Issue 
EPJ Appl. Metamat.
Volume 7, 2020



Article Number  7  
Number of page(s)  14  
DOI  https://doi.org/10.1051/epjam/2020007  
Published online  11 December 2020 
https://doi.org/10.1051/epjam/2020007
Review
Recent advances and perspectives on spacetime coding digital metasurfaces
^{1}
State Key Laboratory of Millimeter Waves, Southeast University, Nanjing 210096, PR China
^{2}
Fields & Waves Lab, Department of Engineering, University of Sannio, Corso Garibaldi 107, 82100, Benevento, Italy
^{*} email: tjcui@seu.edu.cn
^{**} email: vgaldi@unisannio.it
Received:
14
October
2020
Accepted:
13
November
2020
Published online: 11 December 2020
Within the overarching framework of spacetime metastructures, digital metasurfaces based on spatiotemporal coding are emerging as powerful and versatile architectures for complex field manipulations, also in view of their inherently programmable nature. Here, we provide a compact survey of our recent results and ongoing studies in this research area. Examples of field manipulations include harmonic beam steering and/or shaping and programmable nonreciprocal effects. Possible applications are abundant and range from wireless communications to radars and imaging.
Key words: Metasurfaces / spacetime / digital / coding / communications / radars
© L. Zhang et al., published by EDP Sciences, 2020
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The concept of “digital” metastructures was put forward in 2014, independently, by the Groups led by Profs. Della Giovampaola and Engheta [1] and Cui et al. [2]. The basic idea underpinning these types of artificial materials is the reliance on a limited number of elementary inclusions (in the most extreme case, only two), and yet the ability to engineer a variety of complex field manipulations. Whenever applicable, this approach greatly simplifies the design process, as the search space can be effectively explored via discreteoptimization strategies [3]. Moreover, it streamlines the fabrication process, also improving the robustness with respect to the unavoidable tolerances.
Referring to an electromagnetic (EM) metasurface scenario [2], one may think of a set of reflecting elements, such as metallic patches laid on a grounded dielectric substrate, ideally characterized by a unitamplitude reflection coefficient and a quantized phase response that can be univocally associated with a digital coding, for example, 0° and 180° in the simplest binary case, 0°, 90°,180° and 270° in a 2bit case, and so on. In this way, each possible spatial combination of these elements can be equivalently described in terms of a digital coding sequence. To some extent, this concept can also be interpreted as a generalization of the socalled “checkerboard” metasurfaces, featuring periodic distributions of metallic and artificialmagneticconductor elements [4].
The advantages of a digital coding representation become even more evident in the presence of reconfigurable elements which, via the integration of active elements such as diodes or microelectromechanical systems, can implement all required states. In this way, by controlling independently the coding elements via an electronic circuit such as a fieldprogrammable gate array (FPGA), it is possible to attain a completely “programmable” metasurface platform [2]. In such scenario, the digital coding sequence defines a desired field manipulation, in analogy with a software application defining a desired functionality in a computer or a portable electronic device.
This is an extremely fascinating concept, with a wealth of interesting implications and ramifications, which has found a broad variety of applications. At microwave frequencies, particularly worth of mention are those to dynamical beam/polarization manipulations [5,6], reprogrammable holograms [7] and imaging [8], independent control of copolarized amplitude and phase responses [9], scatteringsignature reduction [10,11], and direct transmission of digital messages [12]. Applications to terahertz frequencies have also been actively pursued [13], as well as transmissiontype [14], transmissionreflectionintegrated [15], and acoustic [16] scenarios. Perhaps, the most thoughtprovoking aspect of digital coding metastructures is the bridging between wave physics and information science [17,18]. This leads to a grand vision of “information metastructures” which leverages the programmable nature to implement softwaredefined, selfadaptive, and cognitive functionalities, possibly powered by artificial intelligence [19,20].
