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I was on a plane recently and took a few pictures from the window. I had the polarizer on initially, and observed this circular rainbow effect:

enter image description here.

What is the reason for this?

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    It might not have been what you were after, but it makes for an excellent picture! – FreeMan Dec 17 '18 at 14:04
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A sheet of glass or plastics will typically have internal stresses. For glass and some clear plastics these lead to the birefringence patterns you see, when you shine (partially) polarized light on them and then view them through a polarizing filter. You can try this out by either using two polarizing filters, one in front and one behind, or holding such an object in front of an LCD monitor and putting a filter in front. If your filter came in a clear case, that could be a good test object. You can find out more about the topic by searching for "polarizer" and "stress", "stress analysis", or "birefringence". In fact, the Wikipedia article on birefringence has a lot of information and some example pictures.

In your case the sky is providing a partially polarized light source, the airplane window produces the birefringence patterns, and the polarizing filter makes them visible to the camera.

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    This is one reason commercial pilots are not permitted to wear polarized sunglasses. When you have them on you get these stress patterns all throughout the cockpit windows. (Also you can't read half your LCD instrument.) – user80702 Dec 17 '18 at 0:09
  • @user80702 Almost sounds like a ban would be unnecessary, as nobody would want to wear those.. Then again, if they look cool enough... – Kjeld Schmidt Dec 17 '18 at 12:45
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    The problem is not every instrumental panel or set of windows would reveal these sorts of artefacts, especially early days when such glasses were new, and instruments were mechanical or windows made of different material. So a pilot could have been surprised because they had never thought (or forgot) their shades were special in this manner. Minimizing surprises in the cockpit is sort of a thing. Guidelines for what sort of personal equipment is used in the cockpit is pretty common in aviation. – user31502 Dec 17 '18 at 15:31
  • @user80702 Speaking of LCD instruments, I have always wondered why nobody thinks to design LCDs so that they can be read with polarized sunglasses (it's just a 90 degree turn at most!) - I was quite disappointed the first time I pulled out my laptop in full sunlight and found I couldn't read the screen. – Michael Dec 17 '18 at 15:50
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    @Michael decent displays are designed like this, but normally with a 45 degree turn so they work portrait and landscape. tangentially related, but I've seen similar effects through car windows a few times – Sam Mason Dec 18 '18 at 11:42

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