Illusion Mechanisms with Cylindrical Metasurfaces: A General Synthesis Approach
Mahdi Safari, Hamidreza Kazemi, Ali Abdolali, Mohammad Albooyeh, Filippo Capolino
IIllusion Mechanisms with Cylindrical Metasurfaces: A General Synthesis Approach
Mahdi Safari , Hamidreza Kazemi , Ali Abdolali , Mohammad Albooyeh , ∗ , and Filippo Capolino Department of Electrical and Computer Engineering, University of Toronto, Toronto, Canada Department of Electrical Engineering and Computer Science,University of California, Irvine, CA 92617, USA Department of Electrical Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran corresponding author: ∗ [email protected] explore the use of cylindrical metasurfaces in providing several illusion mechanisms includingscattering cancellation and creating fictitious line sources. We present the general synthesis approachthat leads to such phenomena by modeling the metasurface with effective polarizability tensors andby applying boundary conditions to connect the tangential components of the desired fields to therequired surface polarization current densities that generate such fields. We then use these requiredsurface polarizations to obtain the effective polarizabilities for the synthesis of the metasurface. Wedemonstrate the use of this general method for the synthesis of metasurfaces that lead to scatteringcancellation and illusion effects, and discuss practical scenarios by using loaded dipole antennas torealize the discretized polarization current densities. This study is the first fundamental step thatmay lead to interesting electromagnetic applications, like stealth technology, antenna synthesis,wireless power transfer, sensors, cylindrical absorbers, etc. I. INTRODUCTION
Metasurfaces are surface equivalents of bulk meta-materials, usually realized as dense planar arrays ofsubwavelength-sized resonant particles [1–4]. Since theemergence of this research topic, most studies havebeen carried out to analyze and synthesize infinitely ex-tended planar metasurfaces [5–31]. The synthesis ofsome topologies other than planar, namely cylindricaland spherical ones, has been studied in the past [32–40]but without a general systematic approach. Cylindricaltopologies are among the commonly used structures inengineering electromagnetics and optics. For instance,cloaking, radar cross section reduction, obtaining an ar-bitrary radiation pattern from cylindrical objects, andetc. are only a few problems which involve metasurfacessynthesis in cylindrical coordinates.For the analysis and synthesis of planar metasurfaces,several techniques have been reported as in Refs [9–14, 16–23, 25–28] for example. These works are based onmodeling the metasurfaces with impedance tensors [9–14, 16–20], with equivalent conductivities and reac-tances [21], or with effective surface susceptibility or po-larizability tensors [22, 23, 25–28]. Although the analysisand synthesis of planar metasurfaces are well-establishedand discussed in the literature, to the best of our knowl-edge, such a comprehensive analysis and/or synthesis isnot yet conducted for cylindrical metasurfaces. There-fore, we believe there is a need to develop an extensiveanalysis and synthesis method for assigning prescribedphysical properties to cylindrical metasurfaces.In particular here we aim at developing some illusionmechanisms using cylindrical metasurfaces. To thatend, we present a comprehensive analysis and synthesismethod for cylindrical metasurfaces analogous to whathas been previously performed for the case of planarmetasurfaces in Refs. [26, 27]. Note that the analysis ofplanar metasurfaces has been already extended to con- formal metaurfaces with large radial curvatures (at thewavelength scale) [41]. However, that method was basedon the analysis of planar structures with open bound-aries while a cylindrical metasurface can be generallyclosed in its azimuthal plane and it involves rather differ-ent interaction mechanisms. Our formulation covers themost general case of metasurfaces with dipolar (electricand magnetic) resonant responses including bianisotropiccases [42]. By applying our synthesis approach, we fur-ther conceptually synthesize several examples with inter-esting practical applications, e.g., scattering cancellationfrom conductive and dielectric cylinders and also gen-erating fictitious line sources, appearing away from theoriginal line source.The paper is structured as follows. We first present thetheory for the analysis of infinitely extended (along theaxial direction) cylindrical metasurfaces in Secs. II byproviding the relation between the desired fields at themetasurface boundary and the required surface polariza-tion densities. Next, we exhibit an expansion of the inci-dent (here, plane waves and line sources) and scatteredfields in cylindrical coordinates in Secs. III and IV. Afterthat, we conceptually synthesize two examples of scatter-ing cancellation and an illusion device in Secs. V A, V B,and V C. We further present a practical realization ap-proach by discretizing the continuous required polariza-tion currents in Sec. VI. Finally, we conclude the studyby discussing the influence of the presented method inpractical applications and new possibilities by cylindri-cal metasurfaces in engineering electromagnetic waves.
