The bend at the boundary, and what it is really about

A straight stick pushed into water looks bent. The stick is not bent, the water is not doing anything to the stick, and yet the appearance is completely convincing — convincing enough that it was used for two thousand years as an argument that the senses cannot be trusted.

The senses are fine. Light really does change direction at the surface, by an amount that can be predicted to several decimal places, and the reason it does is that light travels at a different speed in water.

The law, and the constant in it #

Measure both angles from the normal — the line perpendicular to the surface — and the relationship is

$$n_1 \sin\theta_1 = n_2 \sin\theta_2.$$

The constants $n$ are refractive indices: 1.00 for air, 1.33 for water, about 1.5 for common glass, 2.42 for diamond. Light entering a higher index bends toward the normal; leaving it, away.

The sines are the surprising part. A rule involving the angles themselves would be much easier to guess, and it is what Ptolemy assumed when he tabulated refraction in the second century — his tables are accurate at small angles and wrong at large ones, which is exactly the signature of confusing $\sin\theta$ with $\theta$. The correct law waited until the seventeenth century, and it took the wave picture to explain rather than merely state.

Refractive index looks like a material property and is really a ratio of speeds:

$$n = \frac{c}{v},$$

the speed of light in vacuum divided by its speed in the medium. Water’s 1.33 says light moves through water at three-quarters of its vacuum speed. Diamond’s 2.42 says it crawls, at forty percent.

So the law is not really about light bending. It is about light slowing, with the bending as a geometric consequence — which is why the same law, with the same sines, describes sound refracting in the ocean, seismic waves refracting in the mantle, and radio waves refracting in the ionosphere. Nothing in the derivation below is about light.

Why slowing makes it turn #

The clean argument uses wavefronts rather than rays.

Picture a wave arriving at the boundary at an angle. Its wavefronts — the lines joining points in step, the crests — are perpendicular to the direction of travel. Because the front is tilted, one end of it reaches the boundary before the other.

That end starts moving slowly while the other end is still in the fast medium. The front pivots. It is the same reason a marching band wheels around a corner when the outer rank keeps its stride and the inner rank shortens theirs, and the same reason a car pulls to one side when one wheel drops onto gravel.

Working the geometry through gives the sines directly. In one period, the fast side advances $v_1 T$ along the boundary’s direction and the slow side $v_2 T$; the tilt of the front is set by the ratio, and the ratio of the sines of the two angles equals the ratio of the speeds. Snell’s law falls out with no further assumptions:

$$\frac{\sin\theta_1}{\sin\theta_2} = \frac{v_1}{v_2} = \frac{n_2}{n_1}.$$

One consequence deserves separating out, because it is the source of endless confusion. The frequency does not change at the boundary — the far side is being driven by the near side, and it must oscillate at the rate it is driven. So with $v = f\lambda$ and $f$ fixed, the whole change lands on the wavelength. Light entering glass has its wavelength shortened by a factor of $n$, and its colour, which is set by frequency, is untouched.

The same law from an entirely different idea #

There is a second derivation, and it is strange enough to be worth the detour.

Fermat proposed in 1662 that light travels between two points by the path that takes the least time. Not the shortest path — the quickest. In a uniform medium those are the same thing and the principle says nothing interesting. At a boundary they differ, and the principle becomes a real prediction.

The setup is the lifeguard problem. A swimmer is in trouble offshore and to one side. The lifeguard runs faster than they swim, so the quickest route is not the straight line: it is worth running further along the beach to shorten the swim. Minimise the total time over the entry point and the condition that comes out is Snell’s law, with running and swimming speeds in place of light speeds.

That two arguments this different — one about wave crests pivoting, one about minimising a total — give the identical formula is not a coincidence, and the connection took another two centuries to make precise. The stationary-time path is where nearby paths agree in phase and therefore reinforce; every other path has neighbours that cancel it. Fermat’s principle is interference stated as a variational principle, and it is the ancestor of the least-action formulations that run through the whole of modern physics.

It also comes with a caution that is usually omitted. The correct statement is stationary time, not least time — a concave mirror can produce a path that is a maximum among its neighbours. The principle picks out where the derivative vanishes, which is a weaker and more useful claim than picking out a minimum.

Coming back the other way #

Reverse the ray and the law is unchanged: $n_1\sin\theta_1 = n_2\sin\theta_2$ does not care which side is which. But the consequences are wildly asymmetric, because going from dense to rare bends the ray away from the normal, and an angle that bends away can run out of room.

