There are two cases I'm asking about. The square in the middle is a cube beam splitter in the same orientation for both cases. We are looking at the beam splitter from the top. In the first case, the beam comes in from the left and half gets transmitted and half gets reflected downwards.

Polarizingbeam splitterprinciple

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How does a beam splitter workphysics

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there is actually a difference between the two cases other than the diection of the reflected beam. Cube beamslpitters are constructed using two parts glued together using a special cement. The mirror coating is done on one of the two parts (look at the picture from Thorlabs). If this is the case, then you should use such that the beam hits the mirror coating before hitting the cement as marked on the beamsplitter.

How does a beam splitter workdiagram

So, now changing only the beam direction in case 2, with the beam splitter in the same orientation, will the beam split like I've drawn? Does the beam splitter work if the laser comes from opposite directions?

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A beam splitter works like a mirror that transmits part of the light. So there is always part of light that goes directly through without changing the direction. The rest gets reflected from the diagonal, which represents the reflecting surface. Therefore, beam coming from the left will be reflected downwards (as is shown in your case 1) but the beam coming from the right will be reflected upwards (case 2). In case the beam comes from top (or bottom), part is always going through, i.e. down (up) and the other part gets reflected like on a mirror -- to the right (left).