The angle past which light cannot leave

Every mirror is a compromise. A silvered surface returns about ninety-five percent of the light and absorbs the rest; the best dielectric coatings reach ninety-nine point nine, at considerable expense, over a narrow band of wavelengths and angles.

There is one arrangement that returns all of it. No coating, no metal, nothing deposited on anything — just a boundary between two transparent materials, approached from the denser side at a steep enough angle. And the reason it works is that the equation describing what should happen instead has no solution.

The equation runs out #

Snell’s law says $n_1\sin\theta_1 = n_2\sin\theta_2$. Going from glass into air, $n_1 = 1.5$ and $n_2 = 1.0$, so

$$\sin\theta_2 = 1.5\sin\theta_1,$$

and the refracted angle is always the larger of the two. Push the incidence up and the refracted ray swings toward the surface, faster than the incident ray swings toward it.

At $\sin\theta_1 = 1/1.5 = 0.667$ — an incidence of 41.8° — the refracted angle reaches ninety degrees and the ray is running along the boundary. Past that, the equation asks for $\sin\theta_2 > 1$, and there is no angle whose sine exceeds one.

This is not an approximation failing or a subtlety being glossed over. It is a genuinely empty solution set, and the physics has to do something else. What it does is put everything into the reflected ray.

$$\theta_c = \arcsin\!\left(\frac{n_2}{n_1}\right).$$

The failure of the equation is worth being precise about, because it is not a breakdown of the physics. Snell’s law is a statement about matching the phase of a wave along a boundary, and beyond the critical angle there is no propagating wave in the second medium whose phase can be matched. The law has not become wrong; it has run out of solutions of the kind it was written to find, and the solution that exists instead decays rather than travels.

The transition between those two figures is not gradual in the way most physical transitions are. Below the critical angle a transmitted ray exists and carries most of the energy; above it, none does. In between, over the last degree or two, the transmitted fraction falls to zero — the Fresnel equations describe the run-up, and they show the reflectivity climbing steeply and then locking at one.

Only one direction has this #

The effect exists in one direction across the boundary and not the other, and that asymmetry is worth stating plainly.

Light going from rare to dense bends toward the normal, so the refracted angle is always smaller than the incident one, and a smaller angle can never run out of room. Light entering glass at ninety degrees — grazing along the surface — enters at 41.8° and gets in. There is no critical angle for entering.

So the boundary is a one-way trap. Light gets in easily from any direction and can only get out through a cone of directions around the normal. That cone is what a fish sees looking up: the entire hemisphere of sky above the water, compressed into a circle about 97° wide, with the surface outside that circle acting as a mirror showing the lake bottom. Snell’s window is the name for it, and it is one of the few places where an everyday scene is a direct picture of a trigonometric limit.

The three figures are the same computation at three indices, and the trend across them is the whole of gemstone cutting. In diamond, most rays that enter strike a facet beyond 24° and are reflected rather than transmitted; they bounce internally several times, taking a long path through a strongly dispersive material, and emerge through the top separated into colours. The brilliant cut is an arrangement of facet angles designed to make that sequence happen as often as possible. A glass imitation with a critical angle of 42° leaks most of its light out of the back, and looks dull for a reason that can be calculated.

The three figures also illustrate a habit worth keeping. A curve plotted across a parameter says more than three separate ray diagrams would, because the critical angle appears as a feature of the curve rather than as an annotation on it — much as the runaway in the lens equation shows up as an asymptote and is invisible in any single ray construction.

Getting in is not the same as getting out #

Comparing that figure with the one above it is the clearest statement of the asymmetry. Snell’s law is symmetric under reversal and its consequences are not, because bending toward the normal cannot run out of angles and bending away can.

The consequence is a light trap. Anything that gets into a high-index medium tends to stay, bouncing until it happens to strike a surface within the escape cone or until absorption removes it. That is why a stack of clear glass sheets looks white at the edges, why a crack inside an ice cube is visible as a bright sheet, and why light injected into a fibre stays there. It is also why a fluorescent screen loses most of its light sideways into the substrate rather than out toward the viewer — a problem that costs display engineers a great deal of effort.

Light in a pipe #

The most consequential application is a hair-thin thread of glass.

An optical fibre is a core of higher-index glass surrounded by a cladding of slightly lower index. Light entering within a certain cone strikes the core-cladding boundary beyond the critical angle, reflects totally, strikes the far side, reflects again, and continues — following the fibre around bends, for kilometres, with losses measured in a fraction of a decibel per kilometre.

