Abstract

Natural particles such as ice crystals in cirrus clouds generally are not pristine but have additional microroughness on their surfaces. A two-dimensional finite-difference time-domain (FDTD) program with a perfectly matched layer absorbing boundary condition is developed to calculate the effect of surface roughness on light scattering by long ice columns. When we use a spatial cell size of 1/120 incident wavelength for ice circular cylinders with size parameters of 6 and 24 at wavelengths of 0.55 and 10.8 μm, respectively, the errors in the FDTD results in the extinction, scattering, and absorption efficiencies are smaller than ∼0.5%. The errors in the FDTD results in the asymmetry factor are smaller than ∼0.05%. The errors in the FDTD results in the phase-matrix elements are smaller than ∼5%. By adding a pseudorandom change as great as 10% of the radius of a cylinder, we calculate the scattering properties of randomly oriented rough-surfaced ice columns. We conclude that, although the effect of small surface roughness on light scattering is negligible, the scattering phase-matrix elements change significantly for particles with large surface roughness. The roughness on the particle surface can make the conventional phase function smooth. The most significant effect of the surface roughness is the decay of polarization of the scattered light.

© 2004 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. Part III: Light scattering by irregular ice crystals,” J. Atmos. Sci. 52, 818–837 (1995).
    [CrossRef]
  2. J.-M. Perrin, J.-P. Sivan, “Scattering and polarization of light by rough and porous interstellar grains,” Astron. Astrophys. 247, 497–504 (1991).
  3. D. B. Vaidya, R. Gupta, “Extinction by porous silicate and graphite grains,” Astron. Astrophys. 328, 634–640 (1997).
  4. K. Chamaillard, J.-P. J. Lafon, “Statistical approach of the effect of roughness on the polarization of light scattered by dust grains,” J. Quant. Spectrosc. Radiat. Transfer 70, 519–528 (2001).
    [CrossRef]
  5. E. M. Purcell, C. P. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 196, 705–714 (1973).
    [CrossRef]
  6. A. Mugnai, W. J. Wiscombe, “Scattering of radiation by moderately nonspherical particles,” J. Atmos. Sci. 37, 1291–1307 (1980).
    [CrossRef]
  7. J. I. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of light by stochastically rough particles,” Appl. Opt. 19, 4088–4095 (1989).
    [CrossRef]
  8. K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577–601 (1996).
    [CrossRef]
  9. A. Battaglia, K. Muinonen, T. Nousiainen, J. I. Peltoniemi, “Light scattering by Gaussian particles: Rayleigh-ellipsoid approximation,” J. Quant. Spectrosc. Radiat. Transfer 63, 277–303 (1999).
    [CrossRef]
  10. W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, G. Videen, “Light scattering by Gaussian particles: a solution with finite-difference time-domain technique,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 1083–1090 (2003).
    [CrossRef]
  11. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).
  12. 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]
  13. 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]
  14. W. Sun, Q. Fu, Z. 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]
  15. 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]
  16. W. Sun, N. G. Loeb, Q. Fu, “Finite-difference time-domain solution of light scattering and absorption by particles in an absorbing medium,” Appl. Opt. 41, 5728–5743 (2002).
    [CrossRef] [PubMed]
  17. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  18. D. E. Merewether, R. Fisher, F. W. Smith, “On implementing a numeric Huygen’s source in a finite difference program to illustrate scattering bodies,” IEEE Trans. Nucl. Sci. NS-27, 1829–1833 (1980).
    [CrossRef]
  19. A. Taflove, Computational Electrodynamics: the Finite-Difference Time Domain Method (Artech House, Boston, Mass., 1995).
  20. G. Lazzi, O. P. Gandhi, “On the optimal design of the PML absorbing boundary condition for the FDTD code,” IEEE Trans. Antennas Propag. 45, 914–916 (1996).
    [CrossRef]
  21. G. Videen, W. S. Bickel, “Light-scattering Mueller matrix for a rough fiber,” Appl. Opt. 31, 3488–3492 (1992).
    [CrossRef] [PubMed]
  22. G. Videen, W. S. Bickel, “Light-scattering Mueller matrix from a fiber as a function of MgO contamination,” Appl. Opt. 30, 3880–3885 (1991).
    [CrossRef] [PubMed]

2003 (1)

W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, G. Videen, “Light scattering by Gaussian particles: a solution with finite-difference time-domain technique,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 1083–1090 (2003).
[CrossRef]

2002 (1)

