A Discrete Fourier Transform-Based Framework for Analysis and Synthesis of Cylindrical Omega-bianisotropic Metasurfaces
AA Discrete Fourier Transform-Based Framework for Analysis and Synthesis ofCylindrical Omega-bianisotropic Metasurfaces
Gengyu Xu, ∗ George V. Eleftheriades, and Sean V. Hum
The Edward S. Rogers Sr. Department of Electrical and Computer Engineering,University of Toronto,Toronto, Ontario M5S 3H7, Canada (Dated: July 22, 2020)This paper presents a framework for analyzing and designing cylindrical omega-bianisotropicmetasurfaces, inspired by mode matching and digital signal processing techniques. Using the discreteFourier transform, we decompose the azimuthally varying omega-bianisotropic surface parametersas well as the electric and magnetic field distributions into orthogonal modes. Then, by invokingappropriate boundary conditions, we set up systems of algebraic equations which can be rearrangedto either predict the scattered fields of prespecified metasurfaces, or to synthesize metasurfaces whichsupport arbitrarily stipulated field transformations. The proposed framework facilitates the efficientevaluation of electromagnetic field distributions that satisfy local power conservation, which is one ofthe key difficulties involved with the design of passive and lossless scalar metasurfaces. It representsa promising solution to circumvent the need for active components, controlled power dissipation, ortensorial surface polarizabilities in many state-of-the art conformal metasurface-based devices. Todemonstrate the robustness and the versatility of the proposed technique, we design several devicesintended for different applications and numerically verify them using finite element simulations.
I. INTRODUCTION
Electromagnetic metasurfaces are devices of subwave-length thickness consisting of two-dimensional arrange-ments of scatterers (meta-atoms) with engineered elec-tric and/or magnetic polarizabilities and subwavelengthdimensions [1, 2]. Due to their extraordinary abilityto efficiently manipulate various aspects of electromag-netic waves, metasurfaces have been leveraged in a va-riety of practical applications. For example, beam redi-rectors and splitters [3–6], flat lenses [7], frequency fil-ters [8, 9], polarization transformers [10, 11] and electro-magnetic cloaks [12–15] have all been successfully demon-strated. Of the various reported metasurface-based de-vices, those exhibiting omega-bianisotropy have beenparticularly fascinating, due to their abilities to real-ize highly sophisticated field transformations with nearperfect efficiencies [16–18]. Their robustness stems fromthe cross-coupling between the electric and magnetic re-sponses, which represents an additional degree of freedomthat can be engineered at will.Much of the research efforts so far in the area of meta-surfaces have been devoted to flat planar surfaces. How-ever recently, cylindrical metasurfaces have also gener-ated significant interests due to the potential applica-tions offered by their unique geometries. For instance,the fact that they can completely enclose an object makethem ideal for electromagnetic interference reduction andcloaking [12–15]. They have also been used to createelectromagnetic illusions [19, 20] and conformal anten-nas [21].Wave scattering by cylindrical metasurfaces can beaccurately predicted by the method of moments based ∗ [email protected] on tensorial bianisotropic sheet transistion conditions(BSTCs) [22, 23]. Alternatively, one can analyze thesesurfaces using mode matching [24]. While the latter tech-nique is more suitable for impedance surfaces, it is possi-ble to generalize it for omega-bianisotropic metasurfacessince they can be constructed from multiple concentricimpedance surfaces. By cascading the generalized scat-tering matrices (GSMs) of its constituent layers, a multi-layer omega-bianisotropic metasurface (O-BMS) can beaccurately modelled [25].Although these approaches provide accurate predic-tions for the scattered fields of a cylindrical metasurface,they are inherently analysis techniques which do not of-fer any insights on how to synthesize a surface to performthe desired field transformations. For instance, the GSMof an impedance surface is essentially a black box whichdoes not reveal how its constituent parameters should beengineered.While it is possible to synthesize a cylindrical O-BMSby specifying the desired field distributions everywhereand solving the BSTC equations to obtain the requiredsurface parameters, the resulting designs will be generallyactive and/or lossy. These designs are usually undesir-able as they require active components and/or accuratelycontrolled ohmic losses to implement in practice. A tech-nique to identify field distributions that correspond topassive and lossless designs has been reported [15]. How-ever, it is based on numerical optimization of the spatialdistribution of electromagnetic fields, which can be diffi-cult.In this article, we present a mode-matching frameworkfor analysis and synthesis of circular cylindrical O-BMSsinspired by concepts in digital signal processing. Thefields everywhere as well as the bianisotropic parametersof the metasurface are decomposed into their consituentmodes using the discrete Fourier transform. Then, by in- a r X i v : . [ phy s i c s . c l a ss - ph ] J u l voking the BSTCs, we can set up systems of algebraicequations which can be easily solved to either modelthe scattering behaviour of complex O-BMSs, or to syn-thesize devices that can perform stipulated field trans-formations. In contrast with previous mode-matchingmethods, our formulation inherently accounts for the po-tentially bianisotropic polarizabilities of the metasurface,without needing to decompose it into several constituentparts. This facilitates the derivation of closed-form syn-thesis equations. Furthermore, the algebraic formulationsimplifies the process of identifying passive and losslessfield distributions, leading to very practical metasurfacedesigns.Using the proposed method, we design and investigateseveral passive and lossless cylindrical O-BMS-based de-vices. In order to validate the designs, they are eachsimulated numerically in COMSOL Multiphysics. II. THEORY
The concepts presented in this paper can be applied toany scalar circular cylindrical omega-bianisotropic meta-surfaces. However, as will be explained shortly, someof the steps throughout the analysis and synthesis willrequire slight problem-specific modifications. In this sec-tion, for illustrative purposes, we will investigate the caseof an internally excited O-BMS. Generalization to otherconfigurations will be discussed subsequently when theyarise.The problem under consideration is depicted in Fig. 1.The metasurface, characterized by its φ -dependent sur-face electric impedance Z se ( φ ), surface magnetic admit-tance Y sm ( φ ) and magnetoelectric coupling coefficient K em ( φ ), forms a closed cylindrical cavity of radius α . Inthis paper, we only consider two-dimensional problemswhich are invariant in the z direction. For sake of sim-plicity, let us assume the fields are transverse magneticwith respect to the z -axis (TM z ), meaning (cid:126)E = ˆ zE z . Ex-tension to transverse electric (TE z ) configurations ( (cid:126)H =ˆ zH z ) is straightforward. E r || E i || + metasurface Z se ( ϕ ),Y se ( ϕ ),K em ( ϕ ) ρ α sources ϕ (x)(y)(z) H r || H i || + E t || H t || Ɛ r1 Ɛ r2 FIG. 1. Geometric configuration of the internally excited ci-cular cylindrical O-BMS
