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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].

A diffraction order is the integer $m$ labeling a principal constructive interference direction of a diffraction grating: adjacent elements differ in path length by $m$ wavelengths. In the normal-incidence transmission geometry described above,Positive and negative diffraction orders lie on opposite sides of the forward normal. The condition expresses the range of the sine of a real angle, so it also limits the propagating diffraction orders.

For $\lambda =600\mathrm{nm}$ and $d=2\mu \mathrm{m}$, $\sin {\theta }_m=0.3m$. Taking the inverse sine givesThese are seven allowed propagating diffraction orders. Order $m=4$ would require $\sin {\theta }_4=1.2$, and $m=-4$ would require $-1.2$, so neither has a propagating far-field direction. The allowed diffraction orders are equally spaced in $\sin \theta$, while their actual angles become farther apart toward grazing propagation.

Each principal peak has finite width. For the ideal uniform coherent array, the nearest zeros around order $m$ obeyThus the one-sided distance from the order direction to the first zero is $\Delta (\sin \theta )=\lambda /(Nd)$. Near the first diffraction order in our example, $N=10$ puts the zeros at $0.27$ and $0.33$, while $N=40$ puts them at $0.2925$ and $0.3075$. The respective one-sided widths are $0.03$ and $0.0075$; the full zero-to-zero widths are $0.06$ and $0.015$. These are calculated coherent peak widths, with the slit envelope omitted, and do not describe a measured instrument response. ^[ref].

Four times as many illuminated slits therefore give a fourfold narrowing in this coordinate without moving the diffraction order. That sharper directional selection helps distinguish nearby wavelengths in a spectrograph. Actual performance also depends on illumination, optical coherence, and source width; the separate slit-limited resolving power of a grating describes the contribution of the entrance slit.

Fraunhofer diffraction is the far-field approximation for diffraction: contributions from an aperture reach a selected observation direction with approximately linear path-dependent wave phases. Adding their complex amplitudes gives a Fourier transform of the field across the aperture. It also describes the appropriate rear focal plane of an ideal optical lens under coherent plane-wave illumination. The diffraction grating examples use this regime. ^[ref]; ^[ref].

For a slit of width $a$, wavelength $\lambda$ and screen distance $L$, sufficient conditions near the forward direction areThe last two statements are equivalent: the important comparison is with $a^2/\lambda$, rather than just with the slit width. For a finite rectangular aperture, check the corresponding condition for both transverse dimensions. The single-slit diffraction calculation treats the transverse cut associated with width $a$. ^[ref].

A short explanation is that the distance from slit coordinate $x$ to an observation point at distance $R$ and angle $\theta$ isThe linear term supplies the phase difference; the quadratic term supplies a wave phase error of order $a^2/(\lambda R)$, since the wavenumber is $2\pi /\lambda$. Near the forward direction $R\approx L$, so the stated condition makes this error small. This is why a distant screen permits the simple single-slit diffraction formula. ^[ref].

Single-slit diffraction is the spreading of light transmitted by one slit. Consider an ideal rectangular slit of width $a$, illuminated normally by a uniform plane wave with a single wavelength $\lambda$ in the surrounding medium. This monochromatic wave has uniform wave amplitude and equal incident wave phases across the slit, providing the required optical coherence. In the scalar Fraunhofer diffraction model, the normalized angular wave intensity isHere $\theta$ is measured from the forward normal in the plane across the slit width. The sinc function uses the unnormalized convention $\mathrm{sinc}u=\sin u/u$. The limit makes the central value one. This ideal envelope neglects additional direction-dependent propagation and detector factors; near the forward direction it describes the normalized screen irradiance. ^[ref].

The noncentral zeros satisfyThe restriction follows from the range of the sine of a real angle. The first minima have $\sin \theta =\pm \lambda /a$ when $\lambda /a\le 1$. If $a<\lambda$, this model has no noncentral zero in the forward hemisphere; equality puts the first zeros at grazing directions, outside the small-angle approximation and off any finite perpendicular screen. ^[ref].

For $0<\lambda /a<1$, define the central width as the separation between the two first minima. Its angular width and its width on a perpendicular screen areThe angular approximation uses radians and requires $\lambda /a\ll 1$; the screen approximation uses $y\approx L\theta$. The tangent relation is exact straight-line geometry, even though the underlying Fraunhofer diffraction field is approximate. This width is distinct from the full width at half maximum.

For $\lambda =500\mathrm{nm}$, $a=0.1\mathrm{mm}$ and $L=1\mathrm{m}$, the positive first minimum isThe dimensionless far-field parameter is $a^2/(\lambda L)=0.02$. Doubling the slit width to $a=0.2\mathrm{mm}$ gives $W\approx 5.00002\mathrm{mm}$ and parameter $0.08$; both curves below use the Fraunhofer diffraction approximation. Choosing $L=2\mathrm{m}$ halves those parameters and doubles both screen widths.

