# Beam splitter that splits two incoming beams differently

ב"ה Is there any way to build a beam splitter that will split two incoming beams differently? Say, beam coming from the left (x axes) will transfer 70% and reflect 30% while beam coming from bottom (y axes) will transfer 40% and reflect 60%? I would like to create a interferometer like this: BS1 splits 50/50 (R/T) but BS2 will split path A 70/30 and path B 40/60. Now assuming their is a phase shift between path A and B we will get the following results when we send single photons:

1. if only path A is open then detector Dx will count 70% of the photons and Dy 30%.
2. if path B is open then Dx will count 60% of the photons and Dy 40%.
3. if both paths are open then we will get interference between the two paths but in proportion to the different R/T ratios.

is their any way to construct a interferometer like that?

• Can you arrange to have different polarizations for the two beams? Dec 5, 2022 at 21:08
• Thanks. I explained a little better what I want to achieve Dec 6, 2022 at 19:28

It is reasonably easy to show, see Montgomery Section 9.10 that in a 4-port directional coupler (port numbers $$k=1,2,3,4$$) if ports are to be matched that is all $$S_{kk}=0$$ and say ports $$1,3$$ are isolated, that is $$S_{13}=0$$ then if the junction is also lossless $$\mathbf {S} \cdot \tilde {\mathbf{S}^*} = \mathbf I$$ and reciprocal $$\mathbf {S} = \tilde {\mathbf{S}}$$ then you must also have
$$S_{12}=S_{34}$$, $$S_{14}=S_{23}$$ and $$|S_{12}|^2+|S_{24}|^2=1$$ Therefore the type of coupler you postulate cannot be lossless, or matched or properly isolated.
• Given the scattering matrix $\mathbf S$ of a system, then it is called reciprocal if $\mathbf S = \tilde {\mathbf S}$ (tilde denotes transpose), passive if $\mathbf I-\mathbf S \tilde {\mathbf S^*}$ positive definite; reactive (passive lossless) if $\mathbf I=\mathbf S \tilde {\mathbf S^*}$ . All amplifiers are non-reciprocal but they are also active (non-passive), the most common non-reciprocal elements are based on magnetic effects such ferrite, these are passive and to some extent lossy, too. My practical knowledge is below <100GHz, not in photonics; for real advice ask somebody in that. Dec 11, 2022 at 19:54