Illustrated Physics
physics, drawn
Optics

The angle at which total reflection stops being total

Coat the back face of a glass prism with fifty nanometres of gold and total internal reflection still returns almost all the light — except at one angle, where the reflection of one polarisation falls nearly to zero. The light has gone into an electron wave bound to the far surface of the gold, and because that wave's angle depends on whatever touches the gold, the missing reflection has become one of the most sensitive balances in biochemistry.
15 min read 5 figures What happens at the edgeMade into an instrument

Assumes: The reflection that happens where the glass is not · The frequency below which nothing gets in

The angle past which light cannot leave makes a perfect mirror out of a change of speed: beyond the critical angle, light inside glass is reflected completely at the boundary with a less dense medium. The reflection that happens where the glass is not adds the qualification that makes everything interesting. The reflection is total only because nothing on the far side takes the energy the light sends across the boundary as an evanescent wave. Bring a second piece of glass within a wavelength and the evanescent wave tunnels into it. The reflection is total by default, not by necessity.

Among the things that can take the evanescent wave’s energy, one is special. A metal surface carries a wave of its own — a sheet of electrons oscillating back and forth along the surface, with a field that decays away on both sides — and that wave can be driven by light only in a very particular way. Total internal reflection supplies exactly that way. At one angle the reflection of one polarisation almost vanishes, and the angle depends so sensitively on whatever lies against the metal that a layer of protein a few nanometres thick moves it measurably.

A dip in a perfect mirror

The arrangement is named after Erich Kretschmann, who used it in 1968: a glass prism, a gold film a few tens of nanometres thick evaporated on its back face, and the sample — water here — beyond the gold. Light enters the prism and strikes the gold from inside the glass.

The angle at which a total reflection stops being total. Reflectance of light at 633 nm inside a BK7 prism whose face carries a 50 nm gold film with water beyond it, against the angle of incidence, for the two polarisations, from the three-layer Fresnel sum in complex arithmetic. Past the critical angle for glass and water, 61.62°, s-polarised light is reflected almost completely, as total internal reflection requires, apart from what the thin gold absorbs. p-polarised light is reflected too — except near one angle, where the reflectance falls to 0.014 at 71.52°. The dispersion relation of a surface plasmon on gold against water puts its resonance at 71.67°; the difference is the film's finite thickness and the prism on its other side pulling the resonance slightly.
Fig. 1 Reflectance at 633 nm inside a BK7 prism whose face carries a 50 nm gold film with water beyond, against the angle of incidence, for both polarisations, from the three-layer Fresnel sum. Beyond the critical angle for glass and water, 61.62°, s-polarised light is reflected almost completely apart from what the gold absorbs. p-polarised light is reflected too, except near one angle, where the reflectance falls to 0.014 at 71.52°. A surface plasmon on gold against water has its resonance at 71.67° by its own dispersion relation.

Below the critical angle for glass and water, both polarisations are partly reflected and partly transmitted into the water through the thin gold. Beyond it, transmission into the water as a travelling wave is impossible, and s-polarised light — its electric field along the surface — is reflected almost entirely; what is missing is the small fraction the gold absorbs. p-polarised light, with its field partly perpendicular to the surface, is reflected almost entirely too, until the angle approaches 71.5 degrees. There its reflectance drops, within a few degrees, to 1.4 per cent.

Nothing has been transmitted: the water still cannot carry a travelling wave at that angle. The light has been absorbed, and the absorption happens at one angle only, which is the signature of a resonance.

That only one polarisation is affected is the first clue to what is resonating. Whatever takes the energy must be driven by a field component perpendicular to the surface, because that is what p-polarised light has and s-polarised light lacks. A field perpendicular to a metal surface pushes electrons towards or away from it, piling up surface charge; s-polarised light, whose field lies along the surface and across the plane of incidence, can slide electrons sideways but cannot bunch them along the direction the light is travelling. The resonance is a wave of bunched charge, and only one polarisation can bunch it.

The wave the light is feeding

The resonance is with a surface plasmon. The frequency below which nothing gets in explains why a metal’s free electrons make it reflective: below their plasma frequency they respond faster than the light’s field changes and cancel it, which gives the metal a negative permittivity — gold’s is about −12-12 at 633 nanometres. A boundary between a medium of negative permittivity and one of positive permittivity can carry a wave of its own: a ripple of electron density travelling along the surface, with an electric field that decays exponentially into both the metal and the water. It is a relative of the wave a surface is enough to hold and of the channel with no walls, a wave confined not by a guide but by the boundary itself.

