Illustrated Physics
physics, drawn
Optics

The angle past which light cannot leave

Snell's law asks for the sine of an angle greater than one, and no such angle exists. What happens instead is a perfect mirror made out of nothing but a change of speed.
11 min read 6 figures Where the model stopsThe same equation again

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 critical angle for n = 1.5 into n = 1Refracted angle against incident angle. It rises faster than the incident angle and reaches 90° at 41.8°, beyond which no refracted ray exists at all.020406080020406080incident angle (degrees)critical angle 41.8°beyond this, nothing emergesdashed: no bending at all
Fig. 1 Refracted angle against incident angle for light trying to leave glass. The curve reaches ninety degrees at a finite incidence and simply stops, because past that point Snell’s law asks for the sine of an angle to exceed one.

The equation runs out

Snell’s law says n1sin⁡θ1=n2sin⁡θ2n_1\sin\theta_1 = n_2\sin\theta_2. Going from glass into air, n1=1.5n_1 = 1.5 and n2=1.0n_2 = 1.0, so

sin⁡θ2=1.5sin⁡θ1,\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⁡θ1=1/1.5=0.667\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⁡θ2>1\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.

θc=arcsin⁡ ⁣(n2n1).\theta_c = \arcsin\!\left(\frac{n_2}{n_1}\right).

Refraction from n = 1.5 into n = 1A ray crossing a boundary between media of refractive index 1.5 and 1, bending by the amount Snell's law requires.30°48.6°n = 1.5n = 1some light always reflects as well
Fig. 2 Light leaving glass below the critical angle. A refracted ray exists, bent away from the normal, and a faint partial reflection accompanies it — as it accompanies every boundary crossing.
Total internal reflection at 50°The angle of incidence exceeds the critical angle, so no ray emerges: all the light is reflected back.50°n = 1.5n = 1past the critical angle: nothing gets through
Fig. 3 The same boundary at fifty degrees, past the critical angle. There is no transmitted ray at all, and the whole of the light returns into the glass.

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 critical angle for n = 1.33 into n = 1Refracted angle against incident angle. It rises faster than the incident angle and reaches 90° at 48.8°, beyond which no refracted ray exists at all.020406080020406080incident angle (degrees)critical angle 48.8°beyond this, nothing emergesdashed: no bending at all
Fig. 4 The same relationship for water. A smaller index ratio gives a larger critical angle — 48.8° — so water traps light less effectively than glass does.
The critical angle for n = 2.42 into n = 1Refracted angle against incident angle. It rises faster than the incident angle and reaches 90° at 24.4°, beyond which no refracted ray exists at all.020406080020406080incident angle (degrees)critical angle 24.4°beyond this, nothing emergesdashed: no bending at all
Fig. 5 Diamond, at index 2.42. The critical angle falls to 24.4°, so a ray inside a diamond must be aimed almost straight out to escape at all.

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

Refraction from n = 1 into n = 1.5A ray crossing a boundary between media of refractive index 1 and 1.5, bending by the amount Snell's law requires.65°37.2°n = 1n = 1.5some light always reflects as well
Fig. 6 Light entering glass at a very steep incidence. However grazing the arrival, the refracted ray is well inside the glass — entering has no critical angle at all.

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.

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.

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.

Where this goes next

40°25.4°n = 1n = 1.5some light always reflects as well The bend at the boundary, and what it is really about Light changes direction when it changes speed. Snell's law is the geometry of that statement, and it can be derived without knowing anything about light at all. FFobjectimageu = 2.44 fv = 1.69 fmagnification -0.69 What a lens is doing, and why three rays are enough A lens bends every ray that reaches it. The construction uses three, because three are all that can be drawn without calculation — and any three that meet prove all the rest do. sourcesourcesolid: waves arrive in stepbetween them they arrive opposed When two waves meet, they simply add Waves pass through each other unchanged and their displacements add point by point. From that one impoverished-sounding rule comes interference, beats, and the evidence that light is a wave at all.