II. PROBLEM FORMULATION ANDBOUNDARY CONDITIONS
An electromagnetic metasurface is commonly under-stood as a composite layer composed of infinite numberof subwavelength-sized inclusions which are densely ar- a r X i v : . [ phy s i c s . a pp - ph ] M a y ranged over a surface and is capable of manipulating thewave front in a desired fashion (see e.g. Refs. [3]). Thisdefinition includes also the case of non-planar conformalsurfaces. Here, we study a class of metasurfaces whenthe inclusions are arranged to form a cylindrical surfacewhich is closed in the azimuthal plane and is infinitelyextended in the axial direction. Note that in the limitingcase when the cylinder curvature is infinitely large, thecurrent problem transitions to the case of planar metasur-faces which is widely studied (see e.g. Refs. [16, 21–28]).However, in sharp contrast with planar metasurfaces, ina cylindrical configuration waves bounce inside the cylin-drical metasurface and multiple reflections form the innerpart of the cylindrical boundaries must be considered inthe analysis.Let us consider a metasurface composed of resonant in-clusions that form a cylindrical surface with radius a [seeFig. 1]. The cylindrical surface is assumed to be infinitely z , − − , + + , e m I I a (a) , − − , + + PM (b) z , e m I I x x i H i E i H i E FIG. 1: (a) A cylindrical metasurface composed ofnonidentical inclusions which is infinitly extended in theaxial direction (here z axis) and is exposed to incidencesfrom inside (i.e., infinite line sources with electric andmagnetic current amplitudes I e and I m ) and outside(i.e., a plane wave). (b) The electromagnetic modelingof the problem in (a). The metasurface is replaced bytwo surface polarization densities P and M that providethe same electromagnetic response as the problem in(a).extended along the axial direction (here z -direction).Moreover, we assume that the inner and outer spacesof the cylindrical surface are filled with two differentisotropic materials with the permittivity (cid:15) ± and perme-ability µ ± , respectively (see Fig. 1, the “+” sign refers tothe material properties of the outside medium whereasthe “ − ” sign corresponds to the inside one). We illu-minate the proposed metasurface either from outside orfrom inside. Due to the external/internal illumination,the effective surface electric and magnetic polarizationdensities P (with the unit of Asm − ) and M (with theunit of Vsm − ) are, respectively, induced on the meta-surface boundary (see e.g. [22–27]). Next, similar toRefs. [22, 23, 25–27], we express the metasurface responsein terms of effective electric, magnetic, magnetoelectric,and electromagnetic polarizability tensors ¯¯ α ee , ¯¯ α mm ,¯¯ α em ,and ¯¯ α me , respectively. These surface polarizability ten- sors relate the polarization surface-density vectors P and M to the incident electric and magnetic fields E i and H i through the constitutive relations [42] P = ¯¯ α ee · E i + ¯¯ α em · H i , (1) M = ¯¯ α me · E i + ¯¯ α mm · H i . (2)The superscript “i” denotes “incident” fields, and the ef-fective surface polarizability tensors take into account ofthe effect of multiple reflections inside the surface, i.e., ofall the interactions among all the inclusions with them-selves. The main goal in our synthesis problem is to findthe required effective polarizability tensors ¯¯ α for a de-sired wave manipulation. We note that in some otherstudies, the impedance or susceptibility tensors are themain unknowns of the sysntehis problem [17–20, 25], ver-sus the effective polarizability tensors as in this currentinvestigation.Similarly to what was done for planar metasurfaces [25–27], we start from the boundary conditions for cylindri-cal metasurfaces which read (see Eqs. (4.3a)–(4.3d) inRef. [43] and also Appendix A for a detailed derivationand the constitutive relations) E + z − E − z = − (cid:15) ∂P ρ ∂z + jωM φ , (3) E + φ − E − φ = − a(cid:15) ∂P ρ ∂φ − jωM z , (4) H + z − H − z = − µ ∂M ρ ∂z − jωP φ , (5) H + φ − H − φ = − aµ ∂M ρ ∂φ + jωP z , (6)Here the + and − signs in the superscripts correspondto the total fields outside and inside of the metasurfaceboundary (i.e. ρ = a ), respectively, and the implicit timedependence e jωt is assumed, with ω being the angularfrequency. Moreover, E φ,z , H φ,z , P ρ,φ,z , and M ρ,φ,z arecorrespondingly the field and surface polarization densitycomponents in cylindrical coordinates. Furthermore, µ and (cid:15) are the free space permeability and permittivity,respectively. Indeed, Eqs. (3)–(6) relate the jump ofthe tangential components of the electric and magneticfields across the metasurface sheet to the required sur-face electric and magnetic polarization densities. Theright hand sides of Eqs. (3) and (4) represent the total equivalent tangential magnetic surface polarization den-sities contributing to the discontinuity of the tangentialelectric field, whereas the right hand sides of Eqs. (5)and (6) represent the total equivalent tangential electricsurface polarization densities contributing to the discon-tinuity of the tangential magnetic field [44–46].Next, for simplicity we only consider cases with vanish-ing normal polarization components P ρ , and M ρ in theboundary conditions (3)–(6) to simplify the analysis andsynthesis without loosing any freedom in the manipula-tion of a desired electromagnetic field [47]. Indeed, basedon the Huygens principle, or more precisely from theequivalence theorem [59], it can be shown that a desiredfield in a given volume can be fully engineered know-ing the tangential (with respect to the metasurface sur-face) surface polarization components on the metasurfacesheet [45, 46]. As a result, the boundary conditions forthe total field (3)–(6) simplify to E + z − E − z = + jωM φ , (7) E + φ − E − φ = − jωM z , (8) H + z − H − z = − jωP φ , (9) H + φ − H − φ = + jωP z . (10)In the next step, since cylindrical structures are inher-ently periodic with respect to the azimuthal coordinate φ ,we express the general form of the electromagnetic fields(inside and outside of the closed metasurface boundary)and the surface polarization densities on the metasurfaceas Fourier series E ( ρ, φ, z ) = n = ∞ (cid:88) n = −∞ E n ( ρ, z ) e jnφ , (11) H ( ρ, φ, z ) = n = ∞ (cid:88) n = −∞ H n ( ρ, z ) e jnφ , (12) P ( φ, z ) = n = ∞ (cid:88) n = −∞ P n ( z ) e jnφ , (13) M ( φ, z ) = n = ∞ (cid:88) n = −∞ M n ( z ) e jnφ , (14)respectively, where E n ( ρ, z ) and H n ( ρ, z ) are the coef-ficients of the electromagnetic fields in the Fourier se-ries whereas P n ( z ) and M n ( z ) are the coefficients corre-sponding to the electric and magnetic surface polariza-tion densities on the metasurface sheet.Before the next step and with the goal of practical imple-mentation of the problem, we add one more level of sim-plification to our problem. That is, we consider no varia-tion of the fields and polarization densities along the axialdirection (here, z -direction) as we provide two examplesin this study. It should be noted that the general synthe-sis of three-dimensional problems is a simple generaliza-tion of the z -invariant case studied here. Therefore, ap-plying (11)–(14) into the boundary conditions (7)–(10), assuming no z variation, and considering ∂∂φ → jn , theboundary conditions for each Fourier coefficient due tothe orthogonality of different n -indexed harmonics of thetotal field at ρ = a read E + z,n − E − z,n = + jωM φ,n , (15) E + φ,n − E − φ,n = − jωM z,n , (16) H + z,n − H − z,n = − jωP φ,n , (17) H + φ,n − H − φ,n = + jωP z,n . (18)Equations (15)–(18) are the final forms of the boundaryconditions with the assumed simplifications and will beexploited hereafter. III. INCIDENT FIELDS
We consider two different scenarios: the illuminationsource either outside or inside the metasurface boundary[see Fig. 1]. Based on our examples in Sec. V, we considerthe plane wave excitation for the case of outside illumi-nation whereas we discuss the infinite line source for thecase of illumination from inside. However, we note thatit is possible to consider any other excitation types basedon the same analysis and procedure that follows.
