Diffraction Gratings Revision Notes | A Level AQA Physics
The diffraction grating equation is d sin θ = nλ, where d is grating spacing, n the order and λ the wavelength. Free AQA notes and 14 questions answered.
Updated 12 Sept 2026 · Revise Right Team
Diffraction Gratings is part of Refraction, diffraction and interference in A Level AQA Physics.
7 min
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What this lesson covers
The diffraction grating equation is d sin θ = nλ, where d is grating spacing, n the order and λ the wavelength. Free AQA notes and 14 questions answered.
Describe how a plane transmission diffraction grating produces a pattern of maxima at normal incidence
Derive the grating equation $d\sin\theta = n\lambda$ from the condition for constructive interference
Calculate the angle, order, or wavelength using $d\sin\theta = n\lambda$
Explain why diffraction gratings produce sharper, more widely spaced maxima than a double slit
Key ideas to keep in view
Use this lesson to stay inside Refraction, diffraction and interference while connecting the detail back to the wider Waves topic.
Diffraction grating: An optical component with a large number of equally spaced parallel slits that diffracts incident light into a series of sharp intensity maxima at specific angles. Used when precise wavelength measurement or spectral analysis is required.
Order of diffraction (n): An integer (0, 1, 2,...) that labels each principal maximum produced by a diffraction grating. The zeroth order (n = 0) is the straight-through beam; higher orders appear at increasing angles on either side.
Grating spacing (d): The centre-to-centre distance between adjacent slits in a diffraction grating, measured in metres. Calculated from the number of slits per metre: $d = 1/N$ where $N$ is the number of slits per metre.
Principal maximum: A direction of constructive interference produced by a diffraction grating, occurring when the path difference between adjacent slits is an exact integer number of wavelengths. Appears as a bright, sharp fringe.
Revision notes
Read the core explanations from Diffraction Gratings before testing yourself from memory.
1. Why Diffraction Gratings Matter
A double slit gives you two sources of coherent light — and that produces an interference pattern. Now imagine hundreds or thousands of slits, all equally spaced. The result is something far more powerful: a diffraction grating that splits light into razor-sharp beams at precise, measurable angles.
That precision is what makes diffraction gratings indispensable in science. Astronomers use them to identify elements in distant stars. Physicists use them to measure wavelengths to extraordinary accuracy. The underlying physics is the same constructive interference you already know from Young's double-slit experiment — scaled up dramatically.
2. How a Diffraction Grating Works
Picture a diffraction grating held perpendicular to an incoming beam of monochromatic light — this is called normal incidence, meaning the light hits the grating at 90° to its surface. Each slit acts as a new source of secondary wavelets (Huygens' principle). These wavelets spread out and overlap, and in certain directions they interfere constructively to produce bright maxima.
The key difference from a double slit is the sheer number of slits. With only two slits, the maxima are broad and the minima between them are not very dark. With thousands of slits, each maximum becomes extremely narrow and intense, because any slight deviation from the constructive-interference angle causes destructive interference from the many other slits. The pattern is sharper and the maxima are more widely separated — both features that make precise measurement possible.
The grating is described by the number of slits (or lines) per millimetre, often written as, for example, 600 lines mm$^{-1}$. To use the grating equation you need the grating spacing $d$ — the distance between adjacent slits — which you obtain by taking the reciprocal: $d = 1/N$, where $N$ is the number of slits per metre. Always convert lines per mm to lines per metre before calculating $d$.
3. Deriving the Grating Equation
The derivation starts with two adjacent slits separated by distance $d$. When light arrives at normal incidence and diffracts at angle $\theta$ to the normal, the wavefront from the lower slit travels an extra distance compared with the wavefront from the upper slit before they meet at a distant screen. That extra distance is the path difference.
Draw a perpendicular from the upper slit to the ray from the lower slit. The right-angled triangle formed has hypotenuse $d$ (the slit separation) and the path difference as the opposite side. Trigonometry gives: path difference $= d\sin\theta$.
Constructive interference occurs whenever the path difference equals a whole number of wavelengths. Setting $d\sin\theta = n\lambda$, where $n$ is any integer, gives the condition for a principal maximum of order $n$. This argument applies equally to every adjacent pair of slits across the entire grating, which is why all slits reinforce simultaneously at the same angle — producing the sharp, bright maxima characteristic of a grating.
Notice that $\sin\theta$ can never exceed 1. This places a hard limit on the number of orders that can exist: the maximum order is the largest integer $n$ for which $n\lambda \leq d$. Any value of $n$ that would require $\sin\theta > 1$ is physically impossible — that order does not appear.
4. Using the Grating Equation — Worked Examples
The equation $d\sin\theta = n\lambda$ connects four quantities. In an exam you will be given three and asked to find the fourth. The worked examples below cover the two most common question types: finding an angle and finding a wavelength. Work through each step before revealing the next — this is where the marks are won.
