He describes the light source as monochromatic. This is what makes it coherent. To get quasi-monochromatic light from a non-laser source, you have to filter out most frequencies, and only keep a small portion of the frequency spectrum. That is typically still wider than for light from a laser, but the coherence length can be made longer than the distance-differences in your optical instruments. And that is enough for treating the light as quasi-monochromatic.
If a laser would work better, than I am all for a better experiment. That said, it would now be interesting to me to compared the difference between using a laser vs. filtered light source.
I don't think it matters to your point, but just to be clear, I am not detecting an interference pattern in my experiment, but measuring the number of photons or accumulated intensity.Let me first descibe it in Feynman's photon picture. Assume that each photon would have exactly one frequency, and nicely form a coherent superposition with itself. If different photons have slightly different frequencies, then their interference patterns are slightly different. And the sum of those interference patterns no longer shows sharp interference peaks, but appears washed out. But to see an interference pattern at all, you must look at the image produced by many photons, so all you can see is the washed out pattern.
If your light has the frequency f, and you switch it on for t seconds, then you end up with an frequency distribution in the range f +- 1/t. If f is big compared to 1/t, then it makes sense to talk of quasi-monochromatic light. And if c*t is big compared to the distance-differences in your optical instruments, then treating the light as monochromatic should work well.
So if I make a more practical version of my experiment and use a 200 m mirror and assume red light (700 nm, 430e15 cycles / second) and:
put the source in the middle of the mirror, 100 m from the mirror
put the grating at the far left end of the mirror
put the detector at the far right end of the mirror, 100 m from the mirror
Then as a rough calculation:
The time for light to reach the grating is about 470 ns.
The time for light to reflect off the grating and go past the source is about 840 ns.
The time for light to reflect off the grating and reach the detector is about 1200 ns.
The time for light to take the shortest path, reflect off the mirror and reach the detector is 750 ns.
In this scenario, I would turn on the light source and look at the data from detector for up to 840 ns or 1200 ns.
So f = 430 x 10^15 cycles / second is much greater than (1 / t) = 1.2 x 10 ^ 6 and by your classification would be considered quasi-monochromatic light.
And c*t would be 252 m which seems fairly large compared to what I imagine the size of the optical instruments are and so by your classification would also be considered quasi-monochromatic light.
So given your guidance, I believe it would be justified to treat the light in this more practical scenario as monochromatic.