).
[CrossRef] [PubMed]

W. Sun, Q. Fu, Z. Chen, “Finite-difference time-domain solution of light scattering by dielectric particles with a perfectly matched layer absorbing boundary condition,” Appl. Opt. 38, 3141–3151 (1999).
[CrossRef]

W. Sun, Q. Fu, “Finite-difference time-domain solution of light scattering by dielectric particles with large complex refractive indices,” Appl. Opt. 39, 5569–5578 (2000).
[CrossRef]

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[CrossRef] [PubMed]

G. H. Goedecke, S. G. O’Brien, “Scattering by irregular inhomogeneous particles via the digitized Green’s function algorithm,” Appl. Opt. 27, 2431–2437 (1988).
[CrossRef] [PubMed]

A. J. Baran, P. Yang, S. Havemann, “Calculation of the single-scattering properties of randomly oriented hexagonal ice columns: a comparison of the T-matrix and the finite-difference time-domain methods,” Appl. Opt. 40, 4376–4386 (2001).
[CrossRef]

S. C. Hill, G. Videen, W. Sun, Q. Fu, “Scattering and internal fields of a microsphere that contains a saturable absorber: finite-difference time-domain simulations,” Appl. Opt. 40, 5487–5494 (2001).
[CrossRef]

P. Yang, B.-C. Gao, W. J. Wiscombe, M. I. Mishchenko, S. E. Platnick, H.-L. Huang, B. A. Baum, Y. X. Hu, D. M. Winker, S.-C. Tsay, S. K. Park, “Inherent and apparent scattering properties of coated or uncoated spheres embedded in an absorbing host medium,” Appl. Opt. 41, 2740–2759 (2002).
[CrossRef] [PubMed]

IEEE Trans. Antennas Propag. (2)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

C. L. Britt, “Solution of electromagnetic scattering problems using time domain techniques,” IEEE Trans. Antennas Propag. 37, 1181–1191 (1989).
[CrossRef]

IEEE Trans. Electromagn. Compat. (1)

K. R. Umashankar, A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,” IEEE Trans. Electromagn. Compat. 24, 397–405 (1982).
[CrossRef]

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

J. Quant. Spectrosc. Radiat. Transfer (2)

F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 775–824 (2003).
[CrossRef]

T. Wriedt, U. Comberg, “Comparison of computational scattering method,” J. Quant. Spectrosc. Radiat. Transfer 60, 411–423 (1998).
[CrossRef]

Part. Part. Syst. Charact. (1)

T. Wriedt, “Using the T-matrix method for light scattering computations by nonaxisymmetric particles: superellipsoids and realistically shaped particles,” Part. Part. Syst. Charact. 19, 256–268 (2002).
[CrossRef]

Other (8)

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

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

W. J. Wiscombe, “Mie scattering calculation,” NCAR Tech. Note TN-140+STR (National Center for Atmospheric Research, Boulder, Colo., 1979).

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

P. Yang, K. N. Liou, “Finite difference time domain method for light scattering by nonspherical particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, S. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), Chap. 7, pp. 173–221.
[CrossRef]

A. Taflove, S. C. Hagness, Computational Electromagnetics, 2nd ed. (Artech House, Boston, Mass., 2000).

K. S. Kunz, R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, Boca Raton, Fla., 1993).

S. A. Schelkunoff, Electromagnetic Waves (Van Nostrand, New York, 1943).

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Figures (11)

Fig. 1
Fig. 1

Geometry of light scattering by a spherical particle and the coordinate system used.

Fig. 2
Fig. 2

Phase functions P 11 as a function of scattering angle computed by Mie theory, the SIM, and VIM for different size parameters for a grid size of λ/20. The refractive index is m = 1.0891 + 0.18216i, which is for ice crystals at a 10.8-μm wavelength. Also shown are the relative errors of each integration method.

Fig. 3
Fig. 3

Same as Fig. 2 except the refractive index is m = 7.1499 + 2.914i, which is for water at a 3.2-cm wavelength.

Fig. 4
Fig. 4

Case of size parameter X = 15 as in Fig. 3, but with a grid size of λ/40.

Fig. 5
Fig. 5

Phase matrix elements -P 12/P 11, P 33/P 11, and -P 43/P 11 as a function of scattering angle calculated by Mie theory, the SIM, and VIM. The refractive index is m = 1.0891 + 0.18216i. The size parameter is X = 30 and a grid size λ/20 is used. Also shown are the absolute errors of each integration method.

Fig. 6
Fig. 6

Same as Fig. 5 but for m = 7.1499 + 2.914i.

Fig. 7
Fig. 7

Extinction efficiencies as a function of size parameter computed by Mie theory, the SIM, and VIM for a grid size of λ/20. The refractive index is m = 1.0891 + 0.18216i. Also shown are the relative errors of the results.

Fig. 8
Fig. 8

Same as Fig. 7 but the refractive index is m = 7.1499 + 2.914i.

