Nonreciprocal Nongyrotropic Magnetless Metasurface
Sajjad Taravati, Bakhtiar A. Khan, Shulabh Gupta, Karim Achouri, Christophe Caloz
aa r X i v : . [ phy s i c s . c l a ss - ph ] A ug Nonreciprocal Nongyrotropic MagnetlessMetasurface
Sajjad Taravati , Bakhtiar A. Khan , Shulabh Gupta , Karim Achouri ,and Christophe Caloz Department of Electrical Engineering and Poly-Grames research center, ´Ecole Polytechnique de Montr ´eal,Montr ´eal, Qu ´ebec, Canada Department of Electrical and Computer Engineering, Concordia University, Montr ´eal, Qu ´ebec, Canada Department of Electronics, Carleton University, Ottawa, Ontario, Canada
August 30, 2016
Abstract — We introduce a nonreciprocal nongyrotropicmagnetless metasurface. In contrast to previous nonrecip-rocal structures, this metasurface does not require a bias-ing magnet, and is therefore lightweight and amenable tointegrated circuit fabrication. Moreover, it does not induceFaraday rotation, and hence does not alter the polarizationof waves, which is a desirable feature in many nonrecip-rocal devices. The metasurface is designed according toa Surface-Circuit-Surface (SCS) architecture and leveragesthe inherent unidirectionality of transistors for breakingtime reversal symmetry. Interesting features include trans-mission gain as well as broad operating bandwidth and an-gular sector operation. It is finally shown that the metasur-face is bianisotropic in nature, with nonreciprocity due tothe electric-magnetic coupling parameters, and structurallyequivalent to a moving uniaxial metasurface.
Over the past decade, metasurfaces have spurred hugeinterest in the scientific community due to their unique opticalproperties [1]–[3]. Metasurfaces may be seen as the two-dimensional counterparts of volume metamaterials [4]–[10],where Snell’s law is generalized by the introduction of anabrupt phase shift along the optical path, leading to effectssuch as anomalous reflection and refraction of light [11] anda diversity of unprecedented wave transformation functionali-ties [12].The vast majority of the metasurfaces reported to date arerestricted to reciprocal responses. Introducing nonreciprocityrequires breaking time reversal symmetry. This can be accom-plished via the magneto-optical effect [13]–[18], nonlinear-ity [19]–[21], space-time modulation [22]–[28] or metamate-rial transistor loading [29]–[32]. However, all these approachessuffer from a number of drawbacks. The magneto-opticalapproach requires bulky, heavy and costly magnets [15]. Thenonlinear approach involves dependence to signal intensityand severe nonreciprocity-loss trade-off [33]. The space-timemodulation approach implies high design complexity, espe-cially for a spatial device such as a metasurface. Finally, thetransistor-based nonreciprocal metasurfaces reported in [30],[32] are intended to operate as Faraday rotators, whereasgyrotropy is undesired in applications requiring nonreciprocity without alteration of the wave polarization, such as for instanceone-way screens, isolating radomes, radar absorbers or illusioncloaks.We introduce here the concept of a nonreciprocal nongy-rotropic magnetless metasurface and demonstrate a simplethree-layer Surface-Circuit-Surface (SCS) implementation ofit. In the proposed metasurface, time reversal symmetry is bro-ken by the presence of unilateral transistors in the circuit partof the SCS structure, which is appropriate in the microwaveand millimeter-wave regime. A space-time modulated versionof the structure, although nontrivial, may also be envisionedfor the optical regime. The metasurface is shown to work forall incidence angles and to provide gain. It is finally shown tobe structurally equivalent to a moving uniaxial metasurface.
