Skip to main content

Finite-difference time-domain solution of light scattering by arbitrarily shaped particles and surfaces

  • Chapter
  • First Online:
Light Scattering Reviews, Vol. 6

Abstract

The scattering and absorption of electromagnetic waves by irregularly shaped particles and arbitrary surfaces occur in the atmosphere, ocean, and optical devices. In this chapter, we present the finite-difference time-domain (FDTD) method [1–6] that can be used to calculate light scattering by arbitrary particles and surfaces. The FDTD technique is a numerical solution to Maxwell’s equations and is formulated by replacing temporal and spatial derivatives in Maxwell’s equations with their finite-difference equivalences. This method can be accurately applied to general electromagnetic structures including arbitrary particles and surfaces. The FDTD technique has been successfully applied to calculate light scattering and absorption by particles of different shapes in free space [5] and in absorbing medium [6]. Recently, an advanced FDTD model to calculate the interaction of electromagnetic radiation with arbitrary dielectric surfaces has been developed [7]. In the following sections, these FDTD light-scattering models are reviewed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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).

    ADS  Google Scholar 

  2. A. Taflove and M. E. Brodwin, Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations, IEEE Trans. Microwave Theory Tech. MTT-23, 623–630 (1975).

    Article  ADS  Google Scholar 

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

    MATH  Google Scholar 

  4. 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).

    Article  ADS  Google Scholar 

  5. W. Sun, Q. Fu, and 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).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  7. W. Sun, H. Pan, and G. Videen, General finite-difference time-domain solution of an arbitrary EM source interaction with an arbitrary dielectric surface, Appl. Opt. 48, 6015–6025 (2009).

    Article  ADS  Google Scholar 

  8. G. Mie, Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen, Leipzig, Ann. Phys. 330, 377–445 (1908).

    Article  Google Scholar 

  9. G. Videen, D. Ngo, and M. B. Hart, Light scattering from a pair of conducting, osculating spheres, Opt. Commun. 125, 275–287 (1996).

    Article  ADS  Google Scholar 

  10. D. Petrov, Y. Shkuratov, G. Videen, Analytic light-scattering solution of two merging spheres using Sh-matrices, Opt. Comm. 281, 2411–2423 (2008).

    Article  ADS  Google Scholar 

  11. S. Asano and G. Yamamoto, Light scattering by a spheroidal particle, Appl. Opt. 14, 29–49 (1975).

    ADS  Google Scholar 

  12. M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, T-matrix computations of light scattering by non-spherical particles. A review, J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).

    Article  ADS  Google Scholar 

  13. Lord Rayleigh, The dispersal of light by a dielectric cylinder, Philos. Mag. 36, 365–376 (1918).

    Google Scholar 

  14. J. R. Wait, Scattering of a plane wave from a circular dielectric cylinder at oblique incidence, Can. J. Phys. 33, 189–195 (1955).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. A. Mugnai and W. J. Wiscombe, Scattering from non-spherical Chebyshev particles, Appl. Opt. 25, 1235–1244 (1986).

    Article  ADS  Google Scholar 

  16. D. Petrov, Y. Shkuratov, G. Videen, Analytical light-scattering solution for Chebyshev particles, J. Opt. Soc. Am. A 24, 1103–1119 (2007).

    Article  MathSciNet  ADS  Google Scholar 

  17. M. I. Mishchenko, L. D. Travis, and A. Macke, Scattering of light by polydisperse, randomly oriented, finite circular cylinders, Appl. Opt. 35, 4927–4940 (1996).

    Article  ADS  Google Scholar 

  18. D. Petrov, Y. Shkuratov, G. Videen, The influence of corrugation on light-scattering properties of capsule and finite cylinder particles: Sh-matrices analysis, J. Quant. Spectrosc. Radiat. Transfer 109, 650–669 (2008).

    Article  ADS  Google Scholar 

  19. D. Petrov, Y. Shkuratov, G. Videen, Sh-matrices method as applied to light scattering by circular cylinders, J. Quant. Spectrosc. Radiat. Transfer 109, 1474–1495 (2008).

    Article  ADS  Google Scholar 

  20. H. Laitinen and K. Lumme, T-matrix method for general star-shaped particles: first results, J. Quant. Spectrosc. Radiat. Transfer 60, 325–334 (1998).

