Pseudo-Random Sequence (PRS) (Space)Time-Modulated Metasurfaces

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Introduction
After hardly a decade of intensive worldwide research, metasurfaces have become a revolutionary advance in microwave, terahertz and optical technologies ( ), and this area is far from being exhausted at the time of this writing. Metasurfaces may be considered as dramatic generalizations of Fresnel-zone plate reflectors ( ), reflect/transmit arrays ( ), frequency/polarization selective surfaces ( ), diffraction gratings ( ) and spatial light modulators ( ), given the unprecedented versatility provided by their bianisotropic degrees of freedom ( ); alternately, they may be considered as the two-dimensional reductions of voluminal metamaterials ( ), with the benefits of lower form factor, lower loss, along with -amazingly! -far greater wave transformation capabilities. They have already led to a myriad of applications with unprecedented functionality or/and performance, including refraction and wavefront transformation ( ; ; ; ; ), absorption ( ; ; ), polarization transformation ( ), hologrpahy ( ; ), analog computing ( ), optical force carving ( ; ), and cloaking ( ; ; ; ).
While the vast majority of the metasurfaces reported until were static, i.e., invariant in time, the past lustrum has witnessed an explosion of interest for time-modulated and spacetime-modulated metasurfaces. In these metasurfaces, the time dimension is introduced as an extra degree of freedom to manipulate electromagnetic waves, both by breaking the fundamental bounds of linear time-invariant systems and by bringing about the possibility to engineer the temporal spectrum of waves in addition to just their spatial spectrum ( ; ). These time/spacetime-modulated metasurfaces have quickly given rise to their own range of applications, extending those of static metasurfaces, with the main applications including simplified front-end wireless communication ( ; ; ), spatial frequency conversion ( ; ; ), and generalized nonreciprocitity ( ; ; ; ; ; ), and this field is currently evolving at a spectacular pace.
Bianisotropy is a weak form of spatial dispersion (or spatial nonlocality) occurring in complex media, whereby the response of the medium at a point of space does not depend on the excitation only at this point but also in the neighbourhood of it ( ; ; ). It is the dominant form of spatial dispersion in subwavelength structures, such as typical metamaterials and metasurfaces, but mesocopic structures, whose unit cells are not much smaller than the wavelength, involve higher-order spatial dispersion, with even more degrees of freedom that could be leveraged for further engineering opportunities ( ).
The time/spacetime-modulated metasurfaces mentioned in the applications of the previous paragraph are based on modulation waveforms that are fully deterministic. Specifically, these waveforms are either sinusoidal or exhibiting shapes that are causally related by the required metasurface operation. Such metasurfaces induce relatively light alterations, typically limited to harmonic generation and frequency conversion, of the spectrum of the waves they process. We present here an overview of a different class of time/spacetime-modulated, namely metasurfaces whose modulation is a Pseudo-Random Sequence (PRS), typically of staircase shape and periodic nature. In contrast to their deterministic-modulation counterparts, these metasurfaces induce major alterations of the temporal spectrum of the waves that they process. As a result, they offer complementary applications, such as electromagnetic stealth ( ), direction of arrival (DoA) estimation ( ), and spatial multiplexing ( ).
The paper is organized as follows. Section presents a general description of time-modulated metasurfaces, their operation principle, and their modulation regimes. Then, Sec. establishes the key properties of PRS modulation waveforms in terms of autocorrelation, power spectral density, and orthogonality. Next, Sec. overviews the fundamental operations of PRS time-modulated metasurfaces, namely spectrum spreading, interference suppression. and signal selection. This concludes the theoretical part. From that point, the paper goes on with a practical part. Section discusses technological aspects for microwave implementations, specifically the realization of a binary metaparticle and the design of a fully addressable modulation array. Then, Sec. presents four applications, namely electromagnetic stealth, secured communication, direction of arrival estimation, and spatial multiplexing. Finally, Sec. provides conclusions and invokes possible future developments of the field.

