The drop that cannot hold the light it lets in

The angle past which light cannot leave found that light inside a dense material meeting its surface at more than the critical angle — 48.6° for water, 41.8° for glass, 24.4° for diamond — cannot get out at all: Snell’s law asks for an angle with a sine greater than one, and there is none, so the light is reflected completely. The window a diamond is cut inside found that total reflection inside a gem is what sends its light back to the eye, and the corner that sends light home used it to make a reflector with no silvering.

A raindrop is water and it is bright, and the natural assumption is that its brightness — and the rainbow — come from the same total reflection: sunlight enters, bounces off the inside of the back of the drop perfectly, and comes out towards the observer. That assumption is wrong, and it is wrong for a reason that has nothing to do with water and everything to do with spheres. No ray of light that enters a sphere from outside is ever totally reflected inside it.

Every chord meets the surface at the same angle #

Follow a ray entering a sphere. It arrives at some height $b$ above the line through the centre, in units of the radius, so it strikes the surface at an incidence angle $i$ with $\sin i = b$. It refracts to an angle $r$ with $\sin r = \sin i/n$. Inside, it travels in a straight line to another point on the surface.

The triangle formed by that chord and the two radii to its ends is isosceles — both radii are the same length — so the chord makes the same angle with the radius at both ends. The ray meets the inside of the surface at exactly the angle at which it entered, $r$. If it is partly reflected there, the reflected ray starts a new chord with the same angle at its far end, and so on: every internal reflection of the ray happens at the same angle $r$.

So the question of whether the ray is ever totally reflected reduces to whether $r$ ever exceeds the critical angle $\theta_c$, with $\sin\theta_c = 1/n$. But $\sin r = \sin i/n \le 1/n$, because $\sin i$ cannot exceed one. The refraction angle of any ray that enters is at most the critical angle, and equal to it only for a ray that grazes the sphere at $b = 1$ and so carries no light in. Every internal reflection is partial. At every one, most of the light leaves.

The boundary is reached and never crossed #

The figure makes the argument visible for three materials. Each curve rises from zero, for a ray hitting the drop dead centre, to exactly the critical angle at the edge. The two meet only at the grazing ray. A diamond sphere would be no different: its refraction angles never exceed 24.4°, its critical angle. It is the same function in each case, $\arcsin(b/n)$, compared with its own value at $b = 1$.

The result is easy to check with a glass marble and a laser pointer in a dark room. Shine the pointer into the marble from any direction and a spot of light emerges from the far side every time, with fainter spots where the beam has bounced once or twice inside and leaked out; there is never a direction in which the marble swallows the beam and glows with light it will not release.

That a flat slab behaves the same way is no coincidence. A ray entering one face of a slab with parallel faces meets the opposite face at its refraction angle too, and also cannot be totally reflected there. A sphere is a slab whose faces are everywhere parallel to each other in the sense that matters: the angle a ray makes with the surface is conserved from one meeting to the next. Both shapes conserve the quantity that decides total reflection, and both let out everything they let in.

What makes the result inevitable is time reversal. A ray of light can be run backwards: if a path from outside to some state inside the drop exists, the reversed path from that state to outside exists too. A ray that entered from outside therefore has a route out — the reversal of its route in — and cannot be trapped for ever. That argument does not by itself forbid a single total reflection along the way, but in a sphere and a slab the conserved angle forbids it at every step.

A cylinder seen from the side #

The same argument applies to a long glass rod or an optical fibre lit from the side. In the plane across the rod, every chord again meets the surface at its refraction angle, and the component of a ray’s direction along the rod is unchanged by every reflection, so a ray entering through the curved side can never meet that side at more than the critical angle and leaks out as it went in. A fibre traps light only because the light enters through its flat end, travelling nearly along its axis: then it meets the curved wall at a grazing angle that the side could never have produced. The cone a fibre will accept measured the range of directions at the end face that end up trapped. Light shone onto the side of a fibre goes straight through it, which is why coupling light in from the side needs a grating etched into the core or a tight bend that breaks the geometry.

