TABLE OF CONTENTS In the next Chapter we consider the problem of a source distribution exciting a dielectric slab waveguide and we solve it directly as we deal with a simple twodimensional field. In a three-dimensional case, where TE and TM waves may coexist, it is advantageous to investigate first the method of solution. To this end, it is expedient to formulate the problem in a somewhat abstract manner. Let us consider a source f, a forcing field, which creates a certain unknown field u. The latter must satisfy some linear relationships, i.e. Maxwell's equations and appropriate boundary conditions, which we can briefly summarise as a linear operator L. Therefore, our problem is to find a solution to the following equation
| (2.343) |  |
A numerical solution of the above problem may be obtained by applying the Rayleigh Ritz approach, or the method of moments, as described previously in Section 2.4. However, whenever possible, it is often advantageous to find an analytical solution of our problem. To this end we may follow two different approaches [12, p.134]: we could make use of the spectral representation of our operator, or we could use directly the Green's function pertaining to our problem. Let us consider first the solution by means of a spectral representation.
In this case we imagine we know the eigenvalues ( ?) and eigenfunctions ( ?) of our operator L, satisfying the homogeneous equation
| (2.344) |  |
As always, due to...
