Surface Susceptibility Synthesis of Metasurface Holograms for creating Electromagnetic Illusions
MMANUSCRIPT DRAFT 1
Surface Susceptibility Synthesis of MetasurfaceHolograms for creating Electromagnetic Illusions
Tom. J. Smy, Scott A. Stewart, and Shulabh Gupta
Abstract — A systematic approach is presented to exploit therich field transformation capabilities of Electromagnetic (EM)metasurfaces for creating a variety of illusions using the con-cept of metasurface holograms. A system level approach tometasurface hologram synthesis is presented here, in which thehologram is co-designed with the desired object to be projected.A structured approach for the classification of the creation ofEM illusions is proposed for better organization and tractabilityof the overall synthesis problem. The deliniation is in termsof the initial incident (reference) illumination of the objectto be recreated (front/back-lit), the position of illusion (poste-rior/anterior), and the illumination used to create the illusion(front/back). Therefore the classification is based on the specificrelationship between the reference object to be recreated, theobserver measuring the object, the orientation and placement ofthe reference and illumination field, and the desired placement ofthe metasurface hologram creating a virtual image. In the papera general design procedure to synthesize metasurface hologramsis presented based on Integral Equations (IE) and GeneralizedSheet Transition Conditions (GSTCs), where the metasurfacehologram is described as zero thickness sheet with tensorialsurface susceptibility densities. Several selected configurations arechosen to illustrate various aspects of the hologram creation in2D, along with a novel numerical technique to artificially reverse-propagate the scattered fields, required in the synthesis process.Finally, the impact of the metasurface size and the illuminationfield strength on the quality of the reconstructed scattered fieldsis also discussed.
Index Terms — Electromagnetic Metasurfaces, Holograms, Ef-fective Surface Susceptibilities, Boundary Element Method(BEM), Generalized Sheet Transition Conditions (GSTCs),Method of Moments (MoM), Field Scattering, ElectromagneticIllusions.
I. I
NTRODUCTION
Electromagnetic (EM) metasurfaces are 2D arrays of sub-wavelength resonating particles, where control of the spatialdistribution and EM properties of the individual particlesallows the scattered fields to be engineered with unprecedentedcontrol of both reflection and transmission, and with completepolarization control [1], [2]. Metasurfaces have been proposedfor achieving a variety of wave transformation functionalitiesincluding – cloaking, waveform generation, and lensing [3]–[7], This has resulted in many proposed thin sub-wavelengthsurface devices across the EM spectrum, with a large numberof structural designs and topologies using metals, dielectricsand other exotic materials [8], [9]. There has been also recent
This work was supported and funded by the Department of NationalDefence’s Innovation for Defence Excellence and Security (IDEaS) Program.T. J. Smy, S. A. Stewart, and S. Gupta, are with the Depart-ment of Electronics, Carleton University, Ottawa, Ontario, Canada. Email:[email protected] development in the application of metasurface concepts tosystem-level applications, such as next generation wirelessnetworks for 5G/6G, where such surfaces act as smart re-flectors in large Radio Frequency environments to achieveintelligent wave propagation [10]–[12]. In such cases, themetasurfaces act as part of an overall system and thereforemust be co-designed and integrated with the rest of the systemcomponents.One such application is an
Hologram . Holograms are well-known in optics where the spatial (and possibly temporal)information of an arbitrary object is encoded onto the surface(typically photographic plates) [13], [14]. This is a two stepprocess, where the information about scattering properties ofan object of interest is first recorded using a given referencebeam and modulated onto a given surface. Once the informa-tion is recorded, the encoded surface, when illuminated witha reconstructing beam, projects an illusion of the object. Withincreasing sophistication of encoding capability, more complexillusions can naturally be created. This process essentiallyexploits the wave-transformation capability of a surface –manipulating the reference beam electromagnetically. Meta-surfaces therefore naturally represent a powerful platform tocreate sophisticated EM illusions with complete control overthe scattered fields with respect to both complex amplitudeand polarization. Consequently, these surfaces can also beused to create metasurface holograms , due to their advancedinformation encoding capability [15], [16].In this work, a systematic description of metasurface holo-grams is presented and rigorous procedures are defined to de-sign and synthesize these metasurfaces for achieving a desiredEM illusion. To enable a general treatment of this problem,practical metasurfaces can conveniently be modeled as zerothickness sheets characterized using frequency dependent elec-tromagnetic surface susceptibility tensors ¯¯ χ ( ω ) [17]–[20]. TheEM fields around the metasurface then can be described usingGeneralized Sheet Transition Conditions (GSTCs) [21]. Thespatial distribution of surface susceptibilities of the metasur-face ¯¯ χ ( r ) dictates the scattered (and thus total) fields producedby the metasurface when illuminated by an incident field.Therefore, the key design objective in creating metasurfacebased illusions is to synthesize the spatially varying surfacesusceptibilities, ¯¯ χ ( r ) , to project the desired scattered fields ata given design frequency, ω to the observer location, identicalto that of a real object.Many metasurface synthesis and analysis problems usingsurface susceptibilities have been reported in the literature. Intypical frequency-domain metasurface synthesis procedures,arbitrary incident and desired scattered fields may be spec- a r X i v : . [ phy s i c s . c o m p - ph ] F e b ANUSCRIPT DRAFT 2 ified, which in conjunction with the GSTCs, can be usedto numerically solve, or optimize, for the required surfacesusceptibilities. For planar surfaces, surface susceptibilities canbe analytically computed, for instance see [18], [22]. Whenperforming a synthesis the fields must be specified at themetasurface location and not anywhere in space. While suchmethods are general in nature, their efficacy rests on a physi-cally meaningful specification of the EM fields. On the otherhand, metasurface analysis typically involves numerical com-putations where the GSTCs are coupled into bulk Maxwell’sequations using a variety of standard numerical techniquesbased on Finite-Difference and Finite Element methods [23]–[25], and Integral-Equation (IE) techniques [22], [26]–[30].Given that the field scattering from a metasurface hologrammay need to be evaluated for electrically large domains, IE-GSTC methods are computationally efficient choices and, aswill be shown later, are well suited for metasurface synthesisfor holographic applications.Given this context, a general methodology of synthesizingand designing metasurface holograms is presented in this workusing the IE-GSTC method from a system level perspective.The metasurface hologram design problem is systemicallydefined for various geometrical relationships between thedesired object illusion, a reference beam, the observer locationand the illuminating beam. The 2D IE-GSTC based numeri-cal platform is further developed to synthesize metasurfacesusceptibilities with an integrated approach, where the desiredfields, specified anywhere in space and not necessarily at themetasurface, are generated using a system level descriptionand fed-back into the metasurface design. A novel wave-propagation technique (a numerical inverse propagation) isfurther proposed for accurate determination of the desiredscattered fields in certain specific scenarios. This ensures aphysically meaningful field specification for arbitrary shapedmetasurface designs and avoid ill-posed metasurface synthesisproblems. Consequently, the proposed IE-GSTC based numer-ical platform computes spatially varying surface susceptibili-ties, ¯¯ χ ( r ) and is demonstrated to project complex EM illusionsin a variety of physical scenarios.The paper is structured as follows. Sec. II defines the overallproblem of metasurface holograms from a functional point-of-view in the context of observer and field illuminations,along with basic properties of the general EM metasurfaces.Sec. III presents the IE-GTSC approach: developing the fieldequations to be applied to various hologram situations, thediscretized form of the proposed IE-GSTC field solver, andthe numerical procedure for surface susceptibility synthesis.Sec. IV shows several results demonstrating the metasurfacehologram operation. Sec. V discusses some practical aspects ofmetasurface hologram designs such as the effect of finite-sizesurfaces and the field illuminations strengths on reconstructingthe desired scattered fields. Finally, conclusions are providedin Sec. VI. II. M ETASURFACE H OLOGRAMS
A. Principle of Creating Illusions
