Illustrated Physics
physics, drawn
Optics

The mirror that lights a tenth of a micrometre

Total internal reflection sends back every photon, and it leaves behind a field in the medium beyond that dies away within a fraction of a wavelength. Put a living cell on a coverslip and reflect a laser inside the glass, and that field lights only the hundred nanometres of the cell pressed against the glass: the membrane, and the proteins moving in it. Nothing else glows. The same field works in reverse. A molecule close to the glass sends part of its own light into angles no ray from the water could reach, and how much it sends there says how close it is.
14 min read 5 figures Made into an instrumentWhat happens at the edge

Assumes: The reflection that happens where the glass is not · The angle past which light cannot leave

The reflection that happens where the glass is not found that total internal reflection is total in its energy and not in its geography. Every photon comes back, but the light reaches a short way into the medium beyond before it turns around: the beam re-emerges displaced, and a field sits on the far side of a boundary that no ray crosses. That essay used the field to explain a displacement. The angle at which total reflection stops being total used it to couple into electron waves on a thin film of gold, making a sensor.

This essay uses the field for the simplest thing it can do, which is to illuminate. A layer of light a tenth of a micrometre thick, lying against a glass surface, cannot be made with lenses: no focus is that thin over a whole field of view. Total internal reflection makes one automatically, over the whole area of the reflection, and biology found a use for it at once. Then the same field turns out to work backwards. A molecule sitting inside it sends light into the glass at angles that no ray from the water could reach, and that light is a measure of where the molecule is.

A field that ends in a hundred nanometres

When light travelling in glass strikes a boundary with water beyond the critical angle, it is totally reflected. In the water it leaves a field that oscillates along the surface and decays away from it exponentially. The intensity falls by a factor of ee over a depth

d=λ4πn12sin⁡2θ−n22,d = \frac{\lambda}{4\pi\sqrt{n_1^2\sin^2\theta - n_2^2}},

with λ\lambda the vacuum wavelength, n1n_1 and n2n_2 the indices of the glass and the water, and θ\theta the angle of incidence in the glass.

A sheet of light a tenth of a micrometre thick. The intensity of the evanescent field in water beyond a glass coverslip (index 1.518 against 1.33), against distance from the glass, for 488 nm light totally reflected at 62°, 65°, 70°, 75°, s-polarised, relative to the light arriving in the glass. The critical angle is 61.2°. At 62° the field at the surface is 3.79 times the incident intensity and falls by e in 234 nm; at 65° the field at the surface is 3.07 times the incident intensity and falls by e in 110 nm; at 70° the field at the surface is 2.01 times the incident intensity and falls by e in 75 nm; at 75° the field at the surface is 1.15 times the incident intensity and falls by e in 63 nm. Ordinary illumination (dashed) lights the whole depth of a cell equally; totally reflected light lights only the layer against the glass, which is where a cell's membrane and the proteins in it are.
Fig. 1 The evanescent intensity in water beyond a coverslip (1.518 against 1.33), relative to the incident intensity, for 488 nm light totally reflected at 62°, 65°, 70° and 75°, s-polarised. The critical angle is 61.2°. The surface intensity is 3.79, 3.07, 2.01 and 1.15 times the incident, and it falls by ee in 234, 110, 75 and 63 nm. Ordinary illumination (dashed) lights every depth equally.

The figure plots that field for blue laser light at 488 nanometres, reflected inside a standard glass coverslip with water beyond, at four angles past the critical angle of 61.2 degrees. At 65 degrees the intensity at the surface is three times the incident intensity and falls by ee in 110 nanometres, so by 300 nanometres from the glass it has fallen to a fifteenth of its surface value. Ordinary illumination from below, drawn dashed for comparison, lights every depth equally.

A cell a few micrometres thick lying on the glass is therefore lit in its bottom tenth of a micrometre and nowhere else. That region holds the cell’s membrane where it touches the glass, the proteins embedded in it, the small vesicles that fuse with the membrane to release their contents, and the adhesion structures by which the cell grips its surface. Label a protein with a fluorescent tag and illuminate the cell this way, and only the copies within the sheet glow. The fluorescence from the rest of the cell, which in ordinary illumination swamps the faint signal from the membrane, is simply absent.

