Passive Lossless Huygens Metasurfaces for Conversion of Arbitrary Source Field to Directive Radiation
IIEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 11, NOVEMBER 2014 1
Passive Lossless Huygens Metasurfaces forConversion of Arbitrary Source Field to DirectiveRadiation
Ariel Epstein,
Member, IEEE, and George V. Eleftheriades,
Fellow, IEEE
Abstract —We present a semi-analytical formulation of theinteraction between a given source field and a scalar Huygensmetasurface (HMS), a recently introduced promising conceptfor wavefront manipulation based on a sheet of orthogonalelectric and magnetic dipoles. Utilizing the equivalent surfaceimpedance representation of these metasurfaces, we establishthat an arbitrary source field can be converted into directiveradiation via a passive lossless HMS if two physical conditionsare met: local power conservation and local impedance equal-ization. Expressing the fields via their plane-wave spectrum andharnessing the slowly-varying envelope approximation we obtainsemi-analytical formulae for the scattered fields, and prescribe thesurface reactance required for the metasurface implementation.The resultant design procedure indicates that the local impedanceequalization induces a Fresnel-like reflection, while local powerconservation forms a radiating virtual aperture which follows thetotal excitation field magnitude. The semi-analytical predictionsare verified by finite-element simulations of HMSs designedfor different source configurations. Besides serving as a flexibledesign procedure for HMS radiators, the proposed formulationalso provides a robust mechanism to incorporate a varietyof source configurations into general HMS models, as wellas physical insight on the conditions enabling purely reactiveimplementation of this novel type of metasurfaces.
Index Terms —metasurfaces, Huygens sources, wavefront ma-nipulation, plane-wave spectrum.
I. I
NTRODUCTION E LECTRICALLY thin sheets with repetitive metallic in-clusions or exclusions have been used extensively inthe past in antenna applications to control the properties ofreflected or transmitted power, e.g. its direction, phase, orpolarization [1]–[6]. Such surfaces have received increasingattention lately, as part of the intensive research in the fieldof optical and microwave metamaterials, in an attempt to har-ness ideas from bulk metamterial explorations to design low-profile components with extraordinary wavefront manipulationcapabilities [7]–[10]. In contrast to bulk metamaterials, wheresubwavelength elements are combined to form a volumetricentity with prescribed local response to electromagnetic fields,in metamaterial sheets, or metasurfaces, these subwavelengthatomic units are confined to a region with subwavelengththickness. This geometrical difference should decrease signif-icantly fabrication complexity of metasurfaces and also loss-
The authors are with the Edward S. Rogers Sr. Department of Electricaland Computer Engineering, University of Toronto, Toronto, ON, Canada M5S2E4 (e-mail: [email protected]; [email protected]).Manuscript received March 27, 2014; revised August 27, 2014. related problems; however, it requires development of new de-sign methodologies, as the interaction of electromagnetic fieldswith metasurfaces is naturally described via effective boundaryconditions [11]–[13], as opposed to effective permeabilitiesand permittivities (or effective wave equations), more suitablefor modelling volumetric metamaterials [14]–[16].In particular, it was recently recognized that as metasurfacesact as sources of tangential field discontinuities, they can bemodelled by a distribution of electric and magnetic surfacecurrents, prescribed by the equivalence principle [17, pp. 575-579]. Hence, in principle, for a given incident field, a desirableelectromagnetic field distribution in space can be achieved byengineering the surface to induce currents that would producethe required tangential fields on both of its facets.Approximating the required continuous surface currents bya dense distribution of electric and magnetic dipoles, passivesurfaces implementing plane-wave refraction were demon-strated [18], [19]. The elementary sources were formed by sub-wavelength inductive and capacitive elements that producedthe suitable magnitudes of the current in response to the excit-ing incident plane-wave. Simultaneously, it was demonstratedthat also active elementary sources may be utilized to introducedesirable field discontinuities, e.g., to implement a cloakingdevice based on the same equivalence principle [19]–[21].As these surfaces were composed of orthogonal electric andmagnetic dipoles engineered to induce unidirectional radiation,i.e. acting as Huygens sources [17, pp. 653-660], [22], theywere named Huygens metasurfaces (HMS) [18].In addition to plane-wave refraction and cloaking, recentreports proposed designs of Huygens metasurfaces whichimplement beam shaping, transmission or reflection coeffi-cient engineering and polarization manipulation (using tensorHuygens metasurfaces) [18], [23]–[25]. Although the designmethodologies differ between the various authors, they all relyon the fact that if the dimensions of the unit cells and their spa-tial arrangement obey certain conditions, the metasurface canbe modelled by effective electric and magnetic polarizabilitydistributions, which are translated to position-dependent sheetboundary conditions [8], [11], [12], [26]. These, in turn, canbe equivalently described as surface impedance and surfaceadmittance matrices relating the electric and magnetic fieldcomponents at the two facets of the metasurface [18], [19].Following this approach greatly simplifies device design,as it facilitates the development of simple circuit models toHuygens metasurfaces [21]. Moreover, as the effective surface (cid:13) a r X i v : . [ phy s i c s . op ti c s ] S e p EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 11, NOVEMBER 2014 2 impedance and admittance matrices are directly related to the locally averaged polarizabilities of the elementary scatterers,we may assess the local equivalent surface impedance of aunit cell by simulating or measuring the effective impedanceof an infinite periodic array of such identical unit cells (localperiodicity approximation) [7], [11], [12], [26]–[28].Indeed, this modelling approach was used in recent demon-strations of HMSs: given the incident field and the desirabletransmitted (or reflected) field, the required boundary condi-tions to support the field discontinuities can be formulated,resulting in the required surface impedance and admittancematrices (or electric and magnetic polarizability distributions).However, for realizing the Huygens metasurfaces, it isdesirable for the elements implementing the required polariz-ability distributions to be passive and lossless, i.e. the surfaceimpedance and admittance to possess pure imaginary values.Implementing impedance or admittance sheets with nonva-nishing real parts requires engineering of gain or loss ele-ments, thereby complicating greatly the design and realization.Nonetheless, following the simplistic methodology in whichthe discontinuities between the desirable and incident fields aredirectly translated into surface impedances and admittances byno means guarantees the passivity of the resultant metasurface(See, e.g., [18], [19]). In fact, in [18] the design proceduredid not consider the passivity limitation, however the resultantcomplex surface impedance and admittance were such thatthey could be approximated by purely imaginary functions,leading to a well-functioning, passive and lossless, prototype.The reasons for that encouraging outcome were not analyzedtherein, though.In addition, almost all Huygens metasurfaces presentedin the literature to date were designed to be excited by aplane-wave or a beam propagating towards the surface inhomogeneous medium [18], [19], [23], [26]. Nevertheless,to facilitate the development of realizable antenna devicesbased on Huygens metasurfaces it would be necessary toextend the current design techniques to enable excitation of themetasurface by localized (impulsive) sources, or waveguidedmodes. A step in that direction was made by Holloway etal. [29], where a line source excitation was considered, butonly for metasurfaces with constant polarizability density;as demonstrated by [18], [19], allowing the polarizabilitydensities to vary along the metasurface could be beneficial,providing more degrees of freedom for the design.In this work, we derive from first principles simple rules fordesigning passive lossless Huygens metasurfaces producingdirective radiation to a prescribed angle when excited by agiven (arbitrary) source field. Decomposing the fields to theirplane-wave spectrum and generalizing the approach presentedin [19] for plane-wave excitation, we show that satisfying twophysical conditions is sufficient to guarantee that the desirablefunctionality can be achieved by purely reactive surfaces: localpower conservation across the surface, and local impedanceequalization of the fields on both sides of the metasurface.Enforcing these conditions locally, i.e. at each point on thesurface, leads to a complementary set of simple expressionsfor the surface impedance and surface admittance, as wellas facilitates the semi-analytical evaluation of the reflected