    Article  ADS  Google Scholar 

  21. D. Petrov, Y. Shkuratov, G. Videen, An analytical solution to the light scattering from cube-like particles using Sh-matrices, J. Quant. Spectrosc. Radiat. Transfer 111, 474–482 (2010).

    Article  ADS  Google Scholar 

  22. Lord Rayleigh, On the light from the sky, its polarization and colour, Phil. Mag. 41, 107–120, 274–279 (1871).

    Google Scholar 

  23. V. G. Farafonov, Light-scattering by spheroidal particles in quasi-static approximation, Opt. Spektrosk. 77, 455–458 (1994).

    Google Scholar 

  24. P. Yang and K. N. Liou, Geometric-optics integral-equation method for light scattering by non-spherical ice crystals, Appl. Opt. 35, 6568–6584 (1996).

    Article  ADS  Google Scholar 

  25. P. Barber and C. Yeh, Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies, Appl. Opt. 14, 2864–2872 (1975).

    ADS  Google Scholar 

  26. B. T. Draine, The discrete-dipole approximation and its application to interstellar graphite grains, Astrophys. J. 333, 848–872 (1988).

    Article  ADS  Google Scholar 

  27. B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp. 31, 629–651 (1971).

    Article  MathSciNet  ADS  Google Scholar 

  28. A. Bayliss and E. Turkel, Radiation boundary conditions for wave-like equations, Commun. Pure Appl. Math. 33, 707–725 (1980).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. G. Mur, Absorbing boundary condition for the finite-difference approximation of the time-domain electromagnetic-field equations, IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).

    Article  Google Scholar 

  30. Z. Liao, H. L. Wong, B. Yang, and Y. Yuan, A transmitting boundary for transient wave analyses, Sci. Sin. 27, 1063–1076 (1984).

    MATH  Google Scholar 

  31. R. L. Higdon, Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation, Math. Comp. 47, 437–459 (1986).

    MathSciNet  MATH  Google Scholar 

  32. C. E. Reuter, R. M. Joseph, E. T. Thiele, D. S. Katz, and T. Taflove, Ultrawideband absorbing boundary condition for termination of wave guide structures in FD-TD simulations, IEEE Microwave and Guided Wave Lett. 4, 344–346 (1994).

    Article  Google Scholar 

  33. J. P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comp. Phys. 114, 185–200 (1994).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. D. S. Katz, E. T. Thiele, and A. Taflove, Validation and extension to three dimensions of the Berenger PML absorbing boundary condition for FD-TD meshes, IEEE Microwave and Guided Wave Lett. 4, 268–270 (1994).

    Article  Google Scholar 

  35. Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, A perfectly matched anisotropic absorber for use as an absorbing boundary condition, IEEE Trans. Antennas Propaga. 43, 1460–1463 (1995).

    Article  ADS  Google Scholar 

  36. S. D. Gedney, An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices, IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  38. D. E. Merewether, R. Fisher, and 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).

    Article  ADS  Google Scholar 

  39. K. Umashanker and A. Taflove, A novel method to analyze electromagnetic scattering of complex objects, IEEE Trans. Electromagn. Compat. EMC-24, 397–405 (1982).

    Article  ADS  Google Scholar 

  40. W. C. Mundy, J. A. Roux, and A. M. Smith, Mie scattering by spheres in an absorbing medium, J. Opt. Soc. Am. 64, 1593–1597 (1974).

    Article  ADS  Google Scholar 

  41. P. Chylek, Light scattering by small particles in an absorbing medium, J. Opt. Soc. Am. 67, 561–563 (1977).

    Article  ADS  Google Scholar 

  42. C. F. Bohren and D. P. Gilra, Extinction by a spherical particle in an absorbing medium, J. Colloid Interface Sci. 72, 215–221 (1979).

    Article  Google Scholar 

  43. M. Quinten and J. Rostalski, Lorenz-Mie theory for spheres immersed in an absorbing host medium, Part. Part. Syst. Charact. 13, 89–96 (1996).

    Article  Google Scholar 

  44. A. N. Lebedev, M. Gartz, U. Kreibig, and O. Stenzel, Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films, Eur. Phys. J. D 6, 365–373 (1999).

