Shixiong Yin, Emanuele Galiffi, Gengyu Xu, Andrea Alù
IMAGE LICENSED BY INGRAM PUBLISHING
Recent years have witnessed a surge of interest in exotic electromagnetic (EM) wave propagation in time-varying systems. An interesting concept is the one of a temporal interface, the time-analogue of a spatial interface, formed by an abrupt and uniform change of the EM properties of the host medium in time. A time interface scatters the incident wave in a dual fashion compared to its spatial counterpart, conserving momentum but transforming frequency and exchanging energy with the wave. This article provides an overview of the wave-scattering features induced by time interfaces from an antennas and propagation engineering perspective. We first discuss the dualities of wave propagation across spatial and temporal interfaces, by recasting the telegrapher equations in a linear time-variant (LTV) transmission line (TL). Then, we introduce the scattering matrix to describe temporal scattering processes, and derive the conditions for reciprocity and momentum conservation. Understanding temporal scattering through the lens of conventional microwave engineering theory may facilitate the analysis and implementation of next-generation metamaterials encompassing temporal degrees of freedom.
Scattering is at the basis of most EM wave phenomena, from the red glow we see at sunset to the multipath propagation in wireless communications. From a physics standpoint, it emerges at spatial discontinuities where the impedances of the two involved media are mismatched. As a result, a portion of the impinging wave is rerouted to different directions in space, as in the case of the reflection and refraction phenomena illustrated in Figure 1(a). Morgenthaler envisioned in [1] that a wave can also be scattered by a discontinuity in time, as shown in Figure 1(b), at which the entire medium undergoes an abrupt change of its constitutive parameters. Dual to its spatial counterpart, two scattered waves are launched at this time interface, propagating in the new medium. The time-refracted signal travels forward, while the time-reflected one travels backward in space by conjugating its phase, hence time-reversing the incident signal.
Figure 1. (a) Reflection and refraction at a spatial interface. (b) Temporal scattering at a time interface.
In the past decades, temporal reflection and refraction phenomena have attracted broad attention from numerous communities interested in wave physics, including microwave [1], [2], [3], optics [4], mechanics [5], plasma [6], and quantum physics [7]. Many promising applications leveraging time interfaces have been proposed, e.g., the inverse prism [8], broadband impedance matching, energy trapping and absorption [9], [10], [11], [12], [13], [14], temporal aiming [15], and phase conjugation [16]. An extended recent review on time-varying media can be found in [17]. In this article, we focus on wave scattering at temporal discontinuities, drawing analogies with familiar concepts and terminologies in the antennas and propagation community. In the “Temporal TL Model” section, we derive the telegrapher equations for the charge and flux waves in a homogenous LTV TL and we examine the duality between spatial and temporal reflection coefficients, and between energy and momentum conservation at spatial and temporal interfaces. In the “Temporal Network Analysis” section, we introduce the duality of the scattering matrix for temporal interfaces, and we formulate temporal network analysis. To exemplify its broad applicability, we investigate cascaded temporal networks, reciprocity constraints, and momentum conservation conditions. Finally, we conclude by envisioning opportunities that temporal scattering may offer in various applications.
As a prominent model in microwave engineering, the TL model replaces the minute description of fields with terminal quantities in circuits, by translating Maxwell’s equations into the telegrapher equations. Figure 2(a) shows a conventional microwave TL in the form of a parallel-plate waveguide carrying a transverse EM (TEM) mode and featuring a spatial inhomogeneity. The nonzero field components ${E}_{x}$ and ${H}_{y}$ satisfy ${\partial}_{z}{E}_{x} = {-}{\mu}{(}{z}{)}{\partial}_{t}{H}_{y}$ and ${\partial}_{z}{H}_{y} = {-}{\varepsilon}{(}{z}{)}{\partial}_{t}{E}_{x}$, where ${\varepsilon}$ and ${\mu}$ are the time-invariant permittivity and permeability of the medium between the conductors, respectively. Introducing the voltage across the plates ${V}\,{\propto}\,{E}_{x}$ and the current on the plates ${I}\,{\propto}\,{H}_{y}$, we obtain the telegrapher equations [18]: \begin{align*} \frac{\partial{V}}{\partial{z}} & = {-}{L}{\left({z}\right)} \frac{\partial{I}}{\partial{t}} \\ \frac{\partial{I}}{\partial{z}} & = {-}{C}{\left({z}\right)} \frac{\partial{V}}{\partial{t}} \tag{P1} \end{align*}
Figure 2. (a) A TL featuring a spatial interface. (b) A TL featuring a temporal interface, at which the medium is abruptly switched in time and uniformly in space.
where ${C}\,{\propto}\,{\varepsilon}$ and ${L}\,{\propto}\,{\mu}$ are the capacitance and inductance per unit length, and their z dependence indicates the two media before and after the interface. Since the system is linear and time-invariant (LTI), the frequency ${\omega}$ is conserved, while the wavenumber ${k}{(}{z}{)} = {\pm}\,{\omega} \sqrt{{L}{(}{z}{)}{C}{(}{z}{)}}$ is different in the two regions.
