Trying to understand the Hong-Ou-Mandel effect (Wikipedia link) I got a bit lost with regards to what the reflection phases mean for the experiment and the kind of beam splitter required.
Instead of single photons, let's say that there are two coherent beams (A and B) and a beam splitter plate with a dielectric coating on the bottom (as suggested from Wikipedia link). Beam A is reaching the beam splitter from above, beam B from below. If beam A is reflected or refracted, there is no phase shift. If beam B gets reflected it gets a phase of $\pi$, and a phase of $0$ if transmitted.
- Beam A gets reflected and beam B is transmitted. In that case none of the beams gets a relative phase and both beams constructively interfere.
- Beam A and B get transmitted. Nothing interesting happens, the beams just exchanged sides.
- Beams A and B are reflected off the beam splitter. Nothing interesting happens, each went back their way. With the caveat that beam B obtains a relative phase of $\pi$.
- Beam A is transmitted and beam B gets reflected. This case is confusing me, as beam A goes through the beam splitter, no phase added, and beam B gets reflected with a phase of $\pi$. So in principle the beams cancel as they interfere destructively.
In the Hong-Ou-Mandel experiment, only (1) and (4) contribute, while (2) and (3) cancel out. But that's not what I am getting by looking at the phases. Note that, if (2) and (3) indeed cancel out, the beams should still interfere destructively in (4), so (1) would be the only possible result.
Another possibility is that the beam splitter in this experiment is some kind of symmetrical cube beam splitter, where the reflective surface is inside a crystal. In that case no relative phase is gained by the outgoing beams, but then it is even more difficult to see why relative phases from reflection matter here.
What am I missing in this setup? What can be understood from the nature of the beam splitter and the phases of reflection? Can this effect only be understood using second quantization and the reflection phases do not matter here?