    Article  ADS  Google Scholar 

  45. Q. Fu and W. Sun, Mie theory for light scattering by a spherical particle in an absorbing medium, Appl. Opt. 40, 1354–1361 (2001).

    Article  ADS  Google Scholar 

  46. I. W. Sudiarta and P. Chylek, Mie-scattering formalism for spherical particles embedded in an absorbing medium, J. Opt. Soc. Am. A 18, 1275–1278 (2001).

    Article  ADS  Google Scholar 

  47. Q. Fu and W. Sun, Apparent optical properties of spherical particles in absorbing medium, J. Quan. Spectro. Rad. Transfer 100, 137–142 (2006).

    Article  ADS  Google Scholar 

  48. M. I. Mishchenko, Electromagnetic scattering by a fixed finite object embedded in an absorbing medium, Opt. Express 15, 13188–13202 (2007).

    Article  ADS  Google Scholar 

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

    Google Scholar 

  50. G. H. Goedecke and S. G. O’Brien, Scattering by irregular inhomogeneous particles via the digitized Green’s function algorithm, Appl. Opt. 27, 2431–2438 (1988).

    Article  ADS  Google Scholar 

  51. E. Zubko, D. Petrov, Y. Grynko, Y. Shkuratov, H. Okamoto, K. Muinonen, T. Nousiainen, H. Kimura, T. Yamamoto, and G. Videen, Validity criteria of the discrete dipole approximation, Appl. Opt. 49, 1267–1279 (2010).

    Article  ADS  Google Scholar 

  52. W. Sun, T. Nousiainen, K. Muinonen, Q. Fu, N. G. Loeb, and 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).

    Article  Google Scholar 

  53. D. L. Schuler, J.-S. Lee, D. Kasilingam, and G. Nesti, Surface roughness and slope measurements using polarimetric SAR data, IEEE Transactions on Geoscience and Remote Sensing, 40, 687–698 (2002).

    Article  ADS  Google Scholar 

  54. S. Gomez, K. Hale, J. Burrows, and B. Griffiths, Measurements of surface defects on optical components, Meas. Sci. Technol. 9, 607–616 (1998).

    Article  ADS  Google Scholar 

  55. H. Lin and J. Zhu, Characterization of nanocrystalline silicon films, Proc. SPIE, 4700, 354–356 (2002).

    Article  ADS  Google Scholar 

  56. A. Angell and C. Rappaport, Computational modeling analysis of radar scattering by clothing covered arrays of metallic body-worn explosive devices, Progress In Electromagnetics Research PIER 76, 285–298 (2007).

    Article  Google Scholar 

  57. Lord Rayleigh, The Theory of Sound (MacMillan, London, 1896).

    MATH  Google Scholar 

  58. U. Fano, The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves), J. Opt. Soc. Am. 31, 213–222 (1941).

    Article  ADS  Google Scholar 

  59. S. O. Rice, Reflection of electromagnetic waves from slightly rough surfaces, Commun. Pure Appl. Math. 4, 351–378 (1951).

    Article  MathSciNet  MATH  Google Scholar 

  60. S. O. Rice, Reflection of EM from Slightly Rough Surfaces (Interscience, New-York, 1963).

    Google Scholar 

  61. C. Eckart, The scattering of sound from the sea surface, J. Acoust. Soc. Am. 25, 66–570 (1953).

    Google Scholar 

  62. H. Davies, The reflection of electromagnetical waves from rough surfaces, Proc. IEE (London) 101, 209–214 (1954).

    Google Scholar 

  63. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon Press, Oxford, England, 1963).

    MATH  Google Scholar 

  64. A. K. Fung and G. W. Pan, An integral equation method for rough surface scattering, in Proceedings of the International Symposium on multiple scattering of waves in random media and random surfaces, 701–714 (1986).

    Google Scholar 

  65. A. K. Fung, Z. Li, and K. S. Chen, Backscattering from a randomly rough dielectric surface, IEEE Trans. Geosci. and Remote Sens. 30, 356–369 (1992).

    Article  ADS  Google Scholar 

  66. A. K. Fung, Microwave Scattering and Emission Models and their Applications (Artech House, Norwood, MA, 1994).

    Google Scholar 

  67. L. Tsang, J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves: Numerical Simulations (John Wiley, New York, 2001).