In conventional digital coding metasurfaces, the coding sequence is typically fixed in time and changed only whenever there is a need to switch among different functionalities. However, with the mounting interest in spacetime metastructures [21–24] characterized by spatially and temporally variant properties, it makes sense to exploit the temporal dimension as well, by applying a dynamic switching of the coding sequence. This is the key idea behind spacetime coding digital metasurfaces [25], which extend and generalize the concepts of timemodulated arrays [26] and phaseswitched screens [27,28], and have been successfully applied to several scenarios including harmonic beam steering and shaping [29], nonreciprocal effects [30], harmonic manipulations [31–33], scatteringsignature control [34], wireless communications [35,36], and analog computing [37].
In what follows, we provide a survey of our previous results and ongoing investigations in this fastpacing research area. Accordingly, starting from the basic concepts and related theoretical aspects (Sect. 2), we then illustrate some recent advances and applications (Sect. 3), and finally discuss the challenges and perspectives (Sect. 4).
2 Basic idea and theory of spacetime coding digital metasurfaces
Referring to [25] for more details, we briefly summarize the concept of spacetime coding digital metasurfaces, which is schematized in Figure 1a. We consider a digital, programmable metasurface comprising a 2D arrangement of M × N identical elements (yellow patches) whose reflection coefficients can be controlled electronically via positiveintrinsicnegative (PIN) diodes so as to assume a unit amplitude and a set of quantized phases. In the simplest binary case, the response is switched via a diode between two possible states (e.g., inphase and outofphase reflection), associated with the 0/1 bits. We assume that the switching is controlled in space and time by an FPGA, according to a given coding (represented by the 3D matrix of red and green dots).
In [25], an approximate analytical model was put forward for the scattered field due to a normally incident timeharmonic illumination with exp(j2πf _{ c } t _{ c }) dependence. Such model essentially relies on an adiabatic approximation of the physicaloptics model developed in [2] for conventional (timeinvariant) digital coding metasurfaces, and holds for modulation frequencies much smaller than the EM frequency f _{ c } [38,39]. Accordingly, the farfield timedomain scattered field is expressed as(1)with E(θ, ϕ) denoting the codingelement farfield pattern at the center frequency f _{ c }, θ and ϕ the conventional spherical angles, d _{ x } and d _{ y } the interelement spacings along the x and y directions, respectively, and the center operational wavelength (with c being the corresponding speed of light). Moreover, Γ _{ pq }(t) denotes the timemodulated reflection coefficient of the generic (p, q)th element, for which we assume a periodic behavior with period T _{0}, viz.,(2)where denotes a periodic rectangularpulse function, defined in each period as(3)
L is the length of the timecoding sequence, and is the reflection coefficient (computed at the center frequency) pertaining the (p,q)th coding element within the interval (n − 1)τ ≤ t ≤ nτ. By expanding in a Fourier series the timemodulated reflection coefficient in equation (2), and substituting it in equation (1), the farfield scattering patterns for the generic mth harmonic frequency f _{c} + mf _{0} can be expressed as(4)with the equivalent complex amplitudes given by(5)
Interestingly, these equivalent amplitudes are linear combinations (with complexvalued coefficients) of the quantized reflection coefficients, and thus may exhibit some finer dynamics. For instance, considering a 1bit (binary) coding, it is actually possible to synthesize equivalent amplitudes characterized by four quantized phase states, that is, an equivalent 2bit coding, and so on (see [25] for details). However, it is apparent from equation (5) that this reasoning does not apply at the center frequency (m = 0), at which only the original coding states can be synthesized.
In essence, the design procedure entails the optimization of the spacetime coding in equation (2) in order to synthesize the desired equivalent amplitudes in equation (5), so as to perform simultaneous field manipulations in both space and frequency domains, as described by equation (4).
In what follows, we will illustrate several representative examples of application.
Fig. 1 Harmonic beam steering (adapted from [25]). (a) Conceptual illustration. (b) Optimized spacetimecoding matrix. (c, d) Equivalent amplitudes and phase distributions, respectively, at the various harmonic frequencies. (e, f) Simulated and measured, respectively, scattering patterns at various harmonic frequencies. 
