Space-Time-Coding Digital Metasurfaces: Principles and Applications

Space-time-modulated metastructures characterized by spatiotemporally varying properties have recently attracted great interest and become one of the most fascinating and promising research fields. In the meantime, space-time-coding digital metasurfaces with inherently programmable natures emerge as powerful and versatile platforms for implementing the spatiotemporal modulations, which have been successfully realized and used to manipulate the electromagnetic waves in both the spectral and spatial domains. In this article, we systematically introduce the general concepts and working principles of space-time-coding digital metasurfaces and provide a comprehensive survey of recent advances and representative applications in this field. Specifically, we illustrate the examples of complicated wave manipulations, including harmonic beam control and programmable nonreciprocal effect. The fascinating strategy of space-time-coding opens the door to exciting scenarios for information systems, with abundant applications ranging from wireless communications to imaging and radars. We summarize this review by presenting the perspectives on the existing challenges and future directions in this fast-growing research field.

In the initial studies, digital coding is defined in the spatial domain and fixed in the temporal domain, in which case the space-domain-coding (SDC) digital metasurfaces can only manipulate the spatial distribution of EM waves , such as the far-field scattered beams and the near-field patterns. By extending the digital coding from the spatial domain to the temporal domain, timedomain-coding (TDC) digital metasurfaces can control the spectral distribution of reflected waves by dynamically switching the time-coding sequences [109]. In 2018, the general theory of space-time-coding (STC) digital metasurfaces was originally put forward by Zhang et al. [110]. The constitutive parameters (e.g., reflection phases) of the STC digital metasurfaces are jointly encoded in space and time, which can control the EM waves in both the spectral and spatial domains. That is to say, one can simultaneously manipulate the harmonic distribution and propagation direction of reflected waves [110]. The STC digital metasurfaces extend and generalize the concepts of "phaseswitched screens" [111] and "time-modulated arrays" [112] and have been applied successfully to harmonic beam control [110], scattering reduction [110], nonreciprocal effect [113], multibit phase generation [114], terahertz harmonic manipulations [115], spread-spectrum camouflaging [116], analog computing [117], and wireless communications [118].
In this article, firstly, we briefly introduce the general concepts and working principles of the SDC and STC digital metasurfaces (Section 2). Subsequently, we present some recent advances and representative applications of the STC digital metasurfaces (Section 3). Finally, we discuss the perspectives on the existing challenges and future research directions of the STC digital metasurfaces in the conclusion part (Section 4).

General Concept and Working Principles of STC Digital Metasurfaces
In this section, we briefly introduce the fundamental concept of digital coding and programmable metasurfaces [57]. Figure 1(a) displays the conceptual schematic of a 1-bit digital coding metasurface, which is composed of two distinct coding elements "0" and "1" with a phase difference of π. It can be seen that the digital coding metasurfaces can regulate the reflected EM beams by designing different coding patterns. By embedding active devices such as diodes into the coding elements, programmable metasurfaces are further developed to control the EM fields in real time. Figure 1(b) shows the geometry and reflection phases of the 1-bit programmable element, which exhibits coding states "0" and "1" when the PIN diode is switched between "OFF" and "ON," respectively. The coding states of programmable metasurfaces can certainly be extended from 1 bit to multibits. For example, a 2-bit programmable metasurface can be realized by integrating more PIN diodes into each coding element and thereby has four coding states "00," "01," "10," and "11" with the discrete phases of 0, π/2, π, and 3π/2, respectively, and higher-bit programmable metasurfaces can be similarly characterized and realized. Figure 1(c) shows the flow diagram of the 1-bit programmable metasurface controlled by an FPGA, which can attain different scattered beams by altering different coding sequences of "000000," "111111," "010101", and "001011," as depicted in Figure 1(d).
Next, we present the general concept and working principles of STC digital metasurfaces. Referring to Ref. [110] for details, the STC digital metasurface usually encompasses a 2D array of M × N programmable elements, which is schematically illustrated in Figure 2(a). The programmable elements represented by yellow patches have identical structures, and each of them is integrated with active devices such as positive-intrinsic-negative (PIN) diodes. By loading different biasing voltages to the active devices, reflection coefficients of programmable elements can be dynamically tailored with a set of quantized amplitudes or phases. For the 1-bit phase-encoding case, the response of each element is switched between two coding states, namely, digital bits "0" and "1" corresponding to the in-phase and out-of-phase reflections, respectively. We assume that the coding state is electronically switched in both the spatial and temporal domains via an FPGA, in accordance with the 3D STC matrix represented by red and green dots in Figure 2(a), in which each element is periodically time-modulated with a set of timecoding sequences. Combined with space modulation, the STC digital metasurface can simultaneously manipulate the spectral (harmonic distribution) and spatial (propagation direction) characteristics of the scattered waves.
As previously assumed in [110], an adiabatic approximate model was put forward to represent the scattered fields of STC digital metasurfaces under the incidence of a normal plane wave with the time-harmonic dependence exp ðj2πf c t c Þ. This analytical model depends on the physical optics approximation that was developed for conventional SDC metasurfaces [57]. By assuming that the time-modulated frequency f 0 is much smaller than the EM carrier frequency f c , the far-field scattering pattern of the STC digital metasurface in the time domain is approximately written as where Eðθ, φÞ is the far-field scattering pattern of coding elements at the central frequency f c ; λ c = c/f c is the central operating wavelength (with c denoting the speed of light in vacuum); θ and φ are the elevation and azimuth angles, respectively; and d x and d y represent the element spacing. Moreover, Γ pq ðtÞ represents the time-varying reflection coefficient of the ðp, qÞ th element, which is assumed to be a function with the period T 0 and defined as a linear superposition of pulse functions over one period: where U n pq ðtÞ is a shifted rectangular pulse function and defined in one period as where L denotes the length of the time-coding sequence, τ = T 0 /L denotes the pulse width of U n pq ðtÞ, and Γ n pq = A n pq exp ðjφ n pq Þ denotes the reflection coefficient of the ðp, qÞ th coding element at the central frequency f c within the interval ðn − 1Þτ ≤ t ≤ nτ, with φ n pq and A n pq representing the phase and amplitude, respectively. By decomposing the periodic function Γ pq ðtÞ into the Fourier series Γ pq ðtÞ = ∑ +∞ m=−∞ a m pq exp ðj2πmf 0 tÞ, the Fourier coefficients a m pq can be further expressed as By substituting equation (4) into equation (1), the frequency-domain far-field pattern of the STC digital metasurface at the harmonic frequency f c + mf 0 can be further expressed as It is worth mentioning that these complex-valued coefficients a m pq in equation (4) can be synthesized to produce equivalent amplitudes and phases through linear combinations of the reflection coefficient Γ n pq ; that is to say, we can individually regulate the equivalent amplitudes and phases of each coding element at a specific frequency by designing Γ n pq in the time-coding sequences. Hence, for an arbitrary 3D STC matrix represented by equation (2), we can calculate the equivalent excitations (via equation (4)) and scattering patterns (via equation (5)) of the STC digital metasurface at any harmonic frequencies of interest.