The index difference is deliberately small, often under one percent, which makes the critical angle close to ninety degrees and admits only rays travelling nearly straight along the fibre. That sounds like a limitation and is the design goal: rays at different angles take different path lengths, so a pulse launched as a sharp spike arrives smeared out. Restricting the accepted cone restricts the smearing, and single-mode fibre restricts it to one path by making the core only a few wavelengths across — at which point the ray picture stops applying and the correct description is a waveguide mode.

A waveguide mode is a standing wave across the fibre and a travelling wave along it, and the requirement that a whole number of half-wavelengths fit across the core is what makes the allowed modes discrete. A fibre and a guitar string are solving the same boundary-value problem; the fibre simply propagates along the direction the string does not have.

The idea was demonstrated in 1841 by Daniel Colladon, who guided sunlight along a jet of water pouring from a tank, to considerable applause. It became a technology only when glass was made pure enough for the light to survive the journey, which took until 1970.

What the perfect mirror costs #

Nothing in physics returns a hundred percent of anything for free, and the charge here is not levied on the reflection — which really is total — but on what can be put into the trap in the first place.

The trap works because rays inside the fibre strike the boundary at steep angles. That is the same statement as: only rays arriving within a narrow cone at the end face will end up at steep angles inside. The width of that cone is the fibre’s numerical aperture, $\sqrt{n_1^2 - n_2^2}$, and for the small index differences that make a good fibre it is small — about 0.2 for ordinary multimode fibre, an acceptance half-angle of 11.5°, and about 0.14 for single-mode.

The cost of a narrow acceptance cone is that most of the light from an ordinary source cannot be got in. And no lens fixes it, because the product of area and solid angle is conserved through any optical system: squeezing light into a smaller spot necessarily spreads it over a wider range of angles, and the fibre rejects exactly what the squeezing produced. A light-emitting diode radiating into a hemisphere from a square millimetre cannot be coupled efficiently into a nine-micron core by any arrangement of glass whatsoever. A laser, which starts with the angles already small, can. Fibre communication waited for the laser as much as it waited for pure glass, and the reason is a conservation law rather than an engineering shortfall.

The second charge is paid by any fibre that accepts more than one path. A ray bouncing at the steepest permitted angle travels further than one going straight down the middle, by a factor of $n_1/n_2$, so a sharp pulse arrives spread out. For a step-index fibre with a one percent index difference the spread is about 49 nanoseconds per kilometre — which caps a ten-kilometre link at a couple of megabits per second, and has nothing to do with the light being lost. It arrives; it simply arrives at different times.

Both fixes cost something in turn. Grading the index across the core, so the outer rays travel faster where they travel further, cuts the spread by two orders of magnitude and requires a manufacturing process that controls composition continuously across a hair’s width of glass. Making the core small enough to permit one path only — the single-mode solution, and the one that carries the internet — reduces the core to about nine microns, at which point every splice and connector must align two fibres to a fraction of a micron. The entire connector industry exists to pay this particular bill.

The angle, used as an instrument #

A quantity that depends sharply on a material property is a measuring device waiting to be built, and the critical angle has been one since the nineteenth century.

Put a drop of liquid on a prism of known high index and illuminate the boundary from every direction at once. Rays arriving past the critical angle for that pair of media are totally reflected; rays arriving below it are partly transmitted and lost. Looking into the returning light, the field of view is divided into a bright region and a dark one, separated by a line at exactly the critical angle. Reading the position of that line gives $n_2$, because everything else in $\sin\theta_c = n_2/n_1$ is known.

That is an Abbe refractometer, and the reason it is a good instrument is the reason the transition is interesting: the boundary is a discontinuity rather than a gradual fade, so the eye is being asked to locate an edge rather than to judge a brightness. Edges can be located far more precisely than intensities, and the instrument reaches four decimal places in refractive index from a drop of liquid and a few seconds of work.

The applications are almost entirely outside optics. Sugar concentration in fruit juice, grape must and honey is read directly off a refractometer scale, because dissolved sugar raises the index in a way that has been calibrated once and for all. So is the salinity of aquarium water, the protein content of blood serum, and the state of cutting fluid in a machine shop. A limit in Snell’s law, drawn as a curve that stops, is sold in agricultural supply catalogues.

The wave that is there anyway #

The ray picture says nothing crosses the boundary. The wave picture says something does, and it is measurable.