3 Survey of recent advances and applications
3.1 Harmonic beam steering/shaping
An interesting application of the above theory is harmonic beam steering, intended as the deflection of different scattered harmonics toward specific directions in space (see the conceptual illustration in Fig. 1a) [25]. In principle, this can be attained via a simple timegradient scheme (see, e.g., Figs. 2a and 2b in [25]), which induces different phasegradient distributions in the equivalent amplitudes in equation (5). However, this simple scheme also tends to generate imbalanced amplitude distributions [25]. To overcome this limitation, Figure 1b shows a spacetime 1bit coding matrix optimized (via a binary particleswarm algorithm [40]) so as to equalize the amplitude of the scattered harmonics. In this specific case, an 8 × 8element metasurface is considered, and timecoding sequences have a length of L = 10. Moreover, since the desired steering effect is along one dimension only, the same timecoding sequence is applied to all elements in each column, so that the 3D coding matrix can be effectively flattened to a 2D form. For an interelement spacing , Figures 1c and 1d show the corresponding equivalent amplitude and phase distributions, respectively, from which the amplitude equalization and the appearance of phase gradients at the harmonic frequencies are evident. As anticipated, we note that, in spite of the 1bit coding utilized, the equivalent phase distributions exhibit a finer (2bit) quantization, with the exception of the center frequency. This results in the numerically computed scattering patterns are shown in Figure 1e, which display the expected steering effects and equalized amplitudes.
To experimentally validate these results, a microwave prototype operating at a center frequency of 10 GHz was fabricated and characterized. Referring to reference [25] for details, the coding elements consist of metallic patches laid on a grounded dielectric substrate, with an embedded PIN diode to switch between the 1/0 bits (i.e., out of phase responses). Along the lines of the simulated example, the coding elements are grouped in eight columns (each containing eight elements) connected to common biasing lines, and controlled by an FPGA. A modulation frequency f _{0} = 0.5 MHz is assumed, corresponding to a PINdiode switching speed of 5 MHz. Figure 1f shows the measured scattering patterns, which are in fairly good agreement with the simulated results in Figure 1e.
As can be observed from the above example, working with a 1bit coding does not allow to attain beam steering at the center frequency. As previously mentioned, and better detailed hereafter (see Sect. 3.3), this restriction can be lifted by adopting higherbit coding schemes (e.g., 2bit).
In principle, the strategy utilized for steering can also be exploited to realize beam shaping at the harmonic frequencies [25], although there are certain inherent limitations in the joint multifrequency synthesis (see also the discussion in Sect. 4 below). Other interesting applications include the radarcrosssection reduction/control by redistributing the scattered power in both space and frequency domains.
Overall, the above results hold interesting promises for a variety of application scenarios, including wireless communications, radars, and holographic imaging.
Fig. 2 Nonreciprocal reflection effects (adapted from [30]). (a) Conceptual illustration. (b) Spacetimecoding matrix. (c, d) Equivalent amplitudes and phase distributions, respectively, at the various harmonic frequencies. 
3.2 Nonreciprocal reflection effects
One of the most interesting and promising applications of spacetime metastructures is the possibility to break Lorentz reciprocity without the need of magnetic or nonlinear materials. Theoretical [41–43] and experimental [44–46] studies have shown the possibility to attain nonreciprocal reflection or transmission effects via a metasurface imparting a suitable spacetime phase gradient. In a recent study [30], this concept was implemented and demonstrated by relying on a spacetime coding digital metasurface. Referring to Figure 2a for illustration, an architecture conceptually similar to that in Figure 1a is considered, but now assuming a 2bit coding, that is, a fourlevel quantization of the phase response (e.g., 0, 90°, 180°, 270°) associated to the digits 0, 1, 2, 3. Figure 2b shows the spacetime coding matrix that implements the discretized spacetime phase gradient, assuming a 16 × 16element metasurface, and a coding sequence of length L = 4; also in this case, in view of the 1D character of the steering effects, the same coding sequence is applied to all elements in each column. As can be observed from the equivalent amplitude and phase distributions shown in Figure 2c and 2d, respectively, such spacetime coding essentially induces an anomalous reflection, with most of the power concentrated in the m = 1 harmonic order, which also exhibits an effective phase gradient(6)with d = d _{ x } = d _{ y } denoting the constant interelement spacing. Accordingly, as conceptually illustrated in Figure 2a, assuming a plane wave illumination at frequency f _{1} = f _{ c }, impinging obliquely at an angle θ _{1}, the dominant scattering contribution will be at a different frequency f _{2} = f _{ c } + f _{0} and at an angle [30](7)
