Space-Time Medium Functions as a Perfect -Mixer-Amplifier Transceiver

Sajjad Taravati and George V. Eleftheriades The Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, Ontario M5S 3H7, Canada Email: [email protected]

We show that a space-time-varying medium can function as a front-end transceiver, i.e., an antenna-mixer-amplifier. Such a unique functionality is endowed by space-time as- sociated with complex space-time vectors in a subluminal space-time medium. The proposed structure introduces pure up- and down-conversions and with very weak undesired time harmonics. In contrast to other recently proposed space-time mixers, a large frequency up-/down conversion ratio, associated with gain is achievable. Furthermore, as the structure does not operate based on progressive energy transition between the space-time modulation and the incident wave, it possesses a subwavelength thickness (metasurface). Such a multi-functional, highly efficient and compact medium is expected to find various applications in modern systems.

I. INTRODUCTION an antenna per se. Such an interesting functional- ity of the space-time medium is endowed by space- Reception and transmission of electromagnetic waves time surface waves. Other recently proposed space- is the essence of wireless [1,2]. Such time antenna systems are formed by integration a task requires various functions, such as power am- of the space-time-modulated medium with an an- plification, frequency conversion, and wave radiation. tenna [3, 35, 45], and hence suffer from a number These functions have been conventionally performed by of drawbacks, i.e., requiring long structures, low ef- electronic components and electromagnetic structures, ficiency and narrow-band operation. such as transistor-based amplifiers, -based • The proposed antenna-mixer-amplifier introduces diode mixers, and resonance-based antennas. The ever large frequency up- and -down conversion ratios. increasing demand for versatile wireless telecommuni- This is very practical, because in real-scenario wire- cation systems has led to a substantial need in multi- less telecommunication systems, a large frequency functional components and compact integrated circuits. conversion is required, i.e., a frequency conversion In general, each natural medium may introduce a sin- from a /millimeter-wave frequency to an gle function, e.g., a resonator may operate as an antenna intermediate frequency in receivers. In contrast, at a specific frequency. To contrast this conventional ap- recently proposed space-time frequency converters proach, a single medium exhibiting several functions con- suffer from very conversion ratios currently could lead to a disruptive evolution in telecom- (up-/down-converted frequency is very closed to munication technology. the input frequency) [3, 17, 35, 45]. Recently, space-time-modulated media have attracted a surge of scientific interest thanks to their capabil- • Recently proposed space-time systems operate ity in multifunctional operations, e.g., mixer-duplexer- based on a progressive coupling between the inci- antenna [3], unidirectional beam splitters [4], nonrecip- dent wave and the space-time modulation [3, 33, rocal filters [5,6], and signal coding metagratings [7]. 46–48], and hence a thick (compared to the wave- In addition, a large number of versatile and high effi- length) slab is required. In contrast, the proposed ciency electromagnetic systems have been recently re- antenna-mixer-amplifier does not work based on

arXiv:2005.00807v1 [.app-ph] 2 May 2020 ported based on the unique properties of space-time mod- progressive coupling between the incident wave and ulation, including space-time metasurfaces for advanced the space-time modulation and hence is formed by wave engineering and extraordinary control over electro- a thin (sub-) slab. As a result, the pro- magnetic waves [8–25], nonreciprocal platforms [26–32], posed structure is classified among metasurface cat- frequency converters [17, 33], and time-modulated an- egories and is compatible with the compactness re- tennas [34–39]. Such strong capability of space-time- quirements of modern wireless telecommunication modulated media is due to their unique interactions with systems. the incident field [21, 28, 40–44]. This paper reveals that a space-time medium can func- • The proposed medium inherently operates as a tion as a full transceiver front-end, that is, an antenna- band-pass filter, i.e., a pure frequency down- and mixer-amplifier-filter system. Specific related contribu- up-conversions occurs so that the undesired time tions of this study are as follows. harmonics are highly suppressed. Such a property is governed by the subluminal operation of space- • We show that the space-time medium operates as time modulation. 2

Incoming propagating wave at ω 0 Radiated up-converted

wave at ω0=ωIF+ωm

x Input wave z x d Down-converted atωIF and amplified wave z ω at IF =ω0−ωm (a) (b)

