## Abstract

Time-dependent systems do not in general conserve energy, invalidating
much of the theory developed for static systems and turning our
intuition on its head. This is particularly acute in *luminal* space-time crystals, where the structure
moves at or close to the velocity of light. Conventional Bloch wave
theory no longer applies, energy grows exponentially with time, and a
new perspective is required to understand the phenomenology. In this
paper, we identify a new mechanism for amplification: the compression
of lines of force that are nevertheless conserved in number.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

## 1. INTRODUCTION

Energy can be added to electromagnetic waves in several different fashions.
We identify a mechanism, distinct from conventional ones, in which
compression of lines of force is the active ingredient. There are
instances of this in other contexts: a superconductor repels magnets
because the magnetic lines of forces are compressed as they are rejected
by the superconductor. A more dramatic example is the generation of
thousand Tesla magnetic fields by explosive collapse of a copper cylinder
enclosing magnetic lines of force [1]. Here we invoke the concept in the context of amplifying
electromagnetic waves. We show that in some circumstances the number of
lines of force, electric and magnetic, in a time-dependent system is
conserved, and amplification occurs when these lines of force are squeezed
closer together. A short summary of this work can be found as a
“Memorandum” published in *Optica* [2].

In this paper we use a simple model of a time-dependent grating synthetically moving with a uniform velocity ${c_g} = \Omega /g$:

Similar traveling-wave media have been investigated theoretically in the past [4–8]. More recently, they gained renewed interest thanks to their ability to break Lorentz reciprocity without need for an applied magnetic bias, which can be exploited for use as isolators [9–11], as well as their topological [10], non-Hermitian [12], and cloaking [13] features. They have also recently been analyzed and homogenized as effective media [14,15]. The luminal modulation regime considered here has recently been proposed for pulse formation [16,17] and broadband, nonreciprocal amplification [17].

However, a fundamental explanation of the physical mechanism responsible for this amplification has never been developed.

When waves interact with static structures, we have many tools not only for calculating but also for understanding the processes and for conceptualizing a problem before we even begin to calculate. In periodic structures the Bloch wave vector is conserved and, together with frequency, the other conserved quantity, gives a wealth of understanding. Its dispersion with frequency tells us where the bandgaps are and where we are likely to find surface states. Bloch waves are the basis for understanding the topology of the states through such quantities as the Berry phase [18] and Chern number [19]. We stress that these concepts are not merely computational devices but central to our thought processes as tools for understanding and creating such systems.

In this paper we seek to provide a set of tools for understanding time-dependent structures for which the traditional static concepts fail, by identifying a conserved quantity in the form of the number of lines of force contained in the system.

At first sight Eq. (1) would imply a straightforward generalization of Bloch’s theorem, mixing together waves differing by a space-time reciprocal lattice vector,

so that as well as a Bloch wave vector $k$, there is also a Bloch frequency $\omega$, both conserved modulo $({g,\Omega})$. This is a good description of the problem for ${c_g} \ll {c_0}/\sqrt {{\varepsilon _1}{\mu _1}}$, where ${c_0}/\sqrt {{\varepsilon _1}{\mu _1}}$ is the velocity of light in the background medium. Figure 1(A) shows dispersion of light in the background medium, and Fig. 1(B) shows what happens when the grating is turned on: bandgaps open in the normal way, but now there is asymmetry about $k = 0$ due to the breaking of time-reversal symmetry.In the other extreme, ${c_g} \gg {c_0}/\sqrt {{\varepsilon _1}{\mu _1}}$, shown in Fig. 1(C), we may still cling onto the Bloch wave picture, except that the bandgaps are regions of complex $\omega$ rather than of complex $k$. In these gaps waves can lose or gain energy, a process of parametric amplification [20,21].

The focus of our interest will be in the middle of these extremes: the luminal region in which the speed of the grating approaches that of light in the medium. Figure 1(D) shows the catastrophe that occurs when ${c_g} = {c_0}/\sqrt {{\varepsilon _1}{\mu _1}}$: all the forward-traveling waves become degenerate. Whereas the bandgaps formed between forward- and backward-traveling waves dominate scattering outside this regime, it is forward-forward scattering that dominates here. There is a clearly defined range of ${c_g}$ within which the band picture is invalid as already shown in earlier works [8]. In this range, light is not scattered by the structure but is captured and localized, carried along with the structure with velocity ${c_g}$, amplified, and ejected from the medium as a series of pulses as described in our Memorandum [2], a figure from which is shown in Fig. 2. Here a picture in terms of freely propagating waves is meaningless. Instead we look to the basic elements of the electromagnetic field: the lines of force embodied in the $D$ and $B$ fields. Because their velocity varies with position in the structure, there will be a point toward which they migrate, an accumulation point [17,20,21], and here they are compressed, and our new mechanism of amplification comes into play.