If now one considers the timereversal channel (see Fig. 2a), assuming an incidence at frequency f _{2} and angle θ _{2}, the scattered field does not follow the original forward incidence channel, but it is instead characterized by a different frequency f _{3} = f _{ c } + 2f _{0} and a different angle [30](8)
In other words, for oblique incidence (θ _{1} ≠ 0), the timereversal reflected channel and the forwardincidence one are isolated both spectrally and spatially. It is clear from equations (7) and (8) that the angular separation depends critically on the ratio . To give an idea, Figures 3a and 3b show some representative results (forward and timereversal channel, respectively) numerically computed by assuming f_{c} = 5 GHz, f _{0} = 250 MHz, and θ _{1} = 60°; in this case, a sizable angular separation of ∼9° is obtained. However, the assumed modulation frequency corresponds to a diode switching frequency of 1 GHz, which is not feasible with currently available PIN diodes. Figures 3c and 3d show the results for a more realistic parameter configuration (fc = 9.5 GHz, f _{0} = 1.25 GHz, that is, a diode switching speed of 5 MHz), for which the angular separation is <1°. Although such angular separation could not be resolved with the available measurement setup, it was still possible to validate experimentally the frequency separation. To this aim, a microwave prototype with coding elements embedding two PIN diodes was fabricated, so as to attain the required four quantized phase responses (see [30] for details). Figures 3e and 3f show the measured scattering patterns pertaining to the forwardincidence and timereversal channel, respectively, from which a good agreement with the theoretical predictions can be observed.
The above results may found interesting applications to wireless communications and radar systems, for separating the transmitting and receiving channels. It is also worth highlighting that, thanks to the inherently programmable character of the architecture, nonreciprocal effects can also be realized “ondemand.”
Fig. 3 Nonreciprocal reflection effects (adapted from [30]). (a, b) Numerically computed scattering patterns (forwardincidence and timereversal channel, respectively) for the dominant harmonic orders, by assuming f_{c} = 5 GHz, f _{0} = 250 MHz, and θ _{1} = 60°. (c, d) Same as above (but including higherorder harmonics) for a more realistic parameter configuration featuring f_{c} = 9.5 GHz, f _{0} = 1.25 MHz, d = 14 mm and θ _{1} = 34°. (e, f) Corresponding measured results. 
3.3 Arbitrary multibit programmable phases
As previously mentioned, higherbit programmable metasurfaces exhibit lower quantization phase error and therefore enable a more precise control of EM waves. For instance, in reference [25], it was shown that a 2bit coding could be effectively exploited to realize beam steering and beam shaping at the center frequency. So far, 1bit and 2bit programmable metasurfaces have been successfully realized, but it is very challenging to attain higherbit programmable phases via PINdiodebased metasurfaces. Alternative switching strategies (based, e.g., on varactors, MEMS, graphene, liquid crystals) also exhibit limited phase coverage and cannot realize continuous 360° phase modulation.
Recently, in [29], a 2bit programmable metasurface combined with a timecoding approach was proposed, which can attain arbitrary multibit and even quasicontinuous programmable phases, as schematically illustrated in Figure 4a. Specifically, a method of vector synthesis (illustrated in Fig. 4b) was put forward to analyze and synthesize the phase coverages. For instance, 4bit and arbitrarily higherbit phases can be synthesized via a 2bit coding metasurface, simply by manipulating the timecoding sequences. Several numerical examples of beam steering, at both the center and harmonic frequencies, were illustrated to demonstrate the effectiveness of the proposed approach. Figures 4c and 4d show that the phase coverage at the center frequency and the first harmonic frequency can reach 360° with high amplitudes by suitably designing the 2bit timecoding sequences. For experimental validation, an FPGAcontrolled prototype working at microwave frequencies was fabricated and experimentally characterized (see [29] for details). Overall, this approach provides a simple way to design arbitrary multibit programmable metasurfaces without the requirement of complicated structural design and control systems. This may find important applications to highperformance antennas, highresolution imaging, as well as highcapacity wireless communications and radar systems.
