Sajjad Taravati, George V. Eleftheriades
IMAGES LICENSED BY INGRAM PUBLISHING
Static metasurfaces have been shown to be prominent compact structures for reciprocal and frequency-invariant transformations of electromagnetic waves in space. However, incorporating temporal variation to static metasurfaces enables dynamic apparatuses that are capable of 4D tailoring of both the spatial and temporal characteristics of electromagnetic waves, leading to functionalities that are far beyond the capabilities of conventional static metasurfaces. This includes nonreciprocal full-duplex wave transmission, pure frequency conversion, parametric wave amplification, space–time (ST) decomposition, and ST wave diffraction. This article overviews our recent progress on the analysis and functionalities of ST metasurfaces and slabs to break reciprocity. We study different operation regimes of ST metasurfaces, such as scattering and diffraction at ST interfaces, ST sinusoidally varying surfaces and slabs, and transmissive and reflective ST metasurfaces. Additionally, corresponding applications are outlined.
Controlled transformation of electromagnetic fields has advanced drastically in recent years, thanks to the advent and evolution of metamaterials and metasurfaces [1], [2], [3], [4]. Static metamaterials and metasurfaces have allowed for substantial progress in wave engineering applications [5], [6]. Recently, there has been a growing interest in ST periodic (STP) metasurfaces for 4D wave engineering, where adding the temporal variation to static metasurfaces leads to functionalities that are far beyond the capabilities of conventional static metasurfaces. For instance, asymmetric wave transmission can be achieved by spatially asymmetric structures when multiple modes are involved at different ports. On the other hand, nonreciprocal wave transmission is a far more challenging task that requires an external bias field for breaking the time invariance of the structure [7], [8], [9], unilateral-transistor-loaded cells [3], or nonlinear materials. Among these nonreciprocity approaches, ST modulation is of high interest because of its immense capability for affecting the spectrum of electromagnetic waves while breaking the time-reversal symmetry.
ST metasurfaces provide huge degrees of freedom for arbitrary alteration of the wave vector and temporal frequency of electromagnetic waves, leading to advanced 4D wave processing in fields from acoustics and microwaves [8], [10], [11], [12], [13], [14], [15], [16], [17] to terahertz and optics [18], [19], [20], [21], [22]. They represent a class of compact dynamic wave processors that have been recently proposed for extraordinary manipulation of electromagnetic waves. Such 4D compact apparatuses are endowed with unique properties not readily seen in conventional static metamaterials and metasurfaces. ST metasurfaces may take advantage of ST modulation capabilities, including nonreciprocal frequency generation [14], [23], [24], parametric wave amplification [11], [25], [26], and asymmetric dispersion [27], [28], [29]. Frequency generation and nonreciprocity are of particular interest in STP slabs [22], [23], [27], [29], [30], [31], [32], which are endowed by asymmetric periodic electromagnetic transitions in their dispersion diagrams [15], [27], [28], [33]. In practice, the ST modulation is achieved by pumping external energy into the medium [8], [13], [14], [23], [27].
Some of the recently proposed applications of STP metamaterials and metasurfaces include mixer–duplexer–antennas [34], unidirectional beam splitters [13], nonreciprocal filters [35], signal coding metasurfaces [17], ST metasurfaces for advanced wave engineering and extraordinary control over electromagnetic waves [8], [10], [14], [15], [16], [18], [20], [36], [37], [38], [39], [40], [41], [42], [43], [44], nonreciprocal platforms [27], [29], [42], [45], [46], [47], [48], [49], frequency converters [14], [23], [24], [50], time-modulated antennas [51], [52], spectral camouflage metasurfaces [53], [54], antenna–mixer–amplifiers [55], and enhanced-resolution imaging photonic crystals [56]. This diverse and significant capability of ST-modulated (STM) media is due to their unique interactions with the incident field [27], [57], [58], [59].
This article provides a review on the properties, analysis, and dispersion diagrams of ST metasurfaces and their 4D wave-transformation characteristics. We provide a few examples on how such a peculiar 4D wave guidance in ST metasurfaces can lead to new extraordinary electromagnetic apparatuses. As a complementary article, we have provided a review on experimental implementations of ST metasurfaces and their applications and functionalities in [4]. We first present key properties of ST interfaces, including spatial interfaces, temporal interfaces, and ST interfaces. Then, we show that a nonreciprocal metasurface acts as a very thin ST slab, i.e., a moving metasurface. Next, the analysis of general STM metasurfaces is outlined, including the derivation of scattered electromagnetic fields, 4D dispersion diagrams, boundary conditions, and ST decomposition. Then, full-wave finite difference time domain (FDTD) simulations of ST metasurfaces are presented. Finally, illustrative examples are provided to show the peculiar and unique functionalities of ST metasurfaces.
Figure 1 shows the Minkowski ST diagram and its Fourier-transformed pair known as the dispersion diagram. The Minkowski diagram is the most well-known class of ST diagrams and was developed by Hermann Minkowski in 1908. This diagram is generally a 2D graph that depicts events as happening in a universe consisting of one time dimension and one space dimension. In contrast to regular distance–time graphs, here the distance is demonstrated on the horizontal axis and time on the vertical axis. Furthermore, the time and space units are chosen such that an object moving at the speed of light in the background medium follows a 45° angle to the diagram’s axes. The Minkowski ST diagram in Figure 1 is constituted of two light cones, representing propagation of light in the past and future. The two cones have their apexes at the present, where the 3D (x, y, and z) hyperspace exists. Any discontinuity in the 4D ST diagram results in forward and backward waves in space. The analysis and design of ST media and metasurfaces can be substantially eased by understanding the Fourier pair of the Minkowski diagram. ST metasurfaces and slabs can be analyzed as two interfaces sandwiched between two semi-infinite regions in space. Therefore, to best investigate the wave diffraction by an STM grating, we first study the interaction of the electromagnetic wave with space and time interfaces, separately [60], [61], [62]. In general, three different ST discontinuities may be studied as follows.
Figure 1. ST Fourier pair diagrams. STVM: space-time-varying medium.
Figure 2(a) sketches the ST diagram of a spatial interface at ${z} = {z}_{0}$ in the plane ${(}{z},{v}_{b}{t}{)}$ between two media of refractive indexes ${n}_{1}$ and ${n}_{2},$ intrinsic impedances ${\eta}_{1}$ and ${\eta}_{2},$ and phase velocities ${v}_{1} = {v}_{b}{/}{n}_{1}$ and ${v}_{2} = {v}_{b}{/}{n}_{2},$ respectively. Here, ${v}_{b}$ represents the velocity of the wave in the background medium. This figure shows scattering of forward and backward fields and conservation of energy and momentum for different scenarios. The temporal axis of the Minkowski ST diagram is scaled with the speed of light and is labeled with ${v}_{b}\text{t}$ for changing the dimension of the addressed physical quantity from time to length, in accordance to the dimension associated to the spatial axes labeled z.
