Omega-bianisotropic metasurface for converting a propagating wave into a surface wave
Vladislav Popov, Ana Díaz-Rubio, Viktar Asadchy, Svetlana Tcvetkova, Fabrice Boust, Sergei Tretyakov, Shah Nawaz Burokur
OOmega-bianisotropic metasurfacefor converting a propagating wave into a surface wave
Vladislav Popov ∗ SONDRA, CentraleSup´elec, Universit´e Paris-Saclay, F-91190, Gif-sur-Yvette, France
Ana D´ıaz-Rubio, Viktar Asadchy, and Svetlana Tcvetkova
Department of Electronics and Nanoengineering, Aalto University, P. O. Box 15500, FI-00076 Aalto, Finland
Fabrice Boust
SONDRA, CentraleSup´elec, Plateau de Moulon, 3 rue Joliot-Curie, F-91192 Gif-sur-Yvette, France andONERA - The French Aerospace Lab, 91120 Palaiseau, France
Sergei Tretyakov
Department of Electronics and Nanoengineering, Aalto University, P. O. Box 15500, FI-00076 Aalto, Finland
Shah Nawaz Burokur
LEME, UPL, Univ Paris Nanterre, F92410 Ville d’Avray, France
Although a rigorous theoretical ground on metasurfaces has been established in the recent yearson the basis of the equivalence principle, the majority of metasurfaces for converting a propagatingwave into a surface wave are developed in accordance with the so-called generalized Snell’s law beinga simple heuristic rule for performing wave transformations. Recently, for the first time, Tcvetkovaet al. [Phys. Rev. B 97, 115447 (2018)] have rigorously studied this problem by means of a reflectinganisotropic metasurface, which is, unfortunately, difficult to realize, and no experimental results areavailable. In this paper, we propose an alternative practical design of a metasurface-based con-verter by separating the incident plane wave and the surface wave in different half-spaces. It allowsone to preserve the polarization of the incident wave and substitute the anisotropic metasurface byan omega-bianisotropic one. The problem is approached from two sides: By directly solving thecorresponding boundary problem and by considering the “time-reversed” scenario when a surfacewave is converted into a nonuniform plane wave. We develop a practical three-layer metasurfacebased on a conventional printed circuit board technology to mimic the omega-bianisotropic response.The metasurface incorporates metallic walls to avoid coupling between adjacent unit cells and ac-celerate the design procedure. The design is validated with full-wave three-dimensional numericalsimulations and demonstrates high conversion efficiency.
I. INTRODUCTION
Surface waves propagate along an interface and expo-nentially decay away from it being localized on the sub-wavelength scale. Historically, investigation of surfacewaves started from the discovery of Zenneck waves atradio frequencies and study of optical Wood’s anomaliesthat were explained by the excitation of surface waves [1].The basic system that supports propagation of surfacewaves is represented by two semi-spaces filled with ametal and a dielectric [1]. In optical and infrared do-mains, the effect of strong field localization of surfacewaves (or surface plasmon polaritons) is used in manyapplications only a few of which are listed below. Spechtet al. developed near-field microscopy technique thatharnesses surface plasmon polaritons (SPPs) and allowsone to significantly overcome the diffraction resolutionlimit [2]. On-chip SPPs-based high-sensitivity biosensorplatforms were implemented and commercialized [3, 4]. ∗ [email protected] Surface-enhanced Raman scattering is attributed to ex-citation of SPPs [5, 6]. Application of SPPs in integratedphotonic circuits enables further miniaturization in com-parison to silicon-based circuits [7] and allows one to ap-proach the problem of size-compatibility with integratedelectronics [8].At lower frequencies (THz or microwaves), metals be-have like a perfect electric conductor (PEC) what doesnot allow a surface wave to penetrate in the metallic re-gion but extends it over long distances in a dielectric.Fortunately, the localization degree can be significantlyincreased by making use of artificial structures as it wasdemonstrated in Refs. [9–14]. Properties of surface wavesexcited on a structured interface can be controlled by en-gineering the interface. A surface wave propagating alonga periodically structured interface is called as a spoofsurface plasmon polariton (SSPPs) and mimics opticalSPPs. SSPPs allow one to significanty expand the fre-quency range of SPPs applications. For instance, SSPPscan be used in integrated microwave photonics [15–17].Metasurfaces (or thin two-dimensional equivalent ofmetamaterials) represent a fruitful tool for manipulationof surface