Consider an object of arbitrary shape and material com-position (dielectric or metal), subject to an incident refer- ence wave, ψ ( r , ω ) , as shown in Fig. 1(a). The referencewave interacts with the object producing propagating scatteredfields, ψ s ( r , ω ) . We now place an observer located to theleft of the object measuring the scattered fields ψ s ( r , ω ) within a certain field-of-view. Assume that the incident fieldsare propagating from left of the object, so that from theperspective of the observer, it appears as a Front-Lit Object .By measuring and analyzing the incoming left propagatingscattered fields, the observer perceives (or detects) the presenceand properties of the object, such as its geometrical shape andmaterial characteristics. If, on the other hand, the referencewave illuminates the object from the right side of the domain,the observer measures both the scattered and the referencefields (as both are left propagating), including the shadowproduced by an object. This is termed as a
Back-Lit Object .Let us now remove the object, and introduce an artificialsurface, i.e. a metasurface hologram, as shown in Fig. 1(b)at r = r m . The metasurface is excited with an illumination field, ψ i ( r , ω ) , which may or may not be the same as thereference field ψ ( r , ω ) . Can this surface be engineered torigorously recreate the scattered fields produced by the originalobject ψ s ( r , ω ) within the field-of-view of the observer?In such a case, the real physical object or a virtual object produced by an artificial metasurface are indistinguishablefrom the perspective of the observer. This creation of a falseperception by the metasurface hologram will be referred to asan Electromagnetic Illusion .There exist several possibilities in the placement of themetasurface hologram, which while irrelevant from the per-spective of the observer, is important from the practical designpoint of view and impacts how the metasurface may later besynthesized. If the metasurface is located between the objectand observer, we refer to it as a Posterior Illusion , otherwise,when the metasurface is behind the illusion, we refer to itas an
Anterior Illusion . In case of a posterior illusion, thevirtual object is formed behind the metasurface, while in caseof anterior, it is formed in front of the metasurface. Anothervariable of importance is how the metasurface is excited, i.e.the relative location and direction of the
Illuminating Field .The illuminating field, in general, could be entirely differentfrom the incident field that was used to determine the desiredscattered fields from the real object. If it strikes the surfacefrom the same side as the observer, it is referred to as
FrontIllumination , else, as
Back Illumination .With this description, the various illusion scenarios maybe termed as: Front/Back-Lit Posterior/Anterior Illusions withFront/Back Illumination. The front/back-lit configuration de-termines which fields the observer detects – either scatteredfields or total fields. The Posterior or Anterior position ofthe metasurface determines how the desired fields are to beconstructed to form the virtual image (with a finite physicalseparation from the surface) and are numerically propagatedto the metasurface location. This is important since the meta-surface synthesis requires fields infinitesimally close to thesurface region. It will be shown later that unlike the posterior The field ψ ( r ) is a compact notation for a fully vectorial electric andmagnetic field distribution. ANUSCRIPT DRAFT 3 I NCIDENT F IELD O BJECT S CATTERED F IELDS H ORIZON O BSERVER F RONT - LIT O BJECT O BJECT S CATTERED F IELDS H ORIZON B ACK - LIT O BJECT (a) (b) O BSERVER I LLUMINATING F IELD F RONT -L IT P OSTERIOR I LLUSIONWITH F RONT I LLUMINATION I LLUMINATING F IELD M ETASURFACE V IRTUAL O BJECT O BSERVER S HADOW V IRTUAL O BJECT O BSERVER F RONT -L IT A NTERIOR I LLUSIONWITH B ACK I LLUMINATION M ETASURFACE
LLUMINATING F IELD O BJECT ψ s ( r ,ω ) ψ s ( r ,ω ) ψ s ( r ,ω ) ψ s ( r ,ω ) + ψ ( r ,ω ) ψ s ( r ,ω ) ψ s ( r ,ω ) ψ ( r ,ω ) ¯¯ χ ( r m ,ω ) ψ i ( r ,ω ) ψ i ( r ,ω ) ψ i ( r ,ω )¯¯ χ ( r m ,ω ) Fig. 1. Illustration of producing electromagnetic illusions using metasurfaces. a) Front-lit vs back-lit objects where the incident field and the observer areon the same side of the object. b) Anterior and posterior illusions depending on the relative position of the metasurface and the original object under frontand back illumination cases; the two synthesized metasurfaces are different in these cases. configuration anterior illusions require an unusual inverse fieldpropagation as an initial step before metasurface synthesiscan be performed. Finally, the front vs back illuminationchoice will determine whether a fully reflective metasurface isrequired or a transmissive one. This has important implicationsfor practical realization of the synthesized metasurface, wherecompared to reflective ones, the transmission-type metasurfacerequires both reflection and transmission control.
B. Metasurface Descriptions
In order to generate arbitrarily complex and fully-vectorialscattered fields ψ s ( r ) from another equally arbitrary illumi-nating field ψ i ( r ) , the metasurface hologram located at r = r m (cid:54) = r must be capable of general EM wave transformationswith complete wave control. Also, it should be noted, thatwhile the prescribed fields may be arbitrary and complex, theyare completely physical, making the metasurface synthesisproblem a well-posed physically meaningful problem. Thisis due to the fact that the desired scattered fields are firstcomputed using physical objects under well-defined incident fields, and the metasurface is illuminated with a physical fielddescription as well.The general wave transformation capability of physicalEM metasurfaces can be described by expressing them asmathematical space discontinuities (zero thickness interfaces),owing to their their sub-wavelength thick nature and charac-terizing their EM wave interaction using electric and magneticpolarizabilities. To correctly model zero-thickness sheets, [31]introduced Generalized Sheet Transition Conditions (GSTCs)which were later applied to metasurface problems by [21],[32]. GSTCs (frequency-domain) relate the tangential EMfields around the metasurface to its tangential and normalsurface polarization response as, ˆn × ∆ E T = − jωµ M T − ˆn × ∇ || (cid:18) P n (cid:15) (cid:19) (1a) ˆn × ∆ H T = jω P T − ˆn × ∇ || M n , (1b)where ˆn is the normal vector to the surface, ∆ ψ T = ( ψ + − ψ − ) is the difference between the fields across the surface; { M T , P T } , { M n , P n } are the average tangential and normal ANUSCRIPT DRAFT 4 magnetic and electric surface polarizability densities of thesurface. The polarization densities can be seen as a responseof the metasurface to the average fields, related through thesurface susceptibility densities as [17], P T = (cid:15) ¯¯ χ ee E T,av + ¯¯ χ em √ µ(cid:15) H T,av (2a) M T = (cid:15) ¯¯ χ mm H T,av + ¯¯ χ me (cid:112) (cid:15)/µ E T,av , (2b)where ψ av = ( ψ + + ψ − ) / is the average field at themetasurface, and ¯¯ χ is a general × tensor accounting forvarious microscopic EM characteristics of the metasurface.Eqs. 1 and 2 together rigorously model the EM interaction withthe metasurface, while Eq. 2 captures its field transformationcapabilities via 36 variables inside the tensors.Therefore, the metasurface hologram synthesis problem ofFig. 1 can now be defined in terms of the determinationof these unknown susceptibility tensors needed to producethe desired field configuration across the surface. This fieldconfiguration is, of course, determined by the various incidentand scattered EM fields needed to create the illusion.III. M ODELLING A PPROACH
A. Propagation and Problem Formulation
To elucidate the metasurface hologram synthesis problem,consider, for simplicity, the 2D case of Fig. 2, where allfield interactions with the metasurface happen in the x − y plane and there is no field variation along z , and ∂/∂z = 0 .Consider a reference simulation first, where the first task isto compute the desired scattered fields from a given objectfor the specified reference wave, as shown in Fig. 2(a). Thisis achieved by using the Integral Equation (IE) form of theMaxwell’s equations and the creation of field propagators.The EM fields radiated into free-space from electric andmagnetic current sources, { J , K } , can be generally expressedas [33], [34]: E s ( r ) = − jωµ ( L J )( r , r (cid:48) ) − ( R K )( r , r (cid:48) ) (3a) H s ( r ) = − jω(cid:15) ( R K )( r , r (cid:48) ) + ( R J )( r , r (cid:48) ) , (3b)where the various field operators are given by: ( L J )( r , r (cid:48) ) = (cid:90) (cid:96) [1 + 1 k ∇∇· ][ G ( r , r (cid:48) ) J ( r (cid:48) )] d r (cid:48) ( R J )( r , r (cid:48) ) = (cid:90) (cid:96) ∇ × [ G ( r , r (cid:48) ) J ( r (cid:48) )] d r (cid:48) ( L K )( r , r (cid:48) ) = (cid:90) (cid:96) [1 + 1 k ∇∇· ][ G ( r , r (cid:48) ) K ( r (cid:48) )] d r (cid:48) ( R K )( r , r (cid:48) ) = (cid:90) (cid:96) ∇ × [ G ( r , r (cid:48) ) K ( r (cid:48) )] d r (cid:48) . The operators
L{·} and