Figure 2. ST diagrams showing scattering of forward and backward fields and conservation of energy and momentum for different scenarios. (a) Spatial interface, i.e., ${n}{(}{z}{<}{0}{)} = {n}_{1}$ and ${n}{(}{z}{>}{0}{)} = {n}_{2}{.}$ (b) Temporal interface, i.e., ${n}{(}{t}{<}{0}{)} = {n}_{1}$ and ${n}{(}{t}{>}{0}{)} = {n}_{2}{.}$ (c) Subluminal ST interface ${(}{v}_{m}{<}{v}_{b}{),}$ where ${v} = {v}_{m}{.}$ (d) Superluminal ST interface ${(}{v}_{m}{>}{v}_{b}{),}$ where ${v} = {v}{}_{b}^{2}{/}{v}_{m}{.}$
This problem represents the textbook case of electromagnetic wave incidence and scattering from a spatial (static) interface, where ${n}{(}{z}{<}{0}{)} = {n}_{1}$ and ${n}{(}{z}{>}{0}{)} = {n}_{2}{.}$ The boundary conditions are derived by applying the fundamental physical fact that all physical quantities must remain bounded everywhere and at every time to the space and time derivatives in the sourceless Maxwell equations $\nabla\times{E} = {-}\partial{B}{/}\partial{t}$ and $\nabla\times{H} = \partial{D}{/}\partial{t}{.}$ The discontinuity of the tangential components of electric and magnetic fields at ${z} = {z}_{0}$ would result in unbounded and singular electromagnetic fields at the interface, which is not physically possible. Therefore, the tangential components of the electric and magnetic fields must be continuous at a space discontinuity; i.e., $\hat{z}\times{(}{E}_{2}{-}{E}_{1}{)}{|}{}_{{z} = {z}_{0}} = {0}$ and $\hat{z}\times{(}{H}_{2}{-}{H}_{1}{)}{|}{}_{{z} = {z}_{0}} = {0}{.}$ As a result, the wavenumber k changes; i.e., energy is preserved but momentum changes, such that the wavenumbers of the backward reflected wave in region 1 and the forward transmitted wave in region 2 correspond to ${k}_{R} = {-}{k}_{i}$ and ${k}_{T} = {k}_{i}{n}_{2}{/}{n}_{1},$ respectively, whereas the temporal frequencies of the backward reflected and forward transmitted waves are equal to that of the incident wave, i.e., ${\omega}_{R} = {\omega}_{i}$ and ${\omega}_{T} = {\omega}_{i}{.}$
Throughout the article, we use the wave’s time dependence of $\exp{(}{-}{i}{\omega}{t}{)}$ such that with ${\omega}{>}{0}$ the wave travels from left to right if the wavenumber is positive ${(}{k}{>}{0}{)}$ and from right to left if the wavenumber is negative ${(}{k}{<}{0}{).}$ We consider the incident field in region 1 (traveling toward the $ + {z}$ direction) as ${E}_{I} = \exp{[}{i}{(}{k}_{i}{z}{-}{\omega}_{i}{t}{)],}$ the reflection field in region 1 (traveling toward the ${-}{z}$ direction) as ${E}_{R} = {R}\exp{[}{i}{(}{-}{k}_{i}{z}{-}{\omega}_{i}{t}{)],}$ and the transmitted field in region 2 (traveling toward the $ + {z}$ direction) as ${E}_{T} = {T}\exp{[}{i}{(}{k}_{t}{z}{-}{\omega}_{t}{t}{)],}$ where R and T represent the spatial reflection and transmission coefficients, respectively. The total field in region 1 reads ${E}_{1} = {E}_{I} + {E}_{R},$ and the field in region 2 reads ${E}_{2} = {E}_{T}{.}$ We next apply the spatial boundary condition at ${z} = {z}_{0}$ (i.e., ${E}_{1}{|}{}_{{z} = {z}_{0}} = {E}_{2}{|}{}_{{z} = {z}_{0}}$ and ${H}_{1}{|}{}_{{z} = {z}_{0}} = {H}_{2}{|}{}_{{z} = {z}_{0}}{)}$ to determine R and T as ${R} = {(}{\eta}_{2}{-}{\eta}_{1}{)}{/}{(}{\eta}_{1} + {\eta}_{2}{)}$ and ${T} = {2}{\eta}_{2}{/}{(}{\eta}_{1} + {\eta}_{2}{)}$ where ${\eta}_{1} = \sqrt{{\mu}_{1}/{\epsilon}_{1}}$ and ${\eta}_{2} = \sqrt{{\mu}_{2}/{\epsilon}_{2}}{.}$ More details can be found in the supplementary materials available at http://doi.org/10.1109/MAP.2022.3201195.
Figure 2(b) shows the ST diagram of a time interface between two media of refractive indices ${n}_{1}$ and ${n}_{2},$ which is the dual case of the spatial metasurface in Figure 2(a). Here, the refractive index suddenly changes from one value $({n}_{1})$ to another $({n}_{2})$ at a given time throughout all space; i.e., ${n}{(}{t}{<}{0}{)} = {n}_{1}$ and ${n}{(}{t}{>}{0}{)} = {n}_{2}{.}$ The temporal change of the refractive index produces both reflected (backward) and transmitted (forward) waves, which are analogous to the reflected and transmitted waves produced at the spatial interface between two different media in Figure 2(a). The discontinuity of $\text{D}$ and $\text{B}$ at ${v}_{b}{t} = {v}_{b}{t}_{0}$ would result in unbounded and singular $\text{E}$ and $\text{H}$ at the interface, which is not physically possible. Therefore, $\text{D}$ and $\text{B}$ must be continuous at a time interface; that is, ${(}{D}_{2}{-}{D}_{1}{)}{|}{}_{{v}_{b}{t} = {v}_{b}{t}_{0}} = {0}$ and ${(}{B}_{2}{-}{B}_{1}{)}{|}{}_{{v}_{b}{t} = {v}_{b}{t}_{0}} = {0}{.}$ The total charge Q and the total flux ${\psi}$ must remain constant at the moment of the jump from ${n}_{1}$ to ${n}_{2},$ implying that both the transversal and normal components of $\text{D}$ and $\text{B}$ do not change instantaneously, which is different than the static case [shown in Figure 2(a)], where only the normal components of the magnetic field $\text{B}$ and electric field displacement $\text{D}$ are conserved. Specifically, at a time interface, the magnetic field $\text{B},$ the electric field displacement $\text{D},$ and the wavenumber k are preserved. This yields a change in the temporal frequency of the incident wave so that the frequency of the reflected wave in region 2 and the forward transmitted wave in region 2 correspond to ${\omega}_{\hat{R}} = {\omega}_{i}{n}_{1}{/}{n}_{2},$ ${\omega}_{\hat{T}} = {\omega}_{i}{n}_{1}{/}{n}_{2},$ respectively. However, the spatial frequencies of the backward reflected and forward transmitted waves are equal to that of the incident wave, i.e., ${k}_{\hat{R}} = {k}_{\hat{T}} = {k}_{i},$ where momentum is preserved but energy changes. More details can be found in the supplementary materials available at http://doi.org/10.1109/MAP.2022.3201195. We consider ${D}_{1} = {a}_{0}\exp{[}{i}{(}{k}_{i}{z}{-}{\omega}_{i}{t}{)]}$ and ${D}_{2} = {a}_{0}\exp{(}{ik}_{i}{z}{)}\left({\hat{T}\exp{(}{-}{i}{\omega}_{t}{t}{)} + \hat{R}\exp{(}{i}{\omega}_{t}{t}{)}}\right),$ where $\hat{R}$ and $\hat{T}$ represent the temporal reflection and transmission coefficients, respectively. These two coefficients are determined by applying the boundary conditions ${D}_{1}{|}{}_{{v}_{b}{t} = {v}_{b}{t}_{0}} = {D}_{2}{|}{}_{{v}_{b}{t} = {v}_{b}{t}_{0}}$ and ${B}_{1}{|}{}_{{v}_{b}{t} = {v}_{b}{t}_{0}} = {B}_{2}{|}{}_{{v}_{b}{t} = {v}_{b}{t}_{0}},$ as $\hat{R} = {n}_{1}{(}{\eta}_{2}{-}{\eta}_{1}{)}{/}{(}{2}{n}_{2}{\eta}_{1}{)}$ and $\hat{T} = {n}_{1}{(}{\eta}_{1} + {\eta}_{2}{)/(}{2}{n}_{2}{\eta}_{1}{)}{.}$ More details can be found in the supplementary materials available at http://doi.org/10.1109/MAP.2022.3201195.
