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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. ^[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. ^[ref].

Angular colour separation follows immediately. At fixed nonzero diffraction order, increasing the wavelength moves the peak farther from the normal. With $d=2\mu \mathrm{m}$, 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.

For $N$ identical slits with uniform coherent illumination, meaning stable relative wave phases across the array, the normalized wave interference factor isThe 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. ^[ref].

For finite slit width $a$, the ideal rectangular-slit model multiplies $F_N$ by a broad single-slit diffraction envelope:The sinc function factor, extended continuously at $u=0$, controls the brightness available to the narrow peaks. For example, $a=0.5\mu \mathrm{m}$ 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. ^[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. ^[ref].

Dark absorption features in the solar spectrum, mapped and labelled by Joseph von Fraunhofer. They give reproducible wavelength landmarks for spectroscopy and optical measurements. Solar and terrestrial absorbers contribute different features to a ground-observed solar spectrum.

An optical prism separates a white-light beam because its refractive index depends on wavelength. Snell's law therefore gives different emergence directions for different components. With ordinary visible dispersion, violet is deviated more strongly than red. A narrow entrance slit and a collimated beam make a resolved spectrum; a broad overlapping bundle of rays would wash the colours and lines together.

For prism angle $A$ at minimum deviation, the symmetric ray path givesMeasuring a named line's deviation therefore fixes the material's refractive index at a reproducible wavelength.

The Fraunhofer lines are dark absorption features in the solar spectrum, mapped and labelled by Joseph von Fraunhofer. They provide useful spectral landmarks. Measuring a glass's refractive indices at several such lines characterizes optical dispersion, supports comparison of glasses, and helps design an achromatic doublet that brings selected colours to a common focus. Achromatism at two wavelengths does not remove every residual colour error.

The physical explanation is selective absorption. Radiation from the hotter underlying solar layers passes through material whose atoms and ions absorb strongly at their transition wavelengths. The same chemical species can emit at corresponding wavelengths under suitable excitation, connecting solar absorption with laboratory emission spectra. Some lines in a ground-observed spectrum instead arise in Earth's atmosphere, so not every dark feature identifies solar material.

In modern terms, an atomic transition satisfies $\Delta E=h\nu =hc/\lambda$. This explains the reproducible spectral positions rather than treating the dark lines as gaps introduced by a defective prism. Resolving power, line width and source conditions still govern how accurately a particular feature can be measured.