Refraction

Dip a pencil into a glass of water and it seems to snap at the surface. The pencil hasn’t changed; the light coming from it has. When light passes from one transparent material into another, it changes direction. This bending is called refraction, and it is how every lens works, from a pair of reading glasses to the front of your own eye.

#Light slows down in glass

In empty space light travels at about 300 000 kilometres per second, a speed usually called $c$. In air it is almost as fast. In water it travels at only three-quarters of that speed, and in glass at about two-thirds. The ratio of the speed in a vacuum to the speed in a material is the material’s refractive index:

Here are flat wavefronts travelling from air into glass. The slider changes the refractive index of the glass.

The waves arrive at the surface at a steady rate, and they must leave it into the glass at the same rate. If they travel more slowly inside, they have to crowd together: in glass of index 1.5 the wavefronts are only two-thirds as far apart as in air. The colour of light is set by how often its waves arrive, which doesn’t change, so the light inside the glass is still the same colour.

#Why light bends

Now let the light meet the glass at an angle. One end of each wavefront reaches the glass first and slows down, while the rest is still moving at full speed through the air. The wavefront swings round, like a line of people walking arm in arm from pavement onto sand: whoever reaches the sand first slows down, and the whole line wheels toward them.

The rays, always at right angles to the wavefronts, bend toward the normal, the line at right angles to the surface. We measure each ray’s direction by its angle to the normal: θ₁ before the surface, the angle of incidence, and θ₂ after it. Light that meets the surface head-on doesn’t bend at all. The more obliquely it arrives, the more it bends.

#Snell’s law

The amount of bending follows from the geometry of the swinging wavefront. While one end of a wavefront covers a given distance in air, the other covers only $1/n$ of that distance in the glass, and that ties the two angles together:

This is Snell’s law. Drag the start of the ray in the figure below to change its angle, and try a few materials.

Light entering a material of higher refractive index bends toward the normal, and the bigger the jump in index, the more it bends. Notice also the faint ray reflected back into the air. Every surface reflects a little light: about 4% at ordinary glass when the light meets it head-on, around 7% at high-index plastic, and much more at grazing angles. That is why good spectacle lenses carry anti-reflection coatings.

For the small angles we usually meet in clinical optics, $\sin \theta \approx \theta$ (with θ in radians) and Snell’s law becomes simply $n_1{\theta }_1\approx n_2{\theta }_2$. Much of what follows rests quietly on that approximation.

#Total internal reflection

Going the other way, from glass into air, light bends away from the normal. As the angle inside the glass grows, the ray leaving into the air tilts toward the surface, until at the critical angle it skims right along it. Beyond that angle no light can leave at all: all of it is reflected back inside.

The critical angle is 41.0° for crown glass, 46.6° for the cornea and 48.6° for water. Notice how the reflected ray stays faint for most of the way, then brightens quickly in the last few degrees before the critical angle, where it takes all the light.

#Curved surfaces focus light

A flat surface bends every ray of a parallel beam by the same amount, so the beam stays parallel. A curved surface does something far more useful. Each ray meets it at a different angle, so each is bent by a different amount. A surface that bulges toward the light bends the rays near its edge more than those near its centre, just enough to bring them together at a focus.

The material behind this surface has the refractive index of the cornea. With a radius of curvature of 7.8 mm, about that of a human cornea, parallel light close to the axis comes to a focus 28.5 mm behind it. (The outermost rays cross a little sooner, an imperfection we’ll come back to in Chapter 5.) Flatten the surface and the focus moves away; steepen it and the focus moves closer. Curve it the other way, with a negative radius, and the rays spread apart as if they came from a point in front of the surface.

How strongly a surface converges light is its power, in dioptres:

Here $r$ is the radius of curvature in metres, positive when the surface bulges toward the incoming light, that is, when its centre of curvature lies to the right. For the front of the cornea, $F=(1.376-1)/0.0078\approx +48\text{D}$. It is the strongest focusing surface in the eye. The back of the cornea, with aqueous humour behind it, takes away about 6 D, leaving the cornea as a whole with about +42 D, roughly two-thirds of the eye’s total power.

The power depends on the difference in refractive index across the surface. Open your eyes under water and that difference almost vanishes:

Under water the front surface of the cornea keeps only about +5.5 D of its power, and its focus falls a quarter of a metre behind it. The back surface still takes away its 6 D, so the cornea as a whole is left with almost no power at all, and the world turns to a blur. Swimming goggles fix this by putting a layer of air back in front of the cornea. Their flat window has no power of its own; it simply lets the cornea work in air again.

#A surface adds vergence

There is a simpler way to say what a curved surface does. In the last chapter we described light by its vergence, $L=n/l$. Light reaches a surface with some vergence $L$ and leaves it with a new vergence $L^{\prime }$. The surface simply adds its power:

Before the surface $L=n_1/l$, measured in the first material; after it $L^{\prime }=n_2/l^{\prime }$, measured in the second. Written out, the equation is $n_2/l^{\prime }-n_1/l=(n_2-n_1)/r$, the standard result for refraction at a single surface, which holds for rays close to the axis. That is why the refractive index belongs in the vergence: with it included, the surface’s effect is a plain addition, whatever the materials on either side.

Here is a glass surface with a power of +10.00 D. Move the point of light nearer or farther:

With the point 50 cm away, light arrives with $L=-2.00\text{D}$. The surface adds +10.00 D, so it leaves with $L^{\prime }=+8.00\text{D}$: it now converges, toward an image $l^{\prime }=1.5/8=18.8\text{cm}$ inside the glass. Bring the point closer. At 10 cm, $L=-10.00\text{D}$ exactly cancels the surface’s power and the light inside the glass is parallel. Closer still, the light keeps diverging; now it seems to come from a virtual image in front of the surface, a point the light never actually reaches.

This one addition is the heart of the subject. A lens is made of two such surfaces, and in Chapter 4 we’ll see that a thin lens, too, just adds its power to the vergence of the light. Mirrors and the eye follow the same rule.

#A different index for each colour

The refractive index of a material is slightly different for each colour: higher for blue light than for red. So blue light is bent a little more than red at every surface. This is dispersion, and it is why a prism spreads white light into a spectrum, as we’ll see in the next chapter.

How strongly a material disperses light is described by its Abbe number: the higher it is, the less the colours separate. Crown glass has an Abbe number of about 59 and CR-39 about 58. Polycarbonate, popular for its toughness, has only about 30, which is why some wearers notice coloured fringes when they look through the edges of strong polycarbonate lenses.

The eye disperses light too. Across the visible spectrum its power for blue light is about two dioptres greater than for red. Clinicians put a small slice of this to work in the duochrome test, which we’ll meet in the chapter on the eye.

#Recap

• Light travels more slowly in a material. The refractive index is $n=c/v$.
• At a surface the wavefront swings round and the ray bends: $n_1\sin {\theta }_1=n_2\sin {\theta }_2$, toward the normal when entering a material of higher index, away from it when leaving.
• Beyond the critical angle, $\sin {\theta }_c=n_2/n_1$, light cannot leave a dense material: total internal reflection.
• A curved surface has a power $F=(n_2-n_1)/r$ and adds it to the vergence of the light: $L^{\prime }=L+F$.
• The front of the cornea supplies about +48 D in air but only about +5.5 D under water.

#Test yourself

#Further reading

• A. H. Tunnacliffe, Introduction to Visual Optics. Association of British Dispensing Opticians.
• S. H. Schwartz, Geometrical and Visual Optics: A Clinical Introduction. McGraw-Hill.
• D. A. Atchison and G. Smith, Optics of the Human Eye. Butterworth-Heinemann.