This is total internal reflection fluorescence microscopy, developed by Daniel Axelrod in the early 1980s. Its contrast comes entirely from the geometry of the illumination. It made it possible to watch single molecules in living cells — to see a vesicle arrive at the membrane and release its contents, one event at a time, or to follow individual receptor proteins as they diffuse and bind — because a single fluorescent molecule gives off about as much light as the background from a few hundred nanometres of cell, and the evanescent sheet removes that background.

A depth set by an angle

The penetration depth depends on the angle, and it can be chosen by choosing the angle.

The depth an angle chooses. The intensity penetration depth of the evanescent field in water beyond glass (1.518/1.33) against the angle of incidence in the glass, for 488 nm and 640 nm light. It is infinite at the critical angle, 61.2°, falls steeply just past it and levels off: 158 nm at 63°, 99 nm at 66°, 69 nm at 72° for 488 nm, and 53 nm at grazing incidence. Red light reaches further in proportion to its wavelength. A cell's cytoplasm, at an index near 1.37, raises the critical angle to 64.5° and deepens the field at every angle.
Fig. 2 The intensity penetration depth beyond a glass–water boundary against the angle of incidence, for 488 and 640 nm light. It is infinite at the critical angle, 61.2°, and falls steeply past it: 158 nm at 63°, 99 nm at 66°, 69 nm at 72°, and 53 nm at grazing incidence for 488 nm. Red light reaches further in proportion to its wavelength.

The figure plots it against the angle of incidence for two laser wavelengths. At the critical angle the field does not decay at all, because the light is just grazing along the boundary. A degree or two past it the depth has fallen to a few hundred nanometres, and by 66 degrees it is 99 nanometres at 488. After that the curve flattens, and even at grazing incidence it only reaches 53 nanometres. Red light at 640 nanometres reaches further in proportion to its wavelength.

The steep part is where the control is. Instruments steer the laser across the back of a high-aperture objective to set the angle, and a change of a fraction of a degree near the critical angle changes the depth by tens of nanometres. Imaging the same cell at several angles gives several differently weighted views of the region near the glass, and the distances of fluorescent structures from the membrane can in principle be recovered from them. In practice the recovery is hard, because it means undoing a set of exponential weightings — a problem in which small errors in the data produce large errors in the answer — and variable-angle measurements give distances to tens of nanometres only with careful calibration.

The water is also not quite water. A cell’s cytoplasm has an index nearer 1.37, which raises the critical angle to 64.5 degrees and deepens the field at every angle, and the index varies from place to place inside the cell. The depths drawn are for the medium the cell sits in; inside the cell they are somewhat larger, and the exponential is only an approximation where the index changes over the decay length. Light scattered by the cell’s own structures also leaks into the image, so the sheet is not perfectly clean. What survives these corrections is the order of magnitude, a hundred nanometres, and that is what the method relies on.

More light at the surface than was sent

One number in the first figure looks wrong. At 62 degrees the intensity at the surface is 3.79 times the intensity of the light arriving in the glass. The light has been totally reflected, none of it has gone into the water, and yet the field in the water is stronger than the field that arrived.

More light at the surface than was sent. The intensity of the field in the water right at the glass surface, relative to the incident intensity in the glass, against the angle of incidence, for s and p polarisation (glass 1.518, water 1.33). Below the critical angle it is the ordinary transmitted field. At the critical angle it peaks — 4.00 for s and 5.21 for p — because the incident and totally reflected waves add in phase at the surface, and beyond it falls to zero at grazing incidence. Nothing is created: the evanescent field carries no energy away from the surface, and all of the light is reflected. The surface is simply where two waves overlap.
Fig. 3 The field intensity in the water at the surface, relative to the incident intensity in the glass, against the angle of incidence, for s and p polarisation. Below the critical angle it is the ordinary transmitted field. At the critical angle it peaks at 4.00 for s and 5.21 for p, and beyond it falls to zero at grazing incidence.