That wave has a wavevector along the surface, set by the two permittivities: ksp=k0εmεs/(εm+εs)k_{sp} = k_0\sqrt{\varepsilon_m\varepsilon_s/(\varepsilon_m+\varepsilon_s)}. For gold against water it is larger than the wavevector of light in the water can ever be, which is why light arriving from the water side cannot drive it: the plasmon is slower than the light, with more crests per metre along the surface than any light in the water can supply. Light inside the glass prism, at a steep enough angle, has a component along the surface k0npsin⁡θk_0 n_p\sin\theta that can match it, because the glass is denser than the water. At the matching angle the evanescent wave that total reflection sends through the gold has exactly the plasmon’s wavevector and drives it resonantly. For gold on water under BK7 the dispersion relation says 71.67 degrees; the computed dip sits at 71.52, pulled slightly by the film’s finite thickness and the prism beyond it.

The prism’s role is to make the light slow enough along the surface to keep pace with the plasmon. Total internal reflection is not incidental to the effect. It is the only way, short of patterning the surface, to give light a component along the boundary larger than its wavevector in the medium where the plasmon lives — which is also exactly the condition under which the light cannot escape into that medium.

The plasmon’s own numbers follow from the same dispersion relation, and they say what kind of wave it is. Its effective index along the surface is 1.44 — the prism’s index times the sine of the resonance angle — so its wavelength along the gold is 633 divided by 1.44, about 440 nanometres, shorter than the light’s in water. The imaginary part of the same wavevector says how far it travels before the gold absorbs it: about four micrometres on gold at this wavelength, roughly ten of its own wavelengths, and several times further on silver. A plasmon is a wave, with a wavelength and a direction, but a short-lived one, and its shortness of life is the width of the dip.

Handling a complex permittivity is the whole of the arithmetic. The angle that is two angles shows what happens to Snell’s law when the medium absorbs: the transmitted wave’s direction of travel and direction of decay part company, and the “angle” of refraction becomes a complex number. Here the same complex square roots, kj=εj−ε1sin⁡2θk_j = \sqrt{\varepsilon_j - \varepsilon_1\sin^2\theta}, describe the field in the glass, in the gold and in the water, and the reflectance is assembled from them with nothing approximated.

How thick the gold should be

The film has two jobs that pull in opposite directions. It must be thin enough for the evanescent field to reach through it to the far face, where the plasmon lives, and thick enough that the plasmon, once excited, does not simply leak straight back out through the film into the prism.

How thick the gold has to be. The p-polarised reflectance of the glass | gold | water arrangement against angle for gold films of 30, 40, 50, 60, 70 nm. At 30 nm the dip reaches 0.367 at 71.05°; at 40 nm the dip reaches 0.075 at 71.28°; at 50 nm the dip reaches 0.014 at 71.52°; at 60 nm the dip reaches 0.209 at 71.70°; at 70 nm the dip reaches 0.472 at 71.80°. The deepest dip, found by scanning the thickness in half-nanometre steps, is at 47.0 nm, where it reaches zero to four decimal places: there the energy the plasmon loses by leaking back through the film into the prism equals what it loses by heating the gold, and the reflected wave cancels. Thinner films leak too much and give a broad, shallow dip; thicker ones let too little light reach the plasmon at all.
Fig. 2 The p reflectance against angle for gold films of 30, 40, 50, 60 and 70 nm. The dip reaches 0.367 at 30 nm, 0.075 at 40, 0.014 at 50, 0.209 at 60 and 0.472 at 70. Scanning the thickness in half-nanometre steps, the deepest dip is at 47.0 nm, where it reaches zero to four decimal places.

A 30 nanometre film couples the light in easily, but the plasmon couples out just as easily, and the dip is broad and shallow. A 70 nanometre film lets too little of the evanescent field reach the far face, and the dip is shallow again. Between them, at 47.0 nanometres, the dip goes to zero: the reflected wave is exactly cancelled. That is the condition called critical coupling — the rate at which the plasmon loses energy by leaking back into the prism equals the rate at which it loses energy by heating the gold. The reflected light is the sum of what the film reflects directly and what the plasmon re-radiates, and at critical coupling the two are equal and opposite.

The same balance appears wherever a resonator is fed through a partly transparent wall. The layer that makes a reflection vanish cancels a reflection with a quarter-wave layer of the right impedance; critical coupling cancels one by giving the resonator exactly the right leak. In both, nothing is absorbed by the cancellation itself. Every watt of the light that is not reflected here goes into the gold as heat, by way of the plasmon.

