10.1 Huygens' Principle and the Laws of Reflection and Refraction

Drop a pebble into a still pond and a ring of ripples spreads out; a little later each ring has grown larger by the same amount everywhere, because every part of it moves at the same speed. Now let the ripples run from deep water into a shallow patch, where water waves travel more slowly. The part of each ring that reaches the shallows first falls behind, the ring kinks, and the whole wave changes direction. That is refraction — and it happens for exactly the same reason to sound entering water, to earthquake waves crossing layers of rock, to surf turning to face a beach, and to light entering glass.

In Chapter 9 we used rays and Snell's law as given rules. In this section we stop treating light as rays and treat it as a wave. Around 1678 the Dutch physicist Christiaan Huygens proposed a simple geometrical rule for how any wave moves forward: every point on a wavefront acts as a source of small secondary wavelets, and a moment later the new wavefront is the surface that touches all of them. With this one idea we will derive the laws of reflection and refraction, show that the refractive index is a ratio of speeds, $n=c/v$, and discover what happens to the frequency and the wavelength of light when it enters glass. Huygens' rule also predicted that light should travel more slowly in water than in air — the opposite of Newton's particle picture — and the experiment that settled the question is one of the turning points of physics.

Wavefronts and Rays

A source of light makes the electric and magnetic fields around it oscillate (Chapter 8). Every point near the source oscillates with the same frequency, but points at different distances lag behind by different amounts. Joining up all the points that are oscillating in step gives a surface.

Three shapes cover almost every case (see Interactive 1):

1. Spherical wavefronts spread from a point source in a uniform medium: the wave reaches every direction equally fast, so points in the same phase lie on a sphere centred on the source. The rays are radial.

2. Cylindrical wavefronts come from a line source such as a long, narrow illuminated slit or a fluorescent tube. Points in the same phase lie on a cylinder round the line.

3. Plane wavefronts are what a spherical wavefront becomes very far from its source. A small part of a huge sphere is almost flat, so light from the Sun or a distant star — or from a point source placed at the focus of a converging lens — arrives as a plane wave with parallel rays.

The energy a point source emits each second is spread over the area $4\pi r^2$ of an ever-larger sphere, so the intensity of a spherical wave falls as $1/r^2$ and, because intensity is proportional to the square of the amplitude, its amplitude falls as $1/r$. For a cylindrical wave the area grows only as $r$, so the intensity falls as $1/r$. A plane wave in a non-absorbing medium keeps its amplitude.

The distance between successive wavefronts that are each at a crest is one wavelength $\lambda$. Two points one behind the other along a ray, a distance $\Delta x$ apart, differ in phase by $$\Delta\phi=\frac{2\pi}{\lambda}\,\Delta x,$$ while all points on one wavefront have phase difference zero. This simple link between path and phase is the key to every interference calculation in Section 10.2.

Huygens' Principle

Knowing the wavefront now, where will it be a short time $\Delta t$ later? Huygens' answer treats each point of the wavefront as if it were itself a little source.

Apply the rule to the three wavefronts. For a plane wavefront, all the wavelets have the same radius $v\Delta t$ and their centres lie on a plane; the surface touching them is another plane, parallel to the first and a distance $v\Delta t$ ahead. The wave moves straight on — rays remain straight lines in a uniform medium. For a spherical wavefront of radius $R$, the envelope is a sphere of radius $R+v\Delta t$ with the same centre: the wave keeps spreading from the same point. If a plane wave meets a barrier with a gap, only the wavelets from points inside the gap survive; their envelope is flat in the middle but curls round at the edges, so the wave leaks a little into the region that ray optics calls the "geometrical shadow". That leakage is diffraction, the subject of Section 10.3.

Refraction of a Plane Wave

Let a plane wavefront travel in medium 1, where the wave speed is $v_1$, towards a flat boundary with medium 2, where the speed is $v_2$. The rays make an angle $i$ with the normal to the boundary, so the wavefront — perpendicular to the rays — makes the same angle $i$ with the boundary itself. One edge of the wavefront, $A$, touches the boundary first; the other edge, $B$, still has a distance $BC$ to go before it reaches the boundary at $C$.

