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A pair of ideal linear polarizing filters, oriented perpendicular to each other, will be opaque despite the fact that each filter individually only blocks 50% of the light. I would like to extend this to three filters. Obviously this is impossible if we are restricted to linear polarizers, but I cannot determine if it can be done when circular polarizers are included.

Is there a combination of three ideal polarizing filters such that all arrangements of the three completely block transmission of light, while no two of the filters can do so?

Notes:

  • These filters may be complex e.g. if one of the filters is a linear polarizer followed by a quarter-wave plate, that is fine.
  • Assume each filter retains the orientation and direction you specify i.e. if you specify 'a vertical polarizer followed by a quarter-wave plate' the linear polarizer will stay vertical and the light is guaranteed to pass through it first and the quarter-wave plate second.
  • If desired, you can assume a specific polarization of the incident light.
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  • $\begingroup$ How do you define a circular polarizer? $\endgroup$
    – Jagerber48
    Feb 2 at 21:14
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    $\begingroup$ I see no possibility of having 3 ideal linear polarizers, because no pair can be orthogonal and the last pair in the optical path would have to be orthogonal in order for the 3 polarizer train to block the light. I will fire up my optical calculus simulation software, used here, and see what happens with circular polarizers and such, but my gut feeling is your task has no solution. But I am upvoting because it piqued my interest! I will post an answer if I find one. $\endgroup$
    – Ed V
    Feb 3 at 1:00

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If I understand your question you could have three polarizes. Vertical, +45 degrees and -45 degrees. This will block all the light because the last two polarizers are perpendicular to each other. Now take away the -45 degree polarizer and light will pass through the remaining vertical/+45 combination. Edit: if all three polarizers are different then there is no way ALL arrangements will completely block the photons. Even if you have two polarizers that are perpendicular to each other, you will always have a third one that’s not. If it’s placed between the other two polarizers, no matter how they are arranged, then photons will go through. The only way it would work is if all three pull risers or perpendicular to each other which would mean two of them are the same. Then you could get what you were asking for. This would be like the first half of my answer.

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    $\begingroup$ Unfortunately, I seem to have been unclear. I am looking for a set of three filters which will be opaque in any order, but any two will not be opaque. Your set of three will not be opaque in the order (-45, V, +45) and the subset (-45, +45) will be opaque. I will review my question to see if I can make it clearer. $\endgroup$ Feb 2 at 20:28
  • $\begingroup$ @frodoskywalker yes please make it clearer $\endgroup$ Feb 2 at 20:41
  • $\begingroup$ Edited. Is that clearer? $\endgroup$ Feb 2 at 20:49

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