What is the grating equation?

The grating equation mλ = d(sin α + sin β) relates the wavelength λ, grating spacing d, incident angle α, diffracted angle β, and diffraction order m. It determines at which angles constructive interference occurs for a given wavelength.

What is resolving power?

The resolving power R = mN where m is the diffraction order and N is the total number of illuminated grooves. It determines the minimum resolvable wavelength difference Δλ = λ/R. Higher R means finer spectral detail can be resolved.

What is a blazed grating?

A grating where the groove facets are tilted at a specific "blaze angle" to maximize diffraction efficiency at a specific wavelength and order. A 600 lines/mm grating blazed for 500 nm in first order has facets optimized to put most energy into that combination, achieving 60–80% throughput.

How do gratings compare to prisms?

Gratings give higher dispersion and resolving power, work across wider wavelength ranges, but split light into multiple orders (lower throughput per order). Prisms are higher-throughput per output beam but have lower resolving power and nonlinear dispersion. Modern spectroscopy almost always uses gratings.

Why do gratings have multiple orders?

The interference condition is satisfied at every integer m, including 0 (straight-through), ±1, ±2, etc. Each non-zero order diffracts at different angles. Most spectrometers use first order; higher orders give greater dispersion but lower intensity and risk of order overlap.

What causes order overlap?

In second order, wavelength λ appears at the same angle as wavelength λ/2 in first order. So a 700 nm photon in m=1 overlaps with a 350 nm photon in m=2. Solutions: order-sorting filters that block one range, or cross-dispersion with an additional grating/prism.

How does a CD act as a diffraction grating?

CD data tracks are spaced at 1.6 µm (DVD: 0.74 µm, Blu-ray: 0.32 µm). When illuminated by light, the regular spacing causes constructive interference at specific angles, creating a rainbow pattern. This is why CDs have iridescent backs — they're acting as crude reflection gratings for white room light.

Diffraction Grating Calculator

Find diffraction angles for a transmission or reflection grating. Uses the grating equation to determine output angles, angular dispersion, and resolving power for different diffraction orders.

A diffraction grating is the workhorse of modern spectroscopy. It's a surface with a regular pattern of parallel grooves (often 300–3600 grooves per millimeter), each acting as a tiny source of diffracted light. When monochromatic light hits the grating, the waves from each groove interfere — constructively in specific directions, destructively in most others. Different wavelengths interfere constructively at different angles, separating "white" light into its component colors with much higher resolution than a prism can achieve.

The fundamental relationship is the grating equation: mλ = d(sin α + sin β), where m is the diffraction order (integer), λ is wavelength, d is the groove spacing (d = 1/groove density), α is the incidence angle, and β is the diffraction angle. Solve for β to find where each wavelength diffracts. The first-order spectrum (m=1) is the brightest and most-used; higher orders give greater dispersion but lower intensity per order.

This calculator does the grating math: enter the groove density, incidence angle, wavelength, and order — and get the diffraction angle plus resolving power (R = m × N, where N is the total illuminated grooves). Gratings appear everywhere precision wavelength measurement matters: astronomical spectrographs, Raman and fluorescence spectrometers, atomic absorption analyzers, semiconductor laser modules, and even the back of a CD (which acts as a crude reflection grating, splitting room light into the rainbow you see).

Inputs

Groove Density (lines/mm)

Incident Angle (°)

Wavelength (nm)

Diffraction Order m

Total Illuminated Grooves

Number of grooves illuminated by the beam

Results

Diffraction Angle

-9.79°

Angular Dispersion

0.0349 °/nm

Resolving Power

10,000

Diffraction Grating Results

Parameter	Value
Grating Spacing d	1.6667 μm (1666.67 nm)
Diffraction Angle β	-9.7878°
Diffraction Order m	1
Angular Dispersion	0.034885 °/nm
Resolving Power R	10,000
Min Δλ	0.0550 nm
Groove Density	600 lines/mm
Grating Equation	mλ = d(sinα + sinβ)

