Abstract

Due to the inadequate understanding of the scattering properties of nonspherical aerosols, considerable uncertainties still exist in the radiative transfer numerical simulation. To this end, a new scattering model for nonspherical aerosols is established based on Multi-Resolution Time-Domain (MRTD) scheme. The model is comprised of three modules: near field calculation module, near-to-far transformation module and scattering parameters computation module, in which, the near electromagnetic field is calculated by MRTD technique, the near-to-far transformation scheme is performed by volume integral method, and the calculation models for extinction and absorption cross section are directly derived from Maxwell’s curl equations in the frequency domain. To achieve higher computational efficiency, the model is further parallelized by MPI non-blocking repeated communication technique. The accuracy of the scattering model is validated against Lorenz-Mie, Aden-Kerker and T-matrix theories for spherical particles, particles with inclusions and nonspherical particles. At last, the parallel computational efficiency of the MRTD scattering model is quantitatively discussed as well. The results obtained by parallel MRTD scattering model show an excellent agreement with those of the well-tested scattering theories, where the relative simulation errors of the phase function are less than 5% for most scattering angles. In backward directions, the simulation errors are much larger than that in forward scattering directions due to the stair approximation in particle construction. The computational accuracy of the integral scattering parameters like extinction and absorption efficiencies is higher than phase matrix, where the simulation errors of extinction and absorption efficiencies for the particle with a size parameter of 10 achieve −0.4891% and −1.6933%, respectively.

© 2017 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Simultaneously simulating the scattering properties of nonspherical aerosol particles with different sizes by the MRTD scattering model

Shuai Hu, Taichang Gao, Hao Li, Ming Chen, Feng Zhang, and Bo Yang
Opt. Express 25(15) 17872-17891 (2017)

Application of the pseudospectral time-domain method to the scattering of light by nonspherical particles

Guang Chen, Ping Yang, and George W. Kattawar
J. Opt. Soc. Am. A 25(3) 785-790 (2008)

Comparison between the pseudo-spectral time domain method and the discrete dipole approximation for light scattering simulations

Chao Liu, Lei Bi, R. Lee Panetta, Ping Yang, and Maxim A. Yurkin
Opt. Express 20(15) 16763-16776 (2012)

References

  • View by:
  • |
  • |
  • |

  1. K. N. Liou and Y. Takano, “Light scattering by nonspherical particles: Remote sensing and climatic implications,” Atmos. Res. 31(4), 271–298 (1994).
    [Crossref]
  2. K. N. Liou, An Introduction to Atmospheric Radiation (Academic Press, 2003).
  3. H. Iwabuchi and P. Yang, “Temperature dependence of ice optical constants: Implications for simulating the single-scattering properties of cold ice clouds,” J. Quant. Spectrosc. Radiat. Transf. 112(15), 2520–2525 (2011).
    [Crossref]
  4. M. I. Mishchenko, “Electromagnetic scattering by nonspherical particles:A tutorial review,” J. Quant. Spectrosc. Radiat. Transf. 110(11), 808–832 (2009).
    [Crossref]
  5. P. Yang, K.-N. Liou, L. Bi, C. Liu, B. Yi, and B. A. Baum, “On the Radiative Properties of Ice Clouds: Light Scattering, Remote Sensing, and Radiation Parameterization,” Adv. Atmos. Sci. 32(1), 32–63 (2015).
    [Crossref]
  6. H. Yi, X. Ben, and H. Tan, “Transient radiative transfer in a scattering slab considering polarization,” Opt. Express 21(22), 26693–26713 (2013).
    [Crossref] [PubMed]
  7. IPCC: Climate Change 2007, “Intergovernmental Panel of Global Climate Change,” (2007).
  8. T. Cheng, X. Gu, Y. Wu, H. Chen, and T. Yu, “The optical properties of absorbing aerosols with fractal soot aggregates: Implications for aerosol remote sensing,” J. Quant. Spectrosc. Radiat. Transf. 125, 93–104 (2013).
    [Crossref]
  9. T. H. Cheng, X. F. Gu, T. Yu, and G. L. Tian, “The reflection and polarization properties of nonspherical aerosol particles,” J. Quant. Spectrosc. Radiat. Transf. 111(6), 895–906 (2010).
    [Crossref]
  10. Y. Liu, W. P. Arnott, and J. Hallett, “Particle size distribution retrieval from multispectral optical depth:Influences of particle nonsphericity and refractive index,” J. Geophys. Res. 104(D24), 31753–31762 (1999).
    [Crossref]
  11. M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles, Theory, Measurements, and Application (Academic Press, 2000).
  12. H. Iwabuchi and T. Suzuki, “Fast and accurate radiance calculations using truncation approximation for anisotropic scattering phase functions,” J. Quant. Spectrosc. Radiat. Transf. 110(17), 1926–1939 (2009).
    [Crossref]
  13. J.-Q. Zhao and Y.-Q. Hu, “Bridging technique for calculating the extinction efficiency of arbitrary shaped particles,” Appl. Opt. 42(24), 4937–4945 (2003).
    [Crossref] [PubMed]
  14. J.-Q. Zhao, G. Shi, H. Che, and G. Cheng, “Approximations of the scattering phase functions of particles,” Adv. Atmos. Sci. 23(5), 802–808 (2006).
    [Crossref]
  15. P. Yang, H. Wei, H.-L. Huang, B. A. Baum, Y. X. Hu, G. W. Kattawar, M. I. Mishchenko, and Q. Fu, “Scattering and absorption property database for nonspherical ice particles in the near- through far-infrared spectral region,” Appl. Opt. 44(26), 5512–5523 (2005).
    [Crossref] [PubMed]
  16. M. I. Mishchenko and L. D. Travis, “Capabilities and Limitations of a Current Fortran Implementation of the T-Martrix Method for Randomly Oriented,Rotationally Symmetric Scatterers,” J. Quant. Spectrosc. Radiat. Transf. 60(3), 309–324 (1998).
    [Crossref]
  17. M. I. Mishchenko and L. D. Travis, “T-Martrix computations of light scattering by large spheriodal particles,” Opt. Commun. 109(1-2), 16–21 (1994).
    [Crossref]
  18. N. V. Voshchinnikov and V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204(1), 19–86 (1993).
    [Crossref]
  19. H. M. Al-Rizzo and J. M. Tranquilla, “Electromagnetic scattering from dielectrically coated axisymmetric objects using the generalized point-matching technique,” J. Comput. Phys. 119(2), 356–373 (1995).
    [Crossref]
  20. L. Liu, M. I. Mishchenko, B. Cairns, B. E. Carlson, and L. D. Travis, “Modeling single-scattering properties of small cirrus particles by use of a size-shape distribution of ice spheroids and cylinders,” J. Quant. Spectrosc. Radiat. Transf. 101(3), 488–497 (2006).
    [Crossref]
  21. A. Quirantes, “A T-matrix method and computer code for randomly oriented, axially symmetric coated scatterers,” J. Quant. Spectrosc. Radiat. Transf. 92(3), 373–381 (2005).
    [Crossref]
  22. R. F. Harrington, Field Computation by Moment Methods (Macmillan, 1968).
  23. B. T. Draine, “Discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
    [Crossref]
  24. B. T. Draine and P. J. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1499 (1994).
    [Crossref]
  25. 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(1), 162 (1995).
    [Crossref]
  26. C. Liu, R. L. Panetta, and P. Yang, “Application of the pseudo-spectral time domain method to compute particle single-scattering properties for size parameters up to 200,” J. Quant. Spectrosc. Radiat. Transf. 113(13), 1728–1740 (2012).
    [Crossref]
  27. C. Liu, R. L. Panetta, and P. Yang, “The effects of surface roughness on the scattering properties of hexagonal columns with sizes from the Rayleigh to the geometric optics regimes,” J. Quant. Spectrosc. Radiat. Transf. 129, 169–185 (2013).
    [Crossref]
  28. S. Y. Dai, “Application of Wavelet and Multiresolution Time domain (MRTD) on Electromagnetic scattering,” (Xidian University, 2007).
  29. Y. W. Cheong, Y. M. Lee, K. H. Ra, J. G. Kang, and C. C. Shin, “Wavelet-Galerkin scheme of time dependent inhomogeneous electromagnetic problems,” IEEE Microw. Guided Wave Lett. 9(8), 297–299 (1999).
    [Crossref]
  30. L. Massy, N. Pena, and M. M. Ney, “Dispersion characteristics and stability for an arbitray MRTD scheme with variable mesh,” Int. J. Numer. Model.: Electron. Netw. Dev. Fields 23(6), 470–491 (2010).
    [Crossref]
  31. Q. Gao, Q. Cao, and J. Zhou, “Application of Total-Field/Scattered-Field Technique to 3D-MRTD Scattering Scheme,” in International Forum on Information Technology and Applications(2009), pp. 359–362.
  32. 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(10), 2072–2085 (1996).
    [Crossref]
  33. M. Krumpholz and L. P. B. Katehi, “MRTD: New Time-Domain Schemes Based on Multiresolution Analysis,” IEEE Trans. Microw. Theory Tech. 44(4), 555–571 (1996).
    [Crossref]
  34. E. M. Tentzeris, A. Cangellaris, L. P. B. Katehi, and J. Harvey, “MultiresolutionTime-Domain(MRTD)Adaptive Schemes Using Arbitrary Resolutions of Wavelets,” IEEE Trans. Microw. Tech. 50(2), 501–516 (2002).
    [Crossref]
  35. Y. Liu, Y. Chen, X. Xu, and Z. Liu, “Implementation and analysis of the perfectly matched layer with auxiliary differential equation for the multiresolution time-domain method,” Wuli Xuebao 62, 034101 (2013).
  36. D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. (Pearson Education, Inc, 2014).
  37. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, Inc, 1983).
  38. H. C. van der Hulst, Light Scattering by Small Particles (Dover Publications, 1981).
  39. Y. Liu, Y. Chen, and P. Zhang, “Parallel implementation and application of the MRTD with an efficient CFS-PML,” Prog. Electromagnetics Res. 143, 223–242 (2013).
    [Crossref]

