Victor Pacheco-Peña, Nader Engheta
IMAGE LICENSED BY INGRAM PUBLISHING
While wave-matter interactions can be tailored in space via spatial inhomogeneities in material parameters, temporal and spacetime media are becoming increasingly popular as they may enable the full control of electromagnetic (EM) waves in four dimensions (4D). Here, expanding our previous work on the effective medium concept of temporal media, we develop more general effective medium theories that combine spatial multilayered media with temporal multistepped changes of the permittivity. This work represents a combination of spatial and temporal inhomogeneities in a single structure. As the temporal modulation is applied within certain spatial multilayers (not all the layers), our approach may relax the need for temporally modulating the whole medium where the wave travels while still achieving frequency conversion (a key feature of temporal multistepped media). The theoretical formulation and closed-form expressions for the effective permittivity of such spacetime effective media are presented. The proposed structure is studied via numerical simulations, demonstrating the possibility of designing spacetime effective media using a combination of temporally and spatially modulated materials. The increased degrees of freedom provided by our approach may open new possibilities for manipulating wave-matter interaction in 4D.
Controlling EM wave propagation at will has been of great interest within the scientific community for many years [1], [2]. The notion of metamaterials and metasurfaces can be considered an important paradigm in this context [3], [4], [5], [6], [7], [8]. This is due to the fact that they can enable an arbitrary manipulation of the EM properties of media, allowing the design of artificial materials with permittivity ${(}\varepsilon{)}$ and/or permeability ${(}{\mu}{)}$ responses not easily available in nature, such as near-zero [9], [10], [11], [12], [13] and negative values [14], [15], [16], [17], [18], [19]. Importantly, the scenarios where metamaterials and metasurfaces can be implemented have increased over the years, enabling the scientific community to report their use in applications such as computing [20], [21], [22], [23], antennas [24], [25], [26], [27], [28], circuits [29], [30], [31], and invisibility cloaking devices [32], [33], [34], [35], among others, within frequency ranges spanning from microwaves to terahertz up to the optical regime [36], [37], [38], [39], [40], [41].
The applications of metamaterials and metasurfaces have been explored in the spatial domain (time-harmonic scenario) where the EM wave-matter interaction is achieved by means of introducing spatial (x, y, z) inhomogeneities along the path where EM waves propagate. In recent years, however, the extra degree of freedom of time (t) has been introduced, allowing the scientific community to increase the ability to manipulate EM waves from three dimensions (3D, space) to four dimensions (4D, spacetime) [42], [43]. Two of the early studies in the field of spacetime EM media date back to the last century, when Morgenthaler and Fante explored the propagation of a monochromatic EM wave within a medium where its relative value of ${\varepsilon}_{\text{r}}$ was rapidly changed in time from ${\varepsilon}_{\text{r1}}$ to ${\varepsilon}_{\text{r2}}$ (all values larger than one ${\varepsilon}_{\text{r1}}$, ${\varepsilon}_{\text{r2}}\,{≥}\,{1}$) at ${t} = {t}_{0}$ [i.e., ${\varepsilon}_{\text{r}}{(}{t}{)}$ modeled as a single-step function] [44], [45]. Such a rapid change of ${\varepsilon}_{\text{r}}{(}{t}{)}$ (with a fall/rise time much shorter than the period T of the incident monochromatic EM wave) is now understood to be the temporal analog of a spatial boundary between two media with different values of ${\varepsilon}_{\text{r}}$. In these works, it was demonstrated theoretically how such a temporal boundary can produce a wave traveling forwards (FW wave) and a wave traveling backwards (BW wave). These initial studies have inspired the scientific community to further study the manipulation of EM wave propagation within spatiotemporal media, allowing their application in exciting and exotic scenarios, such as antireflection temporal coatings and filters [46], [47], [48]; non-Foster temporal media [49]; Fresnel drag [50]; phase conjugation [51]; temporal anisotropy for inverse prism [52] and temporal aiming [53]; generalization of the Kramers-Kroning relations [54] and causality considerations [55]; temporal meta-atoms [56], [57]; energy accumulation [58]; and amplification and lasing [59], among others [60]. In addition to these applications, as metamaterials and metasurfaces can be designed to emulate artificial media having effective EM responses (${\varepsilon}_{\text{reff}}$ and ${\varepsilon}_{\text{reff}}$; i.e., effective media), recent studies have also shown how 4D metamaterials can be exploited to enable temporal [61] or spatiotemporal effective media [62], demonstrating the richness of spacetime media for an arbitrary and full control of EM waves.
