Diffraction Gratings

From the grating equation to echelle spectrographs — how periodic structures separate light by wavelength.

Hold a CD under a lamp and tilt it. The rainbow that sweeps across the surface is diffraction: the tightly spaced data tracks (1600 nm pitch, about 625 lines per millimeter) split white light into its colors the same way a prism does, but through interference instead of refraction. The same physics, scaled up to precision-ruled optics, is how every spectrometer in every chemistry lab, telescope, and telecom network separates light by wavelength. In fact, you probably own three diffraction gratings right now: a CD, a DVD, and a Blu-ray disc. We will compute their diffraction angles later and discover why Blu-ray players need blue lasers.

This tutorial covers how diffraction gratings work, from the grating equation that predicts where each color goes, through angular dispersion and resolving power that determine how well a grating separates nearby wavelengths, to the blaze condition that concentrates light into a useful order.

The grating equation

A diffraction grating is a surface with periodic grooves spaced a distance $\mathrm{\Lambda }$ apart (the period). When a plane wave hits the grating at angle ${\theta }_i$ from the normal, each groove scatters light in all directions. In most directions the scattered wavelets cancel. But at certain special angles ${\theta }_m$, wavelets from adjacent grooves arrive in phase, and the scattered light adds constructively. These are the diffraction orders.

The condition for constructive interference is:

where $m=0,\pm 1,\pm 2,\dots$ is the order number. This is the grating equation, and it is the single most important formula in this tutorial.

For $m=0$, the equation gives ${\theta }_m=-{\theta }_i$: the zeroth order is just specular reflection (or straight-through transmission). No wavelength dependence, no dispersion. The useful orders are $m=\pm 1,\pm 2$, and so on.

An order propagates (carries power to the far field) only when $\mathrm{\mid }\sin {\theta }_m\mathrm{\mid }\le 1$. When $\sin {\theta }_m$ exceeds 1, the order becomes evanescent: it exists as a surface wave that decays exponentially away from the grating. The number of propagating orders depends on the ratio of period to wavelength. A fine grating (small $\mathrm{\Lambda }$) with few propagating orders is like a prism. A coarse grating (large $\mathrm{\Lambda }$) with many orders is like an echelle spectrograph.

Worked example: 600 lines per millimeter at normal incidence

A 600 l/mm grating has period $\mathrm{\Lambda }=\mathrm{1,000,000}\mathrm{/}600=1666.7\text{nm}$. At normal incidence (${\theta }_i=0$), the grating equation simplifies to $\sin {\theta }_m=m\lambda \mathrm{/}\mathrm{\Lambda }$.

For green light at 550 nm:

The third orders barely squeak in at 82° from the normal. They are technically propagating, but at such extreme angles that the diffraction efficiency is negligible.

Now the key point: different wavelengths diffract to different angles. In $m=1$, blue light (400 nm) goes to 13.9° and red light (700 nm) goes to 24.8°. The grating has spread white light into a spectrum spanning about 11 degrees. This is exactly what the rainbow on a CD does: the track pitch (1600 nm) is similar to our 600 l/mm grating (1667 nm), so the diffraction angles are nearly the same.

Angular dispersion

How quickly does the diffraction angle change with wavelength? Take the derivative of the grating equation with respect to $\lambda$:

This is the angular dispersion. It tells you how many degrees (or radians) of angular spread you get per nanometer of wavelength change. Two features stand out.

First, dispersion increases with order $m$. Second order has twice the angular spread of first order. An echelle grating operating in $m=41$ has 41 times the dispersion of a first-order grating with the same period.

Second, dispersion increases as ${\theta }_m$ approaches 90° ($\cos {\theta }_m$ approaches zero). Grazing-angle orders have enormous dispersion, but they also have low efficiency and high aberrations, so this is rarely exploited in practice.

For our 600 l/mm grating at normal incidence, $m=1$, $\lambda =550$ nm (${\theta }_m=19.3\mathrm{°}$):

To put this in perspective: the sodium D doublet (589.0 and 589.6 nm, separated by 0.6 nm) would appear 0.022° apart. Whether you can actually resolve the two lines depends not on dispersion alone, but on resolving power.

Resolving power

The resolving power $R$ of a grating sets the smallest wavelength difference it can separate:

where $N$ is the total number of illuminated grooves. This is one of the cleanest formulas in optics. It says that resolving power depends on only two things: the order and the number of grooves. Period, wavelength, angle of incidence: none of them matter directly.

This is worth pausing on. A cheap 600 l/mm grating and an expensive 1200 l/mm grating with the same width have identical resolving power in the same order, even though the 1200 l/mm grating spreads the spectrum wider. The fine grating has more angular dispersion, but it also has half the grooves per unit width. The two effects cancel exactly. What actually determines resolution is how many grooves the light sees, not how closely spaced they are.

A 25 mm wide grating at 600 l/mm has $N=\mathrm{15,000}$ grooves. In first order, $R=\mathrm{15,000}$, so $\delta \lambda =550\mathrm{/}\mathrm{15,000}=0.037\text{nm}$. That easily resolves the sodium D doublet (0.6 nm separation, needing $R>982$).