Let the time taken by $B$ to reach $C$ be $\tau$, so $BC=v_1\tau$. During the same time the secondary wavelet sent into medium 2 from $A$ has grown to radius $v_2\tau$. A point of the boundary halfway between $A$ and $C$ is reached halfway through the interval, so its wavelet has had only half as long to grow, and so on: the wavelets shrink steadily to zero radius at $C$. The forward envelope is therefore the straight line from $C$ that just touches the wavelet from $A$, at the point $E$, with $AE=v_2\tau$. That line, $CE$, is the refracted wavefront, and the refracted ray $AE$ is perpendicular to it.

Triangles $ABC$ and $AEC$ are both right-angled (at $B$ and at $E$) and share the hypotenuse $AC$. The angle of the incident wavefront to the boundary is $\angle BAC=i$; the angle of the refracted wavefront to the boundary is $\angle ACE=r$, which is also the angle the refracted ray makes with the normal. So $$\sin i=\frac{BC}{AC}=\frac{v_1\tau}{AC},\qquad \sin r=\frac{AE}{AC}=\frac{v_2\tau}{AC},$$ and dividing, $$\frac{\sin i}{\sin r}=\frac{v_1}{v_2}.$$ The ratio is a constant for the two media, whatever the angle — that is Snell's law, now derived rather than assumed. If $v_2\lt v_1$, then $r\lt i$: the ray bends towards the normal on entering the slower medium.

What happens to frequency and wavelength? In time $\tau$ the incident wavefront moves forward by $v_1\tau$ and the refracted one by $v_2\tau$, and every point of the boundary is set oscillating by the incident wave and passes the same oscillation on to medium 2. The boundary cannot pile up or lose crests, so the number of crests arriving per second equals the number leaving per second: the frequency does not change. Since $v=\nu\lambda$ in each medium, the wavelength changes in proportion to the speed.

Going from a denser to a rarer medium. Now $v_2\gt v_1$: the wavelet from $A$ grows faster than $B$ moves, so $AE\gt BC$ and $r\gt i$ — the ray bends away from the normal. As $i$ grows, $AE=AC\sin i\cdot\dfrac{v_2}{v_1}$ eventually equals $AC$, giving $r=90^\circ$; this happens at the critical angle, $\sin i_c=\dfrac{v_1}{v_2}=\dfrac{n_2}{n_1}$. For larger $i$ the wavelet from $A$ is bigger than $AC$ itself, no line from $C$ can touch it, and there is no refracted wave at all: the light is totally internally reflected, as in Chapter 9 — now explained by the wave picture.

Reflection of a Plane Wave, and Wavefronts through Prisms, Lenses and Mirrors

Huygens' construction works just as well for reflection. Let a plane wavefront $AB$ fall on a plane mirror at angle of incidence $i$, with edge $A$ touching the mirror first. While edge $B$ travels the distance $BC=v\tau$ to the mirror, the wavelet from $A$ spreads back into the same medium, so it has the same speed and reaches radius $AE=v\tau$. The reflected wavefront is the line $CE$ from $C$ touching that wavelet (and all the smaller ones from points between $A$ and $C$).

The right triangles $ABC$ (right angle at $B$) and $CEA$ (right angle at $E$) share the hypotenuse $AC$ and have $BC=AE$. They are congruent, so $\angle BAC=\angle ECA$: the incident and the reflected wavefronts make equal angles with the mirror. Since the rays are perpendicular to the wavefronts, the rays make equal angles with the normal.

Prisms, lenses and mirrors in the wave picture. Huygens' principle gives every result of ray optics a second, very physical meaning: a lens or a mirror works by reshaping wavefronts.

1. A thin prism. A plane wavefront falls on a prism. The part of it that crosses the thick base spends longer in the slow glass than the part that crosses near the thin apex, so it falls behind. The emerging wavefront is still plane, but tilted towards the base: the light is deviated towards the base.

2. A convex lens. The centre of a convex lens is thicker than its edges, so the middle of a plane wavefront is delayed the most. The emerging wavefront is a sphere curving round a point beyond the lens — it converges to the focus.

3. A concave mirror. The edges of a plane wavefront reach the mirror first and are reflected first, so after reflection the wavefront is again a converging sphere, centred on the focus.

In each case every point of the emerging wavefront is in the same phase, so the light takes the same time from one wavefront to the next along every ray. For a lens this means the time from a point object to its image is the same along every ray: the central ray travels a shorter distance but more of it in slow glass; the edge ray travels further but mostly in fast air. That equality of time is why all the rays from the object arrive at the image in step.

Exercises