Diffraction grating

A diffraction grating is a series of very many parallel slits, all the same distance $d$ (the grating pitch) apart. When light passes through it, the contributions of the countless slits add up, and constructive interference occurs only in quite precise directions. The principal maxima occur at angles satisfying:

Key formula — Grating maxima

$\ev{d\,\sin\theta_n = n\,\lambda \qquad n = 0, \pm 1, \pm 2, \dots}$

Unlike the single slit, the maxima of a grating are narrow and sharply separated: the more slits contribute, the sharper the maximum becomes. Since the angle $\theta_n$ depends on the wavelength, different colours are deflected by different angles: the grating acts as a “spectroscope” that disperses light into its component colours.

Example — Joseph Fraunhofer's grating

A grating with $500$ lines/mm has pitch $d = 1/(500\cdot 10^3) = 2\cdot 10^{-6}\;\text{m}$. Yellow sodium light ($\lambda = 589\;\text{nm}$) produces the first maximum at: $\sin\theta_1 = \lambda/d = 589\cdot 10^{-9}/(2\cdot 10^{-6}) \approx 0.29 \;\Rightarrow\; \theta_1 \approx 17^\circ$ With white light the same grating produces a continuous spectrum: each $\lambda$ has its own distinct angle. This is the method Fraunhofer first used to discover the spectral lines of the Sun (Halliday, Resnick and Walker 2014).

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Topics: Onde Concepts: Diffrazione

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