Illustrated Physics
physics, drawn
Waves

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.
17 min read 7 figures Only some values fitWhat happens at the edge

Assumes: The pipe that will not carry a low note · The angle past which light cannot leave

The rung below this one established that a hollow pipe has a lowest note. Confining a wave across its direction of travel fixes one component of its wavevector, and below the frequency at which that component uses up the whole of it there is nothing left to travel with. No length of pipe helps and no amount of power helps: the mode does not exist.

The critical angle for n = 1.468 into n = 1.462. Refracted angle against incident angle. It rises faster than the incident angle and reaches 90° at 84.8°, beyond which no refracted ray exists at all.
Fig. 1 A real fibre’s two glasses, whose indices differ by four parts in a thousand. The critical angle is 84.8°, so only rays within about five degrees of the axis are guided — and that narrow cone is the whole acceptance of the fibre. A hollow pipe would hold light by having walls; this holds it by having almost no difference at all between two materials, and the smallness is the point rather than a limitation.

An optical fibre has no walls. Its core is glass and its cladding is glass, the two differ in refractive index by about half a per cent, and light is free to cross between them at any angle it likes. It nevertheless guides — and, unlike the pipe, it guides at every frequency, with no cutoff for its lowest mode at all. The difference between the two cases is worth taking apart carefully, because it is the difference between a boundary condition and a potential well.

What holds the light in

At an interface between a faster and a slower medium there is an angle past which no refracted ray exists.

What holds the light in is a refusal rather than a barrier. A ray crossing from glass into air bends away from the normal because the wave speeds up, and as the incident angle grows the refracted one runs out of room first — it reaches ninety degrees while the incident ray still has somewhere to go. Past that point Snell’s law has no solution and nothing is transmitted. There is no mirror anywhere in a fibre, and nothing is absorbed at the boundary; the light simply has nowhere else to be.

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 one, reaching ninety degrees at the critical angle and having nowhere to go beyond it. Past that angle the reflection is total — not strong, total — and the transmitted amplitude has become an evanescent field that carries no energy away.

For a fibre with a core index of 1.4630 and a cladding index of 1.4570, the critical angle is 83.7 degrees from the normal, which is 6.3 degrees from the axis. Any ray within that cone of the axis is trapped; any ray outside it leaks out at every bounce and is gone within a metre. That cone is the numerical aperture, and it is small — which is why launching light into a fibre is a precision operation and why a fibre bent too tightly loses its light. It is also the reason the whole subject is one of angles rather than of intensities: what a boundary does to a wave depends on the impedance step and on the angle, and here the step is tiny and the angle does all the work.

Why the reflection is total without a mirror

Nothing at the boundary obstructs anything. Both media are transparent, and the wave does penetrate the cladding — it simply does so in a form that cannot carry energy away.

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 How far past the boundary the field reaches beyond the critical angle. The wavenumber in the second medium is imaginary, so the field is a decaying exponential rather than an oscillation; its 1/e depth is a fraction of a wavelength at steep incidence and diverges as the critical angle is approached from above.

The reflection is total because a decaying exponential transports no net energy — the electric and magnetic fields are ninety degrees out of phase, so the time-averaged Poynting flux is zero. Nothing has been stopped; the far side has been offered a solution it cannot use.

That evanescent tail is not a technicality. It is where a guided mode partly lives, and it is the reason a second piece of glass brought within a wavelength of the boundary lets the light straight through — the tail reaches it, finds a medium in which the wavenumber is real again, and resumes travelling.

Counting the modes

A ray at some angle inside the core bounces between the two boundaries, and after two bounces it is travelling parallel to itself again. For a mode to exist, the round-trip phase must come back to itself modulo 2π2\pi — the transverse resonance condition — which selects a discrete set of angles.

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 3 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 85.2° it is half at 1141 nm, at 86.5° it is half at 787 nm, at 88° it is half at 475 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. 4 Counting what fits, from the other side. Two guided rays at slightly different angles are two solutions of the same transverse resonance condition, and only the angles for which the wave returns in phase across the core survive — the same condition that picks out a string’s notes. How many there are is what separates a single-mode fibre from a multimode one, and it is set by how many half-wavelengths fit across the core.