An incident wave ${\left({V}_{1}^{+},{I}_{1}^{+}\right)}$ impinging on the spatial interface at ${z} = {0}$, as shown in Figure 2(a), is partially reflected ${\left({V}_{1}^{-},{I}_{1}^{-}\right)}$ and transmitted ${\left({V}_{2}^{-},{I}_{2}^{-}\right)}$ with different wavelengths but identical frequencies. The input-side characteristic impedance ${Z}_{1} = {V}_{1}^{+} / {I}_{1}^{+} = {-}{V}_{1}^{-} / {I}_{1}^{-}$ and the load-side impedance ${Z}_{2} = {V}_{2}^{-} / {I}_{2}^{-}$ determine the spatial reflection coefficient ${\Gamma} = {V}_{1}^{-} / {V}_{1}^{+}$ [18] through: \[{\Gamma} = \frac{{Z}_{2}{-}{Z}_{1}}{{Z}_{2} + {Z}_{1}} \tag{2} \] which can be derived from Kirchhoff’s voltage and current laws, and is consistent with Fresnel scattering coefficients at the interface.
Dually, we can consider a parallel-plate waveguide filled by a spatially homogeneous but time-varying permittivity ${\varepsilon}{\left({t}\right)}$ and permeability ${\mu}{\left({t}\right)}$. Maxwell’s curl equations for the TEM wave read ${\partial}_{t}{D}_{x} = {-}{\mu}^{{-}{1}}{\left({t}\right)}{\partial}_{z}{B}_{y}$ and ${\partial}_{t}{B}_{y} = {-}{\varepsilon}^{{-}{1}}{\left({t}\right)}{\partial}_{z}{D}_{x}$, where ${D}_{x} = {\varepsilon}{\left({t}\right)}{E}_{x}$ and ${B}_{y} = {\mu}{\left({t}\right)}{H}_{y}$ if the material dispersion is negligible. Dual to a spatial boundary with vanishing thickness and infinite extent, an ideal temporal interface is formed when the distributed capacitance and/or inductance experience a discontinuity in time, as shown in Figure 2(b). The equivalent circuit quantities now become the electric charge per unit length ${Q} = {C}{\left({t}\right)}{V}$ and the magnetic flux per unit length ${\Phi} = {L}{\left({t}\right)}{I}$, yielding the telegrapher equations with time-varying characteristic parameters [2]: \begin{align*} \frac{\partial{Q}}{\partial{t}} & = {-}\frac{1}{{L}{\left({t}\right)}} \frac{\partial{\Phi}}{\partial{z}} \\ \frac{\partial{\Phi}}{\partial{t}} & = {-}\frac{1}{{C}{\left({t}\right)}} \frac{\partial{Q}}{\partial{z}}{.} \tag{3} \end{align*}
The wavenumber ${k}$ in this LTV system is preserved due to translational symmetry in space, hence it must be the same before and after the temporal interface, while the frequency follows ${\omega}{(}{t}{)} = {\pm}{k} / \sqrt{{L}{(}{t}{)}{C}{(}{t}{)}}$, and it abruptly changes after ${t} = {0}$. An incident wave denoted by ${\left({\Phi}_{i}^{+},{Q}_{i}^{+}\right)}$ is scattered by such an ideal temporal interface—at which ${L}$ and ${C}$ abruptly change to different values uniformly across the entire TL—inducing two counterpropagating waves: one propagates forward ${\left[{\left({\Phi}_{f}^{+}{,}{Q}_{f}^{+}\right)}\right]}$, while the other is coupled to the negative frequency ${-}{\omega}_{f}$, hence it is time-reflected and propagates backward in space ${\left[{\left({\Phi}_{f}^{{-}}{,}{Q}_{f}^{-}\right)}\right]}$. Physically, the time-reflected wave corresponds to the “echo” of a signal produced by the time interface, played backward in time while keeping the original spatial profile.