Recent Advances and Representative Applications
With the aid of the fundamental theory introduced above, we can utilize the STC digital metasurfaces to simultaneously manipulate EM waves in both the spectral and spatial domains (see the conceptual illustration in Figure 2(a)). Furthermore, by elaborately designing the STC matrix, information encoding and processing can be carried out not only in the spatial domain but also in the temporal domain. In this section, we mainly consider the phase modulation scheme, in which the amplitude A n pq of each coding element is uniform and the phase φ n pq is switched among different coding states according to the STC matrix. The STC strategy significantly expands the application range of conventional meta-surfaces, leading to many promising applications in radars, imaging, and wireless communications [119]. In what follows, we will focus on the recent advances and representative applications of the STC digital metasurfaces.

Harmonic Beam
Control. An attractive application inspired by the STC digital metasurfaces is harmonic beam steering, in which different harmonic beams point to different spatial directions [110], as illustrated in Figure 2(a). Actually, a simple time gradient STC matrix (see Figure 2(b) in [110]) can also be used to realize harmonic beam steering. Different equivalent phase gradients that emerge at different harmonic frequencies explain the essence of harmonic beam steering. However, this simple scheme suffers from the issue of imbalanced power distributions. To solve this issue, the binary particle swarm optimization (BPSO) algorithm is exploited to optimize the STC matrix in [110] so as to equalize power levels at different frequencies. In this case, we consider an STC digital metasurface with 8 × 8 half-wavelength elements and the 1-bit time-coding sequences with the length L = 10. Since the beam steering is designed in the onedimensional plane, each column of elements has the same time-coding sequences so that the 3D coding matrix can be simplified to a 2D matrix (see Figure 2(c)). Figure 2(d) shows the corresponding equivalent amplitudes and phases at different frequencies, from which the balanced amplitude and gradient phase distributions are evident. Accordingly, the harmonic scattering patterns are numerically calculated via equation (5) and shown in Figure 2(e), in which the main beams at different frequencies have uniform power levels and point to different directions in space. To experimentally verify this design, a prototype of a 1-bit STC digital metasurface working around the frequency of 10 GHz was manufactured by the printed circuit board (PCB) technology (see [110] for details), as displayed in Figure 2(b). When the PIN diode inserted in the coding element is switched between "ON" and "OFF" states, the reflection coefficient can obtain a phase difference of 180°. For the one-dimensional beam steering, eight coding elements in each column have the same biasing voltages via an FPGA hardware control board, which provides eight control voltages according to the STC matrix in Figure 2(c). In this experiment, the time-modulated frequency is assumed to be f 0 = 0:5 MHz, and the corresponding switching rate of PIN diodes is 5 MHz. The measured far-field patterns agree well with the theoretical predictions, thereby validating the effect of harmonic beam steering.
In this example, the STC digital metasurface is the same as the previous example, but the length of time-coding sequences is L = 8. Figure 2(f) shows eight sets of 2-bit time-coding sequences, which are carefully selected to achieve equivalent 3-bit responses at the center frequency. The programmable metasurface is divided into eight parts accompanied by the rotated distribution of those timecoding sequences. Figure 2(g) shows the corresponding equivalent amplitudes and phases at the center frequency f c , in which the equivalent 3-bit phases exhibit a spiral-like distribution and can generate a vortex beam with orbital angular momentum (OAM). The corresponding far-field scattering patterns are shown in Figure 2(h), from which a typical hollow-center profile of the vortex beam can be clearly observed. Overall, this STC scheme provides a new pathway to realize multibit programmable metasurfaces (see also the detailed discussion in Section 3.4 below), which gets rid of the complex metasurface layout and control circuit system, and further achieves precise field manipulation in both the spatial and spectral domains.

Reduction of Scattering.
With the rapid development of radar and stealth technologies, how to reduce the radar cross-section (RCS) of a target has always been a pivotal issue. So far, many methods have been exploited for RCS reduction. For example, radar-absorbing materials are widely used to absorb the EM energy, whereas low-scattering metasurfaces using the phase cancellation method can disperse the incident waves to many directions in the spatial domain. Besides, the phase-switched screen was proposed to redistribute the scattered power to odd harmonics in the frequency domain [111]. Against this background, another interesting application enabled by the STC digital metasurface is to reduce RCS [110]. This novel STC mechanism can be used to effectively redistribute the scattered energy in both the spectral and spatial domains, which completely differs from the conventional methods and brings out a better performance of RCS reduction.
We first consider a BPSO-optimized space-coding shown at the top of Figure 3(a), its corresponding backscattered power is distributed uniformly in all possible directions in the space domain, and the maximum of backscattered power is reduced by~12.7 dB by comparison with the reference metallic plate, as shown in the top-left corner of Figure 3(b). And then, we further apply a time-coding sequence "10011010" to this BPSO-optimized space-coding, forming a 3D STC matrix shown at the bottom of Figure 3(a). In this case, the incident power can be redistributed more uniformly in both the spatial and spectral domains, and the maximum of scattered power at different harmonics is further decreased by~8.82 dB compared with the case of only space-coding in Figure 3(a). The corresponding scattering patterns at several harmonics are shown in the red dashed box in Figure 3(b), and the maximum of scattered power at different harmonics is decreased by~21.52 dB compared with the reference metallic plate (see Figure 5(f) in [110]). Moreover, Figure 3(c) also shows the maxima of scattering patterns from the -50 th to the +50 th harmonic frequencies, corresponding to the 3D STC matrix in Figure 3(a). It can be observed that the maximum intensity gradually decreases with the increase of the harmonic order and drops significantly with respect to the first few harmonics. Generally, with the aid of the STC strategy, the scattered energy can be excellently suppressed in both the spatial and spectral domains, which ensures a more effective and robust effect of RCS reduction. Other related applications of scattering control also include spectral camouflage by introducing random time-modulated signals [116].