R{·} produce a field response atlocation r due to a distribution of current sources located at r (cid:48) . This can be conveniently expressed in a matrix form as: (cid:20) E s ( r ) H s ( r ) (cid:21) = (cid:20) − jωµ L ( r , r (cid:48) ) − R ( r , r (cid:48) ) − jω(cid:15) L ( r , r (cid:48) ) + R ( r , r (cid:48) ) (cid:21) (cid:20) J ( r (cid:48) ) K ( r (cid:48) ) (cid:21) which can be further simplified as, F s ( r ) = P ( r , r (cid:48) ) C ( r (cid:48) ) (4) where, C ( r (cid:48) ) = (cid:2) J ( r (cid:48) ) K ( r (cid:48) ) (cid:3) (cid:62) F s ( r ) = (cid:2) E s ( r ) H s ( r ) (cid:3) (cid:62) are the current source and radiated field vectors respectivelyand {·} (cid:62) denotes a matrix transpose. The P ( r , r (cid:48) ) propagatormatrix thus relates the current sources at r (cid:48) to the EM fieldsproduced by them at an arbitrary location r . Finally, G ( r , r (cid:48) ) inside Eq. 3, represents the Green’s function, which for a 2Dcase is given by the 2 nd Hankel function, G ( r , r (cid:48) ) = H (2)0 ( r ) = J ( r , r (cid:48) ) − iY ( r , r (cid:48) ) , (5)where J and Y are the Bessel functions of the 1st and 2ndkind and the function represents outwardly propagating radialwaves.Consider again the illumination of an object producingscattered fields as in Fig. 2a. These scattered fields canalternatively be produced by a set of equivalent currents C ( r o ) on the surface of the object, which are determined by Eq. 4. Byspecifying the appropriate boundary conditions at the knownobject surface, these unknown currents can be easily solvedusing standard IE solvers for a specifed reference field [33],[34].To obtain the desired scattered fields to be later recon-structed by the metasurface hologram, let us introduce a Horizon Plane at r h , which is always located between theobject ( r o ) and the observation plane ( r ). Its utility willbecome clear shortly. The scattered fields at this horizon, F s ( r h ) can be calculated by forward-propagating the fieldsusing Eq. 4 giving, F ref. s ( r h ) = P ( r h , r o ) C ( r o ) . (6)Thus, F ref. s ( r h ) represents the desired reference fields that ourmetasurface hologram must reconstruct in the absence of theobject. This completes the first task.Next, let us remove the object and introduce a metasur-face described in terms of its surface susceptibilities ¯¯ χ ( r m ) ,which is excited with an arbitrary illumination field F i ( r ) , asillustrated in Fig. 2(b). Dropping the perpendicular terms andassuming scalar susceptibilities, for simplicity, Eq. 1 can becombined with Eq. 2 to give, ˆn × ∆ E T = − jωµ ( (cid:15)χ mm H T,av + χ me (cid:112) (cid:15)/µ E T,av ) (7a) ˆn × ∆ H T = jω ( (cid:15)χ ee E T,av + χ em √ µ(cid:15) H T,av ) , (7b)where ∆ ψ T = E ( r m + ) − E ( r m − ) , and ψ av = { E ( r m + ) + E ( r m − ) } / , are expressed in terms of total fields just beforeand after the metasurface. Since, the metasurface and thehorizon are not in general co-located (i.e. r m (cid:54) = r h ), thehorizon fields F ref. s ( r h ) must now be used to determine theaverage fields around the metasurface, F ( r m ) . The relationshipbetween them depends on whether a Posterior or an Anteriorillusion is desired. B. Anterior vs Posterior Illusions
Let us take the case of front-lit object and front illuminationfor explaining the procedures for relating the fields at the
ANUSCRIPT DRAFT 5 V IRTUAL O BJECT O BJECT (b)(a) (c) V IRTUAL O BJECT J , K Scattered F s ( r ) Incidence F ( r ) Illumination F i ( r ) Illumination F i ( r ) Horizon F s ( r h ) Horizon F s ( r h ) Observation F s ( r ) Observation F s ( r ) Observation F s ( r ) Metasurface ¯¯ χ ( r m ) Metasurface ¯¯ χ ( r m ) xy Forward-propagated F s ( r − m ) F s ( r + m ) = 0 Reverse-propagated F s ( r − m ) Reference
Simulation
Posterior
Illusion
Anterior
Illusion
Fig. 2. Methodology to achieve various field transformations using a metasurface to achieve an EM illusion. a) Determination of the required scatteredfields of the desired illusion object. b) Same scattered fields produced by inserting a metasurface in the absence the object, for the Posterior and Anterior (c)positions, for a given illumination field. horizon and the metasurface. For the posterior illusion, themetasurface is located in front of the object, as shown inFig. 2(b). In this case, a judicious choice is to place themetasurface directly at the horizon, so that r m = r h . Thissimplifies Eq. 7, so that the total fields around the surface aregiven by, F ( r m − ) = F i ( r m ) + F ref. s ( r m ) (8a) F ( r m + ) = 0 , (8b)where zero field is arbitrarily enforced on the right-half ofthe metasurface . All the fields in Eq. 7 are now known, andthe unknown surface susceptibilities can now be easily deter-mined to complete the hologram synthesis. With the knownsusceptibilities and the illuminating field, the metasurface willcorrectly reconstruct F ref. s ( r h ) everywhere beyond the horizon,so that the observer (also located beyond the horizon) willperceive an illusion of the original object located at its originalposition. This completes the hologram field specification forthis case and the synthesis of the surface can be performed asdescribed in Sec. III-D.The relationship between F ref. s ( r h ) and F ( r m ) is howevermore complex for the anterior illusion case. See Fig. 2(c),where now the metasurface is located behind the object, i.e. r m (cid:54) = r h . In this case, the horizon fields must be reverse prop-agated to the metasurface location. This reverse propagation isnot the usual physical forward-propagation of the fields, buta purely numerical exercise, where horizon wave-fronts aremathematically propagated back to the metasurface locationwhile maintaining the original flow of EM power towardsthe observer on the left of the horizon. This technique ofreverse-propagation requires a few intermediate steps, beforethe metasurface is ready for synthesis.To reverse-propagate the horizon fields, the equivalenceprinciple is invoked. The horizon is first represented as a In this paper, the observer is always assumed to be on the left of theobject. This has no impact on reconstructing the desired fields at horizon, butimpacts the required surface susceptibility distribution of the metasurface. surface with unknown equivalent currents C ( r h ) , so that F s ( r h + ) = P ( r h + , r h ) C ( r h ) = F ref. s ( r h ) (9a) F s ( r h − ) = P ( r h − , r h ) C ( r h ) = 0 , (9b)where P ( r h ± , r h ) are the self-propagator operators, and thefields on the left of the horizon are fixed to zero. In principle,the unknown equivalent horizon currents can be obtained byinverting Eq. 9(a) giving C ( r h ) = P ( r h + , r h ) − F ref. s ( r h ) .However, in practice, this is not a robust method and therelated matrices are not well-behaved due to an ill-definition ofthe problem. To improve the robustness of the computation weformulate the problem as a set of unknown surface currents, C ( r h ) , and scattered fields, F s ( r h ± ) . Then in addition toEq. 9, the equivalent currents, C ( r h ) are enforced to betangential to the horizon with zero normal components and thetangential field components across the horizon are set equal tothe reference fields at the horizon, F ref. s ( r h ) . These relationscan be expressed conveniently in the matrix form as: P ( r h + , r h ) − I ∅ P ( r h − , r h ) ∅ − I ˆn · ∅ ∅∅ ˆn × − ˆn × C ( r h ) F s ( r h + ) F s ( r h − ) = ∅∅∅ ˆn × F ref. s ( r h ) (10)where ˆn × {·} extracts the tangential components of the ar-gument vector. This solution of this system significantly im-proves the computation of C ( r h ) by ensuring that its normalcomponents are zero and the interface boundary conditions areapplied using the tangential fields only.The fields produced by the equivalent sources, C ( r h ) , mustnow be reverse-propagated to the metasurface location, r = r m . To achieve this operation, the field propagator of Eq. 4is modified so that the propagation is reversed in a temporalsense, F ref. s ( r m ) = P r ( r m , r h ) C ( r h ) , (11)where the operator P r ( r m , r h ) is formulated using an alter-native Green’s function involving a Henkel’s function of the ANUSCRIPT DRAFT 6
Specify Object& Reference Field, F ( r ) Desired Scattered Fieldsat Horizon, F ref. s ( r h ) Compute EquivalentCurrents, C ( r o ) Field Propagator,Eq. 6 PosteriorIllusion Metasurfaceat Horizon r m = r h AnteriorIllusion MetasurfaceBehind Object r m = r h Front-Lit F ref. s ( r h ) := F ref. s ( r h ) Back-Lit F ref. s ( r h ) := F ( r h ) + F ref. s ( r h ) Front Illumination F s ( r m − ) = F i ( r m ) + F ref. s ( r m ) F s ( r m + ) = 0 Back Illumination F s ( r m − ) = F ref. s ( r m ) F s ( r m + ) = F i ( r m ) Compute ¯¯ χ GSTCs,Eq. 7 Reverse-Propagation , F ref. s ( r m ), Eq. 10, 11 Reconstruct Fieldsat Observer F ( r ) BEM Solver, Eq. 19 [26, 30]Metasurface Hologram
Fig. 3. Complete flowchart summarizing the design flow to synthesize a metasurface hologram for the general case of Front/Back-Lit Posterior/AnteriorIllusion with Front/Back Illumination problem of Fig. 2, where the observer is assumed to be always located in the left-half region of the metasurface. first kind: G ( r , r (cid:48) ) = H (1)0 ( r , r (cid:48) ) = J ( r , r (cid:48) ) + jY ( r , r (cid:48) ) . (12)This function represents inwardly propagating radial wavesand with respect to fields generated by surface currents is,of course, a non-physical time reversed solution to Maxwell’sequations. However, in this case it is useful as a mathematicaltool. Once the desired fields are reverse-propagated to themetasurface, the total fields around the metasurface can beformed as F s ( r m − ) = F i ( r m ) + P r ( r m , r h ) C ( r h ) (13a) F s ( r m + ) = 0 , (13b)which when substituted inside the GSTCs of Eq. 7 can nowbe solved for the unknown surface susceptibilities of themetasurface hologram (see Sec III-D).The complete design flow chart for a general Front/Back-Lit Posterior/Anterior Illusions with Front/Back Illuminationproblem is illustrated in Fig. 3, summarizing various metasur-face synthesis scenarios. This process also shows the final con-firmation that the synthesized susceptibilities indeed producesdesired fields beyond the horizon at the observer location r in which a BEM-GSTC framework is used to determine theoutput fields for the specified illuminating field used in themetasurface design [26], [30]. The boundary element method(BEM) is a numerical computational method of solving linearpartial differential equations which have been re-formulated asdescritized integral equations (IE). C. Discretization - BEM Framework