A. Plane wave illumination from outside
For generality, we consider the superposition of twoplane waves as external illumination: one with a z -polarized electric field i.e., E iTM = ˆ z E e − jβ + x , (19)and the other with a z -polarized magnetic field i.e., H iTE = ˆ z H e − jβ + x , (20)which are traveling along the + x direction (normal tothe metasurface axis z ) and impinging on the cylin-drical metasurface. While the former is called TM z –transverse magnetic– polarization, the latter is calledTE z –transverse electric [see Fig. 1]. Here, β + is the prop-agation constant in the medium outside of the meta-surface boundary, whereas E and H are the incidentelectric and magnetic field amplitudes, respectively. Anyplane wave which is propagating along the x directionis decomposable into these two modes. For example,let us consider a plane wave hitting the metasurface asshown in Fig. 1, with its electric field component mak-ing an angle θ with the z -axis. In this case, we have η + H = E tan θ , where η + = (cid:112) µ + /(cid:15) + is the intrinsicwave impedance of the corresponding medium.Next, to solve the problem of scattering from cylindri-cal structures, it is convenient to express the fields ofEqs. (19) and (20) in terms of the cylindrical wave func-tions. It can be shown that the TM z incident plane wavewith the electric field (19) can be expressed as [48] E iTM = ˆ z E n = ∞ (cid:88) n = −∞ j − n J n ( β + ρ ) e jnφ , (21)and that with TM z polarization has magnetic field (20)represented as H iTE = ˆ z H n = ∞ (cid:88) n = −∞ j − n J n ( β + ρ ) e jnφ . (22)Here J n is the n -th order Bessel function and ρ is theradial position. As a result, the total incident electricfield which is the superposition of both the TE and TMincident plane waves reads E i = 1 jω(cid:15) + ∇ × H iTE + E iTM = n = ∞ (cid:88) n = −∞ j − n (cid:20) η + H (cid:18) ˆ ρ nJ n ( β + ρ ) β + ρ + ˆ φ jJ (cid:48) n ( β + ρ ) (cid:19) +ˆ z E J n ( β + ρ )] e jnφ , (23)where ˆ ρ , ˆ φ , and ˆ z are the unit vectors in cylindrical coor-dinates and J (cid:48) n is the derivative of the n -th order Besselfunction with respect to the argument β + ρ . B. Line source illumination from inside
Here we consider the superposition of two infinitely ex-tended electric and magnetic line sources located at thecenter of the cylindrical metasurface as excitation source.The current amplitude of the electric and magnetic linesources are assumed to be I e and I m , respectively. Simi-larly to the previous case of plane wave illumination fromoutside, here the electric line source creates a TM z fieldwhile the magnetic one generates a TE z field. The elec-tric and magnetic fields of the two kinds of sources are,respectively, given by [48] E iTM = − ˆ z I e β − ω(cid:15) − H (2)0 ( β − ρ ) , (24)and H iTE = − ˆ z I m β − ωµ − H (2)0 ( β − ρ ) , (25) where H (2)0 is the 0-th order Hankel function of the secondkind. Therefore the total incident electric field is E i = 1 jω(cid:15) − ∇ × H iTE + E iTM = − β − (cid:104) ˆ φ jI m H (2)0 (cid:48) ( β − ρ ) + ˆ z η − I e H (2)0 ( β − ρ ) (cid:105) . (26)Note that the sources could be located also elsewhereinside the cylindrical metasurface boundary e.g. at ρ (cid:48) away from the origin, and in this case one would needto apply the substitution ρ = | ρ − ρ (cid:48) | , where ρ is theobservation point.In the next steps, we present the general forms of thetotal fields due to the induced sources on the metasurfaceboundary, i.e., due to the secondary sources. IV. FIELDS IN CYLINDRICAL COORDINATES
Generally, the fields inside ( E in ) and outside ( E out ) ofthe cylindrical metasurface boundary generated by theinduced currents (scattered fields) can be expressed as E in = n = ∞ (cid:88) n = −∞ (cid:20) ˆ ρ nb n J n ( β − ρ ) β − ρ + ˆ φ jb n J (cid:48) n ( β − ρ ) + ˆ z a n J n ( β − ρ ) (cid:105) e jnφ , (27)and E out = n = ∞ (cid:88) n = −∞ (cid:34) ˆ ρ nd n H (2) n ( β + ρ ) β + ρ + ˆ φ jd n H (2) n (cid:48) ( β + ρ ) + ˆ z c n H (2) n ( β + ρ ) (cid:105) e jnφ , (28)respectively. Note that the scattered field expressions inEqs. (27) and (28) do not include the fields produced byeither the inner line source or the incident plane wave,and only consider the fields generated by the inducedcurrents on the cylindrical