Collecting the geometry into one dimensionless number gives

V=2πaλncore2−nclad2,V = \frac{2\pi a}{\lambda}\sqrt{n_{\text{core}}^2 - n_{\text{clad}}^2},

which is the core radius measured in wavelengths, multiplied by the numerical aperture. The number of guided modes in a slab is about 2V/π2V/\pi, and a fibre is single-moded when VV is below 2.405 — a number that is the first zero of a Bessel function and arrives from the circular geometry rather than from the physics.

For a standard telecommunications fibre at 1,550 nanometres, that puts the core radius at about four micrometres, which is why the fibre in a transatlantic cable is a hair of glass with a thread of glass down the middle of it.

The one mode that is always there

Here the index guide and the hollow pipe part company, and the reason is a statement about differential equations rather than about optics.

The transverse problem — the profile of the field across the guide — is a one-dimensional eigenvalue problem with a well in the middle: the core is a region where the transverse wavenumber can be real and the cladding a region where it cannot. That is the same equation as a particle in a potential well.

One mode is always there, however small the guide, and the reason is a theorem about one dimension. A symmetric well in one dimension has at least one bound state whatever its depth and width — there is no threshold — because a shallow state can lower its energy by spreading out arbitrarily far, and in one dimension that spreading costs arbitrarily little kinetic energy. In three dimensions it would cost too much, which is why a shallow three-dimensional well can have no bound state at all and a fibre always guides something.

So the fundamental mode of a slab or a fibre never unbinds. Reduce the index contrast toward zero and the mode does not vanish; it spreads, further and further into the cladding, until its field extends over many core diameters and it is a guided mode in name only — but it is still guided, and it is still lossless.

That is a genuinely different situation from a pipe, and the distinction is worth stating in one sentence. A hollow guide’s cutoff comes from a boundary condition, which either can or cannot be satisfied; an index guide’s confinement comes from a well, which always has a state.

What a graded index does instead

Nothing requires the index to change in a step.

How steep a gradient has to be to matter. The index gradient above a hot road against height, on a logarithmic scale, with two constants drawn across it for comparison. At the surface the road's gradient is 6.00e-4 per metre and it dies away over 5 cm. The lower line is the standard atmosphere's, 2.62e-8 per metre, computed from the barometric law and a 6.5 K/km lapse rate rather than quoted, and it points the other way — the index falls with height in the free atmosphere and rises with height above the road, which is why one mirage appears below the object and the other above it. The middle line is the Earth's own curvature, 1.57e-7 per metre. The road beats it by a factor of 3823; the atmosphere reaches 17% of it, which is the whole of why the optical horizon is further off than the geometric one.
Fig. 5 A refractive index that varies smoothly with position, and the rays it produces. There is no boundary anywhere, no reflection anywhere, and the ray path is continuously curved: at every point it bends toward the slower side, which is the same bending that makes a mirage, applied on purpose.
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.
Fig. 6 A graded-index rod, in which every ray launched at a different angle returns to the axis at the same place. A parabolic index profile makes the transverse motion harmonic, and a harmonic oscillator’s period does not depend on its amplitude — so all the rays arrive together, and a rod of the right length is a lens with two flat faces.

That last result is one of the neatest reuses of an old idea in this collection. The transverse ray equation in a parabolic index profile is exactly the equation of a harmonic oscillator, whose period is independent of amplitude, so a graded-index multimode fibre nearly abolishes the modal dispersion that a step-index one suffers from: rays that take longer paths travel through slower regions less, and the delays almost cancel.

What guiding is worth

The alternative to guiding is letting the wave spread.

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 = 1.468 to n = 1.462, against angle of incidence past the 84.82° 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 85° it is 2516 nm, at 86° it is 1039 nm, at 87.5° it is 754 nm, at 89° it is 673 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. 7 What guiding is worth is best seen as the absence of spreading. An unguided wave loses intensity to geometry — its energy spread over a growing area — and a guided one does not, because its cross-section is fixed by the guide. What is left is absorption, and the field reaching into the cladding is where some of that absorption happens. Over a hundred kilometres the difference between losing intensity to geometry and losing it only to absorption is the difference between a signal and nothing at all.

A wave spreading spherically loses a factor of a hundred in intensity for every factor of ten in distance. A wave in a fibre loses 0.2 decibels per kilometre — a factor of ten in fifty kilometres — and the loss is absorption and Rayleigh scattering rather than geometry. That single fact is the whole reason the fibre replaced the microwave relay.