As indicated, the internal and the external regions ofthe O-BMS have relative permittivities (cid:15) r and (cid:15) r re-spectively.The fields enclosed by the cylinder consist of the in-cident electric and magnetic fields { E i || , H i || } in additionto the reflected fields { E r || , H r || } . The fields external tothe cylinder are just the transmitted fields { E t || , H t || } .The subscript “ || ” denotes the transverse components ofthe fields, which are the components of interest in thepresent configuration. For TM z -polarized field distribu-tions, these correpond to the z -component of the electricfields and the φ -component of the magnetic fields.In accordance with the cylindrical coordinate system,the fields inside and outside of the metasurface cavity canbe written as series of cylindrical modes described by E iz ( ρ, φ ) = ∞ (cid:88) p = −∞ e ip H (2) p ( k ρ ) H (2) p ( k α ) e − jpφ ,H iφ ( ρ, φ ) = ∞ (cid:88) p = −∞ e ip Y iin,p ( ρ ) H (2) p ( k ρ ) H (2) p ( k α ) e − jpφ ,E tz ( ρ, φ ) = ∞ (cid:88) p = −∞ e tp H (2) p ( k ρ ) H (2) p ( k α ) e − jpφ ,H tφ ( ρ, φ ) = ∞ (cid:88) p = −∞ e tp Y tin,p ( ρ ) H (2) p ( k ρ ) H (2) p ( k α ) e − jpφ ,E rz ( ρ, φ ) = ∞ (cid:88) p = −∞ e rp J p ( k ρ ) J p ( k α ) e − jpφ ,H rφ ( ρ, φ ) = ∞ (cid:88) p = −∞ e rp Y rin,p ( ρ ) J p ( k ρ ) J p ( k α ) e − jpφ ,k { , } = √ (cid:15) r { , } k o , (1)where k o is the free space wave number. H (2) p ( · ) denotesthe p th order Hankel function of second kind and J p ( · )denotes the p th order Bessel function. The coefficients e { i,t,r } p are the complex amplitudes of the p th mode ofthe incident, transmitted and reflected electric fields eval-uated at ρ = α .The p th modes of E { i,t,r } z can be related to the p th modes of H { i,t,r } φ by the modal wave admittances Y { i,t,r } in,p ( ρ ), where the subscript “ in, p ” indicates thatthese are the p th mode admittances for an internallyexcited metasurface. As will be revealed in Sec. III,the fields of an externally excited surface have differentmodal wave admittances, indicated by their distinct sub-scripts.By invoking the linearity of Maxwell’s equations andthe orthogonality of the cylindrical modal wave functions,the expressions for Y { i,t,r } in,p ( ρ ) can be found to be Y iin,p ( ρ ) = j √ (cid:15) r η o H (2) (cid:48) p ( k ρ ) H (2) p ( k ρ ) ,Y tin,p ( ρ ) = j √ (cid:15) r η o H (2) (cid:48) p ( k ρ ) H (2) p ( k ρ ) ,Y rin,p ( ρ ) = j √ (cid:15) r η o J (cid:48) p ( k ρ ) J p ( k ρ ) . (2)The derivatives in (2) are evaluated with respect to theentire arguments of H (2) p ( · ) and J p ( · ).The exact values of the coefficients e { i,t,r } p can be ob-tained by performing a Fourier transform on the cor-responding fields along a closed circle centered at theorigin. In a similar manner, the surface properties Z se ( φ ) , Y sm ( φ ) and K em ( φ ) can be decomposed into theirconstituent Fourier harmonics because they are also 2 π -periodic. This idea forms the basis of our analysis andsynthesis methods, described in Sec. II A and Sec. II Brespectively. A. Analysis of Internally Excited CircularCylindrical Omega-Bianisotropic Metasurfaces
We first present a method for evaluating the scat-tered fields from internally excited circular cylindricalO-BMSs. We begin by discretizing the surface into N equally sized unit cells. The surface properties can besampled at the centers of these cells, forming N × Z se , ¯ Y sm and ¯ K em whose entries are¯ Z se [ n ] = Z se (cid:12)(cid:12) φ =( n − φ , ¯ Y sm [ n ] = Y sm (cid:12)(cid:12) φ =( n − φ , ¯ K em [ n ] = K em (cid:12)(cid:12) φ =( n − φ , (3)where ∆ φ = 2 π/N is the azimuthal extent of one unitcell.We can also sample the tangential fields on the internaland external facets of each unit cell to obtain N × E tz [ n ] = E tz (cid:12)(cid:12) ρ → α + ,φ =( n − φ , ¯ E { i,r } z [ n ] = E { i,r } z | ρ → α − ,φ =( n − φ , ¯ H tφ [ n ] = H tφ | ρ → α + ,φ =( n − φ , ¯ H { i,r } φ [ n ] = H { i,r } φ | ρ → α − ,φ =( n − φ . (4)Alternatively, since all vectors in (3) and (4) correspondto quantities that are 2 π -periodic in φ , it is possibleto represent them in terms of their projections ontothe Fourier basis by performing N-point dicrete Fouriertransforms (N-DFTs) via multiplications with the N- DFT matrix W N : ˆ E { i,r,t } z = W N ¯ E { i,r,t } z , ˆ H { i,r,t } z = W N ¯ H { i,r,t } z , ˆ Z se = W N ¯ Z se , ˆ Y sm = W N ¯ Y sm , ˆ K em = W N ¯ K em . (5) W N is the N-DFT matrix whose elements are W N [ n ][ m ] = 1 N e j πN (cid:0) n − N +12 (cid:1)(cid:0) m − (cid:1) . (6)We refer to ˆ E { i,r,t } z and ˆ H { i,r,t } φ as the modal field vectors,since their entries represent the amplitudes of the consti-tuient cylindrical modes of E { i,r,t } z and H { i,r,t } φ , evaluatedat ρ = α . On the other hand, ˆ Z se , ˆ Y sm and ˆ K em are themodal property vectors whose entries represent the am-plitudes of the Fourier harmonics constituting Z se , Y sm and K em .The rows of W N , as defined by (6), are arranged suchthat the center entries of the modal vectors correspondto the fundamental ( p = 0) mode. The first and thelast entries correspond to the p = − N (cid:44) p − and the p = N − (cid:44) p + modes respectively. As expected fromthe periodic property of the DFT spectrum, the degreeof spatial discretization ( N ) dictates the total number ofindependent modes, i.e., the length of the modal vectors.One advantage of the modal analysis presented hereinis that ˆ E { i,r,t } z can be algebraically related to ˆ H { i,r,t } φ with the help of (2). More specifically, we can writeˆ H { i,t,r } φ = Y { i,t,r } in ˆ E { i,t,r } z , Y { i,t,r } in = diag (cid:104) Y { i,t,r } in,p − ( α ) · · · Y { i,t,r } in,p + ( α ) (cid:105) T , (7)where Y { i,t,r } in are the diagonal modal admittance ma-trices for the incident, transmitted and reflected fields.They can be used to eliminate all magnetic field quanti-ties from the analysis, thereby reducing the number ofunknowns. Due to the different wave admittance ex-pressions, other configurations such as externally excitedsurfaces may require slightly different modal admittancematrices.The original goal of this section is to predict the trans-mitted and reflected fields, given the incident fields andthe O-BMS surface parameters. To this end, we now re-late the field vectors on either side of the metasurfaceby invoking the BSTCs at each of the N unit cells [16].Doing so results in the system of equations12 ( ¯ E tz + ¯ E iz + ¯ E rz ) = − ¯ Z se (cid:12) ( ¯ H tφ − ¯ H iφ − ¯ H rφ ) − ¯ K em (cid:12) ( ¯ E tz − ¯ E iz − ¯ E rz ) ,