metasurface, i.e., secondarysources. Here H (2) n and H (2) n (cid:48) are the n -th order Hankelfunction of the second kind and its derivative with re-spect to the argument, respectively. Moreover, the firsttwo components i.e., ρ and φ components in Eqs. (28)and (27) correspond to the TE z polarized fields whilethe z component corresponds to the TM z polarized fields.Furthermore, due to the nonsingular nature of the scat-tered fields inside and outside we have expanded the fieldsin terms of the suitable Bessel and Hankel functions, re-spectively. The magnetic fields ( H out and H in ) can becalculated using Maxwell’s equations. Note that a n and b n are coefficients of Bessel-Fourier series representingstanding waves inside the metasurface boundary for TM z and TE z waves, respectively, whereas c n and d n are thecoefficients which represent the TM z and TE z waves out-side the metasurface, correspondingly.At this stage we have all tools to synthesize a specific anddesired electromagnetic wave. To summarize, in order tosynthesize a desired wave profile:1. We first expand it in terms of the cylindrical har-monics as given by Eqs. (27) and (28) to obtain theunknown coefficients a n , b n , c n , and d n .2. Next, by applying the given incident fields (23)or (26) and by exploiting the boundary condi-tions (15)–(18), we find the required coefficients P n and M n that provide the required polarization den-sities P and M .3. Finally, when the polarization densities are found,the required effective polarizability tensors ¯¯ α ee ,¯¯ α mm ,¯¯ α em , and ¯¯ α me are retrieved from the consti-tutive relations (1) and (2).As a final remark, note that each polarization densityvector has two tangential components P ϕ , P z and M ϕ , M z providing a total of four functional equations. We recallthat for simplicity we consider metasurfaces that expressonly tangential polarization densities and exclude thosemetasurfaces with normal polarization densities P ρ and M ρ . However, each polarizability tensor in Eqs. (1) and(2) has four tangential polarizability components α ϕϕ , α ϕz , α zϕ , α zz . Therefore, the solution to the synthesis ofa metasurface is not unique since there are two polariza-tion vectors and four polarizability tensors, i.e., there arefour equations with 16 complex unknown polarizabilitycomponents. V. ILLUSTRATIVE EXAMPLES
We present three representative examples to demon-strate the application of the proposed method and toclarify the synthesis approach. Other manipulations ofelectromagnetic waves in cylindrical coordinates shall bepossible using a similar approach.In the first example we consider the application of a meta-surface as a scattering cancellation device [34, 49, 50]for a cylindrical perfect electric conductor (PEC) whenthe system is excited by a plane wave. In the secondexample we investigate an application similar to that ofthe first example with the only difference that the PECcylinder is replaced by a dielectric cylinder. The mainadvantage in using a metasurface for the realization ofa scattering cancellation device compared to previousmethods such as transformation optics (TO) [50, 51]and transmission line method [52] is that one does notneed to use bulky materials with exotic properties whichinevitably take a large space and/or contain high losses.Instead, a simple electromagnetically thin compositesurface which is easier to fabricate and take less space isexploited.In the last example we consider a metasurface thatsurrounds a line source that creates a fictitious linesource outside the cylindrical metasurface boundary.Following Ref. 51, we call this fictitious source the illusory source since an observer outside the metasurfaceboundary sees only a line source which is displacedwith respect to the original source. The realization ofan illusion device using the available approaches suchas TO requires the source to be located in a bulkycomplex medium which makes it mainly impractical.However, in our approach, the real source is simplylocated in free space and we surround it with a thinmetasurface. Therefore, the advantages of our approachcompared to the previous approaches for the design ofelectromagnetic devices for cylindrical structures is two-fold; the proposed approach is both practical and general.