2015 (1)

P. Yang, K.-N. Liou, L. Bi, C. Liu, B. Yi, and B. A. Baum, “On the Radiative Properties of Ice Clouds: Light Scattering, Remote Sensing, and Radiation Parameterization,” Adv. Atmos. Sci. 32(1), 32–63 (2015).
[Crossref]

2013 (5)

H. Yi, X. Ben, and H. Tan, “Transient radiative transfer in a scattering slab considering polarization,” Opt. Express 21(22), 26693–26713 (2013).
[Crossref] [PubMed]

T. Cheng, X. Gu, Y. Wu, H. Chen, and T. Yu, “The optical properties of absorbing aerosols with fractal soot aggregates: Implications for aerosol remote sensing,” J. Quant. Spectrosc. Radiat. Transf. 125, 93–104 (2013).
[Crossref]

C. Liu, R. L. Panetta, and P. Yang, “The effects of surface roughness on the scattering properties of hexagonal columns with sizes from the Rayleigh to the geometric optics regimes,” J. Quant. Spectrosc. Radiat. Transf. 129, 169–185 (2013).
[Crossref]

Y. Liu, Y. Chen, X. Xu, and Z. Liu, “Implementation and analysis of the perfectly matched layer with auxiliary differential equation for the multiresolution time-domain method,” Wuli Xuebao 62, 034101 (2013).

Y. Liu, Y. Chen, and P. Zhang, “Parallel implementation and application of the MRTD with an efficient CFS-PML,” Prog. Electromagnetics Res. 143, 223–242 (2013).
[Crossref]

2012 (1)

C. Liu, R. L. Panetta, and P. Yang, “Application of the pseudo-spectral time domain method to compute particle single-scattering properties for size parameters up to 200,” J. Quant. Spectrosc. Radiat. Transf. 113(13), 1728–1740 (2012).
[Crossref]

2011 (1)

H. Iwabuchi and P. Yang, “Temperature dependence of ice optical constants: Implications for simulating the single-scattering properties of cold ice clouds,” J. Quant. Spectrosc. Radiat. Transf. 112(15), 2520–2525 (2011).
[Crossref]

2010 (2)

T. H. Cheng, X. F. Gu, T. Yu, and G. L. Tian, “The reflection and polarization properties of nonspherical aerosol particles,” J. Quant. Spectrosc. Radiat. Transf. 111(6), 895–906 (2010).
[Crossref]

L. Massy, N. Pena, and M. M. Ney, “Dispersion characteristics and stability for an arbitray MRTD scheme with variable mesh,” Int. J. Numer. Model.: Electron. Netw. Dev. Fields 23(6), 470–491 (2010).
[Crossref]

2009 (2)

M. I. Mishchenko, “Electromagnetic scattering by nonspherical particles:A tutorial review,” J. Quant. Spectrosc. Radiat. Transf. 110(11), 808–832 (2009).
[Crossref]

H. Iwabuchi and T. Suzuki, “Fast and accurate radiance calculations using truncation approximation for anisotropic scattering phase functions,” J. Quant. Spectrosc. Radiat. Transf. 110(17), 1926–1939 (2009).
[Crossref]

2006 (2)

J.-Q. Zhao, G. Shi, H. Che, and G. Cheng, “Approximations of the scattering phase functions of particles,” Adv. Atmos. Sci. 23(5), 802–808 (2006).
[Crossref]

L. Liu, M. I. Mishchenko, B. Cairns, B. E. Carlson, and L. D. Travis, “Modeling single-scattering properties of small cirrus particles by use of a size-shape distribution of ice spheroids and cylinders,” J. Quant. Spectrosc. Radiat. Transf. 101(3), 488–497 (2006).
[Crossref]

2005 (2)

2003 (1)

2002 (1)

E. M. Tentzeris, A. Cangellaris, L. P. B. Katehi, and J. Harvey, “MultiresolutionTime-Domain(MRTD)Adaptive Schemes Using Arbitrary Resolutions of Wavelets,” IEEE Trans. Microw. Tech. 50(2), 501–516 (2002).
[Crossref]

1999 (2)

Y. W. Cheong, Y. M. Lee, K. H. Ra, J. G. Kang, and C. C. Shin, “Wavelet-Galerkin scheme of time dependent inhomogeneous electromagnetic problems,” IEEE Microw. Guided Wave Lett. 9(8), 297–299 (1999).
[Crossref]