12 ( ¯ H tφ + ¯ H iφ + ¯ H rφ ) = − ¯ Y sm (cid:12) ( ¯ E tz − ¯ E iz − ¯ E rz )+ ¯ K em (cid:12) ( ¯ H tφ − ¯ H iφ − ¯ H rφ ) , (8)where (cid:12) denotes element-wise multiplication.In order to take advantage of the simplifications offeredby modal analysis, we transform (8) using the circularconvolution theorem for N-DFT, which says W N ( ¯ U (cid:12) ¯ V ) = ˆ U (cid:126) N ˆ V = ˆU ˆ V . (9)The operation (cid:126) N denotes modulo-N circular convolu-tion. As alluded to by the second equality in (9), we canimplement this operation as a matrix multiplication bytransforming the first operand ˆ U into an N × N circulantmatrix ˆU given by ˆU = W N diag( W − N ˆ U ) W − N (10)Applying N-DFT to both sides of (8) and applying (7)and (9), we obtain a new system of equations
12 ( ˆ E tz + ˆ E iz + ˆ E rz ) = − ˆZ ( Y tin ˆ E tz − Y iin ˆ E iz − Y rin ˆ E rz ) − ˆK ( ˆ E tz − ˆ E iz − ˆ E rz ) ,
12 ( Y tin ˆ E tz + Y iin ˆ E iz + Y rin ˆ E rz )= − ˆY ( ˆ E tz − ˆ E iz − ˆ E rz ) + ˆK ( Y tin ˆ E tz − Y iin ˆ E iz − Y rin ˆ E rz ) , (11)where ˆZ , ˆY and ˆK are N × N circulant matrices formedby ˆ Z se , ˆ Y sm and ˆ K em respectively. This set of equationscan be easily rearranged to solve for the modal trans-mission matrix ˆT in and the modal reflection matrix ˆR in ;they relate ˆ E tz and ˆ E rz to ˆ E iz according toˆ E tz = ˆT in ˆ E iz , ˆ E rz = ˆR in ˆ E iz . (12)The exact solution for the two matrices are ˆT in = ˆt − in,b ˆt in,a , ˆt in,a = (cid:18) I − ˆK − ˆZY rin (cid:19) − (cid:18) I − ˆK − ˆZY iin (cid:19) − (cid:18) Y rin − ˆY + ˆKY rin (cid:19) − (cid:18) Y iin − ˆY + ˆKY iin (cid:19) , ˆt in,b = (cid:18) Y rin − ˆY + ˆKY rin (cid:19) − (cid:18) Y tin + ˆY − ˆKY tin (cid:19) − (cid:18) I − ˆK − ˆZY rin (cid:19) − (cid:18) I + ˆK + ˆZY tin (cid:19) , ˆR in = ˆr − in,b ˆr in,a , ˆr in,a = (cid:18) I + ˆK + ˆZY tin (cid:19) − (cid:18) I − ˆK − ˆZY iin (cid:19) − (cid:18) Y tin + ˆY − ˆKY tin (cid:19) − (cid:18) Y iin − ˆY + ˆKY iin (cid:19) , ˆr in,b = (cid:18) Y tin + ˆY − ˆKY tin (cid:19) − (cid:18) Y rin − ˆY + ˆKY rin (cid:19) − (cid:18) I + ˆK + ˆZY tin (cid:19) − (cid:18) I − ˆK − ˆZY rin (cid:19) . (13) Here, I is the N × N identity matrix. Since the ˆ H tφ andˆ H rφ can be found using (7), we have essentially completedour analysis. B. Synthesis of Internally Excited CircularCylindrical Omega-Bianisotropic Metasurfaces
Within the proposed DFT framework, it is also possi-ble to synthesize the required surface properties for real-izing a certain stipulated field transformation.While the analysis presented in Sec. II A applies togeneral O-BMSs that can contain power gain and/or loss,we will devote our effort henceworth to the synthesis ofpassive and lossless metasurfaces. This is because theyare much easier to implement in practice, requiring onlyreactive components which can be easily realized withetched patterns on printed circuit boards (PCBs).Previously, it was shown that a sufficient condition foran O-BMS to be passive and lossless is that the fieldtransformation being performed satisfies local power con-servation (LPC) [16], which can be described by the equa-tion (cid:60){ ( ¯ E iz + ¯ E rz ) (cid:12) ( ¯ H iφ + ¯ H rφ ) ∗ } = (cid:60){ ¯ E tz (cid:12) ¯ H t ∗ φ } , (14)where ( · ) ∗ denotes complex conjugation. Although com-posite metasurface systems which satisfy global powerconservation represent possible alternatives [26], they aremuch harder to design and implement. Thus, we willstrictly focus on surfaces that satisfy LPC. From (14),it can be inferred that a passive and lossless transmis-sive scalar metasurface, which is one that produces aprescribed transmitted field distribution from a knownincident field, should generate some amount of parasiticreflected fields (termed “auxiliary reflection”). Similarly,a passive and lossless reflective scalar metasurface will re-quire some auxiliary transmission. Naturally, dependingon the type of metasurface to be synthsized, we wouldfirst need to solve for either { ¯ E r , ¯ H r } or { ¯ E t , ¯ H t } . Todo this directly using (14) can be challenging. One wouldneed to solve a system of coupled nonlinear differentialequations since the unknown auxiliary electic fields andauxiliary magnetic fields are related to each other. Inthis work, we use (7) to eliminate ¯ H { i,t,r } φ from the equa-tion, yielding a non-linear algebraic equation which canbe solved numerically with ease. For transmissive meta-surfaces, where ¯ E rz or ˆ E rz is the unknown, we have (cid:60){ ( ¯ E iz + W − N ˆ E rz ) (cid:12) ( ¯ H iφ + W − N Y rin ˆ E rz ) ∗ } = (cid:60){ ¯ E tz (cid:12) ¯ H t ∗ φ } . (15)For reflective metasurfaces, where ¯ E tz or ˆ E tz is the un-known, the equation to solve is (cid:60){ ( ¯ E iz + ¯ E rz ) (cid:12) ( ¯ H iφ + ¯ E rz ) ∗ } = (cid:60){ ( W − N ˆ E tz ) (cid:12) ( W − N Y tin ˆ E tz ) ∗ } . (16)Solving either (15) or (16) gives the complete field dis-tributions everywhere. We can proceed by assessing therequired { ¯ Z se , ¯ Y sm , ¯ K em } which would support these fielddistributions.In Sec. II A, we transformed the known O-BMS param-eters into circulant matrices and solved for the unknownfield modal vectors. Here, we can perform the inverseprocedure by transforming the known field distributionsinto circulant matrices and solving for the unknown sur-face parameter modal vectors. This leads to the equa-tions12 ( ˆ E tz + ˆ E iz + ˆ E rz ) = ˆ E av = − ∆ ˆ H,in ˆ Z se − ∆ ˆ E,in ˆ K em ,
12 ( ˆ H tφ + ˆ H iφ + ˆ H rφ ) = ˆ H av = − ∆ ˆ E,in ˆ Y sm + ∆ ˆ H,in ˆ K em , (17)where ∆ ˆ E,in and ∆ ˆ H,in are the N × N field discontinuitycirculant matrices formed by the vectors ( ˆ E tz − ˆ E rz − ˆ E iz )and ( ˆ H tφ − ˆ H rφ − ˆ H iφ ) respectively.An O-BMS satisfying LPC will have imaginary ¯ Z se and¯ Y sm , as well as real ¯ K em [27]. This allows us to reduce thenumber of unknowns in (17) by invoking the conjugatesymmetry properties for the DFT of real and imaginarysignals: (cid:60){ ¯ V } = 0 = ⇒ F ˆ V = − ˆ V ∗ , (cid:61){ ¯ V } = 0 = ⇒ F ˆ V = ˆ V ∗ , (18)where F is the N × N reversal matrix. Using the factthat FF = I , we can obtain another system of equationfrom (17) as follows:ˆ E av = ∆ ˆ H,in F ˆ Z ∗ se − ∆ ˆ E,in F ˆ K ∗ em , ˆ H av = ∆ ˆ E,in F ˆ Y ∗ sm + ∆ ˆ H,in F ˆ K ∗ em . (19)Combining (17) with the complex conjugate of (19), wehave sufficiently many independent equations to solve forthe unknown surface parameter modal vectors asˆ K em = k − a ¯ k b , k a = − (cid:0) ∆ ∗ ˆ H,in F ∗ (cid:1) − ( ∆ ˆ E,in F ) ∗ − ∆ − H,in ∆ ˆ E,in , ¯ k b = (cid:0) ∆ ∗ ˆ H,in F ∗ (cid:1) − ˆ E ∗ av + ∆ − H,in ˆ E av , ˆ Z se = − ∆ − H,in ˆ E av − ∆ − H,in ∆ ˆ E,in ˆ K em , ˆ Y sm = − ∆ − E,in ˆ H av + ∆ − E,in ∆ ˆ H,in ˆ K em . (20)This concludes the synthesis procedure of internally ex-cited passive and lossless O-BMS. C. Multi-layer Implementations