A. Scattering cancellation from a PEC cylinder
In this first example we consider a PEC cylinder withradius a = 5 λ ( λ is the wavelength of the excitation field)located in free space which is extended infinitely along itsaxis (here the z -axis) and is illuminated by a z -polarizedplane wave propagating along the x -axis as given byEq. (19) (i.e., TM z incidence) with E = 1 Vm − and ω/ (2 π ) = 1 GHz. Our goal is to find a proper meta-surface which covers the PEC cylinder and suppressesthe scattered electromagnetic field by the PEC. In orderto cancel the scattered field, the total electric and mag-netic fields must satisfy E + = E i , H + = H i outside themetasurface, and inside the PEC, E − = 0 and H − = 0,respectively. Therefore, by applying the boundary con-ditions (15)–(18) one finds the required surface polariza-tion density coefficients P n and M n at the boundary.Hence, the desired surface electric and magnetic polar-ization densities P and M at the boundary are obtainedfrom Eqs. (13) and (14) as P = − ˆ z E ωη n =+ ∞ (cid:88) n = −∞ J (cid:48) n ( β a ) j − n e jnφ , (29)and M = ˆ φ E jω n =+ ∞ (cid:88) n = −∞ J n ( β a ) j − n e jnφ . (30)Here, η = (cid:112) µ /(cid:15) is the intrinsic impedance of freespace and β = ω √ µ (cid:15) is the propagation constant ofthe incident wave (i.e., β a = 10 π in this example). Notethat for the proposed TM z incidence we have θ = 0[see Fig. 1]. Indeed for this special case, the required φ -component of P and the z -component of M vanish, i.e., P z and M φ are the sufficient surface polarization densitycomponents at the boundary to achieve the scatteringcancellation. Similarly, for TE z incidence as in Eq. (20),the surface polarization density components P φ and M z are sufficient to suppress the scattered fields. Note thatbased on the reciprocity theorem the scattered fields ofany impressed surface electric polarization density on thesurface of a PEC is zero, therefore, the obtained surfaceelectric polarization density P in Eq. (29) represents theinduced current on the PEC surface due to the incidentplane wave. However, the magnetic polarization density M at the boundary is an impressed polarization den-sity provided by the metasurface to cancel the scatteredfields by the PEC. The surface polarization densities ofEqs. (29) and (30) that synthesize the scattering cancel-lation of a PEC cylinder are depicted in Fig. 2(a) and(b). ( A s m - ) z P ( V s m - ) M (degrees) ReIm (a) (b)(c) (degrees) (degrees) mm ( m ) FIG. 2: Induced surface (a) electric P z and (b)magnetic M ϕ polarization densities of Eqs. (29)and (30) required for scattering cancellation. (c) Thecorresponding effective magnetic surface polarizability α mm ϕϕ desired for scattering cancellation from a PECcylinder with radius a = 5 λ . The values in plots aresimilarly repeated for 180 ◦ ≤ φ ≤ ◦ The last step is to retrieve the required effective polariz-ability tensors from Eqs. (1) and (2) that realize the po-larization density M obtained in Eq. (30). However, inEq. (2) the number of equations are less than the numberof unknown polarizability components. Therefore, the so-lution to this synthesis problem is not unique since thereare multiple polarizability tensors’ sets which lead to thesame magnetic polarization density M , hence, to thedesired electromagnetic field. Here, as previously men-tioned we simplify the synthesis procedure by neglect-ing the polarizability components normal to the meta-surface plane (i.e., ρ -components). Moreover, for thesake of simplicity, we avoid considering realizations thatpossess bianisotropic polarizabilities ¯¯ α em and ¯¯ α me . Fur-thermore, we consider realizations with vanishing cross-component polarizability components, i.e., with zero off-diagonal terms in the polarizability tensor ¯¯ α mm . To sum- marize, (i) P is the induced surface polarization den-sity on the PEC surface and not a surface polarizationdensity impressed on the metasurface; and (ii) the sur-face polarization density M ϕ given by Eq. (30) to pro-vide scattering cancellation is obtained with a realiza-tion of the metasurface with effective surface polarizabil-ity component α mm ϕϕ shown in Fig. 2(c). As it is clear,the effective polarizability component α mm ϕϕ varies withthe angle, i.e., we require a spatially varying magneticcurrent to cancel the scattering from a PEC cylinder byusing a metasurface. For a TE z plane wave incidence,the solution of canceling the scattered field is dual tothe one just mentioned and the metasurface would haveonly surface polarizability component α mm zz . Figures 3(a)and (b) demonstrate the electric field distributions in theabsence and presence of the synthesized metasurface, re-spectively, by using full-wave simulations based on thefinite element method [58]. It is clear from this Figure / x metasurface PEC -60-6 60 -60-6 60 / x i l og / EE -1 (b)(a) / y PEC
FIG. 3: The normalized (to the incident field amplitude E i ) electric field distribution around the proposed PECcylinder illuminated by a plane wave in (a) the absenceand (b) the presence of the proposed scatteringcancellation cylindrical metasurface. The metasurface isshown with a black dashed line.that the by placing the synthesized metasurface aroundthe PEC cylinder, the cylinder’s scattered fields have per-fectly canceled, in other words the metasurface acts as acloaking device for a PEC cylinder. We next examine thecancellation of the scattering (i.e., cloaking) of a dielec-tric cylinder by covering it by a metasurface to furtherdemonstrate the capability of our proposed general ap-proach. B. Scattering cancellation from a dielectriccylinder
