Illustrated Physics
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Guided waves — the series

10 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · waves
  2. 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.

    part 2 · waves
  3. 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

    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.

    part 3 · waves
  4. A bend of 5 mm, and where the field has to give up. The transverse profile of the guided mode, with the core shaded and the radius at which a bend of 5 millimetres would require the field to outrun the cladding marked. Inside the core the field is a cosine; outside it decays, and the decay is what keeps the mode together. Bending the guide imposes a rigid rotation, so the field a distance x from the axis must travel faster in proportion to x — and past 13.5 micrometres it would have to travel faster than the cladding permits. There the field can no longer be evanescent, and what is there radiates away. The amount there is 1.63e-2 of the peak, which is why the loss is negligible until the caustic moves in, and then is not.

    The mode that will not turn a corner

    Bend a waveguide and the field has to go round with it, which means the part furthest from the centre has to travel faster. Past a certain distance it would have to travel faster than the surrounding medium allows, and everything out there radiates away — which is why bend loss is exponential in the radius and arrives all at once.

    part 4 · waves
  5. Two guides, and the two solutions they have. The transverse field of the two modes a pair of identical slab guides supports, against position across them, for guides half a micrometre wide separated by a gap of 300 nanometres at a wavelength of 1550 nanometres. A single guide has one fundamental mode; two guides side by side have two, and neither of them lives in one guide. The symmetric one is a single hump spanning both, the antisymmetric one has a node exactly between them — checked here to be exactly zero rather than nearly so — and they have slightly different propagation constants because the symmetric one has more of its field in the high-index gap region. That difference, computed from the slab's own dispersion condition, is 4.60e-2 per micrometre. Everything else about a coupler follows from it: a wave launched into one guide alone is the sum of the two supermodes in equal parts, they run at different speeds, and the interference between them moves the power from one guide to the other and back. There is no leakage in the account anywhere — only two solutions beating.

    Two tails that swap everything

    Bring two guides close enough for their evanescent tails to overlap and they do not leak a little power into each other. They exchange all of it, and then exchange it back, over a length fixed by the splitting between two modes that belong to neither guide — so a coupler is cut to a length rather than tuned to a ratio, and the length depends exponentially on a gap of a few hundred nanometres.

    part 5 · waves
  6. The light follows one supermode through the crossing. Inside a tapered coupler 800 µm long at 1550 nm. Above: the fraction of the light in the first guide along the coupler, from integrating the coupled-mode equations, and dashed, the share of the first guide in the local supermode the light was launched into. They agree to within 8.3 percentage points along the whole length: the light does not beat between the guides but follows the supermode as that supermode changes from being in the first guide to being in the second. Below: the two supermodes' propagation constants relative to their average, which approach each other and repel across a gap of twice the coupling, 0.084 per micrometre, at the point where the guides are equally wide; dashed, the two single guides' constants, which cross.

    The coupler that does not care about the colour

    Two identical guides side by side swap their light back and forth, and a coupler cut to the length of one swap works perfectly at one wavelength and badly at every other. Make the guides unequal, and sweep the inequality from one sign to the other along their length, and the light no longer swaps — it follows a single mode of the pair as that mode moves from one guide to the other. The transfer is then nearly complete across hundreds of nanometres of wavelength and immune to the widths being made wrong, and it costs length.

    part 6 · waves
  7. One mode that loses less the faster it is driven. The wall loss, in decibels per kilometre, of three modes in a copper pipe 60 mm across, against frequency on logarithmic axes: TE₁₁, the lowest mode, cut off below 2.93 GHz; TM₀₁, cut off below 3.82 GHz; and TE₀₁, cut off below 6.09 GHz. Each loss is infinite at its cutoff and falls just above it. Then TE₁₁ and TM₀₁ turn and rise with frequency, as the square root, the way every conductor's loss does. TE₀₁ keeps falling: 0.51 dB/km at 60 GHz and 0.20 dB/km at 110 GHz, against 21 dB/km for TE₁₁ at 60 GHz.