Having obtained the theoretical O-BMS surface pa-rameters in Sec. II B, we now consider a physical unit cell ρ ϕ (x) (y) (z) Z se [n],Y se [n],K em [n] ρ α = Z o [n]Z m [n]Z i [n] tt Ɛ r Ɛ r Ɛ r2 Ɛ r1 PMC Baffle
FIG. 2. Triple impedance layer unit cell topology used torealize the theoretically derived O-BMS properties. topology that is suitable for practical realization of thederived properties. For planar metasurfaces, unit cellsconsisting of three parallel electric impedance sheets sep-arated by dielectric substrates have often been used [28].It was shown that the unit cell can exhibit omega-bianisotropic response if its three constituent layers areasymmetric with respect to the middle one (i.e. the topand the bottom layers have different impedances). Thistopology is usually preferred because it is highly com-patible with standard PCB fabrication technologies. Thethree impedance sheets can be easily realized using threelayers of etched conductive patterns [29, 30]. In this sec-tion, we consider an analogous curved triple impedancelayer topology, depicted in Fig. 2.Each unit cell of the surface is assigned a set of inner,middle and outer impedance values; they are denoted as¯ Z i [ n ], ¯ Z m [ n ] and ¯ Z o [ n ] respectively, for the n th cell. Thethree impedance layers are supported by two cylindrialdielectric shells with thickness t and relative permittiv-ity (cid:15) r . The inner surface impedances ¯ Z i reside on thecylindrical surface ρ = α .For planar O-BMSs, equivalent transmission-line mod-els have frequently been used to aid the design of themulti-layer unit cells [31]. It is hard to directly adopt thisapproach here because the wave impedances for cylindri-cal waves are dependent on the radial coordinate ρ . Thus,they vary throughout the longitudinal extent of each unitcell, making it difficult to construct a simple equivalenttransmission-line circuit. Instead, we employ a general-ization of the ABCD matrix approach previously used todesign a mode-converting cylindrical O-BMS [32].Before proceeding, we note that the ABCD matrix ap-proach assumes local periodicity. That is, each unit cellis analyzed and designed as if it resides within a homo-geneous surface. An accompanying assumption is thatonly the p = 0 mode is propagating between the mul-tiple impedance layers. In a realistic metasurface, dueto the φ -dependent impedances, local periodicity is notsatisfied and higher order modes can propagate withinthe extent of each unit cell. This can make the ABCDmatrix approach inaccurate, especially if the dielectriclayers are thick [33]. To alleviate this phenomenon, weemploy perfect conducting baffles to shield each unit cellfrom its neighbors. This approach is inspired by a pre-vious work which used baffles to decouple the adjacentunit cells in a planar bianisotropic metasurface [34]. Asdiscussed therein, the baffles act as an array of smallwaveguides which cut off all but the fundamental modelocally within each unit cell. For the TM z configura-tion discussed in this study, perfect magnetic conductor(PMC) baffles are used, as seen in Fig. 2. For TE z con-figurations, perfect electric conductor (PEC) baffles arerequired. While the PEC baffles are more practical sincethey can be fabricated using plated via fences, PMC baf-fles will mainly serve as a numerical tools to aid the val-idation of theoretically synthesized metasurfaces.Assuming that the ABCD matrix approach is valid, wecan apply it to each unit cell individually. The ABCDmatrix M between two concentric cylindrical surfaceswith radii ρ and ρ relates the total electric and mag-netic fields on those surfaces according to (cid:20) E z ( ρ ) H φ ( ρ ) (cid:21) = M (cid:20) E z ( ρ ) H φ ( ρ ) (cid:21) = (cid:20) A BC D (cid:21) (cid:20) E z ( ρ ) H φ ( ρ ) (cid:21) . (21)For instance, the ABCD matrix relating the fields oneither side of a cylindrical sheet with electric impedance Z is M Z = (cid:20) Z (cid:21) . (22)This can be easily derived from the BSTC equations.We can also find the ABCD matrix for the dielectricsubstrates [25]. If the inner and outer radii of the sub-strate are ρ i and ρ o respectively, and the relative permit-tivity is (cid:15) r , then the matrix is given by M sub ( ρ i , ρ o ) = (cid:20) A ( ρ i , ρ o ) B ( ρ i , ρ o ) C ( ρ i , ρ o ) D ( ρ i , ρ o ) (cid:21) = m i m − o , m { i,o } [1][1] = H (2)0 ( √ (cid:15) r k o ρ { i,o } ) , m { i,o } [1][2] = J ( √ (cid:15) r k o ρ { i,o } ) , m { i,o } [2][1] = Y + sub ( ρ { i,o } ) H (2)0 ( √ (cid:15) r k o ρ { i,o } ) , m { i,o } [2][2] = Y − sub ( ρ { i,o } ) J ( √ (cid:15) r k o ρ { i,o } ) , (23)where Y + sub ( ρ ) = j √ (cid:15) r η o H (2) (cid:48) ( √ (cid:15) r k o ρ ) H (2)0 ( √ (cid:15) r k o ρ ) ,Y − sub ( ρ ) = j √ (cid:15) r η o J (cid:48) ( √ (cid:15) r k o ρ ) J ( √ (cid:15) r k o ρ ) . (24)From Fig. 2, we can see that the matrices M sub ( α, α + t ) and M sub ( α + t, α + 2 t ) model the innerand the outer dielectric substrates respectively. Cascad-ing the matrices for the three impedance layers as well asthose for the two dielectric substrates in the appropriateorder, we obtain the overall ABCD matrix for a unit cellas M = M Z i M sub ( α, α + t ) M Z m M sub ( α + t, α + 2 t ) M Z o , (25) where Z i , Z m , Z o are the inner, middle and outerimpedances for that cell respectively.Assuming we have already calculated the O-BMS pa-rameters { Z se , Y sm , K em } for the unit cell following theprocedures in Sec. II B, we can use (8) and (21) to findthe equivalent ABCD matrix parameters of that cell as A = 4 K em + 4 Y sm Z se + 4 K em + 14 K em + 4 Y sm Z se − ,B = 4 Z se K em + 4 Y sm Z se − ,C = 4 Y sm K em + 4 Y sm Z se − ,D = 4 K em + 4 Y sm Z se − K em + 14 K em + 4 Y sm Z se − . (26)If we denote the elements of dielectric substrate ABCDmatrices as M sub ( α + t, α + 2 t ) = (cid:20) a b c d (cid:21) , M sub ( α, α + t ) = (cid:20) a b c d (cid:21) , (27)then the required Z i , Z m and Z o can be found to be Z i = b Bb D − d B + a b d − b b c ,Z m = − b b a b − B + b d ,Z o = − b Ba B − b A − a b d + b c c . (28)Using (26) and (28) on each of the N unit cells givesthe final theoretical implementation of our synthesizedO-BMS. D. Comments on Other MetasurfaceConfigurations
As mentioned earlier, some of the steps in the analysisand synthesis procedures need to be modified accordingto the problem geometry. This requirement stems fromthe fact that the cylindrical wave functions (1) whichconstitute the electric and magnetic fields are problem-specific. Furthermore, the BSTC equations (8) are dif-ferent depending on the location of the source (internalor external). This in turn leads to different analysis (13)and synthesis (20) equations. On the other hand, theABCD matrix formalism presented in Sec. II C does notassume anything about the source location, and thus canbe used for any problem. As we go over detailed designexamples in Sec. III, it will become clear that the overallDFT framework is readily generalizable.