Fig. 4 Arbitrary multibit programmable phases (adapted from [29]). (a) Conceptual illustration. (b) Basic illustration of the vector synthesis method in the complex plane for the generation/analysis of multibit phases. (c, d) Equivalent phase coverage at the center angular frequency ω _{ c } = 2πf _{ c } and the first harmonic ω _{ c } + ω _{0} = 2π (f _{ c } + f _{0}), respectively, for a 2bit timecoding sequences with length L = 8. 
3.4 Nonlinear harmonic manipulations via time coding
Other interesting timevarying architectures can be implemented via the use of varactor (instead of PIN) diodes, which enables more flexibility in the modulation waveform. For instance, in reference [32], a time coding metasurface was proposed that possesses powerful capabilities in manipulating the spectral distribution of EM fields, as conceptually illustrated in Figure 5a. In this prototype study, only temporal coding was considered, assuming a spaceinvariant temporal modulation of the reflection coefficient according to a periodic function Γ (t), with period T _{0}. Accordingly, assuming timeharmonic planewave illumination at frequency f _{ c }, the general theory in equation (1) can be particularized to express the spectrum of the reflected wave E _{ r } in terms of the incident one E _{ i } as follows [32],(9)where δ (⋅) denotes the Dirac delta function, ^{*} is the convolution operation, and a _{ k } represents the complex Fourier coefficient of the kthorder harmonic. As previously mentioned, although the reflection coefficient is defined for a linear timeinvariant system, equation (9) is approximately valid in the adiabatic limit f _{0} ≪ f _{ c } [38,39]. From equation (9), it is possible, in principle, to attain arbitrary harmonic distributions by judiciously designing the temporal modulation for the reflection coefficient Γ (t), which opens up interesting pathways in the spectral shaping of EM waves.
Figure 5b illustrates a practical implementation of a coding element, featuring two rectangular patches linked by a varactor diode acting as a tunable element, controlled via a biasing voltage [32]. Accordingly, narrow slots are etched on the bottom plate so as to bias the diodes through metallic vias. Also in this architecture, an FPGAbased control platform is exploited to generate the required driving signals so as to implement different functionalities in the reflectedwave spectrum.
Figures 5c–5f illustrate some representative experimental results obtained by means of a microwave prototype (see [32] for details). Specifically, Figure 5c shows the measured spectral distributions pertaining to a binary phase modulation associated to the timedomain digital coding 01010101…, with the “0” and “1” bits denoting 0° and 180° phase levels, respectively (corresponding to biasing voltages of 0 and −9 V). Consistently with the theoretical predictions from equation (9), most of the impinging power (at the center frequency) is transferred to higherorder harmonics (most notably, the orders +1 and −1); the small residual at the center frequency is attributable to modeling approximations and uncertainties, as well as fabrication tolerances. As can be observed from the measured scattering patterns in Figure 5e, the harmonic orders +1 and −1 exhibit similar shapes and comparable amplitudes. Figure 5d shows instead the spectral distribution pertaining to a 2bit coding, with four digital states “00”, “01”, “10” and “11” corresponding to reflection phases 0°, 90°, 180° and 270°, respectively (obtained via biasing voltages 0, −6, −9 and −21 V). In this case, due to the timegradient enforced, the spectral symmetry is broken and most of the power is transferred to the harmonic order +1. As apparent from the scattering patterns shown in Figure 5f, the harmonic orders +1 and −1 maintain similar shapes, but now the amplitudes are significantly different.
The above examples illustrate the potential of the proposed approach in controlling the spectral distribution of the reflected EM waves. This concept may find important applications to the development of simplified and compact communication and radar systems, and can be extended from the microwave range to the optical regime and, in principle, to acoustic scenarios.
Fig. 5 Nonlinear harmonic manipulation via time coding (adapted from [32]). (a) Conceptual illustration. (b) Schematic of a coding element featuring a varactor diode. (c, c) Examples of measured spectral distributions, for 1bit and 2bit coding, respectively. (e, f) Corresponding scattering patterns (theoretical and measured) for the harmonic orders +1 and −1. 
