Illustrated Physics
physics, drawn
Concept

Guided waves — where it appears

Waves confined across their direction of travel, so that only a discrete set of transverse patterns propagates. What does the confining decides the behaviour: a wall imposes a lowest frequency below which nothing travels, and a region of slower medium imposes none at all.

Named by 12 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

waves · Guided waves 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.

waves · Guided waves 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.

waves · Guided waves A cone of 14.3°, from two indices and nothing else. On the left, the acceptance cone of a step-index fibre with a core index of 1.4677 and a cladding of 1.4624. A ray entering steeper than the cone reaches the wall inside the critical angle and is refracted out at the first bounce; one inside it is trapped. The sine of the half-angle is √(n₁² − n₂²) = 0.125, which is 7.2° in air. On the right, that number against the fractional index difference between core and cladding. Nothing about the core's diameter appears: a fibre a hundred times thicker accepts exactly the same cone, and takes a hundred times the area's worth of light through it.

The cone a fibre will accept

A fibre takes light from a cone whose half-angle depends on two refractive indices and nothing else — not on how thick it is, not on how long, not on what is shining into it. That single number, squared and multiplied by the core's area, is all the light it will ever carry.

optics · Etendue 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.

waves · Guided waves The étendue, tiled. The phase space of a one-dimensional optical system: position across a twenty-micrometre aperture, against the optical direction cosine n·sinθ, for a system accepting ±0.1. The shaded rectangle is what the beam occupies, and its area — 4 micrometre-radians — is the one-dimensional étendue, the quantity no arrangement of lenses can reduce. Divided by a wavelength of 500 nanometres it is 8, and the rectangle is tiled here with exactly 8 cells of area one wavelength each. That is the whole content of the count: a cell of phase-space area λ is the smallest patch a field can be confined to, because squeezing it in position spreads it in direction by the same relation that gives a slit its diffraction pattern, so an étendue is a number of modes and its conservation is the conservation of a count. The cells are drawn four wide and two high, which is arbitrary: only a cell's area is fixed, not its shape — a beam may be confined tightly in position and loosely in angle or the other way round, and the trade is what an optical system is for.

The invariant that is a count

Étendue is an area times a solid angle, and ray optics gives it no floor — nothing in a ray has a size. Divide it by the square of the wavelength and it becomes a number of modes, its conservation becomes the conservation of a count, and the count has a least value of one. That is where the ray bound hands over to diffraction, and the handover is the same number written two ways.

optics · Etendue Four rays that arrive together. Meridional rays through a fibre whose index falls as the square of the distance from the axis, launched at four angles, drawn against distance along the fibre in millimetres and radius in units of the core radius. Each path is integrated from the ray equation. A steep ray swings out to where the index is lower and travels faster; a shallow one stays near the axis where the index is highest and travels slowest, and in a parabolic profile the two effects cancel: the four rays cross the axis within 0.36 per cent of a pitch of each other, measured off the traced paths rather than assumed. The pitch is 1.11 millimetres and it contains no launch angle, which is the whole of the result. Nothing about what the fibre accepts has changed — the acceptance cone is what the invariant fixed and it is untouched — and everything about when the light arrives has.

The same cone, and a different arrival

The invariant fixes what a guide accepts and says nothing about when it arrives, and the two turn out to be nearly independent. Shaping the index so that the rays which travel furthest also travel fastest cuts the spread in arrival times by a factor of five hundred, at the cost of exactly half the light — and the acceptance cone the invariant governs is untouched throughout.

optics · Etendue 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.

waves · Guided waves 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.

waves · Guided waves 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.

waves · Guided waves 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.

waves · Guided waves The sideways shaking that arrives longest waves first. The transverse ground motion 3000 km from a sudden shear source, synthesised from the computed dispersion of the fundamental Love wave in a 35 km crust, with periods from about 8 to 120 seconds. Nothing arrives before 11.0 minutes, the travel time at the mantle's shear speed. Then the longest periods come in first — 80 s waves at their group velocity of 4.23 km/s, after 11.8 min — and the period shortens steadily through the train, 30 s waves arriving at 14.3 min. The strongest shaking comes last, near 14.8 min, where the group velocity passes through its minimum and waves over a band of periods arrive together. Read off a real seismogram, the arrival time of each period is the crust's dispersion curve, and the crust's thickness with it.

The shaking a uniform Earth could not carry

Early seismographs recorded a large sideways shaking that arrived after the body waves and before the Rayleigh wave, and no theory of a uniform Earth allowed it: a horizontal shear wave at a free surface reflects as itself and nothing holds it there. Augustus Love showed in 1911 that a layer does. A crust slower than the rock beneath traps the shaking between a perfect mirror at the top and total reflection at the bottom, and because the long waves reach deeper and run faster, the order in which the periods arrive is a measurement of how thick the crust is.

waves · Wave motion

Named alongside it

The objects these essays reach for when they reach for this one.

CutoffDispersionNormal modesNumerical apertureEvanescent waveGroup velocityOptical fibreRefractive indexTotal internal reflectionBoundary conditionsElastic wavesStanding waves

All concepts