The reflection that needs no surface

The law that only asks about one component derived refraction from a single requirement: a boundary cannot change the part of a wave that runs along it, because both sides have to agree on the phase at every point of the surface at every instant. The last three words were doing quiet work. Because the surface is still, the two sides agree at every instant, so the frequency on both sides is the same, and every construction built on that argument — the refracted ray, the critical angle, the ray that leaves on the wrong side of the normal — took a boundary at rest for granted.

A boundary can move instead, and its fastest possible motion is not motion at all. Suppose the index of a medium changes everywhere at once: a flash of ionisation through a gas, a voltage applied to every capacitor of a transmission line in the same nanosecond, a pump pulse that alters a film’s index across its whole face. Then the boundary is not a surface in space but an instant in time. The argument that gave Snell’s law applies to it with space and time exchanged, and what comes out is stranger than refraction and in one respect simpler.

Snell’s law with the axes swapped #

At a still surface, a wave’s frequency is conserved because nothing about the boundary changes in time. Its wavenumber is not, because the surface breaks the symmetry of space: on one side the medium is one thing and on the other side another. In the terms of the conservation law a symmetry hands over, a still surface keeps the symmetry of time and breaks the symmetry of space, so it conserves the quantities that belong to time — frequency, and with it energy — and not the ones that belong to space.

A temporal boundary reverses both. The medium is the same everywhere at every moment, so nothing distinguishes one point from another and the wavenumber is conserved. It is different before and after, so the frequency is not.

The two constructions use the same ingredients. Each medium is a pair of lines on a diagram of frequency against wavenumber, $\omega = \pm ck/n$, and each boundary is a line on that diagram along which its conserved quantity is fixed. At a surface the fixed quantity is the frequency, so the line is horizontal; it meets the new medium’s light line at a wavenumber $n_2/n_1$ times the old, which is the shortening of the wavelength on entering glass. At a temporal boundary the fixed quantity is the wavenumber, the line is vertical, and it meets the new medium’s line at a frequency $n_1/n_2$ times the old.

The vertical line also meets the other branch of the new line, the one with negative frequency. A wave with positive wavenumber and negative frequency is a wave travelling the other way. At a still surface the horizontal line meets the old medium’s line at a negative wavenumber too, and that intersection is the reflected wave, travelling back through the medium it came from. In time there is nowhere to go back to. The backward wave of a temporal boundary has to live in the new medium, in the same space as the forward wave, with the same wavelength and the same new frequency.

What cannot jump at an instant #

The construction says which waves may exist, not how strong they are. The amplitudes need boundary conditions, and choosing them is the one step where a temporal boundary is not the spatial one with a letter changed.

At a surface, the continuous quantities are the tangential components of the electric field $E$ and the magnetic field $H$, because integrating Maxwell’s curl equations around a thin loop straddling the surface leaves only those. At an instant, the corresponding integral runs over a thin slice of time, and the quantities that cannot jump are the ones whose rates of change the equations fix: the displacement $D$, whose time derivative is the curl of $H$, and the magnetic induction $B$, whose time derivative is minus the curl of $E$. Neither curl is infinite, so neither field can change in no time. When the permittivity jumps, $E = D/\varepsilon$ jumps with it.

The grid in the figure is the plainest test of that claim available. It stores $D$ and $B$, advances them with the two curl equations, and recovers $E$ by dividing by the permittivity. Switching the permittivity from 1 to 4 — the index from 1 to 2 — changes nothing but that division. At the instant after, the electric field has the shape it had and a quarter of its size, and the magnetic field is untouched.

That pattern is not a wave of the new medium. A travelling wave in index 2 needs a magnetic field twice its electric field, in these units, and the grid now holds a pattern whose magnetic field is four times its electric field. The only way to write it as waves of the new medium is as the sum of one running forward and one running back, and matching $D$ and $B$ gives their amplitudes:

$$\tau = \frac{n_1(n_1+n_2)}{2n_2^2}, \qquad \rho = \frac{n_1(n_1-n_2)}{2n_2^2},$$

found by Morgenthaler in 1958. For a switch from 1 to 2 they are 0.375 and −0.125, and the pulses that separate on the ring have exactly those sizes to three decimal places. The backward pulse is a reflection with no mirror. It is produced by the whole medium at once, and it runs away from wherever the pulse was at the instant of the switch.

An eighth at most, and no limit at all #

The temporal amplitudes resemble the Fresnel amplitudes at normal incidence, which what happens where the medium changes derived as $t = 2n_1/(n_1+n_2)$ and $r = (n_1-n_2)/(n_1+n_2)$. They behave differently in both directions of change.

Raising the index sends back very little. The backward amplitude $n_1(n_2 - n_1)/2n_2^2$ grows from zero, is largest when the index has exactly doubled, and falls again, and its largest value is an eighth. The grid finds 0.118 at a ratio of 1.6, 0.125 at 2 and 0.120 at 2.5. A still surface between the same media reflects a third of the amplitude when the index doubles, and approaches total reflection as the ratio grows; an instant never reflects more than an eighth, however large the jump.