3.5 Independent control of harmonic amplitudes and phases
A major limitation of spacetime coding digital metasurfaces stems from the inherent coupling effects between the amplitude and phase responses at the various harmonic orders, as can be understood from the expression of the equivalent complexvalued amplitudes in equation (5). In [31], an analytical method was proposed for independent control of the harmonic amplitudes and phases via spacetime coding. Referring to the conceptual illustration in Figure 6a, the reflection phase of the coding element is temporally modulated with a periodic square wave between two selected phase states φ _{1} and φ _{2}; moreover, for a more flexible control, a time delay t _{0} is introduced in the reflection coefficient. For simplicity, no spatial coding is assumed for now. Accordingly, the reflectedwave spectrum in equation (9) can be rewritten as(10)with A denoting a constant reflection amplitude. From equation (10), we notice that the amplitude of the harmonics can be tuned by acting on the phase difference (φ _{1} − φ _{2}). Moreover, the time delay t _{0} offers an additional degree of freedom to tailor the phase response, while maintaining the amplitude unchanged, thereby enabling independent control. In addition, since the same time delay yields different phase lags at various harmonics, it can also be exploited to realize simultaneous multiorder harmonic manipulations.
Figure 6b illustrates the geometry of a coding element, featuring rectangular patches loaded via four varactor diodes; by suitably changing the biasing voltages, the phase response can be tuned over a broad range (∼270°) [31]; once again, an FPGA controller is utilized to generate the required coding signals. A prototype working at microwave frequencies was fabricated and characterized for experimental validation. Referring to reference [31] for details, Figure 6c shows the measured amplitudes and phases at various harmonic orders, for three different combinations of the biasing voltages (applied to all coding elements); for simplicity, the varactors in each column share the same biasing voltage, and therefore there are only two control voltages. As can be observed, it is possible to manipulate the amplitude and phase responses in the spectrum by varying the time coding. Next, spatial coding is introduced by independently modulating the columns of the metasurface elements, thereby enabling some beamshaping capabilities. For illustration, Figure 6d shows three representative patterns measured at the harmonic order +1, corresponding to different spatial codings. As can be observed, by acting on the time coding, it is possible to control the scattering amplitude, while maintaining the beam shape. Finally, as illustrated in Figure 6e, by introducing and controlling a time delay t _{0} in the reflection coefficient, it is possible to tailor the scattering patterns at different harmonic orders, in good agreement with the theoretical predictions [31].
The above results open up interesting possibilities for harmonic manipulations, with potential applications ranging from wireless communications to radars.
Fig. 6 Independent control of harmonic amplitudes and phases (adapted from [31]). (a) Conceptual illustration. (b) Schematic of the coding element featuring four varactor diodes. (c) Measured amplitudes and phases at various harmonic orders, for different combinations of the biasing voltages. (d) Measured scattering patterns at the harmonic order +1, for three different spatial codings; patterns of different colors illustrate the possibility to independently control the amplitude by acting on the temporal coding. (e) Representative examples of multiorder harmonic tailoring of the scattering patterns (measured and theoretical). 
3.6 Applications to wireless communications
The capabilities enabled by spacetime coding digital metasurfaces in terms of manipulating the spatial and spectral distributions of EM waves can be exploited to directly embed digital information in microwave carrier signals, without the need of conventional radio frequency circuits. This sets the stage for novel architectures of wireless communication systems [32,35,36,47–49].
For instance, we observe from Figures 5c and 5d that the power can be efficiently transferred from the center frequency to the harmonic orders +1 and −1 via suitably designed temporal codings. One may interpret the incident wave and the timevarying reflection coefficient as the carrier wave and baseband signal, respectively, of traditional communication systems. Accordingly, direct modulation can be attained by establishing a suitable mapping between the baseband information and the reflection coefficient.