Fig. 9
Fig. 9

Distribution of |E x| over the plane of x = 0 for the x-polarization incident wave case. The refractive index is m = 1.0891 + 0.18216i and the size parameter is X = 5. The dimensionless coordinates are Y = 2πy/λ and Z = 2πz/λ, where y and z are the Cartesian coordinates in the direction of ê y and ê z .

Fig. 10
Fig. 10

Same as Fig. 9 but the refractive index is m = 7.1499 + 2.914i.

Fig. 11
Fig. 11

Phase function P 11 as a function of scattering angle computed by Mie theory, the SIM, and VIM for a spherical copper particle at a wavelength of 0.63 μm and a grid size of λ/20. The refractive index is m = 0.56 + 3.01i and the size parameter is X = 20. Also shown are the relative errors of each integration method.

Equations (45)

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

Eix, y, z=eˆx expikz-iωt,
Hix, y, z=eˆy expikz-iωt.
Esr=1k2r2cos φ n=12n+1×inian sin θπncos θξnkr,
Esθ=-1krcos φ n=12n+1nn+1×in-ianτncos θξnkr+bnπncos θξnkr,
Esφ=1krsin φ n=12n+1nn+1×in-ianπncos θξnkr+bnτncos θξnkr,
Hsr=1k2r2sin φ n=12n+1×inibn sin θπncos θξnkr,
Hsθ=-1krsin φ n=12n+1nn+1inanπncos θξnkr-ibnτncos θξnkr,
Hsφ=-1krcos φ n=12n+1nn+1inanτncos θξnkr-ibnπncos θξnkr.
E2=Ei+Es,
H2=Hi+Hs.
E1r=1mk2r2cos φ n=12n+1×inicn sin θπncos θψnmkr,
E1θ=1krcos φ n=12n+1nn+1in-icnτncos θ×ψnmkr+dnπncos θψnmkr,
E1φ=-1krsin φ n=12n+1nn+1in-icnπncos θ×ψnmkr+dnτncos θψnmkr,
n×H2-H1=0,
n×E2-E1=0,
n·H2-H1=0,
n·m22E2-m12E1=0.
Ex=Er sin θ cos φ+Eθ cos θ cos φ-Eφ sin φ,
Ey=Er sin θ sin φ+Eθ cos θ sin φ+Eφ cos φ,
Ez=Er cos θ-Eθ sin θ,
Es,rEs,r=expikr-z-ikrS2S3S4S1Ei,Ei,.
Esr=k2 expikr4πr Vεr-1Er-nˆnˆ·Erexp-iknˆ·rd3r,
S2S3S4S1=Fα,xFα,yFβ,xFβ,ycos φsin φsin φ-cos φ,
Fα,xrFβ,xr=-ik34π Vεr-1αˆ·Erβˆ·Er×exp-iknˆ·rd3r|Eix=1, Eiy=0,
Fα,yrFβ,yr=-ik34π Vεr-1αˆ·Erβˆ·Er×exp-iknˆ·rd3r|Eix=0, Eiy=1.
nˆ=βˆ×αˆ.
Esr=expikr-ikrk24πnˆ×nˆS×Er-nˆ×nˆS×Hrexp-iknˆ·rˆd2r,
Fα,xrFβ,xr=k24π αˆ·Zβˆ·Zexp-iknˆ·rd2r|Eix=1, Eiy=0,
Fα,yrFβ,yr=k24π αˆ·Zβˆ·Zexp-iknˆ·rd2r|Eix=0, Eiy=1,
αˆ·Z=Ez cos φ-Hy sin θ-Hz cos θ sin φ,
βˆ·Z=Ey sin θ+Ez cos θ sin φ+Hz cos φ,
X=S2 cos φ+S3 sin φαˆ+S4 cos φ+S1 sin φβˆ,
Y=S2 sin φ-S3 cos φαˆ+S4 sin φ-S1 cos φβˆ,
Sθ=S2θ00S1θ.
Cext=124πk2ReX·eˆxθ=0+ 4πk2ReY·eˆyθ=0.
Cext=124πk2ReS2+S1θ=0.
Csca=1202π 0π|X|2k2sin θdθdφ+02π 0π|Y|2k2sin θdθdφ.
Csca=πk2 0πS22+S12sin θdθ.
Cabs=Cext-Csca.
Qext=CextG, Qsca=CscaG, Qabs=CabsG,
Isr, θ, φQsr, θ, φUsr, θ, φVsr, θ, φ=Csca4πr2×P111P12/P1100P12/P1110000P33/P11-P43/P1100P43/P11P33/P11IiQiUiVi,
P114π=|S1|2+|S2|22k2Csca,
P12/P11=|S2|2-|S1|2|S2|2+|S1|2,
P33/P11=2 ReS1S2*|S2|2+|S1|2,
P43/P11=2 ImS1S2*|S2|2+|S1|2.

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