Space-Time Modulated Metasurfaces with Spatial Discretization: Free-Space N-path Systems
TTheoretical and Experimental Investigations of Spatio-Temporally ModulatedMetasurfaces with Spatial Discretization
Zhanni Wu, Cody Scarborough, and Anthony Grbic Department of Electrical Engineering and Computer Science,University of Michigan, Ann Arbor, Michigan 48109-2122, USA (Dated: June 12, 2020)A dual-polarized, spatio-temporally modulated metasurface is designed and measured at X-bandfrequencies. Each column of subwavelength unit cells comprising the metasurface can be indepen-dently biased, to provide a tunable reflection phase over a range of 330 ◦ . In this work, the biaswaveform applied to adjacent columns is staggered in time to realize a discretized traveling-wavemodulation of the metasurface. An analytic model for the metasurface is presented that accounts forits discretized spatial modulation. The analysis considers a finite unit cell size and thus provides in-creased accuracy over earlier analysis techniques for space-time metasurfaces that commonly assumecontinuous spatial modulation. Theoretical and experimental results show that for electrically-largespatial modulation periods the space-time metasurface allows simultaneous frequency translationand deflection. When the spatial modulation period on the metasurface is electrically small, newphysical phenomena such as subharmonic frequency translation can be realized. When the spa-tial modulation period of the metasurface is wavelength-scale, simultaneous subharmonic frequencytranslation and deflection can be achieved. For certain incident angles, retroreflective subharmonicfrequency translation is demonstrated. I. INTRODUCTION
Metasurfaces are two dimensional structures texturedat a subwavelength scale to achieve tailored control ofelectromagnetic waves. Developments in tunable elec-tronic components have allowed dynamic control overthe electromagnetic properties of metasurfaces. Devicessuch as varactors, transistors and MEMS [1–3] in ad-dition to 2D and phase change materials [4–6] can beintegrated into metasurfaces to tune their electric, mag-netic and magneto-electric responses. Often, the prop-erties of a metasurface are spatially modulated to shapeelectromagnetic wavefronts and achieve focusing, beam-steering, and polarization control [7–10]. By incorpo-rating tunable elements into their design, the propertiesof metasurfaces can also be modulated in time [11–13].While spatial modulation redistributes the plane-wavespectrum of the scattered field, temporal modulation pro-vides control over the frequency spectrum. Applyingboth spatial and temporal variation is known as spatio-temporal modulation, and has recently been applied tometasurfaces [14–22]. Space-time modulation can simul-taneously allow frequency conversion and beam steeringand shaping. It can also be used to break Lorentz reci-procity and enable magnetless nonreciprocal devices suchas gyrators, circulators and isolators [23–26].Spatio-temporally modulated structures are typicallyanalyzed and designed as continuous surfaces. That is,the unit cell size of the physical structure is assumed tobe deeply subwavelength. However, accounting for thediscretization of the unit cell provides increased accu-racy and can yield useful effects which are not predictedin the continuum limit. For example, when the spatialperiod of the modulation is smaller than a wavelength,subharmonic frequency translation can be achieved in thespecular direction. In this case, scattered waves can radi-
FIG. 1. A spatio-temporally modulated reflective metasur-face. The incident wave (dark red) can be reflected (lightblue) at an angle and frequency determined by the space-timedependence of the bias voltage. ate at a higher order frequency harmonic determined bythe number of unit cells in a subwavelength spatial pe-riod. Such a behavior does not arise from a continuousanalysis of sub-wavelength modulation periodicity, sincehigher-order spatial harmonics introduced by the spatialdiscretization are not considered.In this paper, we demonstrate a spatio-temporallymodulated metasurface consisting of discrete unit cells,as shown in Fig. 1. Varactor diodes are surface-mountedonto the metasurface, acting as tunable capacitances.Each column of unit cells on the metasurface can be in-dependently, temporally modulated, allowing space-timemodulation along one axis. This structure was first pre-sented in [13], where the varactor diodes on each columnwere temporally modulated with the same bias signal.In [13], a sawtooth reflection phase in time was used, re- a r X i v : . [ phy s i c s . a pp - ph ] J un sulting in Doppler-like (serrodyne) frequency translation.Structures of a similar design have been subsequently re-ported in [27].Here, we consider a discretized, traveling-wave mod-ulation in which the capacitance variation of adjacentcolumns is staggered in time. This modulation schemeis reminiscent of N-path networks that have received sig-nificant attention in the circuits community [28–33] as oflate. In the context of electronic circuits, an N-path net-work consists of a set of linear, periodically time-varying(LPTV) signal paths connected to a common input andoutput. Each path includes at least one time-varying cir-cuit component. The time-modulation of adjacent LPTVpaths is staggered in time by T p /N , where T p is the mod-ulation period, and N is the number of paths in thenetwork. The N-path configuration suppresses certainharmonic mixing products. Specifically, for a modula-tion frequency f p = 1 /T p and excitation frequency f ,the only harmonics present at the input and output arethose at f = f + rN f p , where r ∈ Z [28–33]. N-path net-works have attracted widespread attention in the circuitscommunity due to their filtering capabilities [31, 34, 35]and as a method to break time-reversal symmetry andrealize non-reciprocal devices such as circulators [36, 37],and isolators [26, 30]. The non-reciprocal behavior ofN-path networks can find various applications in full du-plex wireless communication and radar. Non-reciprocaldevices are also needed in optical fiber communicationsand the protection of sensitive electronic equipment fromhigh-power microwaves.Periodic space-time modulation of a metasurface im-parts tangential momenta (an impressed wavenumber)onto each frequency harmonic of the scattered field. Thetangential wavenumber of the modulation is given by thespatial modulation period. Provided that each unit cell issub-wavelength, the behavior of the metasurface can bedivided into three regimes based on whether the spatialperiod of the modulation is (1) electrically small (muchsmaller than a wavelength), (2) electrically large (muchgreater than a wavelength), or (3) on the order of a wave-length. When the spatial modulation period is electri-cally small, the columns of the metasurface appear col-located and the behavior approaches that of an N-pathnetwork. The staggered modulation scheme in this caseresults in harmonic cancellation which can be exploitedto achieve subharmonic frequency translation. Reflectedharmonics at frequencies f = f + rN f p ( r ∈ Z ) corre-spond to propagating wavenumbers, while the remainingfrequencies are evanescent. In contrast, when the spatialmodulation period is much larger than a wavelength, thecapacitance variation between adjacent sub-wavelengthcells is reduced. In this limit, the discretized meta-surface approaches a continuous spatiotemporally mod-ulated structure: the incident wave undergoes frequencytranslation [12, 13] and angular deflection. Finally, whenthe spatial period is comparable to the wavelength, bothsubharmonic frequency translation and angular deflec-tion can be simultaneously achieved. At certain incident angles, the metasurface can perform retroreflective sub-harmonic frequency translation.In this paper, the proposed metasurface design willbe explored both in theory and experiment. In Sec-tion II, the semi-analytical procedure for computingthe response of the presented metasurface is discussed.This includes the homogenization of each unit cell, fol-lowed by a treatment of both time and space-time vari-ation. This procedure is carried out for various modu-lation schemes in Section III. Specifically, effects suchas specular subharmonic frequency translation, deflec-tive/retroreflective serrodyne frequency translation, anddeflective/retroreflective subharmonic frequency transla-tion are examined. The results of the theoretical studyare then validated experimentally in Section IV. II. ANALYSIS OF THE TEMPORALLY ANDSPATIO-TEMPORALLY MODULATEDMETASURFACE
The spatio-temporally modulated metasurface is de-picted in Fig. 1. It is a reflective, electrically-tunableimpedance surface [38], consisting of a capacitive sheetabove a grounded dielectric substrate. The capacitivesheet is realized as an array of metallic patches intercon-nected by varactor diodes. It can be modulated in bothspace and time with a bias signal that is applied throughthe metallic vias that penetrate the substrate.A unit cell of the designed metasurface is shown in Fig.2a. The varactor diodes connecting the metallic patchesare biased through the vias located at the edges of theunit cells, while the the via at the central patch is con-nected to ground. The remainder of the biasing networkis shielded behind the ground plane. The biasing net-work and diode orientations allow the reflection phase ofthe metasurface to be independently tuned for two or-thogonal (TE and TM) polarizations. Bias waveforms V xbias ( t, x ) and V ybias ( t, x ), as shown in Fig. 1, control thesheet capacitance for the two orthogonal polarizations.A detailed description of the fabrication and biasing net-work are provided in Section IV. A cross section of themetasurface is shown in Fig. 3 under TE and TM ex-citations. The biasing vias can be seen perforating the (a) 𝐶(𝑡) 𝑘 𝑠𝑧 , 𝑍 𝑠 𝑙 𝑍 (b) FIG. 2. (a) Unit cell of the dual-polarized, spatio-temporallymodulated metasurface. (b) Circuit model for each polariza-tion. 𝑥𝑧 ⊗ 𝑘𝐸 𝐻 ⊗ 𝑘𝐸𝐻 𝐶 𝑇𝐸 (𝑡, 𝑥) 𝑙 (a) 𝑥𝑧 ⊗ 𝑘 𝐻 𝐸 ⊗ 𝑘𝐸𝐻𝑙 𝐶 𝑇𝑀 (𝑡, 𝑥) (b) FIG. 3. Cross sections of the obliquely illuminated time-modulated metasurface, under (a) TE polarization, and (b)TM polarization. dielectric substrate.Here, we derive a semi-analytical procedure for com-puting the response of the metasurface shown in Fig. 1,with a discretized traveling-wave modulation. In Sec-tion II A, the metasurface unit cells are homogenized andrepresented by an equivalent circuit model. The semi-analytical procedure for obtaining the scattered fields inthe presence of time-modulation alone is examined in Sec-tion II B. Building upon this framework, a procedure forcomputing the scattered fields produced by the space-time modulated metasurface is then presented in SectionII C.
A. Homogenization of the spatio-temporallymodulated metasurface
In the analysis that follows, the unit cells of the meta-surface are homogenized. Within each unit cell, themetallic patches interconnected by varactor diodes willbe treated as a capacitive sheet. This sheet can be mod-ulated in time, independently of adjacent unit cells. Thedielectric substrate perforated by vias ( − l < z < (cid:15) r = (cid:15) h (cid:15) h
00 0 (cid:15) zz , (1)where (cid:15) h is the relative permittivity of the host mediumand (cid:15) zz is the effective relative permittivity along thevias. Since the metasurface is electrically thin at theoperating frequency of 10 GHz ( l = 0 . λ = 0 .
508 mm),a local model can be used to describe the wire medium[39]: (cid:15) zz = (cid:15) h (1 − k p k (cid:15) h ) , (2)where k p = 541 .
