Abstract

We applied a discontinuous Galerkin time domain (DGTD) method, using a fourth-order Runga–Kutta time stepping of the Maxwell equations, to the simulation of the optical properties of dielectric particles in two-dimensional (2D) geometry. As examples of the numerical implementation of this method, the single-scattering properties of 2D circular and hexagonal particles are presented. In the case of circular particles, the scattering phase matrix was computed using the DGTD method and compared with the exact solution. For hexagonal particles, the DGTD method was used to compute single-scattering properties of randomly oriented 2D hexagonal ice crystals, and results were compared with those calculated using a geometric optics method. We consider both shortwave (visible) and longwave (infrared) cases, with particle size parameters 50 and 100. In the hexagonal case, scattering results are also presented as a function of both incident and scattering angles, revealing a structure apparently not reported before. Using the geometric optics method, we are able to interpret this structure in terms of contributions from varying numbers of internal reflections within the crystal.

© 2010 Optical Society of America

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References

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  1. T. Wriedt, “A review of elastic light scattering theories,” Part. Part. Syst. Charact. 15, 67–74 (1998).
    [CrossRef]
  2. A. Kokhanovsky, Optics of Light Scattering Media (Wiley, 1999).
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    [CrossRef]
  4. F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 775–824 (2003).
    [CrossRef]
  5. P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
    [CrossRef]
  6. M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
    [CrossRef]
  7. S. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
    [CrossRef]
  8. A. Taflove and S. C. Hagness, Computational Electromagnetics, 2nd ed. (Artech, 2000).
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    [CrossRef]
  10. W. Sun, Q. Fu, and Z. Chen, “Finite-difference time-domain solution of light scattering by dielectric particles with perfectly matched layer absorbing boundary conditions,” Appl. Opt. 38, 3141–3151 (1999).
    [CrossRef]
  11. Q. H. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength,” Microwave Opt. Technol. Lett. 15, 158–165 (1997).
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  12. Q. H. Liu, “The pseudospectral time-domain (PSTD) algorithm for acoustic waves in absorptive media,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 473 (1998).
    [CrossRef]
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    [CrossRef]
  16. I. Perugia and D. Schötzau, “The hp-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell Equations,” Math. Comput. 72, 1179–1214 (2002).
    [CrossRef]
  17. P. Houston, I. Perugia, and D. Schötzau, “Mixed discontinuous Galerkin approximation of the Maxwell operator,” SIAM J. Numer. Anal. 42, 434–450 (2004).
    [CrossRef]
  18. J. Hesthaven and T. Warburton, “Nodal high-order methods on unstructured grids. I. Time-domain solution of Maxwell’s equations,” J. Comput. Phys. 181, 186–221 (2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  24. P. Yang and 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]
  25. H. C. van de Hulst, Light Scattering by Small Particles(Dover, 1981).
  26. P.-O. Persson and G. Strang, “A simple mesh generator in MATLAB,” SIAM Rev. 46, 329–345 (2004).
    [CrossRef]
  27. W. J. Gordon and C. A. Hall, “Construction of curvilinear co-ordinate systems and application to mesh generation,” Int. J. Numer. Methods Eng. 7, 461–477 (1973).
    [CrossRef]
  28. E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 2nd ed. (Springer, 1999).
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    [CrossRef] [PubMed]

2007 (1)

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer 106, 558–589 (2007).
[CrossRef]

2004 (2)

P. Houston, I. Perugia, and D. Schötzau, “Mixed discontinuous Galerkin approximation of the Maxwell operator,” SIAM J. Numer. Anal. 42, 434–450 (2004).
[CrossRef]

P.-O. Persson and G. Strang, “A simple mesh generator in MATLAB,” SIAM Rev. 46, 329–345 (2004).
[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)

J. Hesthaven and T. Warburton, “Nodal high-order methods on unstructured grids. I. Time-domain solution of Maxwell’s equations,” J. Comput. Phys. 181, 186–221 (2002).
[CrossRef]

I. Perugia and D. Schötzau, “The hp-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell Equations,” Math. Comput. 72, 1179–1214 (2002).
[CrossRef]