2001 (1)

K. Chamaillard, J.-P. J. Lafon, “Statistical approach of the effect of roughness on the polarization of light scattered by dust grains,” J. Quant. Spectrosc. Radiat. Transfer 70, 519–528 (2001).
[CrossRef]

2000 (1)

1999 (2)

W. Sun, Q. Fu, Z. 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]

A. Battaglia, K. Muinonen, T. Nousiainen, J. I. Peltoniemi, “Light scattering by Gaussian particles: Rayleigh-ellipsoid approximation,” J. Quant. Spectrosc. Radiat. Transfer 63, 277–303 (1999).
[CrossRef]

1997 (1)

D. B. Vaidya, R. Gupta, “Extinction by porous silicate and graphite grains,” Astron. Astrophys. 328, 634–640 (1997).

1996 (3)

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577–601 (1996).
[CrossRef]

G. Lazzi, O. P. Gandhi, “On the optimal design of the PML absorbing boundary condition for the FDTD code,” IEEE Trans. Antennas Propag. 45, 914–916 (1996).
[CrossRef]

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]

1995 (2)

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]

Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. Part III: Light scattering by irregular ice crystals,” J. Atmos. Sci. 52, 818–837 (1995).
[CrossRef]

1994 (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

1992 (1)

1991 (2)

G. Videen, W. S. Bickel, “Light-scattering Mueller matrix from a fiber as a function of MgO contamination,” Appl. Opt. 30, 3880–3885 (1991).
[CrossRef] [PubMed]

J.-M. Perrin, J.-P. Sivan, “Scattering and polarization of light by rough and porous interstellar grains,” Astron. Astrophys. 247, 497–504 (1991).

1989 (1)

J. I. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of light by stochastically rough particles,” Appl. Opt. 19, 4088–4095 (1989).
[CrossRef]

1980 (2)

A. Mugnai, W. J. Wiscombe, “Scattering of radiation by moderately nonspherical particles,” J. Atmos. Sci. 37, 1291–1307 (1980).
[CrossRef]

D. E. Merewether, R. Fisher, F. W. Smith, “On implementing a numeric Huygen’s source in a finite difference program to illustrate scattering bodies,” IEEE Trans. Nucl. Sci. NS-27, 1829–1833 (1980).
[CrossRef]

1973 (1)

E. M. Purcell, C. P. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 196, 705–714 (1973).
[CrossRef]

1966 (1)

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

Battaglia, A.

A. Battaglia, K. Muinonen, T. Nousiainen, J. I. Peltoniemi, “Light scattering by Gaussian particles: Rayleigh-ellipsoid approximation,” J. Quant. Spectrosc. Radiat. Transfer 63, 277–303 (1999).
[CrossRef]

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Bickel, W. S.

Chamaillard, K.

K. Chamaillard, J.-P. J. Lafon, “Statistical approach of the effect of roughness on the polarization of light scattered by dust grains,” J. Quant. Spectrosc. Radiat. Transfer 70, 519–528 (2001).
[CrossRef]

Chen, Z. Z.

Fast, P.

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577–601 (1996).
[CrossRef]

Fisher, R.

D. E. Merewether, R. Fisher, F. W. Smith, “On implementing a numeric Huygen’s source in a finite difference program to illustrate scattering bodies,” IEEE Trans. Nucl. Sci. NS-27, 1829–1833 (1980).
[CrossRef]

Fu, Q.

Gandhi, O. P.

G. Lazzi, O. P. Gandhi, “On the optimal design of the PML absorbing boundary condition for the FDTD code,” IEEE Trans. Antennas Propag. 45, 914–916 (1996).
[CrossRef]

Gupta, R.

D. B. Vaidya, R. Gupta, “Extinction by porous silicate and graphite grains,” Astron. Astrophys. 328, 634–640 (1997).

Irvine, W. M.

J. I. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of light by stochastically rough particles,” Appl. Opt. 19, 4088–4095 (1989).
[CrossRef]

Lafon, J.-P. J.

K. Chamaillard, J.-P. J. Lafon, “Statistical approach of the effect of roughness on the polarization of light scattered by dust grains,” J. Quant. Spectrosc. Radiat. Transfer 70, 519–528 (2001).
[CrossRef]

Lazzi, G.

G. Lazzi, O. P. Gandhi, “On the optimal design of the PML absorbing boundary condition for the FDTD code,” IEEE Trans. Antennas Propag. 45, 914–916 (1996).
[CrossRef]

Liou, K. N.

