Reflections and transmission of normally-incident plane waves at the boundaries of on-dimensional isotropic inhomogeneous regions with arbitrary permittivity profile
An approximate method is presented for determining the reflection and transmission coefficients at the boundaries of isotropic inhomogeneous regions with one-dimensional arbitrary permittivity profiles in the direction of propagation. The method is based on replacing the permittivity profile by a number of simpler inhomogeneous segments for which exact solutions are available, and using the wave transmission matrix approach to the resulting stack of inhomogeneous slabs. An application of this
... plication of this method, using smooth piecewise-linear profile replacements, is made to several dissipative and non-dissipative transition regions for which the wave equation has exact analytical solutions. Prom a comparison of the approximate and exact results, it appears that the accuracy obtained by using this piecewise-linear method is good provided that the width of each linear segment is maintained less than .125 λ[subscript]sλ[subscript]s being the shortest wavelength along the segment. This method is therefore particularly suited to the treatment of rapidly and moderately varying transition regions; a definite improvement over other available methods which are either more tedious to apply or do not yield accurate results. In the case of slowly varying transitions a simpler approach to the problem, based on the theory of TEM wave propagation in a non-uniformly-loaded parallel-plate transmission line, is considered. An approximate solution is obtainable when the square of the reflection coefficient magnitude is much smaller than unity. The range of validity of the approximation made is assessed by comparing exact and approximate results obtained for certain forms of non-dissipative transitions.