Inspired by the different possibilities and opportunities enabled by the introduction of spacetime metamaterials and the importance of effective medium concepts in 4D, in this work, we present a study of a technique that provides effective medium theories for the combination of the spatial and temporal inhomogeneities. In our recent article, we showed how, in a spatially homogeneous unbounded medium, a rapid and periodic temporal change of ${\varepsilon}_{\text{r}}{(}{t}{)}$ between ${\varepsilon}_{\text{r1}}$ to ${\varepsilon}_{\text{r2}}$ (a periodic change starting at ${t} = {t}_{0}$) can be modeled as an effective temporal permittivity ${\varepsilon}_{\text{reff}}{(}{t}{)}$ that follows a single-step function, with a change of ${\varepsilon}_{\text{r}}$ from ${\varepsilon}_{\text{r1}}$ to ${\varepsilon}_{\text{reff}}$ at ${t} = {t}_{0}$, which was the temporal equivalent of the effective medium theory for the spatial multilayered metamaterials [63]. Here, we build upon this work and explore how such a temporal multistepped effective medium can be combined together with spatial multilayered effective media to achieve a combined effective medium concept for the design of spacetime effective metamaterials. To that end, we consider cases involving spatially subwavelength (horizontal and vertical) multilayers placed within a cavity in which the electric field of the mode is perpendicular to and parallel with these multilayers, respectively. The alternating spatial multilayers are considered to have either a constant or a time-dependent ${\varepsilon}_{\text{r}}{(}{t}{)}$. Different scenarios are studied for the spatial layers having ${\varepsilon}_{\text{r}}{(}{t}{)}$ values, including ${\varepsilon}_{\text{r}}{(}{t}{)}$ modeled as a single step or a periodic function of time [where the ${\varepsilon}_{\text{r}}$ of the medium is initially ${\varepsilon}_{\text{r1}}$ and then is periodically alternated in time (with the temporal periodicity much shorter than the period of the EM signal) between ${\varepsilon}_{\text{r2}}$ and ${\varepsilon}_{\text{r1}}$, starting at ${t} = {t}_{0}$]. It will be shown how these scenarios can be exploited to design combined effective media having an ${\varepsilon}_{\text{r}}{(}{t}{)}$ modeled as single-step functions where ${\varepsilon}_{\text{r}}{(}{t}{)}$ is rapidly changed from ${\varepsilon}_{\text{r1}}$ to ${\varepsilon}_{{\text{reff}}\_{\text{spacetime}}}$ at ${t} = {t}_{0}$. The effect of the temporal and spatial filling fraction is also explored, showing how different values of ${\varepsilon}_{{\text{reff}}\_{\text{spacetime}}}$ can be obtained by modifying the time duration and/or spatial thickness of the temporal multisteps and/or the thickness of the spatial multilayers, respectively.
The schematic representation of our spacetime effective medium concept is shown in Figure 1, where both temporal multistepped media and spatial multilayers are considered. i.e., our technique combines temporal effective and spatial effective medium concepts. Before merging both configurations, let us first focus our attention on Figure 1(a), where the schematic representation of a temporal effective medium is presented. In this scenario, as we reported in [61], a monochromatic wave travels within a spatially homogeneous unbounded medium with a relative permittivity ${\varepsilon}_{\text{r1}}$ for times ${t}\,{<}\,{t}_{0}$ (from now on, all values of ${\varepsilon}_{\text{r}}$ are real, positive, and ${≥}\,{1}$; note that the effective medium concept in temporal media is also applicable to temporal loss/gain functions [64]; here, however, without the loss of generality, we evaluate real-valued ${\varepsilon}_{\text{r}}$). At ${t} = {t}_{0}$, the ${\varepsilon}_{\text{r}}$ of the whole medium is periodically alternated in time between ${\varepsilon}_{\text{r2}}$ and ${\varepsilon}_{\text{r1}}$ with a time period much shorter than the period of the incident signal (temporal multisteps ${≪}\,{T}$). As shown in [61], such a temporal multistepped function of ${\varepsilon}_{\text{r}}{(}{t}{)}$ is the equivalent of having a temporally effective medium modeled as a single step where ${\varepsilon}_{\text{r}}$ is ${\varepsilon}_{\text{r1}}$ for times ${t}\,{<}\,{t}_{0}$, and it is changed to ${\varepsilon}_{{\text{reff}}\_{\text{time}}}$ at ${t} = {t}_{0}$ and kept to this value for ${t}\,{>}\,{t}_{0}$. Such a temporal effective value ${\varepsilon}_{{\text{reff}}\_{\text{time}}}$ can be theoretically calculated as [61] \[{\varepsilon}_{{\text{reff}}\_{\text{time}}} = \frac{{\varepsilon}_{\text{r1}}{\varepsilon}_{\text{r2}}}{{\varepsilon}_{\text{r1}}{\widetilde{\Delta{t}_{2}}} + {\varepsilon}_{\text{r2}}{\widetilde{\Delta{t}_{1}}}} \tag{1} \]
Figure 1. A schematic representation of the proposed spacetime effective media concept based on combined effective medium theories. For times ${t}\,{<}\,{t}_{0}$, the permittivity of the medium filling a cavity is constant ${\varepsilon}_{\text{r}} = {\varepsilon}_{\text{r1}} = {\text{constant}}$ [left panel in (c)]. At ${t} = {t}_{0}$, a spatial multilayered medium is created with some layers having a constant unchanged ${\varepsilon}_{\text{r}} = {\varepsilon}_{\text{r1}} = {\text{constant}}$ [see (a)], while others have a time-dependent permittivity ${\varepsilon}_{\text{r}}{(}{t}{)}$ modeled either as a temporally periodic multistepped medium (with periodicity much shorter than the period of the EM signal) or as a temporal effective medium using a single-step function of ${\varepsilon}_{\text{r}}$ [left and right panels in (b), respectively]. In this setup, the EM signal is excited using an array of dipoles polarized along the y-axis, and they are switched off at ${t} = {t}_{0}$. With this configuration, the equivalent spacetime effective medium generated using the combination of temporal effective and spatial effective medium concepts is shown in (d). (a) Spatial layer A. (b) Spatial layer B. (c) Spatial multilayers. (d) Spatiotemporal effective medium.