81 rad · m − is the plasma wavenumber ofthe wire medium extracted from a full-wave simulationof the unit cell shown in Fig. 2a, and k is the free-spacewavenumber of the incident wave. The anisotropic sub-strate supports TE (ordinary mode) and TM (extraordi-nary mode) polarizations. The normal wavenumber for each polarization in the substrate can be written as, k T Esz = q k (cid:15) h − k x , (3) k T Msz = r k (cid:15) h − k x (cid:15) h (cid:15) zz , (4)where k x is the tangential wavenumber of the incidentwave.Each unit cell can be modeled with the shunt resonatordepicted in Fig. 2b. The circuit model consists of atunable capacitance (representing the capacitive sheet)backed by shorted transmission-line section (represent-ing the grounded dielectric substrate) that acts as aninductance. As a result, the bias voltage applied to thevaractor diodes can be used to tune the reflection phase.The phase range of this topology is 2 π − ∆ φ , where ∆ φ isthe round trip phase delay through the substrate. Detailson the phase range of the realized metasurface are pro-vided in Section IV. In this paper, two different reflectionphase waveforms are considered. A sawtooth reflectionphase with respect to time is studied, which allows serro-dyne frequency translation [12, 13], as well as a sinusoidalreflection phase with respect to time.As mentioned earlier, each column of unit cells can bebiased independently, allowing for space-time modulationalong a single ( x ) axis. As a result, the homogenizedmodel consists of capacitive strips whose widths are givenby the unit cell size d = λ / λ is thewavelength in free space at 10 GHz. The capacitance seenby each polarization can be controlled independently andis uniform over the strip. B. Time modulation of the metasurface: serrodynefrequency translation
It is instructive to first consider the analysis of themetasurface when it is uniformly biased across all unitcells. In this case, there is no spatial variation in the ho-mogenized model and the reflected power will will spreadinto discrete frequency harmonics due to the periodictime variation of the reflection phase. The tangential in-cident and reflected fields above the metasurface ( z = 0 + )can be expanded into frequency harmonics as E inc t = V inc0 e j ( ω t − k x x ) , (5) E ref t = ∞ X m = −∞ V ref m e jmω p t e j ( ω t − k x x ) , (6) H inc t = I inc0 e j ( ω t − k x x ) , (7) H ref t = ∞ X m = −∞ I ref m e jmω p t e j ( ω t − k x x ) , (8)where ω is the radial frequency of the incident wave, ω p is the radial frequency of the modulation, and k x is thetangential wavenumber of the incident wave. The inci-dent and reflected electric field harmonics can be written B i a s C a p ac it a n ce ( p F ) -200-1000100200 R e f l ec ti on ph a s e ( D e g r ee ) (a) -50-40-30-20-1009.99975 10 10.00025 Frequency (GHz) (b) B i a s C a p ac it a n ce ( p F ) -200-1000100200 R e f l ec ti on ph a s e ( D e g r ee ) (c) -50-40-30-20-1009.99975 10 10.00025 Harmonics (d)
FIG. 4. (a) Calculated capacitance modulation of the time-modulated metasurface for TE polarization. (b) Analyti-cal reflection spectrum of the homogenized, lossless, time-modulated metasurface for TE polarization. (c) Calculatedcapacitance modulation of the time-modulated metasurfacefor TM polarization. (d) Analytical reflection spectrum of thehomogenized, lossless, time-modulated metasurface for TMpolarization. in vector form as V inc and V ref respectively. The vector V inc represents the incident tangential electric field andcontains only a single entry ( V inc m = V inc0 δ m ) since the in-cident field is monochromatic. The vector V ref containsall the reflected tangential electric field coefficients V ref m .Based on the detailed derivation in Supplemental Mate-rial I and II, the reflected electric field can be calculatedfor each polarization, V ref = ( Y T X + Y T X ) − ( Y T X − Y T X ) V inc , (9)where the superscript “X” is “E” for TE polarized wavesand “M” for TM polarized waves. Y T X is the free-space tangential wave admittance matrix. It is a diag-onal matrix containing entries of the free-space admit-tances at the corresponding frequency harmonics. Y T X is the input admittance matrix of the time modulatedmetasurface. It is not a diagonal matrix since the time-modulated capacitive sheet introduces coupling betweendifferent harmonics.This analysis procedure can be used to predict the scat-tered field for arbitrary time-periodic modulating wave-forms. Suppose the reflection phase is modulated by asawtooth waveform. In this case, serrodyne frequencytranslation is expected [12]. The frequency of the in-cident wave is f = 10 GHz and the modulation fre- (a) (b) FIG. 5. The staggered modulation scheme of the spatio-temporally modulated metasurface. (a) The designed meta-surface with a spatio-temporal bias. The unit cell size is d = 6 mm and the spatial modulation period is d . (b) Ho-mogenized model of the spatio-temporally modulated meta-surface. The substrate is modeled as a uniaxial, anisotropicmaterial. quency is f p = 25 kHz. The incident wave impinges onthe metasurface at an oblique angle of 25 ◦ . The capaci-tance modulation needed to upconvert the wave to f + f p is calculated in Supplemental Material I, and shown inFig. 4 for each polarization. The calculation includes 141temporal harmonics for the field expansion and 101 tem-poral harmonics for the capacitance modulation. The re-flected spectra for the two orthogonal polarizations reveala Doppler shift to a frequency of f + f p . For TE polar-ization, a 0 .
107 dB conversion loss and 22 .
83 dB sidebandsuppression are achieved. For TM polarization, a 0 . .
64 dB sideband suppressionare achieved. As mentioned earlier, the equivalent cir-cuit model of the unit cell provides a phase range that isslightly less that 2 π (1 . π for TE polarization and 1 . π for TM polarization used in the analysis), resulting inundesired sidebands. For TM polarization, the reflectionphase range is slightly smaller than TE at the obliqueangle of 25 ◦ , resulting a slightly higher conversion loss. C. Space-time modulation of the metasurface
A space-time gradient can be applied to the metasur-face by introducing a time delay between the capacitancemodulation applied to adjacent columns. This staggeredmodulation scheme is shown in Fig. 5. It provides a dis-cretized travelling wave (N-path) modulation. The ca-pacitance modulation on each path is chosen to eitherproduce a sawtooth or sinusoidal reflection phase withrespect to time. In this staggered modulation scheme,there are N columns of subwavelength unit cells withinone spatial modulation period d . This impresses a mod-ulation wavenumber β p = 2 π/d onto the metasurface. N adjacent columns of the metasurface are modulated withbias signals staggered in time by an interval T p /N , where T p = 1 /f p is the temporal modulation period. One canview each column as a path in a N-path network. The ca-pacitance on each path v is related to that of the adjacent (a) (b) FIG. 6. Modulation scheme for the spatio-temporally modu-lated metasurface. The unit cell dimension is d = 6 mm andthe spatial modulation period is d = Nd . (a) A 2-path mod-ulation scheme ( d = 2 d ). (b) A 3-path modulation scheme( d = 3 d ). path by a time delay, C v ( t, x ) = C v − (cid:18) t − T p N , x − dN (cid:19) . (10)Here C v ( t, x ) is pulse function in space, and periodicfunction in time (see Supplemental Material III).In contrast to an N-path circuit, the paths (columns ofunit cells) are not connected to a common input and out-put. Instead, each path is displaced by a subwavelengthdistance d = 6 mm ( d = λ /
5) from its adjacent paths.Examples of 2- and 3-path spatio-temporal modulationschemes are shown in Fig. 6. At any given time, the spa-tial variation of the reflection phase is a discretized saw-tooth (blazed grating) ranging from 0 to approximately2 π over a period d = N d .As shown in Supplemental Materials III and IV, thecapacitance relationship given by Eq. (10) allows thespatio-temporally modulated sheet capacitance to be ex-panded in the following form, C ( t, x ) = ∞ X r = −∞ ∞ X q = −∞ C rq e jq ( ω p t − β p x ) e − jrβ d x , (11)where β p = 2 π/d is the modulation wavenumber, and β d = 2 π/d = N β p is an additional wavenumber whichresults from the discretization of the spatial modulationinto paths (unit cells). The summation over r accountsfor the discontinuity in capacitance at the the boundaryof each path as well as the microscopic variation of capac-itance within the paths (which in this case is constant).The summation over q accounts for the macroscopic ca-pacitance variation over one spatial modulation period d .The sheet capacitance of each path, is capable of gener-ating a staggered sawtooth reflection phase in time, asshown in Fig. 7.The N-path symmetry of the system establishes a rela-tion between the fields on adjacent paths [28]. The totalelectric field must satisfy E ( t, x, y, z ) = e j ( ω TpN − kxdN ) E ( t − T p N , x − dN , y, z ) . (12) (a) (b) FIG. 7. Space-time modulation scheme for the spatio-temporally modulated metasurface. The temporal modula-tion on each path is chosen to generate a sawtooth reflectionphase varying from 0 to 2 π over each period. The N adja-cent paths of the metasurface are modulated by bias signalsstaggered in time by T p /N . (a) A 2-path modulation scheme( d = 2 d ). (b)A 3-path modulation scheme ( d = 3 d ). Eq. (12) is used in Supplemental Materials IV to showthat the fields can be expanded in terms of a modifiedFourier series. At ( z = 0 + ), the tangential incident andreflected fields on the metasurface can be expressed as E inc t = V inc00 e j ( ω t − k x x ) , (13) E ref t = M X r,q = − M V ref rq e − jrβ d x e jq ( ω p t − β p x ) e j ( ω t − k x x ) , (14) H inc t = I inc00 e j ( ω t − k x x ) , (15) H ref t = M X r,q = − M I ref rq e − jrβ d x e jq ( ω p t − β p x ) e j ( ω t − k x x ) . (16)In Eq. (13-16), spatio-temporal harmonic pair ( r, q ) ofthe electromagnetic field on the surface has a tangentialwavenumber k xrq = qβ p + rβ d + k x = ( q + rN ) β p + k x , (17)and a corresponding radial frequency ω rq = ω + qω p , (18)where N is the number of paths within a spatial period ofthe metasurface. It can be seen that the staggered modu-lation between paths impresses a tangential wavenumberof qβ p onto the q th frequency harmonic. The reflectedangle of each harmonic pair is equal to θ rq = arcsin k xrq ω rq /c = arcsin (( q + rN ) β p + k sin θ i ) ω rq /c (19) 𝛽ω 𝑘 𝑥 𝜔 𝜔 + 𝜔 𝑝 𝛽 𝑝 ∙ ∙ Incident harmonic+1 harmonic (a) 𝛽 ω 𝑘 𝑥 𝜔 𝜔 + 𝜔 𝑝 𝛽 𝑝 ∙ ∙ Incident harmonic +1 harmonic (b)
FIG. 8. Graphic representation of the spatial and temporalfrequency shifts, for different modulation wavenumbers. (a)The modulation wavenumber β p is large. (b) The modulationwavenumber β p is small. The coefficients of the incident/reflected electric andmagnetic fields are related by the free-space tangentialwave admittance defined for each spatio-temporal har-monic pair. Meanwhile, the spatio-temporally modu-lated sheet capacitance provides coupling between dif-ferent harmonic pairs.To solve for the scattered field, the incident and re-flected tangential electric field harmonics are once moreorganized into vectors, V inc and V ref . Each entry corre-sponds to a unique spatio-temporal harmonic pair ( r, q ).The reflected electric field can then be calculated for eachpolarization using Eq. (9). A detailed derivation of theentries of the metasurface admittance matrix Y T X andthe free-space tangential admittance matrix Y T X is pro-vided in Supplemental Materials IV.Note that when the unit cell size is infinitesimally small( d (cid:28) d ), the variation of field across a unit cell is negli-gible. Therefore the harmonic pairs that remain are onlythose with r = 0. The capacitance modulation of themetasurface can thus be seen as continuous, C ( t, x ) = ∞ X q = −∞ C q e jq ( ω p t − β p x ) . (20)For such a modulation, the metasurface supports har-monics at frequency f + qf p , with a correspondingwavenumber k x + qβ p . Note that (20) is of the form of atraveling wave, C ( t, x ) = C ( t − x/v p ), where v p = ω p /β p .This case models the continuum limit, in which the spa-tial discretization of the traveling wave bias can be ne-glected. With the modulation waveform (sawtooth re-flection phase in time) shown in Fig. 4a and Fig. 4c,the metasurface converts the incident wave at ( f , k x )to ( f + f p , k x + β p ), as shown in Fig. 8b. When k x + β p > ( ω + ω p ) /c , the metasurface can convert an in-cident wave to a surface wave (as shown in Fig. 8a), pro-vided that the corresponding surface wave is supportedby the metasurface. However, when the correspondingsurface wave is not supported, the metasurface reflectsall the power back in the specular direction at the samefrequency f .More generally, the transverse resonance condition canbe used to identify surface modes supported by the spa-tially discretized metasurface. For harmonic pairs with 𝐶(𝑡 − 1 𝑁 𝑇 𝑝 ) 𝑘 𝑠𝑧 , 𝑍 𝑠 𝐶(𝑡) 𝑘 𝑠𝑧 , 𝑍 𝑠 𝑙 𝑙 𝑍 ⋮ 𝐶(𝑡 −
𝑁 − 1
𝑁 𝑇 𝑝 ) 𝑘 𝑠𝑧 , 𝑍 𝑠 𝑙 𝜔 ⋮ ⋮ 𝜔 𝜔 + 𝑁𝜔 𝑝 𝜔 − 𝑁𝜔 𝑝 𝜔 − 𝑟𝑁𝜔 𝑝 𝜔 + 𝑟𝑁𝜔 𝑝 FIG. 9. Equivalent circuit model of the spatio-temporallymodulated metasurface when the spatial modulation periodis much smaller than the wavelength of radiation. tangential wavenumbers larger than free space ( k xrq >ω rq /c ), the transverse resonance condition can be used tojudge if the corresponding surface waves are supported.det ( Y T X + Y T X ) = 0 , (21)Solving Eq. (21) yields the ω − k x dispersion relationshipfor the supported surface wave. Note that when a surfacewave is supported, the reflection coefficient V ref /V inc inEq. (9) diverges. This is not the case for the incident an-gle and path number N combinations considered in thispaper. Thus, for all the proceeding examples presentedin this paper, a surface wave is not supported by themetasurface.For our metasurface, the unit cell size is chosen to be d = λ /