The parameters $\varepsilon ({x - {c_g}t})$ and $\mu({x - {c_g}t})$ define the impedance,

and refractive index,Variations in the impedance are responsible for backscattering and hence for the bandgaps. A constant $Z$ eliminates back scattering, removing all gaps so that there is no parametric amplification. Conversely, the refractive index is responsible for forward scattering. Because of degeneracy of the forward-traveling waves in a near-luminal system, forward scattering is of central importance, and backscattering a distraction that we shall neglect. This is an exact statement if $Z$ is constant but approximately true if ${\alpha _\varepsilon} \ll 1$, ${\alpha _\mu} \ll 1$. Eliminating all bandgaps and hence all parametric mechanisms allows us to identify our new mechanism in its purest form.

When backscattering is negligible, the lines of force are conserved. Therefore, any gain in energy can only come from compression of the lines into a sharp pulse. If their local density is increased by a factor of $f$, then the local energy density increases by ${f^2}$, and hence there is not only a local increase in energy but also a net increase.

Here we stress that this is an entirely novel insight into an amplification process. We have already identified parametric amplification that occurs when bandgaps open and give rise to complex values of $\omega$ and wave fields that grow in time. No conservation is at work here, simply a uniform addition of lines of force. Next consider a slab of uniform gain medium characterized by a complex refractive index $n = {n_r} - i{n_i}$ so that a wave of frequency $\omega$ injected into the system acquires a complex wave vector,

and as a result the wave amplitude increases exponentially with penetration into the medium. Cleary there is no conservation of lines of force here. Many more are ejected from the far side of the medium than enter.## 2. EQUATION OF MOTION IN THE ABSENCE OF BACKSCATTERING

We start from Maxwell’s equations,

These equations are symmetric under exchange of $({{E_y},Z{H_z}})$, and therefore solutions factorize into the symmetric and antisymmetric:

The plus sign corresponds to forward-traveling, and the minus sign to backward-traveling waves obeying a first-order partial differential equation,

If we assume that $Z$, $\varepsilon$ are both real, and that $\varepsilon$ is independent of frequency and a local operator, then $D$ and $E$ have the same phase, and there is a further factorization of Eq. (10) into real and imaginary parts:

## 3. CONSERVATION OF LINES OF FORCE

Suppose that

where ${\varepsilon _1}$ is independent of time. At $t = 0$ we turn on the space-time dependence and ask how the total number of lines of force in the system changes:This proves the weak form of the theorem, showing that the only way energy can be added to the system is by rearranging the lines of force.

If we make some further assumptions about $\varepsilon$, a stronger form of the theorem can be found. We now assume that $\varepsilon$ has the form

in other words, it has a fixed profile moving with uniform velocity ${c_g}$. We work with new variables defined by where upon Eq. (11) becomesThe fields remain unchanged as they correspond to the original fields observed in the stationary frame. We ask if the number of field lines contained between two points

is constant:Hence, the region between these points constitutes a trap in which, depending on the detailed variation of the profile, the lines of force may be continuously squeezed together, and in the process energy is pumped into the system. This local form of the theorem does not apply to waves traveling in the opposite direction to the profile where the “$+$” sign applies in Eq. (18), and they are not trapped.

## 4. ENERGY DENSITY

Remembering our assumption that $\hat\mu{\mu _0} = {Z^2}\hat \varepsilon {\varepsilon _0}$, the energy density is given by

and obeys the following equation:In the case of a periodic grating,

we can give some physical insight into energy growth.Figure 3 shows the profile of the grating in the moving frame, where ${X_1}$, ${X_2}$ are points where the local velocity of light equals that of the grating. As explained above, the last term in Eq. (21) is responsible for pumping energy into the system and is a maximum where the derivative of $\varepsilon$ is a maximum; in half the cycle this term is negative and takes energy from the system. The penultimate term in Eq. (21) arises from the Poynting vector and redistributes energy within the confines of ${X_1} \lt X \lt {X_2}$. When the ${c_g} = {c_l}$ line coincides with the $X$ axis, the two terms work together to give peak energy density at the point of maximum gain, otherwise the point of peak energy density drifts away from this point. Ultimately, if the grating moves much quicker or much slower than any of the light within, no valid ${X_1}$, ${X_2}$ points can be found, the strong form of our theorem fails, and lines of force can now escape from one period of the grating to another, though still with overall conservation.