Figure 2(c) and (d) depicts the ST diagram of subluminal ${(}{v}_{m}{<}{v}_{b}{)}$ and superluminal ${(}{v}_{m}{>}{v}_{b}{)}$ ST interfaces, where ${n}{(}{z}{/}{v}_{b} + {t}{<}{0}{)} = {n}_{1}$ and ${n}{(}{z}{/}{v}_{b} + {t}{>}{0}{)} = {n}_{2},$ as the combination of the space and time interfaces in Figure 2(a) and (b), respectively. Here, ${v}_{m}$ represents the velocity of the interface, and the slope of the ST interface is ${v}_{b}{/}{v}_{m}{.}$ It may be seen that the ST interface resembles the spatial interface configuration in Figure 2(a) in the region ${n} = {n}_{1}$ and the temporal interface configuration in Figure 2(b) for ${n} = {n}_{2}{.}$
To derive boundary conditions for subluminal and superluminal ST interfaces, we apply the following Lorentz transformations [63], which transform the coordinates of an event from the laboratory (unprimed) frame (x, y, z, t) to the primed frame ${(}{x'},\,{y'},\,{z'},\,{t'}{),}$ comoving with the medium at a constant velocity ${v} = {a}_{z}{v}$ (where ${v}{<}{v}_{b}{)}$ relative to the laboratory (unprimed) frame, in special relativity, as \[{x'} = {x},{y'} = {y},{z'} = {\gamma}{(}{z}{-}{vt}{)} \quad {\text{and}} \quad {t'} = {\gamma}\left({{t}{-}\frac{v}{{v}_{b}^{2}}{z}}\right), \tag{1} \] where ${\gamma} = {1}{/}\sqrt{{1}{-}{(}{v}{/}{v}_{b}{)}{}^{2}}{.}$ Then, by applying the chain rule, that is, $\partial{/}\partial{z} = {(}\partial{/}\partial{z'}{)(}\partial{z'}{/}\partial{z}{)} + {(}\partial{/}\partial{t'}{)(}\partial{t'}{/}\partial{z}{),}$ and $\partial{/}\partial{t} = $${(}\partial{/}\partial{t'}{)(}\partial{t'}{/}\partial{t}{)} + {(}\partial{/}\partial{z'}{)(}\partial{z'}{/}\partial{t}{),}$ we derive Maxwell’s equations in the primed frame as (more details can be found in the supplementary materials available at http://doi.org/10.1109/MAP.2022.3201195) \begin{align*}\frac{\partial}{\partial{z'}}\left({{\gamma}\left[{{E}_{x}{(}{z}{)}{-}{vB}_{y}{(}{z}{)}}\right]}\right) & = {-}\frac{\partial}{\partial{t'}}\left({{\gamma}\left[{{B}_{y}{(}{z}{)}{-}\frac{v}{{v}_{b}^{2}}{E}_{x}{(}{z}{)}}\right]}\right) \tag{2a} \\ {-}\frac{\partial}{\partial{z'}}\left({{\gamma}\left[{{H}_{y}{(}{z}{)}{-}{vD}_{x}{(}{z}{)}}\right]}\right) & = \frac{\partial}{\partial{t'}}\left({{\gamma}\left[{{D}_{x}{(}{z}{)} + \frac{v}{{v}_{b}^{2}}{H}_{y}{(}{z}{)}}\right]}\right){.} \tag{2b} \end{align*}
Consider a medium moving at a constant velocity ${v}_{m} = {a}_{z}{v}_{m}$ (where ${v}_{m}{<}{v}_{b}{)}$ relative to the laboratory (unprimed) frame, as shown in Figure 2(c). Hence, the slopes of the primed frame ${v}_{b}/\text{v}$ and the slope of the interface ${v}_{b}/{v}_{m}$ are equal, which results in ${v} = {v}_{m}$ such that the primed frame is comoving with the interface, i.e., ${v'}_{m} = {0}$ where ${v'}_{m}$ is the velocity of the interface in the primed frame. Maxwell’s equations in the primed frame in (2) provide the following boundary conditions for the subluminal (space-like) ST interface: \begin{align*}\left({{E}_{1}{-}{v}{\mu}_{1}{H}_{1}}\right){|}{}_{{z'} = {z'}_{0}} & = \left({{E}_{2}{-}{v}{\mu}_{2}{H}_{2}}\right){|}{}_{{z'} = {z'}_{0}}\,{\text{and}} \\ \left({{H}_{1}{-}{v}{\epsilon}_{1}{E}_{1}}\right){|}{}_{{z'} = {z'}_{0}} & = \left({{H}_{2}{-}{v}{\epsilon}_{2}{E}_{2}}\right){|}{}_{{z'} = {z'}_{0}}{.} \tag{3a} \end{align*}
The reflection and transmission coefficients at a subluminal ST interface read (more details can be found in the supplementary materials available at http://doi.org/10.1109/MAP.2022.3201195) \begin{align*}{R} = \frac{{\eta}_{2}{-}{\eta}_{1}}{{\eta}_{1} + {\eta}_{2}}\frac{{v}_{1}{-}{v}_{m}}{{v}_{1} + {v}_{m}} \quad {\text{and}} \quad {T} = \frac{2{\eta}_{2}}{{\eta}_{1} + {\eta}_{2}}\frac{{1}{-}\frac{{v}_{m}}{{v}_{1}}}{{1}{-}\frac{{v}_{m}}{{v}_{2}}}, \tag{3b} \end{align*} and the temporal and spatial frequencies of the reflected and transmitted waves read (more details can be found in the supplementary materials available at http://doi.org/10.1109/MAP.2022.3201195) \begin{align*}\begin{gathered}{{\omega}_{R} = {\omega}_{i}\frac{{v}_{1}{-}{v}_{m}}{{v}_{1} + {v}_{m}},{\omega}_{T} = {\omega}_{i}\frac{{1}{-}\frac{{v}_{m}}{{v}_{1}}}{{1}{-}\frac{{v}_{m}}{{v}_{2}}},}\\{{k}_{R} = {k}_{i}\frac{{v}_{1}{-}{v}_{m}}{{v}_{1} + {v}_{m}},{k}_{T} = {k}_{i}\frac{{1}{-}\frac{{v}_{m}}{{v}_{1}}}{{1}{-}\frac{{v}_{m}}{{v}_{2}}},}\end{gathered} \tag{3c} \end{align*} where ${v}_{1} = {v}_{b}{/}{n}_{1}$ and ${v}_{2} = {v}_{b}{/}{n}_{2}{.}$ The pure-space interface is the ${v}_{m} = {0}$ limit of a subluminal ST interface.