Programmable Nonreciprocal Effects and Frequency
Conversions. In the area of EM engineering, breaking reciprocity has attracted research interests owing to its pivotal role in releasing the reciprocal limitation in communications, energy harvesting, and thermal management. Conventional methods based on nonlinear and magnetic materials can attain nonreciprocal effects but have many disadvantages in practice. Besides, time-variant methods have emerged as appealing alternatives to break reciprocity [30-32, 35, 42], in which space-time-modulated metastructures have aroused great attention [38,39,41,44]. In recent years, some theoretical and experimental works based on metasurfaces have been proposed to achieve nonreciprocal effects by introducing a suitable spatiotemporal phase gradient [45,46].
More specifically, a 2-bit STC digital metasurface has recently been leveraged to realize the programmable nonreciprocal effect via the time modulation of the phase responses [113], which enables anomalous reflections together with the high-efficiency frequency conversion. In this specific application, an STC digital metasurface composed of N columns of programmable elements is considered for a 2D scenario (see Figure 3(a)); the reflection phase of each element is modulated by switching the embedded PIN diodes. The STC digital metasurface is illuminated by an obliquely incident plane wave with the time-harmonic dependence exp ðj2πf c t c Þ, transverse magnetic polarization, and incident angle θ i . The reflection coefficient of the p th element is temporally modulated, which can be expressed as a periodic function Γ p ðtÞ = ∑ n=1,⋯,L Γ n p U n p ðtÞ, with U n p ðtÞ representing a rectangular pulse function with a modulation period T 0 . Hence, the phase response of each element can be characterized by a set of time-coding sequences with length L, and the whole metasurface is further represented by an STC matrix. According to the previously introduced theory of STC digital metasurfaces in Section 2, the time-domain far-field scattering pattern in this circumstance can be approximately written as where EðθÞ = cos θ is the far-field scattering pattern (approximated by a cosine function) of the p th programmable element at the center frequency, d denotes the element spacing, and other parameters are the same as those defined in equation (1). By taking the Fourier series expansion of the periodic function Γ p ðtÞ, the frequency-domain far-field patterns of the STC digital metasurface at the harmonic frequency f c + mf 0 can be expressed as where λ r = c/ð f c + mf 0 Þ denotes the wavelength of the m th harmonic wave, and a m p represents the Fourier coefficients of Γ p ðtÞ:   Figure 4(a) schematically illustrates the process of nonreciprocal effects in a two-port reflection system based on the STC digital metasurface. By judiciously engineering the spatiotemporally gradient coding sequence, a plane wave with frequency f 1 obliquely incident from the angle θ 1 will be reflected at the angle θ 2 and frequency f 2 . While for the time-reversal case, a plane wave with frequency f 2 incident from the angle θ 2 will be reflected at the angle θ 3 and frequency f 3 . Due to the spatiotemporal modulation, the angle θ 3 and frequency f 3 are different from the original angle θ 1 and frequency f 1 (i.e., θ 3 ≠ θ 1 , f 3 ≠ f 1 ), which obviously breaks the Lorentz reciprocity [113].
To this end, we further assume that the STC programmable metasurface encompasses 16 columns of elements, which have 2-bit time-coding sequences with the length L = 4. Figure 4(b) displays a microwave prototype of a 2-bit STC digital metasurface. Figure 4(c) shows the 2-bit STC matrix, in which each coding element is controlled by a time gradient phase with an interval of π/2 and the time-coding sequences across the metasurface are progressively shifted by a time interval T 0 /4 so that every two adjacent elements always exhibit a π/2 phase difference in each time interval. The corresponding equivalent amplitudes and phases (calculated via equation (8)) are shown in Figure 4(d), from which we can observe that this spatiotemporal gradient coding essentially leads to an anomalous reflection with incident EM energy mainly concentrated at the positive first harmonic (m = 1).
Accordingly, the effective phase gradient at the +1 st harmonic frequency can be written as Recalling the forward scenario in Figure 4(a), for a plane wave with frequency f c obliquely incident from an angle θ 1 , the dominant reflected power will be at the +1 st harmonic frequency f c + f 0 and at an angle θ 2 , given by When considering the time-reversal scenario in Figure 4(a), for a plane wave with frequency f c + f 0 incident from the angle θ 2 , the dominated reflected waves will not follow the original incident direction in the forward channel but mainly exist at a different frequency f c + 2f 0 and an angle θ 3 , given by The reflected wave in the time-reversal case no longer propagates along the direction of the original incident wave in the forward case and also brings out a frequency shift of 2f 0 . That is to say, for the oblique incidence (i.e., θ 1 ≠ 0), the time-reversal reflection channel and the forward incident channel are isolated in both the spectral and spatial domains. It can be observed from equation (10) that the angular difference jθ 3 − θ 1 j increases with the frequency ratio f 0 /f c and the original incident angle θ 1 .
Here, we consider the first representative example of the nonreciprocal reflection by assuming that f c = 5 GHz, f 0 = 250 MHz, d = λ c /2, and θ 1 = 60 ∘ . In this circumstance, the reflected angles in the forward and time-reversal scenarios are theoretically calculated (via equations (10) and (11)) as θ 2 = 20:40°and θ 3 = 51:93°, respectively. Figure 5(a) shows the numerically computed scattering patterns (via equation (7)) in the forward and time-reversal cases. It can be clearly seen that angular separation between θ 3 and θ 1 reaches a sizable value of ∼9°. For this parameter configuration, the nonreciprocal effect can separate the wave reflections spatially and spectrally. However, the modulation frequency f 0 = 250 MHz of the time-coding sequence in Figure 4(c) corresponds to the switching rate 4f 0 = 1 GHz of PIN diodes, which is not currently available with commercial PIN diodes.