    The loss that falls as the frequency rises

    Every metal conductor loses more at higher frequencies, because the current crowds into a thinner skin and meets more resistance. One wave in a round metal pipe breaks the rule. The TE₀₁ mode's electric field circles the axis and touches the wall nowhere. As the frequency rises it becomes more and more nearly a plane wave running down the middle of the pipe, and it drives less and less current in the metal. Its loss falls with frequency, to about half a decibel per kilometre at 60 GHz in a pipe 60 mm across. In the 1960s it was the best way anyone knew to send a hundred thousand telephone calls across a continent. The catch was that the pipe also carried seven hundred other modes.

    part 7 · waves
  8. What a station hears from one thunderstorm region. The power spectrum of the vertical field from a single region of thunderstorms, received 20° and 90° of arc away, from a sum of the first seven resonant patterns, each with the measured peak frequency and a quality factor of 4 to 6, both taken as inputs. Measured spectra fall off with frequency rather more than this one, because real strokes and the real ionosphere are not as simple. The resonances are so broad that neighbouring peaks run into each other; a quality factor of five means a peak about a fifth as wide as its frequency, and the ringing dies in a few cycles. At 20° every tone is present, the higher ones merging. At 90° the first, third and fifth have vanished, because those patterns have a node a quarter of the world away, and the spectrum is made of the even tones alone. Real stations receive the sum of the world's storms, which move with the time of day, and the heights of the peaks drift with them.

    The tone the whole Earth rings at

    The ground conducts and so does the ionosphere, eighty kilometres up, and between them lies a waveguide the size of the planet. Two walls, unlike a pipe, carry waves of any length, so the guide takes radio waves forty thousand kilometres long — longer than the world is round — and, closed on itself, it rings. Lightning strikes fifty times a second and keeps it ringing at about 7.8, 14.3 and 20.8 hertz. Perfect walls would put the tones at 10.6, 18.4 and 25.9; the difference is the ionosphere, a ceiling that leaks.

    part 8 · waves
  9. Every wave a plate can carry, at every frequency. The phase velocities of the guided waves in an aluminium plate (longitudinal speed 6.32 km/s, shear 3.13), against the product of frequency and thickness in MHz·mm, from the Rayleigh–Lamb equations: symmetric modes, in which the two faces bulge and thin together (light), and antisymmetric ones, in which they bend together (dark). At low frequency only two exist. The symmetric S₀ is a stretching wave at the plate velocity, 5.44 km/s; the antisymmetric A₀ is a bending wave whose speed falls towards zero, 0.96 km/s at 0.1 MHz·mm. At high frequency both approach the 2.92 km/s of a Rayleigh wave on each face. Higher modes enter at cut-off frequencies where the plate's thickness holds a whole number of half-wavelengths of one bulk wave — the first at 1.56 MHz·mm — arriving from infinite phase velocity and slowing as the frequency rises.

    The plate that carries two kinds of sound

    Tap the edge of a sheet of aluminium and the sound that runs along it is not one wave but two, and neither is the sound of the metal. One stretches the sheet and travels at 5.44 km/s, faster than a shear wave and slower than a compression wave in the same metal. The other bends it, travels at a speed that depends on how high the note is and how thick the sheet is, and at low frequency crawls at a few hundred metres a second. Both are made of the same two bulk waves bouncing between the faces, and the two faces together decide everything.

    part 9 · waves
  10. A pulse that comes out of the average. The cross-correlation of the two stations' records against lag, in seconds: for single sources at 0°, 45° and 90° from the line through the stations, each a copy of the noise's own ten-second wavelet shifted to that source's arrival difference — 20.0 s, 14.1 s and 0 s — and the average over all 720 sources round the ring, scaled up 9-fold to share the axes. The single-source correlations are wherever their geometry puts them. The average has peaks at ±20 s, the time a wave takes to cross 60 km at 3 km/s, one for travel from A to B and one from B to A, and little between them. That pulse is the signal a seismometer at B would have recorded had an earthquake happened at A: the Green's function between the stations, recovered from noise that never came from either.

    The echo hidden in random noise

    A seismometer that is not recording an earthquake is not recording silence. It hears the Earth's hum: the pounding of ocean waves on every coast, arriving from all directions as surface waves with periods near ten seconds, indistinguishable from random noise. Two seismometers sixty kilometres apart hear different noise. Yet if their records are correlated and averaged over hours, a pulse appears at a lag of twenty seconds — the time a wave takes to travel from one to the other — exactly as if an earthquake had happened under one station and been recorded at the other. Noise from everywhere contains the echo between any two points, and since 2005 seismologists have mapped the Earth's crust with it.

    part 10 · waves

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