Consider a medium moving at a constant velocity ${v}_{m}{>}{v}_{b}$ relative to the laboratory (unprimed) frame, as shown in Figure 2(d). The proper use of the Lorentz transformation in the superluminal regime ${(}{v}_{m}{>}{v}_{b}{),}$ e.g., to avoid the imaginary Lorentz factor ${\gamma},$ requires setting the primed frame as one where the medium is purely temporal instead of purely spatial. Therefore, as shown in Figure 2(d), the two slopes $\text{v}/{v}_{b}$ and ${v}_{b}/{v}_{m}$ are equal, which results in ${v} = {v}_{b}^{2}{/}{v}_{m},$ such that the frame is costanding with the interface in time. Maxwell’s equations in the primed frame in (2) provide the following boundary conditions for the superluminal (time-like) ST interface: \begin{align*}{\left.{\left({{D}_{1}{-}\frac{v}{{v}_{b}^{2}}{H}_{1}}\right)}\right|}_{{v}_{b}{t'} = {v}_{b}{t'}_{0}} & = {\left.{\left({{D}_{2}{-}\frac{v}{{v}_{b}^{2}}{H}_{2}}\right)}\right|}_{{v}_{b}{t'} = {v}_{b}{t'}_{0}}\,{\text{and}} \\ {\left.{\left({{B}_{1}{-}\frac{v}{{v}_{b}^{2}}{E}_{1}}\right)}\right|}_{{v}_{b}{t'} = {v}_{b}{t'}_{0}} & = {\left.{\left({{B}_{2}{-}\frac{v}{{v}_{b}^{2}}{E}_{2}}\right)}\right|}_{{v}_{b}{t'} = {v}_{b}{t'}_{0}}{,} \tag{4a} \end{align*} and the reflection and transmission coefficients at a superluminal ST interface read (more details can be found in the supplementary materials available at http://doi.org/10.1109/MAP.2022.3201195) \[\hat{R} = \frac{{n}_{1}}{{n}_{2}}\frac{{(}{\eta}_{2}{-}{\eta}_{1}{)}}{2{\eta}_{1}}\left({\frac{{1}{-}\frac{{v}_{1}}{{v}_{m}}}{{1} + \frac{{v}_{2}}{{v}_{m}}}}\right),\,\,{\text{and}}\,\,\hat{T} = \frac{{n}_{1}}{{n}_{2}}\left({\frac{{\eta}_{1} + {\eta}_{2}}{2{\eta}_{1}}}\right)\left({\frac{{1}{-}\frac{{v}_{1}}{{v}_{m}}}{{1}{-}\frac{{v}_{2}}{{v}_{m}}}}\right), \tag{4b} \] where the temporal and spatial frequencies of the reflected and transmitted waves read (more details can be found in the supplementary materials available at http://doi.org/10.1109/MAP.2022.3201195) \begin{align*}{k}_{\hat{R}} & = {k}_{i}\frac{{1}{-}\frac{{v}_{1}}{{v}_{m}}}{{1} + \frac{{v}_{2}}{{v}_{m}}},\quad{k}_{\hat{T}} = {k}_{i}\frac{{1}{-}\frac{{v}_{1}}{{v}_{m}}}{{1}{-}\frac{{v}_{2}}{{v}_{m}}}{.} \\ {\omega}_{\hat{R}} & = {\omega}_{i}\frac{{1}{-}\frac{{v}_{m}}{{v}_{1}}}{{1} + \frac{{v}_{m}}{{v}_{2}}},\quad{\omega}_{\hat{T}} = {\omega}_{i}\frac{{1}{-}\frac{{v}_{m}}{{v}_{1}}}{{1}{-}\frac{{v}_{m}}{{v}_{2}}}{.} \tag{4c} \end{align*}
The pure-time interface is the ${v}_{m} = \infty$ limit of a superluminal ST interface.
The difference between the excitation and response for validation of the symmetry and reciprocity of electromagnetic systems is clarified in Figure 3(a) and (b) [7], [17]. Here, we consider the general case of new frequency generation that occurs in ST metasurfaces. However, the symmetry and reciprocity properties of media are not related to frequency generation such that a symmetric/reciprocal metasurface can either produce new frequencies or not. Figure 3(a) shows the forward and backward problems for the symmetry test of a particular symmetric electromagnetic system, where the backward problem is represented by the spatial inversion of the forward problem; that is, the applied excitation wave (input) of the backward problem must be the spatial inversion of the excitation wave (input) of the forward problem. As a result, for a symmetric system, the output of the backward problem would be exactly the spatial inversion of the output of the forward problem. Otherwise, the system is asymmetric. Figure 3(b) shows the forward and backward problems for the reciprocity test of a particular reciprocal electromagnetic system, where the backward problem is the spatial inversion of the time-reversed forward problem; that is, the applied excitation wave (input) of the backward problem must be the spatial inversion of the output of the forward problem. As a result, for a reciprocal system, the output of the backward problem would be exactly the spatial inversion of the input of the forward problem. Otherwise, the system is nonreciprocal.
Figure 3. A schematic of the experimental setup configurations for validation of the symmetric and reciprocal response of electromagnetic systems. (a) The electromagnetic symmetry of the system is validated, in which the backward problem is the spatial inversion of the forward problem. (b) The electromagnetic reciprocity of the system is validated, in which the backward problem is the spatial inversion of the time-reversed forward problem.
Transistor-loaded metasurfaces can represent an ST interface [2], [3], [11]. Figure 4(a) depicts the operation principle of a nonreciprocal nongyrotropic metasurface. For ${t}{<}{0},$ the metasurface operates as a reflector where the incident wave, traveling along the $ + {z}$ direction, is reflected by the metasurface and travels back along the ${-}{z}$ direction. For ${t}{>}{0},$ the metasurface operates as a nonreciprocal sheet, where a $ + {z}$-directed traveling wave passes through the metasurface with gain and without polarization alteration, whereas a wave traveling along the opposite direction is being reflected by the metasurface. The transmission scattering parameters of the metasurface are not equal, i.e., ${S}_{21}{>}{S}_{12},$ where ${S}_{21} = \mathbf{\psi}{}_{out}^{F}{/}\mathbf{\psi}{}_{in}^{F}{>}{1}$ and ${S}_{12} = \mathbf{\psi}{}_{out}^{B}{/}\mathbf{\psi}{}_{in}^{B}{<}{1}{.}$ It can be shown that such a nonreciprocal nongyrotropic metasurface is equivalent to a moving metasurface [11], as depicted in Figure 4(b).
Figure 4. A nonreciprocal nongyrotropic metasurface. (a) For ${t}{<}{0}$ it operates as a reflective sheet, whereas for ${t}{>}{0}$ it operates as a nonreciprocal transmissive sheet. (b) The ST representation of the nonreciprocal nongyrotropic metasurface.