Worked example: Finding the angle of a diffraction maximum
A diffraction grating has 400 lines mm$^{-1}$. Light of wavelength 589 nm is incident normally. Calculate the angle of the second-order maximum. [3 marks]
$N = 400 \text{ lines mm}^{-1} = 400 \times 10^{3} \text{ lines m}^{-1} = 4.00 \times 10^{5} \text{ lines m}^{-1}$
$d = \frac{1}{4.00 \times 10^{5}} = 2.50 \times 10^{-6} \text{ m}$ — Convert lines per mm to lines per metre first, then take the reciprocal. [M1]
$\lambda = 589 \text{ nm} = 589 \times 10^{-9} \text{ m} = 5.89 \times 10^{-7} \text{ m}$ — Wavelength must be in metres to match the units of $d$.
$2.50 \times 10^{-6} \times \sin\theta = 2 \times 5.89 \times 10^{-7}$
$2.50 \times 10^{-6} \times \sin\theta = 1.178 \times 10^{-6}$ — $n = 2$ for the second order. [M1]
$\sin\theta = \frac{1.178 \times 10^{-6}}{2.50 \times 10^{-6}} = 0.4712$ — Divide both sides by $d$.
$\theta = \arcsin(0.4712) = 28.1°$ — Use the inverse sine function. Give the answer to 3 significant figures to match the data. [A1]
Answer: $\theta = 28.1°$
Worked example: Finding the wavelength from a measured angle
A grating with 600 lines mm$^{-1}$ produces a first-order maximum at 21.0° when illuminated normally. Calculate the wavelength of the light. [3 marks]
$N = 600 \times 10^{3} \text{ lines m}^{-1} = 6.00 \times 10^{5} \text{ lines m}^{-1}$
$d = \frac{1}{6.00 \times 10^{5}} = 1.667 \times 10^{-6} \text{ m}$ — Same conversion as before — lines per mm to lines per metre, then reciprocal. [M1]
$\lambda = \frac{d\sin\theta}{n}$ — Divide both sides of $d\sin\theta = n\lambda$ by $n$.
$\lambda = 5.98 \times 10^{-7} \text{ m} = 598 \text{ nm}$ — Converting to nm makes the answer easier to interpret — visible light runs from roughly 400 nm to 700 nm, so 598 nm (orange-red) is physically sensible. [A1]
Answer: $\lambda \approx 598$ nm
5. Maximum Number of Orders
Because $\sin\theta \leq 1$, the grating equation sets an upper limit on $n$. Rearranging: $n \leq d/\lambda$. The maximum order is the largest integer that satisfies this inequality. This is a common exam calculation — and a common source of errors.
Worked example: Finding the maximum order of diffraction
A grating has a spacing of $2.00 \times 10^{-6}$ m. Light of wavelength 450 nm is used. Find the highest order maximum that can be observed. [2 marks]
$n_{\max} \leq \frac{d}{\lambda} = \frac{2.00 \times 10^{-6}}{4.50 \times 10^{-7}}$
$= \frac{2.00}{0.450} = 4.44...$ — This is the upper bound on $n$. [M1]
$n_{\max} = 4$ — $n$ must be a whole number and $\sin\theta$ cannot exceed 1, so we round down — never up. [A1]
Answer: The highest observable order is $n = 4$.
6. Comparing the Grating with the Double Slit
You already know that Young's double-slit experiment produces an interference pattern. The diffraction grating produces the same type of pattern — but with two important differences that are worth being able to explain and compare in an exam.
Feature
Double slit
Diffraction grating
Number of slits
2
Hundreds to thousands
Width of maxima
Broad, spread out
Narrow, sharp
Intensity of maxima
Relatively dim
Bright (energy concentrated)
Angular separation of orders
Small (slits close together)
Larger (can be designed for wide separation)
Precision of wavelength measurement
Low
High
The reason the maxima are sharper with more slits is rooted in the interference condition. With only two slits, there is a gradual transition from constructive to destructive interference as you move away from a maximum. With thousands of slits, even a tiny deviation from the exact constructive-interference angle means that contributions from slits spread across the grating cancel almost completely — so the maximum collapses to near-zero intensity very rapidly on either side.
7. Applications of Diffraction Gratings
The ability to separate light into its component wavelengths with high precision makes diffraction gratings central to a wide range of scientific instruments. Each application exploits the fact that different wavelengths diffract at different angles, so a mixed-wavelength source produces a spectrum rather than a single bright spot.
Astronomical spectroscopy: Telescopes fitted with diffraction gratings analyse the spectra of stars and galaxies. The precise wavelengths of absorption or emission lines identify which elements are present and reveal the star's temperature, composition, and radial velocity (via Doppler shift).
Atomic emission spectroscopy: In the laboratory, diffraction gratings are used to measure the emission spectra of elements. Each element produces a unique set of spectral lines, acting as a fingerprint for identification.
Laser technology: Gratings are used to select specific wavelengths within laser cavities and to separate or combine laser beams in optical communications.
X-ray crystallography: Crystal planes act as a natural diffraction grating for X-rays (Bragg diffraction). This technique revealed the structure of DNA and is used to determine the atomic structure of materials.
Telecommunications: Wavelength-division multiplexing (WDM) uses gratings to combine and separate different wavelength channels in optical fibre networks, dramatically increasing data-carrying capacity.