waves [18–20] and are not restricted to mere a r X i v : . [ phy s i c s . a pp - ph ] M a y support of the propagation of spoof SPPs. Maci et al.proposed in Ref. [21] a general approach for transform-ing a wavefront of a surface wave by locally engineeringthe dispersion relation with spatially modulated metasur-faces. For instance, a metasurface-based Luneburg lensfor surface waves was demonstrated in Refs. [21, 22]. Spa-tial modulation significantly broadens the range of appli-cations of metasurfaces and allows one to link propagat-ing waves and surface waves. Metasurface-based leaky-wave antennas radiating a surface wave (or more gener-ally, a waveguide mode) into free space were developedin Refs. [23–26]. Vice versa, one can take advantage ofspatially modulated metasurfaces to convert an incidentplane wave into a surface wave [see the schematics inFig. 1(a)], as it was suggested by Sun et al. in Ref. [27].In this case, an excited surface wave is not an eigen-wave and can propagate along a metasurface only underillumination (in contrast to SPPs and SSPPs). How-ever, one can guide out an excited surface wave on aninterface supporting the propagation of the correspond-ing SSPP [27, 28]. It is worth to note, that metasurface-based converters and leaky-wave antennas are not equiv-alent, since the plane-wave illumination is normally uni-form (other designs also consider Gaussian-beam illumi-nation, see, e.g., Ref. [29]), while a plane wave radiated bya leaky-wave antenna can be essentially inhomogeneous,compare Figs. 1(a) and (b).Although a rigorous theoretical ground on metasur-faces has been established in the recent years on the baseof the equivalence principle [30–32], the majority of meta-surfaces for converting a propagating wave into a surfacewave are developed in accordance with the so-called gen-eralized Snell’s law (see, e.g., Refs. [27, 28, 33]). Initially,the generalized Snell’s law was applied to reflect or re-fract an incident wave at arbitrary angles by engineeringthe phase of a scattered wave at each point along a meta-surface in order to create a linear spacial evolution [34].However, in this case the wave impedance of a scatteredwave does not equal to the wave impedance of an incidentwave. It makes the efficiency of the anomalous reflec-tion (refraction) to decrease significantly when the anglebetween the incident and reflected (refracted) wave in-creases (as well as the impedance mismatch) [31, 35, 36].The outcome is even worse when it comes to the con-version of a propagating wave into a surface wave usingthe recipe provided by the generalized Snell’s law. Thewave impedance of the scattered field is imaginary in thiscase (a propagating wave has a real wave impedance) andthe generalized Snell’s law does not and cannot ensurea proper energy transfer between the propagating waveand the surface wave (the amplitude of the surface wavemust increase along a reactive metasurface according tothe energy conservation, as illustrated by Fig. 1 (a)). Asa result, losses have to be added to the system in orderto arrive at a meaningful solution [27], what makes thegeneralized Snell’s law a tool for designing an absorberrather than a converter (in addition to Ref. [27] see alsoRef. [36] where almost perfect absorption is demonstrated FIG. 1. (a) Schematics of a metasurface converting a nor-mally incident plane wave into a transmitted surface wavewith the propagation constant β y and the growth rate α y .(b) Schematics of a metasurface converting a surface waveinto an inhomogeneous plane wave propagating in the normaldirection with the propagation constant β (cid:48) z . by exciting a single near-field mode).Recently, Tcvetkova et al. have for the first time rig-orously studied the problem of conversion of an incidentplane wave into a surface wave with a growing ampli-tude [37] by means of a reflecting anisotropic metasurface(described by tensor surface parameters). The incidentplane wave and the surface wave had orthogonal polar-izations in order to avoid interference resulting in therequirement of “loss-gain” power flow into the metasur-face [35, 36]. Unfortunately, the anisotropic metasurfacewith the required impedance profile is difficult to realize,and no experimental results are available.In this paper, we elaborate on the work done by theauthors of Ref. [37] and propose an alternative practical design of a metasurface-based converter by separating theincident plane wave and the surface wave of the same po-larization in different half-spaces. A similar idea was usedin Ref. [38] for engineering reflection and transmission