The temporal reflection and transmission coefficients can be defined for the charge waves as ${R} = {Q}_{f}^{-} / {Q}_{i}^{+}$ and ${T} = {Q}_{f}^{+} / {Q}_{i}^{+}$, respectively. If no shock current or voltage appears at this time interface ${(}{\partial{Q}} / {\partial{t}}$ and ${\partial{\Phi}} / {\partial{t}}$ are assumed finite), the electric charge density ${Q}$ and magnetic flux density ${\Phi}$ (or equivalently the fields ${D}_{x}$ and ${B}_{y}$) are continuous [1], which yields: \begin{align*}{1} & = {T} + {R} \\ {Z}_{i} & = {Z}_{f}{T} - {Z}_{f}{R}{.} \tag{4} \end{align*}
Note that the two scattered waves both must experience the new characteristic impedance ${Z}_{f} = {\Phi}_{f}^{+} / {Q}_{f}^{+} = {-}{\Phi}_{f}^{-} / {Q}_{f}^{-}$, since they both exist after the time interface. The initial impedance ${Z}_{i} = {\Phi}_{i}^{+} / {Q}_{i}^{+}$ and final impedance ${Z}_{f}$ are related by the ratio of backward to forward wave amplitude ${\rho} = {R} / {T}$ [19] through: \[{\rho} = \frac{{Z}_{f}{-}{Z}_{i}}{{Z}_{f} + {Z}_{i}}{.} \tag{5} \]
Comparing (5) to (2) reveals that, by replacing the spatial reflection coefficient ${\Gamma}$ with the temporal backward-to-forward wave amplitude ratio ${\rho}$, we can employ conventional TL theory formulas for temporal interfaces. It is possible to use a conventional Smith chart to predict the backward-to-forward wave amplitude ratio ${\rho}$ by considering impedance transformations in temporal slabs, rather than in spatial ones. By engineering impedance transformation networks through multiple tailored temporal interfaces, it is then possible to realize the desired momentum spectrum of ${\rho}$, as recently shown for temporal quarter-wave plates [10]. Leveraging the equivalence between ${\Gamma}$ and ${\rho}$, we recently showed in [19] that broadband impedance matching and frequency conversion can be realized by temporally tapered interfaces, in analogy with Klopfenstein tapers in the spatial domain [20].
The frequency-wavenumber duality discussed so far alludes to an analogous relationship between energy and momentum at spatial and temporal interfaces. In a passive TL, the spatial reflection coefficient satisfies ${\left\vert{\Gamma}\right\vert}\,{≤}\,{1}$. Based on the duality between ${\Gamma}$ and ${\rho}$, we similarly conclude that ${\left\vert{\rho}\right\vert} = {\left\vert{R} / {T}\right\vert}\,{≤}\,{1}$, i.e., ${\left\vert{R}\right\vert}\,{≤}\,{\left\vert{T}\right\vert}$, in a “passive” temporal TL. However, we should stress that the total energy in an LTV system is generally not conserved, since a switching event may impart energy to the system. Hence, here passivity actually refers to the fact that the net momentum, rather than the energy, cannot exceed its initial value. In this section, we formally derive the energy-momentum duality between spatial and temporal interfaces.
Across a lossless spatial interface, as shown in Figure 2(a), the conservation of power (energy) demands that the time-averaged net power flow must be conserved: \[{P}_{\text{net,1}} = {P}_{\text{net,2}}\,{\Leftrightarrow}\, \frac{1}{{Z}_{1}}{-} \frac{{\left\vert{\Gamma}\right\vert}^{2}}{{Z}_{1}} = \frac{{\left\vert{\text{T}}\right\vert}^{2}}{{Z}_{2}} \tag{6} \] where T is the spatial transmission coefficient. Yet, discontinuous charge and flux across the spatial interface implies that the linear momentum of the wave is not conserved, and the ratio of the time-averaged net momentum densities ${g} = {\Re}{\left\{{Q}^{\ast}{\Phi}\right\}} / {2}$ in the two media is: \[\frac{{g}_{\text{net,2}}}{{g}_{\text{net,1}}} = \frac{{k}_{2}^{2}}{{k}_{1}^{2}}{.} \tag{7} \]
By contrast, across a charge- and flux-continuous temporal interface, as shown in Figure 2(b), the momentum density is conserved, while the net power flow averaged in space over a wavelength ${P}_{\text{net}} = {\Re}{\left\{{V}^{\ast}{I}\right\}} / {2}$ (more details can be found in the supplementary material, available at https://doi.org/10.1109/MAP.2023.3254486) is modified by a factor: \[\frac{{P}_{{\text{net}},{f}}}{{P}_{{\text{net}},{i}}} = \frac{{\omega}_{f}^{2}}{{\omega}_{i}^{2}}{.} \tag{8} \]
For the energy balance, the reader is referred to [9]. Dual to the power conservation at a spatial interface in (6), we can also write the condition for momentum conservation at a time-interface in terms of net momentum, which reads: \[{g}_{{\text{net}},{i}} = {g}_{{\text{net}},{f}}\,{\Leftrightarrow}\,{Z}_{i} = {Z}_{f}{\left\vert{T}\right\vert}^{2}{-}{Z}_{f}{\left\vert{R}\right\vert}^{2}{.} \tag{9} \]
The derivation of (6) to (9) can be found in the supplementary materials (available at https://doi.org/10.1109/MAP.2023.3254486). These results demonstrate the energy/momentum duality associated with spatial/temporal scattering. It should also be noted that the conservation of wavenumber ${k}$ at a temporal interface (dual of the conservation of ${\omega}$ at a spatial one) does not necessarily ensure conservation of wave momentum (energy). An evident example is that a lossy LTI system does conserve frequency but not energy. Similarly, we can impart or extract wave momentum in a space-invariant (hence k-conservative) system if it exchanges momentum with the environment while maintaining spatial translational symmetry. For instance, recently we realized a TL metamaterial with periodically loaded shunt capacitors, in which a charge-discontinuous temporal interface was realized by switching off the loads and grounding the charges in the capacitors [21]. The inclusion of such “momentum baths” can produce different forms of temporal scattering and interference compared to the more conventional examples in [1] and [10].
