For beam splitters with two incoming beams, using a classical, lossless beam splitter with Ea and Eb each incident at one of the inputs, the two output fields Ec and Ed are linearly related to the inputs through where the 2×2 element is the beam-splitter transfer matrix and r and t are the and along a particular path through the beam splitter, that path being indicated by the subsc. A beam splitter divides incident light into reflected and transmitted beams at a specified R/T ratio. See the Comprehensive Guide for worked examples, SVG diagrams, and full references. It is a crucial part of many optical experimental and measurement systems, such as interferometers, also finding widespread application in fibre optic telecommunications. Beamsplitters are often classified according to their construction: cube or plate. Each mode of the electromagnetic (radiation) field with frequency ω is described math-ematically by a 1D harmonic oscillator with frequency ω. Suppose $a$ goes through a beam-splitter characterized by a parameter $theta$ coupling it to mode $b$, so that first this first interaction we may write the unitary $$U_theta = exp (itheta (a^dagger b + b^dagger a)) $$ (I'm forgetting about relative phases, global signs and what-not; this.
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