ofpropagating plane waves. We demonstrate realistic im-plementation of the converter based on a conventionalprinted circuit board and confirm its high-efficiency per-formance via full-wave 3D simulations.The rest of the paper is organized as follows. In Sec-tion II, we derive impedance matrix of a metasurface-based converter. By means of two-dimensional full-wavenumerical simulations, we verify theoretical findings inSection III and propose a topology of a practically real-izable metasurface. Section IV is devoted to descriptionof the design procedure and verification of the designvia three-dimensional full-wave simulations. Eventually,Section V concludes the paper. II. THEORYA. Impedance matrix of an ideal converter
Consider the conversion of a normally incident planewave (the magnetic field is along the x -axis, see Fig. 1(a)) into a transmitted TM-polarized surface wave. Thenthe corresponding magnetic and electric fields read as (weassume time-harmonic dependency in the form e iωt ) H x ( y, z ) = e ikz , E y ( y, z ) = ηe ikz ,H x ( y, z ) = Ae ( α z + iβ z ) z e ( α y − iβ y ) y ,E y ( y, z ) = − iη ( α z + iβ z ) k Ae ( α z + iβ z ) z e ( α y − iβ y ) y . (1)Indices 2 and 1 denote the fields above and below themetasurface, respectively, k is the free-space wavenum-ber, and η is the free-space impedance. All the parame-ters α and β are greater than zero and obey the dispersionrelation ( α z + iβ z ) + ( α y − iβ y ) = − k . The extinctioncoefficients α z and α y result in the surface wave attenu-ation away from the metasurface and in its growth alongthe metasurface (along the + y -direction).We avoid interference between the incident and scat-tered waves by introducing the latter one only in the bot-tom half-space. Otherwise, the interference would resultin complex power flow distribution, making it difficultto satisfy power conservation conditions locally withoutgain and lossy structures (also discussed below). Thechosen configuration when the incident and scatteredwaves propagate in different half-spaces allows us to dealwith waves of the same polarization.We characterize the metasurface with a 2 × Z ( y ). It allows one to understand the most fun-damental properties of a system disregarding its concretephysical implementation. In terms of an impedance ma-trix, the boundary conditions determining a metasurfacecan be written in the following matrix form (cid:20) E y ( y, E y ( y, (cid:21) = (cid:20) Z ( y ) Z ( y ) Z ( y ) Z ( y ) (cid:21) (cid:20) − H x ( y, H x ( y, (cid:21) . (2) The set of equations (2) serves to find the impedancematrix necessary to perform the transformation givenby Eq. (1). Unfortunately, the desired field distributionEq. (1) does not satisfy these impedance conditions forany reactive metasurface ( ¯¯ Z = − ¯¯ Z † , the symbol † standsfor the Hermitian conjugate). The physical reason forthis conclusion is that the ansatz fields do not satisfythe energy conservation principle for any choice of thesurface-wave parameters [37]. Although negative, it is animportant result: The condition of locally passive meta-surface is a crucial obstacle that does not allow one toperform an ideal conversion of a propagating plane waveinto a growing surface wave. Thus, we omit this require-ment and proceed with a more general impedance matrix¯¯ Z = i ¯¯ X , where ¯¯ X is a real-valued matrix. Substitutingthis ansatz in Eq. (2), one arrives at the following expres-sion for ¯¯ Z ¯¯ Z ( y ) = − iη (cid:34) − α z k + β z k cot[ β y y ] β z k A csc[ β y y ]exp[ − α y y ]csc[ β y y ] A exp[ α y y ] cot[ β y y ] (cid:35) . (3)Since X (cid:54) = X , the impedance matrix (3) correspondsto a nonreciprocal and locally active or lossy metasur-face. Equation (2) has other then ¯¯ Z = i ¯¯ X forms of solu-tions, as it was shown in [37] for an anisotropic metasur-face. However, for any exact solution, one arrives at thesame conclusion: The impedance matrix corresponds toeither reciprocal or nonreciprocal but always locally ac-tive or lossy metasurface. Noteworthy, active and lossyresponses do not necessarily mean that the metasurfacemust locally radiate or absorb electromagnetic waves. Wespeculate that a metasurface possessing strong spatialdispersion can be designed, as it was done in [39, 40]for controlling reflection of propagating waves. Unfortu-nately, the design procedure of such metasurfaces is stillbased on the local periodic approximation [41, 42] whatdoes not allow one to set the near-field found a priori [40]. B. Small growth approximation
