Theory for the beam splitter in quantum optics: quantum entanglement of photons and their statistics, HOM effect

The theory of the beam splitter (BS) in quantum optics is well developed and based on fairly simple mathematical and physical foundations. This theory has been developed for any type of BS and is based on the constancy of the reflection coefficients $R$ (or the transmission coefficient, where $R+T=1$) and the phase shift $\phi$. It has recently been shown that the constancy of these coefficients cannot always be satisfied for a waveguide BS, where $R$ and $\phi$ depend in a special way on photon frequencies. Based on this, this review systematizes the concept of BS in quantum optics into ``Conventional'' and frequency-dependent BS, and also presents the theory of such BS. It is shown that the quantum entanglement, photon statistics at the output ports, and the Hong-Ou-Mandel (HOM) effect for such BS can be very different. Taking into account the fact that the waveguide BS is currently acquiring an important role in quantum technologies due to the possibility of its miniaturization, this review will be useful not only for theoreticians, but also for experimenters.