Wave Propagation in Linear Media | re-examined

Interlude
power and numerical accuracy, both of which can in turn be hard to cope with. Consequently,
particularly in the early days of computers, approximation and expansion methods were still
widely employed to obtain reasonable results within reasonable time.
Scanning the literature, we nd two principal approaches to treat the wave propagation problem
by means of numerical computing. The rst is to formulate the wave integral and then
apply numerics to evaluate the expression. As mentioned before, the integrand is often simplied
or transformed via series expansion to facilitate integration either in terms of convergence
or accuracy. This way the normally in nite integrals may frequently be truncated at some
nite value without introducing too large an error. The second approach is to dispense with
the wave integral itself and start from the di erential equation together with the appropriate
initial and boundary conditions describing the propagation phenomenon. This usually
involves some sort of discretisation of the di erential equation on a properly chosen grid in
order to nd a numerical solution. Such a strategy was for example pursued by Weiland [105],
who used it calculate electromagnetic elds in complex geometric structures. He divided the
area of interest into small elementary cells of essentially arbitrary shape. Out of the many
possible discretisations, he selected one that uses the integral form of the Maxwell equations
such that at the boundaries between the cells, the tangential component oftheelectric eld
and the perpendicular component of the magnetic eld are continuous (which is not necessarily
the case if other discretisation schemes are used). Solving Maxwell's equations is then
reduced to the easier task of solving matrix equations.
The alternative approach of evaluating the wave integrals numerically has been used by a
larger number of authors | at least as far as electromagnetic wave propagation is concerned.
At the beginning of the computer era, Haskell and Case [97] did not really evaluate the integral
directly, but used the familiar method of saddle-point integration to obtain an approximation
for large values of x. This strategy was stimulated by the observation that most solutions at
that time were suitable only for small propagation distances, and they ended up with a solution
consisting of Fresnel integrals that could be computed numerically. As for the question which
part of the integrand is to be expanded, a variety of aswers were attempted. Knop [96], for
example, expanded the sine function de ning the input signal of the transmission line into a
series of Bessel functions and then applied the inverse Laplace transform. Trizna and Weber
[23], on the contrary, expanded the entire integrand also into a series of Bessel functions
that could be integrated numerically. More recently, Wyns et al. [25] used inverse Laplace
transform, but prior to that, approximated the factor e st by a function which then could
be treated by series expansion. As a result, they could integrate the simpli ed functions
analytically and employ a truncated summation to achieve the desired accuracy. Albanese et
al. [106] cleverly sidestepped the entire problem of Fourier transform by regarding a repetitive
input pulse so that the Fourier integral degenerated to the much simpler Fourier sum. Bolda
et al. [74], as the last and most recent example, used the modern method of Fast Fourier
Transform (FFT) to compute the propagation of a Gaussian pulse in an inverted medium.
The term `wave propagation' is deeply linked to a strong and colourful impression of a lm-like
sequence of individual pictures showing the motion of a wave packet. It should thus not come
as a surprise if some authors took up this idea and attempted to generate such lms in order
to bring the term `motion' to optical life. Indeed, there have been such approaches in the
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