The field equations for synthesizing metasurface hologramshave so far been expressed in terms of the continuous spacevariable r . However, for numerical computation, each of theseequations must be spatially discretized for incorporation intoa BEM solver framework.The surface position vector r S and any equivalent currents( J and K ) on an arbitrary shape (such as the bounding regionof a reference object) can be spatially discretized using m meshing elements, such that r → r S and J → J , i.e. r S = (cid:2) r s, r s, . . . r s,m (cid:3) and J = (cid:2) J J . . . J m (cid:3) , forinstance. Similarly, the field propagation operators L{·} and
R{·} , can also be discretized. For example, L = (cid:2) L ( r ) L ( r ) . . . L m ( r ) (cid:3) with L m i ( r ) J i = (cid:90) ∆ (cid:96) i [1 + 1 k ∇∇· ][ G ( r , r s,i ) J i ] d r s,i . The resulting fields anywhere in space, produced by thesediscretized current sources, are obtained using the field prop-agation operators of Eq. 3. If the region where the fields aremeasured is also discretized, r p = (cid:2) r p, r p, . . . r p,n (cid:3) ,we get F s ( r p , r S ) = (cid:20) − jωµ L ( r p , r S ) J ( r S ) − R ( r p , r S ) K ( r S ) − jω(cid:15) L ( r p , r S ) K ( r S ) + R ( r p , r S ) J ( r S ) (cid:21) which further can be written in compact form as F s ( r p , r S ) = P ( r p , r S ) C ( r S ) , with C = (cid:2) J K (cid:3) (cid:62) . (14)If the observation fields are on the surface itself (required whensolving for equivalent currents of the reference object or duringreverse-propagation), then r p = r S , then we have for the twosides of the surface (+/-), F s ( r p = r S + ) = F sS + = P ( r S , r S + ) C ( r S ) = P S + C ( r S ) F s ( r p = r S − ) = F sS − = P ( r S , r S − ) C ( r S ) = P S − C ( r S ) . Defining a surface field configuration S F = (cid:2) F sS + F sS − (cid:3) T and a surface propagator P S = (cid:2) P S + P S − (cid:3) we have, S F = P S C (15)The metasurface GSTCs of Eq. 7 can be explicitly writtenin terms of total E- and H-fields on the left ( {· } ) and right ANUSCRIPT DRAFT 7 ( {· } ) half of the surface as, (cid:20) ¯ N T × ∅ − ¯ N T × ∅∅ ¯ N T × ∅ − ¯ N T × (cid:21) E + H + E − H − = (cid:20) γ me ¯ N T γ mm ¯ N T γ me ¯ N T γ mm ¯ N T γ ee ¯ N T γ em ¯ N T γ ee ¯ N T γ em ¯ N T (cid:21) E + H + E − H − where the surface susceptibility terms are expressed usingauxiliary variables as, γ ee = jχ ee ω(cid:15) , γ me/em = ∓ jχ me/em ω √ µ(cid:15) , γ mm = − jχ mm ωµ . The matrix operator ¯ N T performs the operation of extractingthe two tangential fields at the surface (one in the x − y planeand the other with respect to ˆ z ) obtaining E T from E forevery surface element. The operator ¯ N T × extracts the totaltangent field and then rotates these two fields to implement the ˆn × {·} T operation on every element. The discretized GSTCmatrices can now be expressed compactly as ¯ D T F S F = ¯ G T F S F with (16a) ¯ G T F = (cid:20) γ me ¯ N T γ mm ¯ N T γ me ¯ N T γ mm ¯ N T γ ee ¯ N T γ em ¯ N T γ ee ¯ N T γ em ¯ N T (cid:21) (16b) ¯ D T F = (cid:20) ¯ N T × ∅ − ¯ N T × ∅∅ ¯ N T × ∅ − ¯ N T × (cid:21) . (16c) D. Susceptibility Synthesis
The metasurface surface susceptibility synthesis rests onsolving the GSTCs matrix equation of Eq. 16(a), once all thedesired scattered fields S sF are known and the illuminationfields, S iF , are specified. To extract the unknown surfacesusceptibilities, let us consider scalar susceptibilities, for sim-plicity. This extraction can be conveniently performed byrearranging ¯ G T F of Eq. 16(b), as ¯ G T F = (cid:20) χ me A me χ mm A mm χ me A me χ mm A mm χ ee A ee χ em A em χ ee A ee χ em A em (cid:21) with A ee = j N T ω(cid:15) , A em/me = ± j N T ω √ µ(cid:15) , A mm = − j N T ωµ Considering that the metasurface susceptibilities are dis-cretized over the surface as χ ( r m ) and thus become vectorsof localized susceptibilities ( X ), we can express the right handside of Eq. 16(a), as ¯ G T F S F = X me ◦ ( A me ( E + E )) + X mm ◦ ( A mm ( H + H )) X ee ◦ ( A ee ( E + E )) + X em ◦ ( A em ( H + H )) , where ◦ is the point-wise Hagamard product. Each of theterms, A me ( E + E ) = (cid:20) B me ,xy B me ,z (cid:21) = B me , is a column vector of one component of tangent fields ( xy and z ). If we wish to create a distributed χ vector we can form, ¯ G me X me = X me ◦ A me ( E + E ) where we define a diagonal matrix, ¯ G me = B me,1 . . . B me,2 . . .
00 0 . . .
00 0 0 B me,2N This form allows a very convenient expression of RHS ofEq. 16(a), where the susceptiblity matrix term is explicitlyextracted as ¯ G T F S F = (cid:20) ¯ G me ¯ G mm ∅ ∅∅ ∅ ¯ G ee ¯ G me (cid:21) X em X mm X ee X me = ¯ QX . (17)Finally using Eq. 16, we now have the explicit relationshipfor the spatially varying surface susceptibility matrix as X = ¯ Q − ¯ D T F S F , (18)which can be used directly for metasurface synthesis for agiven S F . E. Solution of the Final Configuration
To confirm the synthesis of the MS a full simulation ofthe surface with appropriate illumination is needed. For asingle surface subject to an illumination we use Eq. 15 todetermine the surface fields, force the normal components ofthe currents on the metasurface to be zero and enforce theinterface conditions prescribed by Eq. 16a. These equationscan be assembled into a final matrix equation, P S − IN DC ∅∅ ( ¯ D T F − ¯ G T F ) CS sF = ∅∅ − ( ¯ D T F − ¯ G T F ) S iF (19)where N DC takes the dot product of the currents for allelements and enforces N DC C = ∅ – setting the normalcomponent of the currents to zero. The surface fields S F havebeen split into two components: 1) the unknown scatteredfields S sF , and 2) the known applied field on the metasurface(reference or the illumination fields, for instance) S iF , so that S F = S sF + S iF . The solution of this equation provides C and S sF and using C , the fields at any point in the simulationdomain can be obtained using a propagation matrix.IV. R ESULTS AND D ISCUSSION
To illustrate the synthesis procedure for metasurface holo-grams and the ability to reconstruct desired fields, let usconsider a few examples. Given that a large number ofillusion configurations are possible, however, only few will be
ANUSCRIPT DRAFT 8 -0.1 -0.05 0 0.05 0.1-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 y ( m ) x (m) Horizon x = 0 Observation x = x Object θ in (a) Re { E ref. z,t ( x, y ) } - Object only -0.1 -0.05 0 0.05 0.1-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 y ( m ) x (m) Horizon x = 0 Observation x = x Object θ in (b) Re { E ref. z,s ( x, y ) } - Object Only -0.1 -0.05 0 0.05 0.1-6-4-202 10 -3 -0.1 -0.05 0 0.05 0.1-402 10 -3 y (m) y (m)Re { χ } Im { χ } χ mm ( y ) χ ee ( y ) (c) Synthesized Surface Susceptibilities -0.1 -0.05 0 0.05 0.1-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 y ( m ) x (m) Metasurface
Observation x = x VirtualObject θ in (d) Re { E ms. z,t ( x, y ) } - Metasurface only -0.1 -0.05 0 0.05 0.1-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 y ( m ) x (m) MetasurfaceHorizon x = 0 Observation x = x VirtualObject θ in (e) Re { E ms. z,s ( x, y ) } - Metasurface only -2-1012-0.1 -0.05 0 0.05 0.100.511.5 y (m) | E z ( , y ) | ( n o r m . ) | E z ( x , y ) | ( n o r m . ) | E ref. z,s ( x ,y ) || E ms. z,s ( x ,y ) | Re { E ref. z,s (0 ,y ) } , Re { E ms. z,s (0 − ,y ) } Im { E ref. z,s (0 ,y ) } , Im { E ms. z,s (0 − ,y ) } (f) Reconstructed Scattered Fields -0.1 -0.05 0 0.05 0.1-0.02-0.0100.010.02-0.1 -0.05 0 0.05 0.1-0.04-0.0200.02 y (m) y (m)Re { χ } Im { χ } χ mm ( y ) χ ee ( y ) (g) Susceptibilities - Curvilinear Metasurface -0.1 -0.05 0 0.05 0.1-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 y ( m ) x (m) MetasurfaceObservation x = x VirtualObject θ in (h) Re { E ms. z,s ( x, y ) } - Curvilinear Metasurface -0.1 -0.05 0 0.05 0.1 Y | E z ( , y ) | ( n o r m . ) | E z ( x ′ , y ) | ( n o r m . ) | E ref. z,s ( x ,y ) || E ms. z,s ( x ,y ) | Re { E ref. z,s (0 ,y ) } , Re { E ms. z,s (0 − ,y ) } Im { E ref. z,s (0 ,y ) } , Im { E ms. z,s (0 − ,y ) } (i) Reconstructed Scattered FieldsFig. 4. Front-lit posterior illusion with front illumination, where the illumination field is identical to the incident field, using a flat (a-f) and a curvilinearmetasurface (g-i), respectively. The simulation parameters are: f = 60 GHz ( λ = 5 mm), x = − λ , x h = x m = 0 . Rectangular PEC object λ × λ centered at x = 7 . λ , metasurface length (cid:96) ms = 120 λ . Both incident and illumination are oblique uniform plane-waves of | E z | = 1 . with 30 ◦ tilt measuredfrom the x − axis. Notation: ψ ref. z,s represents the z − component of the scattered fields ψ for the simulation with the reference object only; ψ ms. z,t represents the z − component of the total fields ψ for the illusion simulation with the metasurface and no object. ANUSCRIPT DRAFT 9 -0.1 -0.05 0 0.05 0.1-0.0200.020.04-0.1 -0.05 0 0.05 0.1-0.04-0.0200.02 y (m) y (m)Re { χ } Im { χ } χ mm ( y ) χ ee ( y ) (a) Synthesized Surface Susceptibilities -0.1 -0.05 0 0.05 0.1-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 y ( m ) x (m) MetasurfaceObservation x = x VirtualObjectHorizon x = 0 (b) Re { E ms. z,t ( x, y ) } - Metasurface only -2-1012-0.1 -0.05 0 0.05 0.100.511.5 y (m) | E z ( , y ) | ( n o r m . ) | E z ( x , y ) | ( n o r m . ) | E ref. z,s ( x ,y ) || E ms. z,s ( x ,y ) | Re { E ref. z,s (0 ,y ) } , Re { E ms. z,s (0 − ,y ) } Im { E ref. z,s (0 ,y ) } , Im { E ms. z,s (0 − ,y ) } (c) Reconstructed Scattered FieldsFig. 5. Front-lit posterior illusion with back illumination, where the illumination field is a plane-wave incidenting from the right-half region. Simulationparameters are: Rectangular PEC object λ × λ centered at x = 7 . λ ; the illumination is a uniform plane wave traveling along − x direction with | E z | = 1 . . Rest of the parameters are same as that mentioned in caption of Fig. 4. presented for better readability and tractability, and to highlightvarious aspects of the proposed synthesis procedures.The simulation setup follows the illustration of Fig. 2, inthe x − y plane, and the frequency of operation is chosen tobe GHz ( λ = 0 . m) where the observer is fixed at x = − λ . The surface meshing is set to λ/ based onproper convergence and the metasurface has a physical extentof λ unless otherwise noted. A. Front-lit Posterior Illusion with Front/Back-illumination
