Wavefronts and Huygens’ principle

Light as a wave: wavefronts

Narrated video. Transcript:

Is light a stream of tiny particles, or a wave? Newton favoured particles, which he called corpuscles. In 1678, the Dutch physicist Christiaan Huygens put forward the wave theory of light.

Both models can explain reflection and refraction. But to explain refraction, the corpuscular model needs light to travel faster in water than in air.

The wave model predicts the opposite: light is slower in water. In 1850, Foucault measured the speed of light in water, and found it slower. The wave model won.

Why, then, does light seem to travel in straight lines? Because its wavelength is tiny, about 0.4 to 0.7 micrometres. That is far smaller than ordinary mirrors and lenses, so we can usually treat light as rays.

Drop a pebble into a still pond. Ripples spread out in circles. Every point on one circle moves up and down in step with every other point on that circle.

A surface on which the wave has the same phase at every point is called a wavefront. The speed at which the wavefront moves outwards is the speed of the wave.

A point source of light sends waves out equally in all directions. Its wavefronts are spheres centred on the source. This is a spherical wavefront.

Far from the source, a small piece of a very large sphere is almost flat. So light from the Sun, or from a distant star, reaches us as a plane wavefront.

The energy of the wave travels perpendicular to the wavefront. Lines drawn in this direction are the rays. For a spherical wave, the rays spread out from the source; for a plane wave, they are parallel.

How flat is flat? A lamp 2 m away lights a lens of radius 2 cm. Across the lens the wavefront bulges by only r²/2R = (0.02)²/4 = 10⁻⁴ m, or 0.1 mm. From the Sun, the bulge is about 10⁻¹⁵ m: perfectly plane.

Huygens’ principle

Narrated video. Transcript:

Suppose we know the shape of a wavefront at time t = 0. Where will it be a short time τ later? Huygens gave a simple geometrical way to find out.

Every point on the wavefront acts as a new source. It sends out secondary wavelets, which spread in all directions with the speed of the wave, v.

From each point, draw a sphere of radius vτ. The surface that touches all these spheres in the forward direction, their forward envelope, is the new wavefront at time τ.

Try it for a point source O. The wavefront is a sphere of radius R. Wavelets of radius vτ from all its points are touched by a sphere of radius R + vτ, still centred on O. The wave stays spherical as it spreads.

For a plane wavefront, the wavelets all touch a plane a distance vτ ahead, parallel to the first. A plane wave stays plane, and its rays stay parallel.

But look behind the wavefront. The wavelets also have a backward envelope: a wave travelling back towards the source. No such backwave is ever observed.

Huygens assumed that the wavelets are strongest in the forward direction and have zero amplitude backwards. This assumption is not satisfactory; a more rigorous wave theory explains why the backwave is absent.

Worked example: a plane wavefront of light in air moves on for τ = 2 nanoseconds. The wavelets grow to vτ = 3 × 10⁸ × 2 × 10⁻⁹ = 0.6 m. So the new wavefront is a plane 0.6 m further on.

In glass of refractive index 1.5, light travels at c/n = 2 × 10⁸ m/s. In the same 2 nanoseconds, the wavelets grow to only 0.4 m. Smaller wavelets in a slower medium: that is the key to refraction.

Try it: build the next wavefront

Interactive simulation. Things to try:

• The orange arc is the wavefront now. Each orange dot is a source of secondary wavelets (blue circles).
• The green arc touches all the wavelets in front: it is the new wavefront. Press “Next wavefront” or Play to repeat.
• Watch the spherical wavefront get flatter as it grows. Then switch to “Plane wave”.
• Tick “Show the backward envelope”: this is the backwave Huygens had to leave out.

Light from a star

Light from a distant star reaches the Earth. What is the shape of the wavefront that arrives, and how are the rays related to it?

1. Spherical; the rays spread out from the Earth
2. Plane; the rays are parallel to each other and perpendicular to the wavefront
3. Plane; the rays lie along the wavefront
4. Cylindrical; the rays are parallel to its axis

Grow the wavefront

A spherical wavefront from a point source has a radius of 30 cm at some instant. Light in the medium travels at 2 × 10⁸ m/s. Using Huygens’ construction, what is the radius of the wavefront 0.5 ns later?

1. 20 cm
2. 30 cm
3. 40 cm
4. 45 cm

Summary

• Radius of a secondary wavelet after time τ: r = vτ
• Spherical wavefront after τ: R → R + vτ
• Speed of light in a medium: v = c/n

• A wavefront is a surface on which the wave has the same phase everywhere. A point source gives spherical wavefronts; far away they become plane.
• Rays are perpendicular to the wavefronts: they show the direction in which energy travels.
• Huygens’ principle: every point of a wavefront is a source of secondary wavelets moving at the wave speed; their forward envelope is the new wavefront.
• The backward envelope is ignored: no backwave is observed.
• The wave model predicts that light is slower in a denser medium, as Foucault found for water.