5. Since the unit cell size of the metasurface isfixed, the total spatial period d = N d can be controlledby changing N , the number of paths. This enables thesame metasurface to achieve different functions depend-ing on the number of paths included in a spatial period. Ifthe modulation period d is electrically small ( N is small),then the spatial modulation wavenumber β p is large. Asdepicted in Fig. 8a, this can lead to a number of higherorder harmonics existing outside the light cone. Since thespatial modulation period is electrically small, the pathscan be viewed as collocated and a N-path circuit modelcan be used to approximate the physical structure. Theequivalent circuit model of the spatio-temporally modu-lated metasurface for d (cid:28) λ is depicted in Fig. 9. If thespatio-temporally modulated metasurface does not sup-port surfaces waves at the operating frequency, power isonly coupled to radiating harmonics: those within thelight cone. Based on Eq.(17), the radiated harmonics arethose with: q + rN = 0 . (22)Since r ∈ Z , Eq. (22) implies that propagating harmonicscorrespond to q = 0 , ± N, ± N . . . . Therefore, the radi-ated reflected wave only contains frequency harmonics at f + rN f p , where r ∈ Z . This phenomena can only be ob-served when the spatial discretization of the metasurfaceis considered. In the continuum limit, sub-wavelengthspatial modulation results in specular reflection at thesame frequency as the incident wave. However, the spa-tial discretization introduces additional spatial harmon-ics (the summation over r in (13-16)) that can coupleto the incident wave. As a result, the metasurface canachieve subharmonic frequency translation. Note thatthe tangential wavenumbers of the radiated subharmonicmixing terms are all equal to that of the incident tangen-tial wavenumber (since k xrq = k x , when q + N r = 0).With the capacitance modulation shown in Fig. 4a and4c, the sawtooth reflection phase on each path enablesthe metasurface to upconvert the frequency to the firstpropagating harmonic pair. In this case, the metasurfaceperforms subharmonic frequency translation from f to f + N f p .When the modulation period d is electrically large ( N is large), the spatial modulation wavenumber β p is small,as depicted in Fig. 8b. When N is a very large value,according to Eq. (S. 39) in the Supplemental Material,the capacitance coefficient C rq is zero for r = 0. Forthis case, the field variation across each unit cell is small,and the capacitance modulation waveform is simplified tothe continuum limit given by Eq. (20). In other words,the metasurface shows a similar performance to one withan infinitely small unit cell size. Serrodyne frequencytranslation to a deflected angle can be achieved usingthe sawtooth waveform given in Fig. 4.In addition, when the modulation period d is on the or-der of a wavelength, the metasurface can simultaneouslyperform subharmonic frequency translation and angulardeflection. The deflected angle of the harmonic pair ofinterest is given by Eq. (19). Setting θ rq = − θ i inEq. (19) yields an expression for the incidence anglesat which retroreflection occurs for a particular spatio-temporal harmonic. The deflective and retroreflectivebehavior of the metasurface is showcased in Section IIIfor various scenarios. III. SCATTERING FROM A SPACE-TIMEMODULATED METASURFACE FORDIFFERENT SPATIAL MODULATION PERIODS
Computed results are given here for various spatio-temporal modulation cases. In Section III A, space-timemodulation schemes are designed to achieve subharmonicfrequency translation. The spatial modulation period iskept electrically small. In Section III B, the spatial mod-ulation period is electrically large such that beam deflec-tion and frequency translation can be achieved simultane-ously. Finally, in Section III C, the spatially modulationperiod is on the order of a wavelength, allowing simulta-neous retroreflection and subharmonic frequency transla-tion. For each of the cases that follow, the conversion lossand sideband suppression for the desired frequency har-monic at a given observation angle are provided in TableI. The table will be referred to throughout this section.In all of the cases studied, the incident signal frequency
TABLE I. Simulated conversion loss and sideband suppressionto desired reflected frequency harmonic f given: N - thenumber of paths, θ i - the incident angle, θ obs - the observationangle, and the temporal phase modulation waveform (eithera sawtooth or a sinusoid). Note that positive values of θ i and θ obs correspond to waves traveling along the positive x direction.Ex. N θ i θ obs Wave-form f ConversionLoss(dB) SidebandSuppression(dB)TE TM TE TM0 1 25 ◦ ◦ saw f + f p ◦ ◦ saw f + 2 f p ◦ ◦ saw f + 3 f p ◦ ◦ saw f + f p ◦ -25 ◦ saw f + f p ◦ ◦ saw f + f p ◦ -39 ◦ saw f + 3 f p ◦ ◦ saw f + f p ◦ -39 ◦ sin f − f p ◦ ◦ sin f + f p is f = 10 GHz. The modulation frequency, f p = 25kHz, which is the maximum frequency which could beexperimentally validated using the available equipment(see Section IV). For each angle of incidence, the capac-itance modulation is calculated based on Eq. (S.4) toachieve the desired time-varying reflection phase. Unlessspecifically stated otherwise, the reflection phase of eachcolumn (path) is a sawtooth function in time. For bothpolarizations, the field is expanded into 141 ×
141 har-monic pairs. The temporal capacitance modulation oneach path is truncated to 101 temporal harmonics.
A. Small spatial modulation period ( | k x ± β p | > k ) In this section, electrically small spatial modulationperiods ( | k x ± β p | > k ) are considered. In this regime,both the +1 ( k x + β p ) and − k x − β p ) spatial harmonicsare outside of the light cone. Since the unit cell size ofthe metasurface is fixed to d = λ /
5, 2- and 3-pathmodulation ( N = 2 ,
3) are chosen to satisfy the smallperiod condition. The incident wave is chosen to impingeon the metasurface with an oblique angle of 25 ◦ . Themodulation schemes for the 2-path ( N = 2) and 3-path( N = 3) examples are shown in Fig. 6 and Fig. 7.As explained in section II C, the metasurface performssubharmonic frequency translation at the specular angle. -50-40-30-20-1009.99975 10 10.00025 Frequency (GHz) (a) -50-40-30-20-1009.99975 10 10.00025
Frequency (GHz) (b) -50-40-30-20-1009.99975 10 10.00025
Frequency (GHz) (c) -50-40-30-20-1009.99975 10 10.00025
Frequency (GHz) (d)
FIG. 10. Analytical reflection spectrum of the homogenized,lossless, spatio-temporally modulated metasurface. (a) 2-path( N = 2) modulation for TE polarization. (b) 3-path ( N = 3)modulation for TE polarization. (c) 2-path ( N = 2) modula-tion for TM polarization. (d) 3-path ( N = 3) modulation forTM polarization.
1. Case 1: Reflective (specular) subharmonic frequencytranslation
For 2- and 3-path modulation, the reflection spectracomputed from Eq. (9) are shown in Fig. 10. As ex-pected, the strongest reflected harmonic is at a frequencyof f + N f p . For the case of subharmonic frequency trans-lation, all the reflected propagating harmonics share thesame tangential wavenumber as the incident wave. Sincethe modulation frequency is much lower than the incidentfrequency, f p (cid:28) f , all the harmonics are at a reflectionangle of 25 ◦ , as depicted in Fig 11. If the modulationfrequency f p is comparable with f , then each of the re-flected, propagating frequency harmonics will have dif-ferent radiated angles due to their substantially differentfree space wavenumbers. 𝑥 𝑧𝜃 𝑓 , 𝑘 𝑥 𝐶 (𝑡,𝑥) 𝑙 𝐶 𝑁 (𝑡, 𝑥)⋯ 𝐶 (𝑡,𝑥) ⋯ 𝜖 𝑟 𝜃 𝑓 + 𝑁𝑓 𝑝 , 𝑘 𝑥 FIG. 11. Subharmonic frequency translation of the spatio-temporally modulated metasurface. For the presented meta-surface, this can be achieved when N = 2 or 3. The spectra for both polarizations, shown in Fig.10, clearly demonstrate subharmonic frequency trans-lation, where reflected harmonics are only radiated at f = f + rN f s with r ∈ Z . Doppler-like frequency transla-tions are observed for both polarizations, where the dom-inant propagating reflected wave is at frequency f + N f p .The conversion loss and sideband suppression for bothpolarizations using 2-path and 3-path modulation areprovided in examples 1 and 2 of Table I. As mentionedearlier, the unit cell provides a phase range that is slightlysmaller than 2 π , resulting in conversion loss and unde-sired sidebands. It can be seen that, as the convertedfrequency harmonic (which is equal to the path number N in this case) is increased, the conversion loss increasesand the sideband suppression decreases. This is becausethe N-path metasurface upconverts the frequency to thefirst propagating harmonic pair. The higher the upcon-verted frequency, the longer this process takes and thelarger the conversion loss due to the formation of side-bands that results from the imperfect reflection phaserange. B. Large spatial modulation period ( | k x ± β p | < k ) In this section, we consider two cases where the mod-ulation period d is larger than the free space wavelength λ ( N is large). In this regime, both the +1 ( k x + β p )and − k x − β p ) spatial harmonics are inside the lightcone. In addition, when N is a large value, the harmonicpairs ( r, q ) with r = 0 dominate (Eq. (S.38)). In thefirst case, the metasurface exhibits serrodyne frequencytranslation to a deflected angle. In the second case, theincident angle is specifically chosen to achieve serrodynefrequency translation in retroreflection.
1. Case 2: Deflective Serrodyne frequency translation
First, let us consider the example shown in Fig. 12a,where a wave is incident at an angle θ = 25 ◦ and thenumber of paths is large, N = 20. From Eq. (17), thetangential wavenumbers of the reflected harmonic pairs 𝑥 𝑧𝜃 𝑓 , 𝑘 𝑥 𝐶 (𝑡,𝑥) 𝑙 𝐶 𝑁 (𝑡, 𝑥)⋯ 𝐶 (𝑡,𝑥) ⋯ 𝜖 𝑟 𝜃 𝑓 + 𝑓 𝑝 ,𝑘 𝑥 + 𝛽 𝑝 (a) 𝑥 𝑧𝜃 𝑓 ,−𝑘 𝑥 − 𝛽 𝑝 𝐶 (𝑡,𝑥) 𝑙 𝐶 𝑁 (𝑡, 𝑥)⋯ 𝐶 (𝑡,𝑥) ⋯ 𝜖 𝑟 𝜃 𝑓 + 𝑓 𝑝 ,−𝑘 𝑥 (b) FIG. 12. The spatio-temporally modulated metasurface per-forming serrodyne frequency translation to a deflected angle.The path number
N >
5. The modulation frequency f p ismuch lower than the incident frequency f . (a) Wave is inci-dent at an oblique angle θ . (b) Wave is incident at an obliqueangle − θ . -50-40-30-20-1009.99975 10 10.00025 Frequency (GHz) (a) -50-40-30-20-1009.99975 10 10.00025
Frequency (GHz) (b)
FIG. 13. Analytical reflection spectrum of the homogenized,lossless, spatio-temporally modulated metasurface, with anincident angle of 25 ◦ .(a) 20-path ( N = 20) modulation forTE polarization. (b) 20-path ( N = 20) modulation for TMpolarization. -50-40-30-20-1009.99975 10 10.00025 Frequency (GHz) (a) -50-40-30-20-1009.99975 10 10.00025
Frequency (GHz) (b)
FIG. 14. Analytical reflection spectrum of the homogenized,lossless, spatio-temporally modulated metasurface, with anincident angle of − ◦ . (a) 20-path ( N = 20) modulation forTE polarization. (b) 20-path ( N = 20) modulation for TMpolarization. are given by k xq | r =0 = qβ p + k sin(25 ◦ )= q k + k sin(25 ◦ ) , (23)given that d = 20 d = 4 λ . The harmonics located insidethe light cone (propagating harmonics) are those with q = 0 , ± , ± , − , − , −
5. For the capacitance variationshown in Fig. 4a and Fig. 4c, the metasurface acts asa serrodyne frequency translator. It upconverts the in-cident wave to the harmonic pair ( r = 0 , q = 1) withfrequency f = f + f p . Note that in this case, each radi-ated harmonic has its own tangential wavenumber, andthus reflects at a different angle given by Eq. (19). Theharmonic of interest ( f + f p ) reflects to θ = 42 ◦ , asshown in Fig. 12a. The reflected spectra for both po-larizations are given in Fig. 13. The conversion loss andsideband suppression for both polarizations are providedin example 3 of Table I.Let’s consider another example (shown in Fig. 12b) 𝛽 ω −𝑘 𝑥 𝜔 𝜔 + 𝜔 𝑝 𝛽 𝑝 ∙ ∙ Incident harmonic +1 harmonic 𝑘 𝑥 (a) 𝑥 𝑧 𝑓 , −𝑘 𝑥 𝐶 (𝑡,𝑥) 𝑙 𝐶 𝑁 (𝑡, 𝑥)⋯ 𝐶 (𝑡,𝑥) ⋯ 𝜖 𝑟 𝜃 𝑓 + 𝑓 𝑝 , 𝑘 𝑥 (b) FIG. 15. (a) Graphic representation of the spatial and tem-poral frequency shift, for a relatively large path number N .(b) Corresponding retroreflection performance for a relativelylarge path number N . where the spatio-temporal modulation of the metasur-face and incident frequency are kept the same, but theincident and reflected angles are swapped. The incidentangle is θ = − ◦ . Each radiated (propagating) har-monic pair of the reflected field has tangential wavenum-ber k xrq = q k − k sin(42 ◦ ) . (24)where q = 0 , ± , , , , ,
6. The metasurface frequencytranslates the incident signal at f to the harmonic pair( r = 0 , q = 1), which is at frequency f = f + f p . The tan-gential wavenumber of the harmonic pair ( r = 0 , q = 1)can be easily calculated as − k sin(25 ◦ ). When modula-tion frequency f p is comparable to incident frequency f ,the reflection angle can differ from θ = − ◦ due to thesignificant change in the free-space wavenumber at thereflected frequency [14], as shown in Supplemental Ma-terial V. However, modulation frequency here is f p = 25kHz, which is far smaller than the incident frequency of f = 10 GHz. As a result, the reflection angle is − ◦ .The reflection spectra for both polarizations are given inFig. 14. From example 4 of Table I, it can be seen thatthe conversion loss and sideband suppression are nearlyidentical to those of the previous example (shown in ex-ample 3 of Table I).