The numbers, for a real fibre

It is worth putting the arithmetic on a real object, because the quantities are extreme in both directions and the extremity is the point.

A standard single-mode fibre has a core radius of 4.1 micrometres and a numerical aperture of 0.13, giving V = 2.2 at 1,550 nanometres — just under the 2.405 that would admit a second mode. The core is therefore about five wavelengths across, and about eighteen per cent of the fundamental mode’s power travels in the cladding rather than in the core it is named after.

The cladding is 125 micrometres across, thirty times the core, and that number is not arbitrary either: the evanescent tail must have died away to nothing before it reaches the coating, or the polymer would absorb the light. A fibre is mostly cladding for the same reason that a coaxial cable is mostly dielectric.

The numbers for a real fibre are worth putting together, because the contrast in them is the whole of guiding. Core and cladding differ in index by four parts in a thousand, so a single encounter with that boundary at normal incidence reflects about two parts in a million — a boundary so weak it is almost not there. At a grazing angle past the critical one, the same boundary reflects everything. One interface, two regimes, and the entire technology lives in the second.

And the loss is 0.2 decibels per kilometre at 1,550 nanometres, which corresponds to the light travelling about twenty kilometres before half of it is gone. In a material that transparent, a block of the glass a kilometre thick would be as clear as a window pane.

Where the ray picture stops

Everything above has been argued twice: once with rays and once with fields, and the two agree on the mode count and the cutoff. They stop agreeing in three places.

Near cutoff a mode is mostly outside the core, and the ray picture, which draws the light inside the glass, has nothing to say about it. The fraction of power in the cladding rises toward one as a mode approaches its cutoff, which is why higher-order modes are stripped by a bend and the fundamental is not.

Partial reflection below the critical angle is not zero, and the ray picture treats it as a clean cut.

Where the ray picture stops is at the boundary itself. The reflected amplitude does not jump from nothing to everything at the critical angle: it rises smoothly and begins well below it, and the two polarisations do not rise identically. A transverse resonance condition has to account for both, which is why a fibre’s two polarisation modes are not quite degenerate — and why a long fibre scrambles polarisation even though nothing in the ray picture distinguishes the two.

And a ray has no phase. The mode count came from a phase condition, and the phase shift on total internal reflection is not zero — it depends on angle and on polarisation, and including it is what turns the naive count into the right one. That phase shift also displaces the reflected beam sideways along the boundary, which is a measurable length and belongs to the same family of facts as the field on the far side of a surface no ray crosses.

Why the window is at 1,550 nanometres

The figure of 0.2 decibels per kilometre was quoted twice above without saying where the wavelength came from. It is not a free choice: it is the bottom of a curve produced by two mechanisms pulling in opposite directions, and it is one of the cleanest examples in engineering of a minimum being found rather than designed.

Scattering falls with wavelength. Glass is a frozen liquid, and the density fluctuations that were present when it solidified are locked in. Light scatters off them exactly as it scatters off the density fluctuations in air, with the same fourth-power dependence — so this is Rayleigh scattering, the same mechanism that makes the sky blue, operating in a solid. It contributes about 0.8 decibels per kilometre at one micrometre and 0.14 at 1.55, and no purification removes it, because there is no impurity involved: the scatterers are the glass’s own frozen disorder.

Absorption rises with wavelength. The silicon–oxygen bond vibrates in the infrared, and the tail of that absorption reaches back toward the near infrared, climbing steeply. It contributes almost nothing at 1.3 micrometres, a couple of hundredths of a decibel per kilometre at 1.55, and it dominates completely beyond 1.7.

One curve falling as the fourth power and another rising exponentially must cross, and the sum has a minimum where they do. That minimum sits at 1,550 nanometres and its value is about 0.15 decibels per kilometre — which is why every long-haul fibre in the world operates there, and why no amount of engineering will move it.

There was a third feature for many years and its removal is a good story in itself. Hydroxyl groups left over from water in the manufacture absorb strongly at 1,383 nanometres, and the overtone put a mountain in the middle of the useful range. Reducing the residual water to parts per billion flattened it, and “low-water-peak” fibre opened the whole span from 1,260 to 1,625 nanometres for wavelength-division multiplexing.