1999 (1)

1998 (2)

T. Wriedt, “A review of elastic light scattering theories,” Part. Part. Syst. Charact. 15, 67–74 (1998).
[CrossRef]

Q. H. Liu, “The pseudospectral time-domain (PSTD) algorithm for acoustic waves in absorptive media,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 473 (1998).
[CrossRef]

1997 (1)

Q. H. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength,” Microwave Opt. Technol. Lett. 15, 158–165 (1997).
[CrossRef]

1996 (3)

P. Yang and 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]

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

1995 (1)

1994 (1)

1991 (1)

A. C. Cangellaris and D. B. Wright, “Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena,” IEEE Trans. Antennas Propag. 39, 1518–1525(1991).
[CrossRef]

1982 (1)

1973 (2)

W. J. Gordon and C. A. Hall, “Construction of curvilinear co-ordinate systems and application to mesh generation,” Int. J. Numer. Methods Eng. 7, 461–477 (1973).
[CrossRef]

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

1971 (1)

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

1966 (1)

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

Cai, Q.

Cangellaris, A. C.

A. C. Cangellaris and D. B. Wright, “Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena,” IEEE Trans. Antennas Propag. 39, 1518–1525(1991).
[CrossRef]

Carpenter, M. H.

M. H. Carpenter and C. A. Kennedy, “Fourth-order 2N-storage Runge–Kutta schemes,” TM 109112, NASA, Langley Research Center, VA, 1994.

Chen, Z.

Cockburn, B.

B. Cockburn, G. Karniadakis, and C.-W. Shu, “The development of discontinuous Galerkin methods,” in Discontinuous Galerkin Methods: Theory, Computation and Applications, B.Cockburn, G.Karniadakis, and C.-W.Shu, eds., Lecture Notes in Computational Science and Engineering (Springer, 2000), Vol.  11, pp. 3–50.
[CrossRef]

Draine, B. T.

Flatau, P. J.

Fu, Q.

Gedney, S. D.

S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

Gordon, W. J.

W. J. Gordon and C. A. Hall, “Construction of curvilinear co-ordinate systems and application to mesh generation,” Int. J. Numer. Methods Eng. 7, 461–477 (1973).
[CrossRef]

Hagness, S. C.

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

Hall, C. A.

W. J. Gordon and C. A. Hall, “Construction of curvilinear co-ordinate systems and application to mesh generation,” Int. J. Numer. Methods Eng. 7, 461–477 (1973).
[CrossRef]

Hesthaven, J.

J. Hesthaven and T. Warburton, “Nodal high-order methods on unstructured grids. I. Time-domain solution of Maxwell’s equations,” J. Comput. Phys. 181, 186–221 (2002).
[CrossRef]

Hesthaven, J. S.

J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods, Algorithms, Analysis, and Applications (Springer, 2008).
[CrossRef]

Hoekstra, A. G.

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer 106, 558–589 (2007).
[CrossRef]

Houston, P.

P. Houston, I. Perugia, and D. Schötzau, “Mixed discontinuous Galerkin approximation of the Maxwell operator,” SIAM J. Numer. Anal. 42, 434–450 (2004).
[CrossRef]

Hovenier, J. W.

M. I. Mishchenko, W. J. Wiscombe, J. W. Hovenier, and L. D. Travis, “Overview of scattering by nonspherical particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Geophysical Applications, M.I.Mishchenko, J.W.Hovenier, and L.D.Travis, eds. (Academic, 2000), pp. 29–60.
[CrossRef]

Kahnert, F. M.

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

Karniadakis, G.

B. Cockburn, G. Karniadakis, and C.-W. Shu, “The development of discontinuous Galerkin methods,” in Discontinuous Galerkin Methods: Theory, Computation and Applications, B.Cockburn, G.Karniadakis, and C.-W.Shu, eds., Lecture Notes in Computational Science and Engineering (Springer, 2000), Vol.  11, pp. 3–50.
[CrossRef]

Kennedy, C. A.

M. H. Carpenter and C. A. Kennedy, “Fourth-order 2N-storage Runge–Kutta schemes,” TM 109112, NASA, Langley Research Center, VA, 1994.