The figure plots the surface intensity against the angle for both polarisations. Below the critical angle it is the ordinary refracted field. At the critical angle it peaks, at exactly four for s-polarised light and at 5.21 for p, and beyond the critical angle it falls, reaching zero at grazing incidence.

Nothing is being amplified. At total reflection the reflected wave has the same amplitude as the incident one, and at the surface the two overlap. The field in the water must match the field just inside the glass, which is the sum of the two waves, and at the critical angle they add in phase. Two waves of equal amplitude in phase make twice the amplitude and four times the intensity, which is the s-polarised value. For p polarisation the field in the water also has a component perpendicular to the surface, and the boundary condition on that component multiplies it by the square of the index ratio, which gives 5.21 here. None of this carries energy into the water: the evanescent field’s energy flows along the surface and back, and averaged over a cycle nothing crosses the boundary.

The practical point is that a molecule in the sheet is excited more strongly than it would be by the same laser shone straight at it, and most strongly near the critical angle, where the sheet is also thickest. Choosing an angle trades the brightness of the sheet against its thinness.

Light sent where no ray could go

Now turn the problem round. A fluorescent molecule in the water, lit by the sheet, emits its own light, and some of it enters the glass and is collected by the objective below. Ray optics says that light from the water can enter the glass only within the critical angle, because refraction maps the whole hemisphere of the water onto a cone in the glass, the reverse of the cone light has to find to get out. No ray from the water arrives in the glass at a steeper angle than 61.2 degrees.

A molecule close to the glass nonetheless sends light there. It is not a ray. It is the molecule’s own near field — the part of its field that stores energy rather than radiating it, which the distance where a field changes its mind found dominating within a fraction of a wavelength of any source. That field contains components that vary faster along the surface than any wave in water can, the same components the fan of plane waves inside every beam found decaying outside the travelling band. In the water they decay. If the glass is close enough for them to reach, they couple into it and propagate, because in the denser glass those variations along the surface are slow enough to travel. They enter the glass beyond the critical angle.

Light sent where no ray could go. The light emitted into the glass per unit solid angle by a fluorescent molecule in the water, averaged over its orientation, against the angle in the glass, at 520 nm, for the molecule 0 nm, 50 nm, 150 nm from the surface, relative to the peak at contact. Below the critical angle the emission is ordinary refracted light and does not depend on the height. Above it is light no ray from the water could reach: the molecule's own near field, which extends into the glass only while it is closer than a wavelength. In contact, 49 per cent of the light entering the glass is beyond the critical angle; at 50 nm, 37 per cent of the light entering the glass is beyond the critical angle; at 150 nm, 20 per cent of the light entering the glass is beyond the critical angle.
Fig. 4 Light emitted into the glass per unit solid angle by a fluorescent molecule in the water, orientation-averaged, against angle in the glass at 520 nm, for the molecule touching the glass, at 50 nm and at 150 nm. Below the critical angle the emission does not depend on height. Beyond it — forbidden to rays — it fades as the molecule moves away: 49, 37 and 20 per cent of the light entering the glass.

The figure computes that emission for a molecule at three heights, averaged over the molecule’s orientation, using a symmetry that makes the calculation short. By reciprocity, which the coupling that is the same both ways met for two loops of wire, the light a source at a point sends in a given direction is proportional to the field that a plane wave arriving from that direction would make at the point. For directions in the glass beyond the critical angle, that field is exactly the evanescent field of total internal reflection drawn in the first three figures. So the molecule’s forbidden emission falls off with its height as the excitation sheet does, and peaks near the critical angle where the sheet is brightest.

Below the critical angle the emission is ordinary refracted light and is the same at every height. Above it, it is large while the molecule touches the glass: 49 per cent of the light entering the glass travels beyond the critical angle. At 50 nanometres the share has fallen to 37 per cent, and at 150 nanometres to 20. Lukosz and Kunz, who worked this out in the 1970s, called it forbidden light, and for a molecule on a surface it is not a small correction. A large share of the light a surface-bound molecule gives off goes into directions that geometric optics says are empty.

A ruler read in angle

That decay is an instrument.