Furthermore, we consider a practical example with the more feasible parameters of f c = 9:5 GHz, f 0 = 1:25 MHz, d = 14 mm, and θ 1 = 34°. Figure 5(b) displays the corresponding scattering patterns in the forward and time-reversal cases, for which the angular difference between θ 3 and θ 1 is <1°. Although this very small angular difference could not be distinguished in space with the current experimental setup, it is hopeful to experimentally observe the frequency isolation in the spectral domain by a spectrum analyzer with high precision. To experimentally verify this example, a prototype of a 2-bit STC digital metasurface working around the frequency of 9.5 GHz was manufactured (see [113] for details), as displayed in Figure 4(b). Figure 5(c) shows the measured scattering patterns in the forward and time-reversal cases, from which the reflected angles and harmonic distributions show great agreement with the theoretical results, thereby providing the first experimental verification of the nonreciprocal effect enabled by spatiotemporally modulated metasurfaces.
It is worth highlighting that the FPGA-controlled STC digital metasurface lays the foundation for the "on-demand" realization of the nonreciprocal effect and frequency conversion in a completely programmable fashion, which may find many promising applications to the mixer, duplexer, optical isolator, unidirectional transmission, Doppler illusion, wireless communication, and radar systems.

Generation of Arbitrary Multibit Programmable Phases.
As aforementioned in Section 3.1, multibit programmable metasurfaces exhibit low phase quantization error and can obtain more precise control of EM waves and thus have a greater capability to realize some sophisticated EM functions. Up to now, programmable metasurfaces with 1-bit and 2-bit phases have been effectively engineered by integrating with active devices and FPGA. However, it is still difficult to realize multibit (i.e., 3-bit or higher-bit) programmable metasurfaces via PIN diodes. Some alternative techniques have also been exploited for programmable metasurfaces, such as varactors, liquid crystal, graphene, and microelectromechanical systems (MEMS), but all those strategies have limited phase coverages and do not allow the continuous 2π phase modulation.
In a recent study [114], a physical programmable metasurface using the 2-bit time-coding strategy was put forward to achieve arbitrary multibit and nearly quasicontinuous phases, as illustrated in Figure 6(a). Figure 6(b) shows the schematic of the vector synthesis method, which was originally proposed to analyze the phase coverages. With the aid of this method, 4-bit and arbitrary multibit phases can be synthesized through a physical 2-bit programmable metasurface. The reflection coefficients (i.e., e jφ 00 , e jφ 01 , e jφ 10 , and e jφ 11 ) of the original 2-bit case are represented by four basic vectors (red arrows) in a complex plane, as shown in Figure 6(b). By carefully selecting the time-coding sequences, the new vectors (green arrows in Figure 6(b)) with arbitrary phases ψ rn can be successfully synthesized by suitably combining the basic vectors [114].  Overall, the time-coding strategy provides a new path to design the arbitrary multibit programmable metasurfaces, without the need for complicated physical design and control circuit systems, which will find potential applications in high-resolution imaging, high-performance antennas, and high-capacity communications and radars. This strategy can also be extended to other frequency regimes (such as terahertz, optics, and acoustics) and generalized to obtain multibit programmable transmission phases.

Nonlinear Harmonic Manipulations via TDC Digital
Metasurfaces. In addition to using PIN diodes, time-varying metastructures can also be implemented via the use of varactors, which provides more control states in the temporal modulation. As a key branch of STC digital metasurfaces, the TDC digital metasurface was put forward to efficiently manipulate the nonlinear harmonic distributions of EM waves [109,120], as illustrated in Figure 7(a). In such a study [109], only time-coding is considered, in which case the space-invariant reflection coefficient of the whole metasurface is temporally modulated in accordance with the same function ΓðtÞ with a period T 0 = 1/f 0 . By assuming that the TDC digital metasurface is normally illuminated by a monochromatic incident plane wave with the time-harmonic form of exp ðj2πf c t c Þ, the basic theory in Section 2 can be generalized to represent the reflected wave E r ð f Þ in the frequency domain as follows: where * is the symbol of convolution operation, E i ð f Þ denotes the incident wave, δðf Þ denotes the Dirac delta function, and a k represents the Fourier coefficient of the periodic function ΓðtÞ at the k th -order harmonic. It can be observed from equation (12) that the reflected spectrum E r ðf Þ is a linear superposition of the EM fields at many harmonic frequencies f c + kf 0 . Generally, the TDC digital metasurface can be used to generate arbitrarily nonlinear harmonic distributions by elaborately customizing the time-modulated waveforms [109], which opens up a new path to realize the EM spectral shaping. Figure 7(b) shows the basic geometry of the TDC digital metasurface, in which each coding element consists of two rectangular patches connected by a varactor. Some representative examples obtained by a varactor-based TDC digital metasurface were presented to illustrate the working principles. Figure 7(c) displays the spectral intensity distributions regarding the phase modulation waveforms with 1-bit ("010101…") and 2-bit ("00-01-10-11…") time-coding sequences. For the 1-bit time-coding sequence "010101…," the incident power is only transferred to the odd harmonics with the symmetric spectral distribution. But for the 2-bit time-coding sequence "00-01-10-11…," most of the incident energy is spread to the +1 st harmonic with the asymmetric spectral distribution. More details on amplitude modulation can be found in [109]. Overall, both the numerical and experimental results demonstrated the capability of the TDC digital metasurfaces in manipulating the EM spectral distribution, which will find interesting applications in the Doppler shift, velocity illusion, and simplified architecture wireless communication systems.
