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When does passing through a polarizer change phase besides just changing polarization?

Consider three cases:

  1. A polarizing beam splitter (PBS) is a clear case. Horizontally polarized light leaves from one output and vertically polarized light leaves from the other output. As one beam is transmitted and the other beam reflected, they differ in phase because reflection introduced a phase shift.

  2. A linear polarizer oriented horizontally: the horizontally polarized component is transmitted, while the vertical component is absorbed.

  3. The same linear polarizer oriented vertically: the vertically polarized component is now transmitted, while the horizontal component is absorbed.

Would light transmitted in cases #2 (H) and #3 (V) be in phase or out of phase? And how can we measure it experimentally?

A related question: Can we say that light transmitted and light absorbed by a linear polarizer have the same phase? Or is this question meaningless because we don't have information about the absorbed part?

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The question is:

A linear polarizer oriented horizontally: the horizontally polarized component is transmitted, while the vertical component is absorbed. The same linear polarizer oriented vertically: the vertically polarized component is now transmitted, while the horizontal component is absorbed. Would light transmitted in cases #2 (H) and #3 (V) be in phase or out of phase? And how can we measure it experimentally?

Assuming the optical thickness of both linear polarizers is the same (i.e., the substrate is not birefringent), then there is no phase shift of one transmitted component relative to the other. This makes sense, because the polarizer in #3 is just the polarizer in #2, rotated by 90 degrees.

Comparing the phase of two beams when one is polarized vertically and one is polarized horizontally is not very difficult. Build a Mach-Zehnder interferometer using nonpolarizing beamsplitters. At the input, provide a beam that is polarized at 45 degrees. In one arm, place the polarizer you want to test, oriented vertically. At the output of the interferometer, place another vertically oriented polarizer. Adjust the interferometer until a minimum in the interference pattern is at a selected spot on a screen. Now rotate both polarizers 90 degrees and see if the interference pattern minimum moves. If it does not move, there is no phase shift between the two orientations. It's a good idea to test it with flat glass plates instead of polarizers first, to make sure the polarizers rotate only on an axis perpendicular to their surface. If they rotate on a skewed axis, their effective optical thickness will change.

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What the polarizer does when it blocks light is to add to the incident a field that is out of phase, so that the sum is darkness. That is what any opaque screen does. It explains why Babinet's principle works.

A transparent material will shift the phase. A quarter-wave plate will shift light polarized along the fast axis π/2 less than the other polarization direction.

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