Light as a Wave: Huygens' Principle

Rays are not the whole story

The chapters on ray optics treated light as rays that travel in straight lines. Rays explain shadows, mirrors, lenses and prisms very well. They cannot explain everything. Light that passes through two narrow slits falls on a screen in bands of brightness and darkness. Two rays can only add light to light; they can never add up to darkness. Waves can.

For a century the question was open. In 1678 Christiaan Huygens argued that light is a wave. In his book Opticks of 1704, Isaac Newton argued that light is a stream of tiny particles. Newton's fame kept the particle picture in favour until Thomas Young's experiments with two slits, between 1801 and 1803. That experiment is the subject of the fourth lesson of this chapter.

What kind of wave?

Light is a transverse wave, like a wave on a string, but no material moves. It is an electromagnetic wave: an electric field and a magnetic field that vary together. The two fields point at right angles to each other and to the direction of travel. The unit on Electromagnetism shows how such a wave arises. This chapter needs only two facts about it.

First, the electric field does the work. It is the electric field that acts on the retina of the eye and on the sensor of a phone camera. Second, the wave has a wavelength \(\lambda\), a frequency \(f\) and a phase, like the waves of the chapters on waves. Visible light runs from about \(400\,\text{nm}\), violet, to about \(700\,\text{nm}\), red. In vacuum, and almost exactly in air, every colour travels at \(c = 3 \times 10^8\,\text{m/s}\).

With \(c = f\lambda\), yellow-orange light of \(600\,\text{nm}\) has a frequency of \(3 \times 10^8 / (600 \times 10^{-9})\) \(= 5 \times 10^{14}\,\text{Hz}\). The eye cannot follow anything that fast. It sees only the average brightness, the intensity, which grows as the square of the amplitude of the electric field. That fact will matter in every lesson that follows.

Wavefronts

Every point on one ripple of the pond is at the same stage of its up-and-down motion, all on a crest. A surface on which the wave has the same phase at every point is a wavefront. On the pond the wavefronts are circles. For light, which spreads in three dimensions, the wavefronts are surfaces.

The shape of the wavefronts depends on the source:

• A point source, such as a tiny bulb, sends out spherical wavefronts centred on it.
• A line source, such as a long lit slit or a nearby tube light, sends out cylindrical wavefronts about the line.
• A very distant source, such as the Sun or a star, gives plane wavefronts. A small piece of a sphere of enormous radius is flat.

A wave travels at right angles to its wavefronts. A ray is a line drawn along that direction, normal to the wavefronts at every point. Spherical wavefronts go with rays that spread out from the source. Plane wavefronts go with parallel rays. So ray optics and wave optics describe the same light. The ray tells where the light goes; the wavefront shows how its phase is laid out.

Huygens' construction

Huygens asked a sharp question. If you know a wavefront now, how do you find it a moment later? His answer used the leaf on the pond. A disturbance that reaches any point makes that point a new source. So every point on a wavefront sends out small waves of its own, called secondary wavelets.

In a time \(t\), each wavelet grows to a sphere of radius \(vt\), where \(v\) is the speed of light in the medium. Fig 1.1 shows the construction for the two commonest shapes. Play it with each chip.

In the plane chip of Fig 1.1, all the wavelets have the same radius, so their envelope is a plane a distance \(vt\) ahead. In the spherical chip, the wavelets start on a sphere of radius \(R\) about S. Their envelope is a sphere of radius \(R + vt\) about S. The construction carries the shape of the wavefront forward. It will also turn a wavefront, wherever part of it is slowed down.

A wavefront meets a surface

Now let a plane wavefront AB arrive at a flat surface at an angle. The end A touches the surface first. The end B still has a distance BC to travel before it reaches the surface at C. In air, at speed \(v_1\), that takes a time \(t = BC/v_1\). During that time, the wavelet from A has been growing.

If the second medium is glass, the wavelet from A grows inside the glass at the smaller speed \(v_2\). In the time \(t\) it reaches a radius \(AD = v_2 t\). The new wavefront in the glass is the tangent from C to this wavelet, the line CD. Play Fig 1.2 and watch the front turn.

The two right triangles in Fig 1.2 give the law of refraction at once. Each shares the side AC. The angle between the wavefront and the surface equals the angle between the ray and the normal. Each pair is turned from the other by \(90°\).

This is Snell's law, and now it comes with a reason. The ratio of the sines is a ratio of speeds, so the refractive index measures how much light slows down. The wave picture predicts that light is slower in glass and water. Newton's particles, to bend towards the normal, would have to go faster. The measurement settled it, as the story below tells.

For reflection, the wavelet from A grows in the same medium, at the same speed \(v_1\). So in the time \(t\), \(AD = BC = v_1 t\). The right triangles ABC and CDA share the side AC and have \(AD = BC\), so they are congruent. Their angles at A and at C are equal, and so the angle of reflection equals the angle of incidence.

Speed, wavelength and frequency in glass

The frequency does not change. In Fig 1.2, one crest reaches the surface for each crest that enters the glass, so crests leave the surface at the rate they arrive. The speed falls by the factor \(\mu\), so the wavelength \(\lambda = v/f\) falls by the same factor. Light of \(600\,\text{nm}\) in air has \(f = 5 \times 10^{14}\,\text{Hz}\). In glass of \(\mu = 1.5\) it travels at \(2 \times 10^8\,\text{m/s}\), with a wavelength of \(400\,\text{nm}\) and the same frequency.

A lens turns a wavefront

A plane wavefront that falls on a convex lens meets glass of different thickness at different heights. The lens is thickest at the middle and thin at the edge. Light is slower in the glass, so the middle of the front is held back the most. The edges get ahead, and the front comes out curved inwards: a sphere that closes in on the focus.

The delays are tiny. Take a lens \(4\,\text{mm}\) thick at the centre and thin at the edge, of glass \(\mu = 1.5\). Light through the centre loses \((\mu - 1) \times 4\,\text{mm}\) \(= 2\,\text{mm}\) of air path, a delay of \(2 \times 10^{-3}/(3 \times 10^8)\) \(\approx 6.7 \times 10^{-12}\,\text{s}\). That is a few thousand periods of the wave, enough to curve the front into a sphere. A concave lens, thicker at the edge, holds back the edges and gives a diverging sphere.

Now return to the pond. Each ripple builds the next, because every point it reaches starts a wavelet of its own, like the bobbing leaf. Far from the splash, a short piece of a big circle is almost a straight line, a plane wavefront. Near the bank the water is shallow and the ripples slow down. The part of a ripple that reaches the shallows first is held back. So the ripple turns, exactly as light turns at the surface of glass.