Diffraction
Objective
To explain the meaning of diffraction, derive the key mathematical relations for single‑slit and grating diffraction, and show how diffraction limits the resolution of optical instruments – covering Cambridge AS & A‑Level Physics (9702) syllabus sections 8.2 & 8.4.
1. Definition and Physical Origin
- Diffraction is the bending and spreading of a wave‑front when it encounters an obstacle or aperture whose dimensions are comparable to the wavelength λ.
- It is a direct consequence of Huygens’ principle: every point on a wave‑front acts as a source of secondary spherical wavelets; the superposition of these wavelets produces the observed pattern.
- The effect is most noticeable when the ratio λ / size of the aperture is not << 1. Hence longer wavelengths and smaller apertures give stronger diffraction.
- In the syllabus the term “diffraction” usually refers to the **Fraunhofer (far‑field) condition**, i.e. the observation screen is at a distance L ≫ ²/λ, so that the rays reaching the screen are essentially parallel.
2. Single‑Slit Diffraction – Derivation of the Minima Condition
Consider a monochromatic plane wave of wavelength λ incident normally on a slit of width . Divide the slit into 2m equally spaced narrow strips (Fig. 1). For a ray making an angle θ with the normal, the path‑difference between the top and bottom of the slit is
$$\Delta = a\sin\theta$$
**Pair‑wise cancellation argument** – each strip in the top half can be paired with a strip in the bottom half that is a distance below it. The two contributions have a phase difference of
$$\delta = \frac{2\pi}{\lambda}\,\frac{a}{2m}\sin\theta = \frac{\pi}{m}\,a\sin\theta/\lambda.$$
When the total path‑difference equals an integer multiple of the wavelength, i.e.
$$a\sin\theta = m\lambda \qquad (m = \pm1,\pm2,\dots)$$
the phase difference between each pair is π, causing complete destructive interference. This is the **condition for minima** in the single‑slit diffraction pattern.
2.1 Angular Width of the Central Maximum and the Rayleigh Criterion
- The first minima occur for m = ±1. Using the small‑angle approximation (sin θ ≈ θ in radians), the half‑angular width of the central bright fringe is
$$\theta_{1} \approx \frac{\lambda}{a}$$
- Therefore the **full angular width** of the central maximum is approximately
$$\Delta\theta_{\text{central}} \approx \frac{2\lambda}{a}$$
- For a circular aperture of diameter D the diffraction pattern is an Airy disc. The first zero of the Bessel function occurs at 1.22 λ/D, giving the **Rayleigh resolution limit**
$$\theta_{\text{Rayleigh}} \approx 1.22\,\frac{\lambda}{D}$$
The factor 1.22 arises from the first zero of the J₁ Bessel function that describes the intensity distribution of a circular aperture.
3. Intensity Distribution (Fraunhofer Approximation)
The intensity as a function of angle θ is
$$I(\theta)=I_{0}\left(\frac{\sin\beta}{\beta}\right)^{2},\qquad
\beta=\frac{\pi a\sin\theta}{\lambda}$$
where I₀ is the maximum intensity on the central axis (θ = 0). The pattern consists of a bright central maximum flanked by weaker side fringes whose positions are given by the minima condition above.
4. Experimental Demonstration – Single‑Slit Set‑up (Fraunhofer)
- Mount a low‑power laser (≈ 5 mW, red λ ≈ 650 nm) on an optical bench.
- Place an adjustable single slit (micrometer‑controlled) a few centimetres from the laser so that the beam is perpendicular to the slit.
- Position a screen (or a calibrated photodiode on a translation stage) at a distance L ≈ 1–2 m (satisfying L ≫ a²/λ).
- Measure the lateral positions yₘ of the first‑order minima on either side of the centre.
For small angles, sin θ ≈ y/L, so the slit width can be calculated from
$$a = \frac{m\lambda L}{y_{m}}$$
Sample Paper 5 Question
A laser of wavelength 632.8 nm illuminates a single slit. The first‑order minima are observed 3.2 cm either side of the central maximum on a screen 1.50 m away. Calculate the slit width.