Figure 7a shows the schematic of a transmitter based on a time coding metasurface, which was proposed in [32]. In this case, a binary frequency shift keying (BFSK) modulation scheme is implemented, which exploits a pair of discrete frequencies to represent the binary digits “0” and “1”. As a first step, the FPGA controller generates a bit stream of the information that needs to be transmitted (a picture, in the schematic). Subsequently, the bit streams are mapped onto corresponding time coding sequences, which are repeated periodically so as to produce a pair of discrete frequencies according to the BFSK scheme. In this way, the digital information is directly embedded in the reflected EM wave, and hence transmitted. Figure 7b shows a microwave prototype that was fabricated and tested for experimental validation (see [32] for details). Figures 7c and 7d show some representative measurements results, demonstrating the possibility to transmit and correctly receive a high resolution picture.
Another interesting example was proposed in reference [35], implementing the quadrature phase shift keying (QPSK) modulation scheme. Referring to [35] for details, the modulated reflection coefficient can be written as the superposition of scaled and shifted pulses(11) (12)where g (t) represents a basic pulse shaping function, T is the period of message symbol, M is a set of constellation points with cardinality M = 4, and Γ _{ m } is the reflection coefficient during one message symbol. The constellation points in M are associated to 2bit digital states “00”, “10”, “11”, and “01”, respectively. By comparison with the previous BFSK system, the QPSK system is more efficient in the information modulation due to the shorter length of the time coding sequence required during each message symbol, and can therefore reach higher transmission rates.
Also in this case, a proofofconcept validation was carried out via a microwave prototype (see [35] for details). Figure 8a shows a photograph of the realized QPSK system for data transmission in an indoor scenario. Figure 8b shows the corresponding received constellation diagrams, for different transmission distances. Realtime video transmission was demonstrated with a high data rate of about 1.6 Mbps.
Overall, the above results confirm the practical feasibility of wireless communication systems based on timecoding metasurfaces, which offer significant simplifications in the architecture and reduced power consumption, without sacrificing the system performance.
Fig. 7 BFSK transmission system based on spacetime coding digital metasurfaces (adapted from [32]). (a) Conceptual illustration. (b) Photographs of the prototype and measurement setup. (c, d) Measurement results with receiving angle of 0° and 30°, respectively. 
Fig. 8 QPSK transmission system based on spacetime coding digital metasurfaces (adapted from [35]). (a) Photograph of prototype and measurement setup in an indoor scenario. (b) Received constellation diagrams for different transmission distances. 
4 Conclusions and outlook
In summary, digital programmable metasurfaces constitute a very powerful and versatile platform for implementing spatiotemporal modulation schemes, which significantly expand and generalize concepts such as timemodulated arrays and phaseswitched screens. In particular, the temporal dimensionality represents a crucial addition within the emerging paradigm of “information metastructures” [19,20], enabling advanced (possibly nonreciprocal) fieldmanipulation capabilities jointly in the spectral and spatial domains. In this paper, we have reviewed some recent advances and applications, including, harmonic beam steering/shaping, nonreciprocal effects, harmonic manipulations, and wireless communications.
Among the most interesting perspectives for these concepts it is worth mentioning the field of “intelligent reflecting surfaces”, which is establishing itself as a pillar for future wireless communication networks [49–53]. Within this framework, spacetime coding digital metasurfaces can effectively replace the required decoding, encoding, and radiofrequency processing blocks, also enabling softwaredefined, selfadaptive, and cognitive functionalities. Further flexibility may be endowed by transmissiontype and transmissionreflectionintegrated platforms, that are currently being explored, together with hybrid (phase/amplitude) modulation schemes.
Other interesting and promising application scenarios include nextgeneration radars, scatteringsignature reduction/control, imaging and processing systems. Also of great interest are applications to terahertz frequencies and other physical domains (e.g., acoustics).
As previously mentioned, one limitation of the basic harmonic beam steering/shaping scheme presented in Section 3.1 is the inherently entangled character of multifrequency syntheses. This stems from the fact that single codingelement response generally affects all harmonic orders, as apparent from the equivalent amplitudes in equation (5). We have shown (see Sect. 3.5) a possible strategy for independent control of amplitude and phase, which enables some multiorder harmonic manipulations. We are currently working on the implementation of more advanced spacetime coding schemes [54], whereby the scattering patterns at different harmonics can be synthesized simultaneously and independently. These schemes essentially rely on more sophisticated temporal coding sequences, which exploit digitalfiltering concepts to separately control the scattering responses at the various harmonic orders. Preliminary numerical and experimental results look encouraging.
