Fundamental properties of beam-splitters in classical and quantum optics
A lossless beam-splitter 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 number-states |n1> and |n2> 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 beam-splitters via annihilation and creation operators a and a{\dag}. A simple application of the Feynman method provides a form of justification for the Bose enhancement implicit in the well-known formulas a|n>=sqrt(n)|n-1> and a{\dag}|n>=sqrt(n+1)|n+1>.