Like in title, how small can diffraction grating be? And, of course be "fully operational". I mean, littles diffraction grating possible that still works as full-sized one.
You don't specify what properties you mean by "works as a full sized one", but I will assume that you want the strong wavelength discriminating power and to be able to run the usual demonstrations.
There are two issues. The first is that the grating can only diffract as much light as falls on it, so small grating will produce dim diffraction patterns. This is not a problem as along as the the source is intense and/or you have a sensitive detector. The second is the geometric limits of the device, and that is what I want to concentrate on.
Diffraction gratings get there power from have "many" uniformly spaced "slits". If we take many to be more than ten, then the width in the analyzing direction of the thing needs to be more than ten times the spacing. That is one limit.
Grating also diffract only in one direction, which means that the size in the non-analyzing direction should be many times the line spacing. We can take "many" to means more than ten again, and you find that you get about the same limit in the non-analyzing direction.
In both cases taking "many" to mean a bigger multiple of the line spacing will get better quality results, so the question eventually comes down to "Just how good do you want it to be?".