The frequency below which nothing gets in

A bound electron pulled by an oscillating field lags behind it a little, and the polarisation it produces is in phase with the driving. That is what makes an ordinary dielectric’s permittivity bigger than one and its refractive index real, which is the whole business of the neighbouring ladder.

Take away whatever is holding the electron — leave it free, as a metal’s conduction electrons are — and the sign flips. A free charge driven at frequency ω has no restoring force to be in phase with, so its displacement is exactly half a turn behind the driving field, its contribution to the polarisation is negative, and

$$\varepsilon(\omega) = 1 - \frac{\omega_p^2}{\omega^2}, \qquad \omega_p = \sqrt{\frac{ne^2}{\varepsilon_0 m}}.$$

The whole of that expression is three lines of mechanics. A free electron driven by $E_0e^{-i\omega t}$ obeys $m\ddot x = -eE$, so $x = eE/m\omega^2$ — no restoring force appears anywhere, so nothing sets a phase except the two derivatives, and the displacement is exactly opposite to the force. The polarisation is $P = -nex = -ne^2E/m\omega^2$, and a permittivity is just $1 + P/\varepsilon_0 E$. Every ingredient is elementary and the minus sign is the only thing in it that matters.

It is worth seeing where a bound electron differs, because it is one term. Give the electron a spring and the equation becomes $m\ddot x + m\omega_0^2 x = -eE$, whose solution has $\omega_0^2 - \omega^2$ in the denominator instead of $-\omega^2$. Below the resonance that denominator is positive and the permittivity is above one; above it, negative, and the permittivity dips below one and can go below zero. So the free-charge case is not a separate theory: it is the bound case with $\omega_0$ set to zero, which puts every frequency above the resonance. A metal is a dielectric whose resonance is at direct current, and everything peculiar about it here follows from being on the far side of a resonance rather than from being a metal.

Below $\omega_p$ the permittivity written above is less than zero, and that is the entire content of this essay. A negative permittivity is not a strongly absorbing medium and not a very dense one; it is a medium in which no propagating wave exists. Nothing has been said yet about reflection, about metals or about the sky — only that a number that is normally between one and ten has changed sign, and the rest of the essay is what follows from it.

The two ends of that line are the same physics with nothing changed but a number, and this essay is about how much falls out of one expression.

The frequency itself has a mechanical reading. Displace every electron in a slab a small distance and leave the ions where they are: the two sheets of charge that result make a uniform restoring field between them, the field is proportional to the displacement, and the electrons fall back through it and overshoot. That is a harmonic oscillator with a spring constant made of nothing but charge density, and its frequency is ωp. So the number is not a property of a wave — it is the rate at which the material itself rings, and a driver below a resonance meets a wall for the same reason, and the wave meets it as a wall for the same reason a driver below a resonance meets one.

The dispersion relation, and the band with no waves in it #

Putting $\varepsilon(\omega)$ into the wave equation gives $\omega^2 = \omega_p^2 + c^2k^2$, which has no real solution for $k$ at any frequency below $\omega_p$.

Two velocities, and neither is the one to worry about. The phase velocity exceeds c throughout, and it diverges at the cutoff, where the crests move infinitely fast and the pattern is the same everywhere at once. Nothing is transported at that speed: the crests are a shape, and a shape can move as fast as arithmetic allows. What carries a signal is the group velocity, which is below c everywhere and goes to zero at the cutoff — a wave right at the cutoff transports nothing at all, which is a more precise statement of what “does not propagate” means than the absence of a real wavenumber.

That product is exact and it is worth pausing on. $v_\phi v_g = c^2$ follows from the shape of the relation alone, and the same shape — a cutoff frequency added in quadrature to a free dispersion — appears in a place with no free charges anywhere in it.

A hollow pipe carrying sound or light has exactly this dispersion relation, with the cutoff set by the pipe’s width rather than by any density. The mechanism is completely different — a wall imposing a boundary condition rather than charges responding — and the algebra is identical. That is worth noticing early, because it means the shape of the relation is not evidence for the mechanism, and the physics has to be established some other way.

The reflection that is exactly total #

The reason a negative permittivity is interesting rather than merely forbidden is what happens at its surface.

A silver mirror reflects 98 per cent because silver absorbs the other two. A plasma below its own frequency reflects all of it, because there is nowhere for the energy to go: no transmitted wave to carry it forward, and no loss mechanism in the model to turn it into heat. That is a different kind of total reflection from a metal’s, and it is the same kind as total internal reflection at a glass surface — an evanescent field that stores energy and gives every joule back.