A ruler read in the angle of the light. The share of the light a fluorescent molecule sends into the glass that travels beyond the critical angle, against the molecule's distance from the surface, at 520 nm, averaged over orientation. At contact it is 49 per cent; it falls to 37 per cent at 50 nm and 20 per cent at 150 nm, and the ratio of beyond-critical to ordinary light halves in 71 nm. Comparing the two bands of angle therefore measures how far each molecule is from the glass, to within tens of nanometres, with no scanning at all — the same decaying field that lit the molecule, now read in reverse.
Fig. 5 The share of a molecule’s light entering the glass that travels beyond the critical angle, against its distance from the glass, at 520 nm. It is 49 per cent at contact, 37 per cent at 50 nm and 20 per cent at 150 nm; the ratio of beyond-critical to ordinary light halves in 71 nm.

The figure plots the forbidden share against the molecule’s height. It falls smoothly from 49 per cent at contact, and the ratio of forbidden to ordinary light halves in 71 nanometres. A microscope that collects the two bands of angle separately — which is possible because they arrive at different radii in the back focal plane of a high-aperture objective — can divide one by the other for each molecule and read its height from the curve, to within tens of nanometres, without scanning anything. The method is called supercritical-angle fluorescence, and it has been used to measure how far proteins sit from a membrane and to make a sheet of detection that complements the sheet of illumination.

The excitation and the emission are the same physics in the two directions. Light arriving from the glass beyond the critical angle reaches into the water only a hundred nanometres, and light leaving a molecule reaches into the glass beyond the critical angle only if the molecule is within a hundred nanometres. Reciprocity guarantees that the two depths match. An instrument that uses both — exciting with the evanescent sheet and collecting only the forbidden light — sees only the first hundred nanometres twice over, and rejects the background more thoroughly than either alone.

The same sheet in other instruments

A field a hundred nanometres deep that reacts to whatever enters it is useful well beyond biology, and the same numbers turn up in devices that have nothing to do with fluorescence.

Press a finger on a glass prism inside which light is totally reflected, and the ridges of the fingerprint, which touch the glass, sit inside the evanescent field while the valleys between them, a few tens of micrometres away, do not. Where a ridge touches, the field finds a medium it can propagate in, the reflection is frustrated, and light leaks into the skin instead of returning. The reflected image shows the ridges dark on a bright background. Optical fingerprint readers work this way, as did an early generation of multi-touch screens that lit a sheet of acrylic from its edge and watched for the points where fingers frustrated the reflection. The contrast is perfect for the same reason the microscope’s is: the field reaches only what touches the surface.

Bring a second piece of glass within a fraction of a wavelength instead of a finger, and the light crosses the gap by the process the wall that is not quite a wall describes for a quantum particle at a barrier: the evanescent field reaches the far side before it has decayed away, and a propagating wave starts again there. The transmitted fraction falls exponentially with the gap, at a rate fixed by the same penetration depth as in the second figure. Beam splitters with an adjustable split, prism couplers that feed light into thin films, and the couplers between neighbouring optical waveguides, where two tails swap everything, all use the tail of a totally reflected wave that reaches something before it ends.

In every case the distance scale is the one drawn at the start: tens to a few hundreds of nanometres, set by the wavelength and by how far past the critical angle the light strikes. The microscope uses the tail to excite. The fingerprint reader uses it to be absorbed, and the coupler uses it to be transmitted. The molecule in the last two figures uses it in the other direction, to emit. They are four readings of one field, which exists because a totally reflected wave has to satisfy the boundary conditions on both sides of a surface it does not cross.

What the model leaves out

The figures use the simplest description that shows the effect, and the real system is more complicated in three ways.

The molecule is averaged. A real fluorescent molecule has a definite orientation, and its emission pattern depends on it strongly: a molecule whose dipole stands perpendicular to the glass sends most of its forbidden light into a ring near the critical angle, while one lying parallel sends more into the ordinary cone. Orientation averaging is right for molecules that tumble quickly, and for those that do not, the pattern in the back focal plane is itself used to measure orientation.