We consider now the problem of scattering cancella-tion from an infinitely long dielectric cylinder with rela-tive permittivity (cid:15) r = 10 and radius a = λ located in freespace and illuminated by an electromagnetic plane waveas in Eq. (19) as in the previous case. There is a majordifference between this case and the previous one with aPEC cylinder. In the previous example, the electromag-netic fields E − and H − inside the cylinder were requiredto be zero. Accordingly, based on the boundary con-ditions (15)–(18), providing scattering cancellation en-forces both electric and magnetic polarization densities P and M to be present at the boundary of the PEC [53–57]. In the present case with a dielectric cylinder, thereare infinite solutions that solve the scattering cancellationproblem depending on the precise form which the equiv-alent principle is applied. Among them, we consider theone which can be realized when we impose that only theelectric polarization density P is non vanishing. There-fore, assuming the magnetic polarization density M to bezero in Eqs. (15) and (16), the electromagnetic field in-side the dielectric cylinder can be easily obtained. Basedon the considered incident polarization, the electric fieldinside the metasurface has a z -component only. More-over, to have no scattered fields out of the metasurface(i.e., E out = 0), we apply the boundary condition (15)using Eq. (21) to obtain the only surviving coefficient ofEq. (27) as a n = E j − n J n ( β a ) / (cid:0) J n ( β √ (cid:15) r a ) (cid:1) . As aresult, the required surface polarization density P is ob-tained from the boundary condition (18) and using theMaxwell-Faraday equation ∇ × E ± = − jωµ H ± at themetasurface boundary (i.e., ρ = a ), leads to P = − ˆ z E ωη n =+ ∞ (cid:88) n = −∞ j − n e jnφ [ J (cid:48) n ( β a ) − √ (cid:15) r J n ( β a ) J n ( β √ (cid:15) r a ) J (cid:48) n ( β √ (cid:15) r a ) (cid:21) . (31)The longitudinal polarization component is plotted inFig. 4(a) as a function of the azimuthal angle ϕ . Next Re Im (a) (b) (degrees) (degrees) -4
0 90 180 ×10 -1
0 90 180 ×10 -12 - ( A s m ) z P ee / ( m ) zz FIG. 4: (a) The required polarization density for theexample of scattering cancellation metasurface for theproposed dielectric cylinder. (b) The correspondingeffective electric polarizability of the designedmetasurface. The dielectric radius and relativepermittivity are a = λ and (cid:15) r = 10, respectively.we derive the polarizabilities that provide the requiredpolarization density. For this simple case only one kindof polarizability is sufficient. Indeed, we assume thatthe bianisotropic polarizabilities ¯¯ α em and ¯¯ α me are absent as well as the cross-component polarizabilities. Then,by using the constitutive relation (1) the required effec-tive electric polarizability α ee zz is derived and depictedin Fig. 4(b). The electric field distributions (assumingplane wave incidence) in the absence and presence of thedesigned metasurface are, respectively, demonstrated inFig. 5(a) and (b) by using full-wave simulations. As it -3 / x metasurface r r -3 / y i l og / EE -0.8 -1 (b) (a) / x FIG. 5: The normalized (to the incident field amplitude E i ) electric field distribusions in (a) the absence and (b)the presence of the synthesized scattering calncellationcylindrical metasurface in the case of a dielectriccylinder. The metasurface is shown with a black dashedline.is clear from the figure, when the metasurface is presentthe scattered field is suppressed outside the metasurfaceboundary. That is, outside the metasurface we observeonly the propagating incident plane wave.In both the last examples, as seen in Figs. 2 and 4,the imaginary part of the required effective polarizabil-ity tensors is negative at some azimuthal positions andpositive for others. Therefore, one may deduce that thedesired metasurface consist of active elements. However,as discussed in [17, 18], the pointing vector of the wavecan switch the direction locally at the metasurface planeand when its direction is opposite to the incident wave,the polarizabilities can be negative. This implies the pos-sibility of realization of the synthesized structures usingonly passive particles. C. An illusory infinite line source
In this example the goal is to create an electromag-netic illusion that a source is moved somewhere else bysurrounding the actual source with a metasurface. Stud-ies on the design of optical devices based on TO to createoptical illusions have been already performed in past re-cent years (see e.g. Ref. 51). However, besides proposingbulky complex media, there was also the restriction thatboth the radiation source and its illusion were locatedinside the engineered medium. These requirements makethe realization of such devices difficult if not impracti-cal and also less useful. However, as we propose here,the introduction of metasurfaces make the realization ofsuch devices more convenient and practical. Here, thegoal is to show how a cylindrical metasurface around aline source is able to modify the source fields to make itlook like it is generated by a fictitious line source out-side the metasurface boundary, translated from the orig-inal source that is not visible anymore. In summary themetasurface cloaks the original line source and generatesan illusory line source at a different location.Let us consider an infinitely long electric line source I e in free space, along the z -axis of a cylindrical coordi-nate system. The electric field of such current source isdescribed by Eq. (24). The goal is to synthesize a cylin-drical metasurface around this line source that creates atotal field that looks like the one generated by a fictitiousline source translated at ( ρ (cid:48) > a, φ (cid:48) ), for an observer out-side the metasurface [see Fig. 7]. Moreover, the distancebetween the metasurface and the fictitious line sourcemust be subwavelength due to energy conservation con-siderations. Indeed, the total flux of the Poynting vec-tor through any closed surface around the illusory sourcemust be zero