Figure 5(a) shows the full-wave simulation results, where the transmission of waves from left to right is allowed and accompanied by power amplification, but the transmission of waves from right to left is prohibited. Figure 5(b) shows a photo of the fabricated metasurface [11]. The metasurface is formed by an array of unit cells. These unit cells are constituted of two microstrip patch elements interconnected through a unilateral transistor, introducing transmission gain in one direction and transmission loss in the other direction. Figure 5(c) shows the measured transmission levels for both directions and for ${t}{<}{0}$and ${t}{>}{0}{.}$ The general case of a bianisotropic medium reads [11] ${\mathbf{D}} = \mathop{\epsilon}\limits^=\cdot{\mathbf{E}} + \mathop{\xi}\limits^=\cdot{\mathbf{H}}$ and ${\mathbf{B}} = \mathop{\zeta}\limits^=\cdot{\mathbf{E}} + \mathop{\mu}\limits^=\cdot{\mathbf{H}}{.}$
Figure 5. Nonreciprocal metasurface. (a) Full-wave (FDTD) electric field distribution for excitations from the left and right (bottom) [11]. (b) An image of the fabricated metasurface [11]. (c) Experimental scattering parameters versus angle at ${f} = {5}{.}{9}{GHz}$ for transmission in a straight line under an oblique angle [11].
The continuity equations of a metasurface may be expressed as \begin{align*}\hat{z}\times\Delta{\boldsymbol{H}} & = {j}{\omega}{\epsilon}_{0}{\mathop{\chi}\limits^=}_{\text{ee}}\cdot{\boldsymbol{E}}_{\text{av}} + {jk}_{0}{\mathop{\chi}\limits^=}_{\text{em}}\cdot{\boldsymbol{H}}_{\text{av}}, \tag{5a} \\ \Delta{\boldsymbol{E}}\times\hat{z} & = {j}{{\omega}{\mu}}_{0}{\mathop{\chi}\limits^=}_{\text{mm}}\cdot{\boldsymbol{H}}_{\text{av}} + {jk}_{0}{\mathop{\chi}\limits^=}_{\text{me}}\cdot{\boldsymbol{E}}_{\text{av}}, \tag{5b} \end{align*} where $\Delta$ and the subscript “av” represent, respectively, the difference of the fields and the average of the fields between the two sides of the metasurface. Equation (5) provides a relation between the electromagnetic fields on the two sides of a metasurface and its susceptibilities in the absence of normal susceptibility components. The constitutive parameters of the metasurface may be represented according to the susceptibilities in (5) as $\mathop{\epsilon}\limits^= = {\epsilon}_{0}{(}\mathop{I}\limits^= + {\mathop{\chi}\limits^=}_{\text{ee}}{),}$ $\mathop{\mu}\limits^= = {\mu}_{0}{(}\mathop{I}\limits^= + {\mathop{\chi}\limits^=}_{\text{mm}}{),}$ $\mathop{\xi}\limits^= = {\mathop{\chi}\limits^=}_{\text{em}}{/}{c}_{0},$ $\mathop{\zeta}\limits^= = {\mathop{\chi}\limits^=}_{\text{me}}{/}{c}_{0}{.}$
We then seek the susceptibilities that provide the nonreciprocal nongyrotropic response of the metasurface by substituting the electromagnetic fields of the corresponding transformation into (5). Such a transformation includes passing a $ + {z}{-} $propagating plane wave through the metasurface and complete absorption of a – z-propagating plane incident wave, yielding ${\mathop{\chi}\limits^=}_{\text{ee}} = {-}{j}{/}{k}_{0}{[}{1}{0}{;}{0}{1}{],}$ ${\mathop{\chi}\limits^=}_{\text{mm}} = {-}{j}{/}{k}_{0}{[}{1}{0}{;}{0}{1}{],}$ ${\mathop{\chi}\limits^=}_{\text{em}} = {j}{/}{k}_{0}{[}{0}{1}{;}{-}{1}{0}{],}$ ${\mathop{\chi}\limits^=}_{\text{me}} = {j}{/}{k}_{0}{[}{0}{-}{1}{;}{1}{0}{]}{.}$ This shows that elements ${\mathop{\chi}\limits^=}_{\text{em}}$ and ${\mathop{\chi}\limits^=}_{\text{me}}$ are the ones that contribute to the nonreciprocity of the metasurface. This form of the susceptibility tensors is identical to that of a moving uniaxial medium [11].
Space, time, and ST interfaces are fundamental parts of electromagnetic structures. In practice, ST sinusoidally periodic structures are substantially more common than ST interfaces as they introduce peculiar and interesting features and physical phenomena, and they are significantly easier to experimentally realize. Such spatiotemporally periodic structures (usually in sinusoidal form) can be theoretically represented as an infinite cascade of ST interfaces [29]. This section provides an analysis on wave propagation in ST metasurfaces. Electromagnetic scattering from an STP slab is depicted in Figure 6(a) [15]. Such a scattering is composed of the reflection and transmission of ST harmonics (STHs). The ST slab is characterized with the wavenumber ${k'}_{0}$ and possesses the ST-varying electric permittivity ${\epsilon}{(}{z},{t}{)}$ and ST-varying magnetic permeability ${\mu}{(}{z},{t}{)}{.}$ The ST-varying slab is sandwiched between two semi-infinite unmodulated media; that is, region 1 is characterized with the wavenumber ${k}_{0} = {\omega}_{0}\sqrt{{\epsilon}_{\text{r},1}{\mu}_{\text{r},1}}{/}{c} = \sqrt{{k}_{x}^{2} + {k}_{z}^{2}}$ and region 3 with the wavenumber ${k''}_{0} = {\omega}_{0}\sqrt{{\epsilon}_{\text{r},3}{\mu}_{\text{r},3}}{/}{c} = \sqrt{{k''}_{x}{}^{2} + {k''}_{z}{}^{2}}{.}$ We assume the STM structure is infinite in the y direction; that is, ${k}_{y} = {k'}_{y} = {k''}_{y} = {0}{.}$ Here, ${\omega}_{0}$ is the temporal frequency of the incident wave, ${\epsilon}_{\text{r},1}$ and ${\epsilon}_{\text{r},3}$ are the electrical permittivities of regions 1 and 3, respectively, and ${\mu}_{\text{r},1}$ and ${\mu}_{\text{r},3}$ are the magnetic permeabilities of regions 1 and 3, respectively.
Figure 6. Wave scattering from an ST slab. (a) ST decomposition resulting from oblique incidence to an STM metasurface [15]. (b) FDTD scheme for numerical computation of the wave propagation and scattering in STP metasurfaces and slabs [13]. (c) ${E}_{y}$ at ${t} = {20}{ns}$ for forward wave incidence (from the bottom) and transmission [15]. (d) ${E}_{y}$ at ${t} = {20}{ns}$ for backward wave incidence (from the top) and transmission [15].