III. RESULTS AND DISCUSSIONS
In this section, we design several passive and losslessO-BMSs using our proposed method and validate themnumerically using finite element simulations in COM-SOL Multiphysics. Without loss of generality, we assumehenceworth that the external region is comprised of air( (cid:15) r = 1). A. Electromagnetic Illusion
In the first example, we design a cylindrical O-BMSwhich transforms the fields radiated by an electric linesource located at the origin to those from a displacedline source. Previously, metamaterials (MTMs) basedon transformation optics (TO) have been leveraged toachieve this effect [35]. However, the practicality ofTO-MTMs is significantly hindered by their bulkinessand complexity. Here, we attempt to achieve the sameelectromagnetic illusion using a metasurface with deeplysubwavelength thickness, which is much easier to fab-ricate and deploy. Although this type of device hasbeen widely reported [19, 22, 23], passive lossless imple-mentations leveraging omega-bianisotropy have not beendemonstrated thus far to the best of our knowledge.To design the illusion metasurface, we first state theincident electric field modal vector asˆ E iz [ n ] = e i δ n, N +12 , (29)where δ i,j is the Kronecker delta and e i can be an arbi-trary complex constant. There is only one non-zero entryin ˆ E iz because an electric line source located at the origincan only excite the 0 th cylindrical mode.The transmitted modal vector can be obtained usingthe addition theorem, which describes a Hankel functioncentered at ( ρ, φ ) = ( ρ (cid:48) , φ (cid:48) ) in terms of a summation ofcylindrical modes centered at the origin [36]: H (2)0 ( k o | (cid:126)ρ − (cid:126)ρ (cid:48) | ) = ∞ (cid:88) p = −∞ J p ( k o ρ (cid:48) ) H (2) p ( k o ρ ) e jp ( φ − φ (cid:48) ) ρ ≥ ρ (cid:48) (30a) ∞ (cid:88) p = −∞ H (2) p ( k o ρ (cid:48) ) J p ( k o ρ ) e jp ( φ − φ (cid:48) ) ρ ≤ ρ (cid:48) (30b)For the case of a virtual source inside the O-BMS cav-ity ( ρ (cid:48) < α ), we can obtain ˆ E tz using (30a) by setting ρ = α : ˆ E tz [ n ] = e t J n ( k o ρ (cid:48) ) H (2) n ( k o α ) H (2)0 ( k o α ) e jnφ (cid:48) . (31)Here, e t is the amplitude of the virtual source field mea-sured at a distance α away from (cid:126)ρ (cid:48) ; it is set to be equalto e i in this study. As an example, we design an il-lusion metasurface with the specifications listed in Ta-ble I. Since this is a transmissive metasurface, we solve TABLE I. Specification for the passive lossless illusion O-BMS f (GHz) N α [m] ρ (cid:48) φ (cid:48) [rad] (cid:15) r (cid:15) r (cid:15) r t [mm]4.4 451 0.15 0.95 α π/ for the required auxiliary ˆ E rz using (15). The numeri-cally obtained solution is shown in in Fig. 3(a). Usingthe complete field distributions ˆ E { i,t,r } z , we obtain themulti-layer implementation for the O-BMS described byFig. 3(b). Notably, we only plot the reactances of eachlayer, since the real parts of the impedances are iden-tially zero. This indicates the synthesized device is trulypassive and lossless. -200 -150 -100 -50 0 50 100 150 200-0.0200.020.04 ReIm z r (a) M u lti - l a y e r I m p l e m e n t a ti on X i X m X o (b) FIG. 3. (a) The solved auxiliary reflected fields for the il-lusion O-BMS. (b) The reactance values for the multi-layerimplementation of the illusion O-BMS.
Next, we construct the metasurface model in COM-SOL Multiphysics, in which the surface reactance lay-ers are implemented using field-dependent surface electriccurrent densities. The source is an electric line currentplaced at the origin. The simulated total (incident plusscattered) electric field distribution is depicted in Fig. 4.Despite the complex interference pattern produced bythe auxiliary internal reflections, we observe clear un-perturbed wavefronts outside of the metasurface cavityemanating from the desired virtual source location.In some scenarios, one may wish to construct a virtualsource outside of the O-BMS cavity ( ρ (cid:48) > α ). Althoughat first it appears that (30b) should be used to obtain FIG. 4. (cid:60){ E z } for the illusion O-BMS. ˆ E tz , we note that such a field distribution cannot actu-ally be realized with an internally excited O-BMS. Thisis because (30b), and indeed the fields of an ideal ex-ternal virtual line source, exhibit converging power flow(towards the origin) for some points on the external faceof the metasurface. This can be seen from the Besselfunctions which constitute the summation in (30b), or itcan be reasoned intuitively by the fact that the Poyntingvector should strictly diverge from ( ρ (cid:48) , φ (cid:48) ). We can cir-cumvent this issue by approximating the desired externalfield distribution using (30a). Following the same synthe-sis procedure, we can obtain an O-BMS whose externalfields imitate those of the desired virtual source for theregion ρ ≥ ρ (cid:48) . However, in the region α < ρ < ρ (cid:48) , thefields will not accurately depict the desired virtual sourcefields. B. General Penetrable Metasurface Cloak
In this example, we design a metasurface cloak whichconceals a cylindrical target from an external incidentwave. Conventionally, this is often achieved with ac-tive Huygens’ metasurfaces (HMS) which radiate someprescribed fields intended to destructively interfere withthe scattered fields from the target [13, 37]. While ef-fective, this type of cloak usually requires complex cir-cuitry to properly control and can suffer from stabilityissues. Alternatively, TO-MTM cloaks have also beendemonstrated [38]. They are highly robust, capable ofconcealing the target from any illumination type withany incident direction. However, just like other MTM-based devices, their bulkiness significantly restricts theirpractical applications.Recently, a type of penetrable metasurface cloak hasbeen demonstrated [14]. It was proposed that if theobject is penetrable by electromagnetic waves, one cansynthesize a metasurface enclosure which induces zero external scattered fields while permitting some internalscattering. By carefully engineering the internal fieldsbased on the incident fields which are known a priori , lo-cal power conservation can be satisfied. Thus, the cloakcan be realized using a passive and lossless O-BMS. Animportant application for this class of cloaking is the re-duction of electromagnetic interference for complex wire-less communications systems in which the incident fielddistributions are known [39].Previous iterations of penetrable O-BMS cloaks as-sume the incident field to be a perfect plane wave. Inthat case, the required internal fields for satisfying localpower conservation can be easily inferred to be anotherplane wave. Here, we extend this concept in two ways.First, we allow the incident fields to take on arbitraryforms. In general, the required internal fields for sat-isfying local power conservation are intricate and lackanalytical descriptions. With our proposed approach, wecan specify the reflected fields to be zero and numericallysolve the local power conservation equation to obtain therequired internal (auxiliary) fields. This straightforwardprocedure enables us to hide the object from more com-plex illuminations.Furthermore, with some slight modifications, we canrealize perfect cloaking for an impenetrable object suchas a PEC cylinder. Although passive and losslessmetasurfaces with such capability have been demon-strated [15], they rely on carefully optimized orthogo-nally polarized surface waves facilitated by tensorial sur-face properties to achieve pointwise power balance. Incontrast, our proposed approach uses a scalar metasur-face which does not rely on any polarization conversion.In that light, our approach is similar to a previous workthat leverages auxiliary transmitted waves to facilitatesatisfaction of local power conservation in a reflectivemetasurface beam splitter [6]. Scalar metasurfaces canbe easier to fabricate in practice, compared to tensorialsurfaces, owing to the simpler geometries of their con-stituent meta-atoms.To develop the penetrable cloak, let us first considerthe case of a general externally excited cylindrical O-BMS enclosing a homogeneous dielectric cylinder, as de- E r || E i || + metasurface Z se ( ϕ ),Y se ( ϕ ),K em ( ϕ ) ρ α sources ϕ (x)(y)(z) H r || H i || + E t || H t || Ɛ r1 Ɛ r2 FIG. 5. Geometric configuration of the externally excitedcylindrical O-BMS picted in Fig. 5. Unlike the problem in Sec. III A, theincident and reflected fields now exist in the external re-gion whereas the transmitted fields exist in the internalregion, meaning the definitions in (4) need to be adjustedaccordingly. Furthermore, the BSTC equations for thisproblem read:12 ( ¯ E iz + ¯ E rz + ¯ E tz ) = − ¯ Z se (cid:12) ( ¯ H iφ + ¯ H rφ − ¯ H tφ ) − ¯ K em (cid:12) ( ¯ E iz + ¯ E rz − ¯ E tz ) ,