2. Case 3: Retroreflective serrodyne frequency translation
Here, we consider the case where the incident angle ischosen to achieve retroreflection. According to Eq. (19),setting θ rq = − θ i yields an expression for the incidenceangles at which retroreflection occurs for the convertedspatio-temporal harmonic. For this case, the modulationwavenumber β p = 2 k x , as shown in Fig. 15. The reflectedwave propagates back to the source with an upconvertedfrequency. The retroreflection angle θ i can be calculatedby solving θ , = − θ i in Eq. (19), θ i = − arcsin β p k = − arcsin λ N d . (25)Here, the number of paths is chosen to be N = 20, andthe retroreflection angle is calculated to be θ i = − . ◦ .0 -50-40-30-20-1009.99975 10 10.00025 Frequency (GHz) (a) -50-40-30-20-1009.99975 10 10.00025
Frequency (GHz) (b)
FIG. 16. Analytical reflection spectrum of the homogenized,lossless, time-modulated metasurface, for an incident angle of7 . ◦ . (a) 20-path ( N = 20) modulation for TE polarization.(b) 20-path ( N = 20) modulation for TM polarization. The calculated reflection spectra for both polarizationsare shown in Fig. 16. The spectra clearly show a Dopplershift to frequency f + f p . The conversion loss and side-band suppression for both polarizations are provided inexample 5 of Table I. Note that in Fig. 16, only theharmonic pair ( r = 0 , q = 1) (at frequency f + f p ) isretroreflective. The reflection angle of other harmonicscan be calculated based on Eq. (19). C. Wavelength-scale spatial modulation period( | k x | + | β p | > k , & || k x | − | β p || < k ) In this section, we consider three cases where the spa-tial modulation period is on the order of the wavelengthof radiation ( | k x | + | β p | > k & || k x | − | β p || < k ). In thisregime, either the +1 ( k x + β p ) or the − k x − β p ) spatialharmonic is inside the light cone. For the fixed unit cellsize of d = λ /
5, 4-path modulation ( N = 4) is chosento satisfy the wavelength-scale period condition. In thefirst case, the metasurface exhibits simultaneous subhar-monic frequency translation and deflection. In the secondcase, the incident angle is specifically chosen to achievesubharmonic frequency translation in retroreflection. Inthe last case, we show that the retroreflective frequencycan be switched by changing the temporal modulationwaveform to sinusoidal.
1. Case 4: Deflective/retroreflective subharmonic frequencytranslation
First, let us consider the example shown in Fig. 17a,where a wave is incident on the metasurface with a pos-itive k x value. According to Eq. (17), the radiated har-monics are those with: q + rN = 0 or − . (26)Eq. (26) implies that the radiated reflected wave containsfrequency harmonics at f + rN f p and f + ( rN − f p , 𝛽 ω 𝜔 𝜔 + 𝜔 𝑝 𝛽 𝑝 ∙ ∙ Incident harmonic +1 harmonic 𝑘 𝑥 -1 harmonic 𝜔 − 𝜔 𝑝 ∙ 𝛽 𝑝 −𝑘 𝑥 (a) 𝛽 ω 𝜔 𝜔 + 𝜔 𝑝 𝛽 𝑝 ∙ ∙ Incident harmonic +1 harmonic 𝑘 𝑥 -1 harmonic 𝜔 − 𝜔 𝑝 ∙ 𝛽 𝑝 −𝑘 𝑥 (b) FIG. 17. Graphic representation of the spatial and temporalfrequency shifts for a path number N = 4. (a) The incidenttangential wavenumber k x is positive. (b) The incident tan-gential wavenumber k x is negative. where r ∈ Z . Under a capacitance variation that gener-ates sawtooth reflection phase, the reflected wave is up-converted to the first radiated frequency harmonic, whichin this case is the harmonic pair ( r = − , q = 3). There-fore, the reflected wave is Doppler shifted to a frequency f + 3 f p . In addition, we choose an incident angle suchthat the wave is retroreflected. As explained in SectionIII B 2, the modulation wavenumber is set to β p = 2 k x (see Fig. 17a). The retroreflection angle can be calcu-lated by setting θ − , = − θ i in Eq. (19), which is 39 ◦ fora path number of N = 4.The calculated retroreflection spectra are shown in Fig.18a and 18c. Doppler-like frequency translation to fre-quency f + 3 f p occurs for the incident angle of 39 ◦ , forboth polarizations. The frequency of interest f +3 f p ex-hibits retroreflective subharmonic frequency translation.The conversion loss and sideband suppression for bothpolarizations are provided in example 6 of Table I. Notethat in Fig. 18a and 18c, only the harmonics representedby a solid line are propagating in the retroreflective di-rection. The harmonics represented by dashed lines arepropagating in the specular direction.Note that, with a wavelength-scale spatial modulationperiod, the performance of the metasurface is direction-dependent. When the incident angle is − ◦ , as shownin Fig. 17b, it is clear that the radiated harmonics arethose with: q + rN = 0 or 1 . (27)In this case, the harmonic pair ( q = 1 , r = 0) is in-side the light cone. Therefore, the metasurface performsserrodyne frequency translation: upconversion to a a fre-quency f + f p . Since β p = 2 k x , the frequency of interest f + f p is also retroreflected. The calculated reflectionspectra are shown in Fig. 18b and 18d. Doppler-likefrequency translation to frequency f + f p is observedfor both polarizations. The conversion loss and side-band suppression for both polarizations are provided inexample 7 of Table I. An illustration of the direction-dependent retroreflective behavior of the metasurface,with a wavelength-scale spatial modulation period, is de-picted in Fig. 19a.1 -50-40-30-20-1009.99975 10 10.00025 Frequency (GHz) (a) -50-40-30-20-1009.99975 10 10.00025
Frequency (GHz) (b) -50-40-30-20-1009.99975 10 10.00025
Frequency (GHz) (c) -50-40-30-20-1009.99975 10 10.00025
Frequency (GHz) (d)
FIG. 18. Analytical retroreflection spectrum of the homog-enized, lossless, spatio-temporally modulated metasurface.The harmonics denoted by solid lines retroreflect. The har-monics denoted by the dashed lines reflect in the specular di-rection. (a) 4-path ( N = 4) modulation for TE polarization,for an incident angle of 39 ◦ . (b) 4-path ( N = 4) modula-tion for TE polarization, for an incident angle of − ◦ . (c)4-path ( N = 4) modulation for TM polarization, for an in-cident angle of 39 ◦ . (d) 4-path ( N = 4) modulation for TMpolarization, for an incident angle of − ◦ . 𝑥 𝑧 𝑓 , −𝑘 𝑥 𝐶 (𝑡,𝑥) 𝑙 𝐶 𝑁 (𝑡, 𝑥)⋯ 𝐶 (𝑡,𝑥) ⋯ 𝜖 𝑟 𝜃 𝑓 + 𝑓 𝑝 , 𝑘 𝑥 𝜃𝑓 , 𝑘 𝑥 𝑓 + 3𝑓 𝑝 , −𝑘 𝑥 (a) 𝑥 𝑧 𝑓 , −𝑘 𝑥 𝐶 (𝑡,𝑥) 𝑙 𝐶 𝑁 (𝑡, 𝑥)⋯ 𝐶 (𝑡,𝑥) ⋯ 𝜖 𝑟 𝜃 𝑓 + 𝑓 𝑝 , 𝑘 𝑥 𝜃𝑓 , 𝑘 𝑥 𝑓 − 𝑓 𝑝 , −𝑘 𝑥 (b) FIG. 19. Retroreflective performance of the spatio-temporallymodulated metasurface. The path number is N = 4, and theretroreflection angle is θ = 39 ◦ . (a) The sheet capacitancegenerates a reflection phase on each column that is a sawtoothwith respect to time. (b) The capacitance modulation on eachcolumn generates sinusoidal reflection phase with respect totime.
2. Case 5: Retroreflective frequency translation with astaggered sinusoidal reflection phase
Here, retroreflection is also achieved using a sheet ca-pacitance that generates a staggered sinusoidal reflectionphase with respect to time on adjacent columns. Thecapacitance modulation waveform is shown in Fig. 20aand Fig. 20c for each polarization. Each column of B i a s C a p ac it a n ce ( p F ) -200-1000100200 R e f l ec ti on ph a s e ( D e g r ee ) (a) -50-40-30-20-1009.99975 10 10.00025 Frequency (GHz) (b) B i a s C a p ac it a n ce ( p F ) -200-1000100200 R e f l ec ti on ph a s e ( D e g r ee ) (c) -50-40-30-20-1009.99975 10 10.00025 Frequency (GHz) (d)
FIG. 20. (a) Calculated capacitance modulation for a sinu-soidal reflection phase versus time for TE polarization. (b)Analytical reflection spectrum of the homogenized, losslesstime-modulated metasurface for TE polarization. (c) Calcu-lated capacitance modulation for a sinusoidal reflection phaseversus time for TM polarization. (d) Analytical reflectionspectrum of the homogenized lossless time-modulated meta-surface for TM polarization. the metasurface generates a sinusoidal reflection phasein time. When all the columns of the metasurface arebiased with the same waveform, the reflection spectratake the form of a Bessel function, as shown in Fig. 20band Fig. 20d. The peak-to-peak modulation amplitudeis chosen to be 276 ◦ to suppress the zeroth harmonic inreflection [26]. Unlike the sawtooth modulation, the si-nusoidal reflection phase excites both +1 ( r = 0 , q = 1)and − r = 0 , q = −
1) frequency harmonics.Fig. 17 shows that for a positive k x incident wavenum-ber, the reflected +1 frequency harmonic is outside ofthe light cone ( | k x + β p | > k ), and the refleted -1 fre-quency harmonic is inside the light cone ( | k x − β p | < k ).Since the +1 frequency harmonic does not radiate and isnot supported by the metasurface as a surface wave, thepower is reflected from the metasurface with a frequencyof f − f p . In other words, for a wavelength-scale spa-tial modulation period, the metasurface supports single-sideband frequency translation with the sinusoidal mod-ulation. When the incident angle is 39 ◦ , the frequency ofinterest f − f p is retroreflective. Again, the retroreflec-tive behavior of the metasurface is directionally depen-dent. When the incident angle is − ◦ , the +1 frequencyharmonic is inside the light cone, while the − -50-40-30-20-1009.99975 10 10.00025 Frequency (GHz) (a) -50-40-30-20-1009.99975 10 10.00025
Frequency (GHz) (b) -50-40-30-20-1009.99975 10 10.00025
Frequency (GHz) (c) -50-40-30-20-1009.99975 10 10.00025
Frequency (GHz) (d)
FIG. 21. Analytical retroreflection spectrum of the homog-enized, lossless, spatio-temporaaly modulated metasurface.The capacitance modulation generates a sinusoidal reflectionphase on each path. The harmonics denoted by the solid linesare propagating in the retroreflective direction. The harmon-ics denoted by the dashed lines are propagating in speculardirection. (a) 4-path ( N = 4) modulation for TE polariza-tion, for an incident angle of 39 ◦ . (b) 4-path ( N = 4) mod-ulation for TE polarization, for an incident angle of − ◦ .(c) 4-path ( N = 4) modulation for TM polarization, for anincident angle of 39 ◦ . (d) 4-path ( N = 4) modulation for TMpolarization, for an incident angle of − ◦ . itance variation shown in Fig. 20a and 20c, the retrore-flection wave is radiated at a frequency f + f p . For thiscase, the direction-dependent retroreflective behavior ofthe metasurface is depicted in Fig. 19b. The calculatedreflection spectra are shown in Fig. 21. As expected,Doppler-like frequency translation to f − f p is observedfor an incident angle of 39 ◦ , and f + f p for incident angleof − ◦ . The conversion loss and sideband suppressionfor both polarizations are provided in examples 8 and 9of Table I.Note that the retroreflection angle for both of the lasttwo cases (with sawtooth and sinusoidal reflection phase)was ± ◦ for a path number of N = 4. By simply chang-ing the temporal modulation waveform, the retroreflec-tion frequency was changed between f − f p and f + 3 f p ,for the same incident angle of 39 ◦ . IV. METASURFACE DESIGN ANDFABRICATION
In this section, a prototype metasurface is describedand measurements are reported for several of the space-time modulation cases described earlier. Details of themetasurface realization, as well as the measurement setupused to characterize its performance, are given in SectionIV A. The static performance of the metasurface undervarious DC bias conditions is presented in Section IV B.Based on this static (DC) characterization, the requiredbias waveform the time-modulated metasurface is deter-mined in Section IV C. This section also includes the mea-sured reflection spectra for time-variation alone. Finally,measured results are given in Section IV D for several ofthe spatio-temporal modulation cases explored theoreti-cally in Section III.