One coincidence decided which window the industry actually settled in. The wavelength of minimum loss is 1,550; the wavelength at which material and waveguide dispersion cancel in a standard fibre is 1,310. For a while the second won, because dispersion was the limiting problem. Then the erbium-doped fibre amplifier arrived, which amplifies optically without converting to electronics — and erbium’s gain band happens to sit at 1,550. Dispersion could be managed by compensating it; a repeater at every span could not.

The loss that turned out to be chemistry

The last section’s numbers make guiding sound inevitable. It was not, and the reason it was not is worth recording, because the obstacle was misdiagnosed for a decade.

In the early 1960s the best optical glass available lost about a thousand decibels per kilometre. That is a factor of 1010010^{100} over a kilometre, which is not a signal in any sense of the word — light entering such a fibre would be gone within twenty metres. The physics of guiding had been understood since the 1920s and was not in question. What everybody assumed was that the loss was intrinsic to glass.

Charles Kao and George Hockham argued in 1966 that it was not. They measured the loss in bulk samples, compared it with what scattering and lattice absorption ought to produce, and concluded that the difference — which was essentially all of it — came from transition-metal impurities: iron, copper, chromium, present at parts per million. Those are removable. Their paper stated a target, twenty decibels per kilometre, chosen because it would allow repeaters far enough apart to beat a copper cable, and argued that fused silica purified to parts per billion would reach it.

The target was met in 1970, by Corning, at seventeen decibels per kilometre, and beaten to four within two years. The figure now is 0.15, which is within about ten per cent of the Rayleigh floor computed above — so the material is very nearly as good as glass can be made, and the remaining loss is the frozen disorder rather than anything anybody put in.

The shape of the episode is what makes it worth telling. A quantity was measured, found to be a hundred thousand times too large, and assumed to be fundamental; the correct move was to compute what it should be and attribute the difference to a cause that could be removed. A measured limit is not a physical limit until somebody has calculated what the physical limit would be, and the gap between the two is where the engineering lives. Kao received the Nobel Prize for it in 2009, forty-three years later, by which time the answer was under every ocean.

Where the model stops

The index difference is treated as small and it usually is. Everything above uses the weakly guiding approximation, in which the two polarisations are degenerate and the modes are nearly transverse. For a high-contrast waveguide — silicon on insulator, where the contrast is a factor of two rather than half a per cent — the approximation fails completely and the modes have to be solved as full vector fields.

Material dispersion has been ignored. The index of glass depends on wavelength, so the guide’s properties do too, and in a real fibre the material dispersion and the waveguide dispersion have opposite signs over part of the spectrum and cancel at 1,310 nanometres. That cancellation is a design choice, not an accident, and it is why long-haul fibre operates where it does.

Nothing here is non-linear. A fibre carrying a watt in a cross-section of fifty square micrometres has an intensity that makes the glass’s index depend on the light in it, and the resulting self-phase modulation, four-wave mixing and soliton propagation are the subject that begins where this one ends.

And bending is out of scope. A bent guide radiates, because the outer part of the mode would have to travel faster than light to keep up with the inner part, and there is a radius below which every guide leaks however good it is. The loss is exponential in the bend radius, which is why the specification for a fibre patch cord names one.

What the pictures cannot show

The mode figures draw a transverse profile as a static curve. A guided mode is a travelling wave along the guide and a standing pattern across it, and no still drawing distinguishes the two directions — which is exactly the distinction the whole subject rests on.

Nor can the ray figures show that a ray is not a thing. A guided mode has one transverse profile; the “ray at 4.2 degrees” that corresponds to it is a construction that reproduces the phase condition, and looking for the ray inside a single-mode fibre is looking for something that is not there.

The same structure, three subjects along

The transverse well is not an optical idea, and three of its other instances are worth naming because each supplies a check on the reasoning.

Sound in the ocean travels through a layer, a kilometre down, where the competing effects of falling temperature and rising pressure give the slowest sound speed in the column. That layer is a graded-index guide with no walls, and a shot fired in it has been heard across an ocean basin. The mechanism is exactly the rod above with the numbers changed.

An electron in a semiconductor heterostructure sits in a well made by a change of band edge rather than of index, and the same theorem applies: a symmetric well in one dimension always binds one state, which is why a quantum well of any depth has a confined level and why the emission wavelength of a laser diode can be tuned by changing a thickness.