with ${\widetilde{\Delta{t}_{1}}} = {\Delta}{t}_{1} / {(}{\Delta}{t}_{1} + {\Delta}{t}_{2}{)} = {1} - {\widetilde{\Delta{t}_{2}}}$ and ${\widetilde{\Delta{t}_{2}}} = {\Delta}{t}_{2} / {(}{\Delta}{t}_{1} + {\Delta}{t}_{2}{)}$ as the temporal filling fractions for each periodic temporal step corresponding to the times where ${\varepsilon}_{\text{r}}$ is either ${\varepsilon}_{\text{r1}}$ or ${\varepsilon}_{\text{r2}}$, respectively. Based on this, the temporal effective medium indeed represents a single temporal boundary where the frequency of the wave inside the unbounded medium is modified from ${f}_{1}$ to ${f}_{2} = {f}_{1} \sqrt{{\varepsilon}_{\text{r1}} / {\varepsilon}_{{\text{reff}}\_{\text{time}}}}$ (considering ${\mu}_{\text{r1}} = {\mu}_{\text{r2}} = {\mu}_{\text{r}}{)}$ and the wavenumber k is not modified [44], [45].
Now, with this review of our previous work on the concept of temporal effective media, we can now focus our attention on the spatial effective media shown on the right-hand side of Figure 1(c) (spatial multilayered media). Here, we consider a monochromatic Ey-polarized TEM mode within a 1D cavity of width ${d} = {\lambda}_{1} / {2}$ [${\lambda}_{1} = {\left({c} / {f} \sqrt{{\varepsilon}_{\text{r1}}}\right)}$, where c is the velocity of light in vacuum] filled with spatially subwavelength horizontal or vertical multilayers. (Note that we study an EM mode in a cavity instead of wave propagation in a multilayer medium in an unbounded region to better observe the field distribution with the structure, without complicating the problem with the presence of unequal forward and backward waves. However, the same approach here proposed for the spacetime effective media can be considered using spacetime multilayers placed outside of a cavity and being illuminated with a plane wave.) Now, as shown in Figure 1(c), the subwavelength spatial multilayers are periodically placed within the cavity with a spatial periodicity ${≪}\,{\lambda}_{1}$. The spatially periodic arranged layers are filled with two different media having a relative permittivity of ${\varepsilon}_{\text{rA}}$ and ${\varepsilon}_{\text{rB}}$ with a filling fraction of ${\widetilde{\Delta{y}_{\text{A,B}}}}$ or ${\widetilde{\Delta{x}_{\text{A,B}}}}$, respectively, depending on the orientation of the spatial layers (horizontal or vertical, respectively). Now, as is known, for the E-field polarized parallel with the y-axis, such horizontal or vertical spatial multilayered media can be modeled as spatial effective media having an effective permittivity of, respectively [63] \begin{align*}{\varepsilon}_{{\text{reff}}\_{\text{hor}}} = \frac{{\varepsilon}_{\text{rA}}{\varepsilon}_{\text{rB}}}{{\varepsilon}_{\text{rA}}{\widetilde{\Delta{y}_{\text{B}}}} + {\varepsilon}_{\text{rB}}{\widetilde{\Delta{y}_{\text{A}}}}} \tag{2a} \\ {\varepsilon}_{{\text{reff}}\_{\text{ver}}} = {\varepsilon}_{\text{rA}}{\widetilde{\Delta{x}_{\text{A}}}} + {\varepsilon}_{\text{rB}}{\widetilde{\Delta{x}_{\text{B}}}} \tag{2b} \end{align*} where ${\widetilde{\Delta{x}_{\text{A}}}} = {\Delta}{x}_{\text{A}} / {(}{\Delta}{x}_{\text{A}} + {\Delta}{x}_{\text{B}}{)} = {1} - {\widetilde{\Delta{x}_{\text{B}}}}$, ${\widetilde{\Delta{x}_{\text{B}}}} = {\Delta}{x}_{\text{B}} / {(}{\Delta}{x}_{\text{A}} + {\Delta}{x}_{\text{B}}{)}$, ${\widetilde{\Delta{y}_{\text{A}}}} = {\Delta}{y}_{\text{A}} / {(}{\Delta}{y}_{\text{A}} + {\Delta}{y}_{\text{B}}{)} = {1} - {\widetilde{\Delta{y}_{\text{B}}}}$, and ${\widetilde{\Delta{y}_{\text{B}}}} = {\Delta}{y}_{\text{B}} / {(}{\Delta}{y}_{\text{A}} + {\Delta}{y}_{\text{B}}{)}$ are the filling fractions for the subwavelength spatial layers considering vertical or horizontal multilayers.