Loeb, N. G.

W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, G. Videen, “Light scattering by Gaussian particles: a solution with finite-difference time-domain technique,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 1083–1090 (2003).
[CrossRef]

W. Sun, N. G. Loeb, Q. Fu, “Finite-difference time-domain solution of light scattering and absorption by particles in an absorbing medium,” Appl. Opt. 41, 5728–5743 (2002).
[CrossRef] [PubMed]

Lumme, K.

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577–601 (1996).
[CrossRef]

J. I. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of light by stochastically rough particles,” Appl. Opt. 19, 4088–4095 (1989).
[CrossRef]

Merewether, D. E.

D. E. Merewether, R. Fisher, F. W. Smith, “On implementing a numeric Huygen’s source in a finite difference program to illustrate scattering bodies,” IEEE Trans. Nucl. Sci. NS-27, 1829–1833 (1980).
[CrossRef]

Mugnai, A.

A. Mugnai, W. J. Wiscombe, “Scattering of radiation by moderately nonspherical particles,” J. Atmos. Sci. 37, 1291–1307 (1980).
[CrossRef]

Muinonen, K.

W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, G. Videen, “Light scattering by Gaussian particles: a solution with finite-difference time-domain technique,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 1083–1090 (2003).
[CrossRef]

A. Battaglia, K. Muinonen, T. Nousiainen, J. I. Peltoniemi, “Light scattering by Gaussian particles: Rayleigh-ellipsoid approximation,” J. Quant. Spectrosc. Radiat. Transfer 63, 277–303 (1999).
[CrossRef]

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577–601 (1996).
[CrossRef]

J. I. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of light by stochastically rough particles,” Appl. Opt. 19, 4088–4095 (1989).
[CrossRef]

Nousiainen, T.

W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, G. Videen, “Light scattering by Gaussian particles: a solution with finite-difference time-domain technique,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 1083–1090 (2003).
[CrossRef]

A. Battaglia, K. Muinonen, T. Nousiainen, J. I. Peltoniemi, “Light scattering by Gaussian particles: Rayleigh-ellipsoid approximation,” J. Quant. Spectrosc. Radiat. Transfer 63, 277–303 (1999).
[CrossRef]

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577–601 (1996).
[CrossRef]

Peltoniemi, J. I.

A. Battaglia, K. Muinonen, T. Nousiainen, J. I. Peltoniemi, “Light scattering by Gaussian particles: Rayleigh-ellipsoid approximation,” J. Quant. Spectrosc. Radiat. Transfer 63, 277–303 (1999).
[CrossRef]

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577–601 (1996).
[CrossRef]

J. I. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irvine, “Scattering of light by stochastically rough particles,” Appl. Opt. 19, 4088–4095 (1989).
[CrossRef]

Pennypacker, C. P.

E. M. Purcell, C. P. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 196, 705–714 (1973).
[CrossRef]

Perrin, J.-M.

J.-M. Perrin, J.-P. Sivan, “Scattering and polarization of light by rough and porous interstellar grains,” Astron. Astrophys. 247, 497–504 (1991).

Purcell, E. M.

E. M. Purcell, C. P. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 196, 705–714 (1973).
[CrossRef]

Sivan, J.-P.

J.-M. Perrin, J.-P. Sivan, “Scattering and polarization of light by rough and porous interstellar grains,” Astron. Astrophys. 247, 497–504 (1991).

Smith, F. W.

D. E. Merewether, R. Fisher, F. W. Smith, “On implementing a numeric Huygen’s source in a finite difference program to illustrate scattering bodies,” IEEE Trans. Nucl. Sci. NS-27, 1829–1833 (1980).
[CrossRef]

Sun, W.

Taflove, A.

A. Taflove, Computational Electrodynamics: the Finite-Difference Time Domain Method (Artech House, Boston, Mass., 1995).

Takano, Y.

Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. Part III: Light scattering by irregular ice crystals,” J. Atmos. Sci. 52, 818–837 (1995).
[CrossRef]

Vaidya, D. B.

D. B. Vaidya, R. Gupta, “Extinction by porous silicate and graphite grains,” Astron. Astrophys. 328, 634–640 (1997).

Videen, G.

