1. Introduction

Deflection of electromagnetic waves, called also blazing, represents conversion of the energy of an incident plane wave to that of nonspecular diffraction order(s), either in reflection, or in transmission, or in both. This scenario of wavefront manipulation has been a focus of research for decades, arising through diffraction of the incident plane wave by a periodic array of identical unit cells [1].

A conventional blazed grating contains a properly designed unit cell [1–6]. Photonic-crystal gratings and metamaterial-based gratings have been suggested, in which the suppression of specular diffraction depends on the frequency dependences of the constitutive parameters [7–11]. More recently, gradient metasurfaces have been investigated for flexible wavefront manipulation, with multiple subwavelength cells combined to create a unit supercell. This unit supercell functions as the unit cell [17,18], wherein a desired phase change occurs from one constituent subwavelength cell to the next in order [19–21] to enable deflection to a desired direction.

Compared to those metasurfaces, modern metagratings yielding deflection still use volumetric (infelicitously called Mie [22]), surface plasmon, or other resonances, but comprise components of one or two types/sizes instead of multiple components in each unit supercell [12–16]. Notably, in the literature on metasurfaces and metagratings, the terms anomalous reflection and anomalous transmission are often used as substitutes for the conventional terms deflection and blazing, even though wave propagation does not contradict [17,18] either the classical grating theory [1,23,24] or macroscopic electromagnetism [25]. The same metastructures that enable strong diffraction may show a myriad of functionality-enabling effects in the sub-diffraction regime, the effects including but not restricted to electromagnetic wave trapping [26,27], Fano resonances [28], and guided-wave-mode resonances [29]. Parenthetically, all nonspecular reflectances and transmittances are null valued in the sub-diffraction regime [1,23,24].

Contemporaneously, dynamically controllable materials have been extensively used to design and investigate on-off switchable or gradually controllable metasurfaces and metagratings. Among the relevant materials are: phase-change materials (PCM) such as $\mbox {VO}_2$, germanium-antimony-tellurium (GST), and InSb [30–34]; transparent conducting oxides such as indium-doped tin oxide [35]; graphene [36,37]; and liquid crystals [38]. In particular, PCMs have recently been proposed for use in meta-atoms of diverse morphologies [34,39–47]. Meta-atom components made of a PCM play a crucial role in tunable and reconfigurable metasurfaces, because they may enable on-off switchable or gradually tunable functionalities. Moreover, not only on-off switching of a selected functionality but also switching between two different functionalities can be achieved for multifunctional operation [19,40,48,49]. Merging wavefront manipulation with the dynamic controllability provided by one or more of the aforementioned materials is attractive.

The fact that all of these materials dissipate electromagnetic energy pre-determines the ways in which they can be used in metastructures. In most cases, they are used as surface components and/or pads/inserts in meta-atoms, whereas their use as materials for volumetric resonators is restricted by dissipation. Although there are recent examples of (arrays of) volumetric resonators made of $\mbox {VO}_2$ [32,50], their prospects for wavefront manipulation remain moot.

From this perspective, the recently proposed simple metagratings comprising arrays of parallel InSb rods look promising for switchable deflection of THz waves in the reflection mode [31]. Therein, switching between specular reflection (0th order) and single-beam (-1st order) deflection is achieved by involving the vacuum state [31] and the high-real-relative-permittivity insulator phase of InSb. The InSb rods electromagnetically disappear when InSb is in the vacuum state [31].

In this paper, we propose and numerically investigate simple lamellar metagratings with one-side corrugations that exhibit thermally controllable deflection in the transmission mode. These metagratings are of either

As a reference, we consider our earlier work in which reflection gratings with on-off switchable deflection were investigated [31]. Here, in contrast, we study the transmission metagratings with one-side corrugations that may exhibit reciprocal asymmetric transmission (AT) by diffraction into non-specular orders. Although phase transition of a component material is commonly considered as the main enabler of switching capability, AT and its tuning/switching are achievable in the studied metagratings, while varying temperature only within the range in which InSb is an insulator with a high real part of the relative permittivity. To compare, with a proper choice of geometric parameters, significant diffraction may appear also when the real part of the relative permittivity of InSb is not high, as shown in Ref. [31]. At the same time, the vacuum state, which was involved to the tunable/switchable deflection in Ref. [31], and the range of a low real relative permittivity remain important in our case for the switchable subdiffractive transmission and reflection regimes.