There is a second way to feed the plasmon, and it makes the role of the evanescent wave explicit. Andreas Otto, also in 1968, put the metal on the far side of a thin gap of air or water instead of on the prism: light totally reflected inside the prism sends its evanescent wave across the gap, and where the gap is a few hundred nanometres the wave reaches the metal with enough strength to drive a plasmon on its near surface. The gap does the job the film does in Kretschmann’s arrangement — too wide and nothing reaches the metal, too narrow and the plasmon leaks back — and the same critical coupling picks out a best width. It is the frustrated total internal reflection of the reflection that happens where the glass is not, with a resonance waiting on the far side of the gap instead of a second piece of glass.

A balance that weighs a molecule

The plasmon’s wavevector depends on the permittivity of what lies against the gold. Change the water’s refractive index slightly — dissolve something in it, or let a layer of molecules settle on the gold — and the matching angle moves.

A dip that moves when anything touches the gold. The p reflectance of the glass | 50 nm gold | sample arrangement for samples of index 1.333 and 1.343, and the angle of the dip against the sample's index from 1.330 to 1.340. The dip moves from 71.52° to 72.99° for an index change of 0.010, and over the whole range at 144° per unit of index. A dip whose position is read to a thousandth of a degree therefore registers a change of index of 7.0 × 10⁻⁶, averaged over the thin layer next to the gold that the plasmon's field reaches — the scale on which molecules binding to the gold are detected.
Fig. 3 The p reflectance with samples of index 1.333 and 1.343 against the 50 nm film, and the dip’s angle against the sample index from 1.330 to 1.340. The dip moves from 71.52° to 72.99° for a change of 0.010, at 144° per unit of index over the range. Read to a thousandth of a degree, it registers a change of index of 7.0 × 10⁻⁶ in the layer the plasmon’s field reaches.

The dip moves by 1.47 degrees for a change of 0.01 in the sample’s index, 144 degrees per unit of index. A dip whose position can be read to a thousandth of a degree — commercial instruments do better — registers a change of index of a few parts in a million. That is the change produced when a thin layer of protein binds to antibodies fixed on the gold, and it is measured continuously, without labelling the molecules with anything. Since the 1990s surface plasmon resonance has been a standard way of measuring how fast and how tightly one biological molecule binds another, and most of those measurements are this dip, moving.

The sensitivity is concentrated at the surface, which is what makes it useful. The plasmon senses only the layer within reach of its evanescent field, so a change in the bulk of the solution far from the gold matters far less than a change in the first hundred nanometres, where binding happens.

Why the sharpest dip is not the one used

Why the sharpest dip comes from the least lossy metal. The p reflectance against angle for films of silver, gold and aluminium between a BK7 prism and water at 633 nm, each at the thickness that gives it its deepest dip, with permittivities from tabulated values: silver, ε = −18.30 + 0.48i, 55 nm, dip at 67.82° and 0.79° wide at half depth; gold, ε = −12.45 + 1.29i, 47 nm, dip at 71.46° and 5.54° wide at half depth; aluminium, ε = −55.90 + 20.80i, 12 nm, dip at 64.78° and 18.38° wide at half depth. The width is set by how fast the plasmon loses energy into the metal, which grows with the imaginary part of the permittivity relative to the real: silver's resonance is the sharpest, aluminium's so broad it is barely a resonance. Gold is used anyway, because silver tarnishes and gold does not.
Fig. 4 The p reflectance for silver, gold and aluminium films between BK7 and water at 633 nm, each at the thickness giving its deepest dip, with tabulated permittivities: silver, ε = −18.30 + 0.48i, 55 nm, dip at 67.82° and 0.79° wide at half depth; gold, ε = −12.45 + 1.29i, 47 nm, at 71.46° and 5.54° wide; aluminium, ε = −55.90 + 20.80i, 12 nm, at 64.78° and 18.38° wide.

The width of a resonance is set by how fast the resonator loses energy, the relation the width that is a lifetime establishes for any oscillator. A plasmon loses energy into the metal through the imaginary part of the metal’s permittivity, the part that describes electrons colliding and turning their oscillation into heat. Silver has the smallest imaginary part relative to its real part at this wavelength, and its dip is less than a degree wide; gold’s is seven times wider; aluminium, lossy at red wavelengths because of a band transition near 800 nanometres, barely has a resonance at all.