Y. Liu, W. P. Arnott, and J. Hallett, “Particle size distribution retrieval from multispectral optical depth:Influences of particle nonsphericity and refractive index,” J. Geophys. Res. 104(D24), 31753–31762 (1999).
[Crossref]

1998 (1)

M. I. Mishchenko and L. D. Travis, “Capabilities and Limitations of a Current Fortran Implementation of the T-Martrix Method for Randomly Oriented,Rotationally Symmetric Scatterers,” J. Quant. Spectrosc. Radiat. Transf. 60(3), 309–324 (1998).
[Crossref]

1996 (2)

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(10), 2072–2085 (1996).
[Crossref]

M. Krumpholz and L. P. B. Katehi, “MRTD: New Time-Domain Schemes Based on Multiresolution Analysis,” IEEE Trans. Microw. Theory Tech. 44(4), 555–571 (1996).
[Crossref]

1995 (2)

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(1), 162 (1995).
[Crossref]

H. M. Al-Rizzo and J. M. Tranquilla, “Electromagnetic scattering from dielectrically coated axisymmetric objects using the generalized point-matching technique,” J. Comput. Phys. 119(2), 356–373 (1995).
[Crossref]

1994 (3)

K. N. Liou and Y. Takano, “Light scattering by nonspherical particles: Remote sensing and climatic implications,” Atmos. Res. 31(4), 271–298 (1994).
[Crossref]

M. I. Mishchenko and L. D. Travis, “T-Martrix computations of light scattering by large spheriodal particles,” Opt. Commun. 109(1-2), 16–21 (1994).
[Crossref]

B. T. Draine and P. J. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1499 (1994).
[Crossref]

1993 (1)

N. V. Voshchinnikov and V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204(1), 19–86 (1993).
[Crossref]

1988 (1)

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

Al-Rizzo, H. M.

H. M. Al-Rizzo and J. M. Tranquilla, “Electromagnetic scattering from dielectrically coated axisymmetric objects using the generalized point-matching technique,” J. Comput. Phys. 119(2), 356–373 (1995).
[Crossref]

Arnott, W. P.

Y. Liu, W. P. Arnott, and J. Hallett, “Particle size distribution retrieval from multispectral optical depth:Influences of particle nonsphericity and refractive index,” J. Geophys. Res. 104(D24), 31753–31762 (1999).
[Crossref]

Baum, B. A.

P. Yang, K.-N. Liou, L. Bi, C. Liu, B. Yi, and B. A. Baum, “On the Radiative Properties of Ice Clouds: Light Scattering, Remote Sensing, and Radiation Parameterization,” Adv. Atmos. Sci. 32(1), 32–63 (2015).
[Crossref]

P. Yang, H. Wei, H.-L. Huang, B. A. Baum, Y. X. Hu, G. W. Kattawar, M. I. Mishchenko, and Q. Fu, “Scattering and absorption property database for nonspherical ice particles in the near- through far-infrared spectral region,” Appl. Opt. 44(26), 5512–5523 (2005).
[Crossref] [PubMed]

Ben, X.

Bi, L.

P. Yang, K.-N. Liou, L. Bi, C. Liu, B. Yi, and B. A. Baum, “On the Radiative Properties of Ice Clouds: Light Scattering, Remote Sensing, and Radiation Parameterization,” Adv. Atmos. Sci. 32(1), 32–63 (2015).
[Crossref]

Cairns, B.

L. Liu, M. I. Mishchenko, B. Cairns, B. E. Carlson, and L. D. Travis, “Modeling single-scattering properties of small cirrus particles by use of a size-shape distribution of ice spheroids and cylinders,” J. Quant. Spectrosc. Radiat. Transf. 101(3), 488–497 (2006).
[Crossref]

Cangellaris, A.

E. M. Tentzeris, A. Cangellaris, L. P. B. Katehi, and J. Harvey, “MultiresolutionTime-Domain(MRTD)Adaptive Schemes Using Arbitrary Resolutions of Wavelets,” IEEE Trans. Microw. Tech. 50(2), 501–516 (2002).
[Crossref]

Carlson, B. E.

L. Liu, M. I. Mishchenko, B. Cairns, B. E. Carlson, and L. D. Travis, “Modeling single-scattering properties of small cirrus particles by use of a size-shape distribution of ice spheroids and cylinders,” J. Quant. Spectrosc. Radiat. Transf. 101(3), 488–497 (2006).
[Crossref]

Che, H.

J.-Q. Zhao, G. Shi, H. Che, and G. Cheng, “Approximations of the scattering phase functions of particles,” Adv. Atmos. Sci. 23(5), 802–808 (2006).
[Crossref]

Chen, H.

T. Cheng, X. Gu, Y. Wu, H. Chen, and T. Yu, “The optical properties of absorbing aerosols with fractal soot aggregates: Implications for aerosol remote sensing,” J. Quant. Spectrosc. Radiat. Transf. 125, 93–104 (2013).
[Crossref]

Chen, Y.

Y. Liu, Y. Chen, X. Xu, and Z. Liu, “Implementation and analysis of the perfectly matched layer with auxiliary differential equation for the multiresolution time-domain method,” Wuli Xuebao 62, 034101 (2013).

Y. Liu, Y. Chen, and P. Zhang, “Parallel implementation and application of the MRTD with an efficient CFS-PML,” Prog. Electromagnetics Res. 143, 223–242 (2013).
[Crossref]

Cheng, G.

J.-Q. Zhao, G. Shi, H. Che, and G. Cheng, “Approximations of the scattering phase functions of particles,” Adv. Atmos. Sci. 23(5), 802–808 (2006).
[Crossref]

Cheng, T.

T. Cheng, X. Gu, Y. Wu, H. Chen, and T. Yu, “The optical properties of absorbing aerosols with fractal soot aggregates: Implications for aerosol remote sensing,” J. Quant. Spectrosc. Radiat. Transf. 125, 93–104 (2013).
[Crossref]

Cheng, T. H.

T. H. Cheng, X. F. Gu, T. Yu, and G. L. Tian, “The reflection and polarization properties of nonspherical aerosol particles,” J. Quant. Spectrosc. Radiat. Transf. 111(6), 895–906 (2010).
[Crossref]

Cheong, Y. W.

Y. W. Cheong, Y. M. Lee, K. H. Ra, J. G. Kang, and C. C. Shin, “Wavelet-Galerkin scheme of time dependent inhomogeneous electromagnetic problems,” IEEE Microw. Guided Wave Lett. 9(8), 297–299 (1999).
[Crossref]

Draine, B. T.

B. T. Draine and P. J. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1499 (1994).
[Crossref]

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

Farafonov, V. G.

N. V. Voshchinnikov and V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204(1), 19–86 (1993).
[Crossref]

Flatau, P. J.

Fu, Q.

Gu, X.

T. Cheng, X. Gu, Y. Wu, H. Chen, and T. Yu, “The optical properties of absorbing aerosols with fractal soot aggregates: Implications for aerosol remote sensing,” J. Quant. Spectrosc. Radiat. Transf. 125, 93–104 (2013).
[Crossref]

Gu, X. F.

