An S-matrix based hybrid method
Gregory A. Kriegsmann
Department of Mathematical Sciences
New Jersey Institute of Technology, Newark, NJ 07102
The use of S-Matrix theory by engineers and scientists to understand and
design microwave systems has roots that are over sixty years old. The
basic idea behind the method is to decompose the scattering domain into
sub-domains and to characterize the scattering physics in each by an
S-Matrix. These are then "connected" using simple matrix algebra to
achieve an accurate approximation of the complete scattering problem.
Of course the method is viable only if an accurate approximation can
made to the S-Matrices in each sub-domain. These approximations
were computed analytically for simple structures and were obtained
experimentally for more realistic and complicated scatterers.
This theory has more recently been used, in conjunction with numerical
algorithms, to produce accurate and efficient hybrid schemes for several
classes of scattering problems that are not amenable to either finite
difference, finite element, or integral equation methods. The novel
feature here is to replace the experimental determination of the S-Matrix
for a sub-domain by an accurate numerical approximation to it.
Two examples will be presented and discussed which illustrate the
methodology. The first arises in microwave heating experiments in single
mode, highly resonant cavities. The highly resonant character of the
problem makes finite difference time domain methods inefficient, due to
the time required to radiate away the "transients" present in the simulation.
Similarly, finite element and integral equation methods give rise to
ill-conditioned algebraic problems. The hybrid method suffers neither
shortcoming. Numerical experiments will be given.
The second example arises in simple photonic structures. If the scatterers
that make up this structure are sufficiently far apart in the direction
of propagation, then S-Matrix theory can be used to construct a very
efficient and accurate numerical method to study this problem. An example
is provided to make this point.
Finally, we note that the accurate description of the scattering physics
in each sub-domain requires a numerical scheme that is not only accurate,
but faithfully reproduces the properties of the scattering matrix. For
example, if the target in the sub-domain is lossless, then the S-Matrix is
unitary. The numerical method must yield this property. If it fails, then
inaccuracies occur. An example is provided to illustrate this feature.