At forty-two degrees inside glass the refracted ray lies flat along the surface. Past that, Snell’s law asks for the sine of an angle to exceed one, and there is no such angle: the light does not emerge. This is total internal reflection, and it has no counterpart going the other way — light entering a dense medium always gets in.

That asymmetry is the reason a fish can see the entire sky compressed into a cone overhead and blackness outside it, and the reason a diamond, with its critical angle of only twenty-four degrees, traps and re-reflects light so effectively that cutting it well is a matter of ensuring most rays strike a facet beyond that angle.

What the bending is for #

Refraction at a flat surface is a curiosity. Refraction at a curved one is an instrument.

A lens does nothing that has not already been described. Each ray strikes a curved surface, bends by whatever Snell’s law requires at the local angle, crosses the glass, and bends again on leaving. The extraordinary part — that all the rays from one point reconverge at another — comes from the shape, which is chosen so that the bending increases with distance from the axis at exactly the right rate.

That the mirror obeys the same equation is a hint about what imaging really requires. Neither refraction nor reflection is essential; what is essential is imposing a delay across the wavefront that grows as the square of the distance from the axis. A lens does it by making light take longer through the thick middle and a mirror by making the middle further away, and a diffraction plate does it by interference with no glass at all.

The colours, which the picture cannot show #

The figure draws one ray, which quietly assumes a single index. Refractive index depends on frequency, and every material’s index is a little higher for blue light than for red.

The difference is small — for crown glass, about 1.514 at the red end of the visible range and 1.523 at the blue. It is enough. Two rays entering together leave at angles differing by a fraction of a degree, and after a second refraction at the far face of a prism the separation is visible on a wall across a room. That is dispersion, it is the whole content of a prism, and Newton’s demonstration that a second prism could recombine the colours rather than adding more of them was the argument that established white light as a mixture rather than a modification.

The same effect is a nuisance in every lens. Different colours focus at different distances, so a simple lens produces coloured fringes around every edge — chromatic aberration, which is corrected by cementing a converging lens of one glass to a diverging lens of another so that the dispersions cancel while the focusing does not. That trick, the achromatic doublet, is why astronomy has refracting telescopes at all.

Dispersion also makes the rainbow. Water droplets refract, reflect internally, and refract again, and the total deviation has a stationary point near forty-two degrees — different by about two degrees between red and violet, which is the angular width of the bow.

Where the model stops #

The ray is the most useful lie in optics, and its domain of validity is easy to state: it works when everything in the problem is much larger than the wavelength.

Diffraction is what happens when it is not. Light passing an obstacle comparable to its wavelength spreads, and no amount of ray tracing predicts it — it is a wave adding to itself, and rays have no mechanism for that. Sound diffracts obviously because its wavelengths are metres long; light diffracts subtly because its are under a micron.

Partial reflection. The figure draws the reflected ray faintly and that faint ray is not decoration. At every boundary, some light reflects and some transmits, in a ratio depending on angle and polarisation. At one particular angle — Brewster’s — the reflected light is completely polarised, which is what polarising sunglasses exploit against glare from water and roads.

A sharp boundary. Snell’s law assumes the index changes abruptly. Where it varies smoothly, the ray curves continuously instead of kinking: this is why a mirage exists, why sound channels in the ocean carry whale song for hundreds of kilometres, and why the setting Sun appears above the horizon when it is geometrically already below it.

Linear response. At laser intensities the index depends on the intensity itself, and a beam can focus itself. Everything above assumes the medium does not notice the light — the same linearity assumption that lets two waves pass through each other unchanged, failing for the same reason.

The ladder from here #

Later rungs on this anchor: the Fresnel equations, which say how much reflects rather than merely that some does. Brewster’s angle and polarisation by reflection. Dispersion measured properly, and the Cauchy and Sellmeier fits. The rainbow’s geometry, including the second bow with its reversed colours. Mirages and continuously varying media. Refraction of seismic waves, and how it mapped the Earth’s core. Negative index materials, where the refracted ray comes out on the wrong side of the normal entirely. And the passage from Snell’s law to the lens, which is nothing but refraction at two curved surfaces in succession.

Snell found the law in 1621 and did not publish. Descartes published it in 1637 without attribution, Fermat derived it from a principle he could not justify, and Huygens explained it from waves in 1690. The law is named after the one who kept quiet.