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Obviously I'm wrong but I'm not sure why. Light bouncing off of two mirrors can interfere constructively or destructively, depending on the phase difference. The phase difference from light bouncing off of two aligned mirrors is random (as it's non coherent light). Adding a constant in the phase difference due to misalignment would still lead to a random difference. So it seems to me the phase shift wouldn't make a difference.

What am I missing?

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The phase difference between light bouncing off two aligned mirrors is NOT random. This is what you are missing.

Non-coherent light only means it is made of many wavelengths at the same time (Think of it as the sum of many « coherent lights »). Interference does happen even for incoherent light, within each wavelength.

Example: oil iridescence on water is interference which works on « incoherent » light.

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