= Diffraction grating
{wiki}
A diffraction grating is a periodic optical structure whose repeated slits or grooves direct <light> into <diffraction orders> through <wave interference>. Its beauty is that many individually broad contributions cooperate to select narrow directions: the spacing determines where the principal peaks occur, while the number of illuminated elements determines how sharply they are selected. This makes a <diffraction grating> useful in a <spectrometer>, where separating <wavelengths> spatially lets us examine an <optical spectrum>. https://ocw.mit.edu/courses/6-007-electromagnetic-energy-from-motors-to-lasers-spring-2011/4835d62777f42e82a40c319defcc80a6_MIT6_007S11_lec28.pdf[MIT 6.007: Multiple-slit diffraction and spectrometer application, slides 5-6]{ref}.
Consider a transmission <diffraction grating> in air, illuminated at normal incidence by a <plane wave>. Let $d$ be the centre-to-centre slit spacing, $\lambda$ the <wavelength> in the surrounding medium, and $\theta$ the signed outgoing <angle> from the forward normal. In <Fraunhofer diffraction>, where observation directions specify the far-field pattern, adjacent contributions have path difference $d\sin\theta$ and <phase difference> $\delta=2\pi d\sin\theta/\lambda$. Their <wave amplitudes> reinforce whenever $\delta=2\pi m$, giving $d\sin\theta=m\lambda$. This normal-incidence geometry should be kept distinct from the incident and reflected <angle> convention in the <grating equation> below. https://www.mit.edu/course/other/viz/EM/visualizations/notes/modules/guide14.pdf[MIT: Interference and diffraction, section 14.8]{ref}.
Angular colour separation follows immediately. At fixed nonzero <diffraction order>, increasing the <wavelength> moves the peak farther from the normal. With $d=2\,\mathrm{\mu m}$, first-order <light> at $450\,\mathrm{nm}$ leaves at $13.003^\circ$, whereas $650\,\mathrm{nm}$ leaves at $18.966^\circ$: a separation of $5.963^\circ$. The zero <diffraction order> stays at $0^\circ$ for both <wavelengths>. Thus <grating dispersion> comes from an orderly dependence of <wave interference> on <wavelength>, rather than an arbitrary assignment of colours to directions.
\Image[/diffraction-grating-orders.png]
{title=Normal-incidence <diffraction orders> at $600\,\mathrm{nm}$ and first-order angular colour separation}
For $N$ identical slits with uniform coherent illumination, meaning stable relative <wave phases> across the array, the normalized <wave interference> factor is
$$
F_N(\theta)=\left|\frac1N\sum_{j=0}^{N-1}e^{ij\delta}\right|^2=\left[\frac{\sin(N\delta/2)}{N\sin(\delta/2)}\right]^2,\qquad \delta=\frac{2\pi d\sin\theta}{\lambda}.
$$
The apparent $0/0$ at a principal peak has limiting value one. A short explanation uses the <geometric series>: equal <wave phases> add $N$ equal <complex amplitudes>; a small directional change distributes those contributions around the complex plane and allows cancellation. With each slit's contribution held fixed, peak <scalar wave intensity> scales as $N^2$. The normalized plots remove that height change to make the narrowing visible. Stable relative <wave phases>, or <optical coherence>, are essential; a sum of independent intensities would lose this array structure. https://farside.ph.utexas.edu/teaching/315/Waveshtml/node93.html[Richard Fitzpatrick, University of Texas: Multi-slit interference, equations 10.38-10.40]{ref}.
For finite slit width $a$, the ideal rectangular-slit model multiplies $F_N$ by a broad <single-slit diffraction> envelope:
$$
\frac{I(\theta)}{I(0)}=\left[\frac{\sin u}{u}\right]^2F_N(\theta),\qquad u=\frac{\pi a\sin\theta}{\lambda}.
$$
The <sinc function> factor, extended continuously at $u=0$, controls the brightness available to the narrow peaks. For example, $a=0.5\,\mathrm{\mu m}$ gives envelope values about $0.811$, $0.405$, and $0.090$ at <diffraction orders> $1$, $2$, and $3$ in the $600\,\mathrm{nm}$ example. If an envelope zero coincides with an allowed <diffraction order>, that peak disappears. A permitted direction therefore need not produce a visible bright spot. https://www.mit.edu/course/other/viz/EM/visualizations/notes/modules/guide14.pdf[MIT: Single-slit envelope and absent interference maxima, sections 14.6-14.7]{ref}.
Historically, <Joseph von Fraunhofer> used fine parallel wires stretched between threaded rods to make <diffraction gratings>. Kenyon's historical apparatus account describes both this construction and a later reconstruction with visible <light>. The example makes the governing idea tangible: regular spacing can turn a simple array of obstacles into an instrument for distinguishing <wavelengths>. https://physics.kenyon.edu/EarlyApparatus/Optics/Diffraction_Grating/Diffraction_Grating.html[Thomas B. Greenslade Jr., Kenyon College: Diffraction grating]{ref}.
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