What is Snell's law of refraction?

Shine a laser from air into water and the beam doesn't carry straight on. At the surface it bends. That bending is called refraction, and Snell's law says exactly how much:

n₁ sin θ₁ = n₂ sin θ₂

The angles are measured from the normal, an imaginary line at right angles to the surface (the dashed line). θ₁ is the angle of the beam coming in, θ₂ the angle going out. Each material has a number n, its refractive index: air 1.000, water 1.333, glass about 1.5, diamond 2.42.

What is n? It tells you how much slower light goes in that material: v = c ÷ n. Light races through air at almost 300,000 km/s, through water at 225,000 and through diamond at only 124,000. Going into a slower material, the beam bends towards the normal. Going into a faster one, it bends away. Think of a lawn roller rolling at a slant from a path onto grass: the wheel that hits the grass first slows first, and the roller swings round.

Two more things happen at every surface. A little light bounces back, the faint reflected beam: about 2% at water, 4% at glass. And going from a slow material to a fast one, there is a critical angle. Tilt the beam past it and no light gets out at all. It all reflects: total internal reflection.

Your brain assumes light travels in straight lines. Refraction breaks that rule, so things under water are not where they look.

The bent straw. Light from the part of a straw under water bends as it leaves the surface, away from the normal. Your eye traces it back in a straight line and sees the straw higher than it really is. The dry part isn't moved, so the straw looks broken right at the surface.

The shallow pool. The same trick makes water look shallower than it is. Looking straight down, the apparent depth ≈ real depth ÷ n, so a 2 m pool looks 1.5 m deep. Looking at a slant it looks even shallower, and the far end seems to rise up. That's why people misjudge pools and rivers. Always check the real depth before you jump.

The prism. n is not quite the same for every colour. In glass, violet light is slowed a little more than red, so it bends a little more. A prism fans white light out into a spectrum. This spreading is called dispersion, and Newton used it in 1666 to show that white light is a mix of all colours.

The rainbow. Every raindrop is a tiny round prism with a mirror at the back. Sunlight bends in, reflects once off the back and bends out. Rays that hit different parts of the drop come back at different angles, but they pile up at about 42° from the point opposite the Sun. Red piles up at 42.4°, violet at about 40.8°. That's why a rainbow is a circle around the shadow of your head, red on the outside.

A convex lens is thicker in the middle than at the edges. Nothing magic happens inside it. Each ray obeys Snell's law once as it enters the curved front face, and again as it leaves the curved back face.

The curve is the trick. Near the edge the surface is tilted, so rays there meet it at a steep angle and bend a lot. Through the middle the surface is almost square-on, so rays bend very little. Get the curve right and rays from one point of an object all meet again at one point behind the lens: a focus. That point is an upside-down image.

A curvier lens bends harder and focuses closer: its focal length f is shorter. For a thin lens, 1/f = (n − 1)(1/R₁ − 1/R₂), and where the image lands follows 1/f = 1/u + 1/v. A camera focuses by moving the lens until the image lands exactly on the sensor (see CameraClear). A phone's tiny lens has f of only about 4 to 6 mm.

Your eye does the same with living parts (see EyeClear). Surprise: the cornea, the clear dome at the front, does about two thirds of the bending, around 42 of the eye's 60 dioptres. The lens behind it only fine-tunes. The cornea is so strong because light jumps from air (n = 1.00) into it (n = 1.38), the biggest change of speed on the way in.

That's why everything is blurry when you open your eyes under water. Water (1.33) is nearly as slow as the cornea, so the cornea hardly bends at all. Goggles fix it by putting air back in front of your eyes.

When light tries to leave a slow material for a faster one, Snell's law runs out of answers past the critical angle. Then every bit of the light reflects back. No mirror is that good: even silver keeps only about 95%. Total internal reflection keeps 100%.

Optical fibre. A glass thread thinner than a hair has a core of slightly slower glass inside a cladding. Light zig-zags down the core, reflecting totally off the cladding again and again, for kilometres. Pulses of infrared light in fibres carry almost all the world's internet, including the undersea cables that land in Mumbai and Chennai. In 1966 Charles Kao worked out that glass pure enough could carry light for kilometres, and won the 2009 Nobel Prize for it.

Diamond sparkle. Diamond's index is so high (2.42) that its critical angle is only 24.4°. A well-cut diamond's back facets are angled so light from the top hits them past that angle, bounces twice and comes back out of the top towards you. Cut the same shape in glass (critical angle 41.8°) and the light leaks out of the back.

Binoculars use glass prisms instead of mirrors. Light meets each face at 45°, past glass's critical angle, and reflects totally. TVs use it too: in an edge-lit LCD (see TVClear), LEDs shine into the edge of a clear acrylic plate, total internal reflection carries the light across, and printed white dots let it out evenly towards the screen.

Mirages on hot roads. On a summer afternoon in India, the road ahead seems to have a shining puddle that is never there when you arrive. The road heats the air just above it. Hot air is thinner, so its index is a tiny bit lower. There's no sharp surface, so the light doesn't bend all at once: it curves smoothly, bending away from the hotter air, until light from the sky skims up off the road and into your eye. You're seeing the sky, upside down, on the road. The change in n is only about 0.00002, so it works only for light grazing the road at less than half a degree.

Bending the wrong way. In every natural material n is positive, and light crosses to the other side of the normal. Around 2000, scientists built metamaterials, arrays of tiny copper rings and wires, with a negative index. Light then bends to the same side of the normal. A flat slab with n = −1 even focuses light like a lens. It has only worked well for microwaves and in labs; at visible light these materials still soak up too much light.

Myth-buster: “Light slows down in glass because it keeps bumping into atoms.” Not really. Between atoms, light always travels at full speed, c. It isn't stopped and delayed at random either: that would scatter light every which way and blur the view through a window. What happens is this. The light's electric field shakes the electrons in the glass, and each shaken electron sends out its own little wave, slightly behind in step. Add all those little waves to the original and you get a wave whose crests move more slowly: c ÷ n. People sometimes say the light is “absorbed and re-emitted”, but nothing is kept: the electrons pass the energy straight on, and the delay is that small step behind. The frequency never changes, so the colour stays the same, but the wavelength shrinks.