Kokhanovsky, A.

A. Kokhanovsky, Optics of Light Scattering Media (Wiley, 1999).

Liou, K. N.

Liu, Q. H.

Q. H. Liu, “The pseudospectral time-domain (PSTD) algorithm for acoustic waves in absorptive media,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 473 (1998).
[CrossRef]

Q. H. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength,” Microwave Opt. Technol. Lett. 15, 158–165 (1997).
[CrossRef]

Mackowski, D. W.

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

M. I. Mishchenko, W. J. Wiscombe, J. W. Hovenier, and L. D. Travis, “Overview of scattering by nonspherical particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Geophysical Applications, M.I.Mishchenko, J.W.Hovenier, and L.D.Travis, eds. (Academic, 2000), pp. 29–60.
[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]

Persson, P.-O.

P.-O. Persson and G. Strang, “A simple mesh generator in MATLAB,” SIAM Rev. 46, 329–345 (2004).
[CrossRef]

Perugia, I.

P. Houston, I. Perugia, and D. Schötzau, “Mixed discontinuous Galerkin approximation of the Maxwell operator,” SIAM J. Numer. Anal. 42, 434–450 (2004).
[CrossRef]

I. Perugia and D. Schötzau, “The hp-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell Equations,” Math. Comput. 72, 1179–1214 (2002).
[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]

Schötzau, D.

P. Houston, I. Perugia, and D. Schötzau, “Mixed discontinuous Galerkin approximation of the Maxwell operator,” SIAM J. Numer. Anal. 42, 434–450 (2004).
[CrossRef]

I. Perugia and D. Schötzau, “The hp-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell Equations,” Math. Comput. 72, 1179–1214 (2002).
[CrossRef]

Shu, C.-W.

B. Cockburn, G. Karniadakis, and C.-W. Shu, “The development of discontinuous Galerkin methods,” in Discontinuous Galerkin Methods: Theory, Computation and Applications, B.Cockburn, G.Karniadakis, and C.-W.Shu, eds., Lecture Notes in Computational Science and Engineering (Springer, 2000), Vol.  11, pp. 3–50.
[CrossRef]

Strang, G.

P.-O. Persson and G. Strang, “A simple mesh generator in MATLAB,” SIAM Rev. 46, 329–345 (2004).
[CrossRef]

Sun, W.

Taflove, A.

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

Toro, E. F.

E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 2nd ed. (Springer, 1999).

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

M. I. Mishchenko, W. J. Wiscombe, J. W. Hovenier, and L. D. Travis, “Overview of scattering by nonspherical particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Geophysical Applications, M.I.Mishchenko, J.W.Hovenier, and L.D.Travis, eds. (Academic, 2000), pp. 29–60.
[CrossRef]

van de Hulst, H. C.

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

Warburton, T.

J. Hesthaven and T. Warburton, “Nodal high-order methods on unstructured grids. I. Time-domain solution of Maxwell’s equations,” J. Comput. Phys. 181, 186–221 (2002).
[CrossRef]

J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods, Algorithms, Analysis, and Applications (Springer, 2008).
[CrossRef]

Waterman, P. C.

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

Wiscombe, W. J.

M. I. Mishchenko, W. J. Wiscombe, J. W. Hovenier, and L. D. Travis, “Overview of scattering by nonspherical particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Geophysical Applications, M.I.Mishchenko, J.W.Hovenier, and L.D.Travis, eds. (Academic, 2000), pp. 29–60.
[CrossRef]

Wriedt, T.

T. Wriedt, “A review of elastic light scattering theories,” Part. Part. Syst. Charact. 15, 67–74 (1998).
[CrossRef]

Wright, D. B.

A. C. Cangellaris and D. B. Wright, “Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena,” IEEE Trans. Antennas Propag. 39, 1518–1525(1991).
[CrossRef]

Yang, P.

Yee, S. K.

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

Yurkin, M. A.

