Diffraction Gratings and Wave Dispersion

A diffraction grating is a surface with many equally spaced parallel slits (or lines) that spreads light into sharp spectra according to d sin θ = mλ, where d is the spacing of the lines and m is the order; it separates colors because each wavelength is diffracted at a different angle, and a grating with more lines gives sharper, brighter maxima. In brief: the grating spacing is d = 1 ÷ (number of lines per meter). Higher orders give larger angles; the maximum possible order is the largest integer less than d ÷ λ. The resolving power is R = λ ÷ Δλ = mN, where N is the number of lines illuminated. Gratings are used in spectrometers to identify elements, in telescopes, and in lasers and CDs (the colors on the surface of a CD come from diffraction).

A grating gives much brighter and sharper spectra than a double slit, and so it is the standard tool for measuring wavelengths. This guide derives the grating equation, explains orders, resolving power and the difference from a prism, compares gratings and prisms in a table, works through numerical examples and corrects common mistakes.

What is the grating equation?

Light of wavelength λ falls at normal incidence on a grating with line spacing d. Light from adjacent slits is in phase in a direction θ when the path difference between neighboring slits, d sin θ, is a whole number of wavelengths:

d sin θ = mλ, m = 0, ±1, ±2, ….

The central maximum (m = 0) is at θ = 0 for all wavelengths. The first-order maximum (m = 1) is at sin θ = λ ÷ d. Because many slits contribute, the intensity maxima are narrow and bright; between them, the light from the many slits cancels, producing nearly dark regions. The maxima become narrower as the number of illuminated slits N increases: the width is proportional to λ ÷ (Nd). A grating is described by the number of lines per millimeter (for example 300, 600 or 1,200 lines/mm), and d is the reciprocal.

How do gratings form spectra?

With white light, each order m ≥ 1 consists of a spectrum: violet (smallest wavelength) is diffracted the least angle, and red the most (the opposite of a prism, where violet is bent the most). The central maximum is white. Higher orders are wider and eventually overlap: order m overlaps order m + 1 when mλ_red > (m + 1)λ_violet. For visible light (400 to 700 nm), the first and second orders do not overlap (700 nm is less than 2 × 400 = 800 nm), but the second and third orders do (2 × 700 = 1,400 nm is greater than 3 × 400 = 1,200 nm). With a light source such as a hot gas (sodium lamp, mercury lamp), a grating shows the characteristic bright lines of the emission spectrum, which identify the element. The angular dispersion, dθ/dλ = m ÷ (d cos θ), is greater for higher orders and finer gratings, so the lines are better separated.

What is resolving power?

The resolving power of a grating, R = λ ÷ Δλ = mN, is the ability to separate two close wavelengths: the smallest wavelength difference Δλ that can be distinguished is Δλ = λ ÷ (mN). A grating with more lines and a higher order resolves finer detail. To resolve the sodium doublet (589.0 nm and 589.6 nm, Δλ = 0.6 nm) in the second order, N ≥ λ ÷ (mΔλ) = 589.3 ÷ (2 × 0.6) = 491 lines must be illuminated. A typical 600 lines/mm grating with a beam width of 1 cm illuminates 6,000 lines, which is more than enough.

Gratings and prisms compared

Worked example: a 600 lines/mm grating

1. Spacing: d = 1 mm ÷ 600 = 1.667 × 10⁻⁶ m = 1.667 µm.
2. First order for sodium light (589 nm): sin θ₁ = λ ÷ d = 589 × 10⁻⁹ ÷ 1.667 × 10⁻⁶ = 0.353, θ₁ = 20.7°.
3. Second order: sin θ₂ = 2 × 0.353 = 0.707, θ₂ = 45.0°.
4. Maximum order: d ÷ λ = 1.667 ÷ 0.589 = 2.83, so the maximum order is m = 2. The third order would need sin θ = 1.06, which is impossible. In total, there are 5 maxima: m = 0, ±1, ±2.
5. Finding the wavelength. The first-order maximum for an unknown light is observed at 25.0°: λ = d sin θ = 1.667 × 10⁻⁶ × sin 25.0° = 1.667 × 10⁻⁶ × 0.4226 = 7.04 × 10⁻⁷ m = 704 nm (red).
6. White light in the first order: violet (400 nm): sin θ = 0.240, θ = 13.9°; red (700 nm): sin θ = 0.420, θ = 24.8°. The spectrum spans about 11° and is nearly linear in wavelength.

Who invented the grating?

David Rittenhouse made the first diffraction grating in 1785, with hairs in two screws. Joseph von Fraunhofer produced precise gratings in the early 1820s and measured wavelengths with them. Henry Rowland made high-quality concave gratings in the 1880s at Johns Hopkins University, with ruling machines of great precision. Today holographic gratings, created by interference of laser beams, are widely used, and gratings are central to spectrometers in chemistry, astronomy and telecommunications (wavelength division multiplexing).

Which mistakes are common?

Mistake 1: Using the number of lines per millimeter as d

d is the reciprocal: d = 1 mm ÷ (lines per mm).

Mistake 2: Expecting violet at the largest angle

In a grating, red is diffracted the most. In a prism, violet is bent the most.

Mistake 3: Forgetting that there is a maximum order

sin θ cannot exceed 1, so m ≤ d ÷ λ.

Mistake 4: Confusing the dispersion and the resolving power

Dispersion is the angular spacing of wavelengths. Resolving power relates to the sharpness, controlled by N.