A. Metasurface design and measurement setup
A unit cell of the dual-polarized, ultra thin (0 . λ )metasurface is shown in Fig. 2a. Varactor diodes(MAVR-000120-1411 from MACOM [40]) are integratedonto the metasurface to act as tunable capacitancesfor two orthogonal polarizations. The biasing networksfor each of these polarizations were printed behind theground plane of the unit cell, as shown in Fig. 22a andFig. 22c. Each bias layer consists of 28 metallic linesthat can independently modulate all 28 columns of themetasurface. A total of 3136 MAVR-000120-1411 var-actor diodes were mounted onto the metasurface. Thevaractor diodes are biased through vias located on thecenter of the metallic patches. A photo of the fabricatedmetasurface is shown in Fig. 22b. The metallic traces ofthe bias layers are routed to four D-SUB connectors edgemounted to the metasurface. Rogers 4003C ( (cid:15) r = 3 . (a) (b) Rogers 4450F
Rogers 4003C
Rogers 4003C
Capacitance layer
Rogers 4450F
Rogers 4003C
Bias layer for 𝑉 𝑏𝑖𝑎𝑠 𝑥 Bias layer for 𝑉 𝑏𝑖𝑎𝑠 𝑦 Ground plane 0.508 mm (c)
FIG. 22. (a) Transparent view of the metasurface prototypewith two bias layers independently controlling each polariza-tion. (b) Photograph of the fabricated metasurface. (c) Crosssection of the fabricated metasurface. FIG. 23. Photograph of the quasi-optical, free-space measure-ment system. and tan δ = 0 . . (cid:15) r = 3 . δ = 0 . . .
726 mm (0 . λ ).The metasurface was experimentally characterized us-ing the quasi-optical Gaussian beam system shown in Fig.23. In the experimental setup, the fabricated metasur-face is illuminated by a spot-focusing lens antenna (SAQ-103039-90-S1). The antenna excites a Gaussian beamwith a beamwidth of 50 mm at the focal length of 10cm. The width of the fabricated metasurface is largerthan 1 . f = 10 GHz was used asthe incident signal. The amplitude of the incident signalimpinging on the metasurface was measured to be − f p = 25 kHz. B. Measurements of a DC biased metasurface:tunable reflection phase
We will first look at the simulated and measured DCperformance of the proposed metasurface. The capaci-tance provided by the varactors ranges from 0 .
18 pF to1 . Frequency (GHz) -20-15-10-50 (a)
Frequency (GHz) -200-1000100200 (b)
Frequency (GHz) -20-15-10-50 (c)
Frequency (GHz) -200-1000100200 (d)
FIG. 24. Simulated reflection coefficient amplitude and phaseof the realized metasurface for a range of varactor capaci-tances. The incident wave is obliquely incident at an angle of25 ◦ . (a) Reflection amplitude for TE polarization. (b) Reflec-tion phase for TE polarization. (c) Reflection amplitude forTM polarization. (d) Reflection phase for TM polarization. pacitance and resistance values of the varactor diode wereextracted as a function of bias voltage from its SPICEmodel [40].The simulated reflection coefficients of the metasurfacefor various varactor capacitance values are given in Fig.24. The incident angle is set to 25 ◦ . At the operatingfrequency of 10 GHz, the reflection phase of the metasur-face can be varied from − . ◦ to 155 ◦ for TE polariza-tion, providing a maximum phase range of 336 . ◦ . ForTM polarization, the reflection phase of the metasurfacecan be varied from − . ◦ to 146 . ◦ , providing a maxi-mum phase range of 328 . ◦ . At the operating frequencyof 10 GHz, the simulated reflection amplitude for bothpolarizations remains greater than − .
41 dB for TE polarization, occurring for a varactorcapacitance of 0 .
313 pF. For TM polarization, the highestreturn loss at 10 GHz is 2 .
47 dB, occurring at a varactorcapacitance of 0 .
30 pF. The metasurface suffers higher4
Frequency (GHz) -20-15-10-50 (a)
Frequency (GHz) -200-1000100200 (b)
Frequency (GHz) -20-15-10-50 (c)
Frequency (GHz) -200-1000100200 (d)
FIG. 25. Measured reflection coefficient amplitude and phaseof the metasurface prototype for a range of bias voltages. Theincident wave is oblique at an angle of 25 ◦ . (a) Reflectionamplitude for TE polarization. (b) Reflection phase for TEpolarization. (c) Reflection amplitude for TM polarization.(d) Reflection phase for TM polarization. loss for TE polarization than TM polarization. This isbecause, at the incident angle of 25 ◦ , the value of the free-space tangential wave impedance for TE polarization iscloser (impedance matches better) to the purely resistiveinput impedance of the metasurface at resonance. Thesimulated cross-polarization behavior of the metasurfaceis lower than −
50 dB for all the orthogonal varactor ca-pacitance combinations.The static (DC biased) performance of the metasur-face was measured under an oblique angle of 25 ◦ . Themeasured TE and TM reflection coefficients under vari-ous bias voltages are given in Fig 25. The bias voltageused in measurement ranged from 0 V to 15 V, providinga varactor capacitance range of 0 .
18 pF to 1 . − . ◦ to 149 . ◦ for TE polarization, providing a phase rangeof 332 . ◦ . For TM polarization, the measured reflectionphase could be varied from − . ◦ to 147 . ◦ , providing aphase range of 324 . ◦ . At resonance, the measured reflec-tion amplitude was found to be much lower than in simu-lation, indicating higher losses in the fabricated metasur-face. This could be attributed to additional ohmic losswithin the diode as well as losses introduced by the tin-ning and soldering procedures used to mount the diodes.Nevertheless, the simulated and measured static perfor-mances of the metasurface are in good agreement. A de- Measured WaveformOptimized Waveform (a)
Frequency (GHz) -50-40-30-20-100 (b)
Measured WaveformOptimized Waveform (c)
Frequency (GHz) -50-40-30-20-100 (d)
FIG. 26. (a) Bias waveform of the time-modulated metasur-face for TE polarization. (b) Measured reflection spectrum ofthe time-modulated metasurface prototype for TE polariza-tion. (c) Bias waveform of the time-modulated metasurfaceprototype for TM polarization. (d) Measured reflection spec-trum of the time-modulated metasurface for TM polarization. tailed comparison between simulation and measurementfor each bias voltage is given in Supplemental MaterialsVI.A harmonic balance simulation with the Keysight ADScircuit solver was used to verify the theoretical analysisand compute reflection spectrum. However, to use theharmonic balance circuit solver, a circuit equivalent of thefabricated metasurface needed to be extracted for eachpolarization. The equivalent circuits are extracted fromfull-wave scattering simulations. A voltage-dependent re-sistance is added to it to account for the added lossesobserved in measurement. The equivalent circuits forthe two polarizations under an oblique incident angle of25 ◦ are given in Supplemental Materials VI. From theequivalent circuits, the capacitance modulation requiredto obtain a given reflection phase versus time dependencecan be obtained. C. Measurements of a time-modulatedmetasurface: serrodyne frequency translation
As discussed in Section II B, if all columns of the meta-surface are biased with the same modulation waveform,providing a sawtooth reflection phase versus time, themetasurface can perform serrodyne frequency transla-5tion. As shown in Fig. 24 and Fig. 25, the reflectionamplitude is not unity due to the loss in the metasur-face. Therefore, the capacitance modulation waveformhad to be numerically optimized. The detailed optimiza-tion process is detailed in Supplemental Materials VII.In the experiment, the optimized waveform was sam-pled at 20 data points per period T p = 40 µ sec, andthe sampled waveform was entered into the D/A con-verter. All channels of the D/A converter were syn-chronized with the same bias waveform. The bias wave-form across several diodes was measured using a differ-ential probe (Tektronix TMDP0200) and Tektronix os-cilloscope MDO3024. The optimized and measured biasvoltage waveforms are shown in Fig. 26a for TE polar-ization and Fig. 26c for TM polarization. The measuredreflection spectrum for an oblique angle of 25 ◦ is shownin Fig. 26b for TE polarization and Fig. 26d for TM po-larization. Both polarizations show serrodyne frequencytranslation to f = f + f p . For TE polarization, a 4 . .
196 dB of sideband suppressionare achieved. For TM polarization, a 3 .
67 dB conver-sion loss and 9 .
86 dB sideband suppression are achieved.For each polarization, the measured reflection spectrumin Fig. 26b and 26d generally agrees with harmonic bal-ance simulations of its extracted circuit models, as shownin Fig. S6 of Supplemental Materials VIII.
D. Measurements of a space-time modulatedmetasurface
As shown in Section III, various functions can berealized by spatio-temporally modulating the metasur-face, including specular subharmonic frequency transla-tion, deflective/retroreflective serrodyne frequency trans-lation, and deflective/retroreflective subharmonic fre-quency translation. Measured results are given here as avalidation of our analysis. Again, the incident frequency f and modulation frequency f p are set to 10 GHz and25 kHz, respectively. Unless stated otherwise, the reflec-tion phase of each column (path) is a sawtooth functionin time. It is optimized as described in SupplementalMaterial VII. In Section IV D 1, the spatial modulationperiod of the metasurface is set to be electrically small( N = 2 , N = 20. Simultaneous beamsteering and frequency translation is demonstrated. InSection IV D 3, the spatial modulation period is on theorder of a wavelength. The number of paths is set to N = 4, and retroreflective subharmonic frequency trans-lation is demonstrated. In Section IV D 4, the spatialmodulation period is still on the order of a wavelength( N = 4); however, a staggered sinusoidal reflection phaseis used to demonstrate retroreflective frequency transla-tion. The measured conversion loss and sideband sup-pression for each of the examples are provided in Table TABLE II. Measured conversion loss to desired reflected fre-quency harmonic f and and sideband suppression given: N -the number of paths, θ i - the incident angle, θ obs - the obser-vation angle, and the temporal phase modulation waveform(either a sawtooth or a sinusoid). Note that positive values of θ i and θ obs correspond to waves traveling along the positive x direction.Ex. N θ i θ obs Wave-form f ConversionLoss(dB) SidebandSupression(dB)TE TM TE TM0 1 25 ◦ ◦ saw f + f p ◦ ◦ saw f + 2 f p ◦ ◦ saw f + 3 f p ◦ ◦ saw f + f p ◦ -25 ◦ saw f + f p ◦ -39 ◦ saw f + 3 f p ◦ ◦ saw f + f p ◦ -39 ◦ sin f − f p ◦ ◦ sin f + f p II, and will be referred to throughout this section.
1. Reflective (specular) subharmonic frequency translation
In this section, electrically-small spatial modulationperiods are considered ( | k x ± β p | > k ). The incidentwave is chosen to impinge on the metasurface with anoblique angle of 25 ◦ . The measured reflection spectrafor 2-path ( N = 2) and 3-path ( N = 3) modulationschemes are given in Fig. 27. The reflection spectraare measured at a reflection angle of θ = 25 ◦ (see Fig.11). The measured spectra for both polarizations clearlydemonstrate subharmonic frequency translation, wherethe only radiated harmonics are those reflected at fre-quencies f = f + rN f s and r ∈ Z . Doppler-like frequencytranslation is observed for both polarizations.The mea-sured conversion loss and sideband suppression for bothpolarizations are shown as examples 1 and 2 in Table II.Compared to the homogenized, lossless metasurface pre-sented in Section III A, the conversion loss and sidebandsuppression degrade more as the number of paths is in-creased. This is attributed to the evanescent harmonicpairs on the surface of the structure. This is discussedfurther in Supplemental Material VIII.
2. Measured deflective serrodyne frequency translation
In this section, the spatial modulation period is cho-sen to be electrically large ( | k x ± β p | < k ). The pathnumber is set to N = 20. For the capacitance variationshown in Fig. 26a and Fig. 26c, the metasurface acts6 Frequency (GHz) -50-40-30-20-100 (a)
Frequency (GHz) -50-40-30-20-100 (b)
Frequency (GHz) -50-40-30-20-100 (c)
Frequency (GHz) -50-40-30-20-100 (d)
FIG. 27. Measured reflection spectrum of the spatio-temporally modulated metasurface prototype. (a) 2-path( N = 2) modulation for TE polarization. (b) 3-path ( N = 3)modulation for TE polarization. (c) 2-path ( N = 2) modula-tion for TM polarization. (d) 3-path ( N = 3) modulation forTM polarization. as a serrodyne frequency translator, and simultaneouslydeflects the wave to a different angle. When the incidentangle is θ = 25 ◦ , the measured reflection spectra at thereflection angle θ = 42 ◦ is shown in Fig. 28. The spec-tra for both polarizations clearly show a Doppler shiftto frequency f + f p . The measured conversion loss andsideband suppression for both polarizations are providedin example 3 in Table II. Note that the refection anglesfor harmonics with frequency f and f + 2 f p are 25 ◦ and67 ◦ respectively. We can see from Fig. 28 that a fractionof the reflected power from these two harmonics was stillcaptured by the finite aperture of the receive antenna dueto its relatively close placement.By simply interchanging the transmitting and receiv-ing antennas, we can measure the reflection spectra forthe case where the incident angle is θ = − ◦ . Themeasured reflected spectra at the reflection angle of θ = − ◦ are given in Fig. 29. Again, the spectra for bothpolarizations clearly show a Doppler shift to a frequency f + f p . The measured conversion loss and sideband sup-pression for both polarizations are shown in example 4 inTable II. The harmonics with frequency f and f + 2 f p are reflected at − ◦ and 9 . ◦ respectively, which arealso captured by the receiving antenna. Note that, thereflected spectra in Fig. 28 and Fig. 29 are almost identi-cal. This is due to the fact that the modulation frequency Frequency (GHz) -50-40-30-20-100 (a)
Frequency (GHz) -50-40-30-20-100 (b)
FIG. 28. Measured reflection spectrum of the spatio-temporally modulated metasurface prototype, with an inci-dent angle of 25 ◦ .(a) 20-path ( N = 20) modulation for TEpolarization. (b) 20-path ( N = 20) modulation for TM po-larization. Frequency (GHz) -50-40-30-20-100 (a)
Frequency (GHz) -50-40-30-20-100 (b)
FIG. 29. Measured reflection spectrum of the spatio-temporally modulated metasurface prototype, with an inci-dent angle of − ◦ . (a) 20-path ( N = 20) modulation forTE polarization. (b) 20-path ( N = 20) modulation for TMpolarization. is far lower than the signal frequency. Otherwise, the re-flection angle would differ from θ = − ◦ , as discussedin Supplemental Material V.