As can be understood from the complex-valued equivalent coefficients a m pq in equation (4), one major limitation of STC digital metasurfaces originates from the inherent strong coupling between the phases and amplitudes of various harmonics. Hence, a recent study on TDC digital metasurfaces was proposed to regulate the amplitudes and phases at different frequencies independently via different timevarying signals [120]. Figure 7(d) shows the conceptual illustration of a reflective TDC digital metasurface, whose reflection phase is periodically modulated in the time domain with a square wave varying between two predesigned phase values ϕ 1 and ϕ 2 (corresponding to the biasing voltages V 1 and V 2 ). To overcome the coupling between the harmonic amplitudes and phases, a time delay t 0 is brought in the time-domain reflection coefficient ΓðtÞ, leading to an additional phase shift −2πkf 0 t 0 with unchanged amplitudes at the k th harmonic frequency [110,120]. After some mathematical derivation, the spectral distribution of reflected waves can be expressed as Reproduced from [109]. (c) The calculated spectral intensities pertaining to phase modulation waveforms with 1-bit ("010101…") and 2bit ("00-01-10-11…") time-coding sequences. Reproduced from [109]. (d) Conceptual illustration of TDC digital metasurfaces for independent manipulation of harmonic phases and amplitudes. Reproduced from [120]. (e) Measured spectral phases and amplitudes at various harmonics, corresponding to three different voltage combinations V 1 /V 2 . Reproduced from [120]. (f) Measured scattering patterns at the +1 st harmonic frequency, corresponding to three sets of spatial coding "00000000," "00001111," and "00110011," respectively. Reproduced from [120].
where A is a constant value of reflection amplitudes and h is an integer. It can be observed from equation (13) that the spectral components of reflected waves only exist at the central frequency (k = 0) and odd-harmonic frequencies (k = ±1, ±3, ±5, ⋯) due to the Fourier transform characteristic of square waves [120]. From equation (13), we can notice that the amplitudes of reflected waves at the central and oddharmonic frequencies can be controlled by tailoring the phase difference ðϕ 1 − ϕ 2 Þ. Besides, the time delay t 0 provides additional degrees of freedom to control the phases at the oddharmonic frequencies, which keeps the amplitude unchanged and thereby enables independent manipulations of harmonic phases and amplitudes. For experimental verification, a microwave prototype of varactor-based TDC digital metasurfaces controlled by FPGA was fabricated and characterized in [120]. Figure 7(e) displays the measured spectral phases and amplitudes at various harmonics, corresponding to three different voltage combinations V 1 /V 2 (applied to the entire metasurface). The amplitude of the +1 st harmonic is associated with the biasing voltage combinations, which can obtain a dynamic range of 25 dB by tuning the external biasing voltages [120]. Next, different time delays 0 and T 0 /2 are assigned to different columns of the metasurface elements for spatial coding "0" and "1," which can be used for beam shaping. Figure 7(f) shows three groups of measured scattering patterns at the +1 st harmonic frequency pertaining to three sets of spatial coding "00000000," "00001111," and "00110011," respectively. It can be obviously observed that the power intensities of scattering patterns are controlled by the voltage combinations V 1 /V 2 , while the beam shapes remain unchanged. The proposed approach has potential applications in wireless communication; more details will be discussed in Section 4.

Frequency Synthesis, Polarization Conversion, and
Convolution Operation. Some other interesting applications of the STC digital metasurfaces include frequency synthesis, polarization conversion, and convolution operation. In [121], an STC digital metasurface integrated with varactors and chip capacitors was proposed to synthesize the ±1 st harmonic frequencies of reflected waves with high efficiency. As conceptually illustrated in Figure 8(a), the reflection phase of the entire metasurface is periodically modulated with a sawtooth waveform in the temporal domain. In some application scenarios, unwanted highorder harmonics need to be suppressed to avoid spectrum pollution. Figure 8(b) shows the time-modulated waveform with 1-bit, 2-bit, and 3-bit quantized phases and continuous sawtooth wave phases, as well as the corresponding harmonic amplitude distributions. We notice that the amplitude of the +1 st harmonic gradually increases as the time-modulated waveform changes from 1-bit discrete phases to continuous phases during one period. In other words, higher quantization bits can be employed to suppress the undesired harmonic components. Actually, the 3-bit quantized phase can be used as a compromise solution for harmonic conversion with an efficiency of~95%. Besides, this high-efficiency harmonic conversion com-bined with the approach of time delay can also be exploited for some beam shaping applications. The measured results presented in [121] agree well with the theoretical analyses, which indicates that the conversion efficiency can reach~88% and the harmonic suppression ratio is~21 dB. This spatial frequency synthesizer based on the STC digital metasurface may find important applications in future wireless communication systems. More recently, some similar designs about frequency translation enabled by time-modulated metasurfaces have also been presented in [54][55][56].
The aforementioned STC and TDC digital metasurfaces can only work in the scenarios of single polarization. In Ref. [122], an anisotropic TDC digital metasurface working at dual polarizations was proposed to realize the programmable polarization conversions at both the fundamental and harmonic frequencies. By applying the theory of nonlinear harmonics introduced in Section 3.5, a general theory of polarization regulation was proposed to synthesize the linear and nonlinear polarization conversions, including the arbitrarily linearly polarized (LP), cross-LP, righthanded circularly polarized (RHCP), and left-handed circularly polarized (LHCP) reflected waves. Due to the dual-polarized structural design of programmable elements (see Figure 4(a) in [122]), the time-varying square wave reflection coefficients Γ xx ðtÞ and Γ yy ðtÞ along two orthogonal directions (i.e., the x-and y-polarized directions) of the metasurface can be independently controlled. As conceptually illustrated in Figure 8(c), the anisotropic TDC digital metasurface is normally illuminated by an incident wave with arbitrary linear polarizations (represented by the orange ray). By adjusting the phases of Γ xx ðtÞ and Γ yy ðtÞ, RHCP, arbitrarily LP, LHCP, and cross-LP reflected waves can be generated, as, respectively, represented by blue ray 1, green ray 2, blue ray 3, and green ray 4. Figure 8(d) shows a polarization conversion diagram with some specific collocations of different x-and y-polarized phases, from which RHCP, LHCP, and cross-LP are successfully synthesized. More theoretical results can be found in [122]. For experimental validation, a microwave prototype of the anisotropic TDC digital metasurface was manufactured. The measured results are in great agreement with the theoretical analysis, verifying the feasibility of the proposed polarization manipulation scheme.
A convolution theorem was first introduced in SDC digital metasurfaces for realizing flexible scattering pattern shifts [90], which can steer a beam pattern to an arbitrary spatial direction. In a recent study [123], a time-domain convolution theorem based on a TDC digital metasurface was proposed to achieve the scattering pattern shift at the harmonic frequencies, as illustrated in Figure 8(e). As previously mentioned, a time delay introduced in the time-domain reflection coefficient can result in an additional phase shift −2πkf 0 t 0 at the k th harmonic frequency [110,120]. Figure 8(f) illustrates an example of nonlinear convolution operation at the +1 st harmonic frequency, as it can be observed that the original dual-beam pattern deviates from the normal axis to the angle θ 2 with negligible distortion. The working process is the same as that in [90], and more results can be found in [123].