But resolving power scales with order. The same 15,000 grooves operating in $m=2$ give $R=\mathrm{30,000}$ ($\delta \lambda =0.018$ nm). In $m=41$ (echelle mode), $R=\mathrm{615,000}$ ($\delta \lambda =0.0009$ nm). This is why high-resolution spectrographs use echelle gratings at high order rather than fine-pitched gratings at low order.

Free spectral range and overlapping orders

There is a catch with higher orders. The grating equation is satisfied for many combinations of $m$ and $\lambda$ simultaneously. Wavelength $\lambda$ in order $m$ arrives at the same angle as wavelength ${\lambda }^{\mathrm{\prime }}=m\lambda \mathrm{/}(m-1)$ in order $m-1$. The free spectral range (FSR) is the wavelength range within a single order before overlap occurs:

In $m=1$, FSR = 550 nm: the entire visible spectrum fits in first order with no overlap. In $m=2$, FSR = 275 nm: second-order blue (400 nm) overlaps with first-order red (800 nm). Without a blocking filter, your detector sees blue and red photons arriving at the same angle and cannot tell them apart.

The echelle case is extreme. At $m=41$, FSR = 13.4 nm. Dozens of orders overlap at every point on the detector. Echelle spectrographs solve this by cross-dispersing with a prism or a low-order grating perpendicular to the echelle, spreading the overlapping orders vertically into a 2D pattern called an echellogram.

The free spectral range of a grating is analogous to the FSR of a Fabry-Perot interferometer: both are periodic in frequency, and both require filtering to separate overlapping orders.

The Littrow configuration

There is a special mounting geometry where the diffracted beam returns exactly along the incoming beam: ${\theta }_m={\theta }_i$. This is the Littrow configuration, and it gives maximum diffraction efficiency for a blazed grating. The Littrow condition simplifies the grating equation to:

For our 600 l/mm grating at 550 nm in $m=1$:

In practice, the spectrometer tilts slightly off Littrow so the detector does not block the incoming beam, but the geometry is close enough that the efficiency advantage holds. Most modern spectrometers use near-Littrow mounting.

The blaze condition

A flat grating with rectangular grooves splits light roughly equally between positive and negative orders. That wastes half the light before you even start. Blazing is the oldest trick in grating design: tilt each groove facet so it acts like a tiny mirror, angled to steer its reflected light toward the order you want. When the mirror angle matches the diffraction angle, almost all the light goes into one order.

The blaze wavelength at Littrow is:

where ${\theta }_B$ is the blaze angle (the tilt of the groove facets). For a 600 l/mm grating blazed at 9.5°, the blaze peak is at 550 nm in $m=1$. At this wavelength, the efficiency can reach 80–90% in the blaze order. Away from the blaze wavelength, the efficiency rolls off following a sinc² envelope: it drops to about 50% at wavelengths roughly 40% above or below the blaze peak.

This is the central design trade-off for a blazed grating. A steep blaze angle gives a blaze peak at longer wavelengths (good for NIR). A shallow blaze gives a peak at shorter wavelengths (good for UV). You cannot optimize for all wavelengths simultaneously.

Worked example: echelle spectrograph

An echelle grating has coarse grooves (79 l/mm, $\mathrm{\Lambda }=\mathrm{12,658}$ nm) and a steep blaze angle (63°). At 550 nm, the Littrow order is:

The resolving power for a 100 mm wide echelle is:

This resolves $\delta \lambda =550\mathrm{/}\mathrm{323,900}=0.0017$ nm, or 1.7 pm. For comparison, the thermal Doppler width of an iron line in the solar photosphere is about 3 pm. An echelle spectrograph can resolve individual spectral line shapes.

The free spectral range is only $550\mathrm{/}41=13.4$ nm, so a cross-disperser is essential. The echellogram from an astronomical spectrograph shows dozens of horizontal stripes (one per order), each covering a 13 nm window, stacked vertically by the cross-disperser.

Gratings you already own

If you do not have a ruled grating in your lab, you have one in your pocket. The data tracks on optical discs act as diffraction gratings:

A CD gives a bright rainbow at moderate angles. A DVD diffracts visible light to steep angles. A Blu-ray disc has tracks so closely spaced that visible light cannot diffract at all: $550\mathrm{/}320=1.72>1$, so the first order is evanescent. Blu-ray players use 405 nm blue-violet lasers precisely because only short wavelengths can resolve the fine track pitch.

The numbers that matter

Here is a workflow for choosing a grating:

The first decision is groove density: this sets how many orders exist and how wide they spread. The second is blaze angle: this determines which wavelength range gets the most light. The third is grating width: more grooves means higher resolving power. Everything else follows from these three choices.

Going further

For a deeper dive into how diffraction efficiency depends on groove profile, polarization, and grating geometry, see the RCWA Simulator, which solves Maxwell's equations directly for periodic structures. The grating equation tells you where the light goes; RCWA tells you how much goes into each order.

Explore the full calculations interactively in the Diffraction Grating Calculator, and look up material refractive indices for transmission gratings in the Material Database.

References

1. Palmer & Loewen, Diffraction Grating Handbook, 8th ed., MKS/Newport (2020).
2. Hecht, Optics, 5th ed., Pearson (2017), Chapter 10.
3. Schroeder, Astronomical Optics, 2nd ed., Academic Press (2000), Chapter 15 (Echelle spectrographs).
4. Hutley, Diffraction Gratings, Academic Press (1982).