Radiate and receive: ω0

ωIF < ωm < ω0

BPF at ω Amplifier ω IF ω IF 0 Received signal at ω ωm IF Modulation:ω ω ω m 0 m Transmitted ω signal at ω 0 ωIF IF BPF at ω0

(c)

FIG. 1. Functionality of the antenna-mixer-amplifier space-time medium. (a) Down-link (reception) state, where a strong transition between a space-wave at ω0 to a space-time surface wave at ωIF = ω0 − ωm occurs. (b) Up-link (transmission) state, where a transition between a space-time surface wave at ωIF to a space-wave at ω0 = ωIF + ωm occurs. (c) Circuital representation of the combined antenna-mixer-amplifier space-wave medium in (a) and (b).

• In contrast to conventional mixers that introduce II. OPERATION PRINCIPLE a significant conversion loss, here a conversion gain may be achieved in the down-link or up-link. In Consider the antenna-mixer-amplifier slab in Fig. 1(a), addition, the radiation from the antenna-mixer- with the thickness of d and a spatiotemporally-periodic amplifier is very directive. electric as

(z, t) = av + m cos(βmz − ωmt), (1)

The paper is organized as follows. SectionII presents with av being the average electric permittivity of the the operation principle of the antenna-mixer-amplifier. background medium, and m representing the modula- Then, Sec.III investigates the theoretical aspects of the tion amplitude. In Eq. (1), ωm denotes the temporal work, and provides an analysis of the wave propagation in frequency of the modulation, and βm is the spatial fre- the antenna-mixer-amplifier space-time medium. Next, quency of the modulation represented by Sec.IV presents the design procedure of the structure, ωm ωm and gives an approximate closed-form solution to calcu- β = = , (2) m v Γ v late the thickness and receive/transmit power gain of the m b structure. SectionV gives the time and frequency domain where vm and vb are the phase velocity of the mod- numerical simulation results for the designed antenna- ulation and the background medium, respectively, and mixer-amplifier. Finally, Sec.VI concludes the paper. Γ = vm/vb is the space-time velocity ratio [28]. 3