    Book  Google Scholar 

  68. M. Saillard and A. Sentenac, Rigorous solutions for electromagnetic scattering from rough surfaces, Waves in Random Media 11, 103–137 (2001).

    Article  MathSciNet  ADS  Google Scholar 

  69. C. Y. Hsieh, A. K. Fung, G. Nesti, A. J. Siber, and P. Coppo, A further study of the IEM surface scattering model, IEEE Trans. Geosci. and Remote Sens. 35, 901–909 (1997).

    Article  ADS  Google Scholar 

  70. A. K. Fung, Z. Li, and K. S. Chen, An improved IEM model for bistatic scattering from rough surfaces, J. Electromagn. Waves and Appl. 16, 689–702 (2002).

    Article  Google Scholar 

  71. K. S. Chen, T. D. Wu, and A. K. Fung, A study of backscattering from multi-scale rough surface, J. Electromagn. Waves and Appl. 12, 961–979 (1998).

    Article  Google Scholar 

  72. F. Mattia, Backscattering properties of multi-scale rough surfaces, J. Electromagn. Waves and Appl. 13, 493–527 (1999).

    Article  MATH  Google Scholar 

  73. P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers (Cambridge Univ. Press, Cambridge, U.K., 1990).

    Google Scholar 

  74. J. M. Jin, The Finite Element Method in Electromagnetics (John Wiley, New York, 1993).

    MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  76. S. B. Singham and G. C. Salzman, Evaluation of the scattering matrix of an arbitrary particle using the coupled dipole approximation, J. Chem. Phys. 84, 2658–2667 (1986).

    Article  ADS  Google Scholar 

  77. W. Sun, G. Videen, B. Lin, and Y. Hu, Modeling light scattered from and transmitted through dielectric periodic structures on a substrate, Appl. Opt. 46, 1150–1156 (2007).

    Article  ADS  Google Scholar 

  78. D. Wu and Y. Zhou, Forward scattering light of droplets containing different size inclusions, Appl. Opt. 48, 2957–2965 (2009).

    Article  ADS  Google Scholar 

  79. G. Videen, Light scattering from a sphere on or near a surface, J. Opt. Soc. Am. A 8, 483–489 (1991); Errata. J. Opt. Soc. Am. A 9, 844–845 (1992).

    Article  ADS  Google Scholar 

  80. E. Fucile, P. Denti, F. Borghese, R. Saija, and O. I. Sindoni, Optical properties of a sphere in the vicinity of a plane surface, J. Opt. Soc. Am. A 14, 1505–1514 (1997).

    Article  MathSciNet  ADS  Google Scholar 

  81. B. R. Johnson, Calculation of light scattering from a spherical particle on a surface by the multipole expansion method, J. Opt. Soc. Am. A 13, 326–337 (1996).

    Article  ADS  Google Scholar 

  82. G. Videen, Light scattering from a sphere behind a surface, J. Opt. Soc. Am. A 10, 110–117 (1993).

    Article  ADS  Google Scholar 

  83. G. Videen, M. G. Turner, V. J. Iafelice, W. S. Bickel, and W. L. Wolfe, Scattering from a small sphere near a surface, J. Opt. Soc. Am. A 10, 118–126 (1993).

    Article  ADS  Google Scholar 

  84. G. Videen, M. M. Aslan, and M. P. Mengüç, Characterization of metallic nanoparticles via surface wave scattering: A. Theoretical framework and formulation, J. Quant. Spectrosc. Radiative Transfer 93, 195–206 (2005).

    Article  ADS  Google Scholar 

  85. R. Borghi, F. Gori, M. Santarsiero, F. Frezza, and G. Schettini, Plane-wave scattering by a perfectly conducting circular cylinder near a plane surface: Cylindrical-wave approach, J. Opt. Soc. Am. A 13, 483–493 (1996).

    Article  MathSciNet  ADS  Google Scholar 

  86. G. Videen and D. Ngo, Light scattering from a cylinder near a plane interface: Theory and comparison with experimental data, J. Opt. Soc. Am. A 14, 70–78 (1997).

