Joint time-frequency analysis on space-time-coding digital metasurfaces

Space-time and time-varying metastructures have attracted a lot of research interest in recent years. On the other hand, digital programmable metasurfaces have also gained great attention owing to their powerful capabilities in controlling electromagnetic (EM) fields and waves in real time, which is very suitable for implementing spatiotemporal modulations in a digital manner. Accordingly, space-time-coding (STC) digital metasurfaces have recently been proposed to realize advanced manipulations of EM wavefronts and digital information, allowing simultaneous control of propagation directions in the space domain and harmonic distributions in the frequency domain. However, their instantaneous responses and the connection between the time- and frequency-domain characteristics have not yet been fully revealed. Here, we present a joint time-frequency analysis method to revisit STC digital metasurfaces, in which the time-domain instantaneous scattering patterns and frequency-domain equivalent excitations are investigated to analyze the spatial-spectral distributions of the modulated waves. This joint time-frequency analysis method helps to better explain the basic working principle of STC digital metasurfaces and is expected to facilitate more applications in wireless communications, radar, imaging, and beamforming.


Introduction
Metamaterials and metasurfaces are artificially engineered structures with subwavelength units, which can be collectively referred to as metastructures. They have attracted wide attention in the past decades due to their unprecedented ability to control electromagnetic (EM) waves [1,2]. The spatial and/or temporal engineering of their parameters, such as the permittivity, permeability, conductivity, and impedance, enables diverse manipulations of EM waves. In recent years, space-time and time-varying metastructures have received a great deal of interest from researchers in the fields of physics, electromagnetism, optics, and materials [3,4]. The earliest studies on time modulation can be traced back to 'time-varying media' [5][6][7] and 'time-modulated arrays' [8][9][10]. Time-varying metastructures provide an extra dimension of time to extend the degree of freedom of EM modulations. Combining space and time modulation, space-time metastructures with spatiotemporally varying parameters can achieve simultaneous and sophisticated manipulations of EM waves in various domains and have been widely studied to produce a wealth of applications, such as nonreciprocity [11][12][13][14][15], Doppler cloak [16], impedance transformers [17], harmonic generations [18], frequency conversion [19,20].
On the other hand, digital coding and programmable metasurfaces have been rapidly developed since they were first proposed in 2014 [21]. The digital representation of metasurfaces can realize simple manipulation of EM waves by switching coding sequences in a discretized manner, which has built a bridge between the physical world and the digital world, leading to the research system of information metasurfaces [22,23]. The digital coding and programmable information metasurfaces have been successfully applied to numerous applications, such as acoustic manipulation [24], scattering control [25], reflect/transmit arrays [26], reprogrammable holograms [27,28], self-adaptive regulation [29], intelligent imaging [30], programmable artificial intelligence machine [31], wireless communication transmitters [32,33], reconfigurable intelligent surfaces [34]. More importantly, digital programmable metasurfaces provide a powerful and versatile platform for implementing space-time modulation, and thus catalyze the development of space-time-coding (STC) digital metasurfaces [35,36]. The phases and/or amplitudes of the STC digital metasurfaces are encoded both in space and time so as to simultaneously manipulate the propagation direction (space domain) and harmonic distribution (frequency domain) of EM waves. STC digital metasurfaces have been applied in many exciting areas such as reciprocal reflection [37], harmonic beam steering [35,38], scattering suppression [35,39], joint multi-harmonic control [40], polarization syntheses [41], analog signal processing [42], direction finding [43], nonuniform periodic modulation [44], multiplexed wireless communication [45].
Although STC digital metasurfaces have been demonstrated in many exotic applications, their instantaneous responses and the connection between the time-and frequency-domain characteristics have not yet been investigated. In this work, we propose a joint time-frequency analysis method to calculate the spatial-spectral characteristics of the STC digital metasurface. Specifically, time-domain instantaneous scattering patterns are presented to analyze the signal waveforms and spectra of reflected waves, and frequency-domain equivalent excitations are calculated to obtain harmonic beam patterns and spectral distributions as well. We reveal that both the time-domain and frequency-domain analysis yield the same results of spatial-spectral distributions, which can help to explain more clearly the working principle of the STC digital metasurface.