All numerical results were obtained by using a coupled-integral-equation technique [51]. Most of these results are presented in the following sections either (i) in the frequency vs. incidence-angle plane at selected temperatures or (ii) in the temperature vs. incidence-angle plane at selected frequencies.

2. Results and discussion

2.1 Metagrating designs

The generic geometries of the lamellar metagratings of types A and B are schematically shown in Fig. 1, with $L$ as the period. The metagratings are assumed to be illuminated by an $s$-polarized plane wave (i.e., with its electric field vector perpendicular to the grating plane) [1,23,24]. It is incident from either

 

Fig. 1. Asymmetric metagratings of types (a) A and (b) B. For both types, the dimensions used for calculations are: $L=62.67~\mu \mbox {m}$, $w=12.485~\mu \mbox {m}$, $h=12.485~\mu \mbox {m}$, and $t=8.681~\mu \mbox {m}$. For the sake of definiteness, forward illumination (i.e., from the left side) is presented in this figure.

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2.2 Constitutive parameters

Each metagrating is either partially or entirely made of InSb, a PCM that has recently been incorporated in various controllable configurations [30,31,42,52–54]. In the terahertz spectral regime, where a temperature-mediated transition from the metallic phase to the insulator phase (or vice versa) occurs for InSb, this material’s relative permittivity $\varepsilon _{\rm InSb}$ obeys the Drude model [30,55] as follows:

Here, $\varepsilon _{\infty }=15.68$ is the high-frequency relative permittivity, $\gamma =\pi \times {10^{11}}~\mbox {rad s}^{-1}$ is the damping constant, $\omega _p=\sqrt {N{q_e^2}/0.015m_e\varepsilon _0}$ is the plasma angular frequency, $q_e=-1.6\times {10^{-19}}$ C is the charge of an electron, $m_e=9.11\times {10^{-31}}$ kg is the mass of an electron, $N=5.76\times {10^{20}}\,T^{3/2}\mbox {exp}(-{\cal E}_g/{2k_{B}T})$ is the intrinsic carrier density (in $\mbox {m}^{-3}$), ${\cal E}_g=0.26$ eV is the band-gap energy, $k_B=8.617\times {10^{-5}}~\mbox {eV K}^{-1}$ is the Boltzman constant, $T$ is the absolute temperature, and $\varepsilon _0=8.854\times 10^{-12}$ F m$^{-1}$ is the free-space permittivity. Throughout the paper, we decorate the variables related to forward illumination and backward illumination, respectively, by $\rightarrow$ and $\leftarrow$.

The direction of the $m$th-order outgoing wave can be found from the grating equation [23] in terms of the angle

Let $\rho _m\in \mathbb {C}$ and $\tau _m\in \mathbb {C}$ denote the reflection and transmission coefficients, respectively, of order $m\in \left \{0,\pm 1,\pm 2,\ldots \right \}$. Order $m=0$ is specular whereas all other orders (i.e., $m\ne 0$) are nonspecular. Reflectances are calculated as ${R_m}=\vert \rho _m\vert ^2\, \mbox {Re}\left [\sin \phi _m\right ]/\sin \theta$ and transmittances as ${T_m}=\vert \tau _m\vert ^2\, \mbox {Re}\left [\sin \phi _m\right ]/\sin \theta$, both being dimensionless. In fact, these quantities (when given in per cent) are $m$th-order diffraction efficiencies, so that none may be larger than unity. Furthermore, their sum cannot exceed unity, in compliance with the principle of energy conservation. The condition $\vert \mbox {sin}\theta +2\pi {m}/kL\vert =1$ sets the angle-dependent threshold frequency at which the $m$th order transforms from either evanescent to propagating or vice versa.

