dorsal/arxiv
View SchemaWave packet propagation by the Faber polynomial approximation in electrodynamics of passive media
| Authors | Andrei G. Borisov, Sergei V. Shabanov |
|---|---|
| Categories | |
| ArXiv ID | physics/0512074 |
| URL | https://arxiv.org/abs/physics/0512074 |
| DOI | 10.1016/j.jcp.2005.12.011 |
Abstract
Maxwell's equations for propagation of electromagnetic waves in dispersive and absorptive (passive) media are represented in the form of the Schr\"odinger equation $i\partial \Psi/\partial t = {H}\Psi$, where ${H}$ is a linear differential operator (Hamiltonian) acting on a multi-dimensional 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 field configuration. The Faber polynomial approximation of the fundamental solution 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 time step is much larger than that in finite differencing schemes, $\Delta t_F \gg \|H\|^{-1}$. 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, $\Delta t_C \sim \|H\|^{-1}$, is exceeded at least in 3000 times in the Faber propagation scheme.
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"abstract": "Maxwell\u0027s equations for propagation of electromagnetic waves in dispersive\nand absorptive (passive) media are represented in the form of the Schr\\\"odinger\nequation $i\\partial \\Psi/\\partial t = {H}\\Psi$, where ${H}$ is a linear\ndifferential operator (Hamiltonian) acting on a multi-dimensional vector $\\Psi$\ncomposed of the electromagnetic fields and auxiliary matter fields describing\nthe medium response. In this representation, the initial value problem is\nsolved by applying the fundamental solution $\\exp(-itH)$ to the initial field\nconfiguration. The Faber polynomial approximation of the fundamental solution\nis used to develop a numerical algorithm for propagation of broad band wave\npackets in passive media. The action of the Hamiltonian on the wave function\n$\\Psi$ is approximated by the Fourier grid pseudospectral method. The algorithm\nis global in time, meaning that the entire propagation can be carried out in\njust a few time steps. A typical time step is much larger than that in finite\ndifferencing schemes, $\\Delta t_F \\gg \\|H\\|^{-1}$. The accuracy and stability\nof the algorithm is analyzed. The Faber propagation method is compared with the\nLanczos-Arnoldi propagation method with an example of scattering of broad band\nlaser pulses on a periodic grating made of a dielectric whose dispersive\nproperties are described by the Rocard-Powels-Debye model. The Faber algorithm\nis shown to be more efficient. The Courant limit for time stepping, $\\Delta t_C\n\\sim \\|H\\|^{-1}$, is exceeded at least in 3000 times in the Faber propagation\nscheme.",
"arxiv_id": "physics/0512074",
"authors": [
"Andrei G. Borisov",
"Sergei V. Shabanov"
],
"categories": [
"physics.comp-ph",
"math.NA",
"physics.optics"
],
"doi": "10.1016/j.jcp.2005.12.011",
"title": "Wave packet propagation by the Faber polynomial approximation in electrodynamics of passive media",
"url": "https://arxiv.org/abs/physics/0512074"
},
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