Time-Modulated Metasurfaces
Figure depicts a generic (space)time-modulated metasurface, assuming a staircase modulation, and its operation principle. A time-harmonic wave, ψ i ptq, of frequency f 0 " ω 0 {p2πq, and hence of period T 0 " 1{f 0 , impinges on the metasurface, which is typically modeled by a bianisotropic susceptibility tensor χ during a fixed-state time interval of the staircase modulation. In order to efficiently process this wave, the metasurface must strongly interact with it; it is therefore designed to be at all times resonant near the frequency of the incident wave (f 0 ). As a result, its impulse response during the n th fixed-state time interval, T f,n , of the modulation, χ Tf,n pt 1 q, rings at the frequency ν 0,n " rω 2 0,n´p γ n {2q 2 s 1{2 , where ω 0,n (« ω 0 ) and γ n are respectively the resonance frequency and the damping of the metasurface during the interval T f,n ( ), and features a time constant of τ dn " 2{γ n and hence a memory time of T d,n " 3τ d,n " 6{γ n .
The metasurface is modulated, via a proper electronic control of its scattering particles, by the modulation waveform mptq (Fig. ). This waveform is typically periodic, with N time intervals of duration T f,n per period and with period T m ( ř N n"1 T f,n " T m ). This wave modulates the metasurface by varying its dispersive (ω) susceptibility in time (t), in possible addition to space (ρ) , i.e., as χ " χpρ, t; ωq. This modulation transforms the incident wave into a scattered wave, ψ s ptq, which may be either a wave that is transmitted through the metasurface (as in the figure) or a wave that is reflected from it. The scattered wave exhibits a transient response during the time T t,n , with T t,n " T 0 , following the transition from the pn´1q th to the n th modulation intervals, as illustrated in Fig. . This transient time is proportional to the dispersion difference between the modulation states pn´1q and n, and is limited to T d,n . If the modulation has a smooth (rather than a staircase) form, which is also quite common, we have T f,n Ñ 0, and T t,n Ñ 0 since the difference between the modulation states pn´1q and n vanishes.
One may distinguish two modulation regimes depending on the ratio between the time scales of the modulation, which is related to the fixed-state time (T f,n ) if the modulation is of staircase form, and of the dispersion of the metasurface, which is related to the transient time (T t,n ). In this paper, we are mostly concerned with the regime pT 0 ! T t,n ! T f,n (case of Fig. ), which we shall refer to as the slow-modulation regime. In this regime, the transient part of the response can be neglected, since it is much shorter than the steady-state part of the modulation states (see ψ s ptq in Fig. ), and the polarization density response of the metasurface can hence be safely written in terms of a multiplication product in the time domain, as P ptq " 0 χpρ, t; ωq¨Eptq (temporal locality), where the transients due to dispersion (ω) may be simply ignored. In contrast, for T t,n " T f,n , or T t,n ą T f,n , which we shall refer to as the fast-modulation regime, the temporal effects due to the modulation and to the dispersion (or memory) of the system are intertwined, T f,n T t , n ( t r a n s i e n t s )

T0
(in cid en t pe rio d) N n x y z Figure : (Space)time-modulated metasurface, processing a harmonic incident electromagnetic wave using a modulation mptq, which will be a staircase periodic Pseudo-Random Sequence (PRS) in this paper. The metasurface is generally dispersive (ω-dependent) and space-varying (ρ " xx`yŷ-dependent) in addition to time-varying (t-dependent), and may be characterized by the susceptibility tensor function χpρ, t; ωq in the slow modulation regime. and the polarization response cannot anymore be expressed in terms of a susceptibility function ( ). In the case of a smooth modulation, the situation is even more complicated. If the modulation is periodic, as in Fig. , the slow-and fast-modulation regimes would rather correspond to the regimes T m ! T d and T m ", ą T d , respectively, where T d would now refer to the maximal dispersion time within one period . Note that the regime T m " T 0 , which belongs the fast regime, is the regime that typically prevails in parametric amplifiers ( ; ).