Note that the specular transmittance ${T_0}^{\rightarrow }={T_0}^{\leftarrow }$, in line with Lorentz reciprocity [56]. Furthermore, $T_{m}^{\rightarrow }{\neq }T_{m}^{\leftarrow }$ in the general case when $m\ne 0$, provided that the metagrating is asymmetric. At the same time, $\phi _{m}^{\rightarrow }=\phi _{m}^{\leftarrow }$ for all $m\in \left \{0,\pm 1,\pm 2,\ldots \right \}$. Therefore, arrows for ${T_0}$ and $\phi _m$ are omitted for most of the cases considered in the paper.

2.3 Controllable transmission-mode features

Let us start with an overview of the basic features that are observed in the transmission mode for various values of $T\in [245,315]$ K, i.e., rather close to normal environmental conditions. In Fig. 2, the computed values of ${T_0}$ and $T_{-1}^{\rightarrow }$ are presented in the $(f,\theta )$-plane for metagrating A (substrate is made of $\mbox {SiO}_2$ with relative permittivity of 2.25). First, rather large regions of transparency with ${T_0}>0.9$ (identified by I in Fig. 2) are observed that include the vacuum state ($\mbox {Re}\left [\varepsilon _{\rm InSb}\right ]=1$). This state occurs in the neighborhood of $f=1.3$ THz ($ka=1.72$), $f=2$ THz ($ka=2.625$), $f=2.82$ THz ($ka=3.7$), and $f=3.14$ THz ($ka=4.121$), respectively, when $T=245$ K, 275 K, 305 K, and 315 K. Since the substrate is made of a low-permittivity material, ${T_0}$ is maximum for $\mbox {Re}[\varepsilon _{\rm InSb}]\neq {1}$ or, strictly speaking, for $\mbox {Re}[\varepsilon _{\rm InSb}]<0$. In particular, ${T_0}$ is maximum when $\mbox {Re}[\varepsilon _{\rm InSb}]\approx {-2.36}$ ($f=1.18$ THz), $\mbox {Re}[\varepsilon _{\rm InSb}]\approx {-3}$ ($f=1.77$ THz), $\mbox {Re}[\varepsilon _{\rm InSb}]\approx {-3.7}$ ($f=2.46$ THz), and $\mbox {Re}[\varepsilon _{\rm InSb}]\approx {-4.53}$ ($f=2.67$ THz), respectively, when $T=245$ K, 275 K, 305 K, and 315 K.

 

Fig. 2. Transmittances in the $(f,\theta )$-plane computed for metagrating A with the dimensions provided in the caption of Fig. 1. (a–d) ${T_0}$ when (a) $T=245$ K, (b) $T=275$ K, (c) $T=305$ K, and (d) $T=315$ K. (e–h) $T_{-1}^{\rightarrow }$ when (e) $T=245$ K, (f) $T=275$ K, (g) $T=305$ K, and (h) $T=315$ K. Neither $T_0$ nor $T_{-1}^{\rightarrow }$ nor the sum of both can exceed unity.

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Additionally, we observe two different categories of narrowband features typical for the sub-diffraction regime. Those in the first category have ${R_0}>0.8$ (so that $T_0<0.2$) and are identified by II in Fig. 2. Those in the second category have ${T_0}>0.8$ (so that $R_0<0.2$) and are identified by III in Fig. 2. Narrowband features of both categories occur for all values of $\theta$ at relatively small values of $\mbox {Re}[\varepsilon _{\rm InSb}]>1$ and are shifted toward higher frequencies due to the temperature-mediated changes in the frequency dependence of $\varepsilon _{\rm InSb}$.

With our calculations confined to the regimes in which $R_m = T_m\equiv 0$ for all $m\notin \left \{0,-1\right \}$, the appearances of the narrowband features in the $(f,\theta )$-plane are determined by two issues. The first is the threshold frequency

In other words, the particular values of $\varepsilon _{\rm InSb}$ needed to obtain the narrowband features in nearly the entire $\theta$-range should be obtained below the threshold frequency for every value of $\theta$ in that range. Indeed, the threshold frequency at moderate and large values of $\theta$ exists for smaller values of $\mbox {Re}[\varepsilon _{\rm InSb}]$ than those enabling the narrowband features for all $\theta \in [0,\pi /2)$; see Figs. 2(c) and 2(d). In each of Fig. 2(c) and 2(d), a narrow trough of low ${T_0}$ spans high values of the incidence angle $\theta$ and frequencies slightly exceeding $f_{\rm th}$. This trough can be considered as somewhat analogous to a Rayleigh–Wood anomaly [1]; note also that $\mbox {Re}[\varepsilon _{\rm InSb}]<1$ in this frequency range.