The light that leaks at every bounce #

Because every internal reflection is partial, the light that enters a drop leaves it in a sequence of stages: most at the first meeting with the far surface, a little after one internal reflection, much less after two, and so on, each order fainter than the last by the same factor, the Fresnel reflectance at the ray’s internal angle. For a ray entering at 0.85 of the radius, four-fifths of the light passes straight through. Eight per cent leaves after one reflection, and that is the light that, concentrated at the angle where the deviation is stationary, forms the primary rainbow that the angle the rainbow has to be placed at 42°. Less than one per cent leaves after two reflections and forms the fainter secondary bow, with its colours reversed.

The rainbow is therefore built from light that a drop has failed to keep. Each bow is fainter than the last by roughly the internal reflectance, a few per cent, which is why the secondary bow is seen often and a third-order bow, behind the observer’s head towards the Sun, only a handful of times in recorded history. The history of the rainbow is a history of getting this right. Theodoric of Freiberg, around 1304, traced light through glass globes filled with water and identified the primary bow as two refractions and one internal reflection, and the secondary as two refractions and two reflections. René Descartes, in 1637, traced ten thousand rays through a sphere by hand and found the angle at which they bunch, which is the deviation the path that does not change found stationary. Neither needed total reflection, and neither could have used it: the reflections that make rainbows are the weak, partial reflections of light off the inside of water, and the bows are faint for exactly that reason. If drops could trap light by total reflection there would be no orders to see: the light would circulate inside until absorbed.

Rays entering near the edge are the exception that is not. Their internal angle approaches the critical angle, where the Fresnel reflectance climbs steeply towards one, and they leave a larger share inside after each bounce — a third or more for the ray at 0.97. They still leak at every bounce, and still get out.

Why road beads are painted behind #

The theorem has a commercial consequence that every road crew relies on. The white lines on roads glow in headlights because they are sprinkled with glass beads, each a small sphere that refracts incoming light to a focus near its back surface, as the lens with no axis found any ball does, and sends some of it back towards its source. A corner cube does the same job by three total reflections, as the corner that sends light home showed. A bead cannot: the light reaching its back surface meets it below the critical angle, as the argument above requires, and only four per cent or so is reflected; the rest leaves through the back and is lost.

So the beads are half-buried in white or reflective paint, or given a hemispherical aluminium coating, to supply by reflection what geometry forbids total reflection from supplying. High-performance retroreflective sheeting uses beads backed with metal for the same reason, and the brightness of a bead-based sign depends as much on its backing as on its glass. The sphere focuses; it cannot reflect for itself.

Light born inside can stay #

The argument depended on the light coming from outside. Light that starts inside a sphere — emitted by a fluorescent molecule, scattered by a speck of dust, sent out by an excited atom — has no reversed entrance path to follow, and its angle with the surface is set by where it starts and which way it goes.

A ray leaving a point at distance $\rho$ from the centre, at an angle $\alpha$ to the radius through that point, keeps a constant value of $\rho\sin\alpha$, the perpendicular distance from the centre to its line — its angular momentum about the centre, in the language of orbits. It meets the surface, at radius one, at an angle whose sine is $\rho\sin\alpha$, and every later reflection is at that same angle. If $\rho\sin\alpha > 1/n$, the ray is totally reflected at every meeting, for ever.

That gives a sharp threshold. A source nearer the centre than $1/n$ of the radius cannot produce a value of $\rho\sin\alpha$ large enough, and all its light escapes, as light entering from outside does. Beyond that radius some of its directions are trapped, more as the source approaches the surface: two-thirds of the light from a source at the surface of a water drop, nine-tenths in diamond. The trapped rays skim round the inside of the surface at a constant angle, never meeting it steeply enough to leave.