Conventional leaky-wave antennas perform the conver-sion of a waveguide mode (e.g., a surface wave) into apropagating wave [43]. It makes one think of the recipro-cal, “time-reversed” process of converting a propagatingwave into a surface wave. We use the quotes to stressthat a wave radiated by a leaky-wave antenna is necessar-ily inhomogeneous , while we are particularly interestedin converting a homogeneous plane wave into a surfacewave. Therefore, these two problems are not equivalent.Nevertheless, in practice there are only finite-size anten-nas and the inhomogeneity can be made arbitrary small(which, however, reduces the radiation efficiency). Let usfind the impedance matrix of a metasurface-based leaky-wave antenna converting a TM-polarized surface wave H x ( y, z ) = Ae ( α z − iβ z ) z e ( α y + iβ y ) y , (4) FIG. 2. Schematics of the COMSOL models used for simulating the conversion with (a) omega-bianisotropic combined sheetand (b) asymmetric three-layer structure. Port 1 launches the normally incident plane wave. Port 2 either launches or acceptsthe surface wave. Port 3 only accepts the excited surface wave. (c–d) Zooming of the three-layer metasurface with metallic walls(implemented with vias in (d)) separating individual unit cells, n is the number of unit cells per a super cell, Z i ( i = 1 , ,
3) isthe electric surface impedance of the corresponding sheet. into an inhomogeneous propagating plane wave with themagnetic field along the x -direction H x ( y, z ) = e − iβ (cid:48) z z + α y y . (5)Here β (cid:48) z = (cid:113) k + α y is the propagation constant of theradiated wave. Figure 1 (b) depicts a schematics of thisprocess. The impedance matrix ¯¯ Z ( y ) is found by solvingthe boundary problem formulated in Eq. (2) and becomessymmetric when A = (cid:112) β (cid:48) z /β z , thus, corresponding to areactive and reciprocal metasurface¯¯ Z ( y ) = − iη − α z k + β z k cot[ β y y ] √ β z β (cid:48) z k csc[ β y y ] √ β z β (cid:48) z k csc[ β y y ] kβ (cid:48) z cot[ β y y ] . (6)Noteworthy, in Ref. [25] Tcvetkova et al. arrived at asimilar impedance matrix for an anisotropic metasurface.The reader is also directed to Ref. [26], where the au-thors consider an omega-bianisotropic metasurface-basedleaky-wave antenna radiating a waveguide mode thatpropagates between the metasurface and a ground plane.In strong contrast with Ref. [26], we employ the conceptof leaky-wave antennas as a tool to approach the problemof converting a uniform plane wave into a surface waveas discussed further.The reciprocity of the impedance matrix (6) allows oneto harness the corresponding metasurface for convertingthe inhomogeneous plane wave at normal incidence (5)into the surface wave (4). Since we are particularly inter-ested in converting a homogeneous plane wave (this is thecase in most practical situations when the source of wavesis in the far zone of the metasurface), the total growth ofthe surface wave amplitude along the length of the meta-surface has to remain small. Mathematically, the smallgrowth condition can be expressed as α y L (cid:28)
1, where L is the total size of the metasurface in the y -direction.Under the condition α y L (cid:28) A = (cid:112) k/β z )converges to the following matrix¯¯ Z ( y ) = − iη − α z k + β z k cot[ β y y ] (cid:113) β z k csc[ β y y ] (cid:113) β z k csc[ β y y ] cot[ β y y ] . (7)Reactive and symmetric impedance matrix (7) representsan approximate solution of the boundary problem (2) andcannot realize exactly the transformation represented byEq. (1) even in case of small (but finite) values of the pa-rameter α y L . Additional waves (not present in Eq. (1))will be excited and play the role of auxiliary waves in theconservation of local normal power flow [38–40]. Fur-thermore, in order to satisfy the small growth condi-tion for a metasurface with the impedance matrix (7),an input surface wave should be excited. Tcvetkova etal. arrived at the same conclusion in Ref. [37]. Indeed,the time-averaged power flow density associated with thesurface wave in Eq. (1) has exponential growth alongthe metasurface that becomes nearly linear under thesmall growth assumption (being non-zero along the wholemetasurface since | α y y | (cid:28)