Now, let us examine Figs. 2(e)–(h) for the results that demonstrate typical characteristics of $T_{-1}^{\rightarrow }$ in the $(f,\theta )$-plane. Strong single-beam deflection occurs in the region identified by IV in Fig. 2, because $T_{-1}^{\rightarrow }>0.7$ therein. The frequency $f>4$ THz in this region for all four values of $T\in [245,315]$ K. Significantly, the $70$-K increase in the temperature from $245$ K to $315$ K results in:

It is worth noting that a significant change in $T_{-1}^{\rightarrow }$ can be obtained at a fixed frequency $f$ and a fixed incidence angle $\theta$ simply by varying the temperature $T$ by as little as 10 K. Volumetric resonances of the $h\times {w}$ dielectric regions to the left of substrate contribute to the resulting transmission/diffraction mechanism, as we confirmed from spatial profiles of the electric field (data not presented here). At the same time, the region of high $T_{-1}^{\rightarrow }$ is seen surrounded by the regions of weak deflection (i.e., $T_{-1}^{\rightarrow }<0.1$). Since high $T_{-1}^{\rightarrow }$ is shown to correspond to a particular range of variation of $\mbox {Re}[\varepsilon _{\rm InSb}]$, the aforementioned weak deflection indicates that $\mbox {Re}[\varepsilon _{\rm InSb}]$ is unsuitable to engender volumetric resonances, which are responsible for high $T_{-1}^{\rightarrow }$ for fixed $w$ and $h$. The value of $\mbox {Re}[\varepsilon _{\rm InSb}]$ needed to obtain high $T_{-1}^{\rightarrow }$ is dependent on temperature, because constant $\mbox {Re}[\varepsilon _{\rm InSb}]$ corresponds to different values of $kw$ and $kh$ at different $T$.

It stands to reason that the observed sensitivity of ${T_0}$ and $T_{-1}^{\rightarrow }$ to temperature can be used to obtain switchability to spatial (angular) filtering (i.e., filtering in the $\theta$-domain) [8,58–60]. Two scenarios are as follows:

The filtering scenarios (i) and (ii) have a notable difference. In scenario (i), there is no deflection as compared with the incidence direction. In second scenario (ii), the transmitted wave of order $m=-1$ is deflected according to Eq. (2), with $\phi _{-1}$ depending nonlinearly on $\theta$.

For more clarity, plots of the transmittances ($T_0$ and $T_{-1}^{\rightarrow }$) and reflectances ($R_0$ and $R_{-1}^{\rightarrow }$) vs. frequency are presented for metagrating A at the selected values of $\theta$ in Figs. S1 and S2 of the Supplement 1 [57]. Moreover, the variations of $R_0$ and $R_{-1}^{\rightarrow }$ in the ($f$, $\theta$)-plane are presented in Fig. S3 of the Supplement 1 [57], similarly to Fig. 2.

Figure 3 is analogous to Fig. 2, but for metagrating B (for which the substrate is also made of InSb). Regions I–IV defined for Fig. 2 also exist in Fig. 3, but two main differences can be noted when both figures are compared. First, the regions of high ${T_0}$ and $T_{-1}^{\rightarrow }$ can be wider for metagrating B than for metagrating A. Second, narrowband features can be observed for both $f<f_{\rm th}(\theta )$ and $f>f_{\rm th}(\theta )$ for metagrating B but the ones for metagrating A exist rather for $f<f_{\rm th}(\theta )$. Some or all of these features are connected with guided-wave propagation [24].