The slab in Fig. 1(a) is obliquely illuminated by a y- Ei , ω 0 polarized incident electric filed under an angle of inci- dence of θi, as θ θ E (x, z, t) = ˆyE ei(kxx+kz z−ω0t), (3) i rn i i ω IF =ω0−ωm x ESW ε = ε + ε β −ω where Ei is the amplitude of the incident wave, and ω0 (z,t) av m cos( m z mt) ω =ω −ω p 2 2 p 2 2 IF 0 m and k0 = kx + kz = (k0 sin(θi)) + (k0 cos(θi)) are z θn Em,ωn respectively temporal and spatial of the inci- Received surface-wave dent wave. Here, k0 = ω0/c, with c being the velocity of ω =ω −ω ESW light in vacuum. IF 0 m θ In the receiving state (down-link) in Fig. 1(a), strong tn transition between a space-wave, with temporal fre- FIG. 2. Operation principle of the antenna-mixer-amplifier quency ω0, to a space-time surface wave, with temporal frequency ω = ω − ω , occurs. In the transmission space-time medium. Transition of the incident space-wave to IF 0 m space-time surface waves at the two boundaries of the medium state (up-link) in Fig. 1(b) a transition between a space- ◦ ◦ (θr,−1 = θt,−1 = 90 ) as well as inside it (θ−1 = 90 )). time surface wave at ωIF to a space-wave at ω0 = ωIF+ωm occurs. Figure 1(c) shows an equivalent circuit representation harmonics outside of this interval represent imaginary of the antenna-mixer-amplifier in Figs. 1(a) and 1(b). k and hence propagate as space-time surface waves Thanks to the unique properties of space-time-modulated z,n along the two boundaries of the slab at z = 0 and z = d. media, that will be described later in this study, such The scattering angles of the space-time harmonics in- a medium introduces the functionality of a highly di- side the space-time-modulated medium read rective antenna, a pure frequency up-converter, a pure frequency down-converter, up-link and down-link filters,     −1 kx −1 k0 sin(θi) and a down-link amplifier. Such a rich functionality has θn = tan = tan . (6) not been experienced in other media unless several com- βz,n βz,0 + nβm ponents and media are integrated, as shown in Fig. 1(c). However, here we show that a single medium can offer Figure2 shows the structure of the antenna-mixer- such versatile and useful operation. amplifier space-time medium. We aim to design the As the slab is periodic in both space and time, the spa- structure such that the first lower space-time harmonic, n = −1, outside the medium scatters along the boundary tial and temporal frequencies of the space-time harmon- ◦ ics inside the structure are governed by the momentum of the medium, i.e., θr,−1 = θt,−1 = 90 . Hence, using conservation law, i.e., Eq. (5), the incident angle reads   γ = β + nβ + iα , (4a) −1 ωm z,n z,0 m z,n θi = sin 1 − . (7) ω0 and the energy conservation law, i.e., In addition, to achieving a strong transition to the n = ωn = ω0 + nωm. (4b) −1 harmonic, the scattered n = −1 harmonic inside the medium should propagate in parallel to the two space- The scattering angles of the space-time harmonics in time surface waves on the two boundaries of the medium ◦ regions 1 and 3 (reflection to the top and transmission at z = 0 and z = d, i.e., θn=−1 = 90 . Thus, using to the bottom of the medium, respectively) may be de- Eq. (6), we achieve 2 2 termined by the Helmholtz relations, i.e., k0 sin (θi) + 2 2 2 2 2 2 2 k0,n cos (θrn) = k0,n and k0 sin (θi) + k0,n cos (θtn) = βz,−1 = 0. (8) 2 k0,n, with θrn and θtn being the reflection and transmis- sion angles of the nth space-time harmonics, reading [7] As a result, the z component of the Wave vector inside the medium is purely imaginary, i.e., γz,−1 = iαz,−1,   −1 sin(θi) whereas the incident field wavenumber kz is purely real. θrn = θt,n = sin , (5) 1 + nωm/ω0 The space-time-modulated medium presents transition from the fundamental harmonic n = 0 to a large (theo- where k0n = ωn/c = (ω0 + nωm)/c. Therefore, given retically infinite, −∞ < n < ∞) number of space-time the common tangential wavenumber, kx = k0 sin(θi) in harmonics. Such a transition is very strong for the lumi- all the regions, the reflection and transmission angles for nal space-time modulation, where the space-time modu- the nth space-time harmonic are equal. Equation (5) lation velocity is close to the background phase velocity, shows that the harmonics in the n-interval [ω0(sin θi − i.e., vm = vb [28, 30]. 1)/ωm, +∞] are scattered at different angles ranging from To prevent generation of strong undesired time har- 0 to π/2 through θi for n = 0. However, the space-time monics, here the space-time-modulated medium operates 4