W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, G. Videen, “Light scattering by Gaussian particles: a solution with finite-difference time-domain technique,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 1083–1090 (2003).
[CrossRef]

G. Videen, W. S. Bickel, “Light-scattering Mueller matrix for a rough fiber,” Appl. Opt. 31, 3488–3492 (1992).
[CrossRef] [PubMed]

G. Videen, W. S. Bickel, “Light-scattering Mueller matrix from a fiber as a function of MgO contamination,” Appl. Opt. 30, 3880–3885 (1991).
[CrossRef] [PubMed]

Wiscombe, W. J.

A. Mugnai, W. J. Wiscombe, “Scattering of radiation by moderately nonspherical particles,” J. Atmos. Sci. 37, 1291–1307 (1980).
[CrossRef]

Yang, P.

Yee, K. S.

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

Appl. Opt. (6)

Astron. Astrophys. (2)

J.-M. Perrin, J.-P. Sivan, “Scattering and polarization of light by rough and porous interstellar grains,” Astron. Astrophys. 247, 497–504 (1991).

D. B. Vaidya, R. Gupta, “Extinction by porous silicate and graphite grains,” Astron. Astrophys. 328, 634–640 (1997).

Astrophys. J. (1)

E. M. Purcell, C. P. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 196, 705–714 (1973).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

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

G. Lazzi, O. P. Gandhi, “On the optimal design of the PML absorbing boundary condition for the FDTD code,” IEEE Trans. Antennas Propag. 45, 914–916 (1996).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

D. E. Merewether, R. Fisher, F. W. Smith, “On implementing a numeric Huygen’s source in a finite difference program to illustrate scattering bodies,” IEEE Trans. Nucl. Sci. NS-27, 1829–1833 (1980).
[CrossRef]

J. Atmos. Sci. (2)

A. Mugnai, W. J. Wiscombe, “Scattering of radiation by moderately nonspherical particles,” J. Atmos. Sci. 37, 1291–1307 (1980).
[CrossRef]

Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. Part III: Light scattering by irregular ice crystals,” J. Atmos. Sci. 52, 818–837 (1995).
[CrossRef]

J. Comput. Phys. (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

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

J. Quant. Spectrosc. Radiat. Transfer (4)

K. Chamaillard, J.-P. J. Lafon, “Statistical approach of the effect of roughness on the polarization of light scattered by dust grains,” J. Quant. Spectrosc. Radiat. Transfer 70, 519–528 (2001).
[CrossRef]

K. Muinonen, T. Nousiainen, P. Fast, K. Lumme, J. I. Peltoniemi, “Light scattering by Gaussian random particles: ray optics approximation,” J. Quant. Spectrosc. Radiat. Transfer 55, 577–601 (1996).
[CrossRef]

A. Battaglia, K. Muinonen, T. Nousiainen, J. I. Peltoniemi, “Light scattering by Gaussian particles: Rayleigh-ellipsoid approximation,” J. Quant. Spectrosc. Radiat. Transfer 63, 277–303 (1999).
[CrossRef]

W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, G. Videen, “Light scattering by Gaussian particles: a solution with finite-difference time-domain technique,” J. Quant. Spectrosc. Radiat. Transfer 79-80, 1083–1090 (2003).
[CrossRef]

Other (1)

A. Taflove, Computational Electrodynamics: the Finite-Difference Time Domain Method (Artech House, Boston, Mass., 1995).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Geometries of the incident direction and the closed rectangular interface of the total field and the scattered field.

Fig. 2
Fig. 2

Geometries of the incident direction, the unit vector in the scattering direction, and the unit vector perpendicular to the scattering direction.

Fig. 3
Fig. 3

Comparison of the elements of the scattering phase matrix from analytic and the FDTD solutions for an infinitely long circular cylinder with a refractive index of 1.4717 + 0.3890i and a size parameter of 24.

Fig. 4
Fig. 4

Cross section of a rough-surfaced column with the maximum deviation from a cylinder, σ = 0.1.

Fig. 5
Fig. 5

Comparison of elements of the scattering phase matrix from the rough-surfaced ice column shown in Fig. 4 computed by the FDTD method with spatial cell sizes of λ/60 and λ/120, where λ denotes the incident wavelength in free space. The size parameter of the column is 24. The refractive index of the column is 1.311. The incident direction is in the reverse x direction (ϕ = 0°).

Fig. 6
Fig. 6

Comparison of elements of the scattering phase matrix from a perfect circular cylinder and randomly oriented rough-surfaced ice columns with σ = 0.1. A refractive index of 1.311 and a size parameter of 6 are used.

Fig. 7
Fig. 7

Same as Fig. 6 but for a size parameter of 24.