Last updated: May 29, 2026

Formula

**Grating equation (transmission and reflection gratings):** mλ = d × (sin α + sin β) Where: - **m**: diffraction order (integer: 0, ±1, ±2, ...) - **λ**: wavelength of light - **d**: groove spacing (distance between adjacent grooves) - **α**: angle of incidence (from grating normal) - **β**: angle of diffraction (from grating normal) **Sign convention**: angles on the same side of the grating normal as the incident beam are positive in some conventions; on the opposite side, negative. Pick one and be consistent. **Solving for diffraction angle:** sin β = (m λ / d) − sin α β = arcsin[(m λ / d) − sin α] If the argument has magnitude > 1, no diffracted beam exists for that order. **Groove spacing from groove density:** d (mm) = 1 / groove_density_per_mm d (nm) = 1,000,000 / groove_density_per_mm For a 600 lines/mm grating: d = 1/600 mm = 1.667 µm = 1667 nm. **Worked example: 600 lines/mm grating, λ = 550 nm, normal incidence (α = 0), first order** d = 1667 nm sin β = (1 × 550) / 1667 − 0 = 0.330 β = 19.27° **Resolving power (chromatic resolution):** R = λ / Δλ = m × N Where N is the total number of illuminated grooves. Higher R means smaller resolvable wavelength difference. For 600 lines/mm grating illuminated over 25 mm width: N = 600 × 25 = 15,000 grooves. At first order (m=1): R = 15,000 → at λ = 550 nm, Δλ = 550/15,000 = **0.037 nm**. This resolution can separate the sodium D doublet (589.0 and 589.6 nm, separation 0.6 nm) easily. **Angular dispersion (how spread out wavelengths are):** dβ/dλ = m / (d × cos β) For larger m or smaller d (denser grating): more angular spread per nm. For our 600 lines/mm at β = 19.27°: dβ/dλ = 1 / (1667 × cos 19.27°) = 6.36 × 10⁻⁴ per nm = 0.036 mrad/nm = 0.0021°/nm. **Free spectral range (where orders don't overlap):** FSR_λ = λ / m So for m=1 at λ = 550 nm: FSR = 550 nm. The first-order spectrum from 1100 to 550 nm doesn't overlap with second-order spectrum (which would put 550 nm at the same angle as 1100 nm in first order). **Common grating types and properties:** | Type | Groove density (lines/mm) | Typical use | |---|---|---| | Visible-range education | 300, 600 | Demos, slit spectrometers | | Astronomy spectrographs | 600–1200 | Stellar spectra, exoplanet detection | | Echelle grating | 30–300, used in high orders | Very high resolution (R > 100,000) | | Concave / aberration-corrected | 600–2400 | Compact spectrometers | | Reflection (ruled) | 600–3600 | Most spectroscopy applications | | Transmission (volume holographic) | varies | DWDM telecom, lasers | | Plane (Echelle) | 79, used in m=100+ | UV/visible high-R spectroscopy | | Blazed (efficiency-optimized) | varies | Highest efficiency at "blaze wavelength" | **Blaze angle** is the angle of the groove facets, designed to put maximum diffracted intensity into a specific order at a specific wavelength. A 600 lines/mm grating blazed for 500 nm in first order has facets tilted to optimize that combination.

How to use this calculator

Enter the groove density in lines/mm (300–3600 typical for spectroscopy).
Enter the angle of incidence (0° for normal incidence is the simplest case).
Enter the wavelength of interest. Try sweeping across visible range (400–700 nm) to see how angles change.
Pick a diffraction order. First order (m=1) is brightest and most common; higher orders give more dispersion.
Read the diffraction angle β and the resolving power R = m × N.
For overlapping orders, check whether your spectrum is in the "free spectral range" of your order.

Worked examples

Astronomy spectrograph design

**Scenario:** A stellar spectrograph uses a 600 lines/mm grating at first order. The beam illuminates 30 mm of the grating. What resolving power, and what's the resolvable wavelength difference at Hα (656.28 nm)? **Calculation:** N = 600 × 30 = 18,000 grooves. R = m × N = 1 × 18,000 = 18,000. Δλ = λ/R = 656.28/18,000 = 0.036 nm. **Result:** Δλ ≈ 0.04 nm resolution at Hα — good enough to detect typical stellar radial velocity Doppler shifts (10–100 m/s would give Δλ of 0.0002–0.002 nm, much less). For exoplanet detection, you'd need a higher-resolution echelle spectrograph (R > 100,000).