We first synthesize a metasurface hologram that creates anillusion of a rectangular PEC (Perfect Electric Conductor)object. The first step is to determine the scattered fields from areal PEC object when excited with a reference field - a uniformplane-wave - incidenting at an angle of θ in as shown in Fig. 4.The reference total and scattered fields are shown in Fig. 4(a)and (b), respectively. The ideal PEC object creates a shadowregion behind it, and produces strong scattered fields throughits flat faces and sharp corner diffraction. Next, the Horizonis defined at x = 0 , which is an arbitrary choice as long asit is lying to the left of the object. The scattered fields (bothamplitude and phase) are now recorded on the horizon, whichour metasurface hologram must recreate.We then remove the object and a metasurface hologram isintroduced. For posterior illusion, the metasurface is placeddirectly at the horizon at x = 0 . For this first test, theillumination fields are kept identical to the reference fields. Themetasurface susceptibilities are next computed using the Hori-zon fields computed earlier assuming a planar configuration,and it is found that it is sufficient to use only the electric andmagnetic surface susceptibilities, χ ee and χ mm , to recreate thefields. They are shown in Fig. 4(c) as a function of space. Theircomplex nature suggests that the metasurface must produce aspatially varying transformation of both amplitude and phase.Then using the standard BEM-GSTC technique of [26], theresponse of this synthesized metasurface is simulated when excited with an illumination field. The resulting total andscattered fields are shown in Fig. 4(d-e). The scattered fieldsof Fig. 4(e) produced by the metasurface hologram must becompared with that of the object only in Fig. 4(b). The meta-surface hologram successfully recreates the scattered fields onthe left of the horizon, while producing zero fields on theother side as imposed in the synthesis steps. The 1D recreatedscattered fields at the metasurface location are compared withthe desired fields at the Horizon and shown to be perfectlysuperimposed in Fig. 4(f), as expected. Furthermore, at theobserver located at x = x , there is an excellent matchbetween the recreated scattered fields and the original fieldsof the object. Therefore, from the perspective of the observer,the metasurface hologram is perceived as the original PECrectangular object – a virtual object image is formed behindthe metasurface.The synthesized surface susceptibilities strongly dependon the illumination fields and the shape of the metasurface.The topography of the metasurface is not limited to planarconfigurations. For instance, if a curvilinear metasurface ispreferred, the scattered fields from the object can also berecreated, however, a different set of surface characteristicsare needed. Fig. 4(g) shows one example of the synthesizedsusceptibilities with its corresponding scattered fields shown inFig. 4(h). As before, this curvilinear metasurface recreates thedesired scattered fields perfectly everywhere to the left of thehorizon as shown in Fig. 4(i). However, this time the surfacesusceptibilities of Fig. 4(g) are significantly different from theones of a planar metasurface of Fig. 4(c). In particular, thecurvilinear metasurface exhibits alternating loss-gain regions,compared to the purely passive susceptibilities of the planarmetasurface. This illustrates that the choice of metasurfaceshape is an important design parameter to be considered inpractical hologram designs.Now let us consider back-illumination of the metasurface,in this case, the metasurface hologram is excited from the ANUSCRIPT DRAFT 10 -0.1 -0.05 0 0.05 0.1-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 y ( m ) x (m) Metasurface x = x m Observation x = x Horizon x = 0 Object θ in (a) Re { E ref. z,t ( x, y ) } - Object only -0.1 -0.05 0 0.05 0.1-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 y ( m ) x (m) Metasurface x = x m Observation x = x Horizon x = 0 Object θ in (b) Re { E ref. z,s ( x, y ) } - Object only -0.1 -0.05 0 0.05 0.1-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 y ( m ) x (m) Metasurface x = x m Observation x = x Horizon x = 0 B a c k - p r o p a g a t e d (c) Re { E bck z,s ( x > , y ) } - Reverse-propagation -0.1 -0.05 0 0.05 0.1-505 10 -3 -0.1 -0.05 0 0.05 0.1-505 10 -3 y (m) y (m)Re { χ } Im { χ } χ mm ( y ) χ ee ( y ) (d) Synthesized Surface Susceptibilities -0.1 -0.05 0 0.05 0.1-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 y ( m ) x (m) Metasurface x = x m Observation x = x Horizon x = 0 VirtualObjectMetasurface (e) Re { E ms. z,s ( x, y ) } - Metasurface only -1-0.500.51-0.1 -0.05 0 0.05 0.1 Y | E z ( , y ) | ( n o r m . ) | E z ( x , y ) | ( n o r m . ) | E ref. z,s ( x ,y ) || E ms. z,s ( x ,y ) | Re { E bck. z,s ( x m ,y ) } , Re { E ms. z,s ( x m − ,y ) } Im { E bck. z,s ( x m ,y ) } , Im { E ms. z,s ( x m − ,y ) } (f) Reconstructed Scattered FieldsFig. 6. Front-lit anterior illusion with front illumination using the reverse-propagation technique, with a Gaussian incident field. The simulation parametersare: Parametrized PEC object centered at x = 7 . λ , x h = 0 , x m = 15 λ , incident field is a Gaussian-wave at ◦ from x − axis, with | E z | = 1 . and widthof λ , illumination is a uniform plane-wave propagating along + x -axis with | E z | = 1 . and (cid:96) ms = 120 λ . -0.1 -0.05 0 0.05 0.1-505 10 -3 -0.1 -0.05 0 0.05 0.1-10-505 10 -3 y (m) y (m)Re { χ } Im { χ } χ mm ( y ) χ ee ( y ) (a) Synthesized Surface Susceptibilities -0.1 -0.05 0 0.05 0.1-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 y ( m ) x (m) Metasurface x = x m Observation x = x Horizon x = 0 VirtualObjectMetasurface (b) Re { E ms. z,t ( x, y ) } - Metasurface only -1-0.500.51-0.1 -0.05 0 0.05 0.100.10.20.30.4 y (m) | E z ( , y ) | ( n o r m . ) | E z ( x , y ) | ( n o r m . ) | E ref. z,s ( x ,y ) || E ms. z,s ( x ,y ) | Re { E bck. z,s ( x m ,y ) } , Re { E ms. z,s ( x m − ,y ) } Im { E bck. z,s ( x m ,y ) } , Im { E ms. z,s ( x m − ,y ) } (c) Reconstructed Scattered FieldsFig. 7. Front-lit anterior illusion with back illumination using the reverse-propagation technique, where the illumination is a Gaussian wave from the righthalf-space and different from the incident field. The simulation parameters are: Parametrized PEC object centered at x = 7 . λ , incident field is a Gaussian-wave at ◦ from x − axis, with | E z | = 1 . and width of λ , illumination field is a Gaussian-wave at ◦ from x − axis, with | E z | = 1 . and width of λ . ANUSCRIPT DRAFT 11 right side of the surface - a uniform plane wave of normalincidence (this is, of course, a case of illumination fieldbeing different from the reference field). Fig. 5(a) shows thesynthesized susceptibilities with its corresponding scatteredfields shown in Fig. 5(b). The reference object’s scatteredfields are perfectly recreated left of the horizon, with a virtualobject image formed behind the metasurface. The 1D plotcomparison of Fig. 5(c) confirms the perfect field creationeverywhere including that at the observer. Compared to thefront illumination, where the metasurface was transformingthe illumination fields in reflection, the back illuminationrequires a transmission-type metasurface, with zero reflection,i.e. a perfectly matched metasurface. Such a metasurface istypically realized using a
Huygens’ source configuration, andis generally difficult to implement compared to reflection-typesurfaces.It should be noted that in all these examples, once themetasurface hologram is synthesized for a given illumination- front or back - the illusion is produced perfectly in the entireregion of space left of the horizon, irrespective of the locationand extent of the observer. However, we should be aware thatthe complexity of the metasurface design rests on the spatialvariation of the surface susceptibilities, which is controlled bythe choice of the reference object, metasurface shape and theillumination conditions, thereby producing virtually an infinitenumber of configuration possibilities. The hologram designerwill need to make judicious choices to achieve practicallyrealizable illusion configurations. For instance, in practicea purely passive metasurface may be desired compared toa loss-gain metasurface which requires an actively poweredmetasurface, as some of these above results demand.
B. Front-lit Anterior Illusion with Front/Back-illumination
We shall now deal with the case of an Anterior illusion,where the metasurface hologram is placed behind the objectat x m . Fig, 6(a-b) shows the total and scattered fields generatedby a parameterized curvi-linear reference PEC object, whichis excited by a Gaussian beam from the bottom left of theobject producing a front-lit object. Complex scattered fieldsare produced along with a shadow behind the object. Since,the object is in between the horizon and the metasurface, thehorizon fields at x h must be mathmatically reverse-propagatedto the metasurface location, before the susceptibility synthesiscan be performed.This reverse-propagation of the horizon fields is shown inFig. 6(c). In this step, the object is removed, and followingthe procedure of Sec. III-B, the fields are reverse propagatedto the desired metasurface location at x m towards the right.The fields are naturally zero on the left of the horizon. Thereverse propagation creates an artificial field distribution whichappears to focus the horizon fields before diverging againat the metasurface location. The resulting fields E bck. z,s (andother associated fields and components) are now the desiredscattered fields that the metasurface hologram must recreate inthe absence of the reference object under specified illuminationfields.Next, for specified illumination fields (front illuminatednormally incident uniform plane-wave) and the reverse- propagated horizon fields, the metasurface surface suscepti-bilities are synthesized, as shown in Fig. 6(d). The resultingscattered fields are further shown in Fig. 6(e) along with thefields comparisons at the metasurface and observer locationwith the reference fields. The scattered fields from the meta-surface at the metasurface are perfectly recreated, while anexcellent reconstruction of the fields at the observer locationis seen with slight ripples.A similar field reconstruction is observed when the metasur-face is re-synthesized for back illumination, where a uniformplane-wave is normally incident from the right, as shown inFig. 7. As expected, for a matched metasurface, the electricand magnetic surface susceptibilities are balanced emulatinga Huygens’ source configuration. Moreover, in both the frontand back illumination case, the synthesized susceptibilities arefound to be purely lossy. Finally, in both cases, the virtualimage of the object is formed in front of the metasurface. Itshould be noted that, while the horizon location is arbitrary, itis limited to the left-most extent of the object, beyond whichthe illusion is perfectly created. C. Back-lit Anterior Illusion with front-illumination
The last example of this section is a case of back-litanterior illusion with front illumination. In this case, thesame parametrized object of Fig. 6 and 7, is excited with areference Gaussian beam from the back of the object. Thecomputed total and scattered fields from the object are shownin Fig. 8(a) and (b). Compared to all the previous front-litcases, the observer this time measures not only the scatteredfields, but also the reference fields. Consequently, the totalfields are captured at the horizon. These total fields are thenreverse propagated to the desired metasurface location behindthe object (anterior illusion), as shown in Fig. 8(c).Next for the specified illumination fields - a front illumi-nating Gaussian beam - the metasurface is synthesized, andthe resulting susceptibilities are shown in Fig. 8(d). The finalfields scattered from the metasurface are shown in Fig. 8(e)along with the field comparisons at the metasurface and theobserver location in Fig. 8(f). In this case, the scattered fieldsfrom the metasurface must be compared to the total fieldsof the reference object, i.e. Fig. 8(a), in the region left ofthe horizon. As expected, a near-perfect reconstruction of thereference fields from the metasurface hologram is observedthroughout the observation region.These near-perfect results may be compared to the previouscase of a front-lit object (Fig. 6 and 7), where slight rippleswere observed in the recreated fields. While the object isthe same in both cases excited with a Gaussian beam, thetwo reference fields are quite different due to front vs backlit conditions. This further suggests that the nature of thereference fields to be recreated by the hologram impacts theaccuracy of the reconstructed fields.V. P
RACTICAL A SPECTS OF M ETASURFACE H OLOGRAMS
A. Effect of Finite-sized Metasurfaces
So far, the metasurface has been treated as an essentiallyideally large surface, separating the two half regions. However,
ANUSCRIPT DRAFT 12 -0.1 -0.05 0 0.05 0.1-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 y ( m ) x (m) Metasurface x = x m Observation x = x Horizon x = 0 Object θ in (a) Re { E ref. z,t ( x, y ) } - Object only -0.1 -0.05 0 0.05 0.1-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 y ( m ) x (m) Metasurface x = x m Observation x = x Horizon x = 0 Object θ in (b) Re { E ref. z,s ( x, y ) } - Object only -0.1 -0.05 0 0.05 0.1-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 y ( m ) x (m) Metasurface x = x m Observation x = x Horizon x = 0 B a c k - p r o p a g a t e d (c) Re { E bck z,s ( x < , y ) } - Reverse-propagation -0.1 -0.05 0 0.05 0.1-6-4-202 10 -3 -0.1 -0.05 0 0.05 0.1-6-4-202 10 -3 y (m) y (m) Re { χ } Im { χ } χ mm ( y ) χ ee ( y ) (d) Synthesized Surface Suceptibilities -0.1 -0.05 0 0.05 0.1-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 y ( m ) x (m) Metasurface x = x m Observation x = x Horizon x = 0 VirtualObjectMetasurface (e) Re { E ms. z,s ( x, y ) } - Metasurface Only -2-1012-0.1 -0.05 0 0.05 0.100.51 y (m) | E z ( , y ) | ( n o r m . ) | E z ( x , y ) | ( n o r m . ) | E ref. z,s ( x ,y ) || E ms. z,s ( x ,y ) | Re { E bck. z,s ( x m ,y ) } , Re { E ms. z,s ( x m − ,y ) } Im { E bck. z,s ( x m ,y ) } , Im { E ms. z,s ( x m − ,y ) } (f) Reconstructed Scattered FieldsFig. 8. Back-lit anterior illusion with front illumination using the reverse-propagation technique, where the illumination is a Gaussian-wave from the lefthalf-space. The simulation parameters are: Parametrized PEC object centered at x = 7 . λ , incident Gaussian-wave at − ◦ from x − axis, with | E z | = 1 . and width of λ propagating along − x , illuminating Gaussian-wave at ◦ from x − axis, with | E z | = 1 . and width of λ propagating along + x . practical metasurfaces are finite-sized in nature, and this finiteextent of the metasurface may have important consequences onthe quality of scattered field recreation using our holograms.The key distortions are the presence of secondary diffractionfrom the surface edges and spilling over of the illuminationfields around the metasurface, and reducing the field-of-viewof the illusion, in some cases.Let us take the previous example of back-lit anterior il-lusion with front illumination case, where the length of themetasurface hologram is changed. Fig. 9(a) shows the caseof a metasurface of length λ , modeled with a dielectric(non-scattering) boundary on each side. The 2D total fieldsclearly show the strong penetration of the back illuminationtowards the left of the horizon. Fig. 9(a) also shows the 2Dfield comparison at the metasurface and the observer comparedto the ideal case of a large metasurface. It is clear that therecreated fields significantly differ from the desired fields, withsome resemblance near the center of the observer only. Largefield oscillations are present on either side of the metasurfacedue to the illumination field spilling around the metasurface.This could also be attributed to any surface waves reflected off the edge discontinuities of the surface and forming standing-wave type fields on the surface. As the metasurface is madelarger to λ , the field reconstruction improves as shownin Fig. 9(b), which eventually becomes near-identical to thedesired ones for a length of λ , as shown in Fig. 9(c). Afurther improvement is shown in Fig. 8(f) in which a surfaceof length λ was used. B. Effect of Illumination Field Strength
It is clearly evident throughout all these examples, thatthe synthesized metasurface susceptibilities strongly dependon the nature and configuration of the illumination fields –how the metasurface hologram is illuminated. In general, thesynthesized surfaces exhibit varied regions of loss-gain charac-teristics to enable the recreation of the desired fields along witha large variation in the susceptibility values. Such surfaces arepractically challenging to implement due to design complexityrequiring active surfaces and purely passive surfaces maybedesired. One simple way to influence the passivity of thesurface, is to increase the illumination field strength. This
ANUSCRIPT DRAFT 13 -0.1 -0.05 0 0.05 0.1-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 y ( m ) x (m) Metasurface x = x m Observation x = x Horizon x = 0 VirtualObjectMetasurface -0.1 -0.05 0 0.05 0.1-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 y ( m ) x (m) Metasurface x = x m Observation x = x Horizon x = 0 VirtualObjectMetasurface -0.1 -0.05 0 0.05 0.1-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 y ( m ) x (m) Metasurface x = x m Observation x = x Horizon x = 0 VirtualObjectMetasurface -0.1 -0.05 0 0.05 0.1-2-1012-0.1 -0.05 0 0.05 0.100.511.52 y (m) y (m) | E z ( , y ) | ( n o r m . ) | E z ( x , y ) | ( n o r m . ) | E ref. z,s ( x ,y ) || E ms. z,s ( x ,y ) | Re { E bck. z,s ( x m ,y ) } , Re { E ms. z,s ( x m − ,y ) } Im { E bck. z,s ( x m ,y ) } , Im { E ms. z,s ( x m − ,y ) } (a) (cid:96) ms = 20 λ -0.1 -0.05 0 0.05 0.1-2-1012-0.1 -0.05 0 0.05 0.100.511.5 y (m) y (m) | E z ( , y ) | ( n o r m . ) | E z ( x , y ) | ( n o r m . ) | E ref. z,s ( x ,y ) || E ms. z,s ( x ,y ) | Re { E bck. z,s ( x m ,y ) } , Re { E ms. z,s ( x m − ,y ) } Im { E bck. z,s ( x m ,y ) } , Im { E ms. z,s ( x m − ,y ) } (b) (cid:96) ms = 40 λ -0.1 -0.05 0 0.05 0.1-2-1012-0.1 -0.05 0 0.05 0.100.51 y (m) y (m) | E z ( , y ) | ( n o r m . ) | E z ( x , y ) | ( n o r m . ) | E ref. z,s ( x ,y ) || E ms. z,s ( x ,y ) | Re { E bck. z,s ( x m ,y ) } , Re { E ms. z,s ( x m − ,y ) } Im { E bck. z,s ( x m ,y ) } , Im { E ms. z,s ( x m − ,y ) } (c) (cid:96) ms = 80 λ Fig. 9. Effect of metasurface length on the fidelity of the reconstructed scattered fields, using the back-lit anterior front illumination case of Fig. 8. Simulationparameters are: Parametrized PEC object centered at x = 7 . λ , incident Gaussian-wave at − ◦ from x − axis, with | E z | = 1 . and width of λ propagatingalong − x , illuminating plane-wave at ◦ from x − axis, with | E z | = 1 . propagating along + x . could be seen as a practical way to enforce a passive surfaceusing an external control.In all the previous examples, the illumination field strengthwas kept the same as the reference field. Fig. 10 shows theeffect of the illumination strength on the synthesized surfacesusceptibilities of the metasurface hologram using the back-lit anterior illusion with front illumination case of Fig. 8.Fig. 10(a) shows the nominal case of unity amplitude, wherethe synthesized susceptibilities show large peaking values of χ ee and χ mm in several local regions, along with several activeregions on the surface where the fields are amplified. Whenthe illumination strength is increased to 1.25, the susceptibilityvalues significantly improve to lower values, with the surfacebecoming near passive and also producing a better recon-struction of the fields, as shown in Fig. 10(b). With a furtherincrease in the illumination strength to 1.5, the susceptibilityvalues drop to small values, although still exhibiting somelocal regions of gain. However, this time the desired fields arenear-perfectly reconstructed, as seen in Fig. 10(c).This example illustrates the sensitivity of the metasurfacehologram design to illumination field strengths and demon-strates that it is an important parameter to take into account.It should be noted that the increased illumination strength isequivalent to using a lower reference field to compute the desired scattered fields. This disparity in the fields will simplymanifest as an illusion of lower brightness under normalillumination conditions.VI. C ONCLUSIONS
A systematic and structured approach has been presentedto exploit the rich field transformation capabilities of EMmetasurfaces for creating a variety of EM illusions usingthe concept of metasurface holograms. A holistic approachof metasurface hologram synthesis has been undertaken herefrom a system point of view, where the desired fields detectedby the observer are first recreated and then fed back intothe overall metasurface synthesis problem. Considering thecomplexity of the problem and large number of configura-tion possibilities, the approach of classifying them in termsof front/back-lit posterior/anterior illusions using front/backillumination has been adopted for better organization andtractability of the overall synthesis problem. These classifica-tions are based on specific relationships between the referenceobject to be recreated, the observer measuring the object, theorientation and placement of the reference and illuminationfield, and the desired placement of the metasurface hologramcreating a virtual image. Consequently, a general design proce-dure to synthesize metasurface holograms has been proposed
ANUSCRIPT DRAFT 14 -0.1 -0.05 0 0.05 0.1-0.100.10.20.3-0.1 -0.05 0 0.05 0.1-0.04-0.0200.02 y (m) y (m) Re { χ } Im { χ } χ mm ( y ) χ ee ( y ) -0.1 -0.05 0 0.05 0.1-10-505 10 -3 -0.1 -0.05 0 0.05 0.1-0.0100.010.020.03 y (m) y (m) Re { χ } Im { χ } χ mm ( y ) χ ee ( y ) -0.1 -0.05 0 0.05 0.1-0.02-0.0100.01-0.1 -0.05 0 0.05 0.1-0.02-0.0100.010.02 y (m) y (m) Re { χ } Im { χ } χ mm ( y ) χ ee ( y ) -0.1 -0.05 0 0.05 0.1-10-505-0.1 -0.05 0 0.05 0.100.511.5 y (m) y (m) | E z ( , y ) | ( n o r m . ) | E z ( x , y ) | ( n o r m . ) | E ref. z,s ( x ,y ) || E ms. z,s ( x ,y ) | Re { E bck. z,s ( x m ,y ) } , Re { E ms. z,s ( x m − ,y ) } Im { E bck. z,s ( x m ,y ) } , Im { E ms. z,s ( x m − ,y ) } (a) | E ill. z | = 1 . -0.1 -0.05 0 0.05 0.1-2-1012-0.1 -0.05 0 0.05 0.100.51 y (m) y (m) | E z ( , y ) | ( n o r m . ) | E z ( x , y ) | ( n o r m . ) | E ref. z,s ( x ,y ) || E ms. z,s ( x ,y ) | Re { E bck. z,s ( x m ,y ) } , Re { E ms. z,s ( x m − ,y ) } Im { E bck. z,s ( x m ,y ) } , Im { E ms. z,s ( x m − ,y ) } (b) | E ill. z | = 1 . -0.1 -0.05 0 0.05 0.1-2-1012-0.1 -0.05 0 0.05 0.100.51 y (m) y (m) | E z ( , y ) | ( n o r m . ) | E z ( x , y ) | ( n o r m . ) | E ref. z,s ( x ,y ) || E ms. z,s ( x ,y ) | Re { E bck. z,s ( x m ,y ) } , Re { E ms. z,s ( x m − ,y ) } Im { E bck. z,s ( x m ,y ) } , Im { E ms. z,s ( x m − ,y ) } (c) | E ill. z | = 1 . Fig. 10. Effect of illumination strength on the synthesized surface susceptibilities, demonstrated using the case of a back-lit anterior illusion with frontillumination case. Simulation parameters are: Parametrized PEC object centered at x = 7 . λ , incident Gaussian-wave at − ◦ from x − axis, and width of λ propagating along − x , illuminating Gaussian-wave at ◦ from x − axis with width of λ propagating along + x . based on the IE-GSTC method and its EM illusion creationcapabilities has been demonstrated using several examples.The proposed synthesis framework results in the determinationof the spatially varying surface susceptibilities describing theEM properties of the metasurface. The synthesis techniquecombines the integral form of the Maxwell’s equations, and thecorresponding field propagators with the GSTCs describing thefield interaction with zero thickness metasurfaces. It has nextbeen implemented using the BEM approach using a rigorousand compact matrix formulation which is capable of synthe-sizing planar as well as curvilinear metasurface hologramsand for arbitrary specifications of the reference object. Finally,the impact of the metasurface size and the illumination fieldstrength on the quality of the reconstructed scattered fieldshas been discussed with respect to the feasibility of practicalmetasurface holograms.The number of possible configurations and situations forcreating EM illusions using metasurface holograms are vir-tually unlimited. While the handful of examples presentedhere have been strategically chosen to highlight and illustrateseveral of the salient features of the hologram synthesis, thepresented framework represents a flexible test-bed to explorea wider variety of illusion scenarios before undertaking prac-tical demonstrations. It further highlights the unprecedentedcapabilities of EM metasurfaces in achieving very complexwave transformations. While only scalar surface suscepti-bilities have been employed in this work, the BEM-GSTC framework is easily extendable to fully tensorial descriptionsof the surface as was done in [26]. Furthermore, the usageof GSTCs combined with surface susceptibility descriptionof zero-thickness surfaces is also a very efficient tool formodeling complex objects (and not limited to PEC objectsas used here). This makes the proposed numerical frameworkcomplete for handling arbitrarily complex problems using acommon IE-GSTC infrastructure. It should also be finallyremarked that, while a vast number of techniques have beenproposed for designing optical holograms, the proposed tech-nique represents a rigorous full-vectorial field-based approachfor metasurface synthesis, as opposed to methods typicallybased on paraxial approximations at optical frequencies [13].Furthermore, compared to the existing works on metasurfacesynthesis, the system level approach undertaken here where thedesired fields are first computed from physical considerations,results in a metasurface synthesis problem that is likely tobe well-posed, as opposed to the possibility of an otherwisearbitrary and physically disconnected problem. Therefore, thiswork provides a set of important tools to metasurface hologramdesigners for creating a myriad of complex EM illusionsthroughout the EM spectrum.R EFERENCES[1] C. Holloway, E. F. Kuester, J. Gordon, J. O’Hara, J. Booth, and D. Smith,“An overview of the theory and applications of metasurfaces: Thetwo-dimensional equivalents of metamaterials,”
IEEE Antennas Propag.Mag. , vol. 54, no. 2, pp. 10–35, April 2012.
ANUSCRIPT DRAFT 15 [2] H.-T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physicsand applications.”
Reports on progress in physics. Physical Society ,vol. 79, no. 7, p. 076401, 2016.[3] G. Zheng, H. Muhlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang,“Metasurface holograms reaching 80% efficiency,”
Nat. Nanotech. ,no. 43, pp. 308–312, Feb. 2015.[4] Y. Yang, H. Wang, Z. X. F. Yu, and H. Chen, “A metasurface carpetcloak for electromagnetic, acoustic and water waves,”
Sci. Rep. , no. 6,pp. 1–6, Jan. 2016.[5] S. A. Tretyakov, “Metasurfaces for general transformations of electro-magnetic fields,”
Philosophical Transactions of the Royal Society ofLondon A: Mathematical, Physical and Engineering Sciences , vol. 373,no. 2049, 2015.[6] Q. Wang, E. T. F. Rogers, B. Gholipour, C.-M. Wang, G. Yuan, J. Teng,and N. I. Zheludev, “Optically reconfigurable metasurfaces and photonicdevices based on phase change materials,”
Nat. Phot. , vol. 5, no. 10, pp.60–65, Dec 2015.[7] A. Shaltout, A. Kildishev, and V. Shalaev, “Time-varying metasurfacesand lorentz non-reciprocity,”
Opt. Mater. Express , vol. 5, no. 11, pp.2459–2467, Nov 2015.[8] M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener,T. Pertsch, and Y. S. Kivshar, “High-efficiency light-wave control withall-dielectric optical huygens’ metasurfaces,”
Adv. Opt. Mat. , pp. 813–820, May 2014.[9] N. Yu and F. Capasso, “Flat optics with designer metasurfaces,”
Nat.Mat. , vol. 13, April 2014.[10] W. Qingqing and Z. Rui, “Towards Smart and Reconfigurable Envi-ronment: Intelligent Reflecting Surface Aided Wireless Network,” arXive-prints , p. arXiv:1905.00152, Apr 2019.[11] E. Basar, “Large Intelligent Surface-Based Index Modulation: A NewBeyond MIMO Paradigm for 6G,” arXiv e-prints , p. arXiv:1904.06704,Apr 2019.[12] M. Di Renzo, M. Debbah, D.-T. Phan-Huy, A. Zappone, M.-S.Alouini, C. Yuen, V. Sciancalepore, G. C. Alexandropoulos, J. Hoydis,H. Gacanin, J. de Rosny, A. Bounceu, G. Lerosey, and M. Fink, “SmartRadio Environments Empowered by AI Reconfigurable Meta-Surfaces:An Idea Whose Time Has Come,” arXiv e-prints , p. arXiv:1903.08925,Mar 2019.[13] J. Goodman,
Introduction to Fourier Optics . Roberts and CompanyPublishers; 3rd Edition edition, 2004.[14] B. E. A. Saleh and M. C. Teich,
Fundamentals of Photonics , 2nd ed.Wiley-Interscience, 2007.[15] L. Huang, S. Zhang, and T. Zentgraf, “Metasurface holography: fromfundamentals to applications,”
Nanophot. , vol. 7, no. 6, pp. 1169–1190,2018.[16] G.-Y. Lee, J. Sung, and B. Lee, “Recent advances in metasurfacehologram technologies,”
ETRI Journal , vol. 41, no. 1, pp. 10–22, 2019.[17] X. Liu, F. Yang, M. Li, and S. Xu, “Generalized boundary conditionsin surface electromagnetics: Fundamental theorems and surface charac-terizations,”
Appl. Sci. , vol. 9, p. 1891, 2019.[18] K. Achouri, M. A. Salem, and C. Caloz, “General metasurface synthesisbased on susceptibility tensors,”
IEEE Trans. Antennas Propag. , vol. 63,no. 7, pp. 2977–2991, Jul 2015. [19] U. R. Patel, P. Triverio, and S. V. Hum, “A fast macromodeling approachto efficiently simulate inhomogeneous electromagnetic surfaces,” 2019.[20] M. Dehmollaian, G. Lavigne, and C. Caloz, “Comparison of tensorboundary conditions with generalized sheet transition conditions,”
IEEETransactions on Antennas and Propagation , vol. 67, no. 12, pp. 7396–7406, Dec 2019.[21] E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway,“Averaged transition conditions for electromagnetic fields at a metafilm,”
IEEE Trans. Antennas Propag , vol. 51, no. 10, pp. 2641–2651, Oct 2003.[22] T. Brown, C. Narendra, Y. Vahabzadeh, C. Caloz, and P. Mojabi, “On theuse of electromagnetic inversion for metasurface design,”
IEEE Trans.Antennas Propag , pp. 1–1, 2019.[23] Y. Vahabzadeh, N. Chamanara, K. Achouri, and C. Caloz, “Compu-tational analysis of metasurfaces,”
IEEE Journal on Multiscale andMultiphysics Computational Techniques , vol. 3, pp. 37–49, 2018.[24] N. Chamanara, K. Achouri, and C. Caloz, “Efficient analysis of meta-surfaces in terms of spectral-domain GSTC integral equations,”
IEEETrans. Antennas Propag , vol. 65, no. 10, pp. 5340–5347, Oct 2017.[25] S. A. Stewart, T. J. Smy, and S. Gupta, “Finite-difference time-domainmodeling of spacetime-modulated metasurfaces,”
IEEE Trans. AntennasPropag , vol. 66, no. 1, pp. 281–292, Jan 2018.[26] S. A. Stewart, S. Moslemi-Tabrizi, T. J. Smy, and S. Gupta, “Scatteringfield solutions of metasurfaces based on the boundary element methodfor interconnected regions in 2-D,”
IEEE Trans. Antennas Propag ,vol. 67, no. 12, pp. 7487–7495, Dec 2019.[27] S. He, W. E. I. Sha, L. Jiang, W. C. H. Choy, W. C. Chew, and Z. Nie,“Finite-element-based generalized impedance boundary conditions formodeling plasmonic nanostructures,”
IEEE Trans. Nanotechnol. , pp.336–345, Mar. 2012.[28] M. Dehmollaian, N. Chamanara, and C. Caloz, “Wave scattering bya cylindrical metasurface cavity of arbitrary cross section: Theory andapplications,”
IEEE Trans. Antennas Propag , vol. 67, no. 6, pp. 4059–4072, June 2019.[29] S. Kagami and I. Fukai, “Application of boundary-element method toelectromagnetic field problems,”
IEEE Trans. Microw. Theory Tech. ,no. 4, pp. 455–461, Apr. 1984.[30] T. J. Smy, J. Connor, S. A. Stewart, and S. Gupta, “General formulationof the boundary element method (bem) for curvilinear metasurfaces inthe presence of multiple scattering objects,” in ,Mar 2020, pp. 1–2.[31] M. M. Idemen,
Discontinuities in the Electromagnetic Field . JohnWiley & Sons, 2011.[32] C. L. Holloway and E. F. Kuester, “Generalized sheet transition con-ditions for a metascreena fishnet metasurface,”
IEEE Trans. AntennasPropag , vol. 66, no. 5, pp. 2414–2427, May 2018.[33] W. Chew, M. Tong, and B. Hu,
Integral Equation Methods for Electro-magnetic and Elastic Waves . Morgan & Claypool Publishers, 2009.[34] W. C. Gibson,