I have actually seen this, since I worked in a lab with a laser tunable to well into the UV. For tests we would run it through the whole spectrum. In addition, this method is interesting because the output brightness stays roughly the same (the number of photons is mostly conserved).
What happens is that you see the light source shift gradually along the rainbow: green then blue then violet. And then the violet smoothly but quickly fades and then you see nothing.
As another poster mentioned, in theory you should perceive different brightnesses during the sweep, but in practice you’re looking at light “leaking” from the photon splitting crystal in a small chamber with uniform walls, so you don’t really notice it.
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