Now, an interesting question can be asked: What would happen if one combined both the temporal effective and the spatial effective concepts within a single structure if both the spatial and temporal boundaries are merged in a single device? Such a spatiotemporal effective medium technique is the subject of this work, and the schematic representation is shown in full in Figure 1(c), where a monochromatic signal is considered to be present within this cavity. For times ${t}\,{<}\,{t}_{0}$, the relative permittivity ${\varepsilon}_{\text{r}}$ of the whole medium filling the cavity is ${\varepsilon}_{\text{r1}}$ [left panel of Figure 1(c)]. At ${t} = {t}_{0}$, the ${\varepsilon}_{\text{r}}$ of part of the medium is changed in time such that a spatial multilayered structure is created. To do this, we consider that some of the deeply subwavelength spatial layers do not change their ${\varepsilon}_{\text{r}}$ in time [${\varepsilon}_{\text{rA}} = {\varepsilon}_{\text{r1}} = {\text{constant}}$, see Figure 1(a)], while other subwavelength spatial layers are made of materials with a time-dependent ${\varepsilon}_{\text{r}}{(}{t}{)}$, as the one shown in Figure 1(b) (temporal effective medium), i.e., the spatial layers filled with ${\varepsilon}_{\text{r}}{(}{t}{)}$ are those layers with ${\varepsilon}_{\text{rB}}{(}{t}{)}$ where ${\varepsilon}_{\text{rB}}$ is changed from ${\varepsilon}_{\text{r1}}$ for ${t}\,{<}\,{t}_{0}$ to either a temporally periodic multistepped function [(left panel of Figure 1(b)] or an equivalent temporal effective medium [right panel of Figure 1(b)] with ${\varepsilon}_{\text{rB}} = {\varepsilon}_{{\text{reff}}\_{\text{time}}}$ at ${t} = {t}_{0}$. In this context, the structure from Figure 1(c) can behave as a spacetime effective medium having a spacetime effective permittivity ${(}{\varepsilon}_{{\text{reff}}\_{\text{spacetime}}}{)}$, which is a result of the combination of (1) and (2), i.e., the spacetime medium from Figure 1(c) corresponds to a spacetime effective medium modeled as a single-step function where its ${\varepsilon}_{\text{r}}$ is ${\varepsilon}_{\text{r1}}$ for ${t}\,{<}\,{t}_{0}$, and it is changed to ${\varepsilon}_{{\text{reff}}\_{\text{spacetime}}}$ at ${t} = {t}_{0}$ [see the schematic representation in Figure 1(d)], as follows: \begin{align*}{\varepsilon}_{{\text{reff}}\_{\text{spacetime}}} & = \frac{{\varepsilon}_{\text{r1}}{\varepsilon}_{{\text{reff}}\_{\text{time}}}}{{\varepsilon}_{\text{r1}}{\widetilde{\Delta{y}_{2}}} + {\varepsilon}_{{\text{reff}}\_{\text{time}}}{\widetilde{\Delta{y}_{1}}}} \tag{3a} \\ {\varepsilon}_{{\text{reff}}\_{\text{spacetime}}} & = {\varepsilon}_{\text{r1}}{\widetilde{\Delta{x}_{1}}} + {\varepsilon}_{{\text{reff}}\_{\text{time}}}{\widetilde{\Delta{x}_{2}}} \tag{3b} \end{align*} for the horizontal and vertical configurations, respectively, for the y-polarized E-field. Finally, as such a single-step function of ${\varepsilon}_{\text{r}}$ is able to induce a temporal boundary, the frequency of the EM mode within the cavity will be changed from ${f}_{1}$ to ${f}_{2} = {f}_{1} \sqrt{{\varepsilon}_{\text{r1}} / {\varepsilon}_{{\text{reff}}\_{\text{spacetime}}}}$, as expected. In the following sections, we provide a detailed study of the proposed spacetime effective medium concept by exploring examples considering deeply subwavelength vertical and horizontal spatial multilayers filled with constant or time-dependent ${\varepsilon}_{\text{r}}{(}{t}{)}$ modeled as temporal multistepped or single-step temporal functions.
Here, we study our proposed spacetime effective medium concept. The designs are studied numerically considering different configurations, including the combination of single temporal step functions of ${\varepsilon}_{\text{r}}$ and spatial multilayers (horizontal or vertical setups), the combination of temporal multisteps with spatial multilayers, and the effect of the temporal and spatial filling fraction.
All the designs are numerically studied using the time-domain solver of the software COMSOL Multiphysics® [65]. The cavity has a width of ${d} = {\lambda}_{1} / {2}$ along the x. Since we deal with a y-polarized TEM mode in the 1D cavity, for the 2D numerical simulation, we put the perfect electric conductor (PEC) walls on the top and bottom of this cavity to limit the computational domain. For the spatial multilayers, as shown in Figure 1(c), a periodicity of ${0.05}{\lambda}_{1}$ was implemented for both vertical and horizontal spatial configurations, using a total of 10 and 17 spatial periods along the x or y axis, respectively. The cavity was surrounded by PEC. To excite the structure, an array of 17 vertically polarized dipoles was placed ${0.05}{\lambda}_{1}$ away from the left PEC boundary. To better appreciate the effect of the proposed spacetime effective change of ${\varepsilon}_{\text{r}}$ in the signal already present within the cavity, the dipoles are switched off once the ${\varepsilon}_{\text{r}}$ within the cavity is changed at ${t} = {t}_{0}$. The structure was spatially meshed using a triangular mesh type with a maximum and minimum size of ${7.5}\,{\times}\,{10}^{{-}{3}}{\lambda}_{1}$ and ${1.5}\,{\times}\,{10}^{{-}{4}}{\lambda}_{1}$, respectively, with a curvature factor of 0.2 and a resolution of narrow regions of 3 to ensure accurate results. The rapid changes of ${\varepsilon}_{\text{r}}$ in time were implemented using analytical functions having a rise/fall time of ${1.5}\,{\times}\,{10}^{{-}{3}}$ T with a smoothing of two continuous derivatives to ensure the convergence of the results.