Solve the wave equation on both sides with the incidence past critical, and the field in the rarer medium does not vanish. It decays exponentially with distance from the boundary, over a scale of roughly a wavelength, and it carries no energy away — it is called the evanescent wave, and it is the reason “total” reflection is a statement about energy flow rather than about the field being zero.

This is testable, and the test is the most direct evidence that the ray model is a summary rather than the truth. Bring a second glass surface within a fraction of a micron of the first, without touching, and light begins to cross the gap — into a region where, according to rays, it has no business being. The transmitted fraction falls off exponentially with the gap. The effect is called frustrated total internal reflection, and it is the exact optical analogue of quantum tunnelling, obeying the same exponential in the same way for the same mathematical reason.

The evanescent field is also the reason “total” reflection can be described honestly only in wave language. Rays are a summary that works when everything is much larger than a wavelength, and a decay length of one wavelength is precisely the scale at which the summary stops being adequate.

It has ordinary uses. A fingerprint sensor works by frustrating the reflection wherever a ridge touches the glass and leaving it intact over the valleys. Some touchscreens work the same way. And in microscopy, illuminating a sample with only the evanescent field lights up a layer a hundred nanometres thick and nothing behind it, which is how single molecules at a cell membrane are watched without the rest of the cell drowning them out.

The whole sky in a circle #

The escape cone read backwards produces one of the most striking things in ordinary experience, and it is available to anyone who opens their eyes underwater.

Every ray reaching a swimmer’s eye from above the surface has been refracted on the way in, bending toward the normal — and the steepest arrival possible is a ray that came in grazing the surface, which enters at exactly the critical angle. So the entire hemisphere above the water, horizon to horizon, arrives compressed into a cone of half-angle 48.6°: a circle about 97 degrees across, directly overhead.

Outside that circle there is no sky at all. Those directions are beyond the critical angle for light trying to leave, so the surface behaves as a mirror there, and what fills the rest of the field of view is a reflection of the pool floor.

The circle has a name — Snell’s window — and it is exactly the escape cone of this page’s figures, observed from the wrong side. Fish see the world through it, which is why a heron standing outside the cone is invisible to the fish it is watching until the last moment, and why the compressed rim of the window shows the horizon smeared into colour by dispersion — the same geometry that traps light in a diamond, drawn at the scale of a swimming pool.

Where the model stops #

Three assumptions deserve naming, because each is violated somewhere useful.

A perfectly smooth boundary. Total reflection is total only if the surface is clean and flat to well within a wavelength. A scratch, a fingerprint or a drop of water changes the local index and lets light through — which is why a cracked fibre leaks at the crack, and why any liquid on the outside of a prism spoils the effect.

Both media transparent. If the rarer medium absorbs, the evanescent field is absorbed too, and the reflection is no longer total — and absorption is energy leaving the ordered field for the disordered motion of the material, which is one-directional in the usual way. Attenuated total reflectance spectroscopy is built on exactly this: press a sample against a prism, reflect infrared light internally, and the sample’s absorption bands appear in the reflected beam without the beam ever having entered the sample in a ray sense.

A sharp index step. Where the index falls off gradually rather than abruptly, the ray does not reflect at a surface — it curves back continuously. That is how a mirage works, and how sound is trapped in the ocean’s SOFAR channel, and it produces the same trapping without any boundary existing at all. The wavefront picture handles this case better than the ray picture does, since a front pivoting continuously is exactly what a smoothly varying speed produces.

And the deepest limit is the one the evanescent wave already exposed. Total internal reflection is a ray-optics name for something that is not a ray phenomenon. The boundary is not a mirror; the field on both sides is solving one wave equation with one set of boundary conditions, and “all the light comes back” is a statement about the time-averaged energy flux, which is a considerably weaker claim than the picture suggests.

The ladder from here #

Later rungs: the Fresnel equations, and the phase shift on total reflection that differs between polarisations — the basis of the Fresnel rhomb, which converts linear polarisation to circular using nothing but two internal reflections. The Goos–Hänchen shift, in which the reflected beam emerges displaced along the surface, as though it had briefly entered the forbidden medium. Waveguide modes, and the point at which counting reflections stops working. Fibre dispersion and the engineering of pulse shapes. Prisms used as mirrors in binoculars and reflex cameras. Retroreflectors, including the ones left on the Moon. And frustrated reflection as the optical tunnelling problem, which is the cleanest classical rehearsal for a quantum effect that exists.

Kepler described the effect in 1611, in a treatise on the optics of the telescope, and had no way to explain it. The explanation needed a law that would not be published for another twenty-six years.