in the subluminal regime, where 0 < vm < vb, i.e., where ˜k are the Fourier series coefficients of the permit- r tivity. Equation (13) may be cast to the matrix form r as 0 < Γsubluminal < . (9) av + m [K] · [En] = 0, (14) As a result, a pure and precise transition between the fundamental (n = 0) harmonic and the desired (here n = where [K] is the (2N +1)×(2N +1) matrix with elements −1) space-time surface wave harmonic can occur. " 2 2 # kx + (βz,n + iαz,n) Knn = 2 − 0, [(ω0 + nωm)/c] (15) III. THEORETICAL INVESTIGATION Knk = −˜n−k, for n 6= k, A. Analytical solution where 2N + 1 is the number of truncated harmonics. Then, the relation reads Considering the TEy incident field in Eq. (3), the in- cident magnetic field reads det {[K]} = 0, (16) Once the dispersion diagram is formed, for a given in- 1 hˆ F i Hi(x, z, t) = ki × Ei (x, z, t) cident temporal frequency ω , the corresponding wave η1 0 number, i.e., γz,n, inside the slab can be computed. Next, E0 = [−ˆx cos(θ ) + ˆz sin(θ )] ei(kxx+kz z−ω0t), the unknown field amplitudes E in the slab are found by i i η n 1 solving (14) after determining the E satisfying bound- (10) 0 ary conditions. Figure3 shows a qualitative 3D dis- persion diagram of a subluminal space-time modulated where η = pµ / is the characteristic impedance of 1 0 0 medium, composed of an infinite periodic set of double- free space. Since the space-time-modulated medium is cones with apexes at β = −nβ and k = 0 and the periodic in space and time, the electric and magnetic z,0 m x slope Γ . As the slope Γ increases and goes towards unity, fields in the slab may be decomposed to space-time the cones get closer to each other and start touching Bloch-Floquet harmonics, as other cones(harmonics). As a result, stronger transition X E (x, z, t) = ˆy E eiγz,nzei(kxx−ωnt), (11a) between harmonics occurs and yields transitions of the m n power from the fundamental harmonic to a large num- n ber of harmonics. However, this is not what we aim at. and As we are interested in a pure transition to a desired 1 harmonic (n = −1), we require a subluminal space-time X iγz,nz i(kxx−ωnt) Hm(x, z, t) = Hn [−ˆxβz,n + ˆzkx] e e . modulation defined in Eq. (9). k n Figure4 plots the analytical isofrequency dispersion (11b) diagram, computed using (16), for the subluminal regime where Γ = 0.2. We have chosen a specific θ , where the The unknown coefficients in Eqs. (11a) and (11b), i n = −1 harmonic is excited at a point corresponding γ = β + iα , E and H shall be found through z,n z,n z,n n n to β = 0 (i.e., β /β = +1). As a consequence, the following procedure. We first construct and solve the z,−1 z,0 m the n = −1 harmonic scatters along the x-direction, i.e., corresponding wave equation. Next, we fix the source θ = 90◦. Although the real part of the z component of frequency ω and then find the corresponding discrete r,−1 0 the wavevector of the n = −1 harmonic is zero (β = γ solutions, which form the dispersion diagram of the z,−1 z,n 0), its imaginary part is greater than zero, and hence space-time medium. We next apply the spatial bound- γ = iα . ary conditions at the edges of the slab, i.e. at z = 0 and z,−1 z,−1 Figure4 shows that the transition between the for- z = d, for all the (ω , γ ) states in the dispersion di- 0 z,n ward n = 0 and n = −1 harmonics (highlighted with agram. This provides the unknown field coefficients E n magenta arrows) occurs for both reception (down-link) and H inside the slab, as well as the scattered (reflected n and transmission (up-link). This may as well be veri- and transmitted) fields outside the slab, i.e., E and E . R T fied by Eq. (6) as follows. For the down-link, the inci- The homogeneous wave equation reads RX dent wave (n = 0) with the incident angle of θi and RX 2 wavenumber of β = βm makes a transition to the 2 1 ∂ [(z, t)Em(x, z, t)] z,0 ∇ Em(x, z, t) − 2 2 = 0. (12) down-converted harmonic (n = −1), which is attributed c ∂t RX ◦ RX RX RX to θ-1 = 90 , βz,−1 = 0 and γz,−1 = iαz,−1. How- Inserting (11a) and (1) into (12), and using the Fourier ever, for the up-link (transmission state), the incident TX ◦ series expansion for the permittivity, results in wave (n = 0) with the incident angle of θi = 90 and wavenumber of βTX = 0 makes a transition to the up- " 2 2 # ∞ z,0 kx + (βz,n + iαz,n) X converted harmonic (n = +1) with θTX = θRX = θRX, En − ˜kEn−k = 0. (13) +1 0 i 2 TX TX [(ω0 + nωm)/c] k=−∞ which is attributed to βz,+1 = βm and γz,+1 = βm. 5

a pure imaginary wavenumber, that is, γz,−1 = iαz,−1. As a result, a perfect space-time transition from a pure isofrequency(kx , βz ) ω n = -1 diagramatω propagating wave to a pure space-time surface wave is 0 ensured. k n = 0 x Figure 5(d) plots the complex dispersion diagram, i.e., γz,n(ω0) for the medium in Fig. 5(b) and 5(c). This figure n = +1 shows the strong transition between n = 0 and n = −1 harmonics from the kx = 0.218 (corresponding to θi = ◦ backw 15 ) cut. d r a =Γ w lope a S r or d f , B. Approximate closed-form solution βz

To best investigate the efficiency of the antenna-mixer- amplifier, we present an approximate closed-form solu- tion. Such an approximate closed-form solution clearly shows the effect of modulation parameters on the effi- ciency of the proposed antenna-mixer-amplifier medium.