A sharper dip is easier to locate precisely, so silver would make the more sensitive instrument. It is not used, because a silver film exposed to air and to biological solutions tarnishes within days, and the chemistry for attaching molecules to gold through sulphur atoms is simple and robust. The sensor is built from the second-best metal for the resonance and the best metal for everything else, and the design of an instrument is usually a matter of that kind of trade.

The field the resonance piles up

The field the plasmon piles up at the surface. The intensity of the magnetic field just beyond the gold, in the water, relative to the incident light's, against the angle of incidence, for the same 50 nm film. At the resonance, 71.03°, it is 12.6 times the incident intensity, although almost no light is reflected and none is transmitted into the water as a propagating wave: the energy is held in the plasmon and dissipated in the metal. Into the water the intensity falls by a factor of e in 96 nm — set, as for any evanescent wave, by the angle alone — so the resonance senses a layer of the order of a hundred nanometres thick and nothing beyond it.
Fig. 5 The intensity of the magnetic field just beyond the gold, in the water, relative to the incident light’s, against angle, for the 50 nm film. At the resonance, 71.03°, it is 12.6 times the incident intensity, although almost no light is reflected and none is transmitted as a travelling wave. The intensity falls by a factor of e within 96 nm of the gold, a length set by the angle alone.

At resonance the field at the gold’s far surface is more than twelve times as intense as the light arriving in the prism. That is not a contradiction of energy conservation: the plasmon stores energy fed in continuously at the resonance and dissipates it at the same rate, and a resonator driven at its resonance always holds a field larger than the one driving it, by a factor related to its quality — the build-up the frequency that gets an answer derives for a driven oscillator, where the response at resonance exceeds the push by exactly the quality factor. The enhancement is why the same effect, at the rough surfaces and sharp tips of metal nanostructures, can amplify the Raman signal of a single molecule by factors of millions.

The field decays into the water within about a hundred nanometres — ninety-six for the intensity, twice that for the amplitude — and this length, as for any evanescent wave, is fixed by the angle of incidence alone. It sets the thickness of the layer the sensor measures. How far a field gets into metal supplies the other side: into the gold, the plasmon’s field decays within about twenty-five nanometres, which is why the film cannot be much thicker than fifty.

Where the three-layer model stops

Smooth, uniform films. The calculation treats the gold as a slab of uniform permittivity with perfectly flat faces. Real evaporated films are polycrystalline, slightly rough, and their permittivity depends on how they were made; the tabulated values vary between sources by ten per cent in the imaginary part, which changes the dip’s width by about as much.

Bulk permittivities. A film only fifty nanometres thick is thick enough to have the bulk metal’s optical response, just. Much thinner films, of a few nanometres, do not: their electrons scatter off the surfaces and the permittivity changes.

Plane waves. The figures use infinitely wide beams at one exact angle. Instruments focus a wedge of angles onto the film and read the dip off a camera, or scan the wavelength at a fixed angle; both measure the same resonance, with a resolution set by the optics.

A stable sample. The dip moves with anything that changes the index near the gold, and temperature is one of those things: water’s index falls by about one part in ten thousand for every degree it warms. A drift of a hundredth of a degree moves the dip by as much as a change of one part in a million in what is dissolved, so instruments hold the sample’s temperature to millidegrees and compare a measuring spot with a reference spot on the same film.

Classical electrons. The permittivity is a measured number standing for the whole response of the metal’s electrons. For structures only a few nanometres across the plasmon’s field varies on the scale of the electron’s own wavelength and the local description fails, which matters for the smallest plasmonic devices and not for a flat film.

The ripple of charge behind the dip

Every figure here is a reflectance or an intensity against angle, and none draws the plasmon itself — the ripple of charge running along the gold’s far face, compressions and rarefactions of electron density a few hundred nanometres apart, with field lines looping from the positive regions to the negative ones through the water and the metal. Nor do the figures show where the energy goes along the surface: a plasmon excited by a finite beam travels some micrometres along the gold before it dies, re-radiating into the prism as it goes, and the reflected beam is shifted along the surface in consequence — a much larger relative of the shift the reflection that happens where the glass is not measures for bare glass.

