Matrix representations for classical and quantum beam splitters

Matrices provide a practical and elegant tool for describing the transformation properties of beam splitters and waveguide couplers acting on single-mode optical fields. Using a systematic approach, we show how the application of various physical constraints determines the form of the matrix for both classical fields and quantum number states. The goal is to provide a clear explanation of the conditions under which various matrix forms are appropriate to represent four-port couplers and beam splitters. Examples of calculations using the matrices are provided.