Loss in a nondispersive TL can be modeled by a series resistance and shunt conductance [18], corresponding to magnetic and electric conduction loss of the material, respectively. For simplicity, here we consider just the electric conductance per unit length, assuming that the loss is concentrated in the electric response of the material. In a conventional TL, if we assume a harmonic time dependence $\varepsilon\left({t}\right)$ the telegrapher equations with loss can be hence written as: \begin{align*} \frac{dV}{dz} & = {-}{j}{\omega}{LI} \\ \frac{dI}{dz} & = {-}{\left({G} + {j}{\omega}{C}\right)}{V} \tag{10} \end{align*} which corresponds to a complex wavenumber ${k} = {\pm}{\omega} \sqrt{{c}_{r}{LC}}$ with ${c}_{r} = {1} + {G} / {(}{j}{\omega}{C}{)}$. It is common in this TL model to assume ${\omega}$ to be real, which is the quantity conserved at the interface, while ${k}$ is complex-valued. As an example, a voltage wave ${V}\,{\sim}\,{e}^{{-}{jkz}}$ propagating along the ${+}{z}$ direction is plotted in Figure 3(a), showing a decay factor in space due to loss, yet with a steady amplitude at each point in space as time varies (different curves in the figure) due to the temporal variation ${e}^{{j}{\omega}{t}}$.
Figure 3. (a) Voltage waves with complex wavenumber and different temporal phases in a lossy TL within a spatial TL model. (b) Charge waves with complex frequency and different spatial phases in a lossy TL for a temporal TL model.
In a temporal TL in the presence of loss, the telegrapher equations with spatial harmonic dependence ${e}^{{-}{jkz}}$ read: \begin{align*} \frac{{d}{\Phi}}{dt} & = \frac{jk}{C}{Q} \\ \frac{dQ}{dt} & = \frac{jk}{L}{\Phi}{-} \frac{G}{C}{Q}{.} \tag{11} \end{align*}
Here, given the dual nature of the conserved quantities, it is convenient to let the frequency ${\omega}$ assume complex values, given by the dispersion relation ${\omega} \sqrt{{c}_{r}{LC}} = {k}$, while ${k}$ is real. As shown in Figure 3(b), this implies that an incoming signal is uniform in space with dependence ${e}^{{-}{jkz}}$, but it decays in time. Because the electric and magnetic fields of a wave in lossy media are not in phase, the energy exchanges at a temporal interface can exhibit nontrivial features [22]. Consider a time-refracted wave with amplitude ${T}{(}{t}{)}\,{\sim}\,{e}^{{j}{\omega}{t}}$, where ${\Re}{\left\{{\omega}\right\}}\,{>}\,{0}$, living in a material with complex impedance ${Z} = \sqrt{{L} / {\left({c}_{r}{C}\right)}}$. Then, the time-reflected component ${R}{(}{t}{)}\,{\sim}\,{e}^{{-}{j}{\omega}^{\ast}{t}}$ takes negative and complex conjugated frequency and thus experiences the conjugate wave impedance ${Z}^{\ast}$. The space-averaged net power flow \[{P}_{\text{net}}{\left({t}\right)} = \frac{{\Re}{\left\{{c}_{r}^{{-}{1}}\right\}}}{\sqrt{LC}} \frac{{\left\vert{T}\right\vert}^{2}{-}{\left\vert{R}\right\vert}^{2}}{2{C}} = \frac{{\Re}{\{}{V}^{\ast}{I}{\}}}{2} \tag{12} \] is proportional to the Poynting vector averaged in space [22], while the total power flow is the sum of the flows carried by the two waves in opposite directions: \[{P}_{\text{tot}}{\left({t}\right)} = \frac{{\Re}{\left\{{c}_{r}^{{-}{1}}\right\}}}{\sqrt{LC}} \frac{{\left\vert{T}\right\vert}^{2} + {\left\vert{R}\right\vert}^{2}}{2{C}}{.} \tag{13} \]
The space-averaged energy density consists of two parts [22]: \[{w}{(}{t}{)} = {U}_{t}{(}{t}{)} + {U}_{i}{(}{t}{)}{.} \tag{14} \]
The first term ${U}_{t}{\left({t}\right)} = {\Re}{\left\{{1} + {c}_{r}^{{-}{1}}\right\}}{\left({\left\vert{T}\right\vert}^{2} + {\left\vert{R}\right\vert}^{2}\right)} / {\left({2C}\right)}$ is directly proportional to the total power flow, as expected, but the second one ${U}_{i}{\left({t}\right)} = {\Re}{\left\{{T}^{\ast}{R}{\left({1}{-}{c}_{r}^{{-}{1}}\right)}\right\}}$ arises from the interference between the counterpropagating waves, which are nonorthogonal in a lossy TL. As a result, the total power flow in the forward and backward waves after one or more time interfaces in the presence of material loss are not just determined by the energy imparted or subtracted by the switching events, but also by the energy exchanges between the waves as they interact with each other, potentially inducing highly unusual phenomena [22].