12 ( ¯ H iφ + ¯ H rφ + ¯ H tφ ) = − ¯ Y sm (cid:12) ( ¯ E iz + ¯ E rz − ¯ E tz )+ ¯ K em (cid:12) ( ¯ H iφ + ¯ H rφ − ¯ H tφ ) . (32)The new source location implies that the incident and thetransmitted fields are now composed of J p ( · ), while thereflected fields are described by H (2) p ( · ). The wavenumberfor those fields also need to be changed accordingly toreflect the appropriate dielectric constants in the internaland external regions. Combining these observations, themodal admittance matrices can be inferred to be: Y { i,t,r } ex = diag (cid:104) Y { i,t,r } ex,p − ( α ) · · · Y { i,t,r } ex,p + ( α ) (cid:105) T ,Y iex,p ( ρ ) = j √ (cid:15) r η o J (cid:48) p ( k ρ ) J p ( k ρ ) ,Y rex,p ( ρ ) = j √ (cid:15) r η o H (2) (cid:48) p ( k ρ ) H (2) p ( k ρ ) ,Y tex,p ( ρ ) = j √ (cid:15) r η o J (cid:48) p ( k ρ ) J p ( k ρ ) . (33)The subscript “ ex ” indicates that these matrices are validfor externally excited O-BMSs.Following the same convolution-based derivation pre-sented in Sec. II A, we can obtain modal transmissionmatrix ˆ T ex and modal reflection matrix ˆ R ex for this newconfiguration. Although these matrices are not explicitlyused in this paper, we still present their full solutions inthe Supplementary Materials for completeness; they aregiven by (A1) and (A2) in Appendix A respectively.To synthesize an externally excited cylindrical O-BMS,we would again need to solve the appropriate local powerconservation equation (15) or (16) for the required aux-iliary fields. However, now we need to pay attention tosubstitute Y tex in place of Y tin , and Y rex in place of Y rin .After the fields everywhere are known, we use (20) toevaluate { ¯ Z se , ¯ Y sm , ¯ K em } . Due to the flipped locationsof the incident, transmitted and reflected fields, new fielddiscontinuity circulant matrices ∆ ˆ E,ex and ∆ ˆ H,ex needto be used in place of ∆ ˆ E,in and ∆ ˆ H,in , where ∆ ˆ E,ex = − ∆ ˆ E,in , ∆ ˆ H,ex = − ∆ ˆ H,in . (34)Last but not least, the conversion from O-BMS pa-rameters { ¯ Z se , ¯ Y sm , ¯ K em } to three-layer impedance im-plementation { ¯ Z i , ¯ Z m , ¯ Z o } can be done with (26) and(28) without any modifications. TABLE II. Specification for the passive lossless penetrableO-BMS cloak for concealing a dielectric cylinder f (GHz) N α [m] ρ s [m] φ s [rad] (cid:15) r (cid:15) r (cid:15) r t [mm]4.4 451 0.15 0.2 0 3 1 3 0.2 Now, let us apply this general procedure to designa penetrable O-BMS cloak which conceals a dielectriccylinder with permittivity (cid:15) r from an external linesource located at ( ρ, φ ) = ( ρ s , φ s ). We first write theincident field modal vector with (30b):ˆ E iz [ n ] = e i H (2) n ( k o ρ s ) J n ( k o α ) e jnφ s . (35)Again, e i can be set to some arbitrary constant. Sincethe goal is to produce zero external scattered field, thedesired reflected modal vector must be ˆ E rz = 0. As thisis a reflective metasurface, we need to solve (16) for therequired ˆ E tz . Assuming the configuration as described inTable II, we obtain the auxiliary transmitted field modalvector to be that depicted in Fig. 6(a). Using these val-ues, we obtain the three impedance layers depicted inFig. 6(b). Again, the real parts are omitted since theyare identically zero.To validate this design, we first simulate the scatteringfrom the dielectric cylinder without the O-BMS cloak.The total electric field distribution is shown in Fig. 7(a).As seen by the highly perturbed wavefronts, the cylin-der interferes with the source radiation to a significantdegree. Next, we add the O-BMS around the dielectriccylinder and resimulate with the same source. The re-sulting fields are shown in Fig. 7(b). Evidently, there isalmost no scattering from the cylinder observable in theexternal region. The object would appear essentially in-visible to any observer outside of the cylindrical volume.Next, we continue the development of more advancedpenetrable metasurface cloaks by designing a passive andlossless O-BMS which perfectly conceals a PEC cylinderfrom a known source. This can be challenging since thetarget itself does not permit any internal fields, meaningit is generally impossible to satisfy (16). However, if weinsert a small gap between the PEC cylinder and the O-BMS, as depicted in Fig. 8, then both sides of the cloakcan support non-zero fields. Practically, this small gapcan represent an air gap or a dielectric coating around thetarget. There is no inherent restriction on its thickness.We first identify the modal admittance matrices forthis problem. Since the incident and reflected fields ofall externally excited O-BMSs have the same constituentwave functions, we have Y ipec = Y iex , Y rpec = Y rex , (36)where the subscript “ pec ” indicates that these expres-sions are valid for O-BMSs surrounding a PEC cylinder.Due to the inclusion of the PEC, the transmitted fieldsare now described by a linear combination of Bessel ( J p )0and Neumann ( Y p ) functions which satisfy E z ( α (cid:48) , φ ) = 0,where α (cid:48) is the radius of the PEC cylinder. The explicitmodal expansions for the fields can be written as: E tz ( ρ, φ ) = ∞ (cid:88) p = −∞ e tp (cid:34) Y p ( k α (cid:48) ) d p ( k α (cid:48) , k α ) J p ( k ρ ) − J p ( k α (cid:48) ) d p ( k α (cid:48) , k α ) Y p ( k ρ ) (cid:35) e − jpφ ,H tz ( ρ, φ ) = ∞ (cid:88) p = −∞ e tp j √ (cid:15) r η o (cid:34) Y p ( k α (cid:48) ) d p ( k α (cid:48) , k α ) J (cid:48) p ( k ρ ) − J p ( k α (cid:48) ) d p ( k α (cid:48) , k α ) Y (cid:48) p ( k ρ ) (cid:35) e − jpφ ,d p ( r , r ) = Y p ( r ) J p ( r ) − Y p ( r ) J p ( r ) . (37)Recalling that the modal vectors represent the N-DFTof the sampled fields at ρ = α , we can writeˆ E tz = (cid:2) e tp − · · · e tp + (cid:3) T = Γ a ˆ E tz − Γ b ˆ E tz , Γ a = diag (cid:104) Y p − ( k α (cid:48) ) J p − ( k α ) d p − ( k α (cid:48) ,k α ) · · · Y p − ( k α (cid:48) ) J p − ( k α ) d p − ( k α (cid:48) ,k α ) (cid:105) T , Γ b = diag (cid:104) Y p − ( k α ) J p − ( k α (cid:48) ) d p − ( k α (cid:48) ,k α ) · · · Y p − ( k α ) J p − ( k α (cid:48) ) d p − ( k α (cid:48) ,k α ) (cid:105) T , (38) -200 -150 -100 -50 0 50 100 150 200-0.0200.02 ReIm z (a) M u lti - l a y e r I m p l e m e n t a ti on X i X m X o (b) FIG. 6. (a) The solved auxiliary transmitted fields for thepenetrable O-BMS cloak. (b) The reactance values for themulti-layer implementation of the cloak. and ˆ H tz = y tpec,a Γ a ˆ E tz − y tpec,b Γ b ˆ E tz (cid:44) Y tpec ˆ E tz , y tpec,a = j √ (cid:15) r η o · diag (cid:20) J (cid:48) p − ( k α ) J p − ( k α ) · · · J (cid:48) p + ( k α ) J p + ( k α ) (cid:21) T , y tpec,b = j √ (cid:15) r η o · diag (cid:20) Y (cid:48) p − ( k α ) Y p − ( k α ) · · · Y (cid:48) p + ( k α ) Y p + ( k α ) (cid:21) T , (39)where Y tpec is the new transmitted modal admittancematrix for this particular problem.Noting that the BSTCs given by (32) still apply, wecan reuse the expressions (A1) and (A2) to constructthe modal transmission matrix ˆT pec and modal reflec-tion matrix ˆR pec of the present configuration, simply bysubstituting Y { i,t,r } pec in place of Y { i,t,r } ex .As the first step of the synthesis procedure, we find therequired auxiliary ˆ E tz with (16), by replacing Y tin with (a)(b) FIG. 7. (cid:60){ E z } for the dielectric cylinder (a) without theO-BMS cloak; and (b) with the O-BMS cloak. E r || E i || + metasurface Z se ( ϕ ),Y se ( ϕ ),K em ( ϕ ) sources H r || H i || + E t || H t || Ɛ r1 Ɛ r2 ρ α (x)(y)(z) α ' ϕ PEC
