Do Multiple Diffraction Gratings Affect Interference Patterns?

[KDPhysics]

Here is my thinking:

Let's replace the second diffraction grating with a screen. Then, after passing through the first grating, I would expect there to be an interference pattern on the screen. Depending on the distances between the first grating and second, I could have several difference fringes. For sake of simplicity, say that in the diffraction grating I get 3 interference patterns (##m = 0, 1). Then, I would have three beams pointing at the diffraction grating.

Now we go back to the original set up with two gratings. Since the same interference pattern should show up on the second diffraction grating, it would be as if there were 3 laser beams pointing at the grating, and thus you'd have the multi-slit interference pattern for three different beams?
Does this make any sense?

So does adding an extra grating do anything? What about its angle of inclination?
Would varying the distance between the two gratings have any effect on the fringe distance?

 

[tech99]

It looks as if the second grating has a row of slits whose amplitude is multiplied by the pattern of the illuminating source - in this case a course pattern (remember that adjacent maxima have opposite phase by the way). As we change the separation of the two gratings, the second grating will have differing illumination patterns.
The effect on the screen is the product of three patterns:-
1) The wide pattern arising from the close spacing between individual slits.
2) The narrow pattern caused by (half) the overall width of the grating.
3) An intermediate pattern caused by the illumination provided by the first grating.

 

[KDPhysics]

I was thinking that maybe Babinet's principle could be somehow helpful?