To facilitate the analysis of wave scattering at temporal interfaces, especially in the presence of multiple such interfaces, it is convenient to develop a temporal network formalism in analogy with the spatial case. Figure 4(a) shows a conventional LTI two-port network in space, where the incident signals impinge from both sides ${(}{z}\,{<}\,{z}_{1}$ with impedance ${Z}_{1}$ and ${z}\,{>}\,{z}_{2}$ with impedance ${Z}_{2})$, and generate scattered signals in both directions. In contrast, a temporal scattering network, shown in Figure 4(b), does not allow propagation from the future (${t}\,{>}\,{t}_{f}$ with impedance ${Z}_{f}$) to the past (${t}\,{<}\,{t}_{i}$ with impedance ${Z}_{i}$). Once we assume a spatial harmonic dependence ${e}^{{-}{jkz}}$ across the entire medium, a signal that travels backward in space can emulate incidence along the ${-}{t}$ axis by taking the negative frequency ${e}^{{-}{jkz}}$. As opposed to the nomenclature in spatial scattering networks, in temporal scattering we assign a superscript “${\pm}$” to a wave propagating along the ${\pm}{z}$ direction, while the subscripts “i” and “f ” denote the (initial) incident and (final) scattered waves, respectively. For example, the backward-propagating signal ${\left({Q}_{i}^{-},{\Phi}_{i}^{-}\right)}$ in Figure 4(b), which appears to be traveling “back in time,” is generated by an incident wave propagating along ${-}{z}$, dual to ${\left({V}_{1}^{-},{I}_{1}^{-}\right)}$ in Figure 4(a). Similarly, the signal ${\left({Q}_{f}^{-},{\Phi}_{f}^{-}\right)}$ plays the role of a temporally scattered wave rather than a spatially incident one, such as ${\left({V}_{2}^{+},{I}_{2}^{+}\right)}$ in Figure 4(a). This different causality relation is also illustrated by the different signal flows in Figure 4: the coupling between two ports in a spatial network is bidirectional in space [Figure 4(a)], while only forward scattering in time is allowed in a temporal network [Figure 4(b)], and multiple scattering can only be achieved through additional switching processes.
Figure 4. (a) A spatial two-port network. The arrows in the box illustrate the bidirectional scattering between the two ports. (b) A temporal two-port network with characteristic impedance changing from ${Z}_{i}$ at ${t} = {t}_{i}$ to ${Z}_{f}$ at ${t} = {t}_{f}$ uniformly in space. The temporal scattering occurs only forward in time due to causality.
Wave scattering by the spatial network shown in Figure 4(a) can be described by a ${2}\,{\times}\,{2}$ scattering matrix S relating ${T}{(}{t}{)}\,{\sim}\,{e}^{{j}{\omega}{t}}$, Analogously, we can define the temporal scattering matrix for the corresponding time-interface, shown in Figure 4(b), at which the impedance is uniformly changed from ${Z}_{i}$ to ${Z}_{f}$ during the time interval ${t}_{i}\,{<}\,{t}\,{<}\,{t}_{f}$, e.g., being switched abruptly [1], sequentially, or smoothly [19], [23]. The scattering matrix determines the amplitude and phase of the scattered waves for arbitrary excitations from the two directions. We define the temporal scattering matrix in terms of the wave amplitudes of electric charge density: \[{\left(\begin{array}{c}{{Q}_{f}^{+}} \\ {{Q}_{f}^{-}}\end{array}\right)} = {S}{\left(\begin{array}{c}{{Q}_{i}^{+}}\\{{Q}_{i}^{-}}\end{array}\right)},{\text{ where }}{S} = {\left(\begin{array}{cc}{{T}_{l}}&{{R}_{r}}\\{{R}_{l}}&{{T}_{r}}\end{array}\right)}, \tag{15} \] where the subscript ${l}{\left({r}\right)}$ denotes the propagation direction of the incident wave ${+}{z}{\left({-}{z}\right)}$.