A general analysis assumes a metasurface with temporally periodic electric permittivity and magnetic permeability and a general aperiodic/periodic spatial variation [15]. Given the temporal periodicity of the structure, we may decompose these constitutive parameters into time-Floquet waves, as ${\epsilon}{(}{z},{t}{)} = {\Sigma}_{{s} = {-}\infty}^{\infty}{\epsilon}_{{k},{aper}}{(}{z}{)}\exp{(}{is}\Omega{t}{)}$ and ${\mu}{(}{z},{t}{)} = $ ${\Sigma}_{{s} = {-}\infty}^{\infty}{\mu}_{{k},{aper}}{(}{z}{)}\exp{(}{is}\Omega{t}{),}$ where $\Omega$ is the temporal frequency of the modulation, and ${\epsilon}_{\text{k},\text{aper}}(\text{z})$ and ${\mu}_{\text{k},\text{aper}}(\text{z})$ are coefficients of the permittivity and permeability that are computed according to the spatial variation of the structure. The incident electric field is considered as a y-polarized plane wave, expressed as ${E}_{I}{(}{x},{z},{t}{)} = \hat{y}{E}_{0}\exp{[}{i}{(}{k}_{x}{x} + {k}_{z}{z}{-}{\omega}_{0}{t}{)]}$ that impinges on the structure in region 1 under the angle of incidence ${\theta}_{I},$ where ${E}_{0}$ is the electric incident wave magnitude. We then express the electric field inside the metasurface as time-Floquet waves; that is, \[{\bf{E}}_{\text{M}}{(}{x},{z},{t}{)} = \hat{y}\mathop{\sum}\limits_{{n},{p}}{{\bf{E}}_{np}}\left({{A}_{0p}{e}^{i{\beta}_{np}^{ + }z} + {B}_{0p}{e}^{{-}{i}{\beta}_{np}^-{z}}}\right){e}^{{i}\left({{k}_{x}{x}{-}{\omega}_{n}{t}}\right)}{.} \tag{6} \]
Here, ${\omega}_{n} = {\omega}_{0} + {n}\Omega$ is the temporal frequency of the nth temporal harmonic, and ${A}_{{0}{p}}$ and ${B}_{{0}{p}}$ are unknown field coefficients to be found by solving Maxwell’s equations and applying boundary conditions [29]. Next, we apply the ST boundary conditions for electric and magnetic fields at ${z} = {0}$ and ${z} = {L}{;}$ that is, ${E}_{I}{(}{x},{0},{t}{)} + {E}_{R}{(}{x},{0},{t}{)} = {E}_{M}{(}{x},{0},{t}{)}$ and ${E}_{M}{(}{x},{L},{t}{)} = {E}_{T}{(}{x},{L},{t}{)}{.}$ The scattered electric fields in regions 1 and 3 read \begin{align*}{\bf{E}}_{\text{R}} & = \hat{y}\mathop{\sum}\limits_{{n},{p}}{\left[{{E}_{np}\left({{A}_{{0}{p}} + {B}_{0p}}\right){-}{E}_{0}{\delta}_{n0}}\right]}{e}^{{i}\left[{{k}_{x}{x}{-}{k}_{zn}{z}{-}{\omega}_{n}{t}}\right]}, \tag{7a} \\ {\bf{E}}_{\text{T}} & = \hat{y}\mathop{\sum}\limits_{\text{n},\text{p}}{{E}_{np}}\left({{A}_{0p}{e}^{i{\beta}_{np}L} + {B}_{{0}{p}}{e}^{{-}{i}{\beta}_{np}{L}}}\right){e}^{{i}\left[{{k}^{''}_{x}{x}{-}{k}^{''}_{zn}{z}{-}{\omega}_{n}{t}}\right]}, \tag{7b} \end{align*} where ${\delta}_{\text{n}0}$ represents the Kronecker delta and is equal to one if ${n} = {0}$ and zero otherwise.
The most common and practical form of periodic ST modulation is the sinusoidal form as ${\epsilon}{(}{z},{t}{)} = {\epsilon}_{0}{\epsilon}_{r}\left[{{1} + {\delta}_{\epsilon}\sin{(}{qz}{-}\Omega{t}{)}}\right]$ and ${\mu}{(}{z},{t}{)} = {\mu}_{0}{\mu}_{r}\left[{{1} + {\delta}_{\mu}\sin{(}{qz}{-}\Omega{t}{)}}\right],$ where ${\delta}_{\epsilon}$ and ${\delta}_{\mu}$ represent, respectively, the permittivity and permeability modulation strengths. The ST-varying intrinsic impedance of the structure may be represented as [29] ${\eta}{(}{z},{t}{)} = \sqrt{{\mu}{(}{z},{t}{)}{/}{\epsilon}{(}{z},{t}{)}}{.}$ Considering an equilibrated ST modulation, where ${\delta}_{\mu} = {\delta}_{\epsilon},$ the structure acquires an ST-invariant intrinsic impedance ${\eta}{(}{z},{t}{)} = {\eta}_{0}{\eta}_{r}$ that is immune to space and time local reflections. The ST decomposition of the reflected and transmitted STHs is illustrated in Figure 6(a). The reflection and transmission angles of the STHs can be derived by satisfying the Helmholtz relations; that is, ${k}^{2}{\sin}^{2}{(}{\theta}_{I}{)} + {k}_{n}^{2}{\cos}^{2}{(}{\theta}_{\text{R}\text{n}}{)} = {k}_{n}^{2}$ and ${k''}^{2}{\sin}^{2}{(}{\theta}_{I}{)} + {k''}_{n}^{2}{\cos}^{2}{(}{\theta}_{\text{T}\text{n}}{)} = {k''}_{n}^{2}{.}$ Here, ${\theta}_{\text{R}\text{n}}$ and ${\theta}_{\text{T}\text{n}}$ denote the angles of reflection and transmission for the nth STH, which read \[{\theta}_{{\text{r}}{n}} = {\theta}_{{\text{t}},{n}} = {\sin}^{{-}{1}}\left({\frac{\sin({\theta}_{\text{I}})}{{1} + \frac{{n}\Omega}{{\omega}_{0}}}}\right), \tag{8} \] which demonstrates spectral decomposition of the scattered STHs. Considering ${k'} = {k''} = {k}_{0},$ the reflection and transmission angles of the nth harmonic are equal for equal tangential wavenumbers ${k}_{x} = {k}_{0}\sin{(}{\theta}_{I}{)}$ in all of the regions. The STHs ranging from ${\omega}_{0}{(}\sin{(}{\theta}_{I}{)}{-}{1}{)}{/}\Omega$ to $ + \infty$ are reflected and transmitted at angles ranging from 0 to ${\pi}{/}{2}$ through ${\theta}_{I}$ for ${n} = {0},$ while the rest of the STHs are imaginary ${k'}_{znp}^{\pm}$ and propagate as surface waves along the boundary of the metasurface. The scattering angle of the pth mode of the nth STH inside the modulated region is given by ${\theta}_{np}^{\pm} = \tan{}^{{-}{1}}\left({{k'}_{x}/{k'}_{znp}}\right) = $ $\tan{}^{{-}{1}}\left({{k'}_{0}\sin{(}{\theta}_{I}{)}{/}{(}{\beta}_{{0}{p}}^{\pm}\pm{nq}{)}}\right){.}$
To best gain insight into the wave propagation in ST metasurfaces and support the analytical solution, we study ST electromagnetic field propagation and scattering through FDTD numerical computation of Maxwell’s equations. The FDTD scheme for oblique incidence to a general ST permittivity- and permeability-modulated metasurface is depicted in Figure 6(b), where the structure is discretized to ${K} + {1}$ and ${M} + {1}$ spatial and temporal samples, respectively, considering the spatial steps $\Delta{z}$ and temporal steps $\Delta{t}{.}$ Considering ${\epsilon'} = \partial{\epsilon}{(}{z},{t}{)}{/}\partial{t}$ and ${\mu'} = \partial{\mu}{(}{z},{t}{)}{/}\partial{t},$ we obtain \begin{align*}\left.{{H}_{x}}\right|{}_{{j} + {1}{/}{2}}^{{i} + {1}{/}{2}} & = \left({{1}{-}\Delta{t}\frac{\left.{\mu'}\right|{}_{{j} + {1}{/}{2}}^{{i}{-}{1}{/}{2}}}{\left.{\mu}\right|{}_{{j} + {1}{/}{2}}^{i}}}\right)\left.{{H}_{x}}\right|{}_{{j} + {1}{/}{2}}^{{i}{-}{1}{/}{2}} + \frac{\Delta{t}}{\left.{\mu}\right|{}_{{j} + {1}{/}{2}}^{i}\Delta{z}}\left({\left.