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I have a non-polarizing BS and a polarized source (linear horizontal). What polarization states should I observe on the output?

My intuition is that, since a non-polarizing BS is just a half-mirror, the reflection should flip horizontal to vertical, while transmission retains the incoming polarization state (if I ignore noise).

I could test it in the lab, but it is not available right now, so I need to infer it from Physics of the process.

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I just did it quick and dirty: a green laser pointer (532 nm), non-polarizing beam splitter cube, and two calcite polarizers. If the incident light is linearly x polarized, both the transmitted and reflected beams are also linearly x polarized. If the incident light is linearly y polarized, both the transmitted and reflected beams are also linearly y polarized.

With x linearly polarized light in, the reflected beam is also x polarized: the polarizers are both x oriented. The transmitted beam is also x polarized (not shown).

x in and passed by x analyzer

With x polarized in and the polarizer y oriented after the reflection at the beamsplitter cube, little light passes.

x in, but blocked by y analyzer

With y polarized incident light, little light passes through the x oriented polarizer after the reflection at the beamsplitter cube.

y in, but blocked by x analyzer

Finally, with y polarized light in and the y oriented polarizer after the reflection at the beamsplitter cube, light passes through.

y in and passed by y analyzer

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