A quarter-wave coating removes a reflection by interference between two real surfaces, works at one wavelength, and degrades either side of it. The plasma reflection is not like that at all: it is total over an entire band, it involves no second surface, and it happens because there is no propagating wave to transmit into. Cancelling a reflection and having nothing to transmit are opposite situations that both produce a mirror.

Where the reflection actually happens #

Calling it a surface reflection is a convenience. The field does enter the plasma; it simply does not travel.

Which raises a question about where the energy is during the reflection. The evanescent field is not zero, so it holds energy — an amount proportional to the layer thickness — and that energy has to be put there when the wave arrives and taken back when it leaves. It is, and the observable consequence is a phase shift on reflection rather than an amplitude loss: the returning wave is delayed, and the delay corresponds to the time taken to fill and empty the layer. For the ionosphere that delay is a real correction to the timing of a radio echo, and for a metal it is why a mirror’s reflection is not quite at the surface.

And it is not the skin depth of a resistive conductor. The familiar $\delta = \sqrt{2/\mu_0\sigma\omega}$ falls with frequency and comes from diffusion of the field against ohmic loss; it is the low-frequency behaviour of a metal, where collisions dominate. Above the collision rate the metal stops being resistive and becomes a collisionless plasma, and its screening length stops depending on frequency altogether. The two expressions describe the same metal in two regimes, and the crossover is at the collision rate — around 10¹⁴ per second for aluminium, in the infrared.

What the numbers are for #

Two of them are instruments.

The ionosphere as a mirror was a discovery about the sky made with a radio set. Heaviside and Kennelly proposed a conducting layer in 1902 to explain how Marconi’s signal had crossed the Atlantic; Appleton established it in 1924 by sweeping a transmitter’s frequency and watching the interference between the ground wave and the returned one shift as the path length changed. That measurement returns the height of the layer, and repeating it at different frequencies returns the density profile — because each frequency turns round where the local plasma frequency matches it, and the height at which that happens is the answer.

And in a laboratory the relation runs the other way. An electron density is hard to measure directly and a cutoff frequency is easy, so a plasma’s density is routinely read off the highest frequency it reflects — the same square root, inverted. The technique needs no probe inserted, disturbs nothing, and works from the outside, which for a hot plasma is the only kind of measurement available.

And the delay it puts on a signal is measured every second by every satellite receiver. The group velocity below the light line means a wave crossing the ionosphere arrives late, by an amount that depends on frequency: expanding $v_g = c\sqrt{1 - \omega_p^2/\omega^2}$ for $\omega \gg \omega_p$ gives an excess path proportional to the total number of electrons along the line of sight divided by the square of the frequency. For a satellite navigation signal at 1.6 GHz and a quiet daytime ionosphere that is a metre or two; through a disturbed one at low elevation it is tens of metres, which is a very large error in a system whose whole purpose is to do better than a metre.

The repair is the $1/f^2$ itself. Two frequencies are broadcast rather than one, both suffer a delay set by the same electron count, and the difference between their arrival times measures that count and removes it — an ionosphere-free combination, computed from nothing but the two delays and the two frequencies. So the plasma cutoff is not merely an obstacle worked around; its shape is the instrument. The same measurement, accumulated over a network of receivers, is how the electron content of the whole ionosphere is mapped hour by hour, and it is a far better map than a sweep of a transmitter’s frequency ever produced.

There is a detail in it that catches people out. The group is delayed and the carrier phase is advanced, because the phase velocity is above c by as much as the group velocity is below it. Both statements come from the same curve, neither carries a signal faster than light, and a receiver that corrected one with the sign of the other would double its error rather than remove it.

The layer that absorbs is not the layer that reflects #

The ionosphere is usually described as one thing and behaves as at least two, and the difference between them is the difference between a cutoff and a collision.

The reflection above happens where the local plasma frequency matches the wave, which for a medium-wave broadcast is high up in the F region. Below that, at 60 to 90 kilometres, sits a much denser neutral atmosphere carrying a much thinner plasma — a region whose own cutoff is far too low to turn anything back, but where an electron driven by the passing wave collides with a neutral molecule before it has finished a cycle. That is the damping term the idealisation here sets to zero, and it does what damping does: it removes energy from the wave rather than returning it.

So a signal that would be reflected has first to survive a passage through an absorber, twice. By day the absorbing region is ionised by sunlight and a medium-wave transmitter is heard for perhaps a hundred kilometres; at night that ionisation recombines away within an hour or so while the reflecting region above it persists, and the same transmitter is heard across an ocean. Nothing about the mirror changed. What changed was the loss in front of it — which is a clean demonstration that total reflection with no absorption and a real signal path are two different claims.