Significance Statement
While most materials and metamaterials are re-ciprocal, i.e. characterized by symmetric scatteringparameters, nonreciprocal devices (e.g. isolators,circulators, nonreciprocal phase shifter and polar-izers) play a fundamental role in a great diversityof microwave and optical applications. Achievingnonreciprocity requires breaking the time reversalsymmetry of a system using an external “force,such as an externally applied field with a specific(bias) direction. Recently, magnet-less nonrecipro-cal metamaterials have been reported as advanta-geous alternatives to ferromagnetic materials. Weintroduce here a magnet-less nonreciprocal meta-surface that is immune of Faraday rotation andaddresses therefore a novel range of potential ap-plications, that may include for instance novel one-way screens, isolating radomes, radar absorbersand illusion cloaks. Fig. 1: Nonreciprocal nongyrotropic metasurface functionality. O PERATION P RINCIPLE
Figure 1 depicts the functionality of the nonreciprocalnongyrotropic metasurface. A wave traveling along the + z direction, ψ in,1 , passes through the metasurface, possibly withgain, without polarization alteration, from side 1 to side 2.In contrast, a wave traveling along the opposite directionfrom side 2, ψ in,2 , is being absorbed and reflected (stillwithout polarization alteration) by the metasurface and cannot pass through the metasurface from side 2 to side 1. Such ametasurface is nonreciprocal, and may hence be characterizedby asymmetric scattering parameters, S = S , where S = ψ out,2 / ψ in,1 > and S = ψ out,1 / ψ in,2 < . Moreover,the metasurface is nongyrotropic since it does not induce anyrotation of the incident electromagnetic field.To realize such a nonreciprocal and nongyrotropic metasur-face, we employ the three-layer Surface-Circuit-Surface (SCS)architecture represented in Fig. 2. The first surface receivesthe incoming wave from one side of the metasurface andfeeds it into the circuit while the second surface collects thewave exiting the circuit and radiates it to the other side ofthe metasurface. The metasurface is constituted of an arrayof unit cells, themselves composed of two subwavelengthlyspaced microstrip patch antennas interconnected by the circuitthat will introduce transmission gain in one direction andtransmission loss in the other direction.To best understand the impact of the circuit on the meta-surface functionality, first consider the reciprocal unit cell ofFig. 3A, where the interconnecting circuit is a direct connec-tion (simple conducting wire). A conducting sheet is placedbetween the two patches to prevent any interaction betweenthem. Figures 3B and 3C show the Finite Difference TimeDomain (FDTD) response of structure in Fig. 3A. Figure 3Bplots the electric field distribution for wave incidence from Fig. 2: Surface-Circuit-Surface (SCS) metasurface architecturefor the magnetless implementation of the nonreciprocal nongy-rotropic metasurface in Fig. 1.the left and right, with the metasurface being placed at z = 0 .The response is obviously reciprocal. The corresponding pass-bands are apparent in the scattering parameter magnitudesplotted in Fig. 3C.Consider now the nonreciprocal unit cell of Fig. 3D, wherethe interconnecting circuit is a unilateral device, typicallya transistor-based amplifier. Figures 3E and 3F show thecorresponding FDTD response. Figure 3E plots the electricfield distribution for wave incidence from the left and right.When the excitation is from the left, the incoming wave passesthrough the structure, where it also gets amplified, and radiatesto the right of the metasurface. When the excitation is fromthe right, the incoming wave is blocked, namely absorbed andreflected, by the metasurface. The pass-band ( S ≃ ) andstop-band ( S ≃ ) are shown in in Fig. 3F. The explanationof the multiple pass-bands suppression is provided in thesupporting information section. E XPERIMENTAL D EMONSTRATION
Figure 4 shows the realized × metasurface, based on theSCS architecture of Fig. 3D. The metasurface is designed tooperate in the frequency range from 5.8 to 6 GHz. Its thicknessis deeply subwavelength, specifically δ ≈ λ / , where λ is the wavelength at the center frequency, . GHz, of the
