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Wave packet propagation by the Faber polynomial approximation in electrodynamics of passive media. (English) Zbl 1119.78017

Maxwell’s equations for propagation of electromagnetic waves in dispersive and absorptive (passive) media are represented in the form of the Schrödinger equation: \(i\partial_t \Psi = H \psi\), where \(H\) is a linear differential operator (Hamiltonian) acting on a multidimensional vector \(\Psi\) composed of the electromagnetic fields and auxiliary matter fields describing the medium response. In this representation, the initial value problem is solved by applying the fundamental solution \(\exp(-itH)\) to the initial value configuration. The Faber polynomial approximation is used to develop a numerical algorithm for propagation of broad band wave packets in passive media. The action of the Hamiltonian on the wave function \(\Psi\) is approximated by the Fourier grid pseudospectral method. The algorithm is global in time, meaning that the entire propagation can be carried out in just a few time steps. A typical step \(\Delta t\) is much larger than that in finite differencing schemes, \(\Delta t_F \geq | | H| | ^{-1}\). The accuracy and stability of the algorithm is analyzed. The Faber propagation method is compared with the Lanczos-Arnoldi propagation method with an example of scattering of broad band laser pulses on a periodic grating made of a dielectric whose dispersive properties are described by the Rocard-Powels-Debye model. The Faber algorithm is shown to be more efficient. The Courant limit for time stepping, \(\Delta t_C \approx | | H| | ^{-1}\), is exceeded at least.

MSC:

78M25 Numerical methods in optics (MSC2010)
78A40 Waves and radiation in optics and electromagnetic theory
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[1] Taflove, A.; Hagness, S. C., Computational Electrodynamics - The Finite-difference Time-domain Method (2000), Artech House: Artech House Boston, MA · Zbl 0963.78001
[2] Muinonen, K., Waves Random Media, 14, 365 (2004) · Zbl 1114.78338
[3] Faber, G.; Reine, J., Angew. Math., 150, 79 (1920)
[4] Smirnov, V. L.; Lebedev, N. A., Functions of a Complex Variable: Constructive Theory (1968), MIT: MIT Cambridge · Zbl 0164.37503
[5] Huang, Y.; Kouri, D. J.; Hoffman, D. K., J. Chem. Phys., 101, 10493 (1994)
[6] Pollard, W. T.; Friesner, R. A., J. Chem. Phys., 100, 5054 (1994)
[7] Huishinga, W.; Pesce, L.; Kosloff, R.; Saalfrank, P., J. Chem. Phys., 110, 5538 (1999)
[8] Tal-Ezer, H.; Kosloff, R., J. Chem. Phys., 81, 3967 (1984)
[9] Leforestier, C.; Bisseling, R. H.; Cerjan, C.; Feit, M. D.; Friesner, R.; Guldberg, A.; Hammerich, A.; Jolicard, G.; Karrlein, W.; Meyer, H.-D.; Lipkin, N.; Roncero, O.; Kosloff, R., J. Comput. Phys., 94, 59 (1991), and references therein
[10] (Cerjan, C., Numerical Grid Methods and Their Application to Schrödinger’s Equation. Numerical Grid Methods and Their Application to Schrödinger’s Equation, NATO ASI Series, Series C: Mathematical and Physical Sciences, vol. 412 (1993), Kluwer Academic Publishers: Kluwer Academic Publishers London) · Zbl 0810.00026
[11] Phys. Rev. E, 67, 056706 (2003)
[12] Borisov, A. G.; Shabanov, S. V., J. Comput. Phys., 209, 643 (2005)
[13] Boyd, J. P., Chebyshev and Fourier Spectral Methods (1989), Springer-Verlag: Springer-Verlag New York
[14] Kosloff, D.; Kosloff, R., J. Comput. Phys., 52, 35 (1983)
[15] Rung, A.; Ribbing, C. G., Phys. Rev. Lett., 92, 123901 (2004)
[16] Huang, K. C.; Bienstman, P.; Joannopoulos, J. D.; Nelson, K. A.; Fan, S., Phys. Rev. Lett., 90, 196402 (2003)
[17] S.V. Shabanov, Electromagnetic pulse propagation in passive media by path integral methods, a LANL e-preprint, 2003, Available from: <http://xxx.lanl.gov/abs/math.NA/0312296>; S.V. Shabanov, Electromagnetic pulse propagation in passive media by path integral methods, a LANL e-preprint, 2003, Available from: <http://xxx.lanl.gov/abs/math.NA/0312296>
[18] Borisov, A. G.; Shabanov, S. V., J. Comput. Phys., 199, 742 (2004)
[19] Kövari, T.; Pommerenke, Ch., Math. Z., 99, 193 (1967)
[20] Trefethen, L. N., SIAM J. Sci. Stat. Comput., 1, 82 (1980)
[21] Starke, G.; Vagra, R. S., Numer. Math., 64, 213 (1993)
[22] Spijker, M. N., Appl. Numer. Math., 13, 241 (1993)
[23] Ellacott, S. W., Math. Comput., 40, 575 (1983)
[24] Geddes, K. O.; Mason, J. C., SIAM J. Numer. Anal., 12, 111 (1975)
[25] Ebbesen, T. W.; Lezec, H. J.; Ghaemi, H. F.; Thio, T.; Wolff, P. A., Nature (London), 391, 667 (1998)
[26] Wood, R. W., Phys. Rev., 48, 928 (1935)
[27] Pilozzi, L.; D’Andrea, A.; Del Sole, R., Phys. Rev. B, 54, 10763 (1996)
[28] Borisov, A. G.; García de Abajo, F. J.; Shabanov, S. V., Phys. Rev. B, 71, 075408 (2005)
[29] Chow, E.; Lin, S. Y.; Johnson, S. G.; Villeneuve, P. R.; Joannopoulos, J. D.; Wendt, J. R.; Vawter, G. A.; Zubrzycki, W.; Hou, H.; Alleman, A., Nature, 407, 983 (2000)
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