Effective Surface Susceptibility Models for Periodic Metafilms Within the Dipole Approximation Technique

A.I.Dimitriadis,N.V.Kantartzis and T.D.Tsiboukis

1 Introduction

Since the first experimental demonstration and subsequent exploding development of double-negative(DNG)metamaterials after the year 2000,one of the most significant challenges that occurred has been the push of artificial magnetic properties toward optical frequencies.To this end,a large number of homogenization methods has been suggested,in order to validate the effective magnetic response of such artificial structures.However,despite the multitude of successive efforts that have been published in the literature,a unified approach for the unambiguous effective-medium representation of all bulk metamaterials is yet to be developed.In the meanwhile,the 2-D equivalents of bulk metamaterials,frequently designated