To begin with, let us first consider the case shown in Figure 2(a) using horizontal deeply subwavelength spatial multilayers combined with single temporal step functions of ${\varepsilon}_{\text{r}}$. For times ${t}\,{<}\,{t}_{0}$, the ${\varepsilon}_{\text{r}}$ of the whole material filling the cavity is ${\varepsilon}_{\text{r1}} = {1}$. Then, at ${t} = {t}_{0}$, a spatial multilayered medium is created within the cavity via alternating layers with a thickness of ${0.025}{\lambda}_{1}$ (i.e., a filling fraction of 0.5) and permittivity values of ${\varepsilon}_{\text{rA}}$ and ${\varepsilon}_{\text{rB}}$. Here, ${\varepsilon}_{\text{rA}}$ is constant in time (throughout this article, ${\varepsilon}_{\text{rA}} = {\varepsilon}_{\text{r1}} = {\text{constant}}{)}$, while ${\varepsilon}_{\text{rB}}$ is changed in time using a single step from 1 to ${\varepsilon}_{\text{rB}} = {\varepsilon}_{\text{r2}} = {30}$. The schematic representations of the temporal functions for ${\varepsilon}_{\text{rA}}{(}{t}{)}$ and ${\varepsilon}_{\text{rB}}{(}{t}{)}$ are shown on the top and bottom panels in Figure 2(a) (left panels), respectively. By using (3a), such a spatiotemporal effective medium can be modeled as a single-step function where ${\varepsilon}_{\text{r}}$ is ${\varepsilon}_{\text{r1}} = {1}$ for times ${t}\,{<}\,{t}_{0}$, and then, it is rapidly changed to ${\varepsilon}_{{\text{reff}}\_{\text{spacetime}}} = {1.9355}$ at ${t} = {t}_{0}$ [see the schematic representation of ${\varepsilon}_{{\text{reff}}\_{\text{spacetime}}}{(}{t}{)}$ on the left panel of Figure 2(e)]. Note that as we are only modifying the ${\varepsilon}_{\text{rB}}{(}{t}{)}$ using a single-step function, ${\varepsilon}_{{\text{reff}}\_{\text{time}}} = {\varepsilon}_{\text{r2}} = {30}$ in (3). With this setup, the numerical results of the electric field distribution (Ey) at different times considering the case using the single temporal step function of ${\varepsilon}_{\text{rB}}{(}{t}{)}$ together with spatial multilayers along with the results for the case using a spacetime effective medium filling the cavity are shown in Figure 2(b)–(d) and Figure 2(f)–(h). As observed, an excellent agreement is obtained between the results, showing how both structures are indeed equivalent. To further corroborate these results, we recorded the Ey field distribution as a function of time within the multilayered structure (at a spatial location corresponding to a layer filled with ${\varepsilon}_{\text{rA}}$ or ${\varepsilon}_{\text{rB}})$ and within the spacetime equivalent medium, and the results are shown in Figure 2(j) and (m), respectively. From these results, it is clear how the period of the Ey field distribution is the same after applying the temporal changes of ${\varepsilon}_{\text{r}}$ in both configurations.
Figure 2. Combining horizontal spatial multilayers with a temporal step of ${\varepsilon}_{\text{r}}$. (a) The temporal functions of ${\varepsilon}_{\text{r}}$ for the spatial regions with ${\varepsilon}_{\text{rA}}$ and ${\varepsilon}_{\text{rB}}$ are shown on the top and bottom panels, respectively. (e) The generated spacetime effective medium. The numerical results of the electric field (Ey) distribution at different times considering the spacetime structure from (a) and its spacetime effective medium from (e) are presented in (b)–(d) and (f)–(h), respectively. The numerical results of the recorded electric field distribution as a function of time within the spacetime structure from panels (a) and (i) and within the spacetime effective medium from (e) and (l) are shown in (j) and (m), respectively, along with the spectral content for ${t}\,{>}\,{t}_{0}$ inside the cavity in (k) and (n), respectively. For panel (k), the spectral content calculated within a spatial layer filled with ${\varepsilon}_{\text{rA}}{(}{t}{)}$ is shown to better compare the results. The color bars from (b)–(d) and (f)–(h) have a scale of ${\times}\,{10}^{4}$. All these results consider a spatial filling fraction of 0.5, ${\widetilde{\Delta{y}_{\text{A}}}} = {\widetilde{\Delta{y}_{\text{B}}}} = {0.5}$. Arb. units: arbitrary units.
Finally, and to further compare the results, the spectral contents of these signals (for times ${t}\,{>}\,{t}_{0}{)}$ are shown in Figure 2(k) and (n), considering again a spacetime medium and its spacetime effective counterpart, respectively. Note how the frequency in both configurations is changed from ${f} / {f}_{1} = {1}$ to ${f} / {f}_{1} = {0.72}$, in excellent agreement with the theoretical calculations, which predict a change of frequency to ${f} / {f}_{1} = {(}{1} / {1.9355}{)}^{1 / 2} = {0.718}\,{\sim}\,{0.72}$. For completeness, an animation showing the results presented in Figure 2(a)–(h) can be found in the supplementary material. Moreover, a case considering the same setup as in Figure 2 but when ${\varepsilon}_{\text{rB}}{(}{t}{)}$ is changed from ${\varepsilon}_{\text{rB}} = {\varepsilon}_{\text{r1}} = {3}$ to ${\varepsilon}_{\text{rB}} = {\varepsilon}_{\text{r2}} = {1}$ at ${t} = {t}_{0}$ can be found as supplementary material.