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer 106, 558–589 (2007).
[CrossRef]

Appl. Opt. (2)

Astrophys. J. (1)

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

IEEE Trans. Antennas Propag. (3)

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

A. C. Cangellaris and D. B. Wright, “Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena,” IEEE Trans. Antennas Propag. 39, 1518–1525(1991).
[CrossRef]

S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

Q. H. Liu, “The pseudospectral time-domain (PSTD) algorithm for acoustic waves in absorptive media,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 473 (1998).
[CrossRef]

Int. J. Numer. Methods Eng. (1)

W. J. Gordon and C. A. Hall, “Construction of curvilinear co-ordinate systems and application to mesh generation,” Int. J. Numer. Methods Eng. 7, 461–477 (1973).
[CrossRef]

J. Comput. Phys. (1)

J. Hesthaven and T. Warburton, “Nodal high-order methods on unstructured grids. I. Time-domain solution of Maxwell’s equations,” J. Comput. Phys. 181, 186–221 (2002).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Quant. Spectrosc. Radiat. Transfer (3)

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

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

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer 106, 558–589 (2007).
[CrossRef]

Math. Comput. (1)

I. Perugia and D. Schötzau, “The hp-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell Equations,” Math. Comput. 72, 1179–1214 (2002).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

Q. H. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength,” Microwave Opt. Technol. Lett. 15, 158–165 (1997).
[CrossRef]

Part. Part. Syst. Charact. (1)

T. Wriedt, “A review of elastic light scattering theories,” Part. Part. Syst. Charact. 15, 67–74 (1998).
[CrossRef]

Phys. Rev. D (1)

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

SIAM J. Numer. Anal. (1)

P. Houston, I. Perugia, and D. Schötzau, “Mixed discontinuous Galerkin approximation of the Maxwell operator,” SIAM J. Numer. Anal. 42, 434–450 (2004).
[CrossRef]

SIAM Rev. (1)

P.-O. Persson and G. Strang, “A simple mesh generator in MATLAB,” SIAM Rev. 46, 329–345 (2004).
[CrossRef]

Other (8)

E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 2nd ed. (Springer, 1999).

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

M. H. Carpenter and C. A. Kennedy, “Fourth-order 2N-storage Runge–Kutta schemes,” TM 109112, NASA, Langley Research Center, VA, 1994.

B. Cockburn, G. Karniadakis, and C.-W. Shu, “The development of discontinuous Galerkin methods,” in Discontinuous Galerkin Methods: Theory, Computation and Applications, B.Cockburn, G.Karniadakis, and C.-W.Shu, eds., Lecture Notes in Computational Science and Engineering (Springer, 2000), Vol.  11, pp. 3–50.
[CrossRef]

J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods, Algorithms, Analysis, and Applications (Springer, 2008).
[CrossRef]

A. Kokhanovsky, Optics of Light Scattering Media (Wiley, 1999).

M. I. Mishchenko, W. J. Wiscombe, J. W. Hovenier, and L. D. Travis, “Overview of scattering by nonspherical particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Geophysical Applications, M.I.Mishchenko, J.W.Hovenier, and L.D.Travis, eds. (Academic, 2000), pp. 29–60.
[CrossRef]

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

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Figures (10)

Fig. 1
Fig. 1

Grids for the simulation of (a) an infinitely long circular cylinder and (b) a hexagonal cylinder. The size parameters are both 10. (Note: sizes are expressed in dimensional units.)

Fig. 2
Fig. 2

Left panels: nonzero phase matrix elements of an infinitely long circular cylinder with a size parameter of 50 and an incident wavelength of 0.532 μm . Right panels: corresponding relative error of each element of the phase matrix.

Fig. 3
Fig. 3

Same as Fig. 2 except that the size parameter is 100.

Fig. 4
Fig. 4

Same as Fig. 2, except that the incident wavelength is 12 μm .

Fig. 5
Fig. 5

Same as Fig. 4, except that the size parameter is 100.

Fig. 6
Fig. 6

Nonzero scattering phase matrix elements of randomly oriented 2D hexagonal ice crystals with a size parameter of 50 and an incident wavelength of 0.532 μm .

