Illustrated Physics
physics, drawn
Concept

Cutoff — where it appears

The frequency below which a guide carries nothing, imposed by a boundary condition a long wavelength cannot satisfy. It is a property of hard walls rather than of confinement in general: a channel that guides by refraction has no cutoff for its lowest mode at any frequency whatever.

Named by 5 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 One line, 21 decades of density, and every mirror on it. Plasma frequency against electron density, both logarithmic, with the horizontal rules at frequencies a reader already has a feel for. A wave is reflected by everything to the right of where its own rule meets the line and passes through everything to the left. a fluorescent tube at 1.0e+17 m⁻³ cuts off at 2.84 GHz; ionosphere, D layer at night at 1.0e+8 m⁻³ cuts off at 90 kHz; ionosphere, E layer at 1.0e+11 m⁻³ cuts off at 2.84 MHz; ionosphere, F2 layer at 1.0e+12 m⁻³ cuts off at 8.98 MHz; aluminium's conduction electrons at 1.8e+29 m⁻³ cuts off at 3819.9 THz. So the F2 layer turns back a 1 MHz broadcast and lets a 100 MHz one straight out, which is why one of them is heard across an ocean at night and the other stops at the horizon; and aluminium's cutoff sits in the far ultraviolet, which is why it is a mirror for everything visible and a window above 78 nm. The dependence is a square root, so the line has slope one half: a hundredfold denser plasma reflects only ten times the frequency. Arriving at an angle helps by a factor of sec θ — 1.00 at 0°, 1.15 at 30°, 2.00 at 60° — because only the component of the motion along the density gradient has to be turned round.

The frequency below which nothing gets in

Free charges give a medium a permittivity that is negative, and a negative permittivity is not an absorbing medium — it is one in which no wave exists at all. Below that frequency the reflection is total, exactly rather than nearly, because there is no transmitted wave and nothing to absorb. The same expression puts the number at 9 MHz for the ionosphere and 3.8 PHz for aluminium.

astrophysics · Plasma oscillation A graded junction against a quarter-wave layer. Reflectance against wavelength for four ways of joining a medium of index 1 to one of 1.52. The bare interface reflects 4.26 per cent at every wavelength. A single quarter-wave layer of index √(n₀n_s) takes that to zero at 550 nm exactly and rises symmetrically either side, which is the shape of every single-layer coating and every single-section transformer. The remaining curves are graded junctions of 150 nm and 400 nm, built as 120 thin layers whose index climbs geometrically and put through the same matrix product. They do not have a design wavelength at all. Below a cutoff they are flat and negligible; above it they climb steeply toward the bare value, and the cutoff is set by the length: 398 nm for the 150 nm taper, 1062 nm for the 400 nm taper. Roughly, a taper works for every wavelength shorter than about twice its own optical length, which is the statement that a reflection needs a partner a quarter of a wavelength further in to cancel against. The engineering versions of this are everywhere: the horn on a loudspeaker, the moth's eye, the flared transition between two waveguides, and the graded layer that lets an ultrasound probe reach tissue across a hundredfold impedance step.

The taper that matches every note

A quarter-wave layer cancels a reflection at one wavelength and only near it. Spread the same change of impedance over a distance instead, and the reflection vanishes for every wavelength shorter than about twice that distance — not by cancelling one echo against another, but by leaving no step anywhere for an echo to come from.

waves · Impedance The chain's dispersion, and the frequency it stops at. Frequency against wavenumber for a chain of equal masses joined by equal springs, in units where the spacing, the mass and the spring are one. At long wavelength the curve is a straight line through the origin — the chain behaves as a continuous string with a sound speed, and the departure from the line is second order in the wavenumber, which is why a lattice is invisible until the wavelength approaches the spacing. At the zone edge, where neighbouring masses move in exact opposition, the curve flattens: the frequency stops rising, the group velocity falls to zero, and the mode is a standing wave that carries nothing. Above that frequency there is no travelling solution at all.

The frequency a lattice cannot carry

A continuous string carries every note. A row of masses joined by springs does not: there is a highest frequency, set by nothing but the time one mass takes to be pushed back by its neighbours, and above it a disturbance does not travel at all.

waves · Periodic media

Named alongside it

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

Evanescent waveGroup velocityDispersionDispersion relationGuided wavesPhase velocityReflection coefficientRefractive indexStanding wavesAntireflectionBand gapBandwidth

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