For completeness, here we consider a similar configuration as in Figure 2 but now using vertical spatial multilayers. The schematic representation of the temporal functions of ${\varepsilon}_{\text{r}}$ for the spatial regions filled with ${\varepsilon}_{\text{rA}}{(}{t}{)}$ and ${\varepsilon}_{\text{rB}}{(}{t}{)}$ are shown on the left-hand side of Figure 3(a). As in Figure 2, ${\varepsilon}_{\text{rA}}$ is constant in time, while ${\varepsilon}_{\text{rB}}$ is changed from ${\varepsilon}_{\text{rB}} = {\varepsilon}_{\text{r1}} = {1}$ to ${\varepsilon}_{\text{rB}} = {\varepsilon}_{\text{r2}} = {30}$ at ${t} = {t}_{0}$. By using (3b), the spacetime effective medium for times ${t}\,{>}\,{t}_{0}$ has a value of ${\varepsilon}_{{\text{reff}}\_{\text{spacetime}}} = {15.5}$ [see the schematic representation in Figure 3(e)]. With this configuration, the numerical results of the Ey field distribution at different times considering the structure from Figure 3(a) and its spacetime effective counterpart [Figure 3(e)] are shown in Figure 3(b)–(d) and Figure 3(f)–(h), respectively. Again, a good agreement is observed. The numerically evaluated values for Ey for both structures are also shown in Figure 3(j) and (m), respectively, along with the spectral content for times ${t}\,{>}\,{t}_{0}$ in Figure 3(k) and (n), respectively. As observed, the frequency of the signal within the cavity is changed from ${f} / {f}_{1} = {1}$ to ${f} / {f}_{1} = {0.25}$ in both structures from Figure 3(a) and (i) and Figure 3(e) and (l), in agreement with the theoretical calculations, ${f} / {f}_{1} = {(}{1} / {15.5}{)}^{1/2} = {0.254}\,{\sim}\,{0.25}$. Moreover, it is important to note how the temporal signals from Figure 3(j) present some high-frequency harmonics (seen as small ripples), which are due to the multiple reflections of the EM signals excited from each of the spatial layers where the permittivity is changed in time to ${\varepsilon}_{\text{rB}}$. In practical scenarios, if needed, low-pass filters could be implemented to eliminate such higher frequency harmonics. Finally, an animation showing the results presented in Figure 3(a)–(h) can be found in the supplementary material. As in the previous example, in the supplementary material, we have also considered the case with the same setup as in Figure 3 but when ${\varepsilon}_{\text{rB}}{(}{t}{)}$ is changed from ${\varepsilon}_{\text{rB}} = {\varepsilon}_{\text{r1}} = {3}$ to ${\varepsilon}_{\text{rB}} = {\varepsilon}_{\text{r2}} = {1}$ at ${t} = {t}_{0}$.
Figure 3. Combining vertical spatial multilayers with a temporal step of ${\varepsilon}_{\text{r}}$. All the panels from this figure are arranged following the same order as those shown in Figure 2. The schematic representations of the vertical spatial multilayers are shown in (a) and (i), and the schematic representations of the corresponding spacetime effective medium are shown in (e) and (l). The color bars from (b), (c), and (f)–(h) have a scale of ${\times}{10}^{4}$. All these results consider a spatial filling fraction of 0.5, ${\widetilde{\Delta{x}_{\text{A}}}} = {\widetilde{\Delta{x}_{\text{B}}}} = {0.5}$.
Let us now consider the case shown in Figure 4(a), where we combine temporal multisteps of ${\varepsilon}_{\text{r}}$ and spatial multilayers. As in the results discussed in Figures 2 and 3, the ${\varepsilon}_{\text{r}}$ of the whole medium filling the cavity is ${\varepsilon}_{\text{r1}} = {1}$ for ${t}\,{<}\,{t}_{0}$. Then, at ${t} = {t}_{0}$, a spatial multilayered medium is created by alternating horizontal spatial layers with a thickness of ${0.025}{\lambda}_{1}$ (a filling fraction of ${\widetilde{\Delta{y}_{\text{A}}}} = {\widetilde{\Delta{y}_{\text{B}}}} = {0.5}{)}$ with ${\varepsilon}_{\text{r}}$ values of ${\varepsilon}_{\text{rA}}$ and ${\varepsilon}_{\text{rB}}$. As before, ${\varepsilon}_{\text{rA}}$ is constant in time and remains ${\varepsilon}_{\text{rA}} = {\varepsilon}_{\text{r1}} = {1}$. However, ${\varepsilon}_{\text{rB}}$ is periodically changed in time (with a period much shorter than the period of the EM signal, 0.1 T in our case) between ${\varepsilon}_{\text{rB}} = {\varepsilon}_{\text{r2}} = {1.8}$ and ${\varepsilon}_{\text{rB}} = {\varepsilon}_{\text{r1}} = {1}$, starting at ${t} = {t}_{0}$, with a temporal duty cycle (DC) of 0.8 ${(}{\widetilde{\Delta{t}_{2}}} = {0.8}$ and ${\widetilde{\Delta{t}_{1}}} = {0.2}{)}$. Note that here we chose a different temporal DC from 0.5 1) to demonstrate that the DC can also be changed as in the temporal effective medium reported in [61] and 2) to enable the convergence of the numerical simulations. Now, by using (1), such a temporal multistepped medium ${\varepsilon}_{\text{rB}}{(}{t}{)}$ can be modeled as a temporal effective medium using a single-step function of ${\varepsilon}_{\text{r}}$ where ${\varepsilon}_{\text{r}}$ is changed from ${\varepsilon}_{\text{r1}} = {1}$ to ${\varepsilon}_{{\text{reff}}\_{\text{time}}} = {1.55}$ [see the purple line on the bottom panel of Figure 4(a)]. Then, we can simply use (3a) with a spatial filling fraction of 0.5 and calculate the spacetime effective value of ${\varepsilon}_{{\text{reff}}\_{\text{spacetime}}}$ for an effective medium modeled as in Figure 4(d), obtaining ${\varepsilon}_{{\text{reff}}\_{\text{spacetime}}} = {1.2162}$ for times ${t} = {t}_{0}$.