Pseudo-Random Sequence (PRS) Modulation
The term 'pseudo-random sequence' (PRS) refers to a sequence of values that are generated by an algorithm whose properties approximate the properties of a sequence of random numbers. Such a sequence has therefore, per se, a white-noise spectrum. However, we shall consider here that the PRS function is periodic. This assumption does not represent a restriction of generality insofar as an aperiodic function can always be treated as the limit of a periodic function with infinite period. However, periodicity endows the PRS with a rich set of temporal and spectral properties that are crucial in many applications. We shall establish here the characteristics and properties of this modulation, with the help of Fig. . In this paper, we will restrict our attention to binary polar staircase periodic PRS modulations, mptq, of the type plotted in Fig. (a). This PRS oscillates between the levels˘1. It has bits of duration T b and N bits per period, corresponding to the period T p " N T b , with N being generally odd for technological reasons (linear feedback shift register generation) ( ).
The autocorrelation of a function is a measurement of the similarity of this function with delayed copies of One may also distinguish the ultra-slow regime Tm Î T d , which corresponds to a simple regime of reconfigurability (between different operation modes), and, at the other extreme, the ultra-fast regime Tm Ï T d , which prevails in attophysics ( ), but these regimes are not of interest in this paper. Figure  itself versus the delay. In the case of the periodic PRS function mptq in Fig. (a), it is found as ( ) where t 0 is an arbitary time and Λp¨q is the triangular function This function is plotted in Fig. (b). It shares the periodicity (T p ) of mptq, since copies of the PRS that are delayed by an integer number of the period perfectly superimpose with each other. Moreover, it is composed of triangular pulses of width 2T b , corresponding to the correlation of rectangular pulses of width T b , with a small negative offset of´1{N that is due to the excess, by one unity, of the number of´1's compared to the number of`1's, due to the odd nature of N . Note that the larger is N , and hence the more random is mptq, the larger is the spacing between the triangles (T p ), and of course the smaller is the offset |´1{N |; in the limit N Ñ 8 of an aperiodic PRS, s m ptq reduces to the triangle at the origin since there is then no delay anymore that leads to perfect superposition, and in the further limit T b Ñ 0, s m ptq reduces to a unit impulse at the origin, corresponding to a perfectly random wave, which superimposes only with the non-delayed copy of itself.
The Fourier transform of the autocorrelation function is called the Power Spectral Density (PSD). In the case of the PRS mptq in Fig. (a), the PSD is ( ) and is plotted in Fig. (c). This function has a sinc 2 p¨q-form envelope, corresponding to the Fourier transform of the triangular components of s m ptq, whose magnitude, pN`1q{N 2 , depends only on N . Due to the periodicity of s m ptq, with period T p ,s m pf q is a discrete function, with spacing of f p " 1{T p between its impulses; note the small DC component, of magnitude 1{N 2 , due to the imperfect balance between the`1's and´1's of mptq, which quickly falls to zero as the length of the PRS increases. Finally, the bandwidth of s m ptq, defined as the width of the main beam, is Some PRS-modulated metasurface applications involve, as we shall see, a set of PRSs, tm k ptqu, with k " 1, 2, . . . , K, where each of the K PRSs, m k ptq, is of the type of mptq in Fig. (a) while composed of a distinct sequence of`1's and´1's. This set can be designed to be approximately orthogonal as where m p ptq and m q ptq are an arbitrary pair of sequences in the set, and the´1{N result is due to the same reason as before.

Operations of PRS Time-Modulated Metasurfaces
PRS-modulated metasurfaces may perform three fundamental operations, which underpin their applications (Sec. ): A. spectrum spreading, B. interference suppression, and C. row/cell selection. We shall demonstrate here these operations with the help of Fig. .

. Spectrum Spreading
The spectrum spreading operation is depicted in Fig. (a). The incident wave, ψ i ptq, is assumed to have a narrow spectrum, ∆f i , centered at f 0 , and it is modulated by the binary˘1 PRS waveform mptq described in Sec. . In the slow-modulation regime (T 0 ! T t,n ! T f,n ), which is assumed throughout the paper, temporal locality (Sec. ) allows to write the scattered wave as where the symbol '¨' denotes the simple multiplication product, which will be from now on omitted.
The Fourier transform of Eq. ( ) expresses then the Fourier transform of the scattered wave as the convolution of the Fourier transforms of the modulation and of the input wave, namelỹ ψ s pf q "mpf q˚ψ i pf q. ( ) Given the assumed narrow width ofψ i pf q, ∆f i , and the much broader width ofmpf q, 2f b , due to the pseudo-random nature of mptq [ Fig. (a)], this operation results in a responseψ s pf q that is much broader thanψ i pf q [ Fig. (a)]. In other words, the PRS-modulated metasurface spreads out the spectrum of the wave that it processes by the factor 2f b {∆f i , which can be extremely large in practice.
Note that the spreading of the spectrum by the modulation does not decrease the envelope of the power spectral density of the scattered wave, because the spectrum envelope is essentially a translation from zero to the frequency of the incident wave (f 0 ) of the modulation spectrum [ Fig. (a)], whose envelope was shown in Sec. to depend only on the length (N ) of the PRS. However, the spectrum spreading is mptq mptq ψ i ptq accompanied by a dramatic reduction of the spectral power level of the wave, particularly at f 0 where the reduction is by the factor 1{N 2 , but also elsewhere, where the reduction is in the form of the sinc 2 p¨q function with maximum of pN`1q{N 2 , which represents an overall major reduction of the power spectral level of the processed wave.
The spectrum spreading ratio between the scattered wave and the incident wave, 2f b {∆f i , is particularly strong -in fact infinite! -when ψ i ptq tends to pure single-tone harmonic wave, i.e., ψ i ptq " e j2πf0t and ψ i pf q " δpf´f 0 q. In this case, Eq. ( ) reduces indeed tõ ψ s pf q "mpf´f 0 q, ( ) which is simply the Fourier transform of mptq shifted to the frequency of the incident wave, and the scattered wave has therefore exactly the same spectral width asmpf q. .