 

Fig. 3. Transmittances in the $(f,\theta )$-plane computed for metagrating B with the dimensions provided in the caption of Fig. 1. (a–d) ${T_0}$ when (a) $T=245$ K, (b) $T=275$ K, (c) $T=305$ K, and (d) $T=315$ K. (e–h) $T_{-1}^{\rightarrow }$ when (e) $T=245$ K, (f) $T=275$ K, (g) $T=305$ K, and (h) $T=315$ K. Neither $T_0$ nor $T_{-1}^{\rightarrow }$ nor the sum of both can exceed unity.

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Regions identified by I extend over a very wide range of $\theta$ for metagratings of both types, but span a much smaller frequency range for metagrating B than for metagrating A especially for lower values of $\theta$. Therefore, a sharper switching between weak and strong ${T_0}$ is possible with B than with A. As an example, just a $30$-K increase in temperature from $245$ K to $275$ K can reduce the value of $T_0$ at $1.32$ THz by a factor of $38$, as is clear from Figs. 3(a) and 3(b).

Strong single-beam deflection occurs in the region identified by IV in Figs. 3(e)–(g), because $T_{-1}^{\rightarrow }>0.7$ therein. The $60$-K increase in the temperature from $245$ K to $305$ K results in:

Note that ${\rm max}\left \{T_{-1}^{\rightarrow }\right \}$ is higher for metagrating B than for metagrating A. However, scenarios of switching between high and low $T_{-1}^{\rightarrow }$ can be more complicated for metagrating B than for metagrating A, because of the multiple narrowband features.

In addition, plots of $T_0$, $T_{-1}^{\rightarrow }$, $R_0$, and $R_{-1}^{\rightarrow }$ vs. frequency are presented for metagrating B at two selected values of $\theta$ in Figs. S4 and S5 of the Supplement 1 [57]. Finally, the variations of $R_0$ and $R_{-1}^{\rightarrow }$ in the ($f$, $\theta$)-plane are presented in Fig. S6 of the Supplement 1 [57], similarly to Fig. 3.

2.4 Deflection and asymmetric transmission at a fixed frequency

A big advantage of temperature-mediated variability of the relative permittivity of InSb is one more degree of freedom, in addition to variations in the incidence angle, when the frequency is fixed. As two particular goals, we consider

Notably, while the occurrences of high-contrast AT in asymmetric photonic-crystal gratings and few-layer polarization-converting metasurface are pre-determined, respectively, by the dispersion characteristics of electromagnetic surface waves [24] and by the properties of the contributing subwavelength resonance(s) [10,61], high-contrast AT is rather incidental in the case of diffraction gratings. Indeed, although AT is formally allowed by structural asymmetry, a high contrast of AT is not guaranteed. Similarly, asymmetry may manifest itself also in absorption [62,63].

Lorentz reciprocity [56] is known to play an important role for AT. As was demonstrated in Ref. [10] if single-wave deflection (blazing) is achieved, say, within a bounded region in the ($f$, $\theta$)-plane for forward illumination, so that the deflected wave takes (almost) all of the energy of the incident plane wave, then we can rather accurately predict the $\theta$-range of deflection for backward illumination and estimate the overall AT capability with the help of Eq. (2).

Figure 4 presents ${T_0}$, $T_{-1}^{\rightarrow }$, and $T_{-1}^{\leftarrow }$ in the $(T,\theta )$-plane for metagratings A and B at the $f=4.58$ THz. For metagrating A in Figs. 4(a)–(c), quite large regions in the ($T$, $\theta$)-plane are observed in which ${T_0}$ is rather small and deflection connected with the order $m=-1$ is dominant, for forward illumination as well as backward illumination. The regions of high $T_{-1}^{\rightarrow }$ extended nearly over 30 K and $20^{\circ }$ in the ($T$, $\theta$)-plane. For forward illumination, an increase of $T$ by less than 25 K is sufficient to switch between strong deflection and vanishing deflection, at and in the vicinity of $\theta =40^{\circ }$. The same is true for backward illumination, but rather between $\theta =20^{\circ }$ and $30^{\circ }$. Parenthetically, the retroreflection regime ($\phi _{-1}=-\theta$) occurs in Figs. 4(b) and 4(c) within the aforementioned regions. It takes place when $\mbox {sin}\phi _{-1}=-\pi /kL$, which corresponds to $\theta =31.5^{\circ }$ when $f=4.58$ THz.