T. H. Cheng, X. F. Gu, T. Yu, and G. L. Tian, “The reflection and polarization properties of nonspherical aerosol particles,” J. Quant. Spectrosc. Radiat. Transf. 111(6), 895–906 (2010).
[Crossref]

Hallett, J.

Y. Liu, W. P. Arnott, and J. Hallett, “Particle size distribution retrieval from multispectral optical depth:Influences of particle nonsphericity and refractive index,” J. Geophys. Res. 104(D24), 31753–31762 (1999).
[Crossref]

Harvey, J.

E. M. Tentzeris, A. Cangellaris, L. P. B. Katehi, and J. Harvey, “MultiresolutionTime-Domain(MRTD)Adaptive Schemes Using Arbitrary Resolutions of Wavelets,” IEEE Trans. Microw. Tech. 50(2), 501–516 (2002).
[Crossref]

Hu, Y. X.

Hu, Y.-Q.

Huang, H.-L.

Iwabuchi, H.

H. Iwabuchi and P. Yang, “Temperature dependence of ice optical constants: Implications for simulating the single-scattering properties of cold ice clouds,” J. Quant. Spectrosc. Radiat. Transf. 112(15), 2520–2525 (2011).
[Crossref]

H. Iwabuchi and T. Suzuki, “Fast and accurate radiance calculations using truncation approximation for anisotropic scattering phase functions,” J. Quant. Spectrosc. Radiat. Transf. 110(17), 1926–1939 (2009).
[Crossref]

Kang, J. G.

Y. W. Cheong, Y. M. Lee, K. H. Ra, J. G. Kang, and C. C. Shin, “Wavelet-Galerkin scheme of time dependent inhomogeneous electromagnetic problems,” IEEE Microw. Guided Wave Lett. 9(8), 297–299 (1999).
[Crossref]

Katehi, L. P. B.

E. M. Tentzeris, A. Cangellaris, L. P. B. Katehi, and J. Harvey, “MultiresolutionTime-Domain(MRTD)Adaptive Schemes Using Arbitrary Resolutions of Wavelets,” IEEE Trans. Microw. Tech. 50(2), 501–516 (2002).
[Crossref]

M. Krumpholz and L. P. B. Katehi, “MRTD: New Time-Domain Schemes Based on Multiresolution Analysis,” IEEE Trans. Microw. Theory Tech. 44(4), 555–571 (1996).
[Crossref]

Kattawar, G. W.

Krumpholz, M.

M. Krumpholz and L. P. B. Katehi, “MRTD: New Time-Domain Schemes Based on Multiresolution Analysis,” IEEE Trans. Microw. Theory Tech. 44(4), 555–571 (1996).
[Crossref]

Lee, Y. M.

Y. W. Cheong, Y. M. Lee, K. H. Ra, J. G. Kang, and C. C. Shin, “Wavelet-Galerkin scheme of time dependent inhomogeneous electromagnetic problems,” IEEE Microw. Guided Wave Lett. 9(8), 297–299 (1999).
[Crossref]

Liou, K. N.

Liou, K.-N.

P. Yang, K.-N. Liou, L. Bi, C. Liu, B. Yi, and B. A. Baum, “On the Radiative Properties of Ice Clouds: Light Scattering, Remote Sensing, and Radiation Parameterization,” Adv. Atmos. Sci. 32(1), 32–63 (2015).
[Crossref]

Liu, C.

P. Yang, K.-N. Liou, L. Bi, C. Liu, B. Yi, and B. A. Baum, “On the Radiative Properties of Ice Clouds: Light Scattering, Remote Sensing, and Radiation Parameterization,” Adv. Atmos. Sci. 32(1), 32–63 (2015).
[Crossref]

C. Liu, R. L. Panetta, and P. Yang, “The effects of surface roughness on the scattering properties of hexagonal columns with sizes from the Rayleigh to the geometric optics regimes,” J. Quant. Spectrosc. Radiat. Transf. 129, 169–185 (2013).
[Crossref]

C. Liu, R. L. Panetta, and P. Yang, “Application of the pseudo-spectral time domain method to compute particle single-scattering properties for size parameters up to 200,” J. Quant. Spectrosc. Radiat. Transf. 113(13), 1728–1740 (2012).
[Crossref]

Liu, L.

L. Liu, M. I. Mishchenko, B. Cairns, B. E. Carlson, and L. D. Travis, “Modeling single-scattering properties of small cirrus particles by use of a size-shape distribution of ice spheroids and cylinders,” J. Quant. Spectrosc. Radiat. Transf. 101(3), 488–497 (2006).
[Crossref]

Liu, Y.

Y. Liu, Y. Chen, X. Xu, and Z. Liu, “Implementation and analysis of the perfectly matched layer with auxiliary differential equation for the multiresolution time-domain method,” Wuli Xuebao 62, 034101 (2013).

Y. Liu, Y. Chen, and P. Zhang, “Parallel implementation and application of the MRTD with an efficient CFS-PML,” Prog. Electromagnetics Res. 143, 223–242 (2013).
[Crossref]

Y. Liu, W. P. Arnott, and J. Hallett, “Particle size distribution retrieval from multispectral optical depth:Influences of particle nonsphericity and refractive index,” J. Geophys. Res. 104(D24), 31753–31762 (1999).
[Crossref]

Liu, Z.

Y. Liu, Y. Chen, X. Xu, and Z. Liu, “Implementation and analysis of the perfectly matched layer with auxiliary differential equation for the multiresolution time-domain method,” Wuli Xuebao 62, 034101 (2013).

Massy, L.

L. Massy, N. Pena, and M. M. Ney, “Dispersion characteristics and stability for an arbitray MRTD scheme with variable mesh,” Int. J. Numer. Model.: Electron. Netw. Dev. Fields 23(6), 470–491 (2010).
[Crossref]

Mishchenko, M. I.

M. I. Mishchenko, “Electromagnetic scattering by nonspherical particles:A tutorial review,” J. Quant. Spectrosc. Radiat. Transf. 110(11), 808–832 (2009).
[Crossref]

L. Liu, M. I. Mishchenko, B. Cairns, B. E. Carlson, and L. D. Travis, “Modeling single-scattering properties of small cirrus particles by use of a size-shape distribution of ice spheroids and cylinders,” J. Quant. Spectrosc. Radiat. Transf. 101(3), 488–497 (2006).
[Crossref]

P. Yang, H. Wei, H.-L. Huang, B. A. Baum, Y. X. Hu, G. W. Kattawar, M. I. Mishchenko, and Q. Fu, “Scattering and absorption property database for nonspherical ice particles in the near- through far-infrared spectral region,” Appl. Opt. 44(26), 5512–5523 (2005).
[Crossref] [PubMed]

M. I. Mishchenko and L. D. Travis, “Capabilities and Limitations of a Current Fortran Implementation of the T-Martrix Method for Randomly Oriented,Rotationally Symmetric Scatterers,” J. Quant. Spectrosc. Radiat. Transf. 60(3), 309–324 (1998).
[Crossref]

M. I. Mishchenko and L. D. Travis, “T-Martrix computations of light scattering by large spheriodal particles,” Opt. Commun. 109(1-2), 16–21 (1994).
[Crossref]

Ney, M. M.

