# An Energy Conserving Mechanism for Temporal Metasurfaces

Kshiteej J. Deshmukh*  | * Graeme W. Milton*  |<EMAIL_ADDRESS>
###### Abstract

Changing the microstructure properties of a space-time metamaterial while a
wave is propagating through it, in general requires addition or removal of
energy, which can be of exponential form depending on the type of modulation.
This limits the realization and application of space-time metamaterials. We
resolve this issue by introducing a mechanism of conserving energy at temporal
metasurfaces in a non-linear setting. The idea is first demonstrated by
considering a wave-packet propagating in a discrete medium of 1-d chain of
springs and masses, where using our energy conserving mechanism we show that
the spring stiffness can be incremented at several time interfaces and the
energy will still be conserved. We then consider an interesting application of
time-reversed imaging in 1-d and 2-d spring-mass systems with a wave packet
traveling in the homogenized regime. Our numerical simulations show that, in
1-d, when the wave packet hits the time-interface two sets of waves are
generated, one traveling forward in time and the other traveling backward. The
time-reversed waves re-converge at the location of the source and we observe
its regeneration. In 2-d, we use more complicated initial shapes and, even
then, we observe regeneration of the original image or source. Thus, we
achieve time-reversed imaging with conservation of energy in a non-linear
system. The energy conserving mechanism can be easily extended to continuum
media. Some possible ideas and concerns in experimental realization of space-
time media are highlighted in conclusion and in the supplementary information.

††preprint: AIP/123-QED

The spatio-temporal modulation of metamaterial microstructure has led to the
discovery of novel and exciting properties like space-time mirrorsBacot  _et
al._ (2016), chromatic birefringenceAkbarzadeh, Chamanara, and Caloz (2018),
space-time cloakingMcCall  _et al._ (2010), and photonic-time-crystals
(PTCs)Zeng  _et al._ (2017). Their added flexibility in manipulating
electromagnetic, acoustic, and elastic wave propagation has led to an emerging
widespread interest in space-time metamaterials. These materials have an
immense potential for applications in radio and optics for electromagnetic
cloaking, frequency multiplication, bandwidth-matching, and filtering. A
common feature of most space-time media is the nonreciprocal
scatteringTaravati, Chamanara, and Caloz (2017); Caloz  _et al._ (2018) of
waves which can be potentially applied to realize magnetless nonreciprocal
devices. A mechanism of controlling photon flow using effective magnetic
fields obtained by dynamic modulation was shown by Fang et al.Fang, Yu, and
Fan (2012). Introducing a synthetic frequency dimension obtained by time
modulation of refractive index has led to interesting ideas of simulating Weyl
points and nodal loops, and 4-dimensional quantum Hall physicsBoada  _et al._
(2012); Celi  _et al._ (2014); Ozawa  _et al._ (2016); Zhang (2016). A
thorough summary of the research on space-time metamaterials until the year
$\displaystyle 2020$ is provided in a two part paper by Caloz et al.Caloz and
Deck-Léger (2019, 2020), and a review of all the work on this topic can be
found in the excellent book by LurieLurie (2007).

Space-time metamaterials, also called dynamic metamaterials, can have time, or
space-time interfaces at which there is a discontinuous change in their
properties. Some of the models for space-time microstructures are space-time
laminates, space-time checkerboards, and electrical transmission line
structuresCassedy (1967); Lurie and Yakovlev (2016) with the capacitance or
inductance being temporally modulated. In simple space-time laminates and
checkerboards the microstructure properties can be doubly periodic in space
and time with piecewise constant values changing abruptly at space-time
interfaces. A recent application of space-time materials is the idea of an
inverse prismAkbarzadeh, Chamanara, and Caloz (2018). Conventional prisms
decompose the incident white light into different colors on passing through
the prism, i.e. they map the frequency to its wavenumber, when subjected to a
spatial discontinuity and temporal dispersion. In contrast, an inverted prism
is a device that uses a temporal discontinuity and spatial dispersion to map
wavenumber to frequency.

