Diffraction grating

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 spectrometer that resolves spectral lines using the angular dispersion of a diffraction grating. A narrow slit at a collimator focal plane supplies nearly parallel illumination; a telescope or focusing lens forms separate images for different outgoing angles. At normal incidence, the grating equation is $d\sin \theta =m\lambda$. The slit width, illuminated groove count, aberrations and detector sampling jointly determine whether a calculated angular separation is actually resolved.

For a uniformly illuminated grating of $N$ identical slits, width $w$ and pitch $p$, the amplitude is a single-slit diffraction factor times the coherent array sum. At integer order $m$, the array amplitude is $N$, giving the displayed peak intensity at fixed incoming intensity. Changing $w$ can increase the central intensity while suppressing a higher order through its envelope. Renormalizing both patterns to unit central intensity removes the information needed for that comparison.

For $N$ equal narrow slits, adjacent phase increment $\delta$ gives amplitude $\sin (N\delta /2)/\sin (\delta /2)$ relative to one slit. Near a principal maximum, set $u=N\delta /2$; for large $N$ the normalized intensity approaches $(\sin u/u{)}^2$. Approximating the first subsidiary maximum by $u=3\pi /2$ gives principal-to-subsidiary intensity ratio $9{\pi }^2/4$. The exact large-array extremum satisfies $\tan u=u$ in $(\pi ,2\pi )$ and gives a slightly different ratio, about $21.2$. The slit envelope can further modify observed peak heights.

For $N$ equally spaced, coherently illuminated narrow slits at normal incidence, the phasor sum is proportional to $\sin (N\delta /2)/\sin (\delta /2)$, where $\delta =2\pi d\sin \theta /\lambda$. The first zero around the central maximum has $d\sin \theta =\lambda /N$. Principal maxima are removable zero-over-zero limits, not intensity zeros.

A normal-incidence diffraction grating of pitch $D$ has order $m$ at $D\sin \theta =m\lambda$ and angular separation $\Delta \theta \simeq m\Delta \lambda /(D\cos \theta )$. A lens whose axis follows the selected order maps a small angular interval to $\Delta x\simeq f\Delta \theta$. Detector pixels substantially smaller than this separation are needed; two pixels per resolution interval correspond to $t\lesssim \Delta x/2$. If instead detector coordinates obey $x=f\tan \theta$ relative to the original axis, the scale is $\Delta x\simeq f{\sec }^2\theta \Delta \theta$. A numerical pixel limit therefore requires an optical layout and a sampling criterion as well as the resolving power of a diffraction grating.

For $N$ coherently illuminated grating lines used in diffraction order $m$, the Rayleigh criterion yields the displayed ideal resolving power of a diffraction grating. The angular maximum-to-first-zero separation is $\lambda /(ND\cos {\theta }_m)$, while nearby wavelengths are separated by $|m|\delta \lambda /(D\cos {\theta }_m)$. The order must exist, carry appreciable light intensity and remain free from dominant instrumental broadening. Line count alone therefore does not determine whether a particular instrument resolves a specified doublet.

For narrow slits and equally bright mutually incoherent spectral lines, $u_j=\pi p\sin \theta /{\lambda }_j$ and the intensities add as displayed. The Rayleigh criterion uses a main-peak-to-first-zero separation $\delta \lambda \simeq \lambda /(|m|N)$. A slightly sub-Rayleigh doublet can still have two local maxima: the criterion is not a theorem of statistical non-identifiability. photon counts, background, unequal strengths and uncertainty in the instrumental profile govern actual inference.

At normal incidence, an ideal grating of $N$ lines and period $d$ resolves a small wavelength separation $\delta \lambda$ in order $m$ by the Rayleigh condition $|m|N\gtrsim \lambda /\delta \lambda$. Combining this with $|m|\lambda =d|\sin \theta |$ yields $W=Nd\gtrsim {\lambda }^2/(\delta \lambda |\sin \theta |)\ge {\lambda }^2/\delta \lambda$. The final expression is a limiting lower bound as diffraction approaches grazing. Finite usable diffraction angle, unequal line strengths, aberrations and intrinsic linewidth can all demand a larger instrument or a more detailed criterion.

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.

A missing diffraction order occurs when a principal maximum of an array factor coincides with a zero of the individual aperture's diffraction envelope. For identical slits of width $a$ and centre spacing $D$, order $m$ has $\sin {\theta }_m=m\lambda /D$. It is missing if $ma/D$ is a nonzero integer. This is cancellation of field contributions, not loss of the incident wavelength or a failure of the grating equation.

For a reflection diffraction grating with both ray directions measured on the same side of its normal, adjacent grooves have path difference $d(\sin \alpha +\sin \beta )$. Constructive interference requires this to be an integer multiple $m\lambda$. Signed-angle conventions can turn the sum into a difference; the physical path difference is unchanged.

A formal grating order must have a real outgoing angle. For normal incidence, $|m|\lambda \le d$ is necessary. Oblique incidence can access orders outside that normal-incidence range; reflection Littrow geometry has $m\lambda =2d\sin \theta$. The ideal resolving power of a diffraction grating applies only to an order that exists and has sufficient signal.

At fixed incidence angle, differentiating the grating equation gives $d\beta /d\lambda =m/(d\cos \beta )$. A camera with focal length $f$ therefore has local focal-plane dispersion $q=fm/(d\cos \beta )$ near its optical axis. With coordinate $X=f\tan (\beta -{\beta }_0)$, the full derivative is $q=f{\sec }^2(\beta -{\beta }_0)m/(d\cos \beta )$.

At fixed wavelength, the grating equation gives $d\beta /d\alpha =-\cos \alpha /\cos \beta$. Thus an incident angular width is magnified by $\cos \alpha /\cos \beta$ in the dispersion direction. Including the ratio of camera and collimator focal lengths gives the projected slit width. The ESO B&C operating manual, Appendix A explicitly includes this anamorphic factor.

A slit of width $x$ at the focus of a collimator subtends $x/f_{\mathrm{coll}}$. Its projected width is $x{f}_{\mathrm{cam}}\cos \alpha /(f_{\mathrm{coll}}\cos \beta )$. Dividing this width by the grating dispersion gives $\Delta \lambda =xd\cos \alpha /(m{f}_{\mathrm{coll}})$. Thus the spectral resolving power is $R=m\lambda f_{\mathrm{coll}}/(xd\cos \alpha )$. Omitting the anamorphic factor gives a denominator $\cos \beta$ instead; these agree in the Littrow configuration.

In the Littrow configuration the selected diffracted ray retraces the incident ray, so $\alpha =\beta$ and the grating equation becomes $m\lambda =2d\sin \alpha$.

An echelle grating works in high diffraction orders. A separate cross-disperser separates their overlapping optical spectra on a two-dimensional detector.