Representing a beam splitter using matrices
In this paper we discuss theoretical grounds to define elements of a 4x4 matrix to more accurately represent the beamsplitter, fully accounting for transverse polarization modes. Question: Is it possible to express the effect of a simple 50% beamsplitter on photon number states using matrices, such that the output can be computed by matrix calculations rather than manual substitution of equations? To explain the problem, consider a 50% beamsplitter and define: $a_ {1,2}^. Using a systematic approach, we show how the application of various physical constraints determines the form of the matrix for. If we neglect the three-dimensional character of the electromagnetic fields and focus on one-dimensional propagation only, we can regard a beam splitter simply as a dielectric plate, possibly consisting of several y consisting of several layers ropagation along.
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