if such surface does not enclose any partof the metasurface. This would be impossible if the fic-titious line source is located at large distance from themetasurface.We assume the fictitious line source has the same am-plitude I e as the original line source but is located at ρ = ρ (cid:48) , φ = φ (cid:48) . By using the addition theorem (see e.g.,Refs. [48, 59]) the electric field created by such a source,which corresponds to the desired total field outside thecylindrical metasurface, reads E + = − ˆ z I e β ω(cid:15) ∞ (cid:88) −∞ J n ( β ρ (cid:48) ) H (2) n ( β ρ ) e jn ( φ − φ (cid:48) ) . (32)Comparing the above equation with Eq. (28) that repre-sents the total scattered electric field outside the meta-surface, one realizes that the only surviving coefficient inEq. (28) is c n = − I e β ω(cid:15) J n ( β ρ (cid:48) ) . (33)The total electric field inside the cylindrical metasur-face reads E − = − ˆ z I e β ω(cid:15) H (2)0 ( β ρ ) + ˆ z + ∞ (cid:88) −∞ a n J n ( β ρ ) e jn ( φ − φ (cid:48) ) , (34)which is the contribution of the field created by theoriginal line source [see Eq. (24)] plus the field createdfrom the cylindrical metasurface [see Eq. (27)]. Notethat the metasurface shall not create any cross compo-nent scattered field, i.e., the ρ - and φ -components of theelectric field simply vanish in Eq. (27). Now we assumethe fictitious source is located at ρ (cid:48) and φ (cid:48) = 0. More-over, we assume the observer is located at distances thatsatisfy ρ > ρ (cid:48) . There are many possible realizations of such metasurface, and here we consider only realizationswith non vanishing electric surface polarizations densityand M assumed to be zero. As a result of the above con-siderations, and by using the electric fields (32) and (34)in the boundary condition (15), the only surviving coef-ficients in Eq. (27) read a n = − I e β ω(cid:15) H (2)0 ( β a ) J ( β a ) [ J ( β ρ (cid:48) ) − n = 0 − I e β ω(cid:15) H (2) n ( β a ) J n ( β a ) J n ( β ρ (cid:48) ) n (cid:54) = 0 . (35)Finally, by applying the boundary condition (18) andby using the Maxwell-Faraday equation ∇ × E ± = − jωµ H ± at the metasurface boundary (i.e., ρ = a ),we obtain the required electric polarization density P forthe desired fields given by (32) and (34) as P = − ˆ z I e β ω (cid:32) H (2)0 (cid:48) ( β a ) − n =+ ∞ (cid:88) n = −∞ e jnφ (cid:104) J n ( β ρ (cid:48) ) H (2) n (cid:48) ( β a )+ 4 a n ωµ I e J (cid:48) n ( β a ) (cid:21)(cid:19) , (36)where a n is given by Eq. (35). This surface polarizationdensity is used in the constitutive relation (1) to retrievethe required effective surface polarizability tensors. Fig-ures 6(a) and (b), respectively, illustrate the required sur-face polarization density and the needed effective electricsurface polarizability for the proposed problem assuming a = λ and ρ (cid:48) = 4 λ/
3. Similarly to what was done in (degree) (degree) Re Im (a) (b)
0 90 180 ×10 -9
0 90 180 ×10 Sampling points - ( A s m ) z P ee / ( m ) zz FIG. 6: (a) The required polarization density for themetasurface example that creates an infinite illusoryline source. (b) The corresponding effective electricpolarizability of the designed metasurface.the previous example, we again have considered a meta-surface realization that does not involve cross-componentand bianisotropic polarizabilities. Figures 7(a) and (b),respectively, show the full-wave simulation results for theelectric field distributions in the absence and the pres-ence of the engineered metasurface. It is clear from thefield distributions that the original line source is cloaked metasurface line source original line source illusory line source / x / y -50-5 5 / x -1.5-2 -2.5-3 (b)(a) l og / A E E FIG. 7: (a) The normalized [to E A = I e β / (4 ω(cid:15) )]electric field distribution around an infinitely longelectric line source in vacuum. (b) The normalizedelectric field distribution of the original line sourcewhen located at the center of a metasurface that issynthesized to translate the perception of the positionof the original line source for an observer outside thecylindrical metasurface. The matasurface radius is a = λ , the illusory source is translated to the position ρ (cid:48) = 4 λ/ φ (cid:48) = 0, and the field appears arising fromsuch source.and an illusory line source is observed at 4 λ/ ρ (cid:48) , i.e., for ρ > ρ (cid:48) . Such a device mayfind practical applications in stealth technology, wirelesspower transfer systems [60], and also sensing technolo-gies. VI. PRACTICAL REALIZATION
In all the above examples we have performed concep-tual syntheses by considering ideal continuous currentdistributions, i.e., continuous surface polarization densi-ties. However, a metasurface is practically composed ofdiscrete elements that mimic such continuous currents.In this section, we present a practical approach to real-ize such metasurfaces with discrete elements, and applyit to the last example, i.e., the illusory line source. Westart by taking samples along the azimuthal angular di-rection that corresponds to the extrema values of the de-sired continuous current distribution given in Fig. 6(a).According to Fig. 6(a) and considering the whole 360 ◦ azimuth angle, we obtain 20 sampling points. Note thatonly half of the whole 360 ◦ angular span is shown inFig. 6(a) since the other half of the cylinder is symmetricto the one shown and hence it exhibits the same values.Next, each of these required currents can be realized bywires with periodic set of loads located at the azimuthalsampling points. For simulation purposes we restrict ouranalysis to a single periodic unit as shown in Fig. 8 andrepresenting all the others by reflections of two parallelperfectly conducting plates. It is noteworthy that thecoupling between the dipoles in the parallel plate regionare tuned by the loads to obtain the desired currents at x y z ~ PEC line source / x line sourcemetasurfaceillusion − − / y -10-2 -3 l og / A E E (b)(a) FIG. 8: (a) The schematic of an exemplary practicalrealization of the discretized metasurface that createsan illusory line source next to the original one that notvisible. (b) The normalized [to E A = I e β / (4 ω(cid:15) )]electric field distribution for the structure given in (a).the sampling points. In order to find the necessary loadimpedance values we first create a cylindrical array of 20identical short dipole antennas located at each samplingpoint. These points are given by the angular positionsof the extrema values given in Fig. 6(a). Each wire inthe plates scatters and couples to all the others, modify-ing their currents and voltages. Therefore by looking atthe voltage and current at each load we relate all thesequantities with a 20 ×