1. Reception state (down-link)

FIG. 3. 3D-dispersion diagram of the space-time medium, Equations (S7) and (S9) in the supplemental material formed by an infinite number of double cones each of which form a matrix differential equation. For the down-link, representing a space-time harmonic. the initial conditions of E0(0) = Ei and E−1(0) = 0 gives   mk0k−1 IV. DESIGN PROCEDURE E0(z) = Ei cos √ z , (17) 4 γz,−1γz,0

A. Complex Dispersion Diagrams   k−1 r γz,0 mk0k−1 E−1(z) = iEi sin √ z , (18) Figure 5(a) plots the analytical isofrequency dispersion k0 γz,−1 4 γz,−1γz,0 diagram (βz,n(kx) at ω/ωm = 1.363) of the sinusoidally 2 space-time surface wave medium with the electric per- where k−1 = ω−1/c . The optimal transition from the mittivity in (1) for the subluminal regime, i.e., Γ = 0.55 incident field to the space-time surface wave occurs at for m → 0. It may be seen from this figure that for- z = d, where ward n = 0 and n = −1 harmonics are excited very d close to each other, where n = 0 is excited at an angle of E−1(z)|z=d = 0, (19) ◦ dz scattering of θ0 = θi = 15 . However, the n = −1 har- monic is intentionally excited at the angle of scattering √ ◦ yielding |mk0k−1d/(4 γz,−1γz,0)| = π/2 which corre- of θ−1 = 90 . sponds to Figure 5(b) plots the same isofrequency βz,n(kx) dia- √ gram as Figure 5(a) except for a greater modulation am- γz,−1γz,0 d = 2π . (20) plitude of m = 0.45. As a result of non-equilibrium in the mk0k−1 electric and magnetic permitivitties of the medium, sev- eral electromagnetic badgaps appear at the intersections The down-link transition power ratio reads between space-time harmonics [30]. As a consequence, strong coupling between some of the harmonics has oc- |E (z = d)|2 k2 γ ω2 β G = −1 = −1 z,0 = −1 m , (21) curred, e.g. between the n = 0 and n = −1 harmonics. d 2 2 2 |Ei| k0γz,−1 ω0αz,−1 Figure 5(c) plots the complex isofrequency dispersion diagram γz,n(kx) for the medium in Fig. 5(b). This Therefore, for a set of modulation parameters, i.e., ωm, diagram is formed by two different curves, i.e., the αz,−1 and Γ , a down-link power gain may be achieved. real βz,n(kx) and the imaginary αz,n(kx) parts of the For the particular case, with the dispersion diagram in wavenumber. For the sake of clarification, we have in- Figs. 5(c) and 5(d), since αz,−1 << βm (i.e., αz,−1/βm = 2 2 cluded only a few number of harmonics. This figure 0.03 and ω−1/ω0 = 0.071), the down-link power gain ◦ shows that at the excited angle of incidence θi = 15 , reads Gd = 3.75 dB. This power gain emerges from the the n = 0 harmonic introduces a pure real wavenumber, coupling of the space-time modulation power to the inci- i.e., γz,0 = βz,0, whereas the n = −1 harmonic acquires dent wave [4, 40, 49, 50]. 6

Up-link transition from ° n=-1 (at 90 ) to n=0 (at θ i) 0.5 F B ° B θ-1 = θ-1 =90 vg,−1 F B F B F vg,−1 vg,0 vg,0 vg,+1 vg,+1 F ° ° θ0 = θi 90 90 fixedθi B m θ β 0 / 0 x ω k Γ (ω / m+ n) n = -1 0 n = 0

n = +1 Down-link transition from Slope=Γ n=0 (at θ ) to n=-1 (at 90° ) -0.5 i -1.5 -1 -0.5 0 0.511.5

βz,0 / βm

FIG. 4. Analytical isofrequency dispersion diagram for Γ = 0.2 and m =→ 0, showing down-link (reception) and up-link (transmission) electromagnetic transitions between the fundamental propagating harmonic (n = 0) and the space-time surface ◦ wave harmonic (n = −1). Excitation of the medium at θi leads to excitation of the n = −1 at 90 .

