Fundamental properties of beamsplitters in classical and quantum optics

A lossless beamsplitter has certain (complex-valued) probability amplitudes for sending an incoming photon into one of two possible directions. We use elementary laws of classical and quantum optics to obtain general relations among the magnitudes and phases of these probability amplitudes. Proceeding to examine a pair of (nearly) single-mode wavepackets in the numberstates |n(1)> and |n(2)> that simultaneously arrive at the splitter's input ports, we find the distribution of photon-number states at the output ports using an argument inspired by Feynman's scattering analysis of indistinguishable Bose particles. The result thus obtained coincides with that of the standard quantum-optical treatment of beamsplitters via annihilation and creation operators (a) over cap and (a) over cap dagger. A simple application of the Feynman method provides a form of justification for the Bose enhancement implicit in the well-known formulas (a) over cap |n > = root n|n-1 > and (a) over cap dagger|n > = root n +1 |n + 1. (c) 2023 Published under an exclusive license by American Association of Physics Teachers.