As a concrete example, Figure 5(a) illustrates a two-port temporal network consisting of a single temporal interface. In the left panels of Figure 5(a), we assume an incident wave of unitary charge amplitude, propagating along ${+}{z}$. The wave impedance ${Z}_{i}$ is determined by the parameters of a lossy TL as detailed in the “Lossy Temporal TLs” section. After the time interface, with the impedance abruptly switched to its final value, temporal refraction and reflection occur. Their amplitudes ${T}_{l}$ and ${R}_{l}$ can be derived from the temporal boundary conditions given by (4), with the impedance in front of the reflection coefficient being conjugated in the presence of loss, yielding: \[{T}_{l} = {\left({Z}_{f}^{\ast} + {Z}_{i}\right)} / {\left({2}{\Re}{\left\{{Z}_{f}\right\}}\right)},\,{R}_{l} = {\left({Z}_{f}{-}{Z}_{i}\right)} / {\left({2}{\Re}{\left\{{Z}_{f}\right\}}\right)}.\tag{16} \]
Figure 5. (a) A TL with initial characteristic inductance ${L}_{i}$, capacitance ${C}_{i}$ and conductance ${G}_{i}$ supports unitary (charge) waves impinging from opposite directions. After the temporal interface (lighting icon), two counterpropagating waves are generated in both cases with different amplitudes. (b) A temporal multilayer structure consisting of ${N}$ temporal interfaces. The initial impedance ${Z}_{i}$ is switched sequentially in time, through ${Z}_{1}$, ${Z}_{2},{\ldots}$, ${Z}_{{N}{-}{1}}$ to its final value ${Z}_{f}$.
On the other hand, when the incident wave travels from the right and sees impedance ${Z}_{i}^{\ast}$, as shown in the right panels in Figure 5(a), the temporal scattering coefficients become: \[{T}_{r} = \frac{{Z}_{f} + {Z}_{i}^{\ast}}{{2}{\Re}{\left\{{Z}_{f}\right\}}},\,{R}_{r} = \frac{{Z}_{f}^{\ast}{-}{Z}_{i}^{\ast}}{{2}{\Re}{\left\{{Z}_{f}\right\}}}{.} \tag{17} \]
The temporal scattering matrix for the network shown in Figure 5(a) can then be explicitly written as: \[{S} = \frac{1}{{2}{\Re}{\left\{{Z}_{f}\right\}}}{\left(\begin{array}{cc}{{Z}_{f}^{\ast} + {Z}_{i}}&{{Z}_{f}^{\ast}{-}{Z}_{i}^{\ast}}\\{{Z}_{f}{-}{Z}_{i}}&{{Z}_{f} + {Z}_{i}^{\ast}}\end{array}\right)}{.} \tag{18} \]
Another example that has been recently studied is the one of a temporal multilayer structure, in which the impedance of the entire medium is switched sequentially in time, forming the building block of time-metamaterials and photonic time crystals [24], [25], [26]. A temporally layered medium is depicted in Figure 5(b), consisting of ${N}$ time interfaces characterized by temporal scattering matrices ${S}_{m}$ with ${m} = {1},{2},{\ldots},{N}$. During each layer of impedance ${Z}_{m}$, the wave propagation can be described by a scattering matrix ${J}_{m} = {\text{diag}}{\left({e}^{{j}{\omega}_{m}{\tau}_{m}},{e}^{{-}{j}{\omega}_{m}^{\ast}{\tau}_{m}}\right)}$, where ${\omega}_{m}$ is the complex frequency and ${\tau}_{m}$ is the duration of each layer. Since the incident and scattered signals are separated on the two sides of (15), the scattering matrix also behaves as the transfer matrix defined for the amplitudes of counterpropagating waves, which implies that the total scattering matrix of this cascaded temporal network equals the chain-product of the individual scattering matrices: \[{S}_{\text{tot}} = {S}_{N} \mathop{\prod}\limits_{{m} = {1}}\limits^{{N}{-}{1}}{J}_{{N}{-}{m}}{S}_{{N}{-}{m}}{.} \tag{19} \]
With the matrix formalism of temporal scattering, we are now ready to study the general constraints on temporal scattering, such as reciprocity and momentum conservation.
