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

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