The reflection that happens where the glass is not

Past the critical angle, Snell’s law asks for the sine of an angle larger than one, and the ray picture answers that no refracted ray exists. That answer is right about the energy: every joule that arrives leaves again, with a reflectivity of exactly one, which no mirror coating achieves. It is wrong about the geometry, and the way it is wrong can be measured.

The reflected beam does not leave from where it arrived. The lateral displacement of a totally reflected beam along the surface, for the two polarisations, at 550 nm from n = 1.5 into n = 1. Geometrical optics puts the outgoing ray at the point where the incoming one struck. It is not there: it is displaced forward along the surface by a distance comparable with a wavelength, which was measured by Goos and Hänchen in 1947 by reflecting a beam many times and looking at the accumulated offset. The displacement is different for the two polarisations — at 42°, 1361 nm for s and 2991 nm for p, at 45°, 350 nm for s and 560 nm for p, at 50°, 237 nm for s and 261 nm for p, at 60°, 183 nm for s and 127 nm for p, at 75°, 161 nm for s and 79 nm for p — which is why an unpolarised beam comes back slightly split. A displacement is only possible if the light spent time on the far side of a boundary it never crossed, and it is the most direct evidence there is that the evanescent field is a real field rather than a bookkeeping term. — Fig. 1 The lateral displacement of a totally reflected beam along the surface, for the two polarisations, at 550 nm from glass into air. Geometrical optics puts the outgoing ray where the incoming one struck. It is not there — it is displaced forward along the surface by a distance comparable with a wavelength, and the displacement is different for the two polarisations, so an unpolarised beam comes back slightly split.

That displacement is the Goos–Hänchen shift, and it is the most direct evidence there is that the ray picture is a summary rather than the truth. A beam cannot come back from somewhere other than where it went in without having gone somewhere.

The field that is there and carries nothing

The rung below this one established the field on the far side. Solving the wave equation on both sides with the incidence past critical gives, in the rarer medium,

E(z) \propto e^{-\kappa z}, \qquad \kappa = \frac{2\pi}{\lambda_0}\sqrt{n_1^2\sin^2\theta - n_2^2},

a field that decays with distance from the boundary rather than oscillating, and that carries no time-averaged energy away because its electric and magnetic parts are a quarter cycle out of step. “Total” is a statement about energy flow, not about the field being zero — and which component the boundary actually constrains is what decides that.

The critical angle for n = 1.5 into n = 1. Refracted 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. — Fig. 2 The refracted angle against the incident angle for glass into air, running to 90° at 41.8° and stopping. This is the ray picture’s whole account: past the critical angle there is no solution, so there is no ray, so there is nothing there. What replaces it is a field that is not a ray, and the boundary where the two accounts part company is the vertical line.

Draw the same boundary past the critical angle and there is no refracted ray to draw, because Snell’s law has no solution there. Every figure of total internal reflection looks like that, and the empty half of it is what this essay is about: the field beyond the boundary is not zero. What is missing from the drawing is a decaying field rather than nothing at all.

The question this rung is about is how far. The usual sentence is that the evanescent field reaches about a wavelength, and it is a bad sentence, because the depth is not a property of the light or of the materials — it is a property of the angle, and it has a pole.

How far past the boundary the light that does not cross it reaches. The depth at which the field beyond the boundary has fallen to 1/e of its value at the surface, for 550 nm light going from n = 1.5 to n = 1, against angle of incidence past the 41.81° critical angle. The usual sentence — "the evanescent field reaches about a wavelength" — is true in the middle of the range and nowhere near the critical angle, where the depth diverges: the exponent is the square root of sin²θ − (n₂/n₁)², which goes to zero there. The depth is at 42° it is 1017 nm, at 45° it is 248 nm, at 50° it is 155 nm, at 60° it is 106 nm, at 75° it is 83 nm. That is the whole of what makes total internal reflection total: not that the field is zero on the far side, which it is not, but that a purely decaying field carries no time-averaged energy away. — Fig. 3 The depth at which the field has fallen to 1/e of its value at the surface, against angle. The exponent contains √(sin²θ − (n₂/n₁)²), which goes to zero at the critical angle, so the depth diverges there. A fifth of a degree past critical it is 1017 nm; at 45° it is 248 nm; at 75° it is 83 nm. The “about a wavelength” is true over a narrow middle of the range and nowhere near the interesting end.