The total down-link electric field inside the slab reads total up-link electric field inside the slab reads

E(x, z, t) = E(x, z, t) = ! s ! mk0k−1 −i(k x+β z−ω t) k0 iαz,−1 mk0k−1 E cos z e x z,0 0 + −i(kxx+βz,0z−ω0t) i p iEi sin p z e + 4 iαz,−1βz,0 k−1 βz,0 4 iαz,−1βz,0 s ! ! k β  k k  k k −1 z,0 m 0 −1 −i(kxx+iαz,−1z−ω−1t) m 0 −1 −i(kxx+iαz,−1z−ω−1t) iEi sin p z e ,Ei cos p z e . k0 iαz,−1 4 iαz,−1βz,0 4 iαz,−1βz,0 (22) (26)

2. Transmission state (up-link) V. RESULTS

For the up-link, the matrix differential equations (S7) This section investigates the functionality and effi- ciency of the antenna-mixer-amplifier space-time slab and (S9) with the initial conditions of E−1(0) = Ei and using FDTD numerical simulation. We consider E0(0) = 0 yields oblique incidence to the slab and compare the numer-   ical results with the analytical solution provided in mk0k−1 E−1(z) = Ei cos √ z , (23) Secs.IIIA andIVB. Figure 6(a) depicts the structure 4 γ γ z,−1 z,0 of the antenna-mixer-amplifier space-time surface wave medium. A substrate integrated (SIW) is in-   tegrated with the space-time slab to efficiently launch k0 rγz,−1 mk0k−1 E0(z) = iEi sin √ z . (24) and receive space-time surface waves. The slab assumes k γ 4 γ γ −1 z,0 z,−1 z,0 ωm = 2π × 2.2 GHz, m = 0.45, Γ = 0.55, and d = 0.8λ0. A plane wave with temporal frequency ω0 = 2π × 3 GHz The up-link transition power ratio reads is propagating in the +z-direction under an angle of in- ◦ cidence of θi = 15 , and impinges on the slab. 2 2 2 |E0(z = d)| k0γz,−1 ω0αz,−1 Figure 6(b) shows the time domain numerical simula- Gu = 2 = 2 = 2 , (25) |Ei| k−1γz,0 ω−1βm tion result for the receiving state (down-link). It may be seen from this figure that a pure transition from which is the inverse of the down-link transition power the incident space-wave at ω0 to the space-time surface ratio. This shows that the proposed antenna-mixer- wave, propagating along the x-direction, at frequency amplifier introduces a power amplification only in one ωIF = ω0 − ωm = 2π × 0.8 GHz occurs. Furthermore, direction (here down-link), as shown in Fig. 1(c). The it may be seen from Fig. 6(b) that the amplitude of the 7

2 2 F B ° F B ° θ-1 = θ-1 = 90 θ-1 = θ-1 = 90 B F F v v − B v 1 g,−1 g, 1 1 vg,−1 g,−1 ° ° θ = θ =15 m i 15 i β m β / 0 x / 0 k x k

-1 -1 n = -1 n = -1 n = 0 n = 0 -2 -2 -3 -2.5 -2 -1.5 -1 -0.5 0 0.511.5 -2 -1.5 -1 -0.5 0 0.511.5 β β z,0 / m βz,0 / βm (a) (b)

α β 2 n = +1 α β n = 0 1.6 1

BG n = -1 n = 0 1.364 ° θ = n = +1 m i 15 backw β /

0 m 1

x n = 0 BG k a ω r / d n = -1 0 rd ω a w -1 n = 0 0.5 or BG f n = +1 n = -1 n = +1 -2 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.511.5

