Diffraction Gratings Revision Notes | A Level AQA Physics

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]

1. $N = 400 \text{ lines mm}^{-1} = 400 \times 10^{3} \text{ lines m}^{-1} = 4.00 \times 10^{5} \text{ lines m}^{-1}$ $d = \frac{1}{4.00 \times 10^{5}} = 2.50 \times 10^{-6} \text{ m}$ — Convert lines per mm to lines per metre first, then take the reciprocal. [M1]
2. $\lambda = 589 \text{ nm} = 589 \times 10^{-9} \text{ m} = 5.89 \times 10^{-7} \text{ m}$ — Wavelength must be in metres to match the units of $d$.
3. $2.50 \times 10^{-6} \times \sin\theta = 2 \times 5.89 \times 10^{-7}$ $2.50 \times 10^{-6} \times \sin\theta = 1.178 \times 10^{-6}$ — $n = 2$ for the second order. [M1]
4. $\sin\theta = \frac{1.178 \times 10^{-6}}{2.50 \times 10^{-6}} = 0.4712$ — Divide both sides by $d$.
5. $\theta = \arcsin(0.4712) = 28.1°$ — 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]

1. $N = 600 \times 10^{3} \text{ lines m}^{-1} = 6.00 \times 10^{5} \text{ lines m}^{-1}$ $d = \frac{1}{6.00 \times 10^{5}} = 1.667 \times 10^{-6} \text{ m}$ — Same conversion as before — lines per mm to lines per metre, then reciprocal. [M1]
2. $\lambda = \frac{d\sin\theta}{n}$ — Divide both sides of $d\sin\theta = n\lambda$ by $n$.
3. $\lambda = \frac{1.667 \times 10^{-6} \times \sin(21.0°)}{1}$ $\sin(21.0°) = 0.3584$ $\lambda = 1.667 \times 10^{-6} \times 0.3584 = 5.975 \times 10^{-7} \text{ m}$ — $n = 1$ for the first order. [M1]
4. $\lambda = 5.98 \times 10^{-7} \text{ m} = 598 \text{ nm}$ — 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]

1. $\lambda = 450 \text{ nm} = 450 \times 10^{-9} \text{ m} = 4.50 \times 10^{-7} \text{ m}$ — Units must match.
2. $n_{\max} \leq \frac{d}{\lambda} = \frac{2.00 \times 10^{-6}}{4.50 \times 10^{-7}}$ $= \frac{2.00}{0.450} = 4.44...$ — This is the upper bound on $n$. [M1]
3. $n_{\max} = 4$ — $n$ 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.

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.