Quantum mechanical four-dimensional non-polarizing beamsplitter

Abstract Some quantum optics researchers might not realize that classical electromagnetism predicts a $$\mathbf {\pi }$$ π phase shift between S- and P-polarized reflection think the coefficients of transverse modes are independent, or such has no measurable consequences. In this paper, we discuss theoretical grounds to define elements 4x4 matrix represent beamsplitter, accurately accounting for polarization relations them. We also provide experimental evidence confirming representation. From scientific point view, paper addresses non-trivial equivalence fields Fresnel formalism canonical commutation quantized photonic fields. That can be readily verified with simple experiment provides further benefit. The beamsplitter expression derived applied in field computing.