We can find an approximate but accurate solution to Eq. (21) for the case of the periodic grating. Here we give a derivation for the case of a weak modulation of $\varepsilon$, which approximately satisfies the no-backscattering condition. Results for modulating both $\varepsilon$ and $\mu$ can easily be found by modifying Eqs. (23) and (24).

Making the substitution

and assuming that $\alpha$ is small, Eq. (21) can be writtenThis implies that the weight of the gain occurs in the close vicinity of

To calculate the first order, we substitute the zeroth order into the missing term:

Hence,

Note that the expression is an exact solution of Eq. (24) if $\delta = 0$, $gX = 0$, $3\pi /2$. The expression describes formation of a pulse contained within a period of the grating whose position shifts as the grating velocity deviates from the average speed of light as described by $\delta$. The result is independent of the frequency of incident radiation $\omega$, and the modulation frequency $\Omega$ appears only through $g = \Omega /{c_g}$ as a scaling variable for length.

The first term in brackets in Eq. (32) arises from the rate of change of $\varepsilon$ and is a maximum when that rate is a maximum. It gives rise to exponential growth in amplitude. Also, there is a point in the grating where $\sin ({gX}) = 0$ and gain switches to loss. The other terms arise from the flow of energy and substantially change the shape of the pulse that forms and where the pulse forms. Narrowing of the pulses is also exponential as our theory requires to be the case.

If the grating travels at the average velocity of light $\delta = 0$, our approximations are rather accurate, and in fact are exact at $gX = \pi /2$ and at $gX = 3\pi /2$, the maximum loss and maximum gain points in the medium, where only the first term matters. When $\delta \ne 0$, energy drifts away from the point of maximum creation toward the point where the velocity of light is the same as that of the grating. Peak energy density lies between the two. We give a further discussion of accuracy later in the paper when comparing to transfer matrix simulations.

Here is the origin of trapping lines of force to which we alluded earlier. If there is no point within the grating where the local velocity of light is the same as that of the grating, then lines of force escape into the next period and the growth mechanism collapses, and we revert to a Bloch wave description. The condition for the growth mechanism to operate is [3]

In Fig. 4 we show the effect of the three contributions to $U({X,t})$. When ${c_g} = {c_0}/\sqrt {{\varepsilon _1}}$, $\delta = 0$, energy accumulates at the point of maximum growth, and the role of the second term is to sharpen the peak as energy migrates inwards. The third term makes no contribution in this case. In contrast, when the grating is moving faster, ${c_g} \gt {c_0}/\sqrt {{\varepsilon _1}}$, $\delta = 0.02$, the third and second terms play a role both in sharpening the peak and in moving it back along the grating.

We recognize that Eq. (32) implies the following density of lines of force:

This is a very curious time evolution containing a double exponential, the function of which is to narrow the width of the pulse and ensure conformity to the flux conservation law.

## 5. DEPENDENCE OF THE PHASE ON TIME

From Eqs. (16), (22), and (23),

We can retrace the steps taken in deriving $U$. We concentrate not on the maximum rate of growth of $\phi$ but rather that of $\psi$, and find to first order,

Back substituting and making successive approximations, we have

The prefactor ${k_1} = {k_0}/\sqrt {{\varepsilon _1}}$ is the wave vector in the background medium and gives the rate of change of $\phi$ at $\tau = 0$. Compare this equation to our expression for the flux density, Eq. (34), which to the accuracy of our approximations shows that the phase is compressed in the same manner as the lines of force.

Integrating Eq. (40), we find the phase

The phase depends on the modulation frequency $g = \Omega /{c_g}$ as a scaling variable for the length; however, the number of oscillations in one cycle is given by ${k_1}/g$ and so does depend on the input frequency.

## 6. COMPARING THEORY TO TRANSFER MATRIX CALCULATIONS

In Fig. 5 we compare the analytic result with transfer-matrix-based simulations [17]. The first two figures show transmission through slabs of grating of two different thicknesses while the grating travels at the average velocity of light, $\delta = 0$. The choice the input frequency, $\omega = 1$, does not affect transmission intensities, but $\Omega = 0.07$ the modulation frequency dictates the number of oscillations of phase per cycle. The choice of a small ratio $\Omega /\omega$ both serves to provide a rich population of phase oscillations and to demonstrate that modulations can be effective while of a much lower frequency than the waves acted upon.