L. Massy, N. Pena, and M. M. Ney, “Dispersion characteristics and stability for an arbitray MRTD scheme with variable mesh,” Int. J. Numer. Model.: Electron. Netw. Dev. Fields 23(6), 470–491 (2010).
[Crossref]

Panetta, R. L.

C. Liu, R. L. Panetta, and P. Yang, “The effects of surface roughness on the scattering properties of hexagonal columns with sizes from the Rayleigh to the geometric optics regimes,” J. Quant. Spectrosc. Radiat. Transf. 129, 169–185 (2013).
[Crossref]

C. Liu, R. L. Panetta, and P. Yang, “Application of the pseudo-spectral time domain method to compute particle single-scattering properties for size parameters up to 200,” J. Quant. Spectrosc. Radiat. Transf. 113(13), 1728–1740 (2012).
[Crossref]

Pena, N.

L. Massy, N. Pena, and M. M. Ney, “Dispersion characteristics and stability for an arbitray MRTD scheme with variable mesh,” Int. J. Numer. Model.: Electron. Netw. Dev. Fields 23(6), 470–491 (2010).
[Crossref]

Quirantes, A.

A. Quirantes, “A T-matrix method and computer code for randomly oriented, axially symmetric coated scatterers,” J. Quant. Spectrosc. Radiat. Transf. 92(3), 373–381 (2005).
[Crossref]

Ra, K. H.

Y. W. Cheong, Y. M. Lee, K. H. Ra, J. G. Kang, and C. C. Shin, “Wavelet-Galerkin scheme of time dependent inhomogeneous electromagnetic problems,” IEEE Microw. Guided Wave Lett. 9(8), 297–299 (1999).
[Crossref]

Shi, G.

J.-Q. Zhao, G. Shi, H. Che, and G. Cheng, “Approximations of the scattering phase functions of particles,” Adv. Atmos. Sci. 23(5), 802–808 (2006).
[Crossref]

Shin, C. C.

Y. W. Cheong, Y. M. Lee, K. H. Ra, J. G. Kang, and C. C. Shin, “Wavelet-Galerkin scheme of time dependent inhomogeneous electromagnetic problems,” IEEE Microw. Guided Wave Lett. 9(8), 297–299 (1999).
[Crossref]

Suzuki, T.

H. Iwabuchi and T. Suzuki, “Fast and accurate radiance calculations using truncation approximation for anisotropic scattering phase functions,” J. Quant. Spectrosc. Radiat. Transf. 110(17), 1926–1939 (2009).
[Crossref]

Takano, Y.

K. N. Liou and Y. Takano, “Light scattering by nonspherical particles: Remote sensing and climatic implications,” Atmos. Res. 31(4), 271–298 (1994).
[Crossref]

Tan, H.

Tentzeris, E. M.

E. M. Tentzeris, A. Cangellaris, L. P. B. Katehi, and J. Harvey, “MultiresolutionTime-Domain(MRTD)Adaptive Schemes Using Arbitrary Resolutions of Wavelets,” IEEE Trans. Microw. Tech. 50(2), 501–516 (2002).
[Crossref]

Tian, G. L.

T. H. Cheng, X. F. Gu, T. Yu, and G. L. Tian, “The reflection and polarization properties of nonspherical aerosol particles,” J. Quant. Spectrosc. Radiat. Transf. 111(6), 895–906 (2010).
[Crossref]

Tranquilla, J. M.

H. M. Al-Rizzo and J. M. Tranquilla, “Electromagnetic scattering from dielectrically coated axisymmetric objects using the generalized point-matching technique,” J. Comput. Phys. 119(2), 356–373 (1995).
[Crossref]

Travis, L. D.

L. Liu, M. I. Mishchenko, B. Cairns, B. E. Carlson, and L. D. Travis, “Modeling single-scattering properties of small cirrus particles by use of a size-shape distribution of ice spheroids and cylinders,” J. Quant. Spectrosc. Radiat. Transf. 101(3), 488–497 (2006).
[Crossref]

M. I. Mishchenko and L. D. Travis, “Capabilities and Limitations of a Current Fortran Implementation of the T-Martrix Method for Randomly Oriented,Rotationally Symmetric Scatterers,” J. Quant. Spectrosc. Radiat. Transf. 60(3), 309–324 (1998).
[Crossref]

M. I. Mishchenko and L. D. Travis, “T-Martrix computations of light scattering by large spheriodal particles,” Opt. Commun. 109(1-2), 16–21 (1994).
[Crossref]

Voshchinnikov, N. V.

N. V. Voshchinnikov and V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204(1), 19–86 (1993).
[Crossref]

Wei, H.

Wu, Y.

T. Cheng, X. Gu, Y. Wu, H. Chen, and T. Yu, “The optical properties of absorbing aerosols with fractal soot aggregates: Implications for aerosol remote sensing,” J. Quant. Spectrosc. Radiat. Transf. 125, 93–104 (2013).
[Crossref]

Xu, X.

Y. Liu, Y. Chen, X. Xu, and Z. Liu, “Implementation and analysis of the perfectly matched layer with auxiliary differential equation for the multiresolution time-domain method,” Wuli Xuebao 62, 034101 (2013).

Yang, P.

P. Yang, K.-N. Liou, L. Bi, C. Liu, B. Yi, and B. A. Baum, “On the Radiative Properties of Ice Clouds: Light Scattering, Remote Sensing, and Radiation Parameterization,” Adv. Atmos. Sci. 32(1), 32–63 (2015).
[Crossref]

C. Liu, R. L. Panetta, and P. Yang, “The effects of surface roughness on the scattering properties of hexagonal columns with sizes from the Rayleigh to the geometric optics regimes,” J. Quant. Spectrosc. Radiat. Transf. 129, 169–185 (2013).
[Crossref]

C. Liu, R. L. Panetta, and P. Yang, “Application of the pseudo-spectral time domain method to compute particle single-scattering properties for size parameters up to 200,” J. Quant. Spectrosc. Radiat. Transf. 113(13), 1728–1740 (2012).
[Crossref]

H. Iwabuchi and P. Yang, “Temperature dependence of ice optical constants: Implications for simulating the single-scattering properties of cold ice clouds,” J. Quant. Spectrosc. Radiat. Transf. 112(15), 2520–2525 (2011).
[Crossref]

P. Yang, H. Wei, H.-L. Huang, B. A. Baum, Y. X. Hu, G. W. Kattawar, M. I. Mishchenko, and Q. Fu, “Scattering and absorption property database for nonspherical ice particles in the near- through far-infrared spectral region,” Appl. Opt. 44(26), 5512–5523 (2005).
[Crossref] [PubMed]

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(10), 2072–2085 (1996).
[Crossref]

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(1), 162 (1995).
[Crossref]

Yi, B.

P. Yang, K.-N. Liou, L. Bi, C. Liu, B. Yi, and B. A. Baum, “On the Radiative Properties of Ice Clouds: Light Scattering, Remote Sensing, and Radiation Parameterization,” Adv. Atmos. Sci. 32(1), 32–63 (2015).
[Crossref]

Yi, H.