Theory of time-frequency analysis
To illustrate the time-frequency analysis method more clearly, we hereby consider a reflection-type STC digital metasurface composed by M columns of 1 bit coding elements, as shown in figure 1. Each column of coding elements has the same time-modulated signal according to the digital '0/1' time-coding sequence with length L, and their reflection phases can be dynamically switched between in-phase and out-of-phase according to the corresponding coding states. The STC digital metasurface is normally illuminated by a monochromatic plane wave with time-harmonic dependence e jωct . Assuming that the modulation frequency ω 0 is much smaller than the incident wave frequency ω c , the time-modulated reflection coefficient of the mth element can be defined as a superposition of L shifted rectangular pulses within one modulation period represents a periodic rectangular pulse with width τ = T 0 / L, and Γ m l denotes the time-invariant reflection coefficient (at the central frequency ω c ) of the mth coding element within the time interval (l − 1)τ ⩽ t ⩽ lτ . In the case of 1 bit phase modulation, we have Γ m l ∈ {1, −1} associated with the coding state {0,1}. According to our previous studies [35][36][37], the time-domain far-field pattern of the STC digital metasurface as a function of time can be approximately expressed as where E(θ) is the far-field scattering pattern of the coding element, k = ω c /c is the wavenumber at the central frequency ω c (with c denoting the speed of light in vacuum), and d is the spatial period of the coding elements. In the time interval t l ∈ [(l − 1)τ, lτ ], the time-domain instantaneous far-field pattern can be further written as and at a spatial direction θ = θ r , the time-domain far-field of the STC digital metasurface as a function of time can be written as f(θ r , t). By directly expending the periodic function f(θ r , t) into Fourier series (FS) its Fourier coefficient F v (θ r ) is given by which represents the spectral distributions of various frequency components (ω c + vω 0 ) in the corresponding direction of θ = θ r . On the other hand, we can also analyze the frequency response of the time-modulated reflection coefficient Γ m (t) of each coding element in equation (1). By FS expansion of Γ m (t), its Fourier coefficients a m v can be represented as a m v can be regarded as complex-valued equivalent excitations of the mth coding element at the frequency ω c + vω 0 . Therefore, the frequency-domain far-field pattern of the STC digital metasurface at the vth harmonic frequency ω c + vω 0 can be expressed as It can be seen that equations (6) and (8)

Results and discussion
As a proof of principle, the center operating frequency and modulation frequency of the STC digital metasurface are assumed as f c = ω c / 2π = 9.5 GHz and f 0 = ω 0 / 2π = 10 MHz, respectively. Each coding element has the period of d = 14 mm and its scattering pattern is approximated as E(θ) = cos(θ). We consider a two-dimensional (2D) 16 × 8 STC matrix, representing 16 columns of coding elements and 8-interval time-coding sequences, as shown in figure 2(a). This STC matrix is optimized by binary particle swarm optimization algorithm to obtain two pencil beams at two first harmonics, pointing to different directions, respectively. Under the modulation of the optimized STC matrix, the STC digital metasurface can control the propagation direction and spectral distribution of reflected EM waves in the desired way. Figure 2(b) gives a distribution of the frequency-domain complex-valued equivalent excitations a m v corresponding to the STC matrix in figure 2(a), which can be calculated by equation (7). The corresponding time-domain instantaneous far-field patterns within eight time-interval t 1 -t 8 can be calculated by equation (4), as shown in figure 2(c).