Solution
- y₁ = 3.2 cm = 0.032 m, L = 1.50 m, m = 1
- $$a = \frac{\lambda L}{y_{1}} = \frac{632.8\times10^{-9}\times1.50}{0.032}
\approx 2.97\times10^{-5}\,\text{m}=29.7\,\mu\text{m}$$
5. Diffraction Grating (Syllabus 8.4)
A diffraction grating consists of many equally spaced parallel slits (or ruled lines). For Fraunhofer diffraction the condition for **principal maxima** is
$$d\sin\theta = n\lambda \qquad (n=0,\pm1,\pm2,\dots)$$
where d is the grating spacing (inverse of the line density). The angular separation of the maxima is much larger than for a single slit, making gratings ideal for wavelength measurement.
Worked Example
A grating with 500 lines mm⁻¹ is illuminated by white light. The first‑order (n = 1) maximum for the yellow component (λ ≈ 580 nm) appears at an angle of 20.5°. Find the line density of the grating.
First calculate the spacing:
$$d = \frac{n\lambda}{\sin\theta}= \frac{1\times580\times10^{-9}}{\sin20.5^{\circ}}
\approx 1.68\times10^{-6}\,\text{m}$$
Line density = 1/d ≈ 595 lines mm⁻¹, confirming the specification.
6. Diffraction vs Interference – Quick Comparison
| Aspect |
Diffraction |
Interference |
| Origin of pattern |
Wave‑front bending around a single aperture or obstacle (Huygens’ wavelets) |
Superposition of waves from two or more coherent point sources |
| Typical condition |
Feature size ≈ λ (e.g., slit width, aperture diameter) |
Path‑difference = nλ (n integer) |
| Pattern |
Broad central maximum with decreasing side fringes (single slit) or sharp orders (grating) |
Equally spaced bright and dark fringes |
| Wavelength dependence |
More pronounced for longer λ; angular width ∝ λ/size |
Fringe spacing ∝ λ |
7. Practical Activity Checklist (AO3)
- Equipment: laser pointer, micrometer‑adjustable single slit, diffraction grating (optional), screen or photodiode, ruler or vernier, laser‑safety goggles.
- Safety
- Never look directly into the laser beam.
- Wear appropriate laser‑safety goggles.
- Secure all optics on the bench to avoid accidental displacement.
- Procedure
- Align the laser so the beam is perpendicular to the slit.
- Record the slit width set on the micrometer.
- Place the screen at a measured distance L; note the positions of the first‑order minima (±y₁).
- Calculate the experimental slit width using a = λL/y₁ and compare with the micrometer reading.
- Repeat with a diffraction grating to determine the wavelength of a known laser or the grating line density.
- Data recording tips
- Use a table with columns for L, y₁ (both sides), average y, calculated a, % error.
- Include measurement uncertainties (e.g., ±0.5 mm on y, ±1 cm on L) and propagate them to the final result.
8. Cross‑Reference Box
Diffraction concepts re‑appear throughout the syllabus:
- 7 Superposition – Huygens’ principle is an application of superposition of secondary wavelets.
- 8.3 Interference – Multi‑slit interference patterns are the product of diffraction from each slit combined with their mutual interference.
- 20 Electromagnetic Waves – Diffraction of radio waves around buildings illustrates the same principle at much longer λ.
- 21 Astronomy & Cosmology – The diffraction limit (Rayleigh criterion) determines the resolving power of telescopes and the size of the Airy disc in stellar imaging.
9. Everyday Examples
- Faint shadows behind a hair when illuminated by a laser pointer.
- Sound spreading around a doorway – you can still hear someone even when the source is not directly visible.
- Bright edge patterns (Fresnel fringes) seen on the rim of a razor blade in a darkened room.
10. Why Diffraction Matters in A‑Level Physics
- Provides a quantitative test of the wave nature of light.
- Underpins the design and performance limits of optical instruments (microscopes, telescopes, spectrometers).
- Enables students to solve typical exam problems on slit width, grating spacing, and resolving power, meeting AO2–AO4 outcomes.
11. Summary
Diffraction is the bending and spreading of a wave‑front when it meets an aperture or obstacle comparable in size to its wavelength. Using Huygens’ principle and the Fraunhofer (far‑field) approximation we obtain:
- Minima condition for a single slit: a sin θ = mλ.
- Intensity distribution: I(θ)=I₀(sin β/β)² with β=πa sinθ/λ.
- Full width of the central maximum ≈ 2λ/a; for a circular aperture the Rayleigh limit is θ≈1.22 λ/D.
- Grating equation: d sin θ = nλ, giving large angular separation of orders.
Mastery of these concepts, together with practical experimental skills, equips students to meet all required outcomes of the Cambridge AS & A‑Level Physics syllabus.