A key point for the exam: in all these applications, the grating is valuable precisely because it produces sharp, well-separated maxima at angles that depend on wavelength. A broader, less precise instrument (such as a prism) cannot resolve closely spaced spectral lines as effectively.
8. Exam Technique and Common Mistakes
Diffraction grating questions appear regularly on AQA papers and follow predictable patterns. The marks are straightforward to collect if you avoid the traps below — and easy to lose if you do not.
For a 'describe the pattern' question, a complete answer names the central maximum at $\theta = 0$, states that first and second (and higher) order maxima appear symmetrically on both sides, notes that the angle increases with order and with wavelength, and mentions that the maxima are sharp and bright. Each of those points corresponds to a mark.
Bringing it all together: the diffraction grating equation $d\sin\theta = n\lambda$ is derived from the path difference condition for constructive interference between adjacent slits, and it underpins both the precise measurement of wavelengths and the wide range of spectroscopic applications that make diffraction gratings one of the most important tools in modern physics and astronomy.
Sources and content standards
Published by Revise Right. Use the official specification reference to check the requirements for your course. Content updated: 2026-09-12.
The diffraction grating equation is d sin θ = nλ, where d is the grating spacing, θ is the angle of the maximum from the normal, n is the order and λ is the wavelength. It comes from the condition for constructive interference: the path difference between adjacent slits, d sin θ, equals a whole number of wavelengths.
What are diffraction gratings used for?
Diffraction gratings are used in astronomical spectroscopy to identify elements in stars, in atomic emission spectroscopy to measure the spectral lines of elements, in laser technology, in X-ray crystallography and in telecommunications. Each use relies on different wavelengths diffracting at different angles into sharp, well-separated maxima.
Why does a diffraction grating give sharper maxima than a double slit?
A diffraction grating gives sharper, more intense maxima than a double slit because it has hundreds or thousands of slits. Any slight deviation from the exact constructive-interference angle means contributions from slits spread across the grating cancel almost completely, so each maximum is narrow and intense, and the orders are more widely separated.
How do you calculate grating spacing from lines per mm?
To find the grating spacing d, convert the number of lines per mm to lines per metre, then take the reciprocal: d = 1/N. For a grating with 400 lines mm⁻¹, N = 4.00 × 10⁵ lines m⁻¹, so d = 1 ÷ 4.00 × 10⁵ = 2.50 × 10⁻⁶ m.
How do you find the maximum order of a diffraction grating?
The maximum order is the largest whole number n for which n ≤ d/λ, because sin θ can never exceed 1. For d = 2.00 × 10⁻⁶ m and light of 450 nm (4.50 × 10⁻⁷ m), d/λ = 4.44, so the highest observable order is n = 4. Always round down, never up.
How do you derive the diffraction grating equation?
The grating equation comes from two adjacent slits separated by d. Light diffracted at angle θ from one slit travels an extra distance d sin θ compared with the next, and constructive interference needs this path difference to equal a whole number of wavelengths, so d sin θ = nλ for a maximum of order n.
What is normal incidence?
Normal incidence is when the incident light beam travels perpendicular to the plane of a diffraction grating, along the normal, arriving at 90° to its surface. The grating equation d sin θ = nλ is derived and applied under this condition.
How do you calculate the angle of a diffraction grating maximum?
The angle of a grating maximum comes from sin θ = nλ/d. For 400 lines mm⁻¹, d = 2.50 × 10⁻⁶ m; with 589 nm light at second order, sin θ = 2 × 5.89 × 10⁻⁷ ÷ 2.50 × 10⁻⁶ = 0.4712, so θ = 28.1°.
Where is the central maximum of a diffraction grating?
The central maximum of a diffraction grating is at θ = 0, straight ahead of the grating. First, second and higher order maxima appear symmetrically on both sides, at angles that increase with order and with wavelength, and all the maxima are sharp and bright.
Why use a diffraction grating instead of a prism?
A diffraction grating is better than a prism for analysing light because it produces sharp, well-separated maxima at angles that depend on wavelength. A broader, less precise instrument such as a prism cannot resolve closely spaced spectral lines as effectively.
How is X-ray crystallography linked to diffraction gratings?
X-ray crystallography uses crystal planes as a natural diffraction grating for X-rays, known as Bragg diffraction. The technique revealed the structure of DNA and is used to determine the atomic structure of materials.
What happens when white light passes through a diffraction grating?
A mixed-wavelength source such as white light produces a spectrum through a diffraction grating rather than a single bright spot, because different wavelengths diffract at different angles. Since sin θ = nλ/d, longer wavelengths appear at larger angles in each order.
What are the common mistakes in diffraction grating calculations?
The most common diffraction grating errors are not converting lines per mm to lines per metre before taking the reciprocal for d, not converting wavelength from nm to metres, and rounding the maximum order up instead of down.
How does wavelength affect the angle of a diffraction grating maximum?
The angle of a diffraction grating maximum increases with wavelength and with order, since sin θ = nλ/d. Longer wavelengths and higher orders therefore appear at larger angles from the central maximum at θ = 0.