Fig. 3: Unit cell of the metasurface in Fig. 2 with (A,B,C) a direct connection for the circuit, corresponding to a reciprocalmetasurface, and (D,E,F) a unilateral device (typically a transistor) for the circuit, corresponding to a nonreciprocal metasurface.(A,D) Structure. (B,E) Full-wave (FDTD) electric field distribution for excitations from the left and right (bottom). (C,F) Full-wave (FDTD) scattering parameter magnitudes.operating frequency range. More details about the fabricatedmetasurface are provided in the Materials and Methods sec-tion.Figure 5 shows the measured transmission scattering pa-rameters versus frequency for normally aligned transmit andreceive antennas. In the → direction, more than dBtransmission gain is achieved in the frequency range of inter-est, while in the → direction, more than dB transmis-sion loss is ensured across the same range, corresponding toan isolation of more than dB.Two other experiments are next carried out to investigatethe angular dependence of the metasurface response. In thefirst experiment, we fix the position of one antenna normalto the metasurface and rotate the other antenna from to ◦ with respect to the metasurface, as illustrated at thetop of Fig. 6A. The bottom of Fig. 6A shows the measuredtransmission levels for both directions, | S | and | S | . Weobserve that the metasurface passes the wave with gain over abeamwidth of about ◦ from θ = 35 ◦ to ◦ in the → direction and attenuates it by more than dB in the → direction, which corresponds to a minimum isolation of about dB across the aforementioned beamwidth.In the next angular dependence experiment, we rotate two Frequency (GHz) T r a n s m i ss i on M a gn it ud e ( d B ) S S S S isolationmetasurface antenna 1antenna 2 Fig. 5: Experimental scattering parameters versus frequencyfor normal incidence and transmission.antennas rigidly aligned from to ◦ with respect to themetasurface, as illustrated at the top of Fig. 6B. The bottomof Fig. 6B shows the measured transmission levels for bothdirections. We observe that the metasurface passes the wavewith gain over a beamwidth of about ◦ from θ = 25 ◦ Fig. 4: Realized × -cell implementation of the metasurface, where, compared to Fig. 3D, the transistors have been shiftedto the surfaces for fabrication convenience. (A) Exploded perspective view. (B) Photograph with zoom on transistor part andcorresponding biasing network.Fig. 6: Experimental scattering parameters versus angle at f = 5 . GHz for transmission (A) under normal (one side) andoblique (other side) angles, (B) in a straight line under an oblique angle. to ◦ in the → direction and attenuates it by morethan dB in the → direction, which corresponds to aminimum isolation of about dB across the aforementionedbeamwidth.Comparing Figs. 6A and 6B reveals that the transmissiongain is less sensitive to angle in the latter case. This is due tothe fact that the optical path difference between any pair ofrays across the SCS structure is null when the antennas arealigned, as in Fig. 6B, whereas it is angle-dependent otherwise,as in Fig. 6A (path difference ∆ L ).The experimental results presented above show that theproposed nonreciprocal nongyrotropic metasurface works asexpected, with remarkable efficiency. Moreover, it exhibitsthe following additional favorable features. Firstly, it providesgain, which makes it particularly efficient as a repeater device.Secondly, in contrast to other nonreciprocal metasurfaces, thestructure is not limited to the monochromatic regime sincepatch antennas are fairly broadband and their bandwidth canbe enhanced by various standard techniques [34]. Note thatthe structure presented here has not been optimized in thissense but already features a bandwidth of over . Thirdly, themetasurface exhibits a very wide operating angular sector, dueto both aforementioned small or null optical path differencebetween different rays, due to the SCS architecture, andthe inherent low directivity of patch antenna elements. Thereported operating sectors, of over ◦ , are much larger thanthose of typical metasurfaces. Characterization as a Bianisotropic Metasurface