3. Measured retroreflective subharmonic frequencytranslation
In this section, the spatial modulation period is onthe order of the wavelength of radiation ( | k x | + | β p | >k , & || k x | − | β p || < k ). The path number is chosen tobe N = 4. As in Section III C, the retroreflective angleis chosen to be ± ◦ . In experiment, a 3 dB directionalcoupler (Omni-spectra 2030-6377-00) was attached to theantenna in order to measure the retroreflected spectra.Note that the modulation waveform on each column isoptimized with the same procedure given in Supplemen-tal material VII, for an incident angle of 39 ◦ . The mea-sured retroreflection spectra at an oblique angle of 39 ◦ are given in Fig. 30a and 30c for TE and TM polariza-tion, respectively. As expected, frequency translation to7 Frequency (GHz) -50-40-30-20-100 (a)
Frequency (GHz) -50-40-30-20-100 (b)
Frequency (GHz) -50-40-30-20-100 (c)
Frequency (GHz) -50-40-30-20-100 (d)
FIG. 30. Measured retroreflection spectrum of the spatio-temporally modulated metasurface prototype. (a) 4-path( N = 4) modulation for TE polarization, for an incident angleof 39 ◦ . (b) 4-path ( N = 4) modulation for TE polarization,for an incident angle of − ◦ . (c) 4-path ( N = 4) modulationfor TM polarization, for an incident angle of 39 ◦ . (d) 4-path( N = 4) modulation for TM polarization, for an incident an-gle of − ◦ . f +3 f p is observed for both polarizations. The measuredconversion loss and sideband suppression for both polar-izations are shown in example 5 in Table II. Note that,comparing Fig. 30a and 30c to Fig. 18a and 18c, onlythe harmonics in solid lines are captured by the antenna.For an incident angle of − ◦ , the measured retrore-flection spectra are shown in Fig. 30b and 30d for TE andTM polarization, respectively. As expected, the spectrafor both polarizations show a Doppler shift to frequency f + f p . The measured conversion loss and sideband sup-pression for both polarizations are shown in example 6 inTable II. Again, comparing Fig. 30b and 30d to Fig. 18band 18d, only the harmonics in solid lines are capturedby the antenna.
4. Measured retroreflective frequency translation with astaggered sinusoidal reflection phase
In this section, the bias waveform on adjacent columnsgenerates staggered sinusoidal reflection phases. Themodulation waveform on each column is optimized withthe same procedure given in Supplemental material VII,for an incident angle of 39 ◦ . Again, a wavelength-scalespatial modulation period ( N = 4) is used. As in Sec- Frequency (GHz) -50-40-30-20-100 (a)
Frequency (GHz) -50-40-30-20-100 (b)
Frequency (GHz) -50-40-30-20-100 (c)
Frequency (GHz) -50-40-30-20-100 (d)
FIG. 31. Measured retroreflection spectrum of the spatio-temporally modulated metasurface prototype. The bias wave-forms generates a sinusoidal reflection phase. (a) 4-path( N = 4) modulation for TE polarization, for an incident angleof 39 ◦ . (b) 4-path ( N = 4) modulation for TE polarization,for an incident angle of − ◦ . (c) 4-path ( N = 4) modulationfor TM polarization, for an incident angle of 39 ◦ . (d) 4-path( N = 4) modulation for TM polarization, for an incident an-gle of − ◦ . tion III C, the retroreflection angle is ± ◦ . The mea-sured retroreflection spectra at an oblique angle of 39 ◦ are given in Fig. 31a and 31c. As expected, frequencytranslation to f − f p is observed for both polarizations.The measured conversion loss and sideband suppressionfor both polarizations are provided in example 7 in Ta-ble II. The measured retroreflection spectra at an obliqueangle of − ◦ are given in Fig. 31b and 31d. Frequencytranslation to f + f p is observed for both polarizations.The measured conversion loss and sideband suppressionfor both polarizations are provided in example 8 in TableII.Both of the two previous retroreflection cases used 4paths per spatial modulation period for a retroreflectionangle of 39 ◦ (examples 5 and 7 in Table II). The only dif-ference between the two cases was the time-dependence ofthe reflection phase (sawtooth versus sinusoidal). Thus,we have shown that simply changing the temporal mod-ulation waveform, the retroreflection frequency can bechanged. In this case, it changed from f − f p to f +3 f p .8 V. CONCLUSION
We reported a spatio-temporally modulated metasur-face that can simultaneously control the reflected fre-quency and angular spectrum. A proof-of-principle meta-surface was designed and fabricated at X-band frequen-cies. Additionally, a theoretical treatment of the spatio-temporally modulated metasurface was presented whichaccounts for the spatial discretization of the structure.The theoretical treatment provides an accurate model ofthe metasurface as well as insight into the subharmonicfrequency translation possible with subwavelength spa-tial modulation periods.Specifically, when the spatial modulation is electri-cally large, the metasurface exhibits serrodyne frequencytranslation, where the metasurface can upconvert ordowncovert the incident frequency f by the modulationfrequency f p . Meanwhile, tuning the spatial modulationperiod allows the metasurface to steer the reflected beam,and even exhibit retroreflection. When the spatial modu-lation is electrically small, the metasurface exhibits sub- harmonic frequency translation. In this case, all the ra-diated harmonics are reflected in the specular direction.When the spatio modulation period is on the order of awavelength, retroreflective subharmonic frequency trans-lation can be achieved. The retroreflected wave carries afrequency that can be switched by changing the temporalmodulation waveform.The designed metasurface provides a new level of re-configurability. Multiple functions including beamsteer-ing, retroreflection, serrodyne frequency translation, andsubharmonic frequency translation can all be achievedwith one ultra-thin (0 . λ ) metasurface by appropriatelytailoring the space-time modulation waveform. The de-signed metasurface can find various applications in next-generation communication, imaging and radar systems. ACKNOWLEDGMENTS
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Zhanni Wu, Cody Scarborough, Anthony GrbicJune 2020
I Finding the time-modulated sheet capacitance
In this section, the relation between the reflection phase and the sheet capacitance will be derived.A homogenized model of the spatio-temporally modulated metasurface is shown in Fig. 5. Itconsists of a discretized, space-time modulated capacitive sheet over a grounded dielectric substrate.The spatial modulation period of the capacitive sheet is d , and its temporal modulation period is T p = 2 π/ω p . Each spatial modulation period is discretized into N paths (unit cells of width d ) overwhich the sheet capacitance is uniform. At the operating frequency of f = ω / π = 10 GHz, thewave admittance in the substrate for T E and
T M polarizations are, Y T Es = E y − H x = k T Es z µ ω , (S.1) Y T Ms = E x H y = (cid:15) (cid:15) h ω k T Ms z , (S.2)where subscripts s , 0 and z denote the substrate, the operating frequency ω , and the z-componentof the wavenumber, respectively. The relative permittivity of the substrate is (cid:15) h . Note that thesubstrate admittance looking down from z = 0 − is simply a ground plane translated by a distance l , which provides an inductive input reactance (see Fig. 2b) Y T Xsub = − jY T Xs cot( k T Xs z l ) , (S.3)where “X” is either “E” for TE polarized waves or “M” for TM polarized waves. If there is no spatialvariation, the entire capacitive sheet has the same time variation. The sheet capacitance C T X ( t ) isassumed to be a periodic function of time, C T X ( t ) = C T X + ∆ C T X ( t ) . (S.4)where C T X is static capacitance designed to resonate with the inductive reactance given by (S.3) atfrequency f for each polarization. It is given by C T X = Y T Xs cot( k T Xs z l ) ω . (S.5)Since Y T Xs resonates with C T X at frequency f , the reflection phase φ ( t ) at the incident frequencyis fully controlled by ∆ C T X ( t ), φ ( t ) = − ω ∆ C T X ( t ) Y T X , (S.6)1 a r X i v : . [ phy s i c s . a pp - ph ] J un here Y T X is the tangential wave admittance in free space at radial frequency ω for each polariza-tion: Y T E = p k − k x µ ω , (S.7) Y T M = (cid:15) ω p k − k x . (S.8)where k is the free space wavenumber at radial frequency ω , and k x is the tangential wavenumberof the incident wave. In this paper, the reflection phase of each column of the metasurface is eithera sawtooth ( φ ( t ) = ω p t ), or a sinusoidal function ( φ ( t ) = A sin( ω p t )) in time; where ω p is the radialfrequency of the modulation. From Eq. S.6, the capacitance ∆ C T X ( t ) can be found for a desiredtime-varying phase φ ( t ) ∆ C T X ( t ) = − Y T X ω tan (cid:18) φ ( t )2 (cid:19) . (S.9)The capacitance C T X ( t ) is a periodic function in time. Therefore, it can be expressed as a Fourierseries: C T X ( t ) = ∞ X q = −∞ C T Xq e jqω p t , (S.10)where the coefficient C T Xq is equal to C T Xq = 1 T p Z T p C T X ( t ) e − jqω p t dt. (S.11)This Fourier representation of the capacitance is used to find the fields scattered from the metasur-face: a time-modulated capacitive sheet over a grounded uniaxial dielectric substrate. II Calculation of the reflected spectrum from the time-modulated metasurface
The total tangential fields above ( z = 0 + ) and below ( z = 0 − ) the time-modulated capacitive sheettake the form, E t | z =0 + = E t | z =0 − = ∞ X q = −∞ V q e jqω p t e j ( ω t − k x x ) , (S.12) H t | z =0 + = ∞ X q = −∞ I q e jqω p t e j ( ω t − k x x ) , (S.13) H t | z =0 − = ∞ X q = −∞ − jV q Y T Xsq cot( k T Xsqz l ) e jqω p t e j ( ω t − k x x ) . (S.14) Y T Xsq and k T Xszq are the tangential wave admittance and normal ( z -directed) wavenumber in thesubstrate at the frequency ω q = ω + qω p , and are given by k T Eszq = r ω q c (cid:15) h − k x , k T Mszq = s ω q c (cid:15) h − k x (cid:15) h (cid:15) zz , (S.15)2 T Esq = k T Eszq µ ω q , Y T Msq = (cid:15) (cid:15) h ω q k T Mszq . (S.16)If the magnitude of the voltage signal that modulates the varactors comprising the time-modulatedcapacitive sheet is much larger than the incident signal at f , the sheet can be treated as a linear,time-varying capacitance. Therefore, the boundary condition [1] at z = 0 is given by H t | z =0 + − H t | z =0 − = ddt ( C T X ( t ) E t ) . (S.17)Inserting the Fourier series expansion for capacitance from Eq. (S.10) and the field expressions fromEqs. (S.12-S.14) into (S.17) yields I q = − jV q Y T Xsq cot( k T Xsqz l ) + j ( qω p + ω ) ∞ X q = −∞ C T Xq − q V q . (S.18)In matrix form, this can be written as I T X = Y T X V T X , (S.19)where I T X is a vector with the complex coefficients I q of the total tangential magnetic field abovethe metasurface, V T X is a vector with the coefficients V q of the total tangential electric field, and Y T X is the input admittance matrix of the metasurface (looking into the metasurface at z = 0 + )with entries Y T Xqq = jω q C T Xq − q − jδ ( q − q ) Y T Xsq cot( k T Xszq l ) . (S.20)The incident and reflected tangential fields above the metasurface are given by Eq. (5-8). Further,the coefficients of the incident electric and magnetic fields, as well as the reflected electric andmagnetic fields, are related by the free-space tangential admittance: I inc = Y T X V inc , I ref = − Y T X V ref . (S.21)For each polarization, the diagonal admittance matrix Y T X contains entries Y T E qq = δ ( q − q ) q ω q /c − k x µ ω q , (S.22) Y T M qq = δ ( q − q ) (cid:15) ω q q ω q /c − k x . (S.23)From Eq. (S.19) and (S.21), the reflected electric field can be calculated for each polarization usingEq. (9) in the main text. III Finding the discretized, space-time modulated sheet ca-pacitance