1) given by S SW ( y, ≈ η α y y ) (cid:18) β y β z y − z (cid:19) . (8)In order to create the initial power flow (at y = 0) alongthe y -direction in accordance with Eq. (8), the amplitudeof the input surface wave should be equal to (cid:112) k/β z (theamplitude of the excited surface wave (1)). Vice versa,the amplitude of the excited surface wave will be equalto the one of the input surface wave. Since there are twoexcitation sources (incident homogeneous plane wave andinput surface wave), one has to correctly adjust the com-plex amplitude of the input surface wave: It must be inphase with that of the incident plane wave and its mag-nitude must be (cid:112) k/β z times larger. Only under these FIG. 3. (a) Conversion efficiency vs. the total length of the metasurface (expressed in terms of the number of periods) fordifferent growth rates α y of the surface wave, when the Port 2 is on and excites an input surface wave. (b–c) Normalized powerreceived by the Ports (b) 3 and (c) 2 vs. the total length of metasurface, when the Port 2 is listening (no input surface wave).(d–g) Snapshots of the magnetic field for a metasurface with 10 periods, the growth rates are (d), (f) α y = 0 . k and (e),(g) α y = 0 . k . The Port 2 is on in figures (d–e) and off in (f–g). The arrows depict directions of the power flow density.(h) Continuous and (i) discretized components (imaginary parts) of the impedance matrix as functions of the y -coordinate.(j) Conversion efficiency in case of a discretized impedance matrix vs. the number of unit cells per period (total length ofa metasurface is 10 L ) for different growth rates α y of the surface wave, when the Port 2 is on. In all figures metasurface isrepresented by an omega-bianisotropic combined sheet and propagation constant of the surface wave equals β y = 1 . k . conditions nearly all the power of the incident plane waveis transferred to the surface wave. Practically, the adjust-ing procedure can be performed by tuning the power andthe phase of the input surface wave (for instance) whilemeasuring the power of the output surface wave. Theprocedure is over as soon as the maximum of the outputpower is found.In spite of all the limitations listed above, theimpedance matrix (7) seems to be the only possible peri-odic, reactive and reciprocal solution for the conversionproblem (which is formulated by the Eq. (1) and Eqs. (4),(5)). In what follows, we use only the impedance matrixgiven by Eq. (7). III. RESULTS OF 2D SIMULATIONS
In this section we present and analyze results of two-dimensional (2D) full-wave numerical simulations on theconversion of an homogeneous incident plane wave into a surface wave. In the 2D simulations a metasurface wasmodeled by means of boundary conditions as describedin more detail further.
A. Omega-bianisotropic combined sheet
A metasurface characterized by a symmetricimpedance matrix can be realized as a combinedsheet possessing omega-bianisotropic response. Then,an incident wave excites electric J es and magnetic J ms surface polarization currents that result in thediscontinuity of both tangential electric and magneticfields at the metasurface. In the particular case whenthe magnetic field is along the x -direction, the boundary FIG. 4. (a) Conversion efficiency vs. the total length of metasurface (expressed in terms of the number of periods) for differentgrowth rates α y of the surface wave, when the Port 2 excites an input surface wave. (b),(c) Normalized power received by thePorts (b) 3 and (c) 2 vs. the total length of metasurface, when the Port 2 is listening (no input surface wave). Metasurface isrepresented by an asymmetric three-layer structure incorporating metallic walls, the impedance matrix is discretized with fourunit cells per period. conditions read as H x ( y, − H x ( y,
0) = J es ( y ) ,E y ( y, − E y ( y,
0) = J ms ( y ) ,J es = 1 Z es E y + E y K me H x + H x ,J ms = Z ms H y + H y − K me E x + E x . (9)Here Z es and Z ms are, respectively, electric and magneticsurface impedances, K me is the magneto-electric couplingcoefficient. When comparing Eq. (2) with Eq. (9), surfaceimpedances and the coupling coefficient can be expressedin terms of the components of the impedance matrix Z es = 14 (cid:88) a,b =1 Z ab , Z ms = det[ ¯¯ Z ] Z es , K me = Z − Z Z es , (10)where det[ ¯¯ Z ] = Z Z − Z is the determinant of ¯¯ Z .In order to verify theoretical findings and estimate theconversion efficiency, we perform 2D full-wave numericalsimulations with COMSOL MULTIPHYSICS. The meta-surface is represented by electric and magnetic surfacecurrents set in accordance with Eq. (9). A schematics ofthe model is illustrated by Fig. 2(a). Thus, the conversionefficiency is defined as the difference between the outputpower from Port 3 ( P ) and the input power from Port2 ( P ) divided over the power delivered by the incidentplane wave from Port 1 ( P ): ( P − P ) /P .Figure 3(a) validates the small growth approximation.It is seen that the conversion efficiency approaches 1 anddoes not depend on the total length of the metasurfaceup to α y L ∼ .