{{E}_{y}}\right|{}_{{j} + {1}}^{i}{-}\left.{{E}_{y}}\right|{}_{j}^{i}}\right), \tag{9a} \\ {H}_{z}{|}{}_{{j} + {1}{/}{2}}^{{i} + {1}{/}{2}} & = \left({{1}{-}\Delta{t}\frac{{\mu'}{|}{}_{{j} + {1}{/}{2}}^{{i}{-}{1}{/}{2}}}{{\mu}{|}{}_{{j} + {1}{/}{2}}^{i}}}\right){H}_{z}{|}{}_{{j} + {1}{/}{2}}^{{i}{-}{1}{/}{2}}{-}\frac{\Delta{t}}{{\mu}{|}{}_{{j} + {1}{/}{2}}^{i}\Delta{z}}\left({{E}_{y}{|}{}_{{j} + {1}}^{i}{-}{E}_{y}{|}{}_{j}^{i}}\right), \tag{9b} \\ {E}_{y}{|}{}_{j}^{{i} + {1}} & = \left({{1}{-}\frac{\Delta{t}{\epsilon'}{|}_{j}^{i}}{{\epsilon}{|}_{j}^{{i} + {1}{/}{2}}}}\right){E}_{y}{|}{}_{j}^{i} + \frac{\Delta{t}{/}\Delta{z}}{{\epsilon}{|}_{j}^{{i} + {1}{/}{2}}}{.} \left[{\left({{H}_{x}{|}{}_{{j} + {1}{/}{2}}^{{i} + {1}{/}{2}}{-}{H}_{x}{|}{}_{{j}{-}{1}{/}{2}}^{{i} + {1}{/}{2}}}\right){-}\left({{H}_{z}{|}{}_{{j} + {1}{/}{2}}^{{i} + {1}{/}{2}}{-}{H}_{z}{|}_{{j}{-}{1}{/}{2}}^{{i} + {1}{/}{2}}}\right)}\right]{.} \tag{9c} \end{align*}
Figure 6(c) and (d) plots FDTD computation results for nonreciprocal electric field scattering from a sinusoidally STP metasurface for forward and backward wave incidences, respectively. Such an equilibrated STP metasurface exhibits strong frequency generation, ST decomposition, and nonreciprocal wave transmission. In [29], it is shown that an equilibrated STM metasurface with ${\delta}_{\mu} = {\delta}_{\epsilon}{>}{0}$ can be created using the same amount of pumping energy that is required for the creation of a permittivity-modulated metasurface, where ${\delta}_{\epsilon}{>}{0}$ and ${\delta}_{\mu} = {0}{.}$ Additionally, such an equilibrated STP metasurface exhibits a matched intrinsic impedance and, therefore, is immune to local space and time reflections.
The analysis provided in the previous section is applicable for ST metasurfaces operating outside the diffraction regime and is not directly applicable to the diffraction regime of ST metasurfaces. We shall stress that the FDTD numerical simulation scheme provided in the section ”Numerical Computation Scheme” is applicable to all operation regimes of ST media, including the diffraction regime that will be discussed in this section. This section analyzes ST metasurfaces that provide spatial diffractions, where each spatial diffraction order denoted by m comprises temporal diffractions denoted by n [17]. The wave vector diagram in Figure 7(a) represents a powerful tool for studying ST diffractions from STP diffraction gratings. This diagram is constituted based on the phase matching of ST diffractions inside and outside the grating. The parameters of the STP grating are the temporal frequency Ω and the spatial frequency K, with $\Lambda = {2}{\pi}{/}{K}$ being the spatial periodicity of the grating. Here, regions 1 to 3 present the phase velocities ${v}_{r} = {c}{/}{n}_{1},{v'}_{r} = {c}{/}{n}_{\text{av}}$, and ${v''}_{r} = {c}{/}{n}_{3},$ and the wave vectors ${k}_{mn} = {k}_{\text{x},\text{mn}}\hat{x} + {k}_{\text{z},\text{mn}}\hat{z},$ ${k'}_{pmn} = {k'}_{\text{x},\text{pmn}}\hat{x} + {k'}_{\text{z},\text{pmn}}\hat{z},$ and ${k''}_{mn} = {k''}_{\text{x},\text{mn}}\hat{x} + {k''}_{\text{z},\text{mn}}\hat{z},$ respectively. Here, p is the number of the mode inside the grating (these modes only exist inside the grating), m denotes the number of the spatial harmonic, and n represents the number of the temporal harmonic.
Figure 7. A transmissive STP diffractive metasurface. (a) The wave vector diagram for analysis of the ST diffraction based on phase matching of ST diffractions inside and outside the modulated metasurface [17]. (b) FDTD numerical computation of Raman–Nath diffraction from a thin sinusoidal STP metasurface [17].
The diffraction condition for ST diffraction gratings can be found by considering ${|}\sin\left({{\theta}_{mn}}\right){|}\leq{1}{.}$ Hence, the mth propagating diffraction order should satisfy the condition [4] \[{\left|{\frac{\sin{(}{\theta}_{i}{)} + {m}\Delta{k}}{{1} + {n}\Delta{\omega}}}\right|}_{{n} = {m}}\leq{1} \tag{10} \] to be diffracted as a propagating ST diffraction order. Here, $\Delta{k} = {K}{/}{k}_{0}$ and $\Delta{\omega} = \Omega{/}{\omega}_{0}{.}$ The condition ${n} = {m}$ in (10) is due to the fact that in the mth spatial diffraction order the ${n} = {m}{th}$ temporal diffraction order represents the dominant ST diffraction order and possesses the largest amplitude [17]. Figure 7(b) presents the FDTD numerical computation of ST diffraction from an STP diffraction metasurface for a y-polarized electric field incident wave. Here, ${k''}_{\text{x},\text{mn}} = {k''}_{n}\sin{(}{\theta''}_{mn}{)}$ and ${k''}_{\text{z},\text{mn}} = \sqrt{{(}{k''}_{mn}{)}^{2}{-}{(}{k''}_{\text{x},\text{mn}}{)}^{2}} = {k''}_{n}\cos{(}{\theta''}_{mn}{)}$ are the x and z components of the wave vectors in region 3, respectively, where ${k''}_{n} = {k''}_{0} + {n}\Omega{/}{v''}_{r}$ and ${k''}_{0} = {\omega}_{0}{/}{v''}_{r}{.}$ By applying the momentum and energy conservation laws, we obtain ${k}_{\text{x},\text{diff}} = {k''}_{\text{x},\text{mn}} = {k}_{x} + {mK}$ and ${\omega}_{diff} = {\omega}_{0} + {n}\Omega,$ where ${k}_{\text{x},\text{diff}}$ and ${\omega}_{diff}$ denote the x component wave vector and temporal frequency of the diffracted field, respectively. Then, the diffraction angle of the forward and backward ST diffracted orders is obtained as [17] \[\sin\left({{\theta''}_{mn}}\right) = \frac{\sin{(}{\theta}_{I}{)} + {mK}{/}{k}_{0}}{{1} + {n}\Omega{/}{\omega}_{0}}{.} \tag{11} \]
We next calculate the electromagnetic fields inside the STP diffractive metasurface as the superposition of ST modes, denoted by p, such that the electromagnetic field of each mode is represented by ST Bloch–Floquet plane waves, denoted by m and n. The operation regime of diffractive metasurfaces may be categorized in two classes, Bragg regime diffraction of thick STP diffractive metasurfaces and Raman–Nath regime diffraction of thin STP diffractive metasurfaces [17]. Such STP diffractive metasurfaces that are modulated unidirectionally introduce nonreciprocal and angle-asymmetric ST diffractions, as shown in Figure 8(a)–(d) for a reflective STP metasurface.