What the free-charge model leaves out #

Bound charges are still there. Real metals have core electrons and interband transitions, whose response is an ordinary bound-charge one, and those contribute an ordinary positive polarisability on top of the free-electron negative one. The effect is to shift the cutoff: a metal whose bound background has permittivity $\varepsilon_\infty$ reflects up to $\omega_p/\sqrt{\varepsilon_\infty}$ rather than $\omega_p$. Gold and copper are coloured for exactly this reason — an interband transition in the blue removes the reflectance there and leaves the red — and aluminium is grey because it has no such transition in the visible.

Collisions. Adding a damping rate γ makes the permittivity complex, the reflection less than total and the evanescent layer lossy. The size of the effect is γ/ω, which for aluminium in the visible is a few per cent — and that is exactly the four per cent aluminium fails to reflect.

The density is not uniform. The ionosphere’s electron density rises with height, peaks in the F2 layer and falls again, so a radio wave does not meet a surface: it is refracted continuously, as a ray in a graded medium is, and: it climbs into steadily denser plasma until the local plasma frequency matches its own, and turns round there. Arriving at an angle helps, because only the component of the motion along the gradient has to be turned round, and the highest frequency returned goes up by sec θ — which is why a transmitter aiming at the horizon reaches further than one aiming up.

In vacuum the wave speed is fixed by two electrostatic constants and nothing else, and the free charges have not touched that. A plasma changes the permittivity and leaves the permeability alone, so everything in this essay is one of the two constants being modified — which is why the effect disappears at high frequency, where the electrons cannot keep up and the permittivity returns to its vacuum value.

And a laser puts the same line at a third place entirely. Inverting the cutoff condition gives a critical density — the density at which a plasma stops transmitting a given wavelength — of about $10^{27}$ electrons per cubic metre for one-micrometre light, falling as the square of the wavelength. A pulse focused into a solid target burns a plasma off its surface and then has to push through it, and it gets as far as the critical surface and no further: the energy is absorbed and reflected there, and everything beyond it is heated by conduction rather than by light. That single number decides the geometry of every laser-plasma experiment there is, and it is the same square root drawn at the top of this essay, evaluated twenty-one decades along from the ionosphere.

The same argument in three other places #

A forbidden band can also come from periodicity rather than from free charge, and the mechanism is different again: the Bloch condition has no real solution across the gap. Every consequence is the same — total reflection, an evanescent field inside, a band edge — and the three cases together make the point that a stop-band is a statement about the absence of a propagating solution rather than about what is doing the stopping.

What the four cases share is not a mechanism but a piece of algebra: in each, the wave equation reduces to $\mathrm{d}^2\psi/\mathrm{d}x^2 = \kappa^2\psi$ with a real κ, whose solutions are growing and decaying exponentials rather than oscillations. Discarding the growing one on the grounds that the medium is thick leaves a decay, and a decay carries no energy away because its Poynting vector is zero on average. Every consequence follows from that one line, and none of it needs to know what made κ real.

The general statement is that a wave meeting a region where its own wavenumber would be imaginary is turned back without loss, and how deeply it gets in before turning is set by how imaginary the wavenumber is. A plasma below its cutoff, a waveguide below its lowest mode, a photonic crystal inside its gap, a quantum particle at a barrier — the arithmetic is the same in all four, and only the reason the wavenumber went imaginary differs.

And the fourth case is the one that makes it feel less like an analogy. An electron meeting a potential step higher than its own energy has an imaginary wavenumber, penetrates a distance set by how much higher, and is reflected with probability one when the barrier is infinitely thick. That is the wall that is not quite a wall, and the only difference from the figures above is which dispersion relation was substituted in.

One number, checked twice. The plasma frequency computed as √(ne²/ε₀m) and the radio engineer’s rule fₚ = 8.98 kHz × √(nₑ in cm⁻³) are the same expression with the four constants already collapsed into one, and every figure above checks the two against each other before drawing. That is a weak test of physics and a strong test of arithmetic, which is what it is for: four fundamental constants written out by hand is the kind of calculation that goes wrong in the fourth digit and stays wrong.

Where this ladder goes next #

Everything here has treated the free charges as a passive medium — something a wave passes through or fails to. The obvious next rung is what the charges do when nothing passes through: displace them all by a small amount and the restoring field pulls them back, at exactly $\omega_p$, which is why the frequency has that name. It is a genuine oscillation of the material rather than a property of a wave in it, it carries no energy anywhere at long wavelengths, and it is quantised — which turns a bulk oscillation of 10²³ electrons into a particle-like excitation with an energy of 15 electronvolts in aluminium, measurable one at a time as a step in the energy lost by a transmitted electron.