The divergence has a physical reading. At exactly the critical angle the refracted ray runs along the surface and the field on the far side is a propagating wave rather than a decaying one — infinite reach, because the decay constant has gone to zero. Approaching that angle from above, the field’s excursion into the forbidden medium grows without bound, and everything that depends on the excursion grows with it.

There is a way of estimating the depth that makes the divergence unsurprising. The component of the wavevector along the surface is conserved across a boundary — that is the whole content of Snell’s law — and past the critical angle it is larger, on the far side, than the total wavevector the light is allowed to have there. The perpendicular component must therefore square to a negative number, and its magnitude is $\kappa$ . Just past critical the two are nearly equal, the difference of squares is nearly nothing, and the imaginary perpendicular component is tiny — which is a decay so slow that the field reaches a long way. Nothing about a wavelength enters that argument except as the scale that converts an angle into a length.

Why the beam comes back somewhere else

The shift and the depth are the same fact seen twice. A totally reflected wave acquires a phase shift on reflection, and that phase shift depends on the angle of incidence — which means it depends on the transverse wavenumber, which means that different plane-wave components of a real, finite beam are shifted by different amounts.

A beam is a superposition of plane waves with a spread of directions. Give each of them a phase that varies with its direction, and the superposition comes back out displaced, by

D = -\frac{\mathrm{d}\phi}{\mathrm{d}k_x},

which is the same relation that turns a frequency-dependent phase into a group delay for a wave packet. The Goos–Hänchen shift is a group delay in space instead of in time.

Evaluating it gives, for the polarisation with its field parallel to the surface,

D_s = \frac{\lambda_1}{\pi}\,\frac{\sin\theta}{\sqrt{\sin^2\theta - (n_2/n_1)^2}},

with $\lambda_1$ the wavelength inside the dense medium. That square root is the same one as in the decay constant, upside down, so the shift diverges at the critical angle exactly as the depth does.

The reflection coefficients from glass into air tell the rest, provided the right thing is plotted. Past the critical angle the magnitude is one for both polarisations — that is what “total” means — and the phase is neither the same for both nor constant with angle. Everything on this page comes out of the part of those curves a plot of magnitude cannot show, which is why the effect went unnoticed for a century after the magnitude was understood.

It is worth noticing what this makes of the reflection point. In the ray picture there is a point on the surface where the light turns round, and that point is where the incoming ray meets the glass. In the wave picture there is no such point: the incoming and outgoing beams are two beams whose axes, extended into the rarer medium, cross at a depth of about $1/2\kappa$ below the surface. The reflection behaves as though it happened at an imaginary plane inside the forbidden medium, and the shift is what that displaced plane looks like from outside. That construction is exact enough to be used in waveguide design, where the effective thickness of a guide is its physical thickness plus twice the penetration depth.

The two polarisations shift by different amounts, and their ratio is not constant. Near the critical angle the p-polarised shift is larger by $(n_1/n_2)^2$ ; at grazing incidence it is smaller by the same factor. That crossing is measurable, and it is why an unpolarised beam undergoing many total reflections in a thick fibre emerges with its two polarisations separated by a distance that has accumulated one reflection at a time — which is exactly how Goos and Hänchen found the effect in 1947, since a single shift of half a micrometre is not observable and a thousand of them is.

Illuminating a hundred nanometres

The divergence of the penetration depth with angle is not merely a caution about a careless sentence. It is a control knob, and one whole microscopy technique is built on turning it.