Fig. 7
Fig. 7

Same as Fig. 6, except that the size parameter is 100.

Fig. 8
Fig. 8

Contributions of the rays of various orders to the phase function of 2D randomly oriented hexagonal ice crystals (IGOM calculation).

Fig. 9
Fig. 9

(a) Contour of the log of the phase function (DG calculation) with respect to scattering angle and incident angle, for hexagonal particles with a size parameter of 100 and an incident wave length of 0.532 μm . (b) Incident angle ( θ i ) and scattering angle ( θ s ) are defined.

Fig. 10
Fig. 10

Decomposition of the log of the phase function using IGOM. The lower-right panel has the highest order approximation and should be compared with the DG calculation result in Fig. 9.

Tables (1)

Tables Icon

Table 1 Set of DGTD Simulations and Computer Time Consumed a

Equations (44)

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

t = λ 1 c 0 t * , ( x , y , z ) = λ 1 ( x * , y * , z * ) , E = ( Z 0 H 0 ) 1 E * , H = H 0 1 H * ,
ε E s t = × H s + ( 1 ε ) E i t σ ( E s + E i ) ,
μ H s t = × E s .
H x s t = E z s y ,
H y s t = E z s x ,
ε E z s t = H y s x H x s y + ( 1 ε ) E z i t σ ( E z s + E z i ) ,
ε E x s t = H z s y + ( 1 ε ) E x i t σ ( E x s + E x i ) ,
ε E y s t = H z s x + ( 1 ε ) E y i t σ ( E y s + E y i ) ,
H z s t = E y s x + E x s y .
C x s t = E z s y σ z C x s ,
C y s t = E z s x σ x C y s ,
D z s t = H y s x H x s y σ y D z s ,
H x s t = E z s y + ( σ x σ z ) C x s σ y H x s ,
H y s t = E z s x + ( σ y σ x ) C y s σ z H y s ,
E z s t = H y s x H x s y + ( σ z σ y ) D z s σ x E z s .
σ s = σ 0 ( s s i s o s i ) m ,
σ 0 = C ( m + 1 ) ( s o s i ) .
F ( x , y , t ) = i = 1 N p F ( ξ i , t ) l i ( x , y ) ,
F ( x , y , t ) t = i = 1 N p F ( ξ i , t ) t l i ( x , y ) ,
F ( x , y , t ) x = i = 1 N p F ( ξ i , t ) l i ( x , y ) x ,
F ( x , y , t ) y = i = 1 N p F ( ξ i , t ) l i ( x , y ) y ,
N p = N ( N + 1 ) 2 .
ε i = 1 N p E s ( ξ i , t ) t D k l i ( x , y ) l j ( x , y ) d x d y = i = 1 N p H s ( ξ i , t ) × D k [ l i ( x , y ) ] l j ( x , y ) d x d y { i = 1 N p [ n ^ × H s ( ξ i , t ) ( n ^ × H s ( ξ i , t ) ) * ] D k ¯ l i ( x , y ) l j ( x , y ) d s } + i = 1 N p ( 1 ε ( ξ i , t ) ) E i ( ξ i , t ) t D k l i ( x , y ) l j ( x , y ) d x d y i = 1 N p σ ( ξ i , t ) ( E s ( ξ i , t ) + E i ( ξ i , t ) ) D k l i ( x , y ) l j ( x , y ) d x d y ,
i = 1 N p H s ( ξ i , t ) t D k l i ( x , y ) l j ( x , y ) d x d y = i = 1 N p E s ( ξ i , t ) × D k [ l i ( x , y ) ] l j ( x , y ) d x d y + { i = 1 N p [ n ^ × E s ( ξ i , t ) ( n ^ × E s ( ξ i , t ) ) * ] D k ¯ l i ( x , y ) l j ( x , y ) d s } ,