Lorentz reciprocity [27] in the Rayleigh–Carson form implies that the electric field induced at a receiver located at a certain point in space and created by a current source at another location are the same when we interchange the positions of source and receiver (see [28, Chapter 1.10]). Breaking reciprocity can be achieved using a dc magnetic bias in magneto-optical materials [29], [30], and more recently it has been explored using asymmetric nonlinearities [31], [32], [33] and space–time modulation [34], [35], [36]. In a reciprocal LTI system, the spatial scattering matrix is symmetric: ${S}{\left({\omega}\right)}^{T} = {S}{\left({\omega}\right)}$. In the presence of temporal interfaces, however, causality implies different properties for the temporal scattering matrix. If the involved phenomena are reciprocal, Lorentz reciprocity implies that an arbitrary temporal interface scatters a wave incident from one direction in the same manner as an opposite traveling wave. Mathematically, we can conclude from (16) and (17) that ${T}_{l} = {T}_{r}^{\ast}$ and ${R}_{l} = {R}_{r}^{\ast}$ for a reciprocal temporal interface, or expressed compactly in terms of the temporal S matrix: \[{\sigma}_{x}{S}^{\ast} = {S}{\sigma}_{x} \tag{20} \] where ${\sigma}_{x} = {\left({0,1;1,0}\right)}$ is the first Pauli matrix that interchanges the rows (columns) of its succeeding (preceding) matrix; (20) also applies to the cascaded matrices of temporal multilayers, as shown in Figure 5(b). To prove this, we first verify that ${\sigma}_{x}{J}_{m}^{\ast} = {J}_{m}{\sigma}_{x}$ and ${\sigma}_{x}{S}_{m}^{\ast} = {S}_{m}{\sigma}_{x}$ if all the layers are reciprocal. Then, by inserting the involutory Pauli matrix ${\sigma}_{x} = {\sigma}_{x}^{{-}{1}}$, we obtain: \begin{align*}{\sigma}_{x}{S}_{\text{tot}}^{\ast} & = {\left({\sigma}_{x}{S}_{N}^{\ast}\right)}{\sigma}_{x} {\left({\sigma}_{x}{J}_{{N}{-}{1}}^{\ast}\right)}{\sigma}_{x}\,{\cdots}\,{\left({\sigma}_{x}{J}_{1}^{\ast}\right)}{\sigma}_{x}{\left({\sigma}_{x}{S}_{1}^{\ast}\right)} \\ & = {\left[{\left({S}_{N}{\sigma}_{x}\right)}{\sigma}_{x}{\left({J}_{{N}{-}{1}}{\sigma}_{x}\right)}{\sigma}_{x}{\cdots}{\left({J}_{1}{\sigma}_{x}\right)}{\sigma}_{x}{S}_{1}\right]}{\sigma}_{x} \\ & = {S}_{\text{tot}}{\sigma}_{x} \tag{21} \end{align*}
[37] considered more complex scenarios involving anisotropic and nonreciprocal media.
As discussed in the “Energy and Momentum Duality” section, the linear momentum of an EM wave upon temporal scattering in a uniform medium is dual to the energy in a time-invariant spatial scattering scenario. Considering two counterpropagating waves ${\left({\Phi}_{{i},{f}}^{\pm},{Q}_{{i},{f}}^{\pm}\right)}$ as in the temporal scattering network shown in Figure 4(b), the net momentum density of the waves is (more details can be found in the supplementary material, available at https://doi.org/10.1109/MAP.2023.3254486): \[{g}_{{i},{f}} = {\Re}{\{}{Z}_{{i},{f}}{\}}{\left({\left\vert{Q}_{i,f}^{+}\right\vert}^{2}{-}{\left\vert{Q}_{i,f}^{{-}}\right\vert}^{2}\right)}. \tag{22} \]
If there is no external shock current or voltage that alters the total charge or flux in the system, then the net momentum is conserved during the temporal scattering, i.e., ${g}_{i} = {g}_{f}$. Substituting the temporal scattering matrix given by (15), we find: \[{a}{\left\vert{Q}_{i}^{+}\right\vert}^{2}{-}{b}{\left\vert{Q}_{i}^{-}\right\vert}^{2} + {2}{\Re}{\{}{cQ}_{i}^{+}{(}{Q}_{i}^{-}{)}^{\ast}{\}} = {0} \tag{23} \] where ${a} = {\left\vert{{S}_{11}}\right\vert}^{2}{-}{\left\vert{{S}_{21}}\right\vert}^{2}{-}{\Re}{\{}{Z}_{i}{\}} / {\Re}{\{}{Z}_{f}{\}},\,{b} = {\left\vert{{S}_{22}}\right\vert}^{2}{-}{\left\vert{{S}_{12}}\right\vert}^{2}{-}{\Re}{\{}{Z}_{i}{\}} / {\Re}{\{}{Z}_{f}{\}}$, and ${c} = {S}_{11}{S}_{12}^{\ast}{-}{S}_{22}^{\ast}{S}_{21}$. For arbitrary incident waves ${Q}_{i}^{\pm}$, (23) must hold. Thus, all three coefficients should vanish. In particular, the relation ${a} = {b} = {0}$ is dual to the power (energy) conservation condition ubiquitous in spatial scattering, which reads ${\left\vert{{S}_{11}}\right\vert}^{2} + {\left\vert{{S}_{21}}\right\vert}^{2} = {\left\vert{{S}_{22}}\right\vert}^{2} + {\left\vert{{S}_{12}}\right\vert}^{2} = {1}$. The “−” sign in ${a},{b}$ stems from the vectorial nature of momentum, and the impedance ratio is due to our definition of the temporal scattering matrix in terms of electric charge density, which requires renormalization when converting the wave amplitude to its momentum. In analogy with ${S}^{\dagger}{S} = {\bf{1}}$ (identity matrix) for energy conservation in the spatial case, we express the condition ${a} = {b} = {c} = {0}$ in matrix form as: \[{\Re}{\{}{Z}_{f}{\}}{\left({\sigma}_{z}{S}\right)}^{\dagger}{\left({\sigma}_{z}{S}\right)} = {\Re}{\{}{Z}_{i}{\}}{\bf{1}} \tag{24} \] where ${\dagger}$ denotes conjugate transpose and ${\sigma}_{z} = {(}{1},{0}{;}{0},{-}{1}{)}$ is the third Pauli matrix. We have hence obtained a constraint for the scattering matrix consistent with a momentum-conserving temporal network. If the system is also lossless (with real-valued impedances) and reciprocal (${S}_{11} = {S}_{22}^{\ast}$, ${S}_{21} = {S}_{12}^{\ast}$), then (24) simplifies to (9) derived from the temporal boundary conditions.
In summary, we have discussed wave scattering at temporal interfaces from a microwave engineering perspective. Starting from the dual form of the telegraphers’ equations in the presence of time-varying constitutive parameters, we have showcased several space–time dualities, including those between frequency and wavenumber, energy and momentum, and the reflection coefficients associated with spatial and temporal boundaries. We then introduced the temporal scattering matrix formalism, which exhibits dual symmetries to the spatial case, as well as distinct properties stemming from causality. The scattering framework presented in this article can serve as a tool to shed light on more sophisticated temporal scattering processes [38], [39], [40]. We believe that this tool can provide fertile ground for both wave physics and EM engineering. For instance, wave interference in a lossy temporal TL can enhance the field of non-Hermitian physics [22]. From the engineering standpoint, temporal scattering can join forces with space-time and Floquet metamaterials [41], [42] to enhance the growing field of reconfigurable intelligent surfaces [43], implying exciting opportunities for antennas and propagation applications, such as broadband frequency translation [21] and signal amplification.
This work was supported by the Air Force Office of Scientific Research and the Simons Foundation. E.G. acknowledges funding from the Simons Foundation through a Junior Fellowship of the Simons Society of Fellows (855344). More details regarding mathematical details and derivations of a few equations in the main text, including the averaging of energy and momentum density, power and momentum balance, and the net momentum in a lossy, can be found in the supplementary material (available at https://doi.org/10.1109/MAP.2023.3254486).
Shixiong Yin (syin000@citymail.cuny.edu) is a Ph.D. student in the Department of Electrical Engineering, City College of The City University of New York, New York, NY 10031, USA.
Emanuele Galiffi (egaliffi@gc.cuny.edu) is a postdoctoral research fellow with the Photonics Initiative, the Advanced Science Research Center at The City University of New York, New York, NY 10031, USA.
Gengyu Xu (gxu@gc.cuny.edu) is a postdoctoral researcher with the Photonics Initiative, the Advanced Science Research Center at The City University of New York, New York, NY 10031, USA.
Andrea Alù (aalu@gc.cuny.edu) is a distinguished professor at the City University of New York (CUNY), the Founding Director of the Photonics Initiative, the CUNY Advanced Science Research Center, and the Einstein professor of physics at the CUNY Graduate Center, New York, NY 10016, USA; and a professor of Electrical Engineering at the City College of New York, New York, NY 10031, USA.
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Digital Object Identifier 10.1109/MAP.2023.3254486