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# Multiple Slit Diffraction Pattern

In this Demonstration we visualize the diffraction pattern of equally spaced slits of equal width, also known as a diffraction grating. It can be shown that the diffraction pattern is equivalent to the diffraction pattern for delta function slits modulated by the diffraction pattern of a single slit of finite width. The latter thus acts as an envelope, shown by the thick dashed line. Special cases of this system include the single () and double () slits, which appear in introductory physics courses. The horizontal scale is arbitrary and the vertical scale normalized to the peak intensity.

### DETAILS

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

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