A concrete example of a paper using the add-i-to-reflected-part type of beam splitter is the "Quantum Cheshire Cats" paper:

A simple way to prepare such a state is to send a horizontally polarized photon towards a 50:50 beam splitter, as depicted in Fig. 1. The state after the beam splitter is |Psi>, with |L> now denoting the left arm and |R> the right arm; the reflected beam acquires a relative phase factor i.

The figure from the paper:

I also translated the optical system into a similar quantum logic circuit:

Note that I also included the left-path detector they talk about later in the paper, and some read-outs that show (among other things) that the conditional probability of the left-path detector having gone off, given that D1 went off, is indeed 100%. (The circuit editor I fiddle with is here.)

It's notable that my recreation uses gates with different global phase factors (the beam splitter is 1/2-i/2 and 1/2+i/2 instead of 1/sqrt(2) and i/sqrt(2)). It also ignores the mirrors that appear once on both paths. The effect is the same because global phase factors don't matter.

edit My ability to make sign errors upon sign errors is legendary and hopefully fixed.