FIG. 8. Schematic for the penetrable O-BMS cloak for a PECcylinder. Y tpec . Since it is still an externally excited O-BMS, wecan use (20) by replacing ∆ ˆ E,in and ∆ ˆ H,in with ∆ ˆ E,ex and ∆ ˆ E,ex , as is done previously for the dielectric cylin-der cloak. This gives the required O-BMS parameters { ¯ Z se , ¯ Y sm , ¯ K em } . In the last step, we convert these pa-rameters to the three-layer impedance implementation { ¯ Z i , ¯ Z m , ¯ Z o } using (26) and (28) without any modifica-tions.Following the aforementioned procedure, we designan O-BMS cloak with the specifications outlined in Ta-ble III. The solved modal vector for the required auxil-iary ˆ E tz is shown in Fig. 9(a). The corresponding pas-sive and lossless multi-layer implementation is shown inFig. 9(b). Inserting these reactance values into a COM-SOL model and simulating with the designated incidentfields, we obtain the electric field distribution shown inFig. 10(a). Note that for better legibility, the fields insidethe dielectric coating region have been exluded from thisplot. The unperturbed external wavefronts signify thesuccessful concealment of the PEC cylinder. For com-parison, the fields without the O-BMS cloak is shown inFig. 9(b), in which the PEC cylinder casts a significantshadow. As illustrated in Fig. 10(c), the dielectric coat-ing region around the PEC contains high-intensity stand-ing waves caused by the auxiliary fields. The amplitudesof the standing waves can be reduced by increasing thecoating thickness. Nevertheless, this evidently does notimpair the intended functionality of the O-BMS cloak. TABLE III. Specification for the passive lossless O-BMS cloakfor concealing a PEC cylinder f (GHz) N α [m] α (cid:48) [m] ρ s [m] φ s [rad] (cid:15) r (cid:15) r (cid:15) r t [mm]4.4 401 0.1025 0.1 0.2 0 2.2 1 3 0.2 -200 -150 -100 -50 0 50 100 150 200-0.500.5 ReIm z (a) M u lti - l a y e r I m p l e m e n t a ti on X i X m X o (b) FIG. 9. (a) The solved auxiliary transmitted field for thepenetrable O-BMS cloak around a PEC cylinder. (b) The re-actance values for the multi-layer implementation of the cloak.
C. High-gain Cavity-excited Antenna
In the last example, we design a cylindrical O-BMScavity which significantly enhances the directivity of anelectric line source that would otherwise produce unidi-rectional radiations by itself. Previously, low-profile de-signs with rectangular cavities have been presented [16,40]. Here, we extend this idea to a cylindrical topologywhich can be useful in direction-finding/navigation sys-tems or radio beacons.To demonstrate the use of O-BMS in constructinghigh-gain antennas, let us design a surface that colli-mates the cylindrical radiation into a directive beam to-wards some angle φ o . Since the incident fields are againthose produced by a line source placed at the origin, themodal vector is described by (29). The desired transmit-ted fields can be obtained by first identifying an envelopefor its magnitude around the circumference of the cavity.For simplicity, let us assume it to be an azimuthal boxfunction centered at φ = 0, with extent Φ: (cid:12)(cid:12) E tz ( ρ = α + , φ ) (cid:12)(cid:12) = e o < φ ≤ Φ2 , (40a)0 Φ2 < φ ≤ π − Φ2 ,(40b) e o π − Φ2 < φ ≤ π .(40c)Note that a more sophisticated envelope function than(40) might lead to higher maximum directivity. This is2 (a)(b) (c) FIG. 10. (a) (cid:60){ E z } for the PEC cylinder with the O-BMScloak. The fields inside the dielectric coating region is omit-ted here and plotted in (c) instead. (b) (cid:60){ E z } for the PECcylinder without the O-BMS cloak. reserved for future studies.Next, we need to properly phase the output field in or-der to produce the desired directional radiation. This canbe done by sampling the phase of a plane wave travellingtowards φ o : ∠ E tz ( ρ = α + , φ ) = − k o α cos (cid:2) φ − φ o (cid:3) . (41)Equations (40) and (41) can be used in conjunction toobtain the desired transmitted modal vector ˆ E tz . Sincethis is classified as an internally excited transmissivemetasurface, we solve (15) to find the required auxiliaryreflections. Assuming the specifications are as shown inTable IV, we calculate the reflected fields to be thosedepicted in Fig. 11(a). Correspondingly, the reactancevalues of the three-layer implementation of the O-BMSantenna are shown in Fig. 11(b). Interestingly, for theangular range π/ < φ < π/
2, the inner layer has near-zero reactance, meaning that essentially half of the O-BMS cavity behaves as a PEC backing. It is consistentwith intuition, since the amplitude envelope for the trans-mitted fields in that region is set to zero. The outer andmiddle layer reactance values for that region in fact haveno influence on the radiation pattern of the antenna. -200 -150 -100 -50 0 50 100 150 200-0.1-0.0500.050.1
ReIm z r (a) M u lti - l a y e r I m p l e m e n t a ti on X i X m X o (b) FIG. 11. (a) The solved auxiliary reflected fields for thecavity-excited O-BMS antenna. (b) The reactance values forthe multi-layer implementation of the O-BMS antenna.
Next, we numerically validate this design in COMSOL.In Fig. 12, we show the simulated total electric field dis-tribution of the antenna when it is fed by an electric linesource located at the origin. The omnidirectional sourcefield is collimated into a highly directive beam pointedat φ o .The directivity plot of the complete antenna systemis shown in Fig. 13. The maximum 2D directivity is13.4 dBi, a significant improvement over that of the om-nidirectional line source. The half power beam width isapproximately 7.5 ◦ . D. Additional Comments
Although we have only considered TM z -polarized fieldsin this paper, it is easy to see that the proposed frame-work can be readily extended to TE z configurations. Onewould simply need to modify the modal admittance ma-trices in accordance with the cylindrical wave functionsof the TE z fields. Furthermore, it is possible to model TABLE IV. Specification for the cavity-excited O-BMS an-tenna f (GHz) N α [m] φ o [ rad ] Φ [rad] (cid:15) r (cid:15) r (cid:15) r t [mm]4.4 451 0.15 0 π FIG. 12. (cid:60){ E z } for the cavity-excited O-BMS antenna. and design tensorial bianisotropic metasurfaces. For in-stance, while assessing the modal transmission matrices,the number of equations and the number of unknownsin the BSTCs would simply double. Instead of a singletransmission matrix and a single reflection matrix, wewould have two of each, corresponding to co-polarizedand cross-polarized responses. These topics are reservedfor future investigations. IV. CONCLUSION
We presented a mode-matching framework for theanalysis and synthesis of scalar cylindrical O-BMSs basedon the discrete Fourier transform. By decomposing theomega-bianisotropic surface parameters into Fourier har-monics and the electromagnetic fields into cylindricalmodes, we transformed the bianisotropic sheet transition -40 dBidBidBidBidBi
FIG. 13. 2D directivity plot of the cavity-excited O-BMSantenna. conditions into simple equations which can be solved toeither predict or engineer wave scattering from cylindricalmetasurfaces. We also proposed a systematic procedurefor designing passive and lossless cylindrical O-BMSs,which involves enforcement of local power conservationin a straightforward manner. Lastly, to bring these de-vices one step closer to practical realization, we presentmethods to realize the derived O-BMS surface param-eters using a topology consisting of multiple concentricazimuthally varying electric impedance sheets.We designed and investigated several passive and loss-less scalar O-BMS-based devices including illusion meta-surfaces, penetrable metasurface cloaks and high-gainmetasurface antennas. Each device is numerically ver-ified with finite element simulations, confirming the ef-fectiveness of the proposed method.