i = 1 N p [ n ^ × H s ( ξ i , t ) ( n ^ × H s ( ξ i , t ) ) * ] D k l i ( x , y ) l j ( x , y ) d s = s = 1 3 N f p [ n ^ × H s ( ξ i s , t ) ( n ^ × H s ( ξ i s , t ) ) * ] D k l i s ( x , y ) l j ( x , y ) d s ,
i = 1 N p [ n ^ × E s ( ξ i , t ) ( n ^ × E s ( ξ i , t ) ) * ] D k l i ( x , y ) l j ( x , y ) d s = s = 1 3 N f p [ n ^ × E s ( ξ i s , t ) ( n ^ × E s ( ξ i s , t ) ) * ] D k l i s ( x , y ) l j ( x , y ) d s ,
M i j = D k l j ( x , y ) l i ( x , y ) d x d y ,
S i j x = D k l j ( x , y ) x l i ( x , y ) d x d y ,
S i j y = D k l j ( x , y ) y l i ( x , y ) d x d y ,
C i s = D k l j s ( x , y ) l i ( x , y ) d s .
n ^ × H s ( n ^ × H s ) * = 1 Z + Z + n ^ × [ Z + ( H s H s + ) α n ^ × ( E s E s + ) ] ,
n ^ × E s ( n ^ × E s ) * = 1 Y + Y + n ^ × [ Y + ( E s E s + ) + α n ^ × ( H s H s + ) ] ,
Z ± = 1 Y ± = μ ± ε ± .
H ˜ x s t = M 1 S y E ˜ z s + M 1 C F ˜ H x s ,
H ˜ y s t = M 1 S x E ˜ z s + M 1 C F ˜ H y s ,
ε E ˜ z s t = M 1 S x H ˜ y s M 1 S y H ˜ x s + M 1 C F ˜ E z s + ( 1 ε ) E ˜ z i t σ ( E ˜ z s + E ˜ z i ) .
C ˜ x s t = M 1 S y E ˜ z s + M 1 C F ˜ H x s σ z C ˜ x s ,
C ˜ y s t = M 1 S x E ˜ z s + M 1 C F ˜ H y s σ x C ˜ y s ,
D ˜ z s t = M 1 S x H ˜ y s M 1 S y H ˜ x s + M 1 C F ˜ E z s σ y D ˜ z s ,
H ˜ x s t = M 1 S y E ˜ z s + M 1 C F ˜ H x s + ( σ x σ z ) C ˜ x s σ y H ˜ x s ,
H ˜ y s t = M 1 S x E ˜ z s + M 1 C F ˜ H y s + ( σ y σ x ) C ˜ y s σ z H ˜ y s ,
E ˜ z s t = M 1 S x H ˜ y s M 1 S y H ˜ x s + M 1 C F ˜ E z s + ( σ z σ y ) D ˜ z s σ x E ˜ z s ,
F ˜ H x s = 1 Y + Y + { n ^ y ( E z s E z s + ) Y + + n ^ x [ n ^ x ( H x s H x s + ) + n ^ y ( H y s H y s + ) ] ( H x s H x s + ) } , F ˜ H y s = 1 Y + Y + { n ^ x ( E z s E z s + ) Y + + n ^ y [ n ^ x ( H x s H x s + ) + n ^ y ( H y s H y s + ) ] ( H y s H y s + ) } , F ˜ E z s = 1 Z + Z + { [ n ^ x ( H y s H y s + ) + n ^ y ( H x s H x s + ) ] Z + ( E z s E z s + ) } .
P 11 ( s ^ ) = [ | F TM ( s ^ ) | 2 + | F TE ( s ^ ) | 2 ] / 2 , P 12 ( s ^ ) = [ | F TM ( s ^ ) | 2 | F TE ( s ^ ) | 2 ] / 2 , P 33 ( s ^ ) = Re { F TM ( s ^ ) · [ F TE ( s ^ ) ] * } , P 34 ( s ^ ) = Im { F TM ( s ^ ) · [ F TE ( s ^ ) ] * } , F TM ( s ^ ) = k 4 [ n x H y s ( r ) n y H x s ( r ) ( n ^ · s ^ ) E z s ( r ) ] · exp ( i k s ^ · r ) d r , F TE ( s ^ ) = k 4 [ n x E y s ( r ) n y E x s ( r ) + ( n ^ · s ^ ) H z s ( r ) ] · exp ( i k s ^ · r ) d r .

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