Light: the Wave Model

In short: The wave model says light travels as a wave. Young's double slit experiment proved it: light from two slits makes bright and dark bands through interference. Diffraction and polarisation also show that light behaves as a transverse wave.

Two old models of light

Newton's fame meant his model was widely accepted for over 100 years. Two experiments ended that:

• Young's double slit (1801) showed interference, which only waves can do.
• Foucault (1850) showed light is slower in water, as Huygens predicted.

Huygens' principle

Huygens said every point on a wavefront acts as a source of tiny new wavelets. The new wavefront is the line that touches all the wavelets. This explains:

• Refraction: wavelets in the slower medium spread less, so the wavefront bends.
• Diffraction: at a narrow gap, the wavelets spread out in all directions beyond it.

Diffraction

Diffraction is the spreading of a wave as it passes through a gap or around an edge. It is strongest when the gap is about the same size as the wavelength. Narrower gap, more spreading.

Light's wavelength is tiny (400 to 700 nm), which is why we rarely notice light diffracting.

Interference and Young's double slit

When two waves meet, they add together:

• Constructive interference: crest meets crest, giving a bright band. This happens when the path difference is a whole number of wavelengths.
• Destructive interference: crest meets trough, giving a dark band. This happens when the path difference is a whole number of wavelengths plus a half.

$$\text{Bright: }d\sin \theta =m\lambda \text{Dark: }d\sin \theta =\left(m+\frac{1}{2}\right)\lambda$$

• $d$ = distance between the slits
• $\theta$ = angle from the centre line to the band
• $m$ = 0, 1, 2, ... (the order of the band)

For small angles, the spacing between neighbouring bright bands on a screen a distance $L$ away is:

$$\Delta x=\frac{\lambda L}{d}$$

The bands spread out more with a longer wavelength (red more than blue), a bigger distance to the screen, or slits that are closer together.

Diffraction gratings

A diffraction grating has thousands of slits per centimetre. It uses the same equation, $d\sin \theta =m\lambda$, but gives much sharper, brighter and more widely spaced bright lines. That makes gratings ideal for measuring wavelengths in spectroscopes.

If a grating has $N$ lines per metre, the slit spacing is $d=\frac{1}{N}$.

Polarisation

Light is a transverse wave: its electric field vibrates at right angles to the direction of travel. In ordinary (unpolarised) light, the field vibrates in every direction across the beam.

A polarising filter only lets through vibrations in one direction. The light that comes out is plane polarised.

• Unpolarised light through one filter: intensity halves.
• Two filters with axes at right angles (crossed): no light gets through.

Only transverse waves can be polarised, so polarisation is strong evidence that light is a transverse wave. Sound, which is longitudinal, cannot be polarised.

Malus's law

When polarised light of intensity $I_{max}$ meets a filter whose axis is at angle $\theta$ to the light's direction of polarisation:

$$I=I_{max}{\cos }^2\theta$$

Quick summary

• Newton modelled light as particles. Huygens modelled it as waves.
• Young's double slit interference and Foucault's slower speed in water supported the wave model.
• Diffraction is the spreading of waves at gaps, strongest when the gap is close to the wavelength.
• Bright bands where $d\sin \theta =m\lambda$. Band spacing on a screen $\Delta x=\frac{\lambda L}{d}$.
• Diffraction gratings give sharp bright lines, used to measure wavelengths.
• Polarisation shows light is transverse. Malus's law: $I=I_{max}{\cos }^2\theta$.

Quiz