The nonreciprocal nongyrotropic metasurface has been con-ceived from an engineering perspective in the previous sec-tions. We present here an alternative theoretical approach, thatwill reveal the equivalence to a moving medium to be coveredin the next section. Not knowing a priori the electromagneticnature of the metasurface, we start from the most generalcase of a bianisotropic medium, characterized by the followingspectral relations D = ǫ · E + ξ · H , (1a) B = ζ · E + µ · H . (1b)A metasurface can be generally characterized by the fol-lowing continuity equations, ˆ z × ∆ H = jωǫ χ ee · E av + jk χ em · H av , (2a) ∆ E × ˆ z = jωµ χ mm · H av + jk χ me · E av , (2b)which relate the electromagnetic fields on both sides of ametasurface to its susceptibilities and assume here no normalsusceptibility components [35]. In these relations, ∆ and thesubscript ‘av’ denote the difference of the fields and theaverage of the fields between both sides of the metasurface.The susceptibilities in (2) are related to the constitutiveparameters in (1) as ǫ = ǫ ( I + χ ee ) , µ = µ ( I + χ mm ) , (3a) ξ = χ em /c , ζ = χ me /c . (3b) Let us now solve the synthesis problem, which consistsin finding the susceptibilities providing the nonreciprocalnongyrotropic response for the metasurface. This consists insubstituting the electromagnetic fields of the correspondingtransformation into (2), as described in [35]. Specifically, thetransformation consists in passing a normally incident andforward propagating ( + z ) plane wave through the metasurfacewith transmission coefficient T = 1 and fully absorbing anormally incident wave in the opposite direction. The resultreads [3] χ ee = − jk (cid:18) (cid:19) , χ mm = − jk (cid:18) (cid:19) , (4a) χ em = jk (cid:18) − (cid:19) , χ me = jk (cid:18) −
11 0 (cid:19) , (4b)and reveals that nonreciprocity in the metasurface is due tothe electric-magnetic coupling contributions, χ em and χ me . Equivalence with a Moving Metasurface
The expressions in (4) suggest an alternative implementationof the nonreciprocal nongyrotropic metasurface. Indeed, theform of these susceptibility tensors is identical to that ofa moving uniaxial medium [36]. Such a medium, assumingmotion in the z -direction, is characterized by the tensor set ǫ ′ ξ ′ ζ ′ µ ′ ! = ǫ ′ ǫ ′ ǫ z µ ′ µ ′
00 0 0 0 0 µ z , (5)where the primes denote the moving frame of reference andwhere ǫ z and µ z can take arbitrary values. To an observer inthe rest frame of reference, this tensor set transforms to thebianisotropic set ǫ ξζ µ ! = ǫ ξ ǫ − ξ ǫ z − ξ µ ξ µ
00 0 0 0 0 µ z , (6)whose elements are found using the Lorentz transform opera-tion [36] C = L − · C ′ · L , (7)where the matrices C and L are respectively given by C = c ( ǫ − ξ · µ − · ζ ) ξ · µ − − µ − · ζ µ − /c ! (8)and L = γ − β
00 1 0 β /γ β − β /γ , (9) where γ = 1 / p − β , β = v/c , with v the velocity of themedium and c the speed of light in vacuum. (6) is indeedidentical to (4).From this point, one can find the moving uniaxial meta-surface that is equivalent to the nonreciprocal nongrytropicmetasurface. This is accomplished by inserting the specifiedsusceptibilities in (4) into (3), which provides the values in (6),and then solve (7) for C ′ and v . It may be easily verifiedthat one finds ǫ ′ r = µ ′ r = 1 and v = c . This may beinterpreted as follows: the forward propagating wave wouldsimply fully transmit through “moving vacuum” while thebackward propagating wave would never be able to catch-upwith it and, thus, never pass through. For T = 1 , one wouldfind a complex velocity. The fact that this design approach ispractically impossible indicates that the engineering approachis clearly preferable. M ATERIALS AND M ETHODS
The metasurface was realized using multilayer circuit tech-nology, where two . in × . in RO4350 substrates withthickness h = 30 mil were assembled to realize a threemetallization layer structure. The permittivity of the substratesare ǫ = ǫ r (1 − j tan δ ) , with ǫ r = 3 . and tan δ = 0 . at10 GHz. The middle conductor of the structure (Fig. 4A) bothsupports the DC feeding network of the amplifiers and acts asthe RF ground plane for the patch antennas. The dimensionsof the 2 × . in × . in.The connections between the layers are provided by anarray of circular metalized via holes, with 18 vias of 30 mildiameter connecting the DC bias network to the amplifiers,while the ground reference for the amplifiers is ensured by18 sets of 6 vias of 20 mils diameter with 60 mils spacing.The connection between the two sides of the metasurface isprovided by 9 via holes (Fig. 4A), with optimized