Figure 4. Combining horizontal spatial multilayers with temporal multisteps. The temporal functions of ${\varepsilon}_{\text{r}}$ for the spatial regions filled with ${\varepsilon}_{\text{rA}}$ and ${\varepsilon}_{\text{rB}}$ are shown on the top and bottom panels in (a), respectively, and the schematic of the spatial multilayer is shown in (b). The spacetime effective medium is shown in (d) and (e). The numerical results for the recorded Ey as a function of time within the spacetime structure from (a) and (b) and within the spacetime effective medium from (d) and (e) are shown in (c) and (f), respectively. All these results consider a spatial filling fraction of 0.5 and a temporal duty cycle (DC) of 0.8.
With this configuration, the numerical results of the recorded electric field distribution as a function of time within a spatial layer filled with ${\varepsilon}_{\text{rB}}$ and within the corresponding spacetime effective medium are shown in Figure 4(c) and (f), respectively. As observed, an excellent agreement between the results is obtained. Note how the amplitude of the electric field is increased/reduced periodically because of the effect of the periodic temporal multisteps for ${\varepsilon}_{\text{rB}}{(}{t}{)}$, as expected. Note that these “ripples” are of a different nature compared to those shown in Figure 3(j), where the “ripples” were due to the multiple spatial reflections after the ${\varepsilon}_{\text{rB}}{(}{t}{)}$ of the multilayers is changed in time using a single-step function. Hence, the high-frequency harmonics from the results shown in Figure 4(c) are more pronounced than those shown in Figure 3(j) given the dependence of their amplitude with the periodic temporal modulation of ${\varepsilon}_{\text{rB}}{(}{t}{)}$. These variations of amplitude translate into high-frequency harmonics, as shown in [61], which could be eliminated, if needed, using filters, as mentioned before. To guide the eye, vertical dashed green lines have been added to Figure 4(c) and (f), to demonstrate that the period of the signal is the same in both structures from Figure 4(b) and (e) with a period ${t} / {T} = {1.104}$. This value is in agreement with the theoretical calculations that predict a frequency change from ${f} / {f}_{1} = {1}$ to ${f} / {f}_{1} = {(}{1} / {1.2162}{)}^{1 / 2} = {0.906}$, i.e., a theoretical period of the signal of ${\sim}{t} / {T} = {1.103}$. An animation showing the results of Figure 4 can be found in the supplementary material.
The corresponding case of vertical spatial multilayers with temporal multisteps is shown in the supplementary material, where it is shown how a combination of spatial multilayers and ${\varepsilon}_{\text{rB}}{(}{t}{)}$ modulated using temporal multisteps also generates high-frequency harmonics but now due to both the spatial reflections within the multilayers [as shown and discussed in Figure 3(j)] and the temporal multistepped modulation of ${\varepsilon}_{\text{rB}}{(}{t}{)}$ [as discussed in the results of Figure 4(c)].
In the previous section, we have shown the case where the temporal multisteps have a DC different from 0.5. Here, as a final configuration, we carry out a similar study to the one shown in Figure 4 but considering a single-step function for ${\varepsilon}_{\text{rB}}{(}{t}{)}$ while the spatial filling fraction of the spatial multilayer is changed. The schematic representations of the temporal functions for ${\varepsilon}_{\text{rA}}{(}{t}{)}$ and ${\varepsilon}_{\text{rB}}{(}{t}{)}$ are shown on the top and bottom panels of Figure 5(a), respectively. These functions are then implemented in the vertical spatial multilayer shown in Figure 5(b) (note that we are showing only a section within the cavity, but this spatial multilayer medium is filling the whole cavity as in the previous sections). Finally, a filling fraction of ${\widetilde{\Delta{x}_{\text{A}}}} = {0.8966}$ ${(}{\widetilde{\Delta{x}_{\text{B}}}} = {1}{-}{\widetilde{\Delta{x}_{\text{A}}}}{)}$ is implemented. With this configuration, we carried out numerical simulations, and the resulting spectral content of the EM signal for times ${t}\,{>}\,{t}_{0}$ within the cavity is shown on the right panel of Figure 5(b) as the black curve (calculated at a spatial location filled with ${\varepsilon}_{\text{rA}}$). As observed, the frequency is changed from ${f} / {f}_{1} = {1}$ to ${f} / {f}_{1} = {0.5}$, meaning that the numerically obtained spacetime value of ${\varepsilon}_{{\text{reff}}\_{\text{spacetime}}}$ is ${\varepsilon}_{{\text{reff}}\_{\text{spacetime}}} = {4}$ for times ${t}\,{>}\,{t}_{0}$. These results are again in agreement with the theoretical calculations with a value of ${\varepsilon}_{{\text{reff}}\_{\text{spacetime}}}\,{\sim}\,{4}$ [using (3b) with ${\varepsilon}_{{\text{reff}}\_{\text{time}}} = {\varepsilon}_{\text{r2}} = {30}{]}$. For completeness, the spectrum of the EM signal within the cavity for times ${t}\,{>}\,{t}_{0}$ considering the corresponding spacetime effective medium modeled with a single-step function of ${\varepsilon}_{\text{r}}{(}{\varepsilon}_{\text{r1}} = {1}$ for ${t}\,{<}\,{t}_{0}$ and changed ${\varepsilon}_{{\text{reff}}\_{\text{spacetime}}} = {4}$ at ${t} = {t}_{0}{)}$ is also shown as the red dashed line on the right panel of Figure 5(b), demonstrating an agreement between the results. Finally, and to demonstrate the flexibility of the design using the proposed spacetime effective medium concept, we used the same functions for ${\varepsilon}_{\text{rA}}{(}{t}{)}$ and ${\varepsilon}_{\text{rB}}{(}{t}{)}$ as in Figure 5(a) but calculated the required filling fraction for an equivalent horizontal spatial multilayer that can generate the same spacetime effective medium as the one from Figure 5(b). The resulting structure is shown in Figure 5(c) with a filling fraction of ${\widetilde{\Delta{y}_{\text{A}}}} = {0.2241}$ ${(}{\widetilde{\Delta{y}_{\text{B}}}} = {1}{-}{\widetilde{\Delta{y}_{\text{A}}}}{)}$ (left panel) along with the spectral content of the EM signal within the cavity for time ${t}\,{>}\,{t}_{0}$ (right panel). From these results, it is clear how the same spacetime effective medium is obtained with an ${\varepsilon}_{{\text{reff}}\_{\text{spacetime}}}\,{\sim}\,{4}$.