Another interesting demonstration of the effect of a time interface was
demonstrated by Bacot et al.Bacot  _et al._ (2016), where the waves on the
surface of a water bath generated by a source encounter a double time
interface. By providing a vertical jolt to the water bath the effective
vertical acceleration was changed and this corresponds to an energy supply to
the system. The double time interface is referred to as the “instantaneous
time mirror”, which generates time reversed waves or backward propagating
waves and forward propagating waves. These time reversed waves converge to the
shape of the original source, even when it is in a shape as complicated as
that of the Eiffel tower. Although there were existing techniques for time
reversal of waves, they were limited to either monochromatic (single
frequency) waves or they were based on the use of emitters and antennas for
recording, storing, and emitting time reversed waves. Temporal modulation
achieves a new way of time-reversing broadband wavesBacot  _et al._ (2016);
Tanter, Thomas, and Fink (1998).

It is well known Lurie and Weekes (2006); Lurie and Yakovlev (2016) that when
we have a wave propagating in a space-time metamaterial, then changing its
microstructure properties generally requires addition of energy or removal of
energy from the system.

A simple temporal laminate or space-time checkerboard constructed by
temporally periodic switching of properties of a spatial laminate has an
exponential growth of energy in waves traveling through it for certain ranges
of material parametersLurie and Weekes (2006); Lurie (2007); Lurie and
Yakovlev (2016). Hence, depending on the type of modulation and the amplitude
of the wave, this energy exchange can be of exponential form in time. This
seriously limits the realization and application of space-time metamaterials.
In some special geometries called “field pattern materials”, it was shown that
one can achieve stable wave propagation without exponential growthMattei and
Milton (2017). Recently, the growth of energy with time of edge waves in
temporal laminates was shown by Movchan et al.Movchan  _et al._ (2022). Some
lossy space-time metamaterials support wave propagation in special regimes, in
which the waves can travel at constant amplitudes, where the exponential
energy growth due to periodic time-modulation of properties is being
compensated by energy dissipationTorrent, Parnell, and Norris (2018).

The requirement of an energy source can be seen easily by imagining a chain of
springs and masses with a wave propagating through it. The spring stiffness
and/or masses change their values on encountering a time interface. In
presence of the wave, if one wants to increase the stiffness of the springs at
a certain time, then some amount of energy typically needs to be added to the
system. Whereas, to make the springs more compliant (by reducing the
stiffness) some amount of energy is typically released from the system.
Similarly, switching the existing masses to lighter masses will typically
release some kinetic energy, while switching to heavier masses will typically
require adding energy to the system.

The space-time metamaterials considered in this work have properties changing
discontinuously at time interfaces and we will refer to them as temporal
metasurfaces. The energy conserving scheme proposed in this section is
applicable for temporal metasurfaces with waves propagating through them. We
observe that certain material properties can be changed at specific instants
of time without any energy cost. Based on this observation we present our
energy-conserving scheme below. In all the numerical simulations that follow,
we implement a finite difference scheme to solve the equations of motion. All
values are to be considered in SI units. Length and time units in all figures
are denoted by meters ($\displaystyle m$) and seconds ($\displaystyle s$)
respectively. We emphasize that the units are quite arbitrary and can be
rescaled as one wants. The wave propagation speed will of course be altered if
the spatial rescaling differs from the time rescaling, unless the mass units
and spring constants are also rescaled accordingly.

Consider a discreet one dimensional (1-d) spring-mass chain with displacements
inline with the one dimension. Initially, each spring has the same stiffness
$\displaystyle k$ and each mass has the same value $\displaystyle m$. Let
$\displaystyle N$ be the number of springs and masses in the 1-d chain and
$\displaystyle a$ be the lattice constant or length of the unit cell. A
schematic representation of the 1-d unit cell is shown in the supplementary
information in Figure S1a. The displacements of the masses are denoted by
$\displaystyle u_{i}(t)$ and are functions of time $\displaystyle t$. For a
wave propagating through this chain, the equation of motion for the
$\displaystyle n^{th}$ mass is given by,

$\begin{split}m\ddot{u}_{n}=-k(u_{n}-u_{n-1})-k(u_{n}-u_{n+1})\end{split},$
(1)

where, the dot denotes partial derivative with respect to time $\displaystyle
t$.

