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I have a basic question about the effect of transmitting a laser beam through multiple diffraction gratings. Suppose a diffraction grating was used to produce many spots as follows:

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Would adding a second grating of the same pattern result in more spots/maximas? If so, would the resulting number be $n^2$ where $n$ is the original number of spots/maximas? If not, what would the result be?

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You can get your own answer by sending the beam through the grating, then reflecting the multiple beams back through the grating from a mirror. The answer is that each new beam is independently diffracted by the second grating.

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  • $\begingroup$ If my understanding is correct, Bragg's law suggests that the geometry of the pattern only depends on the distances between the bars of the grating. Of course, in my specific case, there aren't "bars" in the gratings but rather loops (FFT). By Bragg's law, won't stacking two identical gratings together be the same as using just one? Thus, the second grating will effectively be redundant. $\endgroup$ – John M. May 6 at 4:13
  • $\begingroup$ Well, it depends what you mean by "stacking". If the gratings are added exactly in registration, you end up with one grating with twice the amplitude. But if they are multiplied, which is more often the case, then you get what I described. The grating you showed is apparently a "crossed grating", made by exposing two gratings at rught angles to each other. The medium is saturated, so you get intermodulation which amounts to multiplication of the two gratings. So, each dot turns first into a row of dots, then each row turns into a column of rows. $\endgroup$ – S. McGrew May 6 at 9:47
  • $\begingroup$ Right. Sorry, what does it mean to add/multiply gratings? I thought addition is when two gratings are in the same orientation whilst multiplication is when two gratings are perpendicular to each other (on the same plane). When you say crossed grating is made by exposing two gratings at right angles to each other, do you mean they're on the same plane but perpendicular to each other or the planes themselves are at right angle? I'd have thought the gratings are on the same plane. Like how Christmas light shower lights do it. $\endgroup$ – John M. May 6 at 10:24
  • $\begingroup$ If multiplication is indeed when two gratings are on the same plane but oriented at right angles to each other, does that mean something like that Christmas light could have a third grating at right angle and produce twice as many more spots? $\endgroup$ – John M. May 6 at 10:34
  • $\begingroup$ Multiplication is a bit more complicated than that. If the two gratings are physically stacked, they act independently, which means they are effectively multiplied. If you stack three (A, B, and C), and A produces 3 spots, B produces 7 spots and C produces 8 spots, then you will obtain at least 3 x 7 x 8 spots. You may get more, because of saturation and overmodulation and reflections between the gratings. The spots will be easier to see as distinct spots if the gratings are crossed. $\endgroup$ – S. McGrew May 6 at 13:21

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