    Article  MathSciNet  ADS  Google Scholar 

  87. G. Videen, Light scattering from a particle on or near a perfectly conducting surface, Opt. Commun. 115, 1–7 (1995).

    Article  ADS  Google Scholar 

  88. G. Videen, Light scattering from an irregular particle behind a plane interface, Opt. Commun. 128, 81–90 (1996).

    Article  ADS  Google Scholar 

  89. P. G. Venkata, M. M. Aslan, M. P. Menguc, and G. Videen, Surface plasmon scattering by gold nanoparticles and two-dimensional agglomerates, J. Heat Transfer-Trans. ASME 129, 60–70 (2007).

    Article  Google Scholar 

  90. T. Wriedt and A. Doicu, Light scattering from a particle on or near a surface, Opt Commun. 152, 376-384 (1998).

    Article  ADS  Google Scholar 

  91. P. Denti, F. Borghese, R. Saija, E. Fucile, and O. I. Sindoni, Optical properties of aggregated spheres in the vicinity of a plane surface, Appl. Opt. 16, 167–175 (1999).

    Google Scholar 

  92. D. W. Mackowski, Exact solution for the scattering and absorption properties of sphere clusters on a plane surface, J. Quant. Spectrosc. Radiative Transfer 109, 770–788 (2008).

    Article  ADS  Google Scholar 

  93. R. Schmehl, B. M. Nebeker, and E. D. Hirleman, Discrete-dipole approximation for scattering by features on surfaces by means of a two-dimensional fast Fourier transform technique, J. Opt. Soc. Am. A 14, 3026–3036 (1997).

    Article  ADS  Google Scholar 

  94. P. Albella, F. Moreno, J. M. Saiz, and F. González, Surface inspection by monitoring spectral shifts of localized Plasmon resonances, Opt. Exp. 16, 12,872–12,879 (2008).

    Article  ADS  Google Scholar 

  95. M. A. Yurkin, A. G. Hoekstra, R. S. Brock, and J. Q. Lu, Systematic comparison of the discrete dipole approximation and the finite difference time domain method for large dielectric scatterers, Opt. Express 15, 17902–17911 (2007).

    Article  ADS  Google Scholar 

  96. A. Doicu, Y. Eremin, and T. Wriedt, Acoustic and Electromagnetic Scattering Analysis Using Discrete Sources (Academic Press, San Diego, 2000).

    Google Scholar 

  97. N. J. Cassidy, A review of practical numerical modeling methods for the advanced interpretation of ground-penetrating radar in near-surface environments, Near Surface Geophysics 5, 5–21 (2007).

    Google Scholar 

  98. R. Holland, Threde: A free-field EMP coupling and scattering code, IEEE Trans. Nuclear Sci. 24, 2416–2421 (1977).

    Article  ADS  Google Scholar 

  99. R. Holland, R. L. Simpson, and K. S. Kunz, Finite-difference analysis of EMP coupling to lossy dielectric structures, IEEE Trans. Electromagn. Compat. 22, 203–209 (1980).

    Article  ADS  Google Scholar 

  100. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time Domain Method (Artech House, Boston, 2005).

    Google Scholar 

  101. W. Sun, N. G. Loeb, G. Videen, and Q. Fu, Examination of surface roughness on light scattering by long ice columns by use of a two-dimensional finite-difference timedomain algorithm, Appl. Opt. 43, 1957–1964 (2004).

    Article  ADS  Google Scholar 

  102. W. Sun, B. Lin, Y. Hu, Z. Wang, Y. Fu, Q. Feng, and P. Yang, Side-face effect of a dielectric strip on its optical properties, IEEE Transac. Geosci. Remote Sens. 46, doi: 10.1109/TGRS.2008.916984 (2008).

    Google Scholar 

  103. B. Saleh and M. Teich, Fundamentals of Photonics (John Wiley, New York, 1991).

    Book  Google Scholar 

  104. J. Durnin, Exact solutions for nondiffracting beams. I. The scalar theory, J. Opt. Soc. Am. A 4, 651–654 (1987).

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Wenbo Sun .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Sun, W. et al. (2012). Finite-difference time-domain solution of light scattering by arbitrarily shaped particles and surfaces. In: Kokhanovsky, A. (eds) Light Scattering Reviews, Vol. 6. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15531-4_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-15531-4_3

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15530-7

  • Online ISBN: 978-3-642-15531-4

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics