The crystal made of moments

The reflection that needs no surface changed the refractive index of a whole medium at one instant and found a wave splitting in two, with a backward wave where no surface existed. The construction behind it was Snell’s law with the axes of space and time exchanged. A surface breaks the symmetry of space and keeps the symmetry of time, so it conserves frequency and changes wavenumber; an instant breaks time and keeps space, so it conserves wavenumber and changes frequency.

That essay ended on the obvious next step. A single surface becomes a crystal when it repeats: a stack of alternating layers refuses a whole band of frequencies and turns into a mirror for them. A single instant should become a crystal when it repeats too, as a medium whose index is swung up and down periodically in time, everywhere at once. What that medium does is not a mirror, and the reason is the same exchange of axes carried one step further.

Every wavenumber is its own oscillator #

In a medium that is uniform in space, however it changes in time, waves of different wavenumbers do not mix. The spatial symmetry guarantees that a wave with wavenumber $k$ stays at wavenumber $k$, so each wavenumber can be followed separately, and each is an oscillator. With the displacement field $D$ and the magnetic field $B$ as the quantities that cannot jump when the medium changes, a wave of wavenumber $k$ obeys

$$\frac{\mathrm{d}^2 D}{\mathrm{d}t^2} + \frac{c^2 k^2}{\varepsilon(t)}\, D = 0,$$

an oscillator whose natural frequency is modulated by the medium. Let the permittivity be modulated as $1 + m\cos\Omega t$. Then each wavenumber is a child on a swing whose length is being changed periodically, and whether its motion stays bounded or grows depends on how its own frequency compares with the modulation’s — which is the swing that is pumped rather than pushed, once for every wavenumber at the same time.

The figures below do not solve that equation in general. For each wavenumber they integrate it across a single period of the modulation, starting from two independent initial conditions, and read off the two-by-two matrix that carries any solution from one period to the next. That matrix’s trace decides everything. If it lies between −2 and 2, the solution rotates from period to period and the wave has a real frequency. If it lies outside, one solution grows by the same factor every period and the other shrinks.

A gap in wavenumber #

For most wavenumbers the wave has a real frequency, and it is close to the unmodulated one, $ck$, folded back into the range between zero and half the modulation frequency — a wave in a medium repeating every $2\pi/\Omega$ in time cannot tell frequencies apart that differ by a multiple of $\Omega$, just as a wave in a crystal cannot tell wavenumbers apart that differ by a multiple of the reciprocal lattice. Near the wavenumber at which the wave’s own frequency is half the modulation’s, the folded branches meet, and at depth 0.2 they do not join. Between wavenumbers 0.475 and 0.521, in units of $\Omega/c$, there is no real frequency at all.

That is the temporal counterpart of a band gap. In a stack of layers the gap is a band of frequencies with no real wavenumber, centred where half a wavelength fits in a period of the stack. Here it is a band of wavenumbers with no real frequency, centred where half an oscillation fits in a period of the modulation. The condition is the same resonance, a wave meeting the repeat twice per cycle of its own, with space and time exchanged.

Growth instead of reflection #

What a missing real frequency means is not the same in the two cases, and this is where the exchange stops being cosmetic. In a stack of layers, a frequency in the gap has a complex wavenumber, so the field decays with depth into the stack: light cannot get in, and is reflected. In a time-modulated medium, a wavenumber in the gap has a complex frequency, so the field changes exponentially in time — and one of the two solutions grows.

At a modulation depth of 0.1, the gap spans 4.8 per cent of its central wavenumber and the fastest-growing wave gains 0.079 in the logarithm of its amplitude per period. At 0.2 the gap spans 10.0 per cent and the growth is 0.159; at 0.4, 20.3 per cent and 0.333. Both scale close to linearly with the depth for small depths, and the growth at the centre of the gap was checked against the small-depth result for a parametrically driven oscillator, an eighth of the depth times the modulation’s angular frequency, to within five per cent at depth 0.05.

The shape of each curve is a dome: zero at the edges of the gap, where the wave’s frequency is just ceasing to be real, largest in the middle. A wave with a wavenumber just inside the edge grows slowly; one at the centre grows fastest. The growth is not a small effect once it starts. At depth 0.4 a wave at the centre of the gap grows by a factor of $e$ every three periods of the modulation.

Followed period by period, the difference is stark. A wave with wavenumber 0.462 of $\Omega/c$, just outside the gap, has its amplitude pushed up and down by the modulation and returns to where it started, over and over. A wave at the centre of the gap, 0.500, grows by a factor of 73 in thirty periods, and the growth traced by direct integration matches the rate the single-period calculation predicts. Nothing in the equation stops it. In a real medium something would — the material’s loss, its nonlinearity, the drive running out — but within the model the wave grows without limit.

The energy has to come from somewhere, and the single switch already said where. Changing the permittivity at fixed displacement changes the field’s energy, and the difference is work done by whatever changes the medium. A single instant does a fixed amount of that work; a periodic modulation, phased right against a wave in the gap, does it every period. A photonic time crystal is an amplifier whose power supply is the modulation itself, and the waves it amplifies are the ones whose own oscillation is in step with the swing.

The two crystals side by side #

The two panels are the same calculation twice. On the left the permittivity repeats in space, and for each frequency the equation is integrated across one spatial period; the Bloch wavenumber that results is drawn against frequency, and between frequencies 0.478 and 0.526 there is no real wavenumber. On the right the permittivity repeats in time, and the roles are reversed. The two gaps are nearly the same width, as they should be for the same depth of modulation, and they lie across different axes.

The comparison makes precise what each crystal conserves and what it does not. A stack of layers is uniform in time, so frequency is conserved and every photon that enters leaves with the frequency it came with; it is the wavenumber that is scrambled, reflected into its negative. A time-modulated medium is uniform in space, so wavenumber is conserved — momentum is conserved — and it is the frequency, and with it the energy, that the medium is free to change. That is why one crystal reflects and the other amplifies. Reflection is what a medium that keeps energy does to a wave it cannot carry; amplification, in pairs of forward and backward waves with equal and opposite momentum, is what a medium that keeps momentum does. The single instant already created light in back-to-back pairs; the crystal does it once every period, in step.

A gap in every direction at once #

The exchange of axes has one more consequence, and it is a gift rather than a cost. A stack of layers has its gap only for light travelling along the stack; tilt the light and the gap moves, because the component of wavenumber along the layers changes what the layers look like. Making a spatial crystal reflect every direction of a given colour requires care, and in one dimension it is possible only for light arriving from a low-index medium — the mirror that works from every direction is exactly that care.

A time-modulated medium is uniform in space, so it has no preferred direction at all. The gap is a band of the magnitude of the wavenumber, and every direction with a magnitude in that band is inside it. The temporal crystal’s gap is complete in direction automatically, and a light source inside it would find every direction of emission affected at once.

The growth itself belongs to a family that runs well beyond optics. A modulation at twice a mode’s frequency pumping that mode is the mechanism by which an internal tide splits into two waves of half its frequency, and it is the resonance that the leading candidates for optical time modulation would have to supply: conducting films whose free electrons behave as a plasma, with a permittivity that crosses zero at a characteristic frequency and changes fastest there. The same parametric resonance appears with a different name in each of those subjects, and the band diagram is the version of it that shows every wavenumber together.

The same physics in an optical parametric amplifier #

The mechanism is not new to optics, and the difference from what exists is instructive. An optical parametric amplifier sends a strong pump beam through a nonlinear crystal, and the pump modulates the crystal’s index at its own frequency, twice that of the signal being amplified. That is a modulation in time. It is also a modulation in space, because the pump is a travelling wave, and so the amplified signal must satisfy a phase-matching condition as well as a frequency condition: only signals travelling in particular directions, at particular wavelengths, keep in step with the pump along the crystal.

A photonic time crystal is the limit in which the pump has an infinite wavelength — the whole medium swings in unison. Then there is no phase-matching condition at all, and every direction is amplified equally for wavenumbers in the gap. That is what makes the idea attractive, and it is also what makes it expensive: a pump that is uniform in space has to modulate every part of the medium at once, with no help from a travelling wave that concentrates its effect where the signal is.

What the construction cannot yet be built for #

The gap sits where half of a wave’s period fits into a period of the modulation, so the modulation must run at twice the frequency of the waves it is meant to affect.

For microwaves at 10 gigahertz the modulation must run at 20 gigahertz, and electronic switches can supply that; a temporal gap in wavenumber was reported in 2023 in a microwave metasurface whose capacitances were switched in unison. For light at the telecommunications wavelength of 1550 nanometres the modulation must run at $3.9 \times 10^{14}$ hertz, and for green light at $1.1 \times 10^{15}$.

The depth matters as much as the rate. The gap’s width and the growth rate are both proportional to the modulation depth, and a useful gap needs the permittivity to swing by a sizeable fraction of itself. The electro-optic effect, which changes an index electronically, reaches changes of a part in ten thousand. The materials that come closest are transparent conducting oxides near the wavelength at which their permittivity crosses zero, where a strong laser pulse can change the index by order one within tens of femtoseconds — the same films that showed time refraction. A period of green light is under two femtoseconds, and swinging such a film up and down repeatedly at that rate, rather than switching it once, has not been achieved and may be beyond what any known material can do.

A material that cannot change as fast as the equation asks #

The medium responds instantly. The equation takes the permittivity to be a number that changes as the modulation dictates. Every real material responds through the motion of its electrons and ions, which have their own frequencies, and a modulation faster than those responses meets a medium that has not yet changed. At optical frequencies that is the central difficulty rather than a correction.

There is no loss. A real medium absorbs, and absorption subtracts a constant rate from the growth. A gap is useful only where the growth rate exceeds the loss rate, which raises the modulation depth needed; for the conducting oxides that are the leading candidates, the loss near the zero-crossing of the permittivity is large.

The modulation is sinusoidal and uniform. A modulation with harmonics opens additional gaps at other wavenumbers, and a modulation that is not uniform in space mixes wavenumbers and brings back a phase-matching condition. The pure construction is the limit of a perfectly uniform, perfectly sinusoidal drive.

Growth is exponential only in the linear model. A wave that grows by a factor of seventy in thirty periods is, after a few hundred periods, large enough to change the medium itself, and every real amplifier saturates. The figures show the linear regime, which is where the gap is defined.

Folded frequencies, and the copy a detector sees #

The band diagrams fold every frequency into the range from zero to half the modulation frequency, because that is all a periodic modulation can distinguish, and the folding makes the curves look like a crystal’s bands. What they hide is which of the folded copies a detector would see: a wave in a time-modulated medium carries sidebands at its frequency plus and minus every multiple of the modulation frequency, with amplitudes the diagram does not draw.

Nor do the pictures show space. Every figure is one wavenumber, or a list of them, with no picture of a field in a medium. A wave packet in a real photonic time crystal would contain a range of wavenumbers, some inside the gap and some outside, and the part inside would grow while the rest merely wobbled — the packet reshaping itself around the gap’s centre as it amplified, which is an evolution in space and time that a curve of growth rate against wavenumber only summarises.

Still open: whether an optical time crystal can exist #

The microwave demonstrations establish that the construction works when the modulation can be supplied. Whether it can be supplied for light is not settled, and the reasons it might not be are specific. A material’s index can only change as fast as the charges that produce it respond, and at optical frequencies the responses fast enough are those of free or nearly free electrons, which are also the ones that absorb. Estimates that combine the required depth, the rate, and the loss of the best candidate materials come out marginal at best, and some analyses conclude that the losses and the heating from a drive strong enough to open a useful gap make an optical photonic time crystal impractical with any known medium.

Proposals that sidestep the difficulty — gaps opened in a narrow band near a material resonance, where a small modulation has a large effect, or structures in which the modulation is concentrated where the light is — are being explored. So are the consequences if one is built: an atom placed inside the gap is predicted to emit into the growing modes differently from how it emits in any static medium, and a laser without a mirror, drawing its gain from the modulation, has been proposed. None of these has been realised at optical frequencies.

The habit worth carrying away is to ask which symmetry a structure breaks and what that frees, which is the bargain Noether’s theorem strikes applied to a material rather than to a law. A repeat in space keeps energy and gives away momentum, so it reflects; a repeat in time keeps momentum and gives away energy, so it amplifies. The gap is the same mathematics in both, and the physics on either side of it is decided entirely by which of the two conserved quantities the medium has agreed to leave alone.