Shine light into a coverslip past the critical angle and the sample sitting on top of it is illuminated only by the evanescent field — so only the part of the sample within a couple of hundred nanometres of the glass sees any light at all. Anything fluorescent there is excited; anything a micrometre away is not. A tip brought that close reads the same field, one atom at a time. The result is an optical section far thinner than any lens can produce, made by choosing an angle rather than by focusing.

The thickness of that section is the depth in this essay’s second figure, and it is adjustable. Working just past the critical angle gives a section approaching a micrometre; working well past it gives sixty or seventy nanometres. Turning a mirror by a degree changes what fraction of a cell is being looked at, which is a strange and useful thing for a microscope to be able to do.

Why it is worth the trouble is a signal-to-background argument. A conventional fluorescence microscope illuminates the whole depth of a sample and collects from the whole depth, so a molecule at the surface is seen against everything above it. Illuminating only the surface removes the background rather than rejecting it, which is a much stronger position: a single fluorescent molecule at a membrane can be watched against essentially nothing, and its position followed as it moves.

It also inherits the limitation. The excitation falls off exponentially with height, so a molecule’s brightness depends on how far it is from the glass — which is a nuisance for counting and a gift for measuring, since the same exponential turns brightness into a height measurement with nanometre resolution.

The gap the light crosses anyway

If the field exists on the far side of the boundary, bringing a second piece of glass close enough to reach into it should let light across. It does.

Light crossing a gap it cannot enter. The fraction of a totally reflected beam that crosses a gap of air between two pieces of glass, against the width of the gap, at 550 nm and 5 angles past the critical angle. At zero gap the two pieces are one piece and everything crosses. As the gap opens the transmission falls off as the square of the evanescent field, which is an exponential in the gap — at 42° it is half at 156 nm, at 45° it is half at 141 nm, at 50° it is half at 122 nm, at 60° it is half at 93 nm, at 75° it is half at 51 nm. The formula is the quantum tunnelling formula: a barrier the wave decays inside, a transmission governed by sinh² of the decay constant times the width, and the same divergence of the decay length as the barrier is made shallow. This is the classical rehearsal of an effect that is usually taught as having no classical analogue, and it was demonstrated by Newton, with two prisms, more than two centuries before anyone needed it to explain a nucleus. — Fig. 5 The fraction of a totally reflected beam that crosses a gap of air between two pieces of glass, against the width of the gap. At zero gap the two pieces are one piece and everything crosses. As the gap opens the transmission falls off exponentially — half at 141 nm for a beam at 45°, half at 51 nm at 75° — and the shape of the falloff is the square of the evanescent field.

The formula is worth writing out, because it is not merely analogous to the quantum one, it is the same expression:

T = \left[1 + \tfrac14\left(\gamma + \frac1\gamma\right)^2\sinh^2(\kappa d)\right]^{-1},

with $\gamma$ the ratio of the decay constant inside the gap to the wavenumber component perpendicular to the surface outside it. Replace “gap of air” with “region where the potential exceeds the energy” and every symbol keeps its role.

A wavefunction crossing a barrier it has not the energy for. An electron of 2 electronvolts meeting a 3 electronvolt barrier 1.2 nanometres wide, with the four matching conditions solved rather than sketched. Left of the barrier the incident and reflected waves add to a standing pattern; inside it the amplitude decays exponentially; to the right a travelling wave continues with amplitude 0.0040 of the incident one, so 0.00163 per cent of the electrons get through. Classically none of them do. — Fig. 6 The quantum problem, solved by matching four boundary conditions in complex arithmetic. Inside the barrier the wavefunction is not oscillating and not zero: it decays exponentially, at a rate set by how far the energy is below the top. Compare the shape with the field profile the optical case has in its gap, and the only difference is the vocabulary.

Transmission against barrier width. The probability that an electron of 2 electronvolts crosses a 3 electronvolt barrier, against how wide the barrier is, on a logarithmic scale. It falls from 7.56e-1 at 0.1 nanometres to 2.09e-6 at 1.4 nanometres. The fall is very nearly a straight line on this axis, which means the transmission is exponential in the width: every extra ångström divides it by about 2.8. — Fig. 7 The quantum transmission against barrier width, on the same sinh² law. In both problems the exponent is the decay constant times the width, in both the transmission is exponentially small once that product exceeds a few, and in both there is a prefactor that matters at small widths and is usually dropped. The two curves in this essay differ in what the axes are called.

