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I'm thinking about how diffraction occurs, and I can't imagine how an aperture would cause light to diffract.

It seems to me that light is diffracting at every single point regardless of the existence of an aperture, so an aperture merely lets one "see" the diffraction by getting filtering out all the other light in the same beam. Is this correct?

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I would say this interpretation is correct depending on what is meant by "diffract". See Huygen's Principle.

That is, every point where light is acts as a source a spherical wave of light. When there is no aperture then the spheres of light coming from a plane of points (plane wave) of light interfere to give another plane of light. The plane wave propagates. When an aperture is in place the interference isn't complete so you get the familiar diffraction patterns.

One objection to this point or characterization might be that one might define diffraction to be what light does when it goes through an aperture. In that case it would not be reasonable to say light is always diffracting. In fact looking at the Wikipedia definition it looks like this is the case.

However, I think your physical intuition is still correct and valuable. The point is that the aperture doesn't change the way light behaves. That is, light follows the wave equation everywhere. The point is just that the aperture changes the nature of the space in which the light is propagating.

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