system
Given the following post, respond as if you're commenting on Reddit:
human
Why does a lens perform a Fourier transform on light?

I've been through the derivation and my understanding is still very much in the 'because that's what the math tells us' territory. 

Is there a more intuitive explanation?
gpt
Maybe this gives you some hand-waving intuation:

Spatial Fourier transform takes position space into wavevector space. A lens focuses parallel bundles of beams into a single point. So everything that was described by the same wavevector beforehand shares the same spatial coordinate in the focal plain of the lense.

So the spatial coordinate after the lens contains the information that was encoded in the wavevector before the lens which in mathematical terms corresponds to a Fourier transformation.