V. ACKNOWLEDGEMENT
The authors would like to thank Nicolas Faria for theinsightful discussions on cylindrical metasurfaces.4 [1] O. Quevedo-Teruel et al. , Journal of Optics , 073002(2019).[2] M. Chen, M. Kim, A. M. Wong, and G. V. Eleftheriades,Nanophotonics , 1207 (2018).[3] C. Pfeiffer and A. Grbic, Phys. Rev. Lett. , 197401(2013).[4] V. S. Asadchy, M. Albooyeh, S. N. Tcvetkova, A. D´ıaz-Rubio, Y. Ra’di, and S. A. Tretyakov, Phys. Rev. B ,075142 (2016).[5] G. Xu, S. V. Hum, and G. V. Eleftheriades, in (2020) pp. 1–5.[6] A. Epstein and G. V. Eleftheriades, Phys. Rev. Lett. ,256103 (2016).[7] M. Chen, A. Epstein, and G. V. Eleftheriades, IEEETrans. Antennas and Prop. , 4678 (2019).[8] G. Xu, S. V. Hum, and G. V. Eleftheriades, IEEE Trans.Antennas and Prop. , 780 (2018).[9] G. Xu, G. V. Eleftheriades, and S. V. Hum, IEEE Trans.Antennas and Prop. , 6033 (2018).[10] T. Niemi, A. O. Karilainen, and S. A. Tretyakov, IEEETrans. Antennas and Prop. , 3102 (2013).[11] M. Kim and G. V. Eleftheriades, Phys. Rev. Applied ,014009 (2020).[12] M. Selvanayagam and G. V. Eleftheriades, Opt. Express , 14409 (2013).[13] P. Ang and G. V. Eleftheriades, Sci. Rep. , 2021 (2020).[14] M. Dehmollaian and C. Caloz, in (Atlanta,GA, USA, 7-12 July, 2019) p. 1323.[15] D.-H. Kwon, Phys. Rev. B , 125137 (2018).[16] A. Epstein and G. V. Eleftheriades, IEEE Trans. Anten-nas and Prop. , 3880 (2016).[17] K. Achouri, M. A. Salem, and C. Caloz, IEEE Trans.Antennas and Prop. , 2977 (2015).[18] V. S. Asadchy, A. Daz-Rubio, and S. A. Tretyakov,Nanophotonics , 1069 (2018).[19] M. Safari, H. Kazemi, A. Abdolali, M. Albooyeh, andF. Capolino, Phys. Rev. B , 165418 (2019).[20] D.-H. Kwon, Phys. Rev. B , 235135 (2020).[21] H. Li, C. Ma, F. Shen, K. Xu, D. Ye, J. Huangfu, C. Li,L. Ran, and T. A. Denidni, IEEE Access , 185264(2019).[22] M. Dehmollaian, N. Chamanara, and C. Caloz, IEEETrans. Antennas and Prop. , 4059 (2019).[23] S. Sandeep and S. Y. Huang, IEEE J. Multiscale Multi-physics Comput. Techn. , 185 (2018).[24] D. B. Zvonimir Sipus, Zoran Eres, Radioengineering ,505 (2019).[25] Z. Sipus, M. Bosiljevac, and A. Grbic, IET Microw. An-tennas Propag. , 1041 (2018).[26] A. H. Dorrah and G. V. Eleftheriades, IEEE AntennasWirel. Propag. Lett. , 1788 (2018).[27] Y. Ra ' di and S. A. Tretyakov, New Journal of Physics , 053008 (2013).[28] J. P. S. Wong, A. Epstein, and G. V. Eleftheriades, IEEEAntennas Wirel. Propag. Lett. , 1293 (2016).[29] M. Chen, A. Epstein, and G. V. Eleftheriades, IEEETrans. Antennas and Prop. , 4678 (2019). [30] E. Abdo-S´anchez, M. Chen, A. Epstein, and G. V.Eleftheriades, IEEE Trans. Antennas and Prop. , 108(2019).[31] G. Xu, S. V. Hum, and G. V. Eleftheriades, in (Atlanta,GA, USA, 7-12 July, 2019) p. 1973.[32] J. Li, A. D´ıaz-Rubio, C. Shen, Z. Jia, S. Tretyakov, andS. Cummer, Phys. Rev. Applied , 024016 (2019).[33] S. W. Marcus and A. Epstein, Phys. Rev. B , 115144(2019).[34] G. Xu, S. V. Hum, and G. V. Eleftheriades, IEEE Trans.Antennas and Prop. , 6935 (2019).[35] J. Yi, P.-H. Tichit, S. N. Burokur, and A. de Lustrac,Journal of Applied Physics , 084903 (2015).[36] C. A. Balanis, Advanced Engineering Electromagnetics,2nd (John Wiley & Sons, Inc., New Jersey, 2012).[37] M. Selvanayagam and G. V. Eleftheriades, IEEE Anten-nas Wirel. Propag. Lett. , 1226 (2012).[38] D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B.Pendry, A. F. Starr, and D. R. Smith, Science , 977(2006).[39] Z. H. Jiang, P. E. Sieber, L. Kang, and D. H. Werner,Advanced Functional Materials , 4708 (2015).[40] A. Epstein, J. Wong, and G. V. Eleftheriades, Nat. Com-mun. , 10360 (2016). Appendix A: Analysis Equations for ExternallyExcited Cylindrical O-BMSs
The modal transmission matrix ˆ T ex and modal reflec-tion matrix ˆ R ex for the externally excited TM z -polarized O-BMS pictured in Fig. 5 are as follows: ˆT ex = ˆt − ex,b ˆt ex,a , ˆt ex,a = (cid:18) I + ˆK + ˆZY rex (cid:19) − (cid:18) I + ˆK + ˆZY iex (cid:19) − (cid:18) Y rex + ˆY − ˆKY rex (cid:19) − (cid:18) Y iex + ˆY − ˆKY iex (cid:19) , ˆt ex,b = (cid:18) Y rex + ˆY − ˆKY rex (cid:19) − (cid:18) Y tex − ˆY + ˆKY tex (cid:19) − (cid:18) I + ˆK + ˆZY rex (cid:19) − (cid:18) I − ˆK − ˆZY tex (cid:19) . (A1) ˆR ex = ˆr − ex,b ˆr ex,a , ˆr ex,a = (cid:18) I − ˆK − ˆZY tex (cid:19) − (cid:18) I + ˆK + ˆZY iex (cid:19) − (cid:18) Y tex − ˆY + ˆKY tex (cid:19) − (cid:18) Y iex + ˆY − ˆKY iex (cid:19) , ˆr ex,b = (cid:18) Y tex − ˆY + ˆKY tex (cid:19) − (cid:18) Y rex + ˆY − ˆKY rex (cid:19) − (cid:18) I − ˆK − ˆZY tex (cid:19) − (cid:18) I + ˆK + ˆZY rex (cid:19) . (A2)To obtain the the modal transmission matrix ˆT pec and modal reflection matrix ˆR pec , for an O-BMS surroundinga PEC cylinder, replace Y { i,t,r } ex with Y { i,t,r } pecpec