Yu, T.

T. Cheng, X. Gu, Y. Wu, H. Chen, and T. Yu, “The optical properties of absorbing aerosols with fractal soot aggregates: Implications for aerosol remote sensing,” J. Quant. Spectrosc. Radiat. Transf. 125, 93–104 (2013).
[Crossref]

T. H. Cheng, X. F. Gu, T. Yu, and G. L. Tian, “The reflection and polarization properties of nonspherical aerosol particles,” J. Quant. Spectrosc. Radiat. Transf. 111(6), 895–906 (2010).
[Crossref]

Zhang, P.

Y. Liu, Y. Chen, and P. Zhang, “Parallel implementation and application of the MRTD with an efficient CFS-PML,” Prog. Electromagnetics Res. 143, 223–242 (2013).
[Crossref]

Zhao, J.-Q.

J.-Q. Zhao, G. Shi, H. Che, and G. Cheng, “Approximations of the scattering phase functions of particles,” Adv. Atmos. Sci. 23(5), 802–808 (2006).
[Crossref]

J.-Q. Zhao and Y.-Q. Hu, “Bridging technique for calculating the extinction efficiency of arbitrary shaped particles,” Appl. Opt. 42(24), 4937–4945 (2003).
[Crossref] [PubMed]

Adv. Atmos. Sci. (2)

P. Yang, K.-N. Liou, L. Bi, C. Liu, B. Yi, and B. A. Baum, “On the Radiative Properties of Ice Clouds: Light Scattering, Remote Sensing, and Radiation Parameterization,” Adv. Atmos. Sci. 32(1), 32–63 (2015).
[Crossref]

J.-Q. Zhao, G. Shi, H. Che, and G. Cheng, “Approximations of the scattering phase functions of particles,” Adv. Atmos. Sci. 23(5), 802–808 (2006).
[Crossref]

Appl. Opt. (2)

Astrophys. J. (1)

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

Astrophys. Space Sci. (1)

N. V. Voshchinnikov and V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204(1), 19–86 (1993).
[Crossref]

Atmos. Res. (1)

K. N. Liou and Y. Takano, “Light scattering by nonspherical particles: Remote sensing and climatic implications,” Atmos. Res. 31(4), 271–298 (1994).
[Crossref]

IEEE Microw. Guided Wave Lett. (1)

Y. W. Cheong, Y. M. Lee, K. H. Ra, J. G. Kang, and C. C. Shin, “Wavelet-Galerkin scheme of time dependent inhomogeneous electromagnetic problems,” IEEE Microw. Guided Wave Lett. 9(8), 297–299 (1999).
[Crossref]

IEEE Trans. Microw. Tech. (1)

E. M. Tentzeris, A. Cangellaris, L. P. B. Katehi, and J. Harvey, “MultiresolutionTime-Domain(MRTD)Adaptive Schemes Using Arbitrary Resolutions of Wavelets,” IEEE Trans. Microw. Tech. 50(2), 501–516 (2002).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

M. Krumpholz and L. P. B. Katehi, “MRTD: New Time-Domain Schemes Based on Multiresolution Analysis,” IEEE Trans. Microw. Theory Tech. 44(4), 555–571 (1996).
[Crossref]

Int. J. Numer. Model.: Electron. Netw. Dev. Fields (1)

L. Massy, N. Pena, and M. M. Ney, “Dispersion characteristics and stability for an arbitray MRTD scheme with variable mesh,” Int. J. Numer. Model.: Electron. Netw. Dev. Fields 23(6), 470–491 (2010).
[Crossref]

J. Comput. Phys. (1)

H. M. Al-Rizzo and J. M. Tranquilla, “Electromagnetic scattering from dielectrically coated axisymmetric objects using the generalized point-matching technique,” J. Comput. Phys. 119(2), 356–373 (1995).
[Crossref]

J. Geophys. Res. (1)

Y. Liu, W. P. Arnott, and J. Hallett, “Particle size distribution retrieval from multispectral optical depth:Influences of particle nonsphericity and refractive index,” J. Geophys. Res. 104(D24), 31753–31762 (1999).
[Crossref]

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

J. Quant. Spectrosc. Radiat. Transf. (10)

C. Liu, R. L. Panetta, and P. Yang, “Application of the pseudo-spectral time domain method to compute particle single-scattering properties for size parameters up to 200,” J. Quant. Spectrosc. Radiat. Transf. 113(13), 1728–1740 (2012).
[Crossref]

C. Liu, R. L. Panetta, and P. Yang, “The effects of surface roughness on the scattering properties of hexagonal columns with sizes from the Rayleigh to the geometric optics regimes,” J. Quant. Spectrosc. Radiat. Transf. 129, 169–185 (2013).
[Crossref]

L. Liu, M. I. Mishchenko, B. Cairns, B. E. Carlson, and L. D. Travis, “Modeling single-scattering properties of small cirrus particles by use of a size-shape distribution of ice spheroids and cylinders,” J. Quant. Spectrosc. Radiat. Transf. 101(3), 488–497 (2006).
[Crossref]

A. Quirantes, “A T-matrix method and computer code for randomly oriented, axially symmetric coated scatterers,” J. Quant. Spectrosc. Radiat. Transf. 92(3), 373–381 (2005).
[Crossref]

H. Iwabuchi and T. Suzuki, “Fast and accurate radiance calculations using truncation approximation for anisotropic scattering phase functions,” J. Quant. Spectrosc. Radiat. Transf. 110(17), 1926–1939 (2009).
[Crossref]

M. I. Mishchenko and L. D. Travis, “Capabilities and Limitations of a Current Fortran Implementation of the T-Martrix Method for Randomly Oriented,Rotationally Symmetric Scatterers,” J. Quant. Spectrosc. Radiat. Transf. 60(3), 309–324 (1998).
[Crossref]

H. Iwabuchi and P. Yang, “Temperature dependence of ice optical constants: Implications for simulating the single-scattering properties of cold ice clouds,” J. Quant. Spectrosc. Radiat. Transf. 112(15), 2520–2525 (2011).
[Crossref]

M. I. Mishchenko, “Electromagnetic scattering by nonspherical particles:A tutorial review,” J. Quant. Spectrosc. Radiat. Transf. 110(11), 808–832 (2009).
[Crossref]

T. Cheng, X. Gu, Y. Wu, H. Chen, and T. Yu, “The optical properties of absorbing aerosols with fractal soot aggregates: Implications for aerosol remote sensing,” J. Quant. Spectrosc. Radiat. Transf. 125, 93–104 (2013).
[Crossref]

T. H. Cheng, X. F. Gu, T. Yu, and G. L. Tian, “The reflection and polarization properties of nonspherical aerosol particles,” J. Quant. Spectrosc. Radiat. Transf. 111(6), 895–906 (2010).
[Crossref]

Opt. Commun. (1)

M. I. Mishchenko and L. D. Travis, “T-Martrix computations of light scattering by large spheriodal particles,” Opt. Commun. 109(1-2), 16–21 (1994).
[Crossref]

Opt. Express (1)

Prog. Electromagnetics Res. (1)

Y. Liu, Y. Chen, and P. Zhang, “Parallel implementation and application of the MRTD with an efficient CFS-PML,” Prog. Electromagnetics Res. 143, 223–242 (2013).
[Crossref]

Wuli Xuebao (1)

Y. Liu, Y. Chen, X. Xu, and Z. Liu, “Implementation and analysis of the perfectly matched layer with auxiliary differential equation for the multiresolution time-domain method,” Wuli Xuebao 62, 034101 (2013).

