Beam splitters- Direction of use - again Can anyone help me understand the effective performance characteristics when sending a beam in to non-entrance faces of a cube beam slitter? (specifically visible non-polarizing ones if it matters)
Answers to a previous question on the topic seemed to conclude the beam could enter any direction however I'm not convinced the exit beam would have identical characteristics regardless of the entrance face.
The sort of products I have been looking at are all constructed with a BS coating on one face and a cement to stick the two halves together. If it did not matter the direction why do they mark Entrance Faces on the cubes?

 A: This is a good question: those marks are mystifying when you meet a beamsplitter for the first time, but I understand from manufacturers that the following can all be reasons for the marks (point 1 is the most common):


*

*When using very high power beams, you want most of the split to happen as soon as the beam meets the first interface, so that half the beam's power is reflected away. The cemented layer is the device's most vulnerable part to absorptive heat damage. If the beam meets the interface before travelling into the cement layer, the latter feels only half of the input beam together with the evanescent field from the reflecting beam. The evanescent field of course cannot heat the cement;

*A beam splitter's full scattering matrix is an $8\times 8$ unitary (for lossless), self-transpose (for reciprocity) matrix: there are four ports with two polarization eigenstates for each. Even if the device doesn't mix polarizations, you've got a $4\times 4$ scattering matrix, which means that, in general, it takes a great many complex numbers to define the device fully (see footnote). Even though most of these numbers will be near to nought, the manufacturers assume that you will use a beam splitter in a certain orientation so that they only need to control / measure a few of these parameters. For some manufacturers and precision applications, the device will therefore only fully meet its specifications when used in the prescribed orientation, although of course it will still behave as a beamsplitter if used in other orientations, assuming that the power is low enough that point 1 isn't an issue;

*Very like point 2: the beamsplitter's four optically active faces have in general three independent orientations. Often two of these faces are controlled more accurately than the third; therefore, if so, the device will only meet the manufacturer's beam parallelism / orthogonality specifications if used in the prescribed orientation.


In summary, the marks really only matter for very high power applications, or, in some cases, precision applications where you are relying on extinction ratio or parallelism / orthogonality specifications. The latter two are often not an issue since most optical designs allow for alignment to correct for them.

A unitary matrix is of the form $U = \exp(H)$, where $H$ is skew-Hermitian ($H = -H^\dagger$). If, further, the matrix is self transpose ($U=U^T \Leftrightarrow H = H^T$), as it is for the scattering matrix of a reciprocal network, this means that $H$ is skew-symmetric and purely imaginary. Thus, a $4\times 4$ scattering matrix of the non-polarization-mixing beamsplitter is defined by ten independent real parameters. If we assume that we are free to define the reference phases of the four ports, this means we are free to adjust $U$ by a transformation of the form $U\mapsto \Lambda\,U\,\Lambda^{-1}$, where $\Lambda$ is a diagonal matrix of phase factors $e^{i\,\varphi}$. So, with this proviso, this means that only six of the ten parameters are independent. That's still quite a few parameters for a production process to control!
For the $8\times 8$ case, the number of independent parameters (with the phasing proviso) is $8+7+6+\cdots+1 - 8 = 28$ independent parameters!