Interference Suppression
The interference suppression operation is depicted in Fig. (b). It corresponds to a scenario where an interfering wave, ψ int ptq, which may typically be a harmonic wave or a narrow-band burst wave, escapes the metasurface and reaches unimpeded a point of space where the signal passed through the metasurface, ψ i ptq, is to be demodulated.
The total wave reaching the point of interest is then ψ r ptq " ψ s ptq`ψ int ptq " mptqψ i ptq`ψ int ptq, ( ) and a receiver possessing there the information of mptq can process this wave as where we used the fact that m 2 ptq " 1 for the assumed unitary bipolar symmetric sequence mptq [ Fig. (a)].
The input wave has thus been demodulated by the receiver [Fig. (b)]. Moreover, Fourier-transforming the final result of Eq. ( ) yieldsψ d pf q "ψ i pf q`ψ int pf q˚mpf q, ( ) which shows that the spectrum of the interfering wave has been spread out, just as the wave incident on the metasurface in the previous subsection, so that its power spectral level has decreased to a very small level Fig. (b). The effect of the interference can then be further reduced, if necessary, to quasi total suppression by applying a band-filter at f 0 so as to eliminate all of its energy contents distributed over the other spectral impulses. .

Row/Cell Selection
The row/cell selection operation is depicted in Fig. (c). This pertains yet to another scenario, where the goal is to select out, at a given point of space where this selection is to be performed, the parts of the incident wave, ψ i ptq, that impinged on the different rows/cells of the metasurface. We shall call the so-defined k th part of the wave ψ ik ptq, where k " 1, 2, . . . , k, . . . , K, for a metasurface modulated in its K columns.
Here, the incident wave may generally be a combination of different waves coming from arbitrary directions of space. The row/cell selection operation can be performed using an orthogonal set of modulation PRSs, tm k ptqu, k " 1, 2, . . . , K, i.e., a set of PRSs whose any pair satisfies the condition ( ).
Note that such a system is actually a space-time modulated system. Indeed, the different rows/cells, even if they happen to have all the same shape, are distinctly modulated, so that the incident wave actually "sees" a spatial modulation in addition to the temporal modulation when impinging on the metasurface.
If the K parts of the incident wave have distinct magnitudes and phases, they may be generically expressed as ψ ik ptq " A k ptqe jpω0t`φ k ptqq , and the global signal scattered by the metasurface is then which represents a row/cell encoded mixture of the incident wave. This scattered wave may then down-For simplicity, we shall mostly restrict here our attention to the case where all the cells of the k th row are addressed by the same modulation, m k ptq, but this may be generalized to the addressing of individual cells, with a modulation m ij ptq, as will be shown in the design of the next section.
( ) If the PRS set tm k ptqu is known, and assuming that A k ptq and φ k ptq are slowly varying function of time with respect to the time scale of the modulation period (T p ) (Figs. and ), this wave may be processed as follows at the point of space where the selection is to be performed: where the orthogonality relation ( ) was used in the last-but-one equality and the approximation N " K was used in the last equality. This result shows that integrating an m r ptq-weighted version of the wave ψ 0 ptq selects out the magnitude and phase of the part ψ i,r ptq of the incident wave, i.e., the part of this wave that impinged on the r th row/cell of the metasurface.