Those circulating rays are the whispering-gallery modes of a sphere, named after the gallery under the dome of St Paul’s Cathedral, where a whisper against the wall travels round the circumference by repeated grazing reflection. The mode that will not turn a corner found that a glass sphere a tenth of a millimetre across can hold light in such a mode with losses limited only by absorption in the glass and scattering from its surface. Light can be put into those modes from outside only by a trick that breaks the conserved angle: bringing a tapered fibre or a prism within a fraction of a wavelength of the sphere, so that the evanescent field the reflection that happens where the glass is not described couples across the gap. Rays alone cannot do it.

Shape, not index, decides #

The same question asked of other shapes gives different answers. A flat sheet traps all internal light except the two narrow cones of directions that meet its faces within the critical angle — three-quarters of the light emitted inside glass — and that trapped light runs along the sheet to its edges, the principle of the luminescent solar concentrator, a coloured sheet that absorbs sunlight, re-emits it inside, and delivers it to cells along its rim. The cone light has to find to get out found the same escape cone limiting how much light a flat LED chip can release, the problem turned round. A cube has six faces and six escape cones, and traps only the directions outside all of them, under a quarter of the light in glass; below an index of about 1.22 its cones overlap enough to cover every direction, and it traps nothing.

None of those numbers is a property of the material alone. The sphere, with the highest index anyone could make, still lets out every ray it let in. Whether light is held is decided by whether the shape conserves the angle at which a ray meets its surface, and by whether the ray began inside or out.

Trapping in a film of water #

The inside-source result explains a familiar effect from the other direction. The sand that darkens when it is wet found two reasons wet sand is darker than dry: the water between the grains reduces the contrast in refractive index, so light scatters less and penetrates deeper, and the thin film of water over the surface traps light by total internal reflection. Light scattered upward inside that film, from the grains beneath, meets the water–air surface at all angles, and the part outside water’s escape cone — most of it — is reflected back down into the sand for another chance to be absorbed. The film is a flat sheet, which traps internal light; a drop is a sphere, which does not trap light coming in; and the difference between a dark wet beach and a bright rainbow is a difference of shape.

Where the clean argument stops #

Drops are not perfect spheres. Raindrops larger than a millimetre are flattened by the air flowing past them, and a flattened drop no longer conserves the internal angle exactly, so an occasional ray can reach the critical angle after several bounces. The effect is small and appears as subtle changes in the brightness of the rainbow’s highest orders, which have been used to measure drop shapes.

Ray optics is an approximation. In a drop only a few wavelengths across, rays are a poor description, and Mie’s exact solution for scattering by a sphere replaces them. Even in a large drop, light can tunnel out of a ray path that touches the critical angle from inside, and light can tunnel in to a whispering-gallery mode that rays say is unreachable; the resonances this produces show up as sharp spikes in the scattering of small, uniform droplets.

Absorption ends everything. Real water absorbs weakly in the visible, and trapped light is eventually absorbed rather than circulating for ever. For a whispering-gallery mode in a high-quality glass sphere that takes microseconds and many kilometres of path; for a drop of muddy water, much less.

Still open: how small a sphere can hold light #

Whispering-gallery resonators are among the best optical cavities ever made — fused-silica spheres and toroids, crystalline discs polished to angstroms — and they are used as filters, sensors able to detect single molecules landing on them, and sources of frequency combs. How small one can be made and still hold light is limited by the leakage that ray optics says is absent: on a strongly curved surface, a mode’s evanescent tail outside the sphere carries away energy by tunnelling, at a rate that grows sharply as the sphere shrinks towards a few wavelengths across. Spheres only a few micrometres in diameter can still hold light for many thousands of round trips, and how close to the wavelength scale a useful resonator can be pushed, and whether materials of very high index or structured surfaces can push it further, is being worked out.

For light coming in from outside there is no such question. A sphere conserves the angle at which a ray meets its surface, and a ray that got in at that angle can always get out at it. Every drop of rain lets go of every ray of sunlight it receives, a few per cent at a time, and the rainbow is the record of the order in which it does so.