In Fig. 5(A) the light has been in the grating for a short time $\tau = 150$, and in Fig. 5(B) for a longer time $\tau = 600$, resulting in a higher narrower pulse. Our compression of lines of force theory predicts that the pulse height scales inversely as the square of the pulse width, which it does. Note the agreement with the transfer matrix calculations. In Fig. 5(C) we add the phase calculated analytically and compare the real part of the electric field with that calculated using the transfer matrix. Derivation of the phase formula follows closely that for the field amplitude. The formula is accurate where the linearization assumption holds good.

Finally, in Fig. 5(D) the calculation in Fig. 5(B) is extended to other grating velocities. Here the analytic result is less accurate [it depends on linearization of $\varepsilon (X)$ about $gX = 3\pi /2$]. The amplification mechanism is on the point of breakdown at $\delta = \pm 0.05$, beyond which values a Bloch wave picture reasserts itself. Note how the peak lags behind the point of maximum gain when $\delta \gt 0$ and conversely when $\delta \lt 0$.

## 7. DISCUSSION

Our idealized model, which assumes no loss and no dispersion, will inevitably be compromised to some extent by the materials available to us.

First let us consider the problem of loss. Adjusting Eq. (10) to include uniform loss,

Another consideration is the practicality of modulating material properties
at very high frequencies. At RF, where a rich variety of material
properties is available, there will be no problem, and several schemes
involving varactors have been proposed [14], but at higher frequencies only weak modulation can be
expected. However, there is a let out clause in that only the *speed* of the modulation need keep pace with the
radiation; the modulation *frequency* $\Omega$ can in principle be much lower provided
that ${c_g} = \Omega
/g$ is fast enough. In lossy systems, limits
are imposed by Eq. (45)
because a small value of $\Omega$ implies a low rate of modulation.

We take the opportunity to note that we can use Eq. (43) to illustrate what happens in a gain medium by reversing the sign of ${\varepsilon _i}$. Instead of loss we now have amplification, but this term contributes uniform amplification and invalidates the flux conserving theorem.

Choosing a value of $\Omega \ll \omega$ helps sidestep the issue of dispersion. When $\varepsilon$ is a function of frequency, each frequency excited has a different velocity and therefore a different accumulation point within the cycle (see Fig. 3), and as a result it does not contribute to give a perfectly coherent peak. In fact, if dispersion is so severe that some frequencies escape the trap entirely, they no longer contribute to amplification. This has been demonstrated computationally in an earlier publication [17] by means of transfer matrix calculations. Most materials have a range of frequencies over which dispersion is small so that by choosing $\Omega \ll \omega$ the system can stay within the gain regime to all practical purposes, provided that losses are modest.

Apart from RF systems, graphene is a promising candidate for terahertz (THz) frequencies. It is known that the conductivity of graphene can be modulated at rates exceeding 100 GHz via both electro-optic [22,23] and all-optical mechanisms [24], and the THz surface plasmons of graphene may be the first excitations to be amplified in this fashion. We have previously suggested double-layer graphene as a possible candidate, since the linearity of the acoustic plasmon mode in this configuration enables the circumvention of the aforementioned dispersion effects [17].

Seemingly related but well-distinct light-amplification mechanisms have recently been proposed, which exploit the use of DC currents in graphene, where the velocity of the carriers can be uncommonly high, and the mechanism has been associated with the phenomenon of Landau damping [25], suggesting a possible common underlying origin, although no charge motion is present in our case.

Although our discussion has been entirely in terms of electromagnetic waves, similar processes will apply to other waves: to water waves and particularly to acoustic waves. In many ways acoustic systems could be much more amenable to realization: frequencies are much lower, removing the problem of modulation speed and introducing the possibility of electronic control [26]. Also, many acoustic systems have extremely low loss and low dispersion, alleviating other difficulties.

## Funding

Engineering and Physical Sciences Research Council (EP/L015579/1, EP/T51780X/1); Fundação para a Ciência e a Tecnologia (CEECIND/03866/2017, UID/EEA/50008/2020); Moore Family Foundation (-).

## Acknowledgment

We thank the following for support: P.A.H. acknowledges funding from Fundação para a Ciencia e a Tecnología and Instituto de Telecomunicaçõoes. E.G. is supported through a studentship in the Center for Doctoral Training on Theory and Simulation of Materials at Imperial College London funded by the Engineering and Physical Sciences Research Council (EPSRC). J.B.P. acknowledges funding from the Gordon and Betty Moore Foundation.

## Disclosures

The authors declare no conflicts of interest.

## Data Availability

No data were generated or analyzed in the presented research.

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