Nowhere near. An interferometer should be able to measure the entire axis though, from start to end, with ~100nm resolution. A dial gauge can only measure a few millimeters, with micrometer resolution.
Okay, I’ll play. How do you count the “pulses” so to speak? Using a photomultiplier tube or photodiode, and then the some counter circuitry or oscilloscope? That is not $100 in the end. If physics fascinates you, oh please do, I’m all for self development and fulfillment. But go into it being an full-fledged experimenter.
Oh I know this isn’t an “off the shelf” kind of solution.
How do you count the “pulses” so to speak?
Photodiode plus some kind of microcontroller. Each bright/dark transition should be 1/4 wavelength, so ~120nm for green light. That means around 8 light/dark transitions per µm, or 8000 per mm. An Arduino should be able to handle 300kHz from a quick Google, so 37.5mm/s, or 2250mm/min.
I have a bunch of these small microcontrollers laying around so maybe an extra $1 for a photodiode and a few passive components to make it digital.