Suppose the $\displaystyle n^{th}$ spring is in equilibrium at a time instant
$\displaystyle t_{1}$, i.e., there is zero stretching or compression of the
spring, $\displaystyle\Delta u(t_{1})=u_{n}(t_{1})-u_{n-1}(t_{1})=0$. At this
instant, the potential energy of the spring is
$\displaystyle\frac{1}{2}k\left(\Delta u(t_{1})\right)^{2}=0$, and the
stiffness can be increased or decreased without adding or removing any energy
from the spring-mass system. Alternatively, at some other instant of time say
$\displaystyle t_{2}$, let the velocity $\displaystyle\dot{u}_{n}(t)$ of the
$\displaystyle n^{th}$ mass be zero, i.e.,
$\displaystyle\dot{u}_{n}(t_{2})=0$. Then at the instant $\displaystyle
t_{2}$, the kinetic energy of this mass is given as,
$\displaystyle\frac{1}{2}m_{n}\dot{u}_{n}^{2}(t_{2})=0$, and we can add or
reduce this mass without any exchange of energy. By considering only a single
spring-mass system, one can easily see that changing the mass when it is at
rest will not change the displacement amplitude but will change the period of
oscillation. Whereas, changing the stiffness of the spring at the equilibrium
position will change both the amplitude and period of oscillation, but the
velocity amplitude remains the same. These serve as our basic mechanisms for
time-modulation of material properties while conserving the total energy. As
the time modulation of properties depends on the specific instants of time, at
which the spring is unstretched or the mass is at rest, this gives rise to the
non-linearity in the problem. A numerical illustration of the energy
conserving scheme in 1-d is shown in Figure S2 of the supplementary
information.

Figure 1: Time interfaces in 1-d. (a) Ratio of energies in the backward
traveling wave ($\displaystyle E_{b}$) and forward traveling wave
($\displaystyle E_{f}$) vs. the change in stiffness ($\displaystyle\Delta k$)
using a single time interface with the abscissa in log scale. The blue (red)
curve represents the 1-d spring mass chain with an initial spring stiffness
value of $\displaystyle k=1$ ($\displaystyle k=10$). (b) Regeneration of the
source in 1-d: The initial disturbance in the form of a Gaussian wave packet
splits into two wave packets: traveling to right and left. After the time
interface we see each of the wave packets split into a forward traveling wave
and a backward traveling wave. The backward traveling waves reconverge to form
the source.

The application of temporal metasurfaces to form time reversed waves discussed
earlier is reconsidered here, but now using energy conserving temporal
metasurfaces instead. In 1-d, the numerical calculations are presented using a
discreet one-dimensional spring-mass system giving our energy conserving
temporal metasurface.

Practically, changing the spring stiffness values is easier than changing the
mass values. The examples below focus on changing the stiffness values at
appropriate times near the time interface. A 1-d spring-mass chain with
$\displaystyle k=1,m=1$ and $\displaystyle a=1$ as described before is
considered here. The initial displacement for the masses is given by a
Gaussian pulse of sufficient width to ensure a homogenized behavior with
negligible dispersion. The initial wave pulse splits into right and left
traveling wave packets with negligible dispersion. A double time interface of
width $\displaystyle\Delta t=3.2$ is generated after time $\displaystyle
t=62.5$, where we change the stiffness of the springs by an amount
$\displaystyle\Delta k=10$ at the first time interface and then at the next
time interface the stiffness is changed to the original value. The stiffness
change at the time interface is carried out according to our energy conserving
mechanism, i.e., the stiffness is changed whenever the springs are neither
stretched nor compressed. Figure 1 shows the space-time ($\displaystyle x-t$)
plot of the displacement of masses as it encounters the double time interface.
The left and right traveling waves hit the time interface and two sets of
waves are generated from each of them; one traveling backward in time as shown
in blue, and the other traveling forward in time as shown in red. These time-
reversed waves form the reconstructed source image.

For a 1-d energy conserving temporal metasurface, Figure 1 shows the ratio of
energy in the backward propagating wave to energy in the forward propagating
wave versus the change in stiffness at a single time interface as measured by
the quantity $\displaystyle(\Delta k+k)/k$. For high values of
$\displaystyle(\Delta k+k)/k$, the energy in the backward traveling waves
($\displaystyle E_{b}$) is seen to be higher than the energy in the forward
traveling waves ($\displaystyle E_{f}$). The blue and red curves, representing
the base stiffness values of $\displaystyle k=1$ and $\displaystyle k=10$
respectively, lie on top of each other, thus indicating that the energy
distribution is independent of base stiffness value.