The space-time capacitance, C ( t, x ), of the capacitive sheet can be expanded as a 2-D Fourier series: C ( t, x ) = ∞ X m = −∞ ∞ X q = −∞ C mq e − jmβ p x e jqω p t , (S.24)3here β p = 2 π/d is the spatial modulation wavenumber and ω p = 2 π/T p is the radial frequency(temporal modulation wavenumber) of the modulation. The coefficients C mq of the 2-D Fourierseries can be calculated as: C mq = 1 dT p Z d Z T p C ( t, x ) e jmβ p x e − jqω p t dtdx. (S.25)As noted in the main text, the capacitive sheet is assumed to be spatially invariant across a givenpath. According to Eq. (10), the capacitance modulation of a path is staggered in time by T p /N with respect to its adjacent path. Therefore, if there are N unit cells in one spatial modulationperiod d (N-path configuration), the sheet capacitance, of a path can be expressed as C v ( t, x ) = ( C T X ( t − ( v − T p N ) , v − N d < x < vN d , otherwise , (S.26)which is a pulse function in space, and periodic function in time (see Eq. (S.10)). The spatio-temporally varying sheet capacitance can then be expressed as C ( t, x ) = N X v =1 C v ( t, x ) . (S.27)The capacitance of path v , C v ( t, x ), can also be expanded as a 2-D Fourier series, C v ( t, x ) = ∞ X m = −∞ ∞ X q = −∞ C vmq e − jmβ p x e jqω p t , (S.28)where C vmq = 1 dT p Z d Z T p C v ( t, x ) e jmβ p x e − jqω p t dtdx = 1 dT p Z vN d v − N d Z T p C T X ( t − ( v − T p N ) e jmβ p x e − jqω p t dtdx. (S.29)The equation above can be used to derive the following relationship between the Fourier coefficientsof the sheet capacitance on adjacent paths, C vmq = C v − mq e jm πN e − jq πN . (S.30)The Fourier coefficents of the overall capacitive sheet, C mq , given by Eq. (S.25), can be found by4umming the capacitance over all the paths and employing Eq. (S.30), C mq = 1 dT p Z dN Z T p C ( t, x ) e jmβ p x e − jqω p t dtdx + . . . + Z d N − N d Z T p C ( t, x ) e jmβ p x e − jqω p t dtdx ! = 1 dT p Z d Z T p C ( t, x ) e jmβ p x e − jqω p t dtdx + . . . + Z d Z T p C N ( t, x ) e jmβ p x e − jqω p t dtdx ! = N X v =1 C vmq = N X v =1 C mq e j πN ( m − q )( v − . (S.31)It is clear from Eq. (S.31) that the coefficient C mq is zero except when m − q = rN, where r ∈ Z . (S.32)Therefore, C mq = ( N C mq , m − q = rN, where r ∈ Z , otherwise . (S.33)Given the staggered modulation of the paths (unit cells), the metasurface functions as an N-pathsystem, and the indices m and q are related by Eq. (S.32). Inserting Eq. (S.32) into Eq. (S.24), thesheet capacitance can be rewritten as, C ( t, x ) = ∞ X r = −∞ ∞ X q = −∞ C q + rN,q e − j ( q + rN ) β p x e jqω p t ∆ = ∞ X r = −∞ ∞ X q = −∞ C rq e jq ( ω p t − β p x ) e − jrβ d x , (S.34)where the wavenumber β d = N β p = 2 π/d is an additional wavenumber resulting from the discretiza-tion of the spatial modulation. The summation over r accounts for the discontinuity in capacitanceat the the boundary of each path as well as the microscopic variation of capacitance within the paths(which in this case is constant). The summation over q accounts for the macroscopic capacitancevariation over one spatial modulation period d .Given Eq. (S.33), the spatio-temporal coefficients of the capacitance variation are given by C r,q = 1 dT p Z d Z T p C v ( t, x ) e j ( q + rN ) β p x e − jqω p t dtdx = N C r,q = NdT p Z d/N Z T p C T X ( t ) e j ( q + rN ) β p x e − jqω p t dtdx = NdT p Z d/N Z T p C T X ( t ) e jq ( ω p t − β p x ) e − jrβ d x dtdx. (S.35)5rom Eq. (S.36), we see that the capacitance C v ( t, x ) along a path is a separable function, C v ( t, x ) = f v ( t ) g v ( x ) , (S.36)where f v ( t ) is the temporal modulation of capacitance C T X ( t ) along path 1. In this paper, f v ( t )generates either a sawtooth reflection phase in time (see Fig. 5a and 5c) or a sinusoidal reflec-tion phase in time (see Fig. 20a and 20c). In addition, g v ( x ) is a function describing the spatialdependence of capacitance along path 1, which is assumed to be a pulse function, g v ( x ) = ( , ( v − dN < x < vdN , otherwise . (S.37)Inserting Eq. (S.36-S.37) into Eq. (S.35), we obtain C rq = NdT s Z dN e j ( q + rN ) β p x dx Z T s f ( t ) e − jqω p t dt = e j π ( q + rN ) N sinc π ( q + rN ) N C
T Xq , (S.38)where C T Xq are the temporal coefficients of the capacitance modulation for a single path, given byEq. (S.11).
IV Calculation of the reflected spectrum from a discretized,space-time modulated metasurface
As a result of the symmetry introduced by the staggered modulation scheme, the tangential fieldabove the metasurface satisfies the following N-path field relation [2]. E t ( t, x ) = e j ( ω TpN − kxdN ) E t ( t − T p N , x − dN ) (S.39)This space-time field distribution on the surface can also be expressed in terms of a modified 2-DFloquet expansion: E t ( t, x ) = E t | z =0 + = E t | z =0 − = ∞ X m = −∞ ∞ X q = −∞ V mq e − jmβ p x e jqω p t e j ( ω t − k x x ) . (S.40)Substituting (S.40) into (S.39), one again finds that (S.32) must hold. As a result, E t ( t, x ) = ∞ X r = −∞ ∞ X q = −∞ V ( q + rN ) q e − j ( q + rN ) β p x e jqω p t e j ( ω t − k x x ) (S.41) E t ( t, x ) = ∞ X r = −∞ ∞ X q = −∞ V rq e − jrβ d x e jq ( ω p t − β p x ) e j ( ω t − k x x ) , (S.42)where the wavenumber β d = N β p = 2 π/d results from the discretization of the spatial modulation.The summation over r accounts for the microscopic field variation along each path (unit cell sizeof length d = d/N ), while the summation over q accounts for the macroscopic field variation over6ne spatial modulation period d . Observing the field expression in Eq. (S.42), it can be concludedthat the staggered modulation between paths impresses a tangential wavenumber of qβ p onto the q th harmonic. The total tangential magnetic field on the spatio-temporally modulated metasurface(see Fig. 3) can be expressed as, H t | z =0 + = M X r,q = − M I rq e − jrβ d x e jq ( ω p t − β p x ) e j ( ω t − k x x ) (S.43) H t | z =0 − = ∞ X r,q = −∞ − jV rq Y T Xsrq cot( k T Xsrqz l ) e − jrβ d x e jq ( ω p t − β p x ) e j ( ω t − k x x ) . (S.44)Inserting Eqs. (S.42), (S.43) and (S.44) into the boundary condition given by Eq. (S.17) yields I rq = j ( qω p + ω ) M X r ,q = − M C T Xr − r ,q − q V r q − jV rq Y T Xsrq cot( k T Xsrqz l ) , (S.45)where Y T Xsrq and k T Xsrqz are the tangential wave admittance and normal wavenumber in the substratefor each spatio-temporal harmonic pair ( r, q ), Y T Esrq = k T Esrqz µ ω rq , Y T Msrq = (cid:15) (cid:15) h ω rq k T Msrqz , (S.46) k T Esrqz = r ω rq c (cid:15) h − k xrq , k T Msrqz = s ω rq c (cid:15) h − k xrq (cid:15) h (cid:15) zz , (S.47)where s , r , q , and z denote the substrate, harmonic pair ( r, q ), and z-component of the wavenumber.We can separate the total tangential field into incident and reflected tangential fields, as given inEqs. (12-15). The coefficients of the incident electric and magnetic field, as well as reflected electricand magnetic field are related by the free-space wave admittance Y T E rq = k T E rqz µ ω rq , Y T M rq = (cid:15) ω rq k T M rqz , (S.48) k T E rqz = r ω rq c − k xrq , k T M rqz = r ω rq c − k xrq . (S.49)In order to simplify the calculation, each harmonic pair ( r, q ) is mapped to one harmonic index α = ( r + M )(2 M + 1) + q + 1 [3], as shown in Table S1. The harmonic mapping allows thetangential fields, given by Eqs. (S.42) and (S.43), to be represented as vectors V T X and I T X ,where each contains (2 M + 1) entries: V α and I α respectively. The boundary condition and free-space admittance given by Eqs. (S.45) and (S.48) can then be written in matrix form. The size ofmetasurface admittance matrix Y T X and free-space tangential admittance Y T X is (2 M + 1) × (2 M + 1) . The reflected electric field can be calculated for each polarization using Eq. (9). Themetasurface admittance matrix Y T X contains entries: Y T Xαα = jω α C T Xα − α − jδ ( α − α ) Y T Xsα cot( k T Xszα l ) . (S.50)7able S1: Harmonic mapping relationship used in the analysis r q α − M − M − M − M + 1 2... ... ... − M M M + 1 − M + 1 − M M + 2 − M + 1 − M + 1 2 M + 3... ... ... M M (2 M + 1) V Theoretical study of the spatio-temporally modulated meta-surface for a high modulation frequency
In this section, we study the spatio-temporally modulated metasurface under a modulation frequency f p that is comparable to the incident frequency f . The modulation frequency is chosen to be f p = 0 . f = 10 GHz. The incident plane wave is obliquely incident at θ = 25 ◦ , and 20 paths per modulation period are chosen. The reflection phase of each columnis a sawtooth function in time, at the incident frequency of f . For both polarizations, the fieldis expanded into 141 frequency harmonics as well as spatial harmonics. The temporal capacitancemodulation on each path is truncated to 101 frequency harmonics.Based on Eq. (23), the harmonics inside the light cone are those with q = 0 , ± , ± , ± , − f + f p ) reflects at an angle of θ = 39 ◦ . The reflectedTE and TM spectra are given in Fig. S1a and Fig. S1b, respectively. For TE polarization, a 1 . .
84 dB sideband suppression are observed. For TM polarization, a 2 . .
25 dB sideband suppression are observed. -50-40-30-20-1008 9 10 11 12
Frequency (GHz) (a) -50-40-30-20-1008 9 10 11 12
Frequency (GHz) (b) -50-40-30-20-1008 9 10 11 12
Frequency (GHz) (c) -50-40-30-20-1008 9 10 11 12
Frequency (GHz) (d)
Figure S1: Analytical reflection spectrum of the homogenized, lossless spatio-temporally modulatedmetasurface. (a) 20-path ( N = 20) modulation for TE polarization, for an incident angle of 25 ◦ .(b) 20-path ( N = 20) modulation for TM polarization, for an incident angle of 25 ◦ . (c) 20-path( N = 20) modulation for TE polarization, for an incident angle of − ◦ . (d) 20-path ( N = 20)modulation for TM polarization, for an incident angle of − ◦ .8hen the incident signal impinges on the metasurface at θ = − ◦ , the propagating reflectedharmonics are those with q = 0 , ± , , , , , , ,
8. The harmonic of interest ( f + f p ) travels witha reflection angle of θ = − ◦ . Note that the reflected angle θ = θ [4]. The reflected spectrumfor TE and TM polarization is given in Fig. S1c and Fig. S1d, respectively. For TE polarization,a 0 .
89 dB conversion loss and 7 .
51 dB sideband suppression are observed. For TM polarization, a0 .
77 dB conversion loss and 8 .
23 dB sideband suppression are observed.
VI Reflection from a DC biased unit cell for an incident angleof ◦ As described in the paper, the tunability of the metasurface is provided by surface-mounted varactordiodes MAVR-000120-1411. In the full-wave simulations, the varactor diode is modeled as a lumpedcapacitance in series with a resistance. The capacitance and resistance values are extracted as afunction of bias voltage from the varactor’s SPICE model [5]. The simulated reflection coefficientof the unit cell for various capacitance values is shown in Fig. 24 for an incident angle of 25 ◦ . Inaddition, the reflection coefficient of the fabricated metasurface is measured for various bias voltagesfor the same angle of incidence, and is shown in Fig. 25. Comparing the simulated and measuredreflection coefficients, we noticed that the varactor capacitance versus bias voltage characteristicgiven by the SPICE model did not accurately match the experimental results. Therefore, thevaractor capacitance versus experimental bias voltage characteristic was obtained by aligning themeasured reflection phase to simulation. In addition, the measured reflection amplitude indicatedthat there was higher loss in measurement than in simulation. The additional loss can be introducedby a higher measured varactor resistance or by the tinning and soldering processes used to mountthe varactor diodes.In order to conduct harmonic balance simulation (see Section VIII) and predict the reflectionspectrum of the metasurface when modulated, a circuit representation of the fabricated metasurfacewas extracted for each polarization. The circuit models are extracted from the full-wave scattering 𝐶 𝑑 𝐶 𝑑 +− 𝐶 𝑔𝑇𝐸 𝐿 𝑝𝑇𝐸 𝑅 𝑇𝐸 𝑙 dc Feed 𝐶 𝑎 𝑇𝐸 𝑍 𝑇𝐸 +− 𝑍 ℎ𝑇𝐸 , 𝑘 ℎ𝑧𝑇𝐸 𝑉 𝑏𝑖𝑎𝑠𝑦 (a) 𝐶 𝑑 𝐶 𝑑 𝐶 𝑔 𝑇𝑀 𝐿 𝑝𝑇𝑀 𝑅 𝑇𝑀 𝑙 𝐶 𝑎𝑇𝑀 𝑍 ℎ𝑇𝑀 , 𝑘 ℎ𝑧𝑇𝑀 𝐿 𝑣 +− 𝑉 𝑏𝑖𝑎𝑠𝑥 dc Feed +− 𝑍 (b) Figure S2: Extracted circuit models for for a unit cell (see Fig. 2a) of the metasurface under anoblique incident angle of 25 ◦ . (a)TE polarization. (b) TM polarization.9able S2: Detailed information an values of the extracted circuit shown in Fig. S2 Z T E Free space tangential wave impedance(TE) under the oblique angle of 25 ◦ ,415 .