Figure 8. A reflective STP diffractive metasurface introducing angle-asymmetric and nonreciprocal ST diffractions. (a) and (c) Wave incidence from the left side (forward problem) [17]. (b) and (d) Wave incidence from the right side (backward problem) for angle-asymmetric and nonreciprocal diffraction demonstration [17]. PEC: perfect electric conductor.
Beam splitters are essential parts of optical and microwave systems. Conventional beam splitters are passive bulky structures that present a reciprocal response and introduce considerable transmission loss. In [13], it was shown that periodic ST modulation can create a unidirectional active beam splitter and amplifier based on coherent electromagnetic transitions emerging from oblique illumination of STP metasurfaces. The proposed one-way beam splitter and amplifier and its functionality are illustrated in Figure 9(a). The operation of this apparatus is based on a unidirectional energy and momentum transition between the incident wave with temporal frequency ${\omega}_{0}$ and the ST modulation with temporal frequency $2{\omega}_{0}$. Then, by setting the angle of incidence of ${\theta}_{I} = {4}\mathop {5}\nolimits^{\circ},$ the fundamental ${(}{n} = {0}{)}$ and first lower ${(}{n} = {-}{1}{)}$ STHs are transmitted under angles of transmissions ${\theta}_{\text{T},0} = {4}\mathop {5}\nolimits^{\circ}$ and ${\theta}_{{T},{-}{1}} = {-}{4}\mathop {5}\nolimits^{\circ},$ respectively. Additionally, the two transmitted STHs with a 45°-angle difference acquire the same temporal frequency as the incident wave ${\omega}_{0},$ which leads to a perfect unidirectional beam splitter and amplifier. The analytical isofrequency dispersion diagram in Figure 9(b) plots the one-way ST coherency between the ${n} = {0}$ and ${n} = {-}{1}\,{\text{STHs}}{.}$
Figure 9. One-way beam splitting by an STP metasurface, where the two forward transmitted beams, $\text{E}{}_{\text{T},0}^{F}$ and $\text{E}{}_{{T},{-}{1}}^{F},$ are amplified. (a) The schematic [13]. (b) The isofrequency dispersion diagram at ${\omega} = {\omega}_{0}$ [13].
The beam splitter is created by an STP slab with spatial modulation ${q} = {2}{k}_{0}{/}\Gamma$ and temporal modulation frequency $\Omega = {2}{\omega}_{0},$ where $\Gamma = {v}_{m}{/}{v}_{b} = {1}{.}{2}{.}$ The temporal modulation frequency $\Omega = {2}{\omega}_{0}$ is chosen to achieve a constructive ST coherency and beam splitting as follows. The transmission angle of the STHs can be determined by satisfying the Helmholtz relation as ${\theta}_{\text{T},\text{n}} = \sin{}^{{-}{1}}\left({{k}_{x}/{k}_{n}}\right) = \sin{}^{{-}{1}}\left({\sin{(}{\theta}_{I}{)}{/}{(}{1} + {2}{n}{)}}\right),$ which reveals that the ${n} = {0}$ and ${n} = {-}{1}\,{\text{STHs}}$ possessing temporal frequency ${\omega}_{0}$ acquire a 90°-angle difference; that is, ${\theta}_{\text{T},0} = {\theta}_{I} = $ 45° and ${\theta}_{{T},{-}{1}} = {-}{\theta}_{I} = {-}{4}\mathop {5}\nolimits^{\circ}{.}$
Consider the metasurface in Figure 10(a) characterized with a sinusoidal STP electric permittivity and thickness d. In the down-link reception state, the space wave with temporal frequency ${\omega}_{0}$ makes a transition to an ST surface wave with temporal frequency ${\omega}_{IF} = {\omega}_{0}{-}\Omega{.}$ In the up-link transmission state, the ST surface wave at ${\omega}_{IF}$ makes a transition to a space wave at ${\omega}_{0} = {\omega}_{IF} + \Omega{.}$ Because of the ST periodicity of the metasurface, the complex spatial frequency of the STHs is given by ${K}_{\text{z},\text{n}} = {k}_{z} + {nq} + {i}{\alpha}_{\text{z},\text{n}},$ and the temporal frequency of the STHs is ${\omega}_{n} = {\omega}_{0} + {n}\Omega{.}$ Additionally, the incident angle is set to ${\theta}_{I} = {\sin}^{{-}{1}}\left({{1}{-}\Omega{/}{\omega}_{0}}\right){.}$ Figure 10(b) shows that a strong electromagnetic transition from the ${n} = {0}$ STH to the ${n} = {-}{1}\,{\text{STH}}$ can be achieved by designing the dispersion band of the structure in such a way that the scattered ${n} = {-}{1}\,{\text{STH}}$ propagates in parallel to the two ST surface waves along the boundaries at ${z} = {0}$ and ${z} = {d}$ ${(}{\theta}_{{n} = {-}{1}} = {9}\mathop {0}\nolimits^{\circ}{).}$ This engineering of the ST modulation leads to ${\beta}_{{z},{-}{1}} = {0}{.}$ This results in a purely imaginary z-component wave vector, that is, ${K}_{{z},{-}{1}} = {i}{\alpha}_{{z},{-}{1}},$ while the incident space wave possess a real wave vector.
Figure 10. Antenna–mixer–amplifier metasurface. (a) Schematic representation showing the down-link and up-link wave transformations [55]. (b) Isofrequency diagram depicting, for ${\epsilon}_{m} ={\rightarrow}{0}$ and ${\Gamma} = {0.2},$ the down-link electromagnetic transition from an incoming space wave to an ST surface wave and the up-link electromagnetic transition from a surface wave to an ST radiating space wave [55].
We presented a comprehensive review of the theory and analysis of wave propagation in ST metasurfaces and examples of extraordinary 4D wave transformations in such media. It is shown that ST metasurfaces are capable of 4D electromagnetic wave transformations, which are significantly more versatile and useful than the 3D wave transformations of conventional spatially variant static metamaterials and metasurfaces. Recent progress on ST metasurfaces for breaking time-reversal symmetry and reciprocity reveals a great potential for applications of these metasurfaces in low-energy and energy-harvesting telecommunication systems and compact and integrated nonreciprocal devices and subsystems.
This research was sponsored by the Natural Sciences and Engineering Research Council of Canada. This article has supplementary downloadable material available at http://doi.org/10.1109/MAP.2022.3201195.
Sajjad Taravati (sajjad.taravati@utoronto.ca) is with the Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 2E4, Canada. His research interests include applied physics, electromagnetics, nonreciprocal magnetless systems, space–time-modulated structures, circuits, and metamaterials and metasurfaces. He is a Senior Member of IEEE.
George V. Eleftheriades (gelefth@ece.utoronto.ca) is a professor in the Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 2E4, Canada, where he holds the Velma M. Rogers Graham Endowed Chair in Engineering. He is a Fellow of IEEE.
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