Cambridge A‑Level Physics 9702 – Interference, Diffraction & Coherence

1. Superposition – the foundation of wave phenomena

The syllabus treats interference as a direct application of the superposition principle:

• When two or more waves occupy the same region of space, the resultant displacement is the algebraic sum of the individual displacements.

This principle also underlies:

• Stationary (standing) waves – nodes and antinodes arise when two travelling waves of the same frequency move in opposite directions and superpose.
• Diffraction – the bending of waves around obstacles can be described by the superposition of secondary wavelets (Huygens’ principle).
• Polarisation – the vector nature of the electric field means that only components with the same orientation add constructively.

2. Interference of Light

2.1 When does interference occur?

• Coherent sources: a fixed (or slowly varying) phase relationship between the waves.
• Identical (or almost identical) frequency ν (or wavelength λ).
• Comparable amplitudes so that intensity variations are detectable.

2.2 Phase difference and path difference

For two monochromatic waves travelling along paths that differ by \(\Delta r\), the phase difference is

\[ \Delta\phi = \frac{2\pi}{\lambda}\,\Delta r = 2\pi\,u\,\frac{\Delta r}{c}. \]

2.3 Resultant intensity (equal amplitudes)

2.4 Constructive and destructive interference

• Constructive (bright fringe): \(\displaystyle \Delta r = m\lambda \;(m=0,1,2,\dots)\) → \(\Delta\phi = 2m\pi\) → \(I_{\max}=4I_0\).
• Destructive (dark fringe): \(\displaystyle \Delta r = \left(m+\tfrac12\right)\lambda\) → \(\Delta\phi = (2m+1)\pi\) → \(I_{\min}=0\).

2.5 Young’s double‑slit experiment

For small angles (\(\sin\theta\approx \tan\theta \approx y/D\)) the fringe spacing is

\[ \Delta y = \frac{\lambda D}{d}. \]

3. Diffraction

3.1 Why diffraction matters

Diffraction is the spreading of a wave when it encounters an obstacle or aperture comparable in size to its wavelength. In the syllabus it is examined under 8.2 and provides the envelope that modulates any interference pattern.

3.2 Single‑slit diffraction – minima and intensity envelope

• Condition for minima (dark bands): \[ a\sin\theta = m\lambda \qquad (m=1,2,3,\dots) \] where \(a\) is the slit width.
• Derivation (brief): Treat the slit as a series of infinitesimal sources; pairs of points separated by \(\tfrac{a}{2}\) cancel when the path‑difference across the slit equals an integer multiple of \(\lambda\).
• Intensity distribution (envelope): \[ I_{\rm slit}(\theta)=I_0\left[\frac{\sin\!\bigl(\pi a\sin\theta/\lambda\bigr)} {\pi a\sin\theta/\lambda}\right]^{2}. \] This is the familiar \(\operatorname{sinc}^2\) pattern.

Worked example – first‑order minimum

Given a slit width \(a=0.10\;\text{mm}\) illuminated by monochromatic light of wavelength \(\lambda=600\;\text{nm}\), find the angle \(\theta_1\) of the first dark fringe.

\[ a\sin\theta_1 = \lambda \;\Longrightarrow\; \sin\theta_1 = \frac{600\times10^{-9}}{0.10\times10^{-3}} = 6.0\times10^{-3}. \] \[ \theta_1 \approx \sin^{-1}(6.0\times10^{-3}) \approx 0.34^{\circ}. \]

3.3 Double‑slit interference with a single‑slit envelope

The intensity on the screen is the product of the double‑slit interference term and the single‑slit diffraction envelope:

\[ I(\theta)=I_0\; \underbrace{\cos^{2}\!\left(\frac{\pi d\sin\theta}{\lambda}\right)}_{\text{interference}} \; \underbrace{\left[\frac{\sin\!\bigl(\pi a\sin\theta/\lambda\bigr)} {\pi a\sin\theta/\lambda}\right]^{2}}_{\text{diffraction}}. \]

Practical tip

To see distinct interference fringes the slit width \(a\) must be small enough that the diffraction envelope does not wash out the individual bright spots. In practice \(a\lesssim 0.2d\) is a useful rule of thumb for A‑Level investigations.

4. Polarisation (brief note)

• Unpolarised light contains waves with random electric‑field directions.
• A polariser transmits only the component of the field parallel to its transmission axis (Malus’s law: \(I=I_0\cos^{2}\theta\)).
• Interference experiments assume the same polarisation for the two beams; otherwise fringe visibility is reduced.

5. Coherence – ensuring a stable interference pattern

5.1 Temporal coherence

• Definition: correlation of the phase of a wave at the same point but at different times.
• Coherence time \(\displaystyle \tau_c \approx \frac{1}{\Deltau} \approx \frac{\lambda^{2}}{c\,\Delta\lambda}\), where \(\Deltau\) (or \(\Delta\lambda\)) is the spectral width of the source.
• Coherence length \(\displaystyle L_c = c\,\tau_c = \frac{c}{\Deltau} = \frac{\lambda^{2}}{\Delta\lambda}\).
• For visible fringes the path‑difference must satisfy \(\Delta r \lesssim L_c\).

5.2 Spatial coherence

• Definition: correlation of the phase across different points of a wavefront.
• Determined by the size of the source (or aperture) relative to its distance from the interferometer.
• For a source of diameter \(S\) at distance \(D\), the angular coherence condition is \[ \theta \approx \frac{S}{D} \;\lesssim\; \frac{\lambda}{d}, \] where \(d\) is the separation of the two interfering points (e.g., the slits).

5.3 Coherence summary table

Aspect	Temporal coherence	Spatial coherence
What is correlated?	Phase of the same point at different times	Phase at different points across a wavefront
Key parameter	Coherence time \(\tau_c\) / length \(L_c\)	Coherence area / source size \(S\)
Effect on interference	Limits the maximum path‑difference \(\Delta r\)	Limits the angular spread over which fringes stay sharp
Typical sources	Lasers (large \(\tau_c\)), narrow‑band LEDs	Point sources, small apertures, well‑collimated lasers

5.4 Practical implications

• Michelson interferometer: The movable mirror must be positioned so that the optical path difference does not exceed \(L_c\); otherwise fringe contrast drops sharply.
• Double‑slit experiment: Choose slit separation \(d\) and source size \(S\) such that \(\displaystyle \frac{S}{D} \lesssim \frac{\lambda}{d}\). If the source is too large the pattern becomes fuzzy.
• Single‑slit diffraction envelope: A wide slit reduces spatial coherence, broadening the envelope and diminishing the visibility of the finer interference fringes.

6. Summary – key points to remember

• Interference follows directly from the superposition principle and requires coherent waves with a fixed phase relationship.
• Path‑difference conditions:
  • Constructive: \(\Delta r = m\lambda\)
  • Destructive: \(\Delta r = (m+\tfrac12)\lambda\)
• Diffraction provides the \(\operatorname{sinc}^2\) envelope that modulates any interference pattern; the single‑slit minima are given by \(a\sin\theta = m\lambda\).
• Polarisation must be the same for the two beams; otherwise fringe visibility is reduced (Malus’s law).
• Temporal coherence (characterised by \(\tau_c\) and \(L_c\)) limits the allowable path difference; spatial coherence (source size) limits the angular range over which clear fringes are observed.
• Controlling both forms of coherence is essential for successful A‑Level practical investigations such as Young’s double‑slit, Michelson interferometer, and diffraction‑limited imaging.