97 Ω Z T M Free space tangential wave impedance(TM) under the oblique angle of 25 ◦ ,341 .
68 Ω Z T Eh
Substrate tangential wave impedance(TE) under the oblique angle of 25 ◦ ,220 .
77 Ω Z T Mh
Substrate tangential wave impedance(TM) under the oblique angle of 25 ◦ ,181 .
34 Ω C T Eg
Extracted pattern capacitance, 0 . C T Mg
Extracted pattern capacitance, 0 .
046 pF C T Ea
Extracted pattern capacitance, 0 .
02 pF C T Ma
Extracted pattern capacitance, 0 .
03 pF L T Ep
Extracted pattern inductance, 0 .
67 nH L T Mp
Extracted pattern inductance, 0 .
61 nH R T E
Extracted voltage-dependent resistance R T M
Extracted voltage-dependent resistance k T Xhz
Substrate normal wavenumber for TE orTM, electric thickness k T Xhz l = 10 . ◦ L v Extracted series inductance of the vias,0 .
11 nH l Substrate thckness, 0 .
508 mm C d Varactor diode SPICE modelTable S3: Varactor (MAVR-000120-1411) capacitance and resistance verses bias voltage character-istic for TE polarization.Varactor capacitance(pF) Additional loss R T E in circuitsimulation (Ω) Bias voltage usedin circuitsimulation (V) Bias voltage used inmeasurement (V)0.4 2.08 4.03 40.329 1.82 5.2 50.284 1.758 6.5 60.25 1.74 7.98 70.22 1.60 9.58 80.202 1.20 11.28 9Table S4: Varactor (MAVR-000120-1411) capacitance and resistance verses bias voltage character-istic for TM polarization.Varactor capacitance(pF) Additional loss R T M used insimulation (Ω) Bias voltage usedin circuitsimulation (V) Bias voltage used inmeasurement (V)0.385 2.066 4.16 40.319 1.95 5.44 50.271 2.00 6.92 60.239 2.17 8.4 70.213 2.39 10.2 80.19 2.12 12.49 910
Frequency (GHz) -20-15-10-50
MeasurementFull-wave simulationCircuit Simulation (a)
Frequency (GHz) -20-15-10-50
MeasurementFull-wave simulationCircuitSimulation (b)
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MeasurementFull-wave simulationCircuitSimulation (c)
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MeasurementFull-wave simulationCircuitSimulation (d)
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MeasurementFull-wave simulationCircuitSimulation (e)
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MeasurementFull-wave simulationCircuitSimulation (f)
Frequency (GHz) -200-1000100200
MeasurementFull-wave simulationCircuit Simulation (g)
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MeasurementFull-wave simulationCircuit Simulation (h)
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MeasurementFull-wave simulationCircuit Simulation (i)
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MeasurementFull-wave simulationCircuit Simulation (j)
Frequency (GHz) -200-1000100200
MeasurementFull-wave simulationCircuit Simulation (k)
Frequency (GHz) -200-1000100200
MeasurementFull-wave simulationCircuit Simulation (l)
Figure S3: Reflection phase and magnitude versus varactor capacitance for the realized metasurfacefrom full-wave simulation, circuit model simulation and measurement under an oblique incidentangle of 25 ◦ . Results are shown for a TE polarization. Frequency (GHz) -20-15-10-50
MeasurementFull-wave simulationCircuit Simulation (a)
Frequency (GHz) -20-15-10-50
MeasurementFull-wave simulationCircuit Simulation (b)
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MeasurementFull-wave simulationCircuit Simulation (c)
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MeasurementFull-wave simulationCircuit Simulation (d)
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MeasurementFull-wave simulationCircuit Simulation (e)
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MeasurementFull-wave simulationCircuit Simulation (f)
Frequency (GHz) -200-1000100200
MeasurementFull-wave simulationCircuit Simulation (g)
Frequency (GHz) -200-1000100200
MeasurementFull-wave simulationCircuit Simulation (h)
Frequency (GHz) -200-1000100200
MeasurementFull-wave simulationCircuit Simulation (i)
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MeasurementFull-wave simulationCircuit Simulation (j)
Frequency (GHz) -200-1000100200
MeasurementFull-wave simulationCircuit Simulation (k)
Frequency (GHz) -200-1000100200
MeasurementFull-wave simulationCircuit Simulation (l)
Figure S4: Reflection phase and magnitude versus varactor capacitance for the realized metasurfacefrom full-wave simulation, circuit model simulation and measurement under an oblique incidentangle of 25 ◦ . Results are shown for a TM polarization.simulations, with an added voltage-dependent resistance R T X to account for the additional lossobserved in measurement. The dependence of R T X is obtained by aligning the measured and sim-ulated reflection amplitudes. The extracted circuit models are shown in Fig. S2. The values of theextracted circuit parameters are shown in Table S2. The varactor diode C d is modeled using theSPICE model for MAVR-000120-1411 varactors. For each varactor capacitance, the corresponding11ias voltages used in circuit simulation (given by the SPICE model) and measurement, as well asthe additional resistances R T X are given in Table S3 and S4 for the TE and TM polarizations. Thevaractor characteristics and additional losses R T X are slightly different for the two polarizations.This is likely due to tolerances in the varactor capacitance and resistance values.The extracted circuits shown in Fig. S2 are simulated with the commercial circuit solver KeysightAdvanced Design System (ADS). Comparisons between full-wave simulation, measurement, andcircuit simulation are shown in S3 and S4 for various capacitance values. The reverse bias voltagevalues used in circuit simulation are given in Table S3 and S4. The circuit simulations agree closelywith full-wave simulations and measurements of the metasurface, confirming the accuracy of thecircuit model shown in Fig. S2.
VII Calculating the optimized bias waveform
In order to achieve serrodyne frequency translation, a bias waveform is needed that generates asawtooth reflection phase, which varies 2 π radians over each modulation period. To obtain the biaswaveform, the following procedure was followed. Using the extracted circuit models shown in Fig.S2, the reflection amplitude and phase were plotted versus bias voltage at 10 GHz, as shown in Fig.S5. The plots show that the bias voltage versus reflection phase curves follow a tangent function.Since the targeted sawtooth reflection phase is linear with respect to time over each modulationperiod, the bias waveform was assumed to be of the following form, V T Xbias ( t ) = A tan( B ( ω p t + C )) + D, for 0 < V T Xbias < , for V T Xbias < , for V T Xbias > . (S.51)over each 40 usec modulation period ( − T p / < t < T p / A = 0 .
51 V, B =0 . C = 0 .
266 rad, D = 5 .
597 V. The optimized values for TM polarization are A = 0 . B = 0 . C = 0 .
339 rad, D = 5 . Bias voltage (V) -20-15-10-50 -200-1000100200 (a)
Bias voltage (V) -20-15-10-50 -200-1000100200 (b)
Figure S5: Reflection coefficient magnitude and phase of the extracted circuit model for the meta-surface as a function of reverse bias voltage. (a)TE polarization. (b) TM polarization.12as assumed to be of the following form, V T Xbias ( t ) = A tan( B ( A sin( ω p t ) + C )) + D, for 0 < V T Xbias < , for V T Xbias < , for V T Xbias > . (S.52)where A = 138 ◦ , as explained in the paper. The fitting parameters A, B, C, and D were againnumerically optimized to suppress the zeroth harmonic in reflection. The optimized waveform usedin simulation is shown in Figs. S7a and S7c. The optimized values for TE polarization are A = 0 . B = 0 . C = 0 .
286 rad, D = 5 .
53 V. The optimized values for TM polarization are A = 0 . B = 0 . C = 0 .
389 rad, D = 5 . VIII Harmonic balance simulation of the extracted circuitmodel
If all the columns of the metasurface are biased with the same waveform, the metasurface’s responsecan be predicted by performing a harmonic balance simulation of a single unit cell’s extracted circuitmodel. Harmonic balance simulations of the circuit model shown in Fig. S2 were performed usingKeysight ADS. The incident signal was set to an amplitude of −
20 dBm at frequency f = 10 GHz.The optimized waveforms V ybias and V xbias were calculated as described in the previous section.When the reflection phase is a sawtooth function in time, the simulated reflection spectra aregiven in Fig. S6b and S6d. The simulation results agree with the measurement results shown in Fig.26. However, the measured results shows higher conversion loss and lower sideband suppression.This can be attributed to the fact that the measured bias waveform is a coarsely sampled version ofthe optimized waveform. The sampling rate of the D/A converter used in experiment is 0 . N branches of time-varying circuits connected to a common port. Each path (column ofmetasurface) is represented by a circuit model shown in Fig. S2. The bias wavefrom of each pathis given in Fig. S6a and S6c; and is staggered in time by T p /N with respect to that of its adjacentpath. The simulated reflection spectra for 2-path and 3-path configurations are shown in Fig. S8.Note that, as the path number N increases, the conversion loss increases as well. This is because theN-path metasurface upconverts the frequency to the first propagating harmonic pair. The higher13he upconverted frequency, the more loss there is in the frequency conversion process. The simulatedresults agree with the measurement results shown in Fig. 27. However, the conversion loss of themeasured results degrade more severely as the path number increases. This is due to the fact thatwhen the metasurface is lossy, the evanescent harmonic pairs on the metasurface consume energy aswell. Those harmonic pairs are not represented in the N-path circuit network, where the N branchesof time-varying circuits are considered perfectly co-located. References [1] R. E. Collin , “Foundations for microwave engineering”,
John Wiley & Sons , ch. 11, pp. 799-830,2007.[2] C. Scarborough, and A. Grbic “Accelerated N-Path Network Analysis Using the Floquet Scat-tering Matrix Method”,
IEEE Transactions on Microwave Theory and Techniques , vol. 64, pp.1248-1259, 2020.[3] L. Tsang, J. A. Kong, and K. H. Ding “Scattering of electromagnetic waves: theories andapplication,” vol. 27, John Wiley & Sons, 2004.[4] A. M. Shaltout, A. Kildishev, and V. Shalaev “Time-varying metasurfaces and lorentz non-reciprocity,”
Optical Materials Express , vol. 5, pp. 2459–2467, 2015.[5] MACOM Technology Solutions,
Solderable GaAs Con-stant Gamma Flip-Chip Varactor Diode,MAVR-000120-1411
Time (usec) (a) -50-40-30-20-1009.99975 10 10.00025
Frequency (GHz) (b)
Time (usec) (c) -50-40-30-20-1009.99975 10 10.00025
Frequency (GHz) (d)
Figure S6: (a) Optimized bias waveform for a TE polarization. (b) Reflection spectrum fromharmonic balance simulation of the extracted circuit shown in Fig. S5a. The bias waveform isgiven in Fig. S6a. (c) Optimized bias waveform for TM polarization. (d) Reflection spectrum fromharmonic balance simulation of the extracted circuit shown in Fig. S5b. The bias waveform is givenin Fig. S6c. 14
20 40 60 80
Time (usec) (a) -50-40-30-20-100 9.9998 10 10.0002
Frequency (GHz) (b)
Time (usec) (c) -50-40-30-20-1009.99975 10 10.00025
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Figure S7: (a) Optimized bias waveform for a TE polarization. (b) Reflection spectrum fromharmonic balance simulation of the extracted circuit shown in Fig. S5a. The bias waveform isgiven in Fig. S7a. (c) Optimized bias waveform for TM polarization. (d) Reflection spectrum fromharmonic balance simulation of the extracted circuit shown in Fig. S5b. The bias waveform is givenin Fig. S7c -50-40-30-20-1009.99975 10 10.00025
Frequency (GHz) (a) -50-40-30-20-1009.99975 10 10.00025
Frequency (GHz) (b) -50-40-30-20-1009.99975 10 10.00025
Frequency (GHz) (c) -50-40-30-20-1009.99975 10 10.00025
Frequency (GHz) (d)