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.

Coming back the other way is not symmetric, and the asymmetry has a limit in it. Plot the refracted angle against the incident one for light leaving glass and the curve climbs faster than the incident angle does — reaching ninety degrees at a finite incident angle, past which there is no solution at all. Going into a denser medium every angle works; coming out of one, they run out.

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.

What the bending is for is imaging, and a lens is nothing but two curved refracting surfaces. Every ray in the figure obeys the same law twice, once at each surface, and the fact that they all arrive at one place is a consequence of the curvature rather than an extra principle. An object two focal lengths out images two focal lengths beyond, the same size and inverted — the one configuration a reader can check without arithmetic.

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.

A mirror does the identical job by reflection, and the comparison is worth making because the two laws are so different. Refraction depends on a ratio of speeds and therefore on wavelength; reflection does not depend on either. That is why a mirror has no chromatic aberration and a lens does, and why every large telescope built since the eighteenth century has used one.

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.

What the slowing costs #

The essay has treated $n = c/v$ as a bookkeeping ratio. It is also a delay, and the delay is charged to anyone who sends anything through glass.

Light in a silica fibre travels at about 204,000 kilometres per second — two-thirds of its vacuum speed, and slower than a radio wave in air by the same factor. A signal from London to New York covers about 6,600 kilometres of cable rather than the 5,600 of the great circle, because cables follow seabeds rather than geodesics, and the round trip takes something over sixty milliseconds. Roughly a third of that is the refractive index, not the distance: the identical route in air would take about forty.

Twenty milliseconds is nothing to a human and a great deal to a machine. The clearest demonstration is the set of microwave relay chains built between Chicago and New York after 2010, which carry a tiny quantity of financial data through the air over a line of towers rather than through the ground in a fibre. The path is shorter, and the light is not being slowed by glass. The gain is about four milliseconds each way, and the towers were built because that was worth paying for.

The same charge appears wherever glass sits in a signal path. Hollow-core fibre, in which the light travels mostly through air inside a microstructured cladding, exists to recover this factor and gives up a good deal of loss performance to do it. Optical delay lines exploit it in the opposite direction, using a spool of fibre when a known delay is wanted and nothing else can produce one so cheaply.

The deeper point is that $n$ is not a description of a bend. The bend is what is seen, and the slowing is what is happening; the whole law was derived above from the speed ratio and the geometry followed. An index is a delay per unit length, and every consequence of refraction — the bending, the shortened wavelength, the critical angle, the focusing power of a lens — is a consequence of some parts of a wavefront being held back relative to others.

Why nothing refracts without absorbing #

There is one more charge, and it is levied by the structure of the theory rather than by any application.

A material slows light because its electrons are pushed by the passing field and re-radiate slightly out of step, and the combined wave advances more slowly than it would in vacuum. That mechanism has a resonance in it: the electrons are bound, they have natural frequencies, and the response depends on how the light’s frequency compares with them. Which is why $n$ depends on colour at all — dispersion is not an imperfection in a material, it is the signature of the material having natural frequencies.

The consequence is a constraint linking two properties that look independent. A medium that responds to light also absorbs light, at frequencies near its resonances, and the size of the refraction across the whole spectrum is mathematically determined by the absorption across the whole spectrum. The relations expressing this are exact, and they say something flatly counterintuitive: a perfectly transparent material with an index different from one is impossible. Anything that bends light absorbs it somewhere.

Ordinary glass is transparent in the visible because its resonances are in the ultraviolet, where it is opaque, and in the infrared, where it is opaque again. The visible window sits between two absorptions, and the index in that window rises steadily toward the blue precisely because the ultraviolet absorption is being approached. Dispersion in the visible is the ultraviolet edge, felt at a distance.

That explains a pattern that otherwise looks like bad luck. High-index glasses have their absorption edge nearer the visible, so they disperse more — index and dispersion rise together, which is why a strongly refracting glass is a strongly colouring one, and why Newton concluded that an achromatic lens was impossible. The escape is narrow: the proportion between index and dispersion differs a little between glass families, and the achromatic doublet lives in that gap. It is a small loophole in a law that otherwise forbids the thing every lens designer wants.

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.

The index that is less than one #

Every number in the table above is greater than one, and it is easy to take that as a rule. It is not. It is a statement about visible light in transparent materials, and the exception is what X-ray astronomy is built on.

The section above traced the index to how the medium’s electrons respond to the driving field, and to the fact that below a resonance they follow it. Above a resonance they lag by half a cycle and respond the other way, which subtracts from the index instead of adding. X-rays are far above every electronic resonance in ordinary matter, so for them $n = 1 - \delta$ with $\delta$ of order $10^{-5}$: every solid is optically thinner than vacuum.

Snell’s law then works in reverse. A ray going from vacuum into glass bends away from the normal, and there is a critical angle for the ray arriving from outside — total external reflection, at grazing incidence within about a degree of the surface.

That is the only way to make an X-ray mirror. A telescope for X-rays cannot point a dish at the sky, because anything striking a surface steeply goes into it and is absorbed; instead the mirrors are nearly parallel to the incoming light, shaped as grazing-incidence cones nested inside one another like a set of barrels, and the photons are steered by a series of glancing deflections. Every X-ray observatory ever flown has that shape, and the reason is one minus sign in the response of an electron driven above its resonance.

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 #

The next rung is the rainbow’s geometry, where this law is run three times through one drop and the answer is an angle nobody chose. After it: 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. 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.