Figure 2: (Multimedia view) Reconstruction of 2 Gaussian waves: The initial
disturbance or source is in the form of 2 Gaussian waves. The first time
interface occurs at time $\displaystyle t=25$. The time-reversed waves formed
due to the double time interfaces re-converge to form the source at
$\displaystyle t=59$.  Figure 3: (Multimedia view) Dog: The initial source is
in the shape of a dog. At the time interface ($\displaystyle t=3$), the
stiffness is changed by an amount $\displaystyle\Delta k/k\approx-0.9$. The
time reversed waves seen in blue color due to the wave amplitude re-converge
to form the source image.

Next, examples of regeneration of sources in the form of simple and complex
shapes/images in 2-d are considered for temporal metasurfaces with energy
conservation. For antiplane (shear) waves propagating in a three-dimensional
(3-d) elastic material that is geometrically independent in one of the
directions, say out-of-plane direction, we can model the shear restoring force
using springs where the spring constants represent the shear modulus. Hence,
in our numerical simulations we effectively model the shear wave propagation
in 3-d by a two-dimensional (2-d) spring-mass system with out-of-plane scalar
displacements. Such a 2-d spring-mass system, with the unit cell as shown in
supplementary Figure S1b, forms our energy conserving temporal metasurface. We
denote the number of unit cells in the $\displaystyle x$ and $\displaystyle y$
coordinate directions by $\displaystyle N_{x}$ and $\displaystyle N_{y}$
respectively. The mass in the $\displaystyle(i,j)^{th}$ unit cell is denoted
by $\displaystyle m_{i,j}(t)$, its displacements denoted by $\displaystyle
u_{i,j}(t)$, and the stiffness values of 2 springs aligned along the
$\displaystyle x$ and $\displaystyle y$ directions are denoted by
$\displaystyle k_{x_{i,j}}(t)$ and $\displaystyle k_{y_{i,j}}(t)$
respectively. This notation indicates that the material properties as well as
the displacements vary both in space and time. The equation governing the
displacement of mass $\displaystyle m_{i,j}(t)$ is given by Equation (2) and
we drop the dependence on $\displaystyle t$ for conciseness of notation.

$\begin{split}&\frac{\partial}{\partial
t}\left(m_{i,j}\frac{\partial{u}_{i,j}}{\partial
t}\right)=-k_{x_{i,j}}({u}_{i,j}-{u}_{i-1,j})\\\
&-k_{x_{i+1,j}}({u}_{i,j}-{u}_{i-1,j})-k_{y_{i,j}}({u}_{i,j}-{u}_{i,j-1})\\\
&-k_{y_{i,j+1}}({u}_{i,j}-{u}_{i,j+1}).\end{split}$ (2)

The initial velocity of the masses is taken to be zero and the 2-d shape is
prescribed as an initial out-of-plane (scalar) displacement for the masses. In
the 2-d examples below, we change the stiffness of the springs at appropriate
times near the time interface.

A simple source taken in the form of two Gaussian functions is given by the
expression $\displaystyle\exp{\left(-({\mathbold
x}-\mathbold{\mu}_{1})^{2}/\sigma_{1}\right)}+\exp{\left(-({\mathbold
x}-\mathbold{\mu}_{2})^{2}/\sigma_{2}\right)}$, with $\displaystyle{\mathbold
x}=(x,y)$ representing the vector of $\displaystyle x$ and $\displaystyle y$
coordinates, $\displaystyle\mathbold{\mu}_{1}=(175,175)$,
$\displaystyle\mathbold{\mu}_{2}=(225,225)$, and
$\displaystyle\sigma_{1}=\sigma_{2}=50$. Figure 2 (Multimedia view) shows the
snapshots of the displacement of masses (along the vertical axis) at
subsequent instants of time. The initial displacement at time $\displaystyle
t=0$, propagates as a wave and encounters the first time interface at
$\displaystyle t=25$. Near the first time interface, the stiffness of the
springs is changed by an amount $\displaystyle\Delta k/k=3$, and then at the
second time interface ($\displaystyle t=25.5$) the stiffness is changed back
to the original value. This generates the time-reversed waves (backward
propagating waves) which converge to form the reconstructed source image as
seen from the snapshot at $\displaystyle t=59$.

Figure 4:  Energy distribution resulting from scattering of a beam in a 2-d
temporal metasurface: An incoming Gaussian wave pulse encounters the temporal
interface at $\displaystyle t=120$ and subsequent snapshots show that energy
is distributed in all directions.