dimensionsof mils for the via diameters, mils for the pad diametersand mils for the hole diameter in the via middle conductor.For the unilateral components, we used 18 Mini-CircuitsGali-2+ Darlington pair amplifiers. The amplifier circuit isshown in Fig. 4B, where C in = C out = C b1 = 4 . pF, C b2 = 1 nF and C b3 = 1 are a set of AC by-pass capacitors. A 4.5-V DC-supply provides the DC signal for the amplifiers throughthe DC network with a bias resistor of R bias = 39 Ohmcorresponding to a DC current of mA for each amplifier.The measurements were performed by a 37369D Anritsunetwork analyzer where two microstrip array antennas wereplaced at two sides of the metasurface to transmit and receivethe electromagnetic wave.R EFERENCES[1] C. L. Holloway, E. F. Kuester, J. Gordon, J. O. Hara, J. Booth,D. R. Smith et al. , “An overview of the theory and applications ofmetasurfaces: The two-dimensional equivalents of metamaterials,”
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Coupled-Structure Resonances Suppression
In this section, we provide the exact analytical solution forthe scattering parameters of the reciprocal and nonreciprocalunit cells in Figs. 3A and 3D. While the solution for theformer case gives more insight into the patch and coupled-structure resonances, plotted in Fig. 3C, the solution for thelatter case explains how the unilateral device suppresses thecoupled-structure resonances, leading to the result of Fig. 3F.Figure S1 shows the general representation of the unfoldedversion of the SCS structures in Figs. 3A and 3D, wherewave propagation through the SCS structure from one sideto the other side of the metasurface is decomposed in fivepropagation regions. Since microstrip transmission lines areinhomogeneous, their wavenumbers depend on the width ofthe structure [34], and therefore the wavenumbers in differentregions are different, i.e. β (= β ) = β (= β ) = β . region 1 η , β region 2 η , β region 3 η , β region 4 η , β region 5 η , β T T T T d d d R R R R R R R Fig. S1: Multiple scattering in the unfolded version of the SCSstructures in Figs. 3A and 3D.The total electric field in the n th region, where n = 1 , . . . , ,consists of forward and backward waves as E n = V + n e − jβ n z + V − n e jβ n z , (S1)where V + n and V − n are the amplitudes of the forward andbackward waves, respectively, and β n is the wavenumber. Itshould be noted that the backward waves, propagating along − z direction, are due to reflection at the different interfacesbetween adjacent regions. Upon application of boundary con-ditions at the interface between regions n and n + 1 , the totaltransmission and total reflection coefficients between regions n and n + 1 are found as as [37] e T n +1 ,n = V + n +1 V + n = T n +1 ,n e − j ( β n − β n +1 ) z − R n +1 ,n e R n +1 ,n +2 e − j β n +1 d n +1 , (S2a) e R n,n +1 = R n,n +1 + e R n +1 ,n +2 e − j β n +1 d n +1 R n,n +1 e R n +1 ,n +2 e − j β n +1 d n +1 . (S2b)where R n,n +1 = ( η n +1 − η n ) / ( η n +1 + η n ) , with η n beingthe intrinsic impedance of region n , is the local reflectioncoefficient within region n between regions n and n + 1 , and R n +1 ,n = − R n,n +1 . The local transmission coefficient fromregion n to region n +1 is then found as T n +1 ,n = 1+ R n,n +1 . The factor e − j ( β n − β n +1 ) z in (S2a) shows that, due to thenonuniformity of structure in Fig. S1, a phase shift correspond-ing to the difference between the wavenumbers in adjacentregions occurs at each interface. The total transmission fromregion 1 to region N is the product of the transmissions fromall interfaces and phase shift inside each region s N,1 = N − Y n =1 e T n +1 ,n e − jβ n d n . (S3)Figure S2A shows the unfolded version of the SCSarchitecture in Fig. 3A where, comparing with the generalrepresentation of the problem in Fig. S1, we denote β = β = β , β = β = β p and β = β t the wavenumbersin the air, in the two patches, and in the interconnectingtransmission line, respectively. We subsequently denote R , = − R , = − R , = R p = ( η p − η ) / ( η p + η ) the reflection coefficient at the inter-face between a patch and the air, and R , = − R , = − R , = R , = R t = ( η t − η p ) / ( η t + η p ) the local reflection coefficient at the interface between a patchand the interconnecting transmission line.The total transmission