CD as a reflection grating

**Scenario:** A CD has data tracks spaced at 1.6 µm (= 625 tracks/mm equivalent). Shine red laser (650 nm) at normal incidence. Where does the first-order diffracted beam go? **Calculation:** d = 1600 nm. sin β = (1 × 650) / 1600 = 0.406. β = 24°. **Result:** First-order diffraction goes off at 24°. That's why a CD acts as a "rainbow mirror" under white light — different wavelengths diffract at different angles, splitting into the visible spectrum. The data spiral itself isn't a perfect grating (it's a single spiral, not parallel lines), but it's regular enough on small scales to act like one for incoherent white light.

Raman spectrometer choice

**Scenario:** Designing a Raman spectrometer for visible light (532 nm excitation). What grating density gives 0.5 nm resolution with a 25 mm beam? **Calculation:** Need R = λ/Δλ = 532/0.5 = 1064. R = m × N → N = 1064 / 1 (first order) = 1064 grooves illuminated. With 25 mm beam, density needed = 1064/25 = 42.6 lines/mm. Way more than needed; a standard 300 lines/mm grating gives N = 7500, R = 7500, Δλ = 0.07 nm — much better than 0.5 nm. **Result:** Even a low-density 100 lines/mm grating exceeds the resolution target. For Raman, the constraint is usually CCD pixel size rather than grating resolution: each pixel covers a finite wavelength range, often 0.1 nm or so. Match the grating dispersion to put resolvable features across multiple pixels.

When to use this calculator

**Use diffraction gratings for:**

- **Astronomical spectroscopy**: stellar classification, exoplanet detection, redshift measurement. - **Raman and fluorescence spectrometers**: chemical analysis, materials characterization. - **Mass spectrometer detectors with light-based readout**. - **DWDM (dense wavelength division multiplexing) telecom**: separate/combine ITU-grid channels. - **Wavelength meters for laser tuning**: high-precision wavelength determination. - **Atomic absorption spectrometers**: quantitative element analysis. - **CD/DVD/Blu-ray optical pickups**: actually use a tracking grating to separate light beams. - **Educational physics demonstrations**: visible-spectrum separation, interference pattern.

**Choosing a grating:**

- **Groove density**: higher = more dispersion but smaller useful spectral range. 600 lines/mm is the all-rounder. - **Blaze angle / blaze wavelength**: optimizes throughput in a specific order. Match to your operating wavelength. - **Substrate**: aluminum (visible), gold (NIR), MgF₂ (UV). - **Type**: reflection (most common), transmission (some specialty), holographic (low scatter, no shadows from ruling errors). - **Size**: bigger grating = more grooves N illuminated = higher resolving power.

**Order overlap problem:**

In second order, λ appears at the same angle as λ/2 in first order. For broad spectra, this causes overlap that confuses analysis. Solutions: order-sorting filters, cross-dispersion (echelle), or limit to first order only.

**Grating efficiency**:

- Ungrooved reflection grating: ~30% in first order, rest in other orders. - Blazed grating: 60–80% in optimized order at blaze wavelength. - Holographic grating: low ghosting and stray light, often 50–70% in first order. - Volume Bragg grating: near 100% diffraction efficiency at one specific wavelength and angle.

**Grating vs prism comparison:**

| Property | Grating | Prism | |---|---|---| | Dispersion | High, linear in angle | Lower, nonlinear | | Resolving power | High (R up to 10⁶) | Lower (R up to ~10⁴) | | Wavelength range | Wide | Limited by transmission | | Order overlap | Issue | None | | Brightness | Lower (light split into orders) | Higher (one beam out) | | Cost | Moderate to high | Generally lower |

Modern spectrometers almost always use gratings except in cases where prism advantages (brightness, no overlap) are crucial.

Common mistakes to avoid

Confusing groove density units. Lines per mm, lines per cm, and lines per inch are all used in different contexts.
Forgetting to convert wavelength to consistent units with d. Both must be in same length units (typically nm or µm).
Using "average" angles for non-normal incidence. Always use the actual incidence angle α in the equation.
Treating higher orders as more useful without considering efficiency drop. Higher m has lower throughput.
Forgetting that "no diffraction" condition (|sin β| > 1) means that order doesn't exist for given λ, α, d.
Overlapping orders without filters. The order-sorting filter is essential in broadband spectroscopy.
Computing resolving power without considering pixel resolution at the detector. The actual instrument resolution is limited by the worse of grating R and detector sampling.

Frequently Asked Questions

Sources & further reading

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