Fig. 8
Fig. 8

Comparison of elements of the scattering phase matrix from a perfect circular cylinder and randomly oriented rough-surfaced ice columns with σ = 0.1. A refractive index of 1.4717 + 0.3890i and a size parameter of 6 are used.

Fig. 9
Fig. 9

Same as Fig. 8 but for a size parameter of 24.

Tables (3)

Tables Icon

Table 1 Single-Scattering Properties of a Cylinder with a Refractive Index of 1.311

Tables Icon

Table 2 Single-Scattering Properties of a Cylinder with a Refractive Index of 1.4717 + 0.3890i

Tables Icon

Table 3 Relative Differences in PMEs between the Rough-Surfaced and Perfect Cylinders at a Scattering Angle of 170° for the Case in Fig. 7

Equations (29)

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

Hxn+1/2i, j+1/2=Hxn-1/2i, j+1/2-Δtμi, jΔsEzni, j+1-Ezni, j,
Hyn+1/2i+1/2, j=Hyn-1/2i+1/2, j+Δtμi, jΔsEzni+1, j-Ezni, j,
Ezn+1i, j=exp-εii, jεri, j ωΔtEzni, j+exp-εii, jεri, j ωΔt/2×Δtεri, jΔs×Hyn+1/2i+1/2, j-Hyn+1/2i-1/2, j-Hxn+1/2i, j+1/2+Hxn+1/2i, j-1/2,
Hxn+1/2i, ja-1/2=Hxn+1/2i, ja-1/21a+Δtμ0Δs Ez,incni, ja,
Ezn+1i, ja=Ezn+1i, ja1c-Δtε0Δs Hx,incn+1/2i, ja-1/2.
Hxn+1/2i, jb+1/2=Hxn+1/2i, jb+1/21a-Δtμ0Δs Ez,incni, jb,
Ezn+1i, jb=Ezn+1i, jb1c+Δtε0Δs Hx,incn+1/2i, jb+1/2.
Hyn+1/2ia-1/2, j=Hyn+1/2ia-1/2, j1b-Δtμ0Δs Ez,incnia, j,
Ezn+1ia, j=Ezn+1ia, j1c+Δtε0Δs Hy,incn+1/2ia-1/2, j.
Hyn+1/2ib+1/2, j=Hyn+1/2ib+1/2, j1b+Δtμ0Δs Ez,incnib, j,
Ezn+1ib, j=Ezn+1ib, j1c-Δtε0Δs Hy,incn+1/2ib+1/2, j.
Hxn+1/2i, j+1/2 =exp-σy*j+1/2Δt/μ0×Hxn-1/2i, j+1/2-1-exp-σy*j+1/2Δt/μ0σy*j+1/2Δs×Ezni, j+1-Ezni, j,
Hyn+1/2i+1/2, j=exp-σx*i+1/2Δt/μ0Hyn-1/2i+1/2, j+1-exp-σx*i+1/2Δt/μ0σx*i+1/2Δs×Ezni+1, j-Ezni, j,
Ezxn+1i, j=exp-σxiΔt/ε0Ezxni, j+1-exp-σxiΔt/ε0σxiΔs×Hyn+1/2i+1/2, j-Hyn+1/2i-1/2, j,
Ezyn+1i, j=exp-σyjΔt/ε0Ezyni, j-1-exp-σyjΔt/ε0σyjΔs×Hxn+1/2i, j+1/2-Hxn+1/2i, j-1/2,
Ezn+1i, j=Ezxn+1i, j+Ezyn+1i, j,
σρ=σmρdf,
R0=exp-2/n+1σmd/ε0c.
σa=k|E0|2s εiξEξ·Eξd2ξ,
σa,TM=k s εiξEzξEz*ξd2ξ,
σa,TE=k s εiξExξEx*ξ+EyξEy*ξd2ξ.
σa=12σa,TM+σa,TE.
EsR=s GR, ξk2II+ξξ·P/ε0d2ξ,
GR, ξ=i3/28πk1/2expik|R-ξ||R-ξ|1/2.
s1=i3/2k28πk1/2s1-εξ/ε0Ez exp-ikr·ξd2ξ,
s2=i3/2k28πk1/2s1-εξ/ε0a·Ex, Eyexp-ikr·ξd2ξ,
σs=02π12s1s1*+s2s2*dα,
Rm=a1+rnm, s=1σ,
ϕm=mπ60+rnm, s=2π180,

Metrics