Wave packet propagation by the Faber polynomial approximation in electrodynamics of passive media

Abstract

Maxwell's equations for propagation of electromagnetic waves in dispersive and absorptive (passive) media are represented in the form of the Schrödinger equation i∂Ψ/∂t = HΨ, where H is a linear differential operator (Hamiltonian) acting on a multi-dimensional vector Ψ 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 Ψ 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, ∆t F ≫ 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, ∆t C ∼ H −1 , is exceeded at least in 3000 times in the Faber propagation scheme.

Periodic nanostructures for photonics

“…Similarly, operator-exponential methods that rely on Chebyshevpolynomial expansions [104] provide very high accuracy for extremely long-time simulations but are only conditionally stable and strictly limited to hard-wall boundary conditions and nonabsorbing materials. A recent modification of this method utilizes a Faber-polynomial expansion technique [105]. Although this modification allows to include losses, the implementation of PMLs still remains an open issue.…”

Finite-element methods for spatially resolved mesoscopic electron transport

“…39 This would allow us to simulate chemical reaction probabilities 40 or the transmission properties of dielectric media. 41 With direct access to the wave function, the scattering matrix can be obtained as a function of the magnetic field, and the zero-temperature direct-current conductivity can be computed using the Landauer-Büttiker relation between quantum transmission and conductance. This provides a complete description of the quantum mechanical features of ballistic transport in a device including the local electron density.…”

A Krylov‐subspace based solver for the linear and nonlinear Maxwell equations

“…It is straightforward to show [25] that the impedance matching requires the use of anisotropic magnetic and dielectric materials that break the skew-symmetry of the time-evolution operator. Thus, recently proposed methods for solving Maxwell's equation which appear to require the said symmetry [26][27][28][29] have so far resisted the implementation of this most important class of ABCs (for recent progress in this active area of research we refer to Ref. [30] in this volume).…”