01. When increasing the growth rate α y (the rest of the parameters are fixed), the conversionefficiency decreases for longer metasurfaces what leads toappearance of spurious scattering in the far-field (com-pare distribution of the power flow density in Figs. 3(d)and (e)).As it was noticed above, the small growth approxima-tion can be strictly valid only when there is an input surface wave from the Port 2. Figures 3(b–c) demon-strates the scenario when the Port 2 is listening. Inbright contrast with the case of Fig. 3(a), the part ofpower of the incident wave coupled to the surface waveincreases (but eventually saturates) for larger values of α y L , compare Figs. 3(a) and 3(b). The difference stemsfrom the normal power flow mismatch at the left end ofthe metasurface occurring in the case when the Port 2is switched off. In the result, surface waves propagatingalong and opposite to the y -axis are excited when there isno an input surface wave as demonstrated by Figs. 3(b)and (c). Moreover, it is seen that for small α y L thepower received by the Port 2 is approximately equal tothe power received by the Port 3 (and a significant por-tion of incident power appears in the far-field as spuriousscattering). Snapshots of the magnetic field depicted inFigs. 3(f) and (g) show the influence of the spurious scat-tering on the field profile and power flow distribution inthe cases of small ( α y = 0 . k ) and large ( α y = 0 . k )growth rates. Specific attention should be paid to theregion above the metasurface: Disturbed normally inci-dent power flow indicates the spurious scattering in thefar-field.Although the portion of incident power transfered tothe surface wave is considerably higher in case there is aninput surface wave, the conversion of a propagating waveinto a surface wave usually assumes absence of any inputsurface wave. At this point one can conclude that meta-surfaces do not represent the best approach to the prob-lem but, however, can perform very efficient enhance-ment of an input surface wave (phase an amplitude ofthe incident plane wave should be accordingly adjustedas discussed in Section II).Practically, it is important to study the influenceof the discretization of a continuous impedance matrixon the performance of a metasurface. The discretizedimpedance matrix is found from the continuous one as¯¯ Z ( y − mod( y, L/n ) + L/n/ n is the number ofunit cells per period. The components of the impedancematrix as functions of y are plotted in Fig. 3(h) for FIG. 5. Topology of the copper (in yellow color) patternsimplementing grid impedances in the three-layer design of themetasurface performing the conversion of a normally incidentplane wave into the surface wave with β y = 1 . k and α y =0 . k at the frequency 10 GHz. There are four unit cellsper period ( L = 2 π/β y ≈ .
57 mm) separated by metallicwalls (illustrated by red rectangles). Thickness of the coppercladding is 35 µ m. Minimal width of copper traces and gapsis 0 .
35 mm. β y = 1 . k and α y = 0 . k . The components (as func-tions of y ) of the corresponding discretized impedancematrix ( n = 4) are shown in Fig. 3(i). Figure 3(j) demon-strates that only in the case of two unit cells per pe-riod there is a drop in the conversion efficiency. Makingthe discretization finer the efficiency quickly grows andreaches the limit of the continuous impedance matrix foras few as n = 4 unit cells per period. This result is veryimportant as allows one to use large unit cell and simplifythe design of a sample. B. Three-layer asymmetric structure
Omega-bianisotropic response can be mimicked withthree grid impedances separated by two dielectric sub-strates [32] as illustrated in Figs. 2(b) and (c). In theCOMSOL model grid impedances are introduced via elec-tric surface currents (in the similar manner with the pre-vious section). From the transmission line (TL) theory,the impedance matrix (7) corresponds to the following
TABLE I. Physical dimensions of the designed metasurface.Parameter w represents the width of the strip/slot for eachinductive/capacitive impedance. Parameter l is the length ofthe meander in the strips or slots. Cell 1 Cell 2 Cell 3 Cell 4Top No meanders w = 1 .