Harmonic Information Transitions and Independent
Control of Dual Harmonics. From the perspective of information technology, the STC digital metasurface can be regarded as an information processer, which can transform the spatiotemporally modulated signals to spatial-spectral information at various harmonics. In a recent study [124], the information transition mechanisms of STC digital metasurfaces were proposed, in which the idea of group extension and independent control of dual-harmonic waves are characterized as two major tools to analyze the harmonic information transitions. Figure 9(a) shows the conceptual illustration of manipulating wave transmission via an STC digital metasurface. Referring to [124] for details, the group extension mechanism is used to expand the phase states of coding elements via the factor q, which can realize more precise control of EM waves without the need for complicated metasurface design. To some extent, this idea is essentially similar to the concept of multibit programmable  Reproduced from [122]. (e) Conceptual illustration of TDC digital metasurfaces for convolution operation. Reproduced from [123]. (f) An illustrative example of nonlinear convolution operation at the +1 st harmonic frequency, as it can be observed that the original dual-beam pattern deviates from the normal axis to the angle θ 2 with negligible distortion. Reproduced from [123].
phases [114], as previously presented in Section 3.4. This analytical method of group theory provides a new perspective to understand equation (4) proposed in [110]. Besides, the independent control of the spectral responses of the STC digital metasurface is also demonstrated. Figure 9(b) shows the schematic of independent control of phase states at two frequencies, in which the same colored lines connect the phase states from the central frequency to the +1 st harmonic frequency. By using the permutation and translation operations in the same time-coding sequence [124], the phases of the output spectral responses can be independently controlled at two frequencies. Actually, the proposed permutation and translation, respectively, work the same as the initial phase and time delay introduced in the time-coding sequences [125], which will be discussed later in this section. Moreover, the information transition efficiencies of the STC digital metasurfaces regarding the aforementioned two mechanisms were also investigated by incorporating Shannon's entropy theory [124]. Figure 9(c) shows the numerically calculated information transition effi-ciency, pertaining to different lengths of the time-coding sequences and the number of input phase states. The theoretical results can be applied to predict the channel capacity of the STC digital metasurfaces, which offers potential guidance for future wireless communications.
In Ref. [125], an STC strategy with controllable initial phases and time delays was proposed to simultaneously control dual harmonics, leading to independent beam control at arbitrary two harmonics. Figure 9(d) shows a conceptual illustration of dual-harmonic manipulation based on an STC digital metasurface. By changing the initial phase ψ 0 and time delay t 0 in the time-modulated phase periodic function, one can independently manipulate beam shaping at two harmonic frequencies. Referring to [125] for more details, the theoretical derivations show that the initial phase ψ 0 has the same effect on the phases of all harmonics but the time delay t 0 induces different phase shifts with respect to harmonic orders, and neither of them has any effect on the amplitudes of harmonics. As an illustrative example, Figure 9(e) shows a spiral-like phase distribution at the +1 st harmonic frequency Length of temporal periodicity Reproduced from [124]. (c) Numerically calculated information transition efficiency, pertaining to different lengths of the time-coding sequences and the number of input phase states. Reproduced from [124]. (d) Conceptual illustration of dual-harmonic manipulation based on an STC digital metasurface. Reproduced from [125]. (e) A spiral-like phase distribution at the +1 st harmonic frequency and a diagonal gradient distribution at the +2 nd harmonic frequency, respectively. Reproduced from [125]. (f) The corresponding combinations of initial phases and time delays. Reproduced from [125]. (g) The corresponding numerically calculated 2D scattering patterns at the +1 st and +2 nd harmonic frequency, respectively. Reproduced from [125]. and a diagonal gradient distribution at the +2 nd harmonic frequency. Figure 9(f) shows the corresponding combinations of initial phases and time delays for this example. The phases of STC digital metasurfaces are temporally modulated by parameters in Figure 9(f), which generates a vortex beam at the +1 st harmonic frequency and a single beam with a deflection angle at the +2 nd harmonic frequency, as shown in Figure 9(g). These numerical examples were also validated by experiments, which demonstrated the effectiveness of dual-harmonic control, leading to potential applications in multiuser wireless communications.
3.8. Joint Multifrequency Beam Shaping and Steering. As previously discussed in Section 3.1, harmonic beam steering at multiple frequencies can be obtained by using brute-force numerical optimization (e.g., BPSO algorithm). However, the applicability of optimization methods is strictly limited by the computational complexity and nonlinear feature.
One limitation of conventional STC schemes is the inherent entanglement of multifrequency synthesis [110], which originates from equation (4) that the equivalent phases and amplitudes at each harmonic frequency are influenced by all the reflection coefficients in the time-coding sequences. Therefore, the basic STC schemes cannot attain the independent and simultaneous syntheses of the prescribed beam patterns at multiple frequencies. To overcome this limitation, a new coding strategy was put forward to achieve joint multifrequency beam control [126]. This strategy depends on the sophisticated time-coding sequences with properly designed and temporally intertwined subsequences, which can efficaciously disentangle the joint multifrequency syntheses. As illustrated in Figure 10(a), an STC digital metasurface combined with the new coding strategy can synthesize independently and simultaneously the desired scattering patterns (e.g., different beam numbers, directions, and shapes) at desired multiple frequencies. In [126], the reflection phases of the STC digital metasurface are quantized by the number S over the interval ð0, 360°Þ. Thus, for a 1-bit case (S = 2), we can obtain Γ ðnmÞ l ∈ f1,−1g corresponding to the digits f0, 1g, while for a 2-bit case (S = 4), we can obtain Γ ðnmÞ l ∈ f1, j,−1, −jg corresponding to the digits f0, 1, 2, 3g, and so forth. Figure 10(b) schematically illustrates the proposed strategy of intertwined coding subsequences. As assumed in [126], the problem is formulated as the joint Q-harmonic syntheses (i.e., orders ν = 0, 1, ⋯, Q − 1). To this end, each timecoding sequence is decomposed into Q sets of intertwined coding subsequences with the length P; thus, the total length of the time-coding sequence is L = PQ in one period T 0 . The time-coding sequence composed of intertwined subsequences can be expressed as where Ω pq are phase shift factors with the same quantization as the original reflection states Γ q+1 . Thus, the equivalent complex-valued amplitudes in equation (4) can be rewritten as where c ν = 1/L ⋅ sinc ðπν/LÞ exp ð−jπν/LÞ governs the decay of the spectrum, and α qν are regarded as digital filter terms and can be expressed as In order to disentangle the multifrequency syntheses, α qν need to be designed by judiciously selecting the phase shift factors Ω pq . Ideally, this would require α qν ∝δ qν withδ qν denoting a reduced Kronecker delta. In this case, each coding subsequence (represented by the same colored square in Figure 10(b)) only basically affects one specific harmonic, thereby realizing joint syntheses at multiple frequencies independently and simultaneously. Referring to [126] for details, this synthesis problem has a very simple solution when the number Q of harmonic frequencies of interest is not greater than the number S of quantized reflection phases, so we should choose P = S and Ω pq = exp ðj2πpq/SÞ in this case.