 

Fig. 4. Transmittances in the $(T,\theta )$-plane computed for metagratings A and B with the dimensions provided in the caption of Fig. 1 when $f=4.58$ THz (i.e., $kL=6.01$). (a) ${T_0}$, (b) $T_{-1}^{\rightarrow }$, and (c) $T_{-1}^{\leftarrow }$ for metagrating A. (d) ${T_0}$, (e) $T_{-1}^{\rightarrow }$, and (f) $T_{-1}^{\leftarrow }$ for metagrating B. Neither $T_{-1}^{\leftarrow }$ nor $T_{-1}^{\rightarrow }$ can exceed unity.

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Although the observed overlapping of the regions of high $T_{-1}^{\rightarrow }$ and $T_{-1}^{\leftarrow }$, as exemplified by Figs. 4(b) and 4(c), may be appropriate for deflection-related applications, it is clearly unsuitable for AT. In line with Ref. [10] , the $\theta$ ranges for high $T_{-1}^{\rightarrow }$, and $T_{-1}^{\leftarrow }$ should be different, in order to enable well-pronounced AT. Lorentz reciprocity manifests itself in geometric similarity of the regions of high $T_{-1}^{\rightarrow }$ and high $T_{-1}^{\leftarrow }$, respectively, in Figs. 4(b) and 4(c). Indeed, the map of $T_{-1}^{\leftarrow }$ in Fig. 4(c) looks like an inverted and slightly stretched version of the map of $T_{-1}^{\rightarrow }$ in Fig. 4(b), consistently with Eq. (2).

Next, Figs. 4(d)–(f) present analogous results for metagrating B. Now, the regions of large $T_{-1}^{\rightarrow }$ and large $T_{-1}^{\leftarrow }$ are located within different $\theta$-ranges, as required for AT. For the strong-deflection regions, $\theta$ varies

In contrast with metagrating A in Figs. 4(a)–(c), narrowband features can play key roles in switching scenarios wherein either $T$ is varied and/or $\theta$ is varied when the frequency is fixed. One can see that a 33.5 K change in $T$ is sufficient for switching between strong deflection and weak deflection, as exemplified at $\theta =52^{\circ }$ in Fig. 4(e).

Examples of AT switchable by changing $\theta$ at constant temperature can be found in Figs. 4(d)–(f). For instance, at $T=280$ K and $f=4.58$ THz, we have

Thus, switching between $T_{-1}^{\rightarrow }=0.895$ and $T_{-1}^{\leftarrow }=0.756$ is possible by switching $\theta$ between $52^{\circ }$ and $12^{\circ }$. The aforementioned geometric similarity of the maps of $T_{-1}^{\rightarrow }$ and $T_{-1}^{\leftarrow }$ resulting from Lorentz reciprocity is observed also in Figs. 4(e) and 4(f). Therefore, metagrating B cannot exhibit retroreflection.

2.5 More about asymmetric transmission

The question remains whether thermally switchable AT can be exhibited by the metagratings A and B, with and without variations in $\theta$. Since high-contrast AT is rather incidental in the case of diffraction gratings, it would not be straightforward to find such a switchable scenario.

Figure 5 presents spectrums of ${T_0}$, $T_{-1}^{\rightarrow }$, and $T_{-1}^{\leftarrow }$ calculated for metagrating B at selected combinations of $T$ and $\theta$. At 275-K temperature, AT is evident at

Although AT with strong efficiency achieved for each of the two opposite illumination directions is possible at close enough frequencies, similarly to that studied earlier for PhC gratings in Ref. [64], the values of $\theta$ (and, therefore, $\phi _{-1}$) are very different in our case.

 

Fig. 5. Transmittances ${T_0}$ (blue solid lines), $T_{-1}^{\rightarrow }$ (red solid lines), and $T_{-1}^{\leftarrow }$ (red dashed lines) vs. $f$, when (a) $T=275$ K and $\theta =60^{\circ }$, (b) $T=275$ K and $\theta =20^{\circ }$, (c) $T=245$ K and $\theta =60^{\circ }$, (d) $T=245$ K and $\theta =20^{\circ }$, and (e,f) $T=245$ K and $\theta =40^{\circ }$, for metagrating B with the dimensions provided in the caption of Fig. 1.