Figure 5. Combining spatial multilayers having different filling fractions with a single temporal step. (a) The temporal functions of ${\varepsilon}_{\text{r}}$ for the spatial regions filled with ${\varepsilon}_{\text{rA}}$ and ${\varepsilon}_{\text{rB}}$ are shown on the top and bottom panels, respectively. (b) The schematic representation of the vertical spatial multilayer with ${\widetilde{\Delta{x}_{\text{A}}}} = {0.8966}$ ${(}{\Delta}{x}_{\text{A}}\,{\sim}\,{0.0448}{\lambda}_{1}{)}$ is shown on the left panel. The spectral content of the recorded Ey field distribution after ${t}\,{>}\,{t}_{0}$ (calculated within a region filled with ${\varepsilon}_{\text{rA}}$) for the vertical spatial multilayer is shown on the right panel of (b) as a black line along with the spectrum for the case using the spacetime effective medium concept (red dashed line). (c) A schematic of a horizontal spatial multilayer emulating the same response as the vertical spatial multilayer from (b) with ${\widetilde{\Delta{y}_{\text{A}}}} = {0.2241}$ ${(}{\Delta}{y}_{\text{A}}\,{\sim}\,{0.0112}{\lambda}_{1}{)}$, left panel, along with the spectrum of the recorded Ey field distribution (calculated within a region filled with ${\varepsilon}_{\text{rA}}$), right panel.
In terms of potential experimental demonstrations of the proposed spacetime effective medium concept, the structure could be implemented by using transmission line techniques via loaded time-dependent circuit elements [66] where these elements could be added periodically in space to emulate the spatial multilayered regions where the ${\varepsilon}_{\text{r}}$ is and is not changed in time $({\varepsilon}_{\text{rB}}$ and ${\varepsilon}_{\text{rA}}$, respectively). This could be more challenging at higher frequencies, such as in the optical domain; however, experimental studies of temporal and spacetime modulated media are an active research topic in various groups worldwide, reporting different configurations [67], [68], [69]. In this context, we are hopeful that our spacetime effective medium concept will be experimentally demonstrated within different frequency ranges in the near future. Potential applications may include frequency modulation, temporal signal processing, and reconfigurable metamaterials and metasurfaces.
In this work, we have studied the combined effective medium concepts by merging the temporal multistepped changes of ${\varepsilon}_{\text{r}}$ and spatial multilayers. The structure consisted of a 1D cavity filled with spatially periodic deeply subwavelength multilayers (either horizontal or vertically arranged) having either a constant permittivity or a temporally changing permittivity. Different scenarios were considered for the regions with a temporal permittivity, including single-step functions and temporally periodic multistepped changes of ${\varepsilon}_{\text{r}}$. Moreover, studies of the effect of changing the spatial filling fraction as well as the temporal DC were presented. The physics behind the proposed spacetime effective medium concept was presented, and the designs were corroborated via numerical simulations, demonstrating an excellent agreement between the simulations and the theoretical designs. These results may open new avenues in the manipulation of wave propagation at will in 4D using spacetime effective media.
This research was sponsored in part by the Newcastle University (Newcastle University Research Fellowship) (to V. P.-P.), and in part by the Simons Foundation/Collaboration on Symmetry-Driven Extreme Wave Phenomena Grant 733684 (to N.E.). This article has supplementary downloadable material available at https://doi.org/10.1109/MAP.2023.3254480, provided by the authors.
Victor Pacheco-Peña (victor.pacheco-pena@newcastle.ac.uk) is an associate professor (senior lecturer) at the School of Mathematics, Statistics and Physics, Newcastle University, NE1 7RU Newcastle Upon Tyne, U.K. His research interests include light-based computing, metamaterials, space-time media, metasurfaces, plasmonics, and photonics.
Nader Engheta (engheta@seas.upenn.edu) is the H. Nedwill Ramsey Professor at the Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104 USA. He received the 2023 Benjamin Franklin Medal in Electrical Engineering and 2020 Isaac Newton Medal and Prize from the Institute of Physics. He is a Life Fellow of IEEE.
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