23 mm 2 meanders w = 0 .
35 mm l = 4 .
48 mm 4 meanders w = 0 .
35 mm l = 1 .
35 mm 2 meanders w = 0 .
35 mm l = 2 .
65 mmMid. 2 meanders w = 0 .
35 mm l = 1 .
23 mm 2 meanders w = 0 .
35 mm l = 1 .
52 mm 2 meanders w = 0 .
35 mm l = 1 .
82 mm 2 meanders w = 0 .
35 mm l = 1 .
52 mmBott. 1 meander w = 0 .
35 mm l = 1 .
49 mm 1 meander w = 0 .
35 mm l = 2 .
27 mm 1 meander w = 0 .
35 mm l = 2 .
60 mm 1 meander w = 0 .
35 mm l = 1 .
10 mm grid impedances [32] Z = η s tan( k s h ) i + η s tan( k s h ) Z + Z det[ ¯¯ Z ] ,Z = − ( η s tan( k s h )) Z det[ ¯¯ Z ] sec( k s h ) − iη s tan( k s h ) Z det[ ¯¯ Z ] ,Z = η s tan( k s h ) i + η s tan( k s h ) Z + Z det[ ¯¯ Z ] , (11)where k s = √ ε s k and η s = η/ √ ε s , ε s is the relativepermittivity of the dielectric substrates (of thickness h each). The TL theory assumes that inside the substratesonly waves with the exp( ∓ ik s z ) spatial dependence prop-agate. This assumption can be strictly valid only forspatially uniform grid impedances. However, it is not thecase of wavefront transforming metasurfaces (and consid-ered metasurface-based converters of propagating wavesinto surface waves) which require spatial modulation ofimpedances. Indeed, closely placed spatially modulatedimpedance sheets also interact via waves propagatingalong the substrates which are not taken into account byEq. (11). In order to reduce the impact of these waves onehas to use very thin and high permittivity substrates [32]which refract the waves closer to the normal direction(and introduce high dielectric losses). Unfortunately, itstill does not allow one to design the grid impedances sep-arately by means of only Eq. (11) (due to the couplingbetween adjacent unit cells). Instead, Eq. (11) providesa coarse approximation which is used as a first step of adesign procedure aimed at obtaining a given impedancematrix.The waves propagating along the substrates can be cutoff by means of metallic walls separating each unit cellfrom the others (in analogy with the idea introduced inacoustics [44]), see Fig. 2(c). Practically, metallic wallscan be implemented as arrays of vias in a multi-layerprinted circuit board. Such design solution allows one touse substrates of arbitrary large thicknesses h and per-form design of a sample considering each grid impedanceseparately. Since a pair of metallic walls represents aparallel plate waveguide inside a unit cell, waves canpropagate with tangential component of wave vector tak-ing the discrete values β m = mπ/d where d = L/n and m = 0 , ± , ± , ... Thus, the finer the discretiza-tion, the less the interaction between the adjacent gridimpedances. However, practically it is easier to increasethe substrate thickness than to decrease the unit cell sizein order to reduce the interaction via higher order spatialharmonics. Figure 4 demonstrates the dependence of theconversion efficiency on the total length of metasurfaceand the growth rate α y when there is and there is no aninput surface wave from the Port 2 [see Fig. 2(b)]. Bycomparing Figs. 3 and 4 one can see that the resultsfor the practical three-layer structure qualitatively re-peat those for omega-bianisotropic combined sheet whilequantitative differences are minor and can be explainedby the impedance mismatch between the Port 2 and thethree-layer metasurface, see Fig. 2(b). IV. SAMPLE DESIGN AND RESULTS OF 3DSIMULATIONS
The next step towards a real metasurface-based con-verter is to implement (by means of metallic patterns)three grid impedances found from Eq. (11). The de-sign is performed at the chosen operating frequency of10 GHz in accordance with requirements of the conven-tional printed-circuit-board technology. On the base ofthe conducted analysis of 2D simulations, we have chosenthe growth rate parameter equal 0 . k , the propagationconstant of the surface wave is 1 . k . Eventually, we vali-date the developed design by comparing the results of 2Dand 3D full-wave numerical simulations for a metasurfaceof total length L = 10 L .The design procedure is based on the commonly usedlocal periodic approximation (see, e.g., Refs. [41, 42]).Each grid impedance is designed separately. It is pos-sible due to incorporation of metallic walls and usageof thick dielectric substrates. Specifically, commerciallyavailable F4BM220 substrates with relative permittivity ε s = 2 . − i − ) and thickness h = 5 mm are used.The topology of the designed grid impedances is depictedin Fig. 5, parameters are specified in Tab. I.In order to validate the design, we exploit the recipro-cal scenario when the metasurface is excited from Port3 and the Port 2 is listening (Port 1 is absent in thisgeometry). We compare 2D and 3D simulations. Theschematics of the model is shown in Fig. 6 (a). In such aconfiguration the metasurface transforms the input sur-face wave from the Port 3 into a propagating wave andbecomes a leaky-wave antenna. Figures 6 (b) and (c)compare the distribution of the magnetic field obtainedin the 2D and 3D simulations, respectively. Figure 6 (d)allows one to see the difference between the magneticfields at the distance λ/