Design
The design of a time-modulated metasurface system involves a mixture of electromagnetics engineering, for the metasurface structure part, and of electronics engineering, for the time modulation part. While the former is equally mature in the microwave and optical regimes ( ), the latter still represents an insurmountable task in the optical regime at the present time. Therefore, we shall discuss here only microwave-regime designs, with the help of the representative example described in Fig. . Figure (a) depicts a unit-cell design for the assumed binary-polar PRS modulation and for individual cell addressing, with polarity reversal following from switching between the opposite sides of the receiving (resonant, and hence λ{2-long) slotted-patch unit-cell particle. Figure (b) shows a simplified version of (a), and plots the corresponding scattering parameters, which exhibit an excellent broadband equalmagnitude and opposite-phase response. Figure (c) shows a modulation array for individual cell addressing, inspired by the Thin-Film Transistor (TFT) matrices used in screen displays ( ) and implementable in standard CMOS technologies ( ), with Field Programmable Gate Array (FPGA) control. Finally, Fig. (d) shows the individual-cell addressing scheme of this array, where the array refresh time, T r , is assumed to be much larger (up to the bit duration T b , see Fig. ) than M T s , with M being the number of rows and T s the row selection time, so that the sequential addressing cycle of all the rows of the array can be, practically, considered simultaneous.

Applications
Figure presents four applications of PRS (space)time-modulated metasurfaces. Figure (a) depicts the application of electromagnetic stealth ( ). Instead of altering the shape of the object to conceal, as typical stealth techniques, this technology covers the object by the PRS time-modulated metasurface and alters its spectrum. The narrow-band signal of the interrogating radar is spread out upon reflection and its spectral power level is reduced below the noise floor of that radar, according to the spectrum spreading principles established in Sec. . , which makes the object undetectable. Such temporal spectrum spreading can be enhanced by spatial dispersion upon adding spatial modulation in the form of alternating PEC and PMC metasurface cells ( ).   receives a jammed, undecipherable version of the message, plus possible interfering burst noise, the intended receiver can safely demodulate the message using the modulation key in his possession while eliminating the interfering noise, according to the interference suppression principles established in Sec. . . If the position of the receiver is known, the metasurface may additionally use spatial modulation in the form of an appropriate phase gradient, possibly dynamic for motion tracking, to specifically radiate in the direction of the receiver, and hence further enhance the safety of the communication link.
Figure (c) describes the application of direction of arrival (DoA) estimation ( ). In contrast to conventional DoA systems, which require an array of independent receive antennas with individual phase detectors, this DoA system offers the advantage of requiring only one pick-up antenna, thanks to the row/cell selection principles established in Sec. . . The wave impinging on the metasurface, under an angle θ (to determine), is modulated at each row of the metasurface by a different PRS from a set of mutually orthogonal PRSs. Then, the wave scattered by the metasurface is picked up by an antenna, down-converted to base-band and stored in a memory. Next, the phase of the waveform part originating from any r th row of the metasurface is determined from the operation corresponding to Eq. ( ), and θ is finally obtained from the so obtained phases between waveforms from adjacent rows, φ i,r and φ i,r`1 , as θ " sin´1rcpφ i,r`1´φi,r q{pω∆dqs, where ∆d is the distance between adjacent rows of the metasurface.
Finally, Fig. (d) shows the application of spatial multiplexing ( ), where the metasurface can operate either as a multiplexer or as a demultiplexer. In the demultiplexing mode, it receives a mixture of Q precoded mptq messages destinated to different users,ψ i ptq " ř Q q"1ψ i,q ptq, whereψ i,q ptq " ψ i,q ptq ř N n"1 e jn∆φq m n ptq with ∆φ q "´k 0 ∆dsinθ q representing the metasurface's inter-row phase gradient that encodes the destination direction, θ q , of the q th message, ψ i,q ptq, and tm k ptqu representing a set of orthogonal PRSs. The metasurface modulates the signalψ i ptq by the same PRS sequences as those used to precode the messages, which results into the scattered wave ψ s pt, θq " ř N r"1 m r ptqψ i ptqe jr∆φpθq , where ∆φpθq " k 0 ∆dsinθ. It may be easily verified that substituting in this expression for ψ s pt, θq the expression forψ i ptq and applying the PRS orthogonality relation ( ) leads to ψ s pt, θq " ř Q q"1 ψ i,q ptq ř N r"1 e jk0r∆dpsinθ´sinθqq . Thus, according to basic antenna array theory ( ), each message ψ i,q ptq is properly radiated into its intended direction, θ q , and the multiplexing operation is therefore accomplished as expected. The reciprocal, multiplexing operation of the system is essentially identical to that of the DoA system presented in the previous paragraph, with multiple angles corresponding to the multiple messages. Using a unique antenna, this metasurface multiplexing system clearly represents a major interest for massive Multiple-Input Multiple-Output (MIMO) systems.