For an array of

slits of width

and equal spacing

, the intensity of the diffracted light on a screen may be neatly expressed as

,

where

is the peak intensity and

is the Chebyshev polynomial of the second kind, which appears when we simplify the ratio

.

The parameter

is a normalized wavenumber. It is related to the actual wavenumber

, the perpendicular distance

from the diffraction grating to the screen on which the pattern is observed, and the distance from the center of the pattern

, by

.

The formula for the intensity is valid within the Fraunhofer diffraction regime, for which

. In this case, the diffraction pattern is equivalent to the Fourier transform of the diffraction grating. This explains why, since an array of finite-width slits is equivalent to the

*convolution* of an array of delta function slits with a single slit, the resulting diffraction pattern is the

*product* of the two corresponding diffraction patterns.

Snapshot 1: for a single, infinitely narrow slit, the diffraction pattern is constant; this is essentially because the Fourier transform of the delta function is constant

Snapshot 2: for multiple infinitely narrow slits, there is an infinitely repeating pattern of peaks and troughs, corresponding to constructive/destructive interference between paths from different slits

Snapshot 3: for a single slit of finite width, the diffraction pattern has the well-known form of a sinc function

Snapshot 4: for multiple slits of finite width, the diffraction is a pattern of peaks and troughs modulated by the sinc function pattern arising from the finite width of each slit