Lowering the index does the opposite, and without limit. As $n_2$ falls, both amplitudes grow as $1/n_2^2$. At a ratio of 0.4 the grid gives a forward wave 4.377 times the incident one and a backward wave of 1.876, where continuity gives 4.375 and 1.875. At a still surface the transmitted amplitude approaches two and never passes it, and the transmitted energy never exceeds the incident, because a surface at rest does no work on the wave. An instant is under no such constraint, and why is the next thing to see.

Energy is not kept, and momentum is #

The energy ratio follows from the continuity conditions in two lines. Before the switch, a travelling wave carries half its energy in the electric field and half in the magnetic. The magnetic energy $B^2/2\mu$ does not change at the instant, since $B$ does not. The electric energy $D^2/2\varepsilon$ changes by the factor $\varepsilon_1/\varepsilon_2 = n_1^2/n_2^2$. So the total becomes $(1 + n_1^2/n_2^2)/2$ of what it was, which is the curve in the figure, and the grid’s summed energy lies on it: 0.625 for a doubling, 0.580 for a ratio of 2.5, 3.63 for a ratio of 0.4. The floor of one half is the magnetic energy, which no change of permittivity can reach.

The energy that appears or vanishes is work done by whatever changes the medium, and a capacitor shows the same bookkeeping at leisure. With its charge fixed, pulling the dielectric out lowers the permittivity and raises the stored energy $Q^2/2C$, and the difference is the work done pulling against the force that draws a dielectric in — the accounting where the energy of a field is makes for a capacitor’s stored energy. Lowering a medium’s index at fixed displacement is the same operation done everywhere in one instant.

Momentum is another matter. The momentum density of the field is proportional to $D \times B$, both factors are continuous, so at the instant of the switch the field’s momentum does not change; and since the medium is uniform afterwards, nothing can exchange momentum with the waves later. The forward and backward waves carry it in opposite directions, and the amounts balance. For a doubling of the index the forward wave carries 1.125 of the original momentum and the backward wave 0.125 the other way, leaving exactly one. A still surface conserves energy and takes momentum — that is the push light gives a mirror — while an instant conserves momentum and does work. Each keeps the quantity its own symmetry protects.

New light, in back-to-back pairs #

The momentum balance has a reading in photons that makes it exact rather than coincidental. A photon’s momentum is $\hbar k$ in the medium, and the switch does not change $k$, so the balance is a count: forward photons minus backward photons after the switch equals the photons there were before. The energy, with each photon’s $\hbar\omega$ reduced by the factor $n_1/n_2$, then fixes the total. For a doubling of the index, every hundred photons become 112.5 running forward and 12.5 running back — twelve and a half new pairs, each with equal and opposite momentum. In general the number of photons is multiplied by $(n_1^2 + n_2^2)/2n_1n_2$, which is never less than one and equals one only when nothing changed.

The switch creates light, in pairs, out of the light already present, and the classical wave calculation found that without mentioning photons. The quantum version of the same calculation goes one step further: a change of the medium abrupt enough creates pairs even when no light is present at all, out of the vacuum’s fluctuations — a relative of the dynamical Casimir effect, in which the boundary conditions on the vacuum are altered quickly enough to shake photons out of it.

A new colour for light already in flight #

The last prediction of the construction is the easiest to state and the most useful: light already in flight changes its frequency by the inverse ratio of the indices, in a linear medium, with nothing moving. The grid, switched from index 1 to 1.5, measures a period of 30.04 before and 45.07 after, in units of the time light takes to cross a cell. The measured ratio also agrees, to a tenth of a per cent, with the ratio the grid’s own discrete dispersion predicts at this resolution, so its agreement with the continuum value is not an accident of the grid.

The measurement needs a care the construction does not warn about. At a fixed point after the switch, the field is the sum of the forward and backward waves, and as the two pulses part, their amplitude ratio at that point changes. That slowly shifts the phase of the sum and moves its zero crossings: timed from the total field, a switch from index 1 to 2 reads wrong by more than half a per cent. The forward wave alone, separated from the backward one using the magnetic field — which points opposite ways relative to $E$ in the two — is a single pulse whose zeros are those of its carrier, and its period is exact.

A change of frequency without motion is a genuinely different tool. Ordinary optics changes a colour by moving something, as in the round-trip shift a moving mirror gives, or with a nonlinear material that mixes light with light and works well only for intense beams. A temporal boundary is linear: every frequency component of a pulse is multiplied by the same factor, so a whole spectrum is translated at once, faint light as well as bright.

Slow switches, plasmas and matched switches #

A slow switch keeps the frequency change and loses the reflection. If the index changes over many periods instead of one instant, the wavenumber is still conserved, since the medium is still uniform, and the frequency still ends at $n_1/n_2$ of its start. But the field follows the changing medium continuously, as the local travelling wave, and the backward wave falls away rapidly once the switching time is longer than a period. The number of photons is then conserved rather than increased — the adiabatic invariant of an oscillator, applied to every mode — so the energy ends at $n_1/n_2$ of its start. For a doubling of the index the slow switch leaves half the energy and the instant leaves five eighths, and the extra eighth is exactly the energy of the twelve and a half pairs.