To show the potential of this new coding strategy, in [126], we consider a 2-bit (S = 4) STC digital metasurface with 16 × 16 programmable elements. For an exact closedform solution, we apply the proposed approach for joint syntheses at Q = 4 harmonic frequencies (i.e., ν = 0, 1, 2, 3) and assume that the length of each intertwined subsequence is P = 4; thereby, the total length of the time-coding sequence is L = PQ = 16 in one period. As an illustrative example, Figure 10(c) shows the synthesized phase patterns for the equivalent complex-valued amplitude a ðnmÞ ν at the first four harmonic frequencies, whereas Figure 10(d) displays the cor-responding 2D scattering patterns. It can be clearly observed that different scattering patterns at the four frequencies of interest are independently synthesized; more illustrative examples about diffuse and OAM-type scattering patterns can be found in [126]. For the experimental proof of concept, the previous prototype of a 2-bit STC digital metasurface in [113] was utilized again to validate the feasibility of the joint multifrequency syntheses. The measured patterns at different frequencies were in good agreement with the numerical simulations.
Overall, the proposed approach of joint multifrequency syntheses remarkably enhances the wavefront manipulation capabilities of STC digital metasurfaces, which paves the way for promising applications in wireless communication, radar, imaging, and some more advanced information systems. It is worth emphasizing that the proposed coding strategy inherits the character of programmability in the digital metasurface platform, and thus, it is expected to dynamically alter the synthesized scattering patterns or redistribute the harmonic spectrum "on demand," by changing the coding sequences preloaded in an FPGA.
3.9. New Architecture Wireless Communication Systems. One of the most exciting and intriguing applications of STC digital metasurfaces is the construction of new architecture wireless communication systems. As a branch of STC digital metasurfaces, the TDC digital metasurfaces only consider the temporal coding, in which no spatial coding is assumed. The above analyses in Section 3.5 show that TDC digital metasurfaces have powerful capabilities in accurately engineering the harmonic amplitudes and phases of the reflected EM waves. The TDC digital metasurface can process the digital information at the interface, in which the baseband signal can be modulated on the carrier wave directly without the need for digital-analog conversion and mixing processes. This direct modulation based on metasurfaces can be realized by establishing a mapping relation between the baseband signals and the time-varying reflection coefficients.
For instance, a new architecture wireless communication system with a binary frequency shift keying (BFSK) modulation scheme was proposed in [109]. The TDC digital metasurface can transfer the power from the carrier frequency to the ±1 st harmonic frequency by varying the time-coding sequences, thereby realizing the BFSK communication system. Figure 11(a) illustrates the metasurface-based transmitter of the BFSK wireless communication system. Referring to [109] for details, the transmitting process mainly contains three steps: firstly, the FPGA baseband module generates a bit stream of the original information (e.g., a photo); subsequently, the bit stream is mapped to different sets of timecoding sequences, which can produce the specific harmonic distributions required by the BFSK scheme; and finally, the incident wave is modulated by the time-coding sequences and the TDC metasurface transmits the modulated EM waves carrying the digital information. The measured results are illustrated at the bottom of Figure 11(a), in which a picture is successfully recovered by a software-defined radio (SDR) receiver with a message transmission rate of 78.125 kbps at the frequency of 3.6 GHz.
To further improve the message transmission rate, a quadrature phase shift keying (QPSK) modulation scheme was proposed for a wireless communication system [95]. This QPSK wireless communication system can smoothly transmit a video with a message transmission rate of 1.6 Mbps at 4 GHz. Figure 11(b) shows the indoor scenario of the QPSK system for video transmission and the corresponding measured constellation diagrams according to different communication distances. Moreover, some highorder modulation schemes such as 8PSK and 16QAM were presented in [96]. By regulating the reflection phases of the TDC digital metasurface, arbitrary constellation diagrams can be synthesized at the +1 st harmonic frequency. Figure 11(c) shows the block diagram of the communication transmitter based on the TDC digital metasurface and the constellation diagram of QPSK, 8PSK, and 16QAM. For experimental validation, a microwave prototype of the TDC digital metasurface working around the frequency of 4.25 GHz was fabricated, as shown in Figure 11(d).
Overall, numerical and experimental results demonstrate the good performance of the new architecture wireless communication systems, which significantly simplify the system architecture and have promising applications in the future 6G communication scenario.