What makes the correspondence exact rather than suggestive is that both problems are the same differential equation with the same boundary conditions. In the optical case the field in the gap satisfies the Helmholtz equation with an effective perpendicular wavenumber that is imaginary; in the quantum case the wavefunction satisfies the time-independent Schrödinger equation with an imaginary wavenumber for the same algebraic reason — the kinetic term has gone negative. Both are second-order equations with continuity of the function and its derivative at each face. The physics being described is entirely different and the mathematics is not merely similar, it is identical, which is why a result derived for one transfers to the other with no work at all.

Newton did the optical version. He pressed a convex lens against a flat plate and observed that the ring of contact was not a ring but a spot — that light crossed where the surfaces were close but not touching — and he described the transmission falling off with the gap. He had no wave theory to explain it and he reported the observation anyway, which is why the effect appears in the Opticks of a man who did not believe in waves.

Built from wavelets, refraction gets its change of direction from a change of wavelet radius, and running the same construction the other way past the critical angle is instructive because it fails. The wavelets in the rare medium cannot be reached by any real front — there is no envelope to draw — and that failure is exactly the point at which a decaying field replaces a propagating one. The geometric method does not give a wrong answer; it gives none, and something has to be put in its place.

There is one substantive difference between the two, and it is worth stating so the analogy is not oversold. In the optical case the “barrier” is a region where a propagating solution exists but not at that transverse wavenumber: the light could travel in the gap perfectly well if it arrived at a different angle, and it is the conservation of the along-surface component that forbids it. In the quantum case the barrier is a region where no propagating solution exists at that energy at all, whatever the direction. The optical barrier is a kinematic prohibition and the quantum one is energetic, and the shared mathematics comes from both prohibitions producing the same sign in the same place.

The delay that appears not to grow

The gap figure has a consequence that caused thirty years of argument, and it is worth setting out because the resolution is instructive rather than merely reassuring.

A pulse crossing the gap arrives late, by a delay that is the derivative of the transmission’s phase with respect to frequency. Compute that delay for a thick gap and something odd happens: it stops depending on the width. Doubling the gap doubles the attenuation and does not double the delay, so the apparent speed of the crossing grows without limit as the gap is widened.

The effect is real and has been measured — in frustrated total internal reflection, in undersized waveguides, and in microwave analogues — and the apparent speeds are several times $c$ . It is not a signal travelling faster than light, and the reason is the same as everywhere else that a superluminal number appears: the quantity being timed is the peak of the transmitted pulse, and the transmitted pulse is not the incident pulse arriving late.

What crosses a thick barrier is an exponentially attenuated remnant, and the barrier attenuates the late part of the pulse more than the early part — because the late part has to be reconstructed from components that were already there. The output is therefore built preferentially from the incident pulse’s leading edge, and its peak sits earlier than a naive comparison expects. Nothing has outrun anything; a shape has been reshaped, with the parts that survived being the parts that arrived first.

The test that settles it is the one that always settles it. Put a sharp front on the pulse — a moment before which the field is exactly zero — and ask when the front appears on the far side. It appears exactly at the light-travel time, never earlier, in every calculation and every measurement. The front is what carries information, and it does not move.

The tail that couples two fibres

The gap calculation has a use that is less exotic than tunnelling and rather more consequential.

A mode travelling in an optical fibre is not confined to the core: it decays exponentially into the cladding, over a distance set by the same square root as everything else in this essay. That is why the cladding has to be there at all, and why it has to be many wavelengths thick — not to guide the light, which the core does, but to keep the evanescent tail away from anything that would absorb or scatter it. A fibre whose cladding is thinned or contaminated loses light with no reflection anywhere having failed.

Bring two fibres close enough for their tails to overlap and the light crosses from one to the other. It does so gradually and completely: power oscillates between the two cores along the length of the interaction, going fully across and fully back, at a rate set by the overlap. Cutting the interaction at the right length gives whatever split is wanted — all of it, half of it, a tenth.

That is a directional coupler, and it is the fibre-optic component that makes networks possible. Every splitter, every tap, every interferometer built in fibre is two guides brought near enough for their evanescent fields to reach each other, and its specification is a length.

The same construction at a much smaller scale is how light is moved between waveguides on a chip, and how a resonator is fed. In each case nothing is in contact, nothing crosses a boundary in a ray sense, and the coupling strength is set by a gap and an exponential — which is to say by the two numbers this essay has been computing throughout.

Where the model stops

Everything here is a single frequency in a steady state. A pulse reflecting past the critical angle also suffers a temporal delay, the Wigner delay, which is the derivative of the same phase with respect to frequency rather than transverse wavenumber. It is of order a femtosecond, and the two delays are not independent: they are two components of one derivative.

The beam is assumed wide. The derivation replaces a beam by a narrow bundle of plane waves and keeps only the first derivative of the phase. A beam only a few wavelengths across has a wide angular spread, the second derivative matters, and the reflected beam comes back distorted as well as displaced — the shift then stops being a well-defined number.

Both media are assumed transparent. If the rarer medium absorbs, the evanescent field is absorbed too and the reflection stops being total. That failure is an instrument: attenuated total reflectance spectroscopy presses a sample against a prism and reads its absorption bands in the reflected beam, without the beam ever having entered the sample in a ray sense — a measurement of something a few hundred nanometres thick, made by light that never went there.

And the surfaces are assumed clean and flat to well within a wavelength. The gap figure is drawn for two perfect planes at a controlled separation, and any real pair of surfaces touches at high points. That is why a fingerprint sensor works: a ridge in contact frustrates the reflection locally, a valley does not, and the image is a map of the gap.

One further limit belongs on the list because it is what the whole essay has been circling. A boundary is an idealisation. Everything here assumes the refractive index changes discontinuously from $n_1$ to $n_2$ across a mathematical plane, and no real interface does that: there is always a transition region of a few atomic layers, and often a great deal more where a surface is polished, coated or contaminated. That region is thin compared with a wavelength, which is why the idealisation works; it is not thin compared with the evanescent depth at grazing incidence, which is why the sharpest measurements of these effects are also measurements of how clean a surface is.

What the pictures cannot show

The evanescent field itself appears in no figure here. What is drawn is its decay length and its consequences — the shift, the transmission across a gap — because the field is not a propagating wave and has no wavefronts to draw. Every picture in this essay is a picture of a number that the field determines.

Nor do the figures show that the reflection remains total while all of this happens. The lateral shift, the excursion into the far medium and the phase change all occur at a reflectivity of exactly one, and a drawing that shows the light going somewhere naturally suggests some of it stayed there. None of it does.

Where the ladder goes next

This ladder began with the angle past which light cannot leave. This rung is about what is on the other side of a boundary nothing crosses. The rungs above it: waveguide modes, where counting reflections stops working and the evanescent tails of the field decide which modes exist at all; the Fresnel rhomb, which converts linear polarisation to circular using nothing but the phase difference between the two curves in the coefficient figure; surface plasmons, where the evanescent field couples to an electron oscillation in a metal film and the reflection stops being total at one particular angle; and near-field microscopy, which uses the evanescent field as an illuminator and beats the diffraction limit because a decaying field has no wavelength to be limited by.

The habit worth carrying away is about what a limit describes. A ray is what a wave looks like when everything is much larger than a wavelength, and the evanescent depth is the scale at which “much larger” fails. Every phenomenon in this essay lives within a few hundred nanometres of a surface, which is exactly where the summary stops summarising — and the summary gives no warning, because it has no length in it to compare anything with.

The reflection that happens where the glass is not

The field that is there and carries nothing