The Aperture Field Distribution

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Outline of the Method Used to Find the Aperture Field Distribution

Decompose the incident field into TE & TM components of the zeroth order Floquet modes (denoted by coefficients A001 & A002 respectively)
Expand the transmitted & reflected fields as an infinite series of Floquet modes with unknown coefficients
Match the boundary conditions at the filter, assuming an infinitely thin, perfectly conducting filter
Substitute the integral representation of the reflection and transmission coefficients into the resulting equation to form an integral equation
Provide faster convergence by expanding the tangential field at the filter into orthonormal functions that span the aperture space (the set of wave guide modes)
Use the orthonormality of the wave guide functions to convert the integral equation into a matrix equation (Galerkin's moment method)
Decide where to truncate the infinite matrix equation
Solve numerically by matrix inversion to determine unknown transmission and reflection coefficients
Use the coefficients to find the aperture field distribution