Other (9)

D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. (Pearson Education, Inc, 2014).

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

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

Q. Gao, Q. Cao, and J. Zhou, “Application of Total-Field/Scattered-Field Technique to 3D-MRTD Scattering Scheme,” in International Forum on Information Technology and Applications(2009), pp. 359–362.

R. F. Harrington, Field Computation by Moment Methods (Macmillan, 1968).

S. Y. Dai, “Application of Wavelet and Multiresolution Time domain (MRTD) on Electromagnetic scattering,” (Xidian University, 2007).

IPCC: Climate Change 2007, “Intergovernmental Panel of Global Climate Change,” (2007).

K. N. Liou, An Introduction to Atmospheric Radiation (Academic Press, 2003).

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles, Theory, Measurements, and Application (Academic Press, 2000).

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 (8)

Fig. 1
Fig. 1 The basic frame of MRTD scattering model for nonspherical aerosol.
Fig. 2
Fig. 2 Correction regions of the field component Ex for the TF-SF Technique in MRTD scheme.
Fig. 3
Fig. 3 The data exchange scheme for parallel MRTD scattering model.
Fig. 4
Fig. 4 Comparison of the scattering phase matrices simulated by MRTD and Lorenz-Mie theory (r = 0.5μm). The left panel presents the variation of phase matrix elements with scattering angle, and the right panel shows the variation of the simulation errors, where the simulation errors of F11 are evaluated by the relative error; for F12/ F11, F34/F11 and F44/F11, the errors are described by absolute errors.
Fig. 5
Fig. 5 Comparison of the scattering phase matrices simulated by MRTD and Lorenz-Mie theory for the spherical particle with a size parameter of 100. The left panel presents the curves of phase matrix elements with scattering angle, and the right panel shows the simulation error.
Fig. 6
Fig. 6 Comparison of the scattering phase matrices simulated by MRTD and Aden-Kerker theory for the concentric sphere. The left panel presents the curves of phase matrix elements with scattering angle, and the right panel shows their simulation errors.
Fig. 7
Fig. 7 Comparison of the phase matrix elements simulated by MRTD and T-matrix method for spheroidal particle. The left panel presents the variation of phase matrix elements with scattering angle, and the right panel shows their relative simulation errors.
Fig. 8
Fig. 8 Comparison of scattering phase matrix elements simulated by MRTD and T-matrix method for circular cylindrical particle. The left panel presents the curves of phase matrix elements with scattering angle, and the right panel shows the variation of their simulation errors.

Tables (4)

Tables Icon

Table 1 Comparison of the integral scattering parameters simulated by Lorenz-Mie theory and parallelized MRTD scattering model.

Tables Icon

Table 2 Comparison of the integral scattering parameters of spheroidal particles simulated by T-matrix theory and parallelized MRTD scattering model.

Tables Icon

Table 3 Comparison of the integral scattering parameters of cylindrical particles simulated by T-matrix theory and parallelized MRTD scattering model.

Tables Icon

Table 4 the parallel computational efficiencies for different number of processors.

Equations (53)