A plasma adds dispersion, and the construction still works. When a gas is ionised suddenly, its permittivity becomes $1 - \omega_p^2/\omega^2$, the form that sets the frequency below which nothing gets in. The index now depends on frequency, so the vertical line has to meet the plasma’s own dispersion curve, $\omega^2 = \omega_p^2 + c^2k^2$, rather than a straight light line. With $k$ conserved, light already inside the gas emerges at $\sqrt{\omega^2 + \omega_p^2}$ — always higher. That up-shift in flash-ionised gases, often called photon acceleration, motivated much of the theory, and it has been measured on probe pulses crossing laser-produced ionisation fronts.

Impedance, not index, decides the reflection. The grid switches only the permittivity. If the permeability is switched in the same proportion, the new medium has the impedance of the old, and the same continuity conditions give no backward wave at all while the frequency still changes by $n_1/n_2$. That is the temporal version of the rule a still surface obeys — nothing reflects between media of equal impedance — and it shows that the reflection and the frequency shift are separate effects which arrive together only because the permittivity was switched alone.

What has been seen #

Temporal reflection was predicted in the 1950s and observed decades later, because the switch has to be fast compared with a period. For water waves that is easy. In 2016 a bath of water was given a sudden vertical jolt, which changes the effective gravity everywhere at once, and the ripples on its surface sent back a time-reversed copy that converged on the point they had spread from — an instantaneous time mirror. For radio-frequency signals it was shown in 2023 in a transmission line whose capacitance was switched along its whole length at once: a signal split into forward and backward parts at a new frequency, as the construction predicts.

At optical frequencies a period is about two femtoseconds, and no known material changes its index by enough that fast. Films of indium tin oxide, near the wavelength at which their permittivity crosses zero, can have their index changed by order one within tens of femtoseconds by an intense pump, and have shown the frequency translation — time refraction. A clean optical time reflection, with a strong backward wave, has not been achieved.

Where an instant stops being an instant #

The switch was instantaneous and uniform. A real switch takes time, and the backward wave is strong only when that time is short beside a period. A switch that sweeps across the medium at finite speed is a moving boundary rather than a temporal one, and conserves a mixture of frequency and wavenumber set by its speed; an instant is the limit in which that speed is infinite.

The medium had no memory. The grid’s permittivity is a number, not a function of frequency, so it responds to the switch at once. Every real medium responds through the motion of charges that take time to follow, and a switch faster than that response meets a medium whose index has not yet changed. The plasma shows how the construction adapts; a general dispersive medium needs the same care at every frequency.

Only the permittivity changed, in one dimension, on a ring. A switch in three dimensions, for waves travelling obliquely, adds a dependence on polarisation, as a surface does. The ring is periodic so that the pulses have nowhere to leave, and that has no effect on what happens at the instant.

The momentum is the field’s own, $D \times B$. It is the quantity that the symmetry of a uniform medium protects, and the long argument over how the momentum of light in matter is shared between the field and the atoms does not disturb the balance here, since the medium is uniform before and after.

And the work done by the switch is not modelled. The grid computes the field’s energy before and after and nothing else. Where that energy comes from — a pump pulse, a voltage source, the shaker that jolts the bath — is the physical price of a temporal boundary, and it is usually the reason one is hard to make.

What the grid pictures leave out #

The pulse figure draws the electric field, and the electric field is the one quantity that jumps. A drawing of $D$ or $B$ would show nothing at all happening at the instant of the switch, which is the truer picture of the continuity and the less informative picture of what a detector measuring $E$ would record.

The frequency figure draws the total field at one cell and reports a period measured from something else, the forward-running part. The trace and the measured quantity differ deliberately, and the difference between them is exactly the drifting phase that makes the obvious measurement wrong.

Still open: a medium that repeats in time #

A single step in space is a surface, and a periodic stack of steps is a crystal whose band gap turns it into a mirror. The temporal dual is a medium whose index is modulated periodically in time: a photonic time crystal. The same construction predicts gaps not in frequency but in wavenumber. For a wavenumber inside the gap no real frequency exists, and a wave there grows exponentially in time, drawing its energy from the modulation — the wave version of the parametric instability that splits an internal tide, with every mode inside the gap pumped at once.

Whether a photonic time crystal can be built at optical frequencies is not known. The width of the gap is proportional to the depth of modulation, and a useful gap needs the index to swing by a large fraction of itself at twice the optical frequency — faster than any known material responds. Microwave versions, built from surfaces loaded with switched circuits, have reported the momentum gap. Whether an optical version is possible at all, or whether the response time of every real medium forbids it, depends on materials that may not exist.

The habit worth carrying away is to ask which symmetry a boundary breaks. A surface breaks space and keeps time, so it keeps frequency and energy; an instant breaks time and keeps space, so it keeps wavenumber and momentum, and every difference between a reflection in space and a reflection in time follows from which of the two was given up.