Time-domain instantaneous responses
For a clear presentation, figure 3 shows the separate display of time-domain instantaneous 2D scattering patterns (in dB form) within different time intervals t 1 -t 8 . Since the STC matrix in figure 2(a) is optimized to obtain two harmonic beams at different spatial directions, the instantaneous scattering patterns within eight time-intervals also exhibit the shape of dual beams. We further calculate the three-dimensional (3D) time-domain amplitude patterns as a function of time via equation (3), in which the main beams are concentrated around ±34 • over time, as shown in figure 4(a). To analyze the spectral response of modulated wave, the time-domain phase patterns are also calculated and shown in figure 4(b). It can be seen that the and arg [f(34 • , t)] respectively exhibit descending and ascending time gradients over one modulation period, which is responsible for the harmonic conversion that will be discussed later.
Moreover, we herein analyze the time-domain signal waveform of the space-time-modulated wave. As previously assumed, the STC digital metasurface is illuminated by a time-harmonic wave with the dependence e jωct , and the amplitudes and phases of the reflected waves are modulated according to the time-domain instantaneous far-field patterns. Figure 5 shows the signal waveforms of reflected waves at three different angles corresponding to figure 4, from which we can see that their envelopes are consistent with the amplitudes in figure 4(a). Actually, time-domain analysis is suitable for evaluating the effect of time modulation on transmitted signals.

Frequency-domain spatial-spectral characteristics
From another point of view, we calculate the frequency-domain complex-valued equivalent excitations a m v by equation (7), which can also be used to analyze the spatial-spectral characteristics of the STC digital metasurface modulated by the matrix in figure 2(a). Figures 6(a) and (b) show the equivalent amplitudes and phases of each coding element from −5th to +5th harmonic frequencies. It can be seen that the STC digital metasurface mainly converts the incident wave at the frequency ω c into the reflected waves at ±1st harmonic frequencies ω c ± ω 0 , and the equivalent phases at ±1st harmonic frequencies exhibit different directions of space-gradient, which can steer the ±1st harmonic beams to different angles, respectively. The STC digital metasurface produces high amplitudes and spatial phase gradients at ±1st harmonic frequencies, which is the essential cause of harmonic beam steering. Frequency-domain analysis is more convenient to calculate the scattering patterns at specific harmonics by using the complex-valued equivalent excitations. The corresponding 2D frequency-domain scattering pattern of the STC digital metasurface from −5th to +5th harmonic frequencies are shown in figures 6(c) and (d), which are calculated by equation (8) according to the equivalent excitations in figures 6(a) and (b). The pencil beams at ±1st harmonic frequencies are steered to the angles of ±34 • , respectively, and the power intensities of scattering patterns at other harmonic frequencies are suppressed to ensure the high efficiency of beam steering at ±1st harmonic frequencies.
We further calculate the 3D frequency-domain amplitude patterns as a function of harmonic orders via equation (8), as shown in figure 7(a). Thus, the spectral distributions F v (θ r ) of the reflected signal at a specific angle θ r can be directly extracted from the 3D frequency-domain amplitude patterns, and their normalized amplitudes at different angles of θ r = −34 • , θ r = 0 • and θ r = 34 • are depicted in figures 7(b)-(d), respectively. Besides, we can also utilize the time-domain amplitude and phase distributions in figures 4(c) and (d) to calculate the corresponding spectra by using the FS. Due to the descending and ascending temporal phase gradients in figure 4(d), the spectra of F v (−34 • ) and F v (34 • ) are concentrated in −1st and +1st harmonic frequencies, respectively, meaning that monochromatic incident wave is converted into ±1st harmonics at the angle of ±34 • with high efficiency. As can be seen from the comparisons in

Conclusions
In summary, we proposed the joint time-frequency analysis method to calculate the spatial-spectral characteristics of the STC digital metasurface. Through theoretical derivation and the illustrative example, we revealed the connection between the time-domain and frequency-domain responses. In the time domain, the instantaneous far-field scattering patterns were presented to analyze the signal waveforms of reflected waves at different angles. In the frequency domain, the complex-valued equivalent excitations were utilized to calculate the harmonic beam patterns. Additionally, we also investigated the temporal phase gradient at specific directions and spatial phase gradient at desired frequencies, which are responsible for the harmonic conversion (frequency domain) and beam steering (space domain), respectively. The proposed method helps researchers gain a deeper understanding of how STC digital metasurfaces can flexibly control EM waves in the time, frequency and space domains, and provides guidance for potential applications in directional modulation, secure wireless communication, radar waveform generation, and advanced beamforming.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.