3.10. Space-and Frequency-Division Multiplexing Wireless Communication Scheme. As previously mentioned, the TDC digital metasurfaces were effectively used to build the new architecture wireless communication systems, which support BFSK, QPSK, 8PSK, and 16QAM. However, spacedomain modulation is not implemented in such systems [95][96][97]109]. In fact, the TDC digital metasurfaces have the limitation of transmitting the same information to receivers at different spatial directions, in which case only the power   levels of the received signals are different. Therefore, undesired users at other locations can still recover the correct information by using a sufficiently sensitive receiver. The STC digital metasurface can lift this restriction by also implementing information encoding in the space domain. On the other hand, multiplexing techniques have been widely used to establish multiple independent channels between transmitters and receivers, which can improve the capacity of communications. Some multiplexing techniques have been developed in the past decades, such as time-division multiplexing (TDM), frequency-division multiplexing (FDM), space-division multiplexing (SDM), and code-division multiplexing (CDM). FDM requires high-performance filters and mixers to divide the frequency range. SDM is usually realized by using a phased array, which is composed of many antennas and radiofrequency (RF) chains, leading to a communication system with high cost and high complexity. The STC digital metasurfaces with simultaneous controls of the spatial and spectral characteristics can be used to overcome the aforementioned limitations and challenges, which have the advantages of low cost, simple structures, and easy fabrications. In [118], a new information encoding scheme based on the STC digital metasurface was proposed to implement both the space-and frequency-division multiplexing techniques in a multichannel wireless communication system. By encoding the optimized STC matrices (see Figure 2 in [118]), different digital messages can be directly transmitted to multiple users simultaneously and independently. As illustrated in Figure 12(a), a multichannel wireless communication system of direct data transmission is established by using an STC digital metasurface. In this case, different data streams are directly routed to designated users (e.g., Users #1, #2, #3, and #4) located in different directions. Each designated user has its own independent receiving channel via a specific frequency, while undesired users located in other directions cannot recover the correct information.
For experimental validation, a dual-channel wireless communication system based on a 2-bit STC digital metasurface was built to transmit two different pictures to two users, as shown in Figure 12(b). Referring to [118] for more details, the system transmitter mainly includes a control platform, a microwave signal generator, and the STC digital metasurface fed by a linearly polarized horn antenna, as shown in the left part of Figure 12(b). Accordingly, the receiving terminals mainly contain two horn antennas, down converters, the SDR receiver, and the postprocessing computer, as displayed in the right part of Figure 12(b). The process of dual-channel direct data transmissions was detailed in [118]. Figures 12(c)   transmission rate (see the inset in Figure 12(b)), validating the feasibility of the space-and frequency-multiplexed information encoding scheme. More results of the dualchannel system with arbitrary user locations as well as the three-channel system can be found in [118]. The STC digital metasurface simultaneously plays the role of information modulation and energy radiation. The STC strategy provides a low-cost solution for realizing SDM and FDM wireless communication, eliminating the requirement for antenna arrays, filters, and mixers. This new wireless communication scheme based on STC digital metasurfaces has low interference between different user channels and also offers the characteristic of directional modulation for secure transmission, which has important applications in secure communications, multicarrier communications, frequencyhopping communications, and future radar systems.

Conclusions and Perspectives
In conclusion, the STC digital metasurfaces provide versatile and powerful platforms for implementing spatiotemporal modulations, which enable advanced EM field manipulations in both the space and frequency domains and open up a new degree of freedom to expand the functionality of information metasurface systems. Owing to the huge advantages of the STC strategy, wave manipulation and information processing can be simultaneously performed in a multidimensional domain, leading to numerous applications in the fields of electromagnetics and information science. In this review article, we have summarized the general concepts and working principles of STC digital metasurfaces and presented their recent progress and representative applications, focusing particularly on the harmonic beam steering/shaping, scattering-signature reduction, programmable nonreciprocal effect, arbitrary multibit programmable phases, nonlinear harmonic manipulations, frequency synthesis, polarization conversion, convolution operation, joint multifrequency syntheses, harmonic information transitions, and new architecture wireless communication systems.
Among the most interesting applications of the STC digital metasurfaces, it is worth mentioning the reconfigurable intelligent surfaces (RISs) [127][128][129][130][131], which are also widely referred to as intelligent reflecting surfaces (IRSs) [132,133]. RISs are currently attracting a lot of attention for the next-generation 6G wireless communication networks. By deploying RISs in the wireless propagation environment, the wireless channels can be reprogrammable and controllable in time, leading to the smart radio environments [128,133]. RISs help improve the signal-to-noise ratio (SNR) via beamforming [132] and achieve signal coverage enhancement or interference suppression at designated receivers. By installing the low-cost and energy-efficient RISs on building walls, transmitted signals will be reflected by the RISs and then propagated along designated directions by controlling the elements' phases and/or amplitudes, thereby establishing supplementary links for wireless communications and information/power transfer systems [128,133]. Within this overarching framework of RIS-assisted wireless networks, the STC digital metasurfaces provide a possibility to remove the complex modulation/encoding and RF processing modules that are usually required in the traditional wireless communication systems and enable a simple implementation of diversity and multiplexing in terms of frequency, space, time, polarization, and pattern [118]. Other promising applications inspired by the STC digital metasurfaces include the next-generation information systems of cognitive radars, microwave computational imaging, smart sensing/recognition, and advanced mathematical operations. Besides, more information theory and signal processing methods can be explored to analyze the physical intentions behind STC coding metasurfaces, which is expected to provide guidance for future information systems.
However, there also exist some challenges that need to pay attention to. The current STC digital metasurfaces are experimentally realized with only phase modulations, where the amplitude and phase-amplitude modulations have not yet been achieved in practice. Thus, one important challenge is the joint phase-amplitude modulation scheme, which is difficult to realize in the practical design of the programmable metasurfaces but has great potential for improving the spectral efficiency of multiple harmonic frequencies [126]. In addition, the current reflection-type STC digital metasurface may encounter the issue of blockage caused by the feeding antenna in some specific scenarios. To avoid the blockage effects, one could design a transmission-type STC digital metasurface with the feeding antenna behind it [76] or a waveguide-fed metasurface architecture [134,135].
Another critical aspect is about the modulation mechanism and switching speed of the programmable character. Most experimental realizations of STC digital metasurfaces to date are based on PIN or varactor diodes, which have limited switching speed and are restricted to the microwave frequency band. In some applications, such as the scenario of the nonreciprocal effect illustrated in Section 3.6, it is crucial to improve the modulation frequency (switching speed) for attaining enough angular separation in the space domain. So far, some faster switching schemes such as graphene [34,136], vanadium dioxide [137,138], and 2D electron gas [139] have been explored for phase/amplitude modulations in the metasurface applications to reach higher-frequency ranges (e.g., terahertz) [140], which are promising for implementing faster spatiotemporal modulation and paving the way for more advanced applications in the future.

Conflicts of Interest
The authors declare no conflicts of interest.