Beam splitter phase shifts: Wave optics approach

We investigate the phase relationships between transmitted and reflected waves in a lossless beam splitter having a multilayer structure, using the matrix approach as outlined in classical optics books. Contrarily to the case of the quantum optics formalism generally employed to describe beam splitters, these matrices are not unitary. In this note we point out the existence of general relations among the elements of the transfer matrix that describes the multilayer beam splitter. Such relations, which are independent of the detailed structure of the beam splitter, fix the phase shifts between reflected and transmitted waves. It is instructive to see how the results obtained by Zeilinger by using spinor algebra and Pauli matrices can be easily derived from our general relations.