Fig. 1. Physical configuration of a Huygens metasurface excited by anarbitrary source situated at x ≤ x (cid:48) < . The formalism applies withoutmodification also to scenarios in which the region x ≤ x (cid:48) is occupied by aninhomogeneous medium, as long as its cross-section remains uniform withrespect to the x axis, i.e the permittivity, permeability and conductivity are afunction of x coordinate only (e.g., plane-stratified media) [30, pp. 183-202]. and transmitted fields. These results enable design of directiveHMS radiators with a wide range of source excitations, thusextending significantly the possible applications. Moreover, thederivation provides clear physical interpretation of the con-ditions required to implement passive lossless HMSs, whichmay also be indicative to equivalent requirements in moregeneralized scenarios (e.g. HMSs which perform other func-tionalities). Altogether, this forms an efficient and powerfultool for HMS engineering, promoting design of novel antennadevices. II. T HEORY
A. Formulation
We consider a 2D configuration ∂/∂y = 0 in which a Huy-gens metasurface situated at x = 0 is excited by an arbitrarycurrent distribution limited to the half-space x < (Fig. 1).The surrounding media is assumed to be homogeneous, withpermittivity (cid:15) and permeability µ , defining the wave impedance η = (cid:112) µ/(cid:15) . Harmonic time dependency of e jωt is assumed(and suppressed), defining the wavenumber k = ω √ (cid:15)µ .The metasurface is characterized by its surface impedance Z se ( z ) and surface admittance Y sm ( z ) , inducing discontinu-ities in tangential magnetic and electric field components, re-spectively, given by the generalized sheet transition conditions(GSTC) as formulated by Kuester et al. [11] Z se ( z ) (cid:126)J s = Z se ( z ) ˆ x × (cid:104) (cid:126)H (cid:12)(cid:12)(cid:12) x → + − (cid:126)H (cid:12)(cid:12)(cid:12) x → − (cid:105) == 12 (cid:104) (cid:126)E (cid:12)(cid:12)(cid:12) x → + + (cid:126)E (cid:12)(cid:12)(cid:12) x → − (cid:105) Y sm ( z ) (cid:126)M s = − Y sm ( z ) ˆ x × (cid:104) (cid:126)E (cid:12)(cid:12)(cid:12) x → + − (cid:126)E (cid:12)(cid:12)(cid:12) x → − (cid:105) == 12 (cid:104) (cid:126)H (cid:12)(cid:12)(cid:12) x → + + (cid:126)H (cid:12)(cid:12)(cid:12) x → − (cid:105) , (1)where (cid:126)J s and (cid:126)M s are the surface currents induced by thetangential electric and magnetic field components, respec-tively, and we assumed the impedance and admittance matricescan be described by scalar quantities. The half-spaces belowand above the metasurface are referred to as region 1 andregion 2, respectively, and we require the sources not to be EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 11, NOVEMBER 2014 3 infinitesimally close to the HMS such that a source-free region x (cid:48) < x < can be defined just below it (Fig. 1).To ensure that we harness all possible degrees of freedomfor the HMS design, we wish to analyze the most generalfield constellation admissible by Maxwell’s equations in our2D scenario. This is achieved by considering both TE andTM polarized fields, by allowing reflections where applicable,and by utilizing the spectral representation of the fields forsemi-analytical formulation. To facilitate the fluent reading ofthe rest of the paper, we include the basic field definitionsin this Subsection. Starting with Maxwell’s equations, thesedefinitions enable us to precisely point out the approximationsmade along the way, guaranteeing the consistency of thederivation and the accurate interpretation of its results.In region 1, thus, we distinguish between the incident fieldand reflected field, while region 2 is populated only by thetransmitted fields (denoted by inc , ref , and trans superscripts,respectively) (cid:126)E ( x, z ) = (cid:26) (cid:126)E inc ( x, z ) + (cid:126)E ref ( x, z ) x < (cid:126)E trans ( x, z ) x > (cid:126)H ( x, z ) = (cid:26) (cid:126)H inc ( x, z ) + (cid:126)H ref ( x, z ) x < (cid:126)H trans ( x, z ) x > , (2)and the incident fields are defined as the fields produced bythe sources in the absence of the HMS .In 2D configurations Maxwell’s equations can be separatedto two decoupled sets of equations related to the trans-verse electric (TE) or transverse magnetic (TM) polarizedfield components. In source-free regions, the TE-polarized( E x = 0 ) nonvanishing field components are ( E y , H x , H z ) ,and Maxwell’s equations can be reduced to a wave equationfor E y and curl equations relating the other components to it, (cid:0) ∇ + k (cid:1) E y = 0 , H x = jkη ∂E y ∂z , H z = − jkη ∂E y ∂x . (3)Analogously, in source-free regions, the TM-polarized ( H x =0 ) nonvanishing field components are ( H y , E x , E z ) , andMaxwell’s equations can be reduced to (cid:0) ∇ + k (cid:1) H y = 0 , E x = − ηjk ∂H y ∂z , E z = ηjk ∂H y ∂x . (4)As this work does not deal with polarization manipulationof the source fields, we may design an HMS directive radiatorfor each polarization independently, namely a TE-HMS anda TM-HMS. The TE-HMS would interact only with E y and H z , therefore the required surface impedance Z se ( z ) wouldbe implemented using scatterers sensitive to electric field inthe y direction (e.g. loaded wires parallel to the y -axis) andthe surface admittance Y sm ( z ) would be implemented usingscatterers sensitive to magnetic field in the z direction (e.g.loaded loops whose axis is parallel to the z -axis) [19]. Onthe other hand, the TM-HMS should interact only with H y and E z , therefore Z se ( z ) and Y sm ( z ) would be composedby scatterers sensitive to electric field in the z direction andmagnetic field in the y direction, respectively. As the elementsimplementing the TE-HMS and the TM-HMS are orthogonal, A formal definition of the incident fields, more suitable in cases the region x < x (cid:48) includes scattering elements (e.g., as we allow in Subsection III-C)will be given in (5)-(6). there should be no coupling between them; thus, ideally,the two HMSs may be combined to a single metasurfacewithout any changes to the independent designs. In view ofthis observation, we formulate the procedure to design anHMS assuming the incident field is TE-polarized; the designrules for TM-polarized excitation can be readily derived byduality (Appendix A). Extension of this work to polarizationmanipulating HMSs [25] will be addressed in a separate report.
1) Spectral Decomposition:
As denoted, when the excita-tion field is TE-polarized, the nonvanishing field componentsare E y ( x, z ) , H z ( x, z ) , and H x ( x, z ) ; the scalar surfaceimpedance only induces electric currents in the y direction;and the scalar surface admittance only induces magnetic currents in the z direction. In view of (3) a general solution forthe fields in the source-free region x > x (cid:48) can be formulatedin the spectral domain as [29]–[32] E inc y ( x, z ) = kη I π ∞ (cid:90) −∞ dk t β e f ( k t ) e − jβx e jk t z E ref y ( x, z ) = − kη I π ∞ (cid:90) −∞ dk t β e Γ ( k t ) e f ( k t ) e jβx e jk t z E trans y ( x, z ) = kη I π ∞ (cid:90) −∞ dk t β e T ( k t ) e − jβx e jk t z , (5)where k t is the transverse wavenumber (associated with thepropagation along z ), and β = (cid:112) k − k t is the longitudinalwavenumber (associated with the propagation along x ); tosatisfy the radiation condition we demand that (cid:61) { β } < .The e left superscript denotes TE-HMS related parametersthroughout the paper, and I is a unit current magnitude.It can be readily verified that all three integrals in (5) satisfythe wave equation of (3), where e f ( k t ) is the source-relatedplane-wave spectrum, e Γ ( k t ) is the reflection coefficient inthe spectral domain, and e T ( k t ) corresponds to the spectralcontent of the transmitted fields. As the sources reside at x ≤ x (cid:48) in region 1, and the scattering metasurface is situated at x = 0 , the general solution in x (cid:48) < x < must consider bothupwards (incident) and downwards (reflected) propagatingwaves. In contrast, as neither sources nor scatterers exist inregion 2, only upwards (transmitted) propagating waves areallowed for x > , as to satisfy the radiation condition.We emphasize that the formal solution presented in (5) isvalid in general only for x > x (cid:48) and that the presence ofthe sources at x ≤ x (cid:48) introduces a discontinuity in the fields,which should be accounted for in that region. Nonetheless,for designing the HMS only the tangential fields at x → ± as formulated in (5) are required [29], and this formulation isvalid regardless of the nature of the source distribution x ≤ x (cid:48) .In fact, (5) applies without modification also to scenarios inwhich the region x ≤ x (cid:48) is occupied by an inhomogeneousmedium, as long as its cross-section remains uniform withrespect to the x axis, i.e the permittivity, permeability andconductivity are a function of x coordinate only (e.g., plane-stratified media) [30, pp. 183-202]. As shall be demonstratedin Subsection III-C, this allows for even a wider range ofexcitation schemes to be investigated using our model. EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 11, NOVEMBER 2014 4
From (5) and the curl equations of (3) we derive thecorresponding tangential component of the magnetic field at x > x (cid:48) , H inc z ( x, z ) = I π ∞ (cid:90) −∞ dk t e f ( k t ) e − jβx e jk t z H ref z ( x, z ) = I π ∞ (cid:90) −∞ dk t e Γ ( k t ) e f ( k t ) e jβx e jk t z H trans z ( x, z ) = I π ∞ (cid:90) −∞ dk t e T ( k t ) e − jβx e jk t z , (6)and (1) can be rewritten as [19] e Z se ( z ) = − E trans y (0 , z ) + (cid:2) E inc y (0 , z ) + E ref y (0 , z ) (cid:3) H trans z (0 , z ) − [ H inc z (0 , z ) + H ref z (0 , z )] e Y sm ( z ) = − H trans z (0 , z ) + (cid:2) H inc z (0 , z ) + H ref z (0 , z ) (cid:3) E trans y (0 , z ) − (cid:2) E inc y (0 , z ) + E ref y (0 , z ) (cid:3) (7)
2) Statement of the Problem:
Given an incident field, wewould like to determine the required variation of the surfaceimpedance and admittance along the metasurface such that • Both the surface impedance and surface admittanceare purely reactive (passive and lossless), namely (cid:60) { Z se ( z ) } = (cid:60) { Y sm ( z ) } = 0 . For that we are willingto allow some reflections from the metasurface. • The transmitted field E trans y ( x, z ) , H trans z ( x, z ) , H trans x ( x, z ) will form a directional radiation to aspecified angle θ with respect to the x axis (Fig. 1). Tothis end we would like to form a virtual aperture on thesurface with surface current having the suitable linearphase variation and as uniform as possible magnitude.In order to satisfy the second demand, we require thatthe spectral content of the transmitted fields e T ( k t ) will belocalized around k t = k t, = − k sin θ . Ideally, e T ( k t ) wouldbe a delta function; however, this can be obtained only foruniform excitation of an infinite HMS, i.e. for plane waves(e.g., as in [19]). When finite sources and metasurfaces areconsidered, the virtual aperture must have a compact supportin space. To facilitate this, we introduce to our formulation aslowly-varying window function e W ( x, z ) , decaying towardsthe edges of the metasurface, which would serve as an en-velope for the desirable linear phase function. If the envelopevariation would be moderate with respect to the required phasevariation, the resulting radiation would be directed as desired.Formally, we define this virtual aperture window function as E trans y ( x, z ) (cid:44) kηI e W ( x, z ) e − jkx cos θ e − jkz sin θ (8)and demand that the HMS would be designed such that theresulting virtual aperture window would form a slowly-varyingenvelope as x → + , namely, (cid:12)(cid:12)(cid:12)(cid:12) ∂∂x e W ( x, z ) (cid:12)(cid:12)(cid:12)(cid:12) x → + (cid:28) | k cos θ e W ( x, z ) | x → + . (9)Applying this constraint on (6) enables us to approximate thetangential magnetic field at x → + as H trans z ( x, z ) ≈ k cos θ I e W ( x, z ) e − jkx cos θ e − jkz sin θ . (10) The condition (9), thus, ensures the tangential fields on thevirtual aperture locally resemble those of a plane-wave towards θ , promoting directive radiation.Substituting (8) and (10) into (7) yields e Z se ( z ) = − η θ e F + ( z ) + e F − E ( z ) e F + ( z ) − e F − H ( z ) e Y sm ( z ) = − cos θ η e F + ( z ) + e F − H ( z ) e F + ( z ) − e F − E ( z ) , (11)where we have defined dimensionless quantities, proportionalto the fields on the lower (minus-sign superscript) and upper(plus-sign superscript) facets of the metasurface, as follows e F − E ( z ) (cid:44) I kη (cid:2) E inc y (0 , z ) + E ref y (0 , z ) (cid:3) e F − H ( z ) (cid:44) I k cos θ (cid:2) H inc z (0 , z ) + H ref z (0 , z ) (cid:3) e F + ( z ) (cid:44) e W (0 , z ) e − jkz sin θ . (12)It should be noted that the fields in our formulationare ”macroscopic” in the sense that they result from av-eraged boundary conditions, applicable for an infinitely-thinhomogenized equivalent surface, exhibiting continuous sheetimpedance and admittance profiles [8]. Hence, they are strictlyvalid only from a finite distance away from the metasurface,where the subwavelength field variations due to the elementsimplementing the HMS become negligible. In general, thisdistance should be larger than both the unit cell thickness andperiodicity, to ensure sufficient decay of corresponding higher-order Floquet modes [12, pp.79-82].This is important, for instance, when interpreting the slowly-varying envelope condition (9), as the limit x → + isonly applicable for the ”macroscopic” fields; nonetheless, asguaranteed by the GSTC derivation [11], [12], adhering tothis constraint when designing the HMS, would result in thedesirable virtual aperture formation at the regions where theequivalent and real physical problems coincide [18], [21],[26], [29]. Practically, for the HMS implementation we utilizeherein (Appendix B), keeping the sources and the observationpoints at least λ/ away from the HMS plane x = 0 , shouldmaintain the model accuracy. B. Sufficient Conditions for Passive Lossless HMS1) Local Power Conservation:
As we require the HMSto be purely reactive, the real power across the metasurfacemust be conserved. However, to obtain sufficient conditionsfor designing passive lossless HMSs we may require a strictercondition to be met, namely, that the real power impinging themetasurface from region 1 is equal locally to the real powertransmitted to region 2, at each point z on the surface.To assess the consequences of this local power conservationcondition we utilize (12) to express the local power densitiesalong the metasurface in region 1 and 2 using the dimen-sionless field quantities. These are given, respectively, by theprojection of the Poynting vector on the x axis as x → − e S − x ( z ) = ˆ x · (cid:104) (cid:126)E ( x, z ) × (cid:126)H ∗ ( x, z ) (cid:105) x → − = kη | I | e F − E ( z ) e F −∗ H ( z ) k cos θ , (13) EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 11, NOVEMBER 2014 5 and as x → + e S + x ( z ) = ˆ x · (cid:104) (cid:126)E ( x, z ) × (cid:126)H ∗ ( x, z ) (cid:105) x → + = kη | I | (cid:12)(cid:12) e F + ( z ) (cid:12)(cid:12) k cos θ , (14)where the asterisk indicates the complex-conjugate operation.The condition for local power conservation is (cid:60) (cid:8) e S − x ( z ) (cid:9) = (cid:60) (cid:8) e S + x ( z ) (cid:9) which reads, using (13)-(14), (cid:12)(cid:12) e F + ( z ) (cid:12)(cid:12) = (cid:60) (cid:8) e F − E ( z ) e F −∗ H ( z ) (cid:9) . (15)Multiplying the rational functions in (11) by the complexconjugate of their respective denominators and plugging inthe local power conservation requirement (15) yields e Z se ( z ) = − η θ e F + ∗ ( z ) e F − E ( z ) − e F + ( z ) e F −∗ H ( z ) (cid:12)(cid:12) e F + ( z ) − e F − H ( z ) (cid:12)(cid:12) e Y sm ( z ) = − cos θ η e F + ∗ ( z ) e F − H ( z ) − e F + ( z ) e F −∗ E ( z ) (cid:12)(cid:12) e F + ( z ) − e F − E ( z ) (cid:12)(cid:12) (16)
2) Local Impedance Equalization:
In view of (16), a suf-ficient condition for the real part of both e Z se and e Y sm tovanish is given by e F − E ( z ) = e F − H ( z ) (cid:44) e F − ( z ) , (17)resulting in a purely imaginary numerator for both fractions.The physical meaning of the latter is revealed by rewriting(17) using (12): the condition locally equalizes the waveimpedance of the fields on the two facets of the metasurface.Indeed, substituting (12) into (17) leads to the local impedanceequalization condition E inc y (0 , z ) + E ref y (0 , z ) H inc z (0 , z ) + H ref z (0 , z ) = E trans y (0 , z ) H trans z (0 , z ) = η cos θ , (18)and we stress that, for an arbitrary source, the existence of areflected field is generally necessary to satisfy this condition.When (17) is satisfied, the local power conservation condi-tion (15) takes a simpler form, namely, (cid:12)(cid:12) e F + ( z ) (cid:12)(cid:12) = (cid:12)(cid:12) e F − ( z ) (cid:12)(cid:12) (cid:44) | e F ( z ) | . (19)Subsequently, we may define the dimensionless field quantitiesabove and below the metasurface as (cid:26) e F + ( z ) (cid:44) | e F ( z ) | e jϕ + ( z ) e F − ( z ) (cid:44) | e F ( z ) | e jϕ − ( z ) , (20)where ϕ ± ( z ) ∈ R . Substituting these definitions into (16)leads to a complementary set of compact expressions for thedesirable surface impedance and surface admittance e Z se ( z ) = − j e Z (cid:20) ϕ − ( z ) − ϕ + ( z )2 (cid:21) e Y sm ( z ) = − j e Y (cid:20) ϕ − ( z ) − ϕ + ( z )2 (cid:21) , (21)where e Z = 1 / e Y = η/ cos θ is the wave impedance of aTE-polarized plane-wave propagating in region 2 at an angleof θ with respect to the x axis, given generally by [30] (cid:126)E t (0 , z ) (cid:44) Z (cid:104) (cid:126)H t (0 , z ) × ˆ x (cid:105) , (22) (cid:126)E t and (cid:126)H t being the tangential components of the fields. As e Z , e Y , ϕ ± ( z ) are all real, the HMS defined via (21) isindeed purely reactive, as required.It is important to note that although it may appear that therequired effect of the HMS is merely to introduce a localphase-shift to the incident field, seemingly allowing for apassive lossless implementation without any reflections, thisis not the case, in general. In order to transform one validelectromagnetic field to another (both satisfying Maxwell’sequations), one has to consider also the variation of the localwave impedance at each point along the metasurface. Asdemonstrated, for example, in [18], transforming both thelocal phase and the local wave impedance of the incidentfields without incurring reflections in (7) gives rise to surfaceimpedance and admittance values with non-vanishing realparts, which should be somehow mitigated if lossless passiverealization is desirable (See ”Supplemental Material” of [18],pp. 3-6). Nonetheless, as suggested in [19] and generalizedherein, introducing another degree of freedom in the formof the reflected fields to the design enables such a control,requiring neither active nor lossy elements.Lastly, we emphasize that both local power conservation(15) and local impedance equalization (17) are required toestablish sufficient conditions for a passive lossless HMSfor our application. Moreover, our derivation results only in sufficient conditions, thus it does not invalidate the possibilitythat other passive lossless designs, not adhering to theseconditions, may achieve similar functionality. C. Explicit Evaluation of the Metasurface Reactance andSusceptance, and the Scattered Fields
Equation (21) prescribes the required variation of the surfacereactance and susceptance to implement the desirable HMS.However, in order to use this formula, we should instruct howthe phases of the fields at x → ± , defined as ϕ ± ( z ) , are tobe evaluated.To that end, we first indicate how to find the reflected andtransmitted fields from the given source field and the require-ment that the power is locally conserved and the impedance islocally equalized. Local impedance equalization (18) requires E inc y (0 , z ) − η cos θ H inc z (0 , z ) == − (cid:20) E ref y (0 , z ) − η cos θ H ref z (0 , z ) (cid:21) , (23)from which, using the spectral representation of the fields (5)-(6), the reflection coefficient can be evaluated as e Γ ( k t ) = k cos θ − βk cos θ + β (24) regardless of the source field. Equation (24) is merely a privatecase of the Fresnel reflection formula for an incident plane-wave at an angle arcsin ( k t /k ) and a transmitted plane-wave atan angle of θ , travelling in media with the same permittivityand permeability [31]. Actually, it is a generalization ofthe reflection coefficient derived in [19] for passive losslessHMSs excited by a plane-wave. However, in the general caseconsidered herein, the source field consists of an infinite EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 11, NOVEMBER 2014 6 number of plane-waves, propagating in different directions.Therefore, the reflection coefficient of (24) actually ensures individual impedance equalization of each of the plane-wavesin the source field spectrum (5)-(6) to that of a plane-wavepropagating in region 2 with an angle of θ with respect tothe x axis.Equation (24) enables evaluation of the reflected fields(everywhere in region 1) via (5)-(6). Combined with thegiven incident field, we may then utilize (12) to evaluate e F − E ( z ) = e F − H ( z ) = e F − ( z ) . Explicitly, this is given by e F − ( z ) = 12 π ∞ (cid:90) −∞ dk t k cos θ + β e f ( k t ) e jk t z , (25)from which the total dimensionless field magnitude | e F ( z ) | and phase ϕ − ( z ) on the lower facet of the HMS are assessed,following (20).The local power conservation condition (19) indicates thatthe dimensionless field magnitudes must be continuous at each z along the metasurface. This facilitates the evaluation of thevirtual aperture window function e W (0 , z ) via (12), namely e W (0 , z ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π ∞ (cid:90) −∞ dk t k cos θ + β e f ( k t ) e jk t z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − jξ , (26)where ξ is an arbitrary (constant) phase shift which may beintroduced to the transmitted fields without affecting neitherthe radiation directivity nor the reflected fields (similarly to[25]).From the definition (8) and the spectral representation (5),the spectral content of the transmitted field can be evaluatedvia the inverse Fourier transform, e T ( k t ) = 2 β ∞ (cid:90) −∞ e W (0 , z ) e − j ( k t + k sin θ ) z dz, (27)using which the transmitted fields everywhere in region 2 maybe computed from (5)-(6).Finally, the transmitted field phase variation on the HMScan be explicitly formulated using (26) and (12) as ϕ + ( z ) = − kz sin θ − ξ (28) regardless of the source fields, and (21) can be rewritten as e Z se ( z ) = − j e Z (cid:20) kz sin θ + ξ + ∠ e F − ( z )2 (cid:21) e Y sm ( z ) = − j e Y (cid:20) kz sin θ + ξ + ∠ e F − ( z )2 (cid:21) , (29)where ∠ e F − ( z ) = ϕ − ( z ) is the phase of the dimensionlessfield parameter given by (25).The design procedure described in this Section is summa-rized in Table I, with references to relevant equation numbersin the text. It should be noted that the fields used to designthe HMS are approximate, valid subject to the slowly-varyingenvelope condition (9); thus, after the design procedure iscompleted, and the predicted transmitted fields have beenevaluated via (27), the satisfaction of (9) should be verifiedto assess the consistency of the theoretical derivation (See,e.g., Subsections III-A,III-B3). III. R ESULTS AND D ISCUSSION
To verify our theory, we follow the design procedureoutlined in Section II and summarized in Table I to designpassive lossless HMS directive radiators for three different(TE-polarized) source excitations: a plane-wave, an electricline source (ELS), and an electric line source positioned infront of a perfect electric conductor (PEC) infinite plane (Fig.2). In the following Subsections we derive the expressionsfor the required surface impedance and surface admittance,and compare the performance of the HMS, predicted bysemi-analytical means, to results of respective finite-elementnumerical simulations, in which the HMS is implementedusing loaded loops and wires (similarly to [19]).
A. Excitation by a Plane-wave
We begin by verifying the consistency of our theory withprevious HMS design formulae derived in [19] for the privatecase of plane-wave excitation. This would also serve as asimple demonstration of the proposed design procedure, withindication of the relevant steps in Table I.In the configuration under consideration, the source field isproduced by a TE-polarized plane-wave travelling at an angle θ i with respect to the x -axis incident upon the HMS (Fig.2(a)). The spectrum of the source field is thus given by e f ( k t ) = 4 πβδ ( k t − k t,i ) , (30)where k t,i = − k sin θ i . To enforce local impedance equal-ization (Table I / Step 1), the reflection coefficient is definedaccording to (24), using which the incident and reflected fields(5)-(6) may be evaluated E inc y ( x, z ) = kηI e − jkx cos θ i e − jkz sin θ i E ref y ( x, z ) = − cos θ − cos θ i cos θ + cos θ i E inc y ( x, z ) (31) H inc z ( x, z ) = kI cos θ i e − jkx cos θ i e − jkz sin θ i H ref z ( x, z ) = cos θ − cos θ i cos θ + cos θ i H inc z ( x, z ) , (32)and it is readily verified that the fields indeed satisfy (23). Thedimensionless total field at x → − is thus given by (25) e F − ( z ) = 2 cos θ i cos θ + cos θ i e − jkz sin θ i , (33)from which we arrive at the desirable output of [Table I / Step1], namely, (cid:12)(cid:12) e F − ( z ) (cid:12)(cid:12) ≡ θ i cos θ + cos θ i , ϕ − ( z ) = − kz sin θ i , (34)and we note that θ i , θ ∈ ( − π/ , π/ .Local power conservation (Table I / Step 2) essentiallyrequires that the magnitude of the virtual aperture windowfunction on the upper facet of the HMS would follow themagnitude of the total fields (incident+reflected) on its lowerfacet. Hence, W (0 , z ) is given by combining (26) with (34),yielding e W (0 , z ) ≡ θ i cos θ + cos θ i e − jξ , (35) EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 11, NOVEMBER 2014 7
TABLE IS
UMMARY OF D ESIGN P ROCEDURE FOR P ASSIVE L OSSLESS H UYGENS M ETASURFACE FOR D IRECTIVE R ADIATION
No. Step Input Output Relevant Equations1 Local impedanceequalization Incident fields: (cid:126)H inc ( x, z ) , (cid:126)E inc ( x, z ) Transmission angle: θ Reflected fields: (cid:126)H ref ( x, z ) , (cid:126)E ref ( x, z ) Total fields at x → − : (cid:12)(cid:12) F − ( z ) (cid:12)(cid:12) e jϕ − ( z ) (5)-(6),(23)-(25)2 Local powerconservation Field magnitude at x → − : (cid:12)(cid:12) F − ( z ) (cid:12)(cid:12) Transmission angle: θ Field magnitude at x → + : W (0 , z ) Transmitted fields: (cid:126)H trans ( x, z ) , (cid:126)E trans ( x, z ) (26)-(27),(5)-(6), (8)-(10)3 Metasurfacereactance Field phase at x → − : ϕ − ( z ) Transmission angle: θ Surface impedance: Z se ( z ) Surface admittance: Y sm ( z ) (20)-(22),(28)-(29)Fig. 2. Physical configurations of Huygens metasurfaces excited by (a) a plane wave forming an angle of θ i with the x -axis (Subsection III-A); (b) anelectric line source situated at ( x, z ) = ( x (cid:48) , carrying a current of (cid:126)J = I δ ( x − x (cid:48) ) δ ( z ) ˆ y (Subsection III-B); (c) the same electric line source positionedin front of a PEC, separated by d from the HMS; θ int is the angle of internal constructive interference between the source and its image (Subsection III-C).The transmitted field radiates towards the direction defined by θ . where ξ is an arbitrary phase, in case any is desirable.Using the last result and (27), the plane-wave spectrum ofthe transmitted wave can be evaluated, e T ( k t ) = 4 πβδ ( k t − k t, ) 2 cos θ i cos θ + cos θ i e − jξ , (36)and subsequently, from (5)-(6), also the transmitted fields, E trans y ( x, z ) = kηI θ i e − jξ cos θ + cos θ i e − jkx cos θ e − jkz sin θ H trans z ( x, z ) = cos θ η E trans y ( x, z ) . (37)This completes Step 2 of the design procedure.Finally, we may use (29) with the phase calculated in (34) todefine the surface impedance and surface admittance requiredto implement the desirable HMS (Table I / Step 3). These aregiven by e Z se ( z ) = − j e Z (cid:20) kz (sin θ − sin θ i ) + ξ (cid:21) e Y sm ( z ) = − j e Y (cid:20) kz (sin θ − sin θ i ) + ξ (cid:21) , (38)which, when ξ = 0 , coincide with the results obtained in [19]for the same configuration (recall e Z = 1 / e Y = η/ cos θ ).As this type of HMS was thoroughly investigated in [19],[21], including demonstration of its performance using nu-merical simulation tools, we would not discuss it here further.It is, however, worthwhile to note that in the context of thedesign procedure formulated in Section II, the virtual aperturewindow function e W ( x, z ) formed by the HMS has infiniteextent in this case. More precisely, from (37) and (8) it follows that e W ( x, z ) is constant. Therefore, the satisfaction of (9) istrivial for any transmission angle ( | θ | < π/ ), indicating thatthe theoretical prediction of the HMS functionality is valid. B. Excitation by an Electric Line Source
Next, we consider the scenario in which the HMS is excitedby an electric line source (cid:126)J = I δ ( x − x (cid:48) ) δ ( z ) ˆ y situated inregion 1, where according to our convention x (cid:48) < (Fig. 2(b)).This configuration is different from the plane-wave excitationscenario discussed in Subsection III-A in three significantaspects, all originate from the localized nature of the source:first, the source introduces discontinuity to the fields at x = x (cid:48) (region 1); second, its spectral representation consists of a widerange of plane-waves; third, its illumination of the HMS isnon-uniform, creating a localized spot on the virtual aperture.In the following Subsubsections, we will emphasize the effectsof these differences on the design procedure.
1) HMS Design:
As mentioned in the previous paragraph,the derivation of the source plane-wave spectrum e f ( k t ) nowinvolves a source condition , requiring the discontinuity ofthe derivative of the characteristic Green’s function at x = x (cid:48) [29]–[32]. Moreover, to satisfy the radiation condition at x →−∞ we should use a different x dependency for the incidentfield in the region x < x (cid:48) than we used for the source-freeregion x (cid:48) < x < , manifesting the fact that the plane-wavespropagate away from the source at all regions. This would be a consequence of the introduction of a (singular) nonhomo-geneous term to the Helmholtz (wave) equation of (3), required if the wholeregion 1 ( x < ) is to be described by this equation. EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 11, NOVEMBER 2014 8
Considering these conditions, the incident and reflectedfields in the whole region 1 ( x < ) may be formulated as E inc y ( x, z ) = kη I π ∞ (cid:90) −∞ dk t β e − jβ | x − x (cid:48) | e jk t z E ref y ( x, z ) = − kη I π ∞ (cid:90) −∞ dk t β e Γ ( k t ) e jβ ( x + x (cid:48) ) e jk t z (39) H inc z ( x, z ) = ± I π ∞ (cid:90) −∞ dk t e − jβ | x − x (cid:48) | e jk t z H ref z ( x, z ) = I π ∞ (cid:90) −∞ dk t e Γ ( k t ) e jβ ( x + x (cid:48) ) e jk t z , (40)where the reflection coefficient remains a free parameter andthe upper and lower signs in (40) should be used when x > x (cid:48) or x < x (cid:48) , respectively; this change of signs establishes therequired discontinuity in the magnetic field at the source posi-tion. Comparing (39)-(40) in the source-free region x (cid:48) < x < with the general form (5)-(6) yields the expression for thesource-related plane-wave spectrum, namely e f ( k t ) = e jβx (cid:48) , (41)which forms the necessary input to begin our design procedure.To enforce local impedance equalization (Table I / Step 1),we define the reflection coefficient according to (24), whichthen enables the evaluation of the incident and reflected fieldsacross the entire region 1. The dimensionless total field at x → − is thus given by (25) e F − ( z ) = 12 π ∞ (cid:90) −∞ dk t k cos θ + β e jβx (cid:48) e jk t z . (42)As a result of the wide spectral content of the source, and thefact that the reflection coefficient required to guarantee localimpedance equalization varies with the transverse wavenumber k t , obtaining an analytical closed-form expression for e F − ( z ) is not trivial as it was for the plane-wave excitation scenario(See (33)). Nonetheless, as the integral of (42) consists of aslowly varying part / ( k cos θ + β ) and an oscillatory part e jβx (cid:48) e jk t z , we may employ asymptotic evaluation techniques(e.g., the steepest-descent-path method [30]) to evaluate it inclosed-form for those evaluation points (on the metasurface)which are in the far field of the source. If, however, thedistance between the evaluation point (on the metasurface)and the source ρ (cid:48) = √ x (cid:48) + z is not very large with respectto the wavelength, the oscillatory part variation is moderateenough such that, in general, e F − ( z ) may be evaluated bystraightforward numerical integration .One way or the other, the magnitude and phase functions( (cid:12)(cid:12) e F − ( z ) (cid:12)(cid:12) and ϕ − ( z ) , respectively) can be calculated from(42), as required in [Table I / Step 1]. These would determine,respectively, the profile of the virtual aperture window function(following (26) [Table I / Step 2]), and the phase compensation In fact, the integrand is bounded on the entire real k t axis due to thereflection coefficient. required by the HMS to ensure the fields on that aperture carrya linear phase (following (28) [Table I / Step 3]).Finally, completing Step 2 and Step 3 of the design pro-cedure, we are able to evaluate the fields at each point inspace, as well as the desirable surface impedance and surfaceadmittance defining the HMS. The latter may be written as ageneralized form of (38), namely, e Z se ( z ) = − j e Z (cid:20) kz (sin θ − sin θ i ( z )) + ξ (cid:21) e Y sm ( z ) = − j e Y (cid:20) kz (sin θ − sin θ i ( z )) + ξ (cid:21) , (43)where we used the definition of the equivalent angle ofincidence θ i ( z ) ϕ − ( z ) = − kz sin θ i ( z ) (44)in analogy to (34). As in Subsection III-A, upon evaluationof the transmitted fields, the satisfaction of the slowly-varyingenvelope condition (9) should be verified to ensure the con-sistency of the design procedure.
2) Virtual Aperture Engineering:
Before we proceed todemonstrate the performance of several ELS-excited HMSdesigns, we refer to the last point mentioned in the openingparagraph of this Subsection. As part of the device engi-neering, it is important to control the shape of the virtualaperture, as it determines to a large extent the width anddirectivity of the transmitted radiation. However, as opposedto the case of plane-wave excitation, in which the metasurfacewas illuminated uniformly across the entire z axis, when finite-energy sources are used, the nature of the resultant virtualaperture is not as easily predicted. The reason for that isthat the profile of the virtual aperture window function isnot constant anymore, and is determined by the total field at x → − , i.e. the sum of the incident and reflected fields; whilethe former is known, the latter is an outcome of the integrationof the individually reflected source plane-waves (24).Nevertheless, if the source is not illuminating the meta-surface at a grazing angle, the reflection coefficient variesrather moderately with k t in the spectral region contributingdominantly to the reflected field. In that case, a good zero-order approximation for the shape of the virtual aperturewindow would be given by the magnitude of the incident field,thus providing a starting point for selecting sources suitable fora desirable virtual aperture design. More than that, the phaseof the incident field will be then a reasonable approximationto ∠ e F − ( z ) , providing an insight on the variation of Z se and Y sm along the HMS (29). If, in addition, the evaluationpoint on the metasurface is in the far-field of the source, theequivalent angle of incidence θ i ( z ) defined in (44) receives anelegant physical interpretation: this is the angle of incidence ofthe ray (”local” plane-wave) incident upon the metasurface inthe neighbourhood of z . Accordingly, (43) can be interpretedas a generalization of (38), where due to the localized source,different points along the metasurface interact with plane-waves having different angles of incidence.
3) Numerical and Semi-analytical Results:
To verify ourformulation, we have designed and simulated three Huygens
EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 11, NOVEMBER 2014 9
Fig. 3. Comparison between the theoretically (semi-analytical) predicted performance and the results of HFSS simulations of Huygens metasurfaces designedto convert the fields produced by an electric line source situated at x (cid:48) = − λ to directive radiation towards (a,d,g,j) θ = 0 ◦ , (b,e,h,k) θ = 30 ◦ , and (c,f,i,l) θ = 60 ◦ . (a)-(c) Required surface reactance X s ( z ) = (cid:61) { Z se ( z ) } (blue dashed line) and surface susceptance B s ( z ) = (cid:61) { Y sm ( z ) } (red solid line)calculated from (43). (d)-(f) Theoretically predicted (blue dashed line) and HFSS-simulated (red solid lines) normalized radiation pattern (dB scale). (g)-(i)Real part of the electric field phasor |(cid:60) { E y ( x, z ) }| as simulated by HFSS. (j)-(l) Theoretical prediction of |(cid:60) { E y ( x, z ) }| (Appendix C). metasurfaces according to the procedure described in thisSubsection (with ξ = 0 ), designated to convert the fieldsproduced by an electric line source at x = x (cid:48) = − λ to directiveradiation towards θ = 0 ◦ , θ = 30 ◦ , and θ = 60 ◦ (Fig. 3).For an HMS stretching from z = − λ to z = 5 λ (totallength L = 10 λ ) the required variation of the surface reactance X s ( z ) = (cid:61) { Z se ( z ) } and surface susceptance B s ( z ) = (cid:61) { Y sm ( z ) } with z is presented in Fig. 3(a)-(c). For clarity,only values of | X s | < η and | B s | < /η are presented.Recalling the physical meaning of the equivalent angle ofincidence (44) we can indeed observe a change of signs in thecotangent argument of (43) when the angles formed between EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 11, NOVEMBER 2014 10
TABLE IIP
ERFORMANCE P ARAMETERS OF L INE S OURCE E XCITED
HMS D
IRECTIVE R ADIATORS C ORRESPONDING TO F IG . 3 θ = 0 ◦ θ = 30 ◦ θ = 60 ◦ HFSS Theory RelativePerformance HFSS Theory RelativePerformance HFSS Theory RelativePerformanceTransmission Efficiency
33% 42% 79% 36% 43% 84% 39% 42% 93%
Half-Power Beam Width . ◦ . ◦
85% 8 . ◦ . ◦
95% 16 . ◦ ◦ Aperture Efficiency
60% 71% 85% 71% 74% 95% 57% 80% 71%
Peak Directivity . . . . . . the source and the evaluation point approaches θ , i.e. in theproximity of z/ | x (cid:48) | = tan θ . This would happen around z = 0 (Fig. 3(a)), z = 0 . λ (Fig. 3(b)), and z = 1 . λ (Fig. 3(c)), for θ = 0 ◦ , θ = 30 ◦ , and θ = 60 ◦ , respectively.Moreover, as the difference between the equivalent anglesand the transmission angle becomes larger, the effective periodof the cotangent should become shorter (43); indeed, if wefocus on the z < region of the metasurface, where theequivalent angles of incidence are mostly negative, we observethat the effective period for θ = 0 ◦ (Fig. 3(a)), is longer thanthat of θ = 30 ◦ (Fig. 3(b)), which is, in turn, longer thanthe one corresponding to θ = 60 ◦ (Fig. 3(c)). The practicalimplication of this observation is that if we desire to harnessthose rays incident upon the metasurface at angles whichdiffer significantly from the desirable transmission angle, weshould anticipate a fast variation of the corresponding surfaceimpedance and admittance, which, in turn, requires smallerdistances between the elements implementing the metasurface.We have implemented the designed HMSs in a commer-cially available finite-element solver (ANSYS HFSS) usingone hundred λ/ -long unit cells comprised of loaded wiresand loops [19], [21], as described in Appendix B. The HMSwas excited by an electric line source carrying I = 1A currentoscillating at a frequency of f = 1 . ; the simulated elec-tric field variation as a function of position, |(cid:60) { E y ( x, z ) }| ,is presented in Fig. 3(g)-(i) for the three HMSs considered.To compare these results with the theoretical predictions, wehave calculated the spectral integrals (5)-(6) in conjunctionwith (24),(26),(27), and (39)-(41), to evaluate the fields inthe region ( x, z ) ∈ ( − λ, λ ) × ( − λ, λ ) , as presentedin Fig. 3(j)-(l). The theoretical plots rely on semi-analyticalapproximations (Appendix C), which assume the HMS is ofinfinite extent to calculate the fields in region 1 ( x < )and on the virtual aperture x → + . Then, to account forthe finite HMS length when evaluating the fields in region 2,the virtual aperture window function W (0 , z ) is truncated at z = ± L/ before utilizing (27). These approximations yieldaccurate results when most of the excitation power interactswith the HMS on its finite extent ( | z | < L/ ).In addition, the steepest-descent-path method is employedto evaluate the radiated power in the far-field regions x → ±∞ using the spectral integrals (5)-(6) and based on the same as-sumptions [30]–[32] (Appendix C). This allows us to comparein Fig. 3(d)-(f) the theoretically predicted far-field radiationpatterns (dashed blue line) with the ones calculated by theHFSS simulation (solid red line); all radiation patterns are normalized to their maximum.Although some discrepancies between the simulation resultsand the theoretical predictions are observed in Fig. 3(d)-(l), itis clear that the designed HMS successfully convert the linesource fields to directive radiation toward the desirable angle.Both the beam-width and the immediate side lobe levels are ina good agreement with the semi-analytical theory. In all threecases, the simulated directivity values outside the main beam(in region 2) are at least below the peak directivity.Importantly, the theoretical calculations indicate that theslowly-varying envelope condition (9) is indeed satisfied for alltransmission angles considered, except at some points towardsthe edge of the metasurface, where the exclusion of diffractioneffects introduce a discontinuity in the fields along the z axis(not shown). Another support for the validity of the theoreticalresults is provided by the fact that the absolute values of thefields as presented in Fig. 3(g)-(i) and Fig. 3(j)-(l) in ηI /λ units, are to scale.Table II concentrates performance parameters calculatedfrom the theoretical and simulated radiation patterns of Fig.3(d)-(f). These include the transmission efficiency, i.e. the ratiobetween the power transmitted to region 2 and the total powerradiated by the source; the half-power beamwidth (HPBW),i.e. the angular difference between the half-power points; theaperture efficiency, i.e. the ratio between the HPBW of theHMS radiation and the HPBW of a uniformly excited aperturewith the same length L [17]; and the (2D) peak directivity [33].For each parameter, Table II indicates its value as calculatedfrom the simulation results and the theoretical predictions,and the respective relative performance, defined as the ratiobetween the two. In consistency with the results presented inFig. 3(d)-(l), the performance parameters also indicate thatthe power radiated into region 2 is successfully funnelledinto a directive beam, with performance comparable with thetheoretical predictions, peak directivity excepted.In view of the theoretically predicted values themselves,two comments are in place. First, we refer to the pre-dicted transmission efficiencies, which are calculated to bearound . Due to the finite length L of the HMS, only arctan ( L/ | x (cid:48) | ) /π of the line source power interacts with theHMS. If we assume minor reflections from the metasurface,and consider L/ | x (cid:48) | = 10 as in our case, this rough estimationleads to a transmission efficiency of , very close to thevalues in Table II. Hence, these values indicate that if theHMS implementation would perfectly match its design, mostof the power interacting with the metasurface is expected EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 11, NOVEMBER 2014 11 to be transmitted to region 2. In other words, the reflectioncoefficient resulting from the enforcement of local impedanceequalization, of which we have little control, should notdeteriorate significantly the HMS performance.The second comment refers to the aperture efficienciespresented in Table II, which do not exceed for alltransmission angles considered, even in the ideal theoreticalscenario. The reason for these values is the utilization of localpower conservation in the HMS design, that coerces the virtualaperture window function to follow the profile of the total(incident+reflected) fields at x → − . As the line source issituated only | x (cid:48) | = λ away from the metasurface in ourconfiguration, most of the incident power is concentrated ina small region near z = 0 . Thus, effectively, the full lengthof the HMS cannot be utilized for radiation, and, in turn, thetheoretical limit for the aperture efficiency is below .To conclude this Subsection, we note that both the resultspresented in Table II and in Fig. 3(d)-(l) indicate that the maindeviations from the theoretical predictions are the increasedreflections from the metasurface in region 1 and the incompleteelimination of the incident field in region 2; both discrepan-cies contribute to a significant difference in peak directivity(Table II). We believe that these differences originate in thefact that the unit cells implementing the HMS have not yetbeen optimized to exhibit the prescribed surface admittanceand impedance accurately over the entire required dynamicalrange. However, the optimization of the metasurface unit cellsrequires specialized treatment, including scattering elementselection [11], [12], and is outside the scope of this paper. C. Excitation by an Electric Line Source in front of a PEC
To further illustrate the versatility of our formulation, weconsider a third excitation configuration, that of an electric linesource positioned in front of a PEC, the latter is separated fromthe HMS by a distance d (Fig. 2(c)); as in Subsection III-B,the line source current is given by (cid:126)J = I δ ( x − x (cid:48) ) δ ( z ) ˆ y .We present this configuration herein for two reasons: first,to demonstrate how scenarios including multiple reflections(more generally, plane-stratified configurations) can be treatedusing the design procedure presented in Section II; second, toprovide an example for virtual aperture engineering via carefulselection of the source excitation.
1) HMS Design:
When region 1 contains not only sourcesbut also scatterers (e.g., abrupt interfaces) the field expres-sions must take into account the boundary conditions inducedby these scatterers, on top of the source condition alreadyencountered in Subsection III-B. The enforcement of theseboundary conditions gives rise to multiple-reflection termsin the spectral response of the fields [30]–[32], which inturn form a dependency between the source-related spectrum e f ( k t ) of (5)-(6) and the reflection coefficient of the HMS.In the configuration considered herein (Fig. 2(c)) the bound-ary conditions introduced by the scatterers at x ≤ x (cid:48) requirethat the tangential electric field vanishes on the PEC, i.e. E y ( − d, z ) = 0 for each z . Combining this requirement withthe source condition at x = x (cid:48) yields the following expressions for the incident and reflected fields (5)-(6) [30]–[32] E inc y ( x, z ) = kη I π ∞ (cid:90) −∞ dk t β e − jβ ( d + x < ) − e jβ ( d + x < ) e jβd − e Γ ( k t ) e − jβd e − jβx > e jk t z E ref y ( x, z ) = − kη I π ∞ (cid:90) −∞ dk t β e − jβ ( d + x < ) − e jβ ( d + x < ) e jβd − e Γ ( k t ) e − jβde Γ ( k t ) e jβx > e jk t z (45) H inc z ( x, z ) = I π ∞ (cid:90) −∞ dk t e − jβ ( d + x < ) ∓ e jβ ( d + x < ) e jβd − e Γ ( k t ) e − jβd e − jβx > e jk t z H ref z ( x, z ) = I π ∞ (cid:90) −∞ dk t ± e − jβ ( d + x < ) − e jβ ( d + x < ) e jβd − e Γ ( k t ) e − jβde Γ ( k t ) e jβx > e jk t z (46)where x < = min { x, x (cid:48) } , x > = max { x, x (cid:48) } , and the upperand lower signs in (46) should be used when x > x (cid:48) or x < x (cid:48) , respectively. As in (40), this change of signs forthe regions below and above the source provides the requireddiscontinuity in the magnetic field at x = x (cid:48) due to the electriccurrent (source condition). We emphasize once more that theformulation of (45)-(46) is valid for any reflection coefficientdependency e Γ ( k t ) , retaining this degree of freedom requiredto employ our design procedure.The fraction in the braces of the integrands (45) accountsfor the reflection from the PEC, and its numerator vanisheswhen x = x < = − d as required. Its denominator correspondsto the multiple reflections taking place between the HMS andthe PEC [31], [32]; consequently, the poles of the integrandscorrespond to guided or leaky modes of this structure [29],[30]. Moreover, although the source-related spectrum is nowdependent on the reflection coefficient, it can be readilyverified that when e Γ ( k t ) = 0 the incident field in (45)-(46) isreduced to the field produced by a line source and its image,positioned symmetrically d + x (cid:48) below the PEC (Fig. 2(c)).As in Subsubsection III-B1, we compare (5)-(6) with (45)-(46) in the source-free region x (cid:48) < x < (i.e., where x < = x (cid:48) and x > = x ) to extract the source-related spectrum. Thisresults in e f ( k t ) = e − jβ ( d + x (cid:48) ) − e jβ ( d + x (cid:48) ) e jβd − e Γ ( k t ) e − jβd . (47)A careful examination of the derivation in Section II revealsthat the dependency of the source-related spectrum in thereflection coefficient does not affect the conditions for localimpedance equalization (Table I / Step 1), as in the source-freeregion, this dependency is the same for all fields (e.g., (45)-(46)). This means that enforcing local impedance equalization(23) on the fields (45)-(46) in that region will result in the sameexpression for the reflection coefficient (24). Utilizing this, theincident and reflected fields can be completely evaluated inregion 1 ( d < x < ), and the dimensionless total field at x → − will thus be given by (25), reading e F − ( z ) = 12 π ∞ (cid:90) −∞ dk t sin [ β ( d + x (cid:48) )] jβ cos ( βd ) − k cos θ sin ( βd ) e jk t z . (48) EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 11, NOVEMBER 2014 12
As the poles of e f ( k t ) and the integrand of e F − ( z ) coincide, the pole contributions to the integral (48) indicate theguided and leaky modal fields in region 1 [29], [30]. However,as these poles are complex ( (cid:61) { k t, pole } (cid:54) = 0 ), their presence donot, in general, introduce significant difficulties to numericalevaluation of the integral. Hence, Step 1 of Table I may becompleted by evaluating (cid:12)(cid:12) e F − ( z ) (cid:12)(cid:12) and ϕ − ( z ) from (48).As in Subsubsection III-B1 we follow (26) [Table I / Step 2]and (28) [Table I / Step 3] to establish the shape of the virtualaperture and the phase compensation of the HMS, respectively,leading to the formulation of the HMS surface impedance andsurface admittance e Z se ( z ) = − j e Z (cid:20) kz (sin θ − sin θ i ( z )) + ξ (cid:21) e Y sm ( z ) = − j e Y (cid:20) kz (sin θ − sin θ i ( z )) + ξ (cid:21) , (49)where we have used again the definition of the equivalent angleof incidence (44). Executing these steps enable the assessmentof the transmitted fields as well.
2) Virtual Aperture Engineering:
As the total field at x → − is now an outcome of multiply-reflected fieldinterference, it seems that controlling the shape of the virtualaperture window function becomes an even more difficulttask. However, the introduction of the PEC reflector actuallyenhances our ability to control this function. As discussed inSubsubsection III-B2, in many cases the total field on the lowerfacet of the HMS can be approximated by the incident field.When the PEC is present, the incident field is created by aninterference between the source and its image. Therefore, bycontrolling the relative position of the source and the PEC,we may affect this interference pattern, and subsequently thevariation of the total field magnitude at x → − .As an example, we may utilize the fact that in the absenceof the HMS, the distance between the source and the PEC, ( d + x (cid:48) ) , is related to the internal angle θ int in which con-structive interference occurs in region 1 (Fig. 2(c)) via [32] ( d + x (cid:48) ) cos θ int = (2 n + 1) λ , n ∈ Z . (50)Formation of two lobes travelling towards ± θ int within region1 should, in general, broaden the effective interaction lengthof the source field with the HMS on its lower facet, which,in turn, may enhance its aperture efficiency. As shall bedemonstrated in the following Subsubsection, (50) can be usedas an initial aperture engineering step, by which the suitableHMS-PEC distance d is determined for given source position x (cid:48) and desirable constructive interference direction θ int .
3) Numerical and Semi-analytical Results:
To verify thedesign procedure for the configuration considered in thisSubsection, we have designed and simulated a Huygens meta-surface according to the prescribed procedure (with ξ = 0 ),designated to convert the fields produced by an electric linesource at x = x (cid:48) = − λ , positioned in front of a PEC at x = − d , to directive radiation towards θ = 0 ◦ (Fig. 4).Following the discussion in Subsubsection III-C2, we aimedat harnessing the PEC reflector to improve the theoreticallimit for aperture efficiency for a line source | x (cid:48) | = λ below Fig. 4. Comparison between the theoretically predicted performance andthe results of HFSS simulations of the HMS designed to convert the fieldsproduced by an electric line source situated at x (cid:48) = − λ in front of a PECat x = − d = − . λ to directive radiation towards θ = 0 ◦ . (a) Requiredsurface reactance X s ( z ) (blue dashed line) and surface susceptance B s ( z ) (red solid line) calculated from (49). (b) Theoretically predicted (blue dashedline) and HFSS-simulated (red solid lines) normalized radiation pattern (dBscale). (c) Real part of the electric field phasor |(cid:60) { E y ( x, z ) }| as simulatedby HFSS. (d) Theoretical (semi-analytical) prediction of |(cid:60) { E y ( x, z ) }| . the HMS, calculated in Subsubsection III-B3 to be for EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 11, NOVEMBER 2014 13
TABLE IIIP
ERFORMANCE P ARAMETERS OF THE
HMS D
IRECTIVE R ADIATOR E XCITED BY A L INE S OURCE IN FRONT OF A
PEC, C
ORRESPONDING TO F IG . 4HFSS Theory Relative PerformanceHalf-Power Beam Width . ◦ . ◦ Aperture Efficiency
93% 93% 100%
Peak Directivity . . θ = 0 ◦ . To that end we positioned the PEC such thatconstructive interference would take place at θ int = 60 ◦ inregion 1, yielding, utilizing (50) with n = 0 , an HMS-PECdistance of d = 1 . λ . Although other values of θ int alsobroaden the effective interaction length of the incident fieldand the HMS, we have found using the semi-analyticallyestimated fields (Appendix C) that θ int = 60 ◦ yields anoptimal result for the λ -long HMS under consideration.Fig. 4(a) presents the surface reactance and surface sus-ceptance required to implement the desirable HMS. As inSubsubsection III-B3, the design procedure output was usedto implement the HMS in ANSYS HFSS (following the sameprocedure and unit cell structure, cf. Appendix B), as well as toassess the predicted reflected and transmitted fields (followingthe semi-analytical approach of Appendix C).Fig. 4(b) compares between the normalized radiation pat-terns evaluated using the HFSS simulation (red solid line)and the semi-analytical theoeretical calculations (blue dashedline), showing a very good agreement between the theory andsimulation for the angular range θ ∈ ( − ◦ , ◦ ) . As impliedby the simulated (Fig. 4(c)) and theoretical (Fig. 4(d)) fieldplots, most of the discrepancy at large angles ( θ > ◦ )originate from the fact that the semi-analytical approximationsneglect the contribution of fields incident at the plane x = 0 outside the metasurface | z | > L/ to the radiation in region 2(Appendix C).Fig. 4(c)-(d) highlight two additional properties of the line-source/PEC configuration. First, these two subfigures indicatein a clear manner that the profile of the virtual aperture x → + indeed follows the total field magnitude at x → − .The two spots formed on the lower facet of the HMS dueto the interference between the line source and its image(Subsubsection III-C2) are clearly translated into two domi-nant beams originating from the same positions on the upperfacet. Second, as discussed briefly in Subsubsection III-C1,the expression for the total field in region 1 (48) containscomplex poles, and indeed the field plots indicate that theexamined structure supports leaky modes. This is particularlypronounced in the theoretical prediction Fig. 4(d), as the semi-analytical formulation used assumes the HMS is of infiniteextent over the z axis when evaluating the fields in region1 (Appendix C), thus facilitating the observed long-rangeguidance of moderately leaking modes.Table III summarizes the performance of the ELS/PECexcited HMS as a directive radiator. In consistency withFig. 4(b), the theoretically predicted and numerically simu-lated HPBW perfectly agree. As anticipated in Subsubsection III-C2, the aperture efficiency has indeed increased from for the ELS excited HMS (Table II) to after introductionof the PEC, due to the utilization of the interference betweenthe source and its image to broaden the incident field profile.An enhanced directivity is also recorded, where we note thatwhen the PEC is present no power is lost due to radiation toregion 1. It should be also noted that the theoretically predicteddirectivity is clearly overestimated, as is does not account forthe fraction of the source power reaching region 2 withoutinteracting with the HMS (i.e., via | z | > L/ , cf. AppendixC), which has non-negligible contribution, as the numericalsimulations reveal (Fig. 4(c)).IV. C
ONCLUSION
We have presented a detailed formulation of a designprocedure for scalar Huygens metasurface directive radiators,applicable for an arbitrary 2D source excitation. Our derivationreveals that satisfaction of two physical conditions is sufficientto guarantee that the designed HMS is passive and lossless:local power conservation and local impedance equalization.By expressing the incident, reflected and transmitted fields viatheir plane-wave spectrum, and utilizing the slowly-varyingenvelope approximation, we have shown that enforcing localimpedance equalization results in a Fresnel-like reflection forthe various spectral components. Furthermore, enforcing localpower conservation dictates that the profile of the virtualaperture forming the transmitted directive radiation follows themagnitude of the total (incident and reflected) excitation fields.At the end of the design procedure, the fields at all regionsmay be assessed semi-analytically, and the required variationof the surface impedance and surface admittance along theHMS is prescribed.We have verified our formulation using three differentsource configurations, showing good agreement between thesemi-analytical predictions and finite-element simulation re-sults executed by implementing the HMS as consecutive unitcells in ANSYS HFSS, yet to be optimized in a dedicatedfuture work. Means to control the virtual aperture via modifica-tion of the source excitation were discussed and demonstratedas well.The proposed design procedure establishes a flexible androbust foundation for the design and verification of novelantennas, allowing exploration of a vast variety of excitationforms via a single uniform formalism. Moreover, the derivationprovides insight regarding the physical requirements to achievepassive lossless Huygens metasurfaces with desirable function-ality, shedding light on previously investigated HMSs as wellas indicating possible directions for future HMS applications.A
PPENDIX AD ERIVATION OF
HMS S
URFACE R EACTANCE AND S USCEPTANCE FOR
TM-P
OLARIZED I NCIDENT F IELDS
For completeness, we provide here the final results of thederivation of the surface reactance required to implement thedual HMS, converting an arbitrary TM-polarized source fieldto directive radiation towards θ (derivable by duality [31]). EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 11, NOVEMBER 2014 14
As denoted in Subsection II-A, when the excitationfield is TM-polarized, the nonvanishing field componentsare H y ( x, z ) , E z ( x, z ) , and E x ( x, z ) ; the scalar surfaceimpedance only induces electric currents in the z direction;and the scalar surface admittance only induces magnetic cur-rents in the y direction. Analogously to (5)-(6), the tangentialmagnetic fields in the source-free region x > x (cid:48) can begenerally formulated in the spectral domain as H inc y ( x, z ) = − k I π ∞ (cid:90) −∞ dk t β m f ( k t ) e − jβx e jk t z H ref y ( x, z ) = k I π ∞ (cid:90) −∞ dk t β m Γ ( k t ) m f ( k t ) e jβx e jk t z H trans y ( x, z ) = − k I π ∞ (cid:90) −∞ dk t β m T ( k t ) e − jβx e jk t z , (51)and the respective tangential electric fields can be derived via(4); the m left superscript denotes TM-HMS related quantities.The scalar surface impedance and admittance may be writtenin analogy to (7) as m Z se ( z ) = 12 E trans z (0 , z ) + (cid:2) E inc z (0 , z ) + E ref z (0 , z ) (cid:3) H trans y (0 , z ) − (cid:2) H inc y (0 , z ) + H ref y (0 , z ) (cid:3) m Y sm ( z ) = 12 H trans y (0 , z ) + (cid:2) H inc y (0 , z ) + H ref y (0 , z ) (cid:3) E trans z (0 , z ) − [ E inc z (0 , z ) + E ref z (0 , z )] . (52)The transmitted tangential fields are defined using a virtualaperture window function m W ( x, z ) , similarly to (8), H trans y ( x, z ) = − kI m W ( x, z ) e − jkx cos θ e − jkz sin θ , (53)which we assume to satisfy the slowly-varying envelopecondition (9) as x → + (with e superscript replaced by m ).The dimensionless field quantities are defined for TM-polarized source field as m F − E ( z ) (cid:44) I kη cos θ (cid:2) E inc z (0 , z ) + E ref z (0 , z ) (cid:3) m F − H ( z ) (cid:44) − I k (cid:2) H inc y (0 , z ) + H ref y (0 , z ) (cid:3) m F + ( z ) (cid:44) m W (0 , z ) e − jkz sin θ (54)using which local impedance equalization and local powerconservation conditions retain the same formulae as in (17)and (19), with the e superscript replaced by m , likewisedefining the analogue of (20).Finally, the surface reactance for the TM case is given by m Z se ( z ) = − j m Z (cid:20) ϕ − ( z ) − ϕ + ( z )2 (cid:21) m Y sm ( z ) = − j m Y (cid:20) ϕ − ( z ) − ϕ + ( z )2 (cid:21) , (55)where m Z = 1 / m Y = η cos θ is the wave impedance of aTM-polarized plane-wave propagating in region 2 at an angleof θ with respect to the x axis (See (22)). Despite the differentwave impedance for TE and TM polarizations, the reflectioncoefficient arising from local impedance equalization remains Fig. 5. Physical configuration of a symmetrical unit cell for HMS imple-mentation. The unit cell is comprised of two identical PEC squared loopsloaded by lumped capacitors (marked in red), and two identical PEC wiresloaded with either lumped capacitors or lumped inductors (marked in blue).The magnetic dipole formed by the loaded loops is oriented parallel to the z -axis while the electric dipole formed by the loaded wires is oriented parallelto the y -axis. the same. Consequently, explicit evaluation of the scatteredfields and the phases ϕ ± ( z ) may be obtained by using (24)-(29) with the e left superscript replaced by m .A PPENDIX BI MPLEMENTATION AND S IMULATION OF THE
HMS
S IN
ANSYS HFSSThe Huygens metasurface designs investigated in Subsub-sections III-B3 and III-C3 were simulated using a commeri-cally available finite-element solver (ANSYS HFSS) to verifythe theoretical predictions (calculated using the continuousequivalent surface impedance and admittance) via comparisonto a more realistic implementation of the HMS.The simulated λ -long HMSs were implemented usingone-hundred λ/ -long unit cells, where the frequency ofthe time-harmonic excitations was f = 1 . . Fig. 5presents the physical configuration of a unit cell, along withits dimensions. Each unit cell is composed of two squarePEC loops, loaded by identical lumped capacitors, and twoPEC wires, loaded either by identical lumped capacitors orby identical lumped inductors, positioned symmetrically withrespect to the center of the cell. The magnetic dipole formedby the loaded loops is oriented parallel to the z -axis while theelectric dipole formed by the loaded wires is oriented parallelto the y -axis, indicating that the unit cell is only sensitive toTE-polarized fields (See Subsubsection II-A1). We simulate a2D environment by placing two PEC surfaces at y = ± λ/ .The surface impedance and admittance of a unit cell for agiven lumped loading is evaluated by simulating the responseof an infinite periodic array of identical unit cells to a nor-mally incident (TE-polarized) plane-wave excitation. Utilizingthe impedance matrix calculated by HFSS for the scatteringproblem, and the circuit model introduced in [21], we extractthe equivalent surface impedance and admittance of the cell.Variation of the lumped loading of the wires and loops induces,respectively, variation of the equivalent surface impedance andadmittance of the cell [12]. This facilitates the generation EEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 11, NOVEMBER 2014 15 of a lookup table, matching pairs of surface reactance andsusceptance to sets of lumped capacitors or inductors. Invokingthe principle of local periodicity, a required polarizabilityvariation such as the ones in Fig. 3(a)-(c) and Fig. 4(a) canbe sampled at a suitable unit-cell-length period ( λ/ in ourcase) and translated using the lookup table into a set of unitcells with prescribed lumped loadings [18], [19], [28].Although the electric and magnetic dipoles are orthogonal,and ideally should exhibit decoupled responses to orthogonalelectric and magnetic fields, we discovered that due to thefinite size of the structures, some coupling does exist. This isespecially pronounced when the lumped loading is such thatthe element is near resonance at the working frequency f .Thus, to enhance the accuracy of the lookup table, we havechosen to simulate the unit cell as a whole and to relate pairsof reactance/susceptance to pairs of lumped elements.A PPENDIX CS EMI - ANALYTICAL P REDICTION OF
HMS P
ERFORMANCE
To establish an effective engineering methodology based onthe design procedure introduced in Section II, it is desirable todevelop computationally efficient tools to estimate the HMSperformance before resorting to numerical simulation tools forfinal optimization. In Subsubsection III-B3 and SubsubsectionIII-C3 we have employed such tools, in the form of semi-analytical formulae, to assess the agreement between simulatedresults of detailed design and the predictions of the moreidealized theory. In this Appendix we describe the detailsregarding the evaluation of these semi-analytical formulae, aswell as the key approximations used.Formally, the theoretical derivation of Section II is validonly for infinitely long HMSs. If the HMS is finite, the prob-lem is no longer uniform (the configuration is not separable),and the spectral analysis employed herein cannot be carriedout. Nevertheless, in practice, if an efficient conversion ofthe source power to directive radiation is to be achieved, theHMS length L should be judiciously chosen such that mostof the interaction between the source power and an infinitelylong HMS takes place over a finite region | z | < L/ of the yz plane. Assuming this is indeed the case ( Assumption 1 ),we may treat the HMS as infinite over the z axis withoutintroducing significant errors with respect to the realistic finite-length implementation. This, in turn, facilitates the utilizationof (25) to evaluate the fields in region 1, disregarding anyeffects arising from the edges of the HMS, e.g. diffraction ora discontinuity in the reflection coefficient.A second assumption essential for the validity of ourderivation is that the magnitude of the total field impingingupon the HMS at x → − , including the reflection inducedby local impedance equalization, varies moderately such thatthe slowly-varying envelope condition (9) is satisfied. Utilizingthis assumption ( Assumption 2 ), we may use (26)-(27) toevaluate the fields on the upper facet of the HMS ( x → + ).However, when evaluating the fields in region 2, we would liketo account for the fact that the HMS is of finite length L , whichmay have significant effects on the radiated fields. Hence, wetruncate the virtual aperture window function W (0 , z ) at the edges of the implemented HMS; practically, the integral of(27) is executed with the infinite limits replaced by ± L/ .This is equivalent to assuming that the fields at the HMSplane x = 0 vanish in the region where the HMS is absent | z | > L/ . Assumption 1 above facilitates this approximation.These assumptions are used to evaluate the spectral contentof the reflected and transmitted fields, and consequently, viathe spectral integrals (5)-(6), the fields everywhere above andbelow the HMS (Fig. 3(j)-(l) and Fig. 3(d)).Nonetheless, when far-field evaluation is required, such asfor the assessment of the radiation pattern, straightforwardspectral integration is problematic due to the highly oscillatingnature of the radiation integrals as x, z → ±∞ . Hence, for theevaluation of far-field radiated power we apply the steepest-descent-path method, yielding closed-form asymptotic approx-imations for the spectral integrals. For the typical spectralintegrals (5)-(6) I ± ( x, z ) = ∞ (cid:90) −∞ dk t β g ( k t ) e ± jβx e jk t z (56)the asymptotic evaluation at a point ( x = r cos θ, z = r sin θ ) where kr → ∞ and θ ∈ ( − π, π ] is given by [30], [32] I ± ( r, θ ) ∼ (cid:114) π kr g ( k t = k sin α ± ) e − jkr e jπ/ (57)where α − = − θ for x > and α + = θ + π for x < ,assuming the sign attached to β in the exponent of (56)satisfies the radiation condition at the respective regions.A CKNOWLEDGMENT
A.E. gratefully acknowledges the support of The Lyon SachsPostodoctoral Fellowship Foundation as well as The Andrewand Erna Finci Viterbi Fellowship Foundation of the Technion- Israel Institute of Technology, Haifa, Israel.R
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