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Notably, exact coincidence of $f$ values can be obtained for two AT scenarios, in which high efficiency is achieved for either the forward or the backward illumination case by means of fine adjustment of temperature and the values of $\theta$ according to Eq. (2), i.e., $\theta ^{\rightarrow }=-\phi _{-1}^{\leftarrow }$ and $\theta ^{\leftarrow }=-\phi _{-1}^{\rightarrow }$.

Similarly to the results in Figs. 5(a) and 5(b) for $T=275$ K, a reversal of the direction of strong transmission in the AT regimes is possible for $T=245$ K at closer frequencies, as shown in Figs. 5(c) and 5(d). At 245-K temperature, AT is achieved at

Notably, transmission contrast (i.e., extent of asymmetry) is insignificant at 275 K for $f\in \left \{4.22,4.3\right \}$ THz in Figs. 5(a) and 5(b). Also, that asymmetry is impractical at 245 K for $f\in \left \{4.45,4.57\right \}$ THz in Figs. 5(c) and 5(d). This is because the extent of asymmetry is moderate in Fig. 5(c) but the magnitude of $T_{-1}^{\leftarrow }$ is lower than desired, the extent of asymmetry is weak in Fig. 5(d). Comparison of the results obtained for the two aforementioned $f$-ranges for both forward and backward illumination indicates the possibility of switching between the regimes of well- and poorly pronounced asymmetry in transmission by varying $T$ at $\theta =\mbox {const}$, at least if using two close values of $f$. However, switching between a well-pronounced AT and near-zero total transmittance (i.e., either $T_0^\rightarrow +T_{-1}^\rightarrow \ll 0.1$ or $T_0^\leftarrow +T_{-1}^\leftarrow \ll 0.1$, as appropriate) simply by changing the temperature remains a challenge at a specific frequency.

Figures 5(e) and 5(f) present the results for $\theta =40^{\circ }$ and $T=245$ K. Although both AT and nearly-symmetric-transmission regimes can be obtained in moderately spaced $f$-ranges, AT is worse than at $\theta =20^{\circ }$ and $60^{\circ }$ in Figs. 5(c) and 5(d).

Although metagratings of type A of the same size show a high potential for deflection, they are typically unsuitable for AT according to our numerical results (not shown). Perhaps, the ratios $w/L$ and $w/h$ have to be optimized to uncover their potential, but that effort lies beyond the scope of this paper. Most importantly, the foregoing results demonstrate that, even if we restrict ourselves to the insulator phase of InSb, variations in temperature are still key enablers of the reported switching of AT.

3. Concluding remarks

To summarize, we proposed and studied two types of simple transmissive metagratings with one-side lamellar corrugations comprising components made of InSb, a thermally controllable phase-change material. Controllable and on-off switchable deflection connected with nonspecular diffraction orders was numerically demonstrated in the terahertz regime, at easily achievable temperatures, and temperature changes between 30 K and 60 K. As desired, significant deflection is possible in the transmission mode in the high-${\rm Re}\left (\varepsilon _{\rm InSb}\right )$ regime, whereas the vicinity of the vacuum state can be used to exploit controllable narrowband features. Temperature variations introduce one more degree of freedom, which allows to obtain a rich variety of transmission/diffraction scenarios even at a (suitably chosen) fixed frequency.

The capability of thermally mediated diffraction in the studied metagratings makes switchable AT possible, whereby weak and strong asymmetries are achieved in transmission at different temperatures. Moreover, the high-transmittance direction is changed for the opposite one at two close frequencies just by varying the incidence angle. The obtained results confirm that InSb has a high potential for wavefront manipulation in the terahertz spectral regime. A much richer variety of switching scenarios may be obtained by optimizing grating shape and dimensions. The proposed metagratings, which are thinner than 1/2.8 of free-space wavelength within the entire frequency range considered, can serve as alternatives to photonic-crystal-based structures, volumetric-metamaterial-based structures, and gradient metasurfaces for the purpose of enabling AT.