Wave propagation in waveguides and periodic structures.
Discrete conservation laws
Gregory Kriegsmann and Darko Volkov
Department of Mathematical Sciences
New Jersey Institute of Technology, Newark, NJ 07102
The objective of our work is to develop numerical schemes
for the propagation of time harmonic waves in a two dimensional
waveguide or period structures assuming that there is a smooth object
inside the propagation medium. We want to calculate reflection and transmission
coefficients relative to that object when it is illuminated by a
propagating wave.
We use the theory of integral equations to obtain highly accurate estimates
for these coefficients. The estimates satisfy conservation laws. We show
that these conservation laws are satisfied at the discrete level. Therefore
they do not indicate how accurate the numerical solution is. We are able
to check for accuracy by different means which we describe.
In a first case we consider a hard object in a hard waveguide, that
is we impose Neumann boundary conditions on the walls of the
waveguide and on the contour of the object.
We choose the wave number in a range that allows only one mode to propagate.
We solve an integral equation
of the second kind to obtain a density function defined on the contour
of the object using Nystrom's method
and the collocation method for weakly singular integral equations.
The density solving the integral equation
is in turn used for the calculation of
transmission and reflection coefficients. These coefficients satisfy discrete
conservation laws.
We provide a rigorous mathematical explanation for that fact.
If the object is symmetric, more conservation laws are satisfied at the
discrete level and we also provide a rigorous mathematical account of
that situation.
We also discuss how we were able to test our codes on problems whose
solution is known.
We then study the propagation of waves in planar periodic structures.
In the modeling of this physical situation we assume that there is a
hard object in a waveguide and that periodic boundary conditions are
satisfied on the walls of the waveguide. We introduce in this situation
another parameter, the incidence angle of the incoming wave.
Our numerical schemes preserve conservation laws in this case too.
Finally we consider the case of a dielectric structure. We suppose that
we have a penetrable object in a soft waveguide where the wave number is
chosen so as to have only one propagating mode. We seek for
a solution to the dielectric equations in the form of a combination
of a single and a double layer
potential. Conservation laws are satisfied
at the limit within 7 decimals of accuracy. In the case of an object
with a complex wave number, we show that our solutions satisfy the relevant
energy conservation law as the number of modes increases.
In the dielectric case too, we
discuss how we were able to test our codes on problems whose
solution is known.