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Interference and Diffraction

Interference and Diffraction — American System Physics Revision Notes & Topical Questions

5 min read 6 parts Lesson 1 of 5 1 quiz

📚 American System Physics Hub / Topical Revision
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Lesson summary

Interference and Diffraction of Light — AMERICAN SYSTEM PHYSICS notes: Explain superposition, double-slit, thin-film, single-slit and grating patterns and use the related equations and the Rayleigh criterion.By the principle of superposition, overlapping waves add.

  • Coherent sources
  • fringe spacing
  • Huygens' principle
  • Worked example 1
  • Worked example 2
  • Worked example 3

Part 1 of 4Superposition and double-slit interference

By the principle of superposition, overlapping waves add. Coherent sources emit waves of the same frequency and wavelength with a constant phase difference, needed for a stable pattern; lasers provide coherent light.

Fringe spacing
The fringe spacing x is the distance between centres of adjacent bright (or dark) fringes.
For a double slit with slit separation d and screen distance L, λ = xd/L (red fringes are wider apart than blue).
Bright fringes occur where path difference = mλ (m = 0, 1, 2 ...); also d sin θ = mλ.
FeatureConstructiveDestructive
Phase relationin phase (0°, 360° ...)antiphase (180°)
Path difference0, λ, 2λ ... = mλ½λ, 1½λ ... = (m + ½)λ
Resultant amplitudesum (maximum)difference (zero if equal)
Lightbright fringedark fringe
Soundloudquiet

Thin film

  • when the film index is higher than its surroundings, reflection is bright when 2t = (m + ½)λfilm, where λfilm = λ/n;
    • there is a phase change of ½λ on reflection from a higher-index medium.
  • Holograms rely on interference.

Part 2 of 4Diffraction at a single slit and resolution

Diffraction is the spreading of waves as they pass through a gap (aperture, width w) or around an edge; dust on lenses can produce diffraction patterns, and diffraction shows light is a wave. Huygens' principle: every point on a wavefront is a source of secondary wavelets, and the new wavefront is the surface tangent to all of them.

Diffraction at a single slit and resolution
The effect is largest when the gap is close to the wavelength; a narrower gap gives more diffraction.
The wavelength and speed do not change; only the direction of travel spreads, so diffraction is not refraction.
The single-slit pattern has a central maximum twice as wide and much brighter than the side maxima, with minima where w sin θ = mλ.

Diffraction limits resolution. By the Rayleigh criterion, two point sources are just resolved when the central maximum of one pattern falls on the first minimum of the other; for a circular aperture of diameter D, θ = 1.22λ/D. Resolving power is the ability to distinguish two close objects. A ripple tank can demonstrate wave diffraction.

Part 3 of 4Diffraction gratings

A grating works like many slits at once.

1
Grating spacing d = 1/N, where N is the lines per metre (500 lines/mm = 5.0 × 105 lines/m, d = 2.0 × 10−6 m).
2
Only at angles where the path difference between neighbouring slits (d sin θ) is a whole number of wavelengths do all waves reinforce,
  • giving the condition d sin θ = mλ and very sharp, bright maxima.
3
A finer grating (smaller d) gives larger angles and fewer orders.
4
With white light the zero order is white (all wavelengths reinforce at θ = 0) and every other order is a spectrum with violet nearest the centre and red farthest out; higher orders may overlap.
5
Gratings measure wavelengths to high precision.
6
An emission line spectrum is a set of bright lines at specific wavelengths from a hot, low-pressure gas, unique to each element.
7
Polarization restricts the vibrations of a transverse wave to one plane, a related property showing light is a transverse wave.

Part 4 of 4Worked examples

  1. Laser light passes through two slits 0.20 mm apart. Fringes on a screen 3.0 m away are 9.5 mm apart. Calculate the wavelength. [3]3 marks

  2. Green light (λ = 5.4 × 10−7 m) passes through slits 0.30 mm apart. Find the fringe spacing on a screen 2.5 m away. [3]3 marks

  3. Two coherent speakers emit sound of wavelength 0.68 m. At P the path difference is 1.02 m. Is P loud or quiet? [2]2 marks

  4. A single slit (w = 0.12 mm) is lit with 600 nm light and the screen is 2.0 m away. How wide is the central maximum? [3]3 marks

  5. A telescope has aperture 0.10 m. Find the smallest angle it resolves for 550 nm light. [2]2 marks

  6. A grating has 400 lines per mm. Find its spacing d. For a grating with d = 2.0 × 10−6 m and λ = 6.0 × 10−7 m, what is the highest visible order? [3]3 marks

Key terms

TermMeaning
Coherent sourcesSources with the same frequency and a constant phase difference.
Fringe spacingDistance between centres of adjacent bright (or dark) fringes.
Huygens' principleEvery point on a wavefront is a source of secondary wavelets; the new wavefront is tangent to them.
Rayleigh criterionTwo sources are just resolved when one central maximum falls on the other's first minimum; θ = 1.22λ/D.
Grating spacing (d)Distance between adjacent slits, d = 1/(lines per metre).
Emission line spectrumBright lines at specific wavelengths from a hot, low-pressure gas, characteristic of the element.
PolarizationRestriction of a transverse wave's vibrations to one plane.

Avoid these mistakes

1
Saying the wavelength grows after a gap — it stays the same; only direction spreads.
2
Saying a bigger gap gives more diffraction — a narrower gap, closer to the wavelength, does.
3
Treating all single-slit bright bands as equal — the central maximum is twice as wide and much brighter.
4
Confusing diffraction with refraction — refraction changes speed; diffraction does not.
5
Forgetting that the zero order with white light is white.

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