# Spacing between primary maxima of $N$-slit diffraction pattern and single-slit envelope

As far as I know, in the double-slit diffraction pattern, the spacing between primary maxima is determined by the equation of double-slit interference pattern, and the intensities of primary maxima are governed by the single-slit envelope. In the double-slit interference pattern, the fringes are equally spaced. So, according to my understanding, the primary maxima of double-slit diffraction pattern should be equally spaced. But, I don't know if this holds for N-slit diffraction pattern.

1. Is my understanding of the double-slit diffraction pattern correct? If not, please explain what's wrong.
2. Are the primary maxima in the N-slit diffraction pattern equally spaced?
3. As the number of slits (N) increases, the intensity of central maximum (and other primary maxima) increases. So, when we talk about the single-slit envelope, the single-slit envelope is the pattern we would see when we leave only 1 slit of the N slits open and diffract the light of the intensity of the central maximum of N-slit diffraction pattern through the open slit? My confusion arose from the image below. When the number of slits increases, the central maximum intensity also increases, so shouldn't the single-slit envelopes for double-slit, triple-slit, ... diffraction patterns also vary in their intensity?

• Take the Fourier transform of the slit structure to see... Mar 16 '20 at 16:03

Two slits is the convolution of a single slit with a pair of Dirac $$\delta$$-functions. So the diffraction pattern will be the product of the FT of the slit with the FT of the $$\delta$$-functions. Accordingly the pattern consists of a cosine (the FT of the pair of $$\delta$$-functions) multiplied by the same sinc function as before. The period of the cosine, and thus the spacing of the zeros, is inversely related to the separation of the slit - move the slits closer together, and the zeros of the diffraction pattern spread out more. They are always periodic though, as they arise from a cosine function.
An $$N$$-slit arrangement can be described as a convolution of the impulse function with an array of $$\delta$$-functions. The diffraction pattern will consist of the product of the sinc function with the FT of the $$\delta$$-functions. If the slits are equally spaced, it is easy to show that the period of this FT will be the same as for the single pair of slits. As $$N$$ gets larger and larger, the FT of the $$\delta$$-functions will approach a Dirac comb. You can see this tendency in the figure you provide; as $$N$$ increases the "wiggles" between the peaks is suppressed.