10 below the metasurface. Sincethe metasurface is designed in accordance with the slowgrowth approximation, not all the power of the surfacewave form the Port 3 is launched as a leaky-wave (ap-
FIG. 6. (a) Schematics of the COMSOL model used for com-paring 2D (three-layer metasurface) and 3D (grid impedancesare substituted by metallic patterns) simulations, WG sectionrepresents surface waveguide implemented as an impedanceboundary condition Z WG = iηα z /k . The Port 2 accepts thesurface wave and the Port 3 excites an input surface wave.(b),(c) Snapshots of the magnetic field for the metasurfaceswith 10 periods in the (b) 2D and (c) 3D simulations, thegrowth rate is 0 . k . The arrows depict directions of thepower flow density. (d) Magnetic field along the metasurface(at the distance λ/
10 below the metasurface) extracted from2D (red curve) and 3D (blue curve) simulations. proximately 50% of power is radiated in the consideredexample). Thus, the surface wave entering the Port 2in Fig. 6 is the equivalent of the input surface wave inFigs. 3 and 4.
V. DISCUSSION AND CONCLUSION
We have theoretically studied the conversion of a nor-mally incident plane wave into a transmitted surface waveby means of a scalar omega-bianisotropic metasurface. Itallows one to decouple the illumination from the scat-tered field without changing its polarization and eventu-ally significantly simplifies the design of a sample. Theproblem has been approached from two sides: By di-rectly solving the corresponding boundary problem andby considering the “time-reversed” scenario when a sur-face wave is converted into a nonuniform plane wave. Inagreement with Ref. [37], we have concluded that the perfect conversion of a uniform plane wave into a trans-mitted surface wave requires the metasurface to exhibitloss-gain response. On the other hand, a surface wavecan be totally radiated into a nonuniform plane wave bya reactive reciprocal metasurface. When imposing thecondition of a slowly growing surface wave, the two ap-proaches lead to the same reactive reciprocal metasurfacewhich can be used for converting a uniform plane waveinto a single surface wave with nearly 100% efficiency.The condition of slow growth requires an input surfacewave to create an initial power flow, which is a necessarycondition to have a metasurface with passive and losslesselements.The theoretical results have been validated throughfull-wave 2D simulations by representing a metasurface as a combined sheet with an omega-bianisotropic response.Next, we have developed a practical three-layer meta-surface based on conventional printed circuit board tech-nology to mimic the omega-bianisotropic response. Themetasurface incorporates metallic walls to avoid couplingbetween adjacent unit cells and accelerate the design pro-cedure. The design has been validated with 2D and3D simulations and demonstrated high conversion effi-ciency. Noteworthy, the three-layer structure is not theonly way to achieve the response prescribed by an asym-metric impedance matrix. Generally, in order to imple-ment omega-bianisotrpic response, one has to considerasymmetric (with respect to the plane z = 0) unit cells.As a concluding remark, metasurfaces may not repre-sent the best solution for the matter at hand and otherstrategies have to be considered. For instance, a re-cently emerged concept of metamaterials-inspired diffrac-tion gratings (or metagratings) have demonstrated un-precedented efficiency in manipulating scattered waveswith sparse arrays (contrary to metasurfaces) of polariz-able particles [45–47]. Due to the sparseness, metagrat-ings inherently possess strong spatial dispersion, what(together with a straightforward design procedure [48])can be beneficial for solving the conversion problem. 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