And a surface wave — a Rayleigh wave on the ground, or a Lamb wave in a plate — is guided by the surface itself, with an amplitude that decays into the bulk. That is the reason an earthquake’s surface waves arrive last and do most of the damage: they are confined to two dimensions and spread as one over distance rather than as two, so they lose amplitude far more slowly than the body waves that arrived before them.

In all three the same two questions decide everything: is there a well, and how many wavelengths across is it.

Where this ladder goes next

The two rungs of this ladder are the two ways of trapping a wave across its direction of travel: a wall, which imposes a condition and produces a cutoff, and a well, which produces a bound state and does not. Everything else about a guide — its mode count, its dispersion, its bend loss, its sensitivity to what is outside it — follows from which of the two it is.

The habit worth carrying away is a question to ask of any confinement. Is this a boundary condition or a potential? A boundary condition is absolute and produces thresholds; a potential is a competition and produces states that can be arbitrarily weakly bound. Mistaking one for the other predicts a cutoff where there is none, which is exactly the error that the phrase the light bounces off the walls commits.

What is left on this ladder is the coupling between guides: two channels brought close enough that each one’s evanescent tail reaches the other exchange power periodically along their length, which makes a directional coupler, a modulator and — at the point where the exchange is complete — a switch.

Part 2 of 6

This essay is one argument about Guided waves. The others:

The lowest note a pipe will carry. The dispersion relation of a guided wave for three cutoffs, in units where the free wave speed is one. Each curve leaves the vertical axis at its own cutoff and bends toward the diagonal, which is the free wave. Above the cutoff the phase velocity is the slope of the line from the origin and always exceeds one, while the group velocity is the slope of the curve and never does: at k = 2 their product is 1.0000, 1.0000, 1.0000, which is one to four decimal places in every case and is an identity rather than a coincidence. Below the cutoff there is no curve, because there is no travelling wave to draw. The pipe that will not carry a low note Part 1 — A wave squeezed sideways acquires a lowest frequency. Below it nothing travels — the field is there, it is large, and it goes nowhere. Above it the guide is dispersive whether or not anything in it is, and the pattern inside runs faster than light while the signal does not. How far into the ground a surface wave goes. The horizontal and vertical displacements of a Rayleigh wave against depth, in wavelengths, at Poisson's ratio 0.25, each divided by the largest displacement anywhere. Both fall off exponentially, and at one wavelength down the vertical component is 19.3 per cent of its surface value — so the wave is confined to a skin about a wavelength thick, and the thickness is set by the wavelength rather than by anything about the material. The horizontal component changes sign at 0.193 wavelengths, which is not a node of a standing wave but a reversal: above that depth the ground moves one way round its elliptical orbit and below it the other. A long-period wave therefore samples deep rock and a short-period one samples only the surface, which is what makes a seismogram's dispersion a measurement of the structure underneath. The wave a surface is enough to hold Part 3 — A pipe guides with walls and a fibre guides with a slower core. A solid needs neither: one free surface binds a wave that is not a bulk wave bouncing but a separate solution, travelling slower than any wave in the material, dying away exponentially into it, and carrying its energy round a circle instead of over a sphere — which is why it is the part of an earthquake that knocks buildings down.

Where this goes next

The lowest note a pipe will carry. The dispersion relation of a guided wave for three cutoffs, in units where the free wave speed is one. Each curve leaves the vertical axis at its own cutoff and bends toward the diagonal, which is the free wave. Above the cutoff the phase velocity is the slope of the line from the origin and always exceeds one, while the group velocity is the slope of the curve and never does: at k = 2 their product is 1.0000, 1.0000, 1.0000, which is one to four decimal places in every case and is an identity rather than a coincidence. Below the cutoff there is no curve, because there is no travelling wave to draw. The pipe that will not carry a low note A wave squeezed sideways acquires a lowest frequency. Below it nothing travels — the field is there, it is large, and it goes nowhere. Above it the guide is dispersive whether or not anything in it is, and the pattern inside runs faster than light while the signal does not. 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. 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. 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.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down.

What this makes readable

Essays that declare this one a prerequisite.

The objects named here

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

Bound stateCutoffDispersionEvanescentGuided wavesModeNumerical apertureRefractive indexStanding wavesTotal internal reflection