coefficient for the reciprocal SCSmetasurface of Fig. S2A, from region 1 to region 5, readsthen S = s = Y n =1 e T n +1 ,n e − jβ n d n , (S4)where e T n +1 ,n , for n = 1 , . . . , is provided by (S2a) with(S2b). In particular, e R = R t + e R , e − j β t d t R t e R , e − j β t d t = R t + R t R p e − j β p d p − ( R t + R p e − j β p d p ) e − j β t d t R t R p e − j β p d p − R t ( R t + R p e − j β p d p ) e − j β t d t (S5)will be used later.After some algebraic manipulations in (S4), the total trans-mission coefficient from the reciprocal SCS structure inFig. S2A is found in terms of local reflection coefficients as S = (1 − R p )(1 − R t ) e j ( β p + β − β t ) d t ( R p R t + e j β p d p ) − ( R t e j β p d p + R p ) e − j β t d t . (S6)The term e − j β t d t in the denominator of this expressioncorresponds to the round-trip propagation through the middletransmission line, whose multiplication by e j β p d p in theadjacent bracket corresponds to the patch-line-patch coupled-structure resonance, with length d p + d t .Figure S2B shows the unfolded version of the nonreciprocalSCS structure in Fig. 3D, where a unilateral device is placed atthe middle of the interconnecting transmission line. Note thatin this case the structure is decomposed in 7 (as opposed to5) regions, with extra parameters straightforwardly followingfrom the reciprocal case. Fig. S2: Wave interference explanation of the responses inFig. 3. (A) Reciprocal case (Figs. 3A, 3B, 3C). (B) Nonrecip-rocal case (Figs. 3D, 3E, 3F).Assuming that the input and output ports of the unilateraldevice are matched, i.e. R = R = 0 , then one has e R = 0 and the backward wave is completely absorbed by the device,i.e. T = e T = 0 while the forward wave is amplified bythe device as T = e T = G . Then, the total reflectioncoefficients at the interface between regions 2 and 3, givenby (S5) in the reciprocal case, reduces to e R = R t . (S7)This relation, compared with the one for the reciprocal case,reveals the suppression of the multiple reflections in the inter-connecting transmission line. The total transmission coefficientfrom the nonreciprocal SCS structure may be found as S = G (1 − R p )(1 − R t ) e − j ( β t d t − β d p ) ( R p R t + e j β p d p ) . (S8)Comparing the denominator of (S8) with that of the recipro-cal case in (S6), shows that the coupled-structure resonances,corresponding to the second term of the denominator, havedisappeared due to the suppression of the multiple reflectionsin the middle transmission line, restricting the spectrum to theharmonic resonances of the two patches, amplified by G .Figure S3A shows the magnitude of the scattering parame-ters for a single isolated patch, where transmission ( | S | = 1 )occurs at the harmonic resonance frequencies of the patch, nf ( n integer), where L p = nλ p / nλ / (2 √ ǫ eff ) with ǫ eff being the effective permittivity [34]. Figure S3B plotsthe magnitude of the scattering parameters of the coupled structure formed by the two patches interconnected by a shorttransmission line of L t = 0 . λ , given by (S6). We see that,in addition to the single patch resonances at f = nv p / (2 L p ) ,extra resonances appear in the spectrum, corresponding to theaforementioned coupled-structure resonances. Increasing thelength of the interconnecting transmission line to L t = 3 λ yields the results presented in Fig. S3C. As expected, morecoupled-structure resonances appear in the response due to thelonger electrical length of the overall structure, while the patchresonances remain fixed. Finally, we place the unilateral devicein the middle of the interconnecting transmission line, stillwith L t = 3 λ . Figure S3D shows the corresponding scatteringparameters, where all the coupled resonances in Fig. S3Chave been completely suppressed due to the absorbtion ofmultiple reflections from the patches by the unilateral device. Itshould be noted that the forward amplification, | S | > , andbackward isolation, | S | ≪ , are due to the nonreciprocalamplification of the unilateral device.Fig. S3: Scattering parameter frequency responses of the struc-tures in Fig. S2. (A) Isolated patch. (B) Structure (reciprocal)in Fig. S2A with L t = 0 . λ c . (C) Same structure (reciprocal)as in (B) except for L t = 3 λ c (D) Structure in Fig. S2B(nonreciprocal) still with L t = 3 λ cc