20 impedance matrix ¯¯ Z in whichthe diagonal elements denote the self impedance of eachantenna and the off-diagonal elements represent the mu-tual couplings between the antennas. The goal is to de-sign the loads at each wire to ensure that the currentdistribution is similar to the one shown in Fig. 6(a). Forconvenience we define the load impedance diagonal ma-trix ¯¯ Z L , which contains the load impedances Z L i at eachsampling points φ i , i = 1 , ...,
20 shown in Fig. 9. Thescattering problem is now posed as an algebraic prob-lem as V = (cid:104) ¯¯ Z + ¯¯ Z L (cid:105) I , where I is a 20 element vectorwith its i -th element I i = jωa P ( φ − φ i ) · ˆ z given by theeffective surface electric polarization density at the sam-pling point φ i [Fig. 6(a)] and V is a voltage (per meter)vector applied to dipoles. Considering identical supply-ing voltage 1 (V / m) for all the antennas, the requiredload impedances are given in Fig. 9. Here we have pro-vided the impedance values for half of the elements sincethe values for the other half is symmetrically distributed[see Fig. 9]. The electric field distribution of such dis-cretized problem is depicted in Fig. 8(b) by using finiteelement method (FEM) implemented in CST. As it isclear from this figure, although the field distribution isnot perfectly cylindrical due to the discretization, theactual line source inside the cylindrical metasurface iscloaked by the metasurface and a fictitious line sourceis inspected by an observer outside the metasurface atelectromagnetic large distances.0 Z L1 jZ L2 Z L3 Z L4 Z L5 Z L6 Z L7 Z L8 Z L9 Z L10 Z L11 Z L1 Z L2 Z L3 j (deg.) 0 19 41 Z L ( W ) -0.11+ j j j Z L7 Z L8 Z L9 j (deg.) 110 127 148 Z L (W ) - + j - + j - + j L4 Z L5 Z L6 j (deg.) 58 74 95 Z L ( W ) - + j + j - + j L10 Z L11 j (deg.) 160 180 Z L ( W ) - + j + j line source illusory line source FIG. 9: The relative positions and load impedancevalues of the dipoles in the proposed cylindricalmetasurface that creates an illusory line source. Here,we show 11 elements while we declare that elementsnumber 2 to 10 are repeated symmetrically in the lowerhalf circle.
VII. DISCUSSION AND CONCLUSIONS
We have investigated the synthesis problem of cylin-drical metasurfaces for general illusion mechanisms, andhave presented a general analysis and synthesis approachby modeling metasurfaces with effective polarizabilitytensors. We have presented three practical examples in-cluding scattering cancellation from PEC and dielectriccylinders as cloaking examples. Furthermore we have alsoshown how to generate an illusion, a translated fictitiousline source, from an original line source by covering sucha line source with a cylindrical metasurface.Besides the mentioned applications, the proposed ap-proach is general and can be applied to many otherproblems. That is, any manipulation of electromagneticwaves including but not restricted to focusing lenses, de-signing antennas with arbitrary radiation patterns, beamforming, etc. Our study opens up possibilities for com-plex electromagnetic wave manipulations using cylindri-cal topologies including stealth technology, antenna syn-thesis problems, wireless power transfer systems, ab-sorbers, etc.
ACKNOWLEDGMENT
The authors would like to thank DS SIMULIA for pro-viding CST Studio Suite that was instrumental in thisstudy, and Prof. C. Simovski and Prof. S. Tretyakov,from Aalto University, Finland, for fruitful discussions.
Appendix A: Boundary conditions
Let us consider the proposed cylindrical metasurfaceof radius ρ = a whose axis coincides with the z -axis (seeFig. 1). We start from the Maxwell’s curl equations withelectric and magnetic polarization densities P v = P δ ( ρ − a ) and M v = M δ ( ρ − a ) located at ρ = a , i.e., ∇ × E = − jω ( µ H + M v ) , (A1) ∇ × H = jω ( (cid:15) E + P v ) , (A2)In the Eqs. (A1) and (A2), µ and (cid:15) are the freespace magnetic permeability and electric permittivity,respectively, and we consider the constitutive relations D = (cid:15) E + P v and B = µ H + M v between the electricdisplacement D , magnetic induction B , electric field E ,magnetic field H , electric polarization density P v , andmagnetic polarization density M v . Next, since the prob-lem has cylindrical symmetry, it is convenient to repre-sent all vectors in cylindrical coordinates as E = E ρ ˆ ρ + E φ ˆ φ + E z ˆ z , (A3) H = H ρ ˆ ρ + H φ ˆ φ + H z ˆ z , (A4) P v = P v ρ ˆ ρ + P v φ ˆ φ + P v z ˆ z , (A5) M v = M v ρ ˆ ρ + M v φ ˆ φ + M v z ˆ z . (A6)By using (A3)–(A6) in the Maxwell curl equations (A1)and (A2) in cylindrical coordinate system we have1 ρ ∂E z ∂φ − ∂E φ ∂z = − jω ( µ H ρ + M v ρ ) , (A7) ∂E ρ ∂z − ∂E z ∂ρ = − jω ( µ H φ + M v φ ) , (A8)1 ρ ∂ ( ρE φ ) ∂ρ − ρ ∂E ρ ∂φ = − jω ( µ H z + M v z ) , (A9)1 ρ ∂H z ∂φ − ∂H φ ∂z = jω ( (cid:15) E ρ + P v ρ ) , (A10)1 ∂H ρ ∂z − ∂H z ∂ρ = jω ( (cid:15) E φ + P v φ ) , (A11)1 ρ ∂ ( ρH φ ) ∂ρ − ρ ∂H ρ ∂φ = jω ( (cid:15) E z + P v z ) . 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