  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.
