Diffraction Grating Equation

The condition $m\lambda = d\sin\theta$ is the same one that governs two slits: light leaves in a bright beam wherever the path difference between neighbouring apertures is a whole number of wavelengths. What a grating adds is not a new condition but sharpness. With two slits, a small deviation from the exact angle puts the pair only slightly out of step and the fringe is broad. With ten thousand slits, the same small deviation accumulates across the ruled width until distant slits are completely opposed, and everything cancels except in a very narrow window around the exact angle. So a grating throws each wavelength into its own thin, bright, well-separated beam, and that is what makes it an instrument rather than a demonstration.

A grating ruled at 600 lines per millimetre has $d = 1/600$ mm $= 1.667$ µm. Green light at 550 nm goes to $\sin\theta = 550/1667 = 0.330$, or 19.3°, in first order. Second order needs $\sin\theta = 0.660$, so 41.3°. Third order would need 0.990, which is 81.9° and barely usable, and a fourth order is flatly impossible because the sine would exceed 1. The number of orders you can obtain is capped by $d/\lambda$, and that cap is a real physical limit rather than a numerical inconvenience.

Joseph von Fraunhofer made the first useful gratings in the 1820s by winding fine wire on a frame, and used them to measure the wavelengths of the dark solar absorption lines he had catalogued — the first absolute measurements of the wavelength of light, and the foundation of astronomical spectroscopy. Henry Rowland's ruling engine at Johns Hopkins in the 1880s produced concave gratings good enough that laboratories worldwide bought them for decades. You have one in your house: the data tracks on a CD are spaced 1.6 µm apart, which is a 625-line-per-millimetre grating by accident, and that is the rainbow flash off the disc. A DVD's finer 0.74 µm pitch spreads the same colours over a wider angle.

Five things to watch. $d$ is the line spacing, and gratings are sold in lines per millimetre — 600 lines/mm means $d = 1.667$ µm, and entering 600 in a length field is the most frequent error here by a distance. Angles are measured from the normal to the grating, not from its surface. Because $\sin\theta$ cannot exceed 1, orders beyond $d/\lambda$ simply do not exist and the solver will say so rather than return a nonsense angle. Fourth, and this one bites in the laboratory: orders overlap. Second-order 400 nm violet emerges at exactly the same angle as first-order 800 nm infrared, because $2 \times 400 = 1 \times 800$, so any real spectrometer needs an order-sorting filter and an unfiltered instrument will show ghost features that get mistaken for spectral lines. Finally, this equation tells you where the beams go and says nothing whatever about how bright they are. That is set by the shape of each groove, and a blazed grating is cut with an asymmetric profile that throws most of the light into one chosen order instead of wasting it in the undispersed zeroth.