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

×H=ε E t +σE, ×E=μ H t σ m H;
ε c = ε 0 m 2 =ε jσ ω ,
E x (r,t)= i,j,k,n= + E i+1/2,j,k ϕx,n ϕ i+1/2 (x) ϕ j (y) ϕ k (z) h n (t);
E y (r,t)= i,j,k,n= + E i,j+1/2,k ϕy,n ϕ i (x) ϕ j+1/2 (y) ϕ k (z) h n (t);
E z (r,t)= i,j,k,n= + E i,j,k+1/2 ϕz,n ϕ i (x) ϕ j (y) ϕ k+1/2 (z) h n (t);
H x (r,t)= i,j,k,n= + H i,j+1/2,k+1/2 ϕx,n+1/2 ϕ i (x) ϕ j+1/2 (y) ϕ k+1/2 (z) h n+1/2 (t);
H y (r,t)= i,j,k,n= + H i,j+1/2,k+1/2 ϕy,n+1/2 ϕ i+1/2 (x) ϕ j (y) ϕ k+1/2 (z) h n+1/2 (t);
H z (r,t)= i,j,k,n= + H i+1/2,j+1/2,k ϕz,n+1/2 ϕ i+1/2 (x) ϕ j+1/2 (y) ϕ k (z) h n+1/2 (t);
h n (t)=h(t/Δtn).
ϕ k (x)=ϕ(x/Δxk+ M 1 ),
εΔxΔyΔz i ' , j ' , k ' = + ( E i ' +1/2, j ' , k ' ϕx,n+1 E i ' +1/2, j ' , k ' ϕx,n ) δ i, i ' δ j, j ' δ k, k ' + σΔxΔyΔz 2 i ' , j ' , k ' = + ( E i ' +1/2, j ' , k ' ϕx,n+1 + E i ' +1/2, j ' , k ' ϕx,n ) δ i, i ' δ j, j ' δ k, k ' =ΔtΔxΔz i ' , j ' , k ' = + H i ' +1/2, j ' +1/2, k ' ϕz, n ' +1/2 δ i, i ' δ k, k ' l=Ls Ls1 a(l) δ j+l, j ' ΔtΔxΔy i ' , j ' , k ' = + H i ' +1/2, j ' , k ' +1/2 ϕy, n ' +1/2 δ i, i ' δ j, j ' l=Ls Ls1 a(l) δ k+l, k ' ,
a(l) d ϕ j+1/2 (x) dx , ϕ jl (x) = 0 ω| ϕ ^ (ω) | 2 sin(ω(l+1/2))dω,
E i+1/2,j,k ϕx,n+1 =CA(m) E i+1/2,j,k ϕx,n +CB(m) l=Ls Ls1 a(l)( H i+0.5,j+l+0.5,k ϕz,n+1/2 /Δy H i+0.5,j,k+l+0.5 ϕy,n+1/2 /Δz) ,
CA(m)= 2εσΔt 2ε+σΔt , CB(m)= 2Δt 2ε+σΔt .
Δt 1 c l=0 Ls1 | a(l) | 1 (Δx) 2 + 1 (Δy) 2 + 1 (Δz) 2 ,
jωεE+σE= s ×H, jωμH σ m H= s ×E;
s i = κ i + σ i α i +jω ε 0 .
1 κ y H z y 1 κ z H y z + ψ exy ψ exz =jωε E x +σ E x ;
ψ exy = σ y ' α y ' +jω ε y ' y H z , ψ exz = σ z ' α z ' +jω ε z ' z H y ;
1 κ y H z y 1 κ z H y z + ψ exy ψ exz =ε t E x +σ E x ;
ε y ' t ψ exy + α y ' ψ exy = σ y ' y H z , ε z ' t ψ exz + α z ' ψ exz = σ z ' z H y ;
E x n+1 =CA(m) E x n +CB(m) ( 1 κ y H z y 1 κ z H y z + ψ exy ψ exz ) | n+1/2 ;
ψ exy n+1 = ψ exy n exp( α y ' ε y ' Δt)+ σ y ' α y ' [ 1exp( α y ' ε y ' Δt) ] y H z | n+1/2 ;
ψ exz n+1 = ψ exz n1 exp( α z ' ε z ' Δt)+ σ z ' α z ' [ 1exp( α z ' ε z ' Δt) ] z H y | n+1/2 .
E i+1/2,j,k ϕx,n+1 =C A i+1/2,j,k E i+1/2,j,k ϕx,n +C B i+1/2,j,k [ 1 κ y Δy l=Ls Ls1 a(l) H i+1/2,j+l+1/2,k ϕz,n+1/2 1 κ z Δz H i+1/2,j,k+l+1/2 ϕy,n+1/2 ] +C B i+1/2,j,k ( ψ exy,i+1/2,j,k ϕx,n+1 ψ exz,i+1/2,j,k ϕx,n+1 );
ψ exy,i+1/2,j,k ϕx,n+1 = b y ψ exy,i+1/2,j,k ϕx,n + a y Δy l=Ls Ls1 a(l) H i+1/2,j+l+1/2,k ϕz,n+1/2 ;
ψ exz,i+1/2,j,k ϕx,n+1 = b z ψ exz,i+1/2,j,k ϕx,n + a z Δz l=Ls Ls1 a(l) H i+1/2,j,k+l+1/2 ϕy,n+1/2 ;
b i =exp( α i ' ε i ' Δt), a i = σ i ' α i ' [ 1exp( α i ' ε i ' Δt) ].
E= E inc + E s , H= H inc + H s ;
s E i+1/2,j,k ϕx,n+1 = s E i+1/2,j,k ϕx,n+1 + Δt εΔy l=Ls Ls1 a (l) s H i+1/2,j+l+1/2,k ϕz,n+1/2 Δt εΔz [ l= K max k Ls1 a (l) s H i+1/2,j,k+l+1/2 ϕy,n+1/2 + l=Ls K max k1 a(l)( tol H i+1/2,j,k+l+1/2 ϕy,n+1/2 inc H i+1/2,j,k+l+1/2 ϕy,n+1/2 ) ],
tol E i+1/2,j,k ϕx,n+1 = tol E i+1/2,j,k ϕx,n+1 + Δt εΔy [ l= J min j Ls1 a (l) tol H i+1/2,j+l+1/2,k ϕz,n+1/2 + l= J min j Ls1 a(l)( s H i+1/2,j+l+1/2,k ϕz,n+1/2 + inc H i+1/2,j+l+1/2,k ϕz,n+1/2 ) ] Δt εΔz [ l= K max k Ls1 a (l) tol H i+1/2,j,k+l+1/2 ϕy,n+1/2 + l=Ls K max k1 a(l)( s H i+1/2,j,k+l+1/2 ϕy,n+1/2 + inc H i+1/2,j,k+l+1/2 ϕy,n+1/2 ) ].
( Δ+ ω 2 ε 0 μ 0 )E=( ω 2 ε 0 μ 0 I+ )P(r),
E(r)= E inc (r)+ V exp(jk| r r ' |) | r r ' | ( k 2 I+ )P( r ' ) d 3 r ' ,
E sca (r)=E(r) E inc (r) = k 2 exp(jkr) 4πr V [ ε( r ' ) ε 0 1 ]{ E( r ' ) e r [ e r E( r ' ) ] }exp(jk e r r ' ) d 3 r ' ,
E sca x (r)= k 2 exp(jkr) 4πr i ' j ' k ' [ ε ¯ ( i ' , j ' , k ' ) ε 0 1 ] { E ¯ x ( i ' , j ' , k ' )sin θ ob cos φ ob Ts } exp(j k x i ' Δx+j k y j ' Δy+j k z k ' Δz)ΔxΔyΔz;
E sca y (r)= k 2 exp(jkr) 4πr i ' j ' k ' [ ε ¯ ( i ' , j ' , k ' ) ε 0 1 ] { E ¯ y ( i ' , j ' , k ' )sin θ ob sin φ ob Ts } exp(j k x i ' Δx+j k y j ' Δy+j k z k ' Δz)ΔxΔyΔz;
E sca z (r)= k 2 exp(jkr) 4πr i ' j ' k ' [ ε ¯ ( i ' , j ' , k ' ) ε 0 1 ] { E ¯ z ( i ' , j ' , k ' )cos θ ob Ts } exp(j k x i ' Δx+j k y j ' Δy+j k z k ' Δz)ΔxΔyΔz;
Ts=sin θ ob cos φ ob E ¯ x ( i ' , j ' , k ' )+sin θ ob sin φ ob E ¯ y ( i ' , j ' , k ' )+cos φ ob E ¯ z ( i ' , j ' , k ' ).
[ E sca φ (r) E sca θ (r) E sca r (r) ]= U T [ E sca x (r) E sca y (r) E sca z (r) ]=[ sin φ i cos φ i 0 cos θ i cos φ i cos θ i sin φ i sin θ i sin θ i cos φ i sin θ i sin φ i cos θ i ][ E sca x (r) E sca y (r) E sca z (r) ].
P ab =Re[ S n s S(r) d 2 r ]=Re[ V S(r) d 3 r ]= 1 2 Re[ V ( E(r)× H * (r) ) d 3 r ],
P ab = 1 2 Re V jω( ε(r)E(r) E * (r)μ(r)H(r) H * (r) )σ(r)E(r) E * (r) d 3 r.
P ab = 1 2 V σ(r)E(r) E * (r) d 3 r.
C ab = P ab P inc = Z 0 V σ(r)E(r) E * (r) d 3 r | E inc | 2 ,
S ext (r)= 1 2 ( E inc (r)× H s * (r)+ E s (r)× H inc * (r) ).
P ext =Re( S S ext (r) n s d 2 r )= 1 2 Re{ V ( E inc (r)× H s * (r)+ E s (r)× H inc * (r) ) d 3 r }.
P ext = 1 2 Re{ V [ jω μ 0 ( H inc (r) H s * (r)+ H inc * (r) H s (r) ) +jω( ε E inc (r) E s * (r)+ ε 0 E inc * (r) E s (r) σ jω E inc (r) E s * (r) ) ] d 3 r }.
P ext = 1 2 { V Im[ ω( ε E inc (r) E s * (r) ε 0 E inc (r) E s * (r)+ σ jω E inc (r) E s * (r) ) ] d 3 r }.
C ext = P ext P inc = Z 0 | E inc | 2 { V Im[ ω( ε ε 0 j σ ω ) E inc (r) E s * (r) ] d 3 r }.
C sc = C ext C ab .
[ E sca E sca // ]= expj(kr+z) jkr S( θ )[ E inc E inc // ]= expj(kr+z) jkr [ S 1 S 4 S 3 S 2 ][ E inc E inc // ],
[ E sca, E sca,// E sca, // E sca,// // ]= expj(kr+z) jkr [ S 1 S 4 S 3 S 2 ][ E inc 0 0 E inc // ].
S p = T 1 / T n ,
η= S p /n.

Metrics