The Diffraction Grating

the aperture — N slits, width a, pitch d

the far screen — the diffraction pattern you'd see

intensity I(sinθ) — closed form vs the direct Fourier transform

Light passing an opening doesn't cast a sharp shadow — it fans out into fringes. In the far field the pattern is the squared Fourier transform of the aperture's shape. One slit gives a sinc; a row of N slits — a grating — multiplies it by a sharp interference comb whose bright orders obey d·sinθ = mλ.

The aperture

Number of slits N 5

Slit width a 1.5 µm

Slit pitch d 6.0 µm

The light

Wavelength λ 550 nm

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Drag the plot to read off any angle. Raise N and the bright orders sharpen (half-width ∝ 1/N) — that's why a grating out-resolves two slits. When the pitch is a whole multiple of the width, some orders vanish: missing orders, killed by the slit's own zero. Switch to white light and each order fans into a spectrum — a grating is a spectrometer.