Diffraction grating

A diffraction grating is a slide with a large number of very fine, equally spaced parallel slits, often hundreds per millimetre. When monochromatic light shines through it, the light diffracts through every slit and the waves interfere, producing a few sharp, bright maxima at particular angles. Those angles are given by the diffraction grating equation, d sin θ = nλ, where d is the spacing between slits, θ is the angle of a maximum from the straight-through direction, n is the order and λ is the wavelength.

The diffraction grating equation

AQA's A-level specification covers a plane transmission diffraction grating at normal incidence, meaning the light hits the grating square on, and includes the derivation of d sin θ = nλ. The equation is in AQA's A-level data and formulae booklet.

Rearranging it

• Wavelength: λ = d sin θ ÷ n
• Angle: θ = sin⁻¹(nλ ÷ d)
• Grating spacing: d = nλ ÷ sin θ

Grating spacing from lines per mm

Gratings are labelled with the number of lines (slits) per millimetre, N. The spacing is one over that, in metres:

A grating with 600 lines per mm has d = 1 × 10⁻³ ÷ 600 = 1.67 × 10⁻⁶ m. More lines per mm means a smaller d, so the maxima are spread out at bigger angles.

Where d sin θ = nλ comes from

Take two neighbouring slits a distance d apart, and the light leaving both at an angle θ to the normal. The ray from the lower slit travels further. Drawing a line from the upper slit at right angles to the rays cuts off the extra distance, the path difference, which is d sin θ.

The waves arrive in phase, and reinforce, when that extra distance equals 0, λ, 2λ or any other whole number of wavelengths. So a bright maximum appears wherever d sin θ = nλ. Because every pair of neighbouring slits has the same path difference, all the hundreds of slits reinforce at once at these angles, and cancel almost everywhere else.

Orders and the diffraction pattern

• The zero order, n = 0, goes straight through, at θ = 0.
• The first order, n = 1, appears on both sides of the zero order at the same angle.
• Higher orders appear at bigger angles, again in pairs.

The maximum order

Since sin θ can never be more than 1, nλ cannot be more than d. So the highest order you can see is:

If d ÷ λ = 3.13, the highest order is 3, not 4, and there are 2 × 3 + 1 = 7 maxima in all: the zero order and three on each side.

Why a grating gives sharper maxima than two slits

Two slits produce maxima at the same angles as a grating with the same spacing, because the path difference argument is the same. But with only two slits, the brightness fades gradually between maxima, giving broad fringes. With hundreds of slits, the light cancels almost everywhere except very close to d sin θ = nλ, so each maximum is sharp and bright, and its angle can be measured precisely. You can see the two-slit version in the Wave Interference Simulator: set it to Double slit and watch where the maxima form as you change the wavelength and the separation.

Try it

Wave Interference Simulator

Overlap waves in a ripple tank to see interference and diffraction

Open the simulatorGo to its page

Diffraction grating worked examples

Example 1: angles of the maxima

Green laser light of wavelength 532 nm shines normally on a grating with 600 lines per mm. Find the angles of the first and second order maxima.

First, d = 1 × 10⁻³ ÷ 600 = 1.667 × 10⁻⁶ m. For the first order:

For the second order, sin θ₂ = 2 × 0.3192 = 0.638, so θ₂ = 39.7°. Note that the second-order angle is not double the first: it is sin θ that doubles.

Example 2: the maximum number of maxima

Using the same laser and grating, how many maxima can be seen?

The highest order is 3, so there are 7 maxima: the zero order plus orders 1, 2 and 3 on each side.

Example 3: finding a wavelength

A grating with 250 lines per mm produces a second-order maximum at 16.0°. Find the wavelength.

d = 1 × 10⁻³ ÷ 250 = 4.0 × 10⁻⁶ m, so:

The wavelength is 551 nm, green light.

Example 4: from measurements on a screen

In a practical, a screen is 1.50 m from a grating with 600 lines per mm. The two first-order maxima are each 0.62 m from the central maximum. Find the wavelength.

The angle comes from the right-angled triangle formed by the grating, the central maximum and a first-order maximum:

The wavelength is about 637 nm, red light. Measuring across from one first-order maximum to the other and halving gives a more precise x than measuring from the centre.

Example 5: how much of a spectrum can you see?

White light falls on the grating with 600 lines per mm. Which colours appear in the third-order spectrum?

A third-order maximum needs 3λ ≤ d, so the longest wavelength that can appear is:

Only wavelengths up to about 556 nm appear in the third order: violet, blue and green. Yellow, orange and red would need sin θ greater than 1, so they are missing. The whole second-order spectrum is visible, because even red light at 700 nm needs only sin θ = 2 × 700 × 10⁻⁹ ÷ 1.667 × 10⁻⁶ = 0.84.

White light and spectra

At θ = 0 nothing is separated: all the colours land in the same place, so the central maximum looks white. In every other order, each wavelength is sent to its own angle, so the light spreads into a spectrum. Violet, with the shortest wavelength, is nearest the centre; red, with the longest, is furthest out. This is why gratings are used to study the light from lamps and stars: each element gives out its own wavelengths, which a grating separates into sharp lines that can be measured. AQA's specification lists observing line spectra with a diffraction grating as part of the energy levels topic.

The diffraction grating required practical

AQA's A-level required practical on interference includes Young's slit experiment and interference by a diffraction grating. A typical method:

1. Shine a laser normally onto the grating, with a screen a measured distance D away (a metre or more).
2. Mark the positions of the maxima on the screen.
3. For each order, measure across from the left maximum to the right one, then halve it to get x.
4. Work out θ from tan θ = x ÷ D, then λ = d sin θ ÷ n.
5. Repeat for each order you can see, and average the results.

Never look into the laser beam or at reflections from the grating or screen. Using a larger D makes the distances bigger, which reduces their percentage uncertainty. The Practical Data Lab helps you process readings and their uncertainties.

Interference only gives a bright maximum where the waves arrive in step. Sort some situations into constructive and destructive, then press Mark to check them in the course.

Common mistakes

• Using lines per mm as d. d = 1 ÷ N. 600 lines per mm is d = 1.67 × 10⁻⁶ m, not 600.
• Forgetting to convert mm to m. 1 ÷ 600 gives d in mm; multiply by 10⁻³ to get metres.
• Doubling the angle for the second order. It is sin θ that doubles, not θ.
• Rounding n_max up. If d ÷ λ = 3.9, the highest order is still 3.
• Using the small-angle fringe formula. The double slit formula w = λD ÷ s only works for small angles. Grating angles are large, so use d sin θ = nλ.
• Calculator in radians. Check it is set to degrees.

Practise waves in a short timed paper

A-level Physics: Year 1 gives you a short timed paper every day, with 8 questions worth 25 to 35 marks across the course, so waves and gratings turn up regularly but not in every paper. The Light and Energy course covers diffraction and interference with interactive models.

Read next

Physics · Waves and lightWave simulator: ripple tank, interference and slitsFree wave simulator: a ripple tank with two sources, a double slit or a single slit, showing interference, diffraction, path difference and fringe spacing.