This next example shows that we can achieve source regeneration (Temporal
image projection) using an energy conserving temporal metasurface even when
the source is in the form of a complicated image. Figure 3 (Multimedia view)
shows the evolution of the source in the form of an image of a dog, into time-
reversed waves and forward propagating waves. The spring stiffness is changed
once by an amount $\displaystyle\Delta k/k\approx-0.9$ near the time
interface. The initial displacement at time $\displaystyle t=0$ is allowed to
evolve for some initial time until $\displaystyle t=3$. After it hits the
single time interface, time-reversed waves are generated and due to the
amplitude are seen in blue color at $\displaystyle t=4$, and eventually they
converge to reconstruct the image of the dog at $\displaystyle t=13.2$.

In the above example, we observe that only a part of the outward propagating
waves are time-reversed to reconstruct the image whereas the forward traveling
waves result in an undesired interference which can be seen as the "noise"
surrounding the reconstructed image. It is desirable to have most of the
energy from the incident wave concentrated in the backward propagating wave as
compared to the forward propagating waves to get an accurate reconstruction of
the source. While Figure 1 shows that we can achieve this in the case of a
single interface by increasing the amount of change ($\displaystyle\Delta k/k$
or $\displaystyle\Delta m/m$) in material properties, it should also be noted
that using a large value for $\displaystyle\Delta k/k$ will cause the wave
front to be more spread out and one may get a blurred reconstructed image.
Similar results for source reconstruction obtained by changing the mass values
are shown in supplementary Figures S4 and S5.

Temporal modulations in an energy conserving temporal metasurface occur at
times neighboring the time interface, resulting in the material being
anisotropic and inhomogeneous for those times. Thus, a Gaussian beam
propagating in a certain direction in an energy conserving temporal
metasurface after encountering a time interface has the total energy scattered
differently in all directions.

A Gaussian beam propagating in a 3-d material is effectively modeled by
considering a Gaussian pulse propagating along a line in a 2-d spring-mass
system. Figure 4 shows the total energy distribution in the spring-mass system
resulting from scattering of the incident wave due to the temporal interface.
An incoming Gaussian pulse from the bottom left corner encounters a temporal
interface at $\displaystyle t=120$ after which the stiffness is changed once
by $\displaystyle\Delta k/k=3$ at appropriate times. In the subsequent
snapshots we observe that, while most of energy is distributed along the line
making an angle of $\displaystyle 45^{\circ}$ to the x-axis (seen in dark
red), some amount of energy is scattered in every other direction. This
illustrates the inhomogeneous nature of the temporal metasurface at times
neighboring the temporal interface.

In conclusion, we have proposed an energy conserving mechanism for temporal
metasurfaces caused by time modulation of material properties. The mechanism
is demonstrated using a discreet spring-mass system in one and two dimensions,
with an application to reconstruction of the source image using time-reversed
waves. For this particular application, the operation is restricted to the
homogenized regime to get a better reconstructed image of the source. However,
the energy conserving mechanism itself is not restricted to the homogenized
regime. In practice, one can change the stiffness of the spring by attaching a
spring in parallel to it between the same points (i.e. a spring of same
length) when the spring is neither stretched nor compressed. Even if one
cannot locate the perfect instant to carry out this operation, the energy
exchange can always be bounded between some limits. With some caution, energy
conserving temporal metasurfaces can be realized easily using electrical
transmission line structures and a brief discussion on this topic can be found
in the supplementary information. The energy conserving mechanism can also be
extended to interfaces at an angle in space-time and to some complicated
geometries in space-time. For waves propagating in continuum media in 1-d, the
modulus can be changed at an instant of time when the stress is close to
$\displaystyle 0$, and the result can be easily generalized to two and three
dimensions. Thus, the energy conserving mechanism can be extended to continuum
systems as well. Experimental realization of space-time metamaterials (with
more than one temporal interfaces) having close to sharp temporal interfaces
that was previously not possible due to the energy exchange requirements, now
seems possible with the proposed scheme.

See supplementary information document for additional details related to this
work.

The authors are grateful to the National Science Foundation for support
through grant DMS-2107926.

## Author Declarations

### Conflict of Interest

The authors have no conflicts to disclose.

### Authors’ Contribution

G. W. Milton suggested the problem and K. J. Deshmukh did all other work.

## Data Availability

Aside from the data available within the article, any other data are available
from the corresponding author upon request.

## References

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