Another critical aspect is related to the modulation frequency, which is inherently limited by the switching speed of the diode element. For instance, in the nonreciprocalreflection scenario illustrated in Section 3.2, the reliance on PIN diodes severely curtailed the angular separation between the forwardincidence and timereversal channels, thereby preventing a complete experimental observation of the phenomenon (see Figs. 3e and 3f). It appears therefore crucial to implement faster switching schemes, also with a view toward applications to higher frequencies (e.g., terahertz). Within this framework, emerging platforms based on graphene [55–57] and vanadium dioxide [58,59] seem very promising [60].
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Cite this article as: Lei Zhang, Jun Yan Dai, Massimo Moccia, Giuseppe Castaldi, Tie Jun Cui, Vincenzo Galdi, Recent advances and perspectives on spacetime coding digital metasurfaces, EPJ Appl. Metamat. 7, 7 (2020)
All Figures
Fig. 1 Harmonic beam steering (adapted from [25]). (a) Conceptual illustration. (b) Optimized spacetimecoding matrix. (c, d) Equivalent amplitudes and phase distributions, respectively, at the various harmonic frequencies. (e, f) Simulated and measured, respectively, scattering patterns at various harmonic frequencies. 

In the text 
Fig. 2 Nonreciprocal reflection effects (adapted from [30]). (a) Conceptual illustration. (b) Spacetimecoding matrix. (c, d) Equivalent amplitudes and phase distributions, respectively, at the various harmonic frequencies. 

In the text 
Fig. 3 Nonreciprocal reflection effects (adapted from [30]). (a, b) Numerically computed scattering patterns (forwardincidence and timereversal channel, respectively) for the dominant harmonic orders, by assuming f_{c} = 5 GHz, f _{0} = 250 MHz, and θ _{1} = 60°. (c, d) Same as above (but including higherorder harmonics) for a more realistic parameter configuration featuring f_{c} = 9.5 GHz, f _{0} = 1.25 MHz, d = 14 mm and θ _{1} = 34°. (e, f) Corresponding measured results. 

In the text 
Fig. 4 Arbitrary multibit programmable phases (adapted from [29]). (a) Conceptual illustration. (b) Basic illustration of the vector synthesis method in the complex plane for the generation/analysis of multibit phases. (c, d) Equivalent phase coverage at the center angular frequency ω _{ c } = 2πf _{ c } and the first harmonic ω _{ c } + ω _{0} = 2π (f _{ c } + f _{0}), respectively, for a 2bit timecoding sequences with length L = 8. 

In the text 
Fig. 5 Nonlinear harmonic manipulation via time coding (adapted from [32]). (a) Conceptual illustration. (b) Schematic of a coding element featuring a varactor diode. (c, c) Examples of measured spectral distributions, for 1bit and 2bit coding, respectively. (e, f) Corresponding scattering patterns (theoretical and measured) for the harmonic orders +1 and −1. 

In the text 
Fig. 6 Independent control of harmonic amplitudes and phases (adapted from [31]). (a) Conceptual illustration. (b) Schematic of the coding element featuring four varactor diodes. (c) Measured amplitudes and phases at various harmonic orders, for different combinations of the biasing voltages. (d) Measured scattering patterns at the harmonic order +1, for three different spatial codings; patterns of different colors illustrate the possibility to independently control the amplitude by acting on the temporal coding. (e) Representative examples of multiorder harmonic tailoring of the scattering patterns (measured and theoretical). 

In the text 
Fig. 7 BFSK transmission system based on spacetime coding digital metasurfaces (adapted from [32]). (a) Conceptual illustration. (b) Photographs of the prototype and measurement setup. (c, d) Measurement results with receiving angle of 0° and 30°, respectively. 

In the text 
Fig. 8 QPSK transmission system based on spacetime coding digital metasurfaces (adapted from [35]). (a) Photograph of prototype and measurement setup in an indoor scenario. (b) Received constellation diagrams for different transmission distances. 

In the text 
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