The Diffraction Grating

A grating is a comb of many fine slits, and shining light through it sharpens the gentle fringes of two slits into a row of needle-thin bright orders, the basis of nearly every spectrometer. The intensity is a product of two patterns: the broad single-slit envelope $(\sin\beta/\beta)^2$ from each slit's own width, multiplied by the many-slit interference factor that spikes wherever all $N$ slits radiate exactly in phase. That happens at the grating equation $d\sin\theta = m\lambda$, the principal maxima, the same condition as two slits but now reinforced by $N$ of them. The new magic is in the sharpness. Between each pair of orders the interference factor passes through $N-1$ perfect zeros and $N-2$ feeble secondary maxima, so the more slits you add the narrower each principal peak becomes, its width shrinking like $1/N$. The scene paints the pattern as you would see it on a screen, a strip of coloured orders above the intensity profile and its envelope, with the orders labelled and a draggable cursor that reads the angle off any point. The bottom panel zooms into the first order so you can count the secondary maxima and watch the peak knife in as $N$ rises. That narrowing is the resolving power $R = mN$, the reason a grating with thousands of lines can split spectral lines a prism would smear together. Slide the slit count, the spacing, and the wavelength, and watch the orders march, spread, and shift in colour.