γ z / βm γ z / βm (c) (d)

FIG. 5. Analytical dispersion diagram of the sinusoidally space-time surface wave medium with the electric permittivity in (1) for subluminal regime, i.e., Γ = 0.55. (a) Isofrequency diagram, i.e., βz,n(kx) at ω/ωm = 1.363, and for m → 0. (b) Same as (a) except for greater modulation amplitude m = 0.45. (c) Isofrequency diagram of the medium in (b), which includes the real and imaginary parts of the γz,n, i.e., βz,n and αz,n, for n = 0 and n = −1. (d) Dispersion diagram, i.e., γz,n(ω0) for the ◦ medium in (a) and (b) for the kx = 0.218 (corresponding to θi = 15 ) cut. received wave is stronger than the amplitude of the in- cal simulation result for the transmission state (up-link). cident wave. Figure 6(c) plots the frequency domain Here, a transition (up-conversion) from the space-time numerical simulation result. This plot clearly shows a surface wave at ωIF to the space-wave at ω0 = ωIF + ωm pure and strong transition (frequency-conversion) from occurs. the incident wave to the down-converted space-time sur- face wave. The 3.5 dB power conversion gain is in agree- Figure 7(b) plots the frequency domain numerical ment with the analytical result in Eq. (21). In addition, simulation result for the transmission state (up-link). the amplitudes of the undesired harmonics are more than This plot clearly shows a pure and strong transition 33 dB lower than the amplitude of the down-converted (frequency-conversion) from the incident wave to the harmonic at ωIF. down-converted space-time surface wave. The 3.52 dB power conversion loss is in agreement with the analytical Figure 7(a) shows the time domain FDTD numeri- result in Eq. (25). In addition, the amplitude of the un- 8

Radiated up-converted wave at ω =ω −ω Incoming 0 IF m propagating wave

at ω0

x Array of grounding vias

z Transmitted d wave at ωIF

Space-time-modulated slab Received SIW waveguide wave at ωIF =ω0−ωm (a)

0 E ω (V/m) 0 1.5 10 ωIF ω0 Incident 0.75 0 Received ) )

0 -10 ωm W z( d B

0 ( -20 2 |

ω E IF | -30

-0.75 -40

27 -50 0 x( ) 10 -1.5 0 1 2 3 0 ω / ω0 (b) (c)

FIG. 6. FDTD simulation results of the antenna-mixer-amplifier space-time surface wave medium with ω0/ωm = 1.363, ◦ m = 0.45, d = 0.8λ0, and θi = 15 . (a) Architecture of the medium. (b) Time- and (c) frequency-domain responses for the down-link transition.

desired harmonics are more than 27 dB lower than the ics. In contrast to the recently proposed space-time mix- amplitude of the down-converted harmonic at ω0. ers, a large frequency ratio between the incident wave frequency and the up-/down-converted wave frequency, with a down- or up-conversion gain, is achievable. Fur- VI. CONCLUSIONS thermore, as the structure does not operate based on pro- gressive energy transition between the space-time modu- We showed that a space-time-varying medium oper- lation and the incident wave, it possesses a subwavelength ates as an antenna-mixer-amplifier. Such a unique func- thickness (metasurface). Such a multi-functional, highly tionality is achieved by taking advantages of space-time efficient and compact medium represents a new class of surface waves associated with complex space-time wave integrated electronic-electromagnetic component, and is vectors in a subluminal space-time medium. The pro- expected to find various applications in modern wireless posed structure is endowed with pure frequency up- and telecommunication systems. down-conversions and very weak undesired time harmon- 9

0 E(V/m) 1.5 10 ω0 ωIF Incident 0 ω 0.75 0 Transmitted ) )

0 -10 W z( d B 0 ( -20 2 | E ωIF | -30 -0.75 -40

27 -50 0 40 0 1 2 3 x( ) -1.5 0 ω / ω0 (a) (b)

FIG. 7. FDTD simulation results of the antenna-mixer-amplifier space-time surface wave medium with ω0/ωm = 1.363, ◦ m = 0.45, d = 0.8λ0, and θi == 15 . (a) and (b) Time-domain and frequency-domain responses for the up-link transition.