The emission rate changes too. Near an interface a molecule’s total rate of emission is altered, because the surface changes the field its own light reflects back onto it. The figures give the angular distribution of the light that enters the glass, not how much light is emitted per second, which also varies with height by tens of per cent.

The cell is not a uniform half-space. Refractive-index variations inside the cell scatter the evanescent field and the emitted light, blurring both the sheet and the ruler, and a coverslip coated with anything — a protein layer, a thin film — changes the depths. Quantitative work calibrates against structures of known height rather than trusting the formula.

Still open: how thin a sheet of light can be made usefully

The evanescent sheet is thin only in one direction, and its thickness has a floor set by the indices: with ordinary glass and water it cannot get much below fifty nanometres at visible wavelengths. Higher-index substrates, such as sapphire or specialised glasses near 1.8, make thinner sheets at the cost of requiring matching objectives and immersion oils. Other approaches — structured illumination, light-sheet microscopes that illuminate from the side, and methods that localise single molecules to a few nanometres in all three dimensions — compete with it and combine with it. Whether evanescent excitation and forbidden-light detection can be combined with those methods to give nanometre axial precision in living cells, without the calibration problems of variable-angle measurements, is an active question, and the answer depends as much on the cell’s own heterogeneity as on the optics.

The habit worth carrying away is to read a field that carries no energy as a field that can still do work. An evanescent field transports nothing across the boundary it clings to, and it excites anything placed inside it; and by reciprocity, a source placed inside it can send light back along the same forbidden paths. Total internal reflection is total for the light and not for the field, and the part that stays behind is a sheet of illumination thinner than any lens can make, which reads, in reverse, as a ruler.

Part 6 of 6

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

Three conditions and the window they leave. For a ray entering a diamond's table straight down, the angle from the normal at which it meets the first pavilion face, the second pavilion face and the top again, against the pavilion angle; diamond's critical angle, 24.4°, is the dashed line. The first two must lie above it, so that the ray is totally reflected; the third must lie below it, so that the ray gets out. The first holds for every pavilion angle drawn. The second fails above 51.9°, where the ray would leak out of the far side of the pavilion. The third fails outside 38.9° to 51.1°, and it is the one that decides. Together they leave a window from 38.9° to 51.1°, and Tolkowsky's 40.75° sits inside it, towards the lower edge. The window a diamond is cut inside Part 5 — A well-cut diamond sends back out of its top most of the light that falls on it, and it does so without a single silvered surface. Light entering the top is totally reflected at one sloping face of the pavilion below, again at the opposite face, and then has to leave through the top — where it must not be totally reflected. Each of those three is a condition on one angle, and together they leave a window a dozen degrees wide inside which a stone looks bright and outside which it looks glassy. The window is wide for diamond and nearly closed for glass, which is why a cutter's skill cannot make glass sparkle.

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. 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. 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. A hemisphere outside is a cone of 16.6° inside. Rays leaving a flat face from a material of index 3.5 into air. Every direction in the hemisphere above the face is reached by a ray from inside a cone of half-angle 16.6°, arcsin(1/n), because refraction at the face fans the cone out; rays steeper than that are reflected back, drawn in the warning colour. The squeeze is exact in the way the invariant requires. The projected solid angle of the inside cone is π sin²θc, which is π/12.25, and multiplied by n² it equals π, the projected solid angle of the whole hemisphere outside — checked on the drawn geometry. So the light that does get out has not gained anything: the radiance inside is n² times higher and the directions available are n² times fewer, and the étendue passing the face is the same on both sides. The cone light has to find to get out Inside a dense material the whole hemisphere of directions outside a flat face shrinks to a narrow cone, and light made inside can leave only if it happens to be travelling within it. For gallium arsenide that is two per cent of the light. Turned round, the same cone keeps light in: a slab of silicon with a rough surface holds the light it admits for fifty-one passes. Both numbers are the n² the optical invariant carries, and neither one breaks it.

The objects named here

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

Critical angleEvanescent waveFluorescence microscopyNear fieldPenetration depthReciprocitySupercritical angle fluorescenceTotal internal reflection