Each curve is normalized to its own $I_0$; the arrows measure between first minima. The calculation uses $\theta =\arctan (y/L)$ before evaluating the sinc function, rather than replacing $\sin \theta$ by $y/L$. The lower diagram shows the exact screen geometry with the vertical scale enlarged. Normalization removes the change in peak brightness when the slit width changes.

The wave interference explanation is especially simple: divide the slit into equal narrow strips and add their complex amplitudes, rather than their wave intensities. With wavenumber $k=2\pi /\lambda$, the continuous slit givesIntegrating the complex exponential gives the endpoint difference; its continuous value at zero is one. Taking the squared complex modulus gives the stated wave intensity. This is the Fourier transform of an interval indicator expressed in angular coordinates. Feynman's continuous-source argument obtains the same pattern by letting the number of sources tend to infinity. ^[ref].

At the positive first minimum, the wave phases span exactly $2\pi$ across the slit. Every strip in one half has a partner $a/2$ away, with equal wave amplitude and phase difference $\pi$, so the pairs cancel. At $\theta =0$, all the wave phases align instead. ^[ref].

The drawing uses $N=8$ equally weighted midpoint samples $x_j/a=-1/2+(j+1/2)/8$, with $j=0,\dots ,7$. At the first minimum their complex amplitudes are a common rotation of the eight roots of unity, so their head-to-tail polygon closes exactly. This finite sum illustrates cancellation; the continuous integral, rather than eight point sources, defines the slit pattern at general angles. The circle shows the continuous range of wave phases; coloured arrows show the eight samples.

A narrower aperture requires a larger angle to accumulate the cancelling phase difference, so narrowing the slit widens the beam. This is a useful limit on geometrical optics: confining light more tightly at an opening increases its subsequent angular spread. At fixed incident irradiance and slit height, reducing $a$ also reduces transmitted power, exposing a practical balance between diffraction-limited resolution and light throughput. In a diffraction grating, this single-slit diffraction envelope modulates the wave interference between different slits.

Spatial optical coherence compares field samples at different positions, usually across a transverse plane at equal time. Two small openings separated by $d$ sample this correlation: recombine their light with nearly matched travel times and measure interferometric visibility near the central interference fringes. A narrow opening in front of a broad-spectrum lamp can improve transverse spatial optical coherence while leaving its short temporal optical coherence. These are separate controls: narrow spectral selection by itself need not make an extended source spatially coherent. MIT's lecture notes illustrate the two-opening test and these contrasting source preparations. ^[ref].

To isolate source width, stipulate a distant, narrowband source with uniform angular brightness on $-\Theta /2\le \alpha \le \Theta /2$, and mutually uncorrelated emission from distinct source directions. Here $\Theta \ll 1$ is the full angular width in radians, $\lambda$ the common wavelength, and the two openings are separated by $d$ along the same transverse direction as the source extent. Neglect amplitude differences across the openings. Each source direction contributes a phase difference approximately $2\pi d\alpha /\lambda$. Averaging its complex cross term givesThe sinc function is extended continuously to one at $q=0$. The sign convention in the exponential depends on which opening is numbered first; the symmetric model gives the same real result either way. Equal detected single-opening light intensities make the magnitude equal to interferometric visibility. A displaced source center adds a deterministic wave phase, shifting the interference fringes without changing their contrast.

The strips show $I/(I_1+I_2)=1+g_{12}^{(1)}(0)\cos \varphi$ with equal single-opening light intensities, a common phase difference coordinate $\varphi$ and no single-slit diffraction envelope. At $q=1/2$, contrast is $2/\pi \approx 0.636620$; at $q=1$, it vanishes. At $q=3/2$, the correlation is $-2/(3\pi )\approx -0.212207$: weak fringes return with bright and dark positions exchanged. For $\lambda =500\mathrm{nm}$ and $d=0.5\mathrm{mm}$, the first zero occurs at $\Theta =1\mathrm{mrad}$. This is a calculated example, not an experimental measurement or a universal source-width law.

A proposed source-width control follows directly from this model: illuminate two fixed openings with a distant adjustable luminous slit of full width $w$ at distance $R$, so $\Theta \approx w/R$ under the small-angle approximation, and hold $d$ and the spectral selection fixed. Increasing $w$ overlays more mutually incoherent, displaced interference fringe patterns. Record profiles over several near-central fringes, measure each opening separately and correct intensity imbalance as in first-order optical coherence. Restrict the recorded region so the path difference stays small compared with the optical coherence length, and resolve the fringes with the detector. Real source nonuniformity, finite opening widths and spectral spread modify the stipulated sinc function law. This protocol is a model-based extension of the verified two-opening method, not a claim that the plotted curves were measured.

Thus an extended incoherent source can give locally excellent wave interference when the sampled separation is small, and almost none when it is larger. The useful scale $d\Theta \sim \lambda$ expresses how source extent limits the aperture over which coherent addition is available. The same angular averaging explains why narrowing a source opening can recover visible interference fringes, even without narrowing its optical spectrum.