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SUPPLEMENTAL MATERIAL

The antenna-mixer-amplifier space-time medium is placed between z = 0 and z = d, and represented by the space-time-varying permittivity of

eq(z, t) = av + m cos(βmz − ωmt), (S1) and µ = µ0. The electric field inside the slab is defined based on the superposition of the m = 0 and m = −1 space-time harmonics fields, i.e.,

−i(kxx+γz,0z−ω0t) −i(kxx+γz,−1z−ω−1t) ES(x, z, t) = E0(z)e + E−1(z)e , (S2) where ω−1 = ω0 − ωm. The corresponding homogeneous wave equation reads ∂2E ∂2E 1 ∂2[ (t, z)E] + = eq . (S3) ∂x2 ∂z2 c2 ∂t2 Inserting the electric field in (S2) into the wave equation in (S3) results in  ∂2 ∂2  h i + E (z)e−i(kxx+γz,0z−ω0t) + E (z)e−i(kxx+γz,−1z−ω−1t) ∂x2 ∂z2 0 −1 2   1 ∂ m m =  + ei(βmz−ωmt) + e−i(βmz−ωmt) (S4) c2 ∂t2 av 2 2    −i(kxx+γz,0z−ω0t) −i(kxx+γz,−1z−ω−1t) × E0(z)e + E−1(z)e , and applying the space and time derivatives, while using a slowly varying envelope approximation (i.e., 2 2 2 2 ∂ E−1(z)/∂z = 0 and ∂ E0(z)/∂z = 0), yields   ∂E0(z) (k2 + γ2 )E (z) − 2iγ e−i(kxx+γz,0z−ω0t) x z,0 0 z,0 ∂z   ∂E−1(z) + (k2 + γ2 )E (z) − 2iγ e−i(kxx+γz,−1z−ω−1t) x z,−1 −1 z,−1 ∂z   (S5) 1 m m = ω2 + (ω − ω )2ei(βmz−ωmt) + (ω + ω )2e−i(βmz−ωmt) E (z)e−i(kxx+γz,0z−ω0t) c2 0 av 2 0 m 2 0 m 0  h m m i + ω2  + (ω − 2ω )2ei(βmz−ωmt) + ω2e−i(βmz−ωmt) E (z)e−i(kxx+γz,−1z−ω−1t) . −1 av 2 0 m 2 0 −1

We then multiply both sides of Eq. (S5) with ei(kxx+γz,0z−ω0t), which gives  ∂E (z) (k2 + γ2 )E (z) − 2iγ 0 x z,0 0 z,0 ∂z   ∂E−1(z) + (k2 + γ2 )E (z) − 2iγ ei(βm−ωmt) x z,−1 −1 z,−1 ∂z   (S6) 1 m m = ω2 + (ω − ω )2ei(βmz−ωmt) + (ω + ω )2e−i(βmz−ωmt) E (z) c2 0 av 2 0 m 2 0 m 0  h m m i + ω2  + (ω − 2ω )2ei(βmz−ωmt) + ω2e−i(βmz−ωmt) E (z)ei(βm−ωmt) , −1 av 2 0 m 2 0 −1

2π R ωm and next, applying 0 dt to both sides of (S6) yields 2 dE0(z) mk0 = i E−1(z). (S7) dz 4γz,0

2π −i(βmz−ωmt) R ωm Following the same procedure, we next multiply both sides of (S6) with e , and applying 0 dt in both sides of the resultant, which reduces to  dE (z) ω2   (k2 + γ2 )E (z) − 2iγ −1 = −1 m E (z) +  E (z) , (S8) x z,−1 −1 z,−1 dz c2 2 0 av −1 12

2 dE−1(z) mk−1 = i E0(z), (S9) dz 4γz,−1

2 where k−1 = ω−1/c . Solving the coupled equations (S7) and (S9) together, we achieve the field coefficients E0(z) and E−1(z) as given in Eqs.(17) and (18) for the reception state and in Eqs.(23) and (24) for the transmission state.