Still open: how small a plasmonic sensor can be

A flat-film sensor averages over a spot tens of micrometres across. Plasmons confined to single metal nanoparticles, whose resonances shift in the same way when molecules bind to them, can in principle detect single molecules, and have done so in favourable cases, but their resonances are broad because small particles radiate and absorb strongly, and the shift one molecule produces is a small fraction of the width. How far the detection limit of plasmonic sensing can be pushed — by particles shaped to concentrate the field in gaps of a nanometre or two, by coupling particles so that their combined resonance narrows, or by reading the resonance through its noise rather than its position — is being actively explored, and the answer depends as much on the chemistry of attaching molecules to exactly the right spot as on the optics.

The habit worth carrying away is to ask of any perfect result what it assumes about its surroundings. Total internal reflection is total because nothing on the far side can accept the evanescent wave, and a resonance on the far side that can accept it turns the perfect mirror into a perfect absorber at one angle. The disappearance of a reflection is then a measurement, and its angle reports on whatever is in the thin layer where the evanescent wave and the resonance meet.

Part 3 of 4

This essay is one argument about Total internal reflection. The others:

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. The reflection that happens where the glass is not Part 2 — Total internal reflection sends back every photon, which is why it is called total. It does not send them back from where they arrived — the beam re-emerges displaced along the surface, by a fraction of a wavelength, and a displacement is only possible if the light spent time on the far side of a boundary it never crossed. Six ways into a corner, each leaving from the opposite side. A corner cube seen along its axis: its hexagonal face, divided into six sectors by the three edges where its mirror faces meet and by their extensions. A ray entering along the axis meets the three faces in an order fixed by the sector it enters: y → x → z, y → z → x, z → y → x, z → x → y, x → z → y, x → y → z. Traced reflection by reflection, every ray leaves travelling exactly back along its arrival direction, to rounding error, and it leaves from the point diametrically opposite where it entered — in the sector whose order is the reverse of its own. The corner that sends light home Part 4 — Three mirrors at right angles send light straight back the way it came, from any direction they can see, because each one reverses one component of its direction. Made from solid glass, a corner reflects by total internal reflection and needs no silver — which is why the reflectors left on the Moon are bare glass — but each of its six sectors then returns a different polarisation, and the returned spot loses three-quarters of its peak.

Where this goes next

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. The reflection that happens where the glass is not Total internal reflection sends back every photon, which is why it is called total. It does not send them back from where they arrived — the beam re-emerges displaced along the surface, by a fraction of a wavelength, and a displacement is only possible if the light spent time on the far side of a boundary it never crossed. A lens with two flat faces. Rays entering a rod whose refractive index falls parabolically from the axis outward, parallel to the axis and at -2.4, -1.2, 0, 1.2, 2.4 mm from it. The ray equation in such a medium is the harmonic oscillator's, so each path is a cosine of the same period whatever height it started at — which is exactly the condition for a focus, and the reason all of them cross the axis together at 12 mm. Nothing is curved anywhere: the faces are flat and the bending is done by the inside of the glass. Cut the rod at a quarter of the period and it images; cut it at half and it relays the beam parallel again, inverted. The period is a property of the profile alone, which is why a rod like this is specified by a length rather than by a curvature. The channel with no walls A pipe will not carry a note below its cutoff, and no length of pipe helps. Replace the walls with nothing but a region where the wave travels slightly slower, and the cutoff disappears — however weak the contrast and however thin the channel, at least one mode is bound. The difference is not a matter of degree; it is the difference between a boundary condition and a potential well. What a real coating leaves behind, and over what range. Reflectance against wavelength for a glass surface of index 1.52 in air, uncoated and with quarter-wave layers of 2 different indices, each a quarter of a wave thick at 550 nm. The ideal index is the geometric mean, 1.2329, and the layer made of it takes the reflectance to zero at the design wavelength exactly. No durable solid has that index: magnesium fluoride at 1.38 is the usual compromise and it leaves 1.26% at the design wavelength against 4.26% bare — a reduction of 3.4× rather than a removal. Both curves rise away from the design wavelength, because the thickness is a quarter of a wave only there, and the useful band is wide but not unlimited: the better coating stays under a quarter of the bare reflectance from 415 to 780 nm. The purple cast of a coated lens is that residual — the ends of the visible reflecting while the middle does not. The layer that makes a reflection vanish A wave meeting a step in impedance reflects, and nothing can be done about the step. Put a third medium between the two, a quarter of a wavelength thick and of exactly the intermediate impedance, and the reflection stops existing — not reduced, cancelled.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

BiosensorCritical couplingEvanescent waveFresnel equationsPermittivityResonanceSurface plasmonTotal internal reflection