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I'm wondering if there's a difference in the interference pattern achieved in the michelson interferometer if the source of light is sending plane waves vs sending diverging waves (for example by putting a diverging lens between the source and the first plate, the one which divides the light).

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Plane waves are a theoretical construct as every aperture produces divergence, so you need an infinite amount of empty space to keep a plane wave plane.

As such every Michelson interferometer deals with diverging waves. These interferometers will take a divergent beam and collimate it before reciprocating the process so that sharp interference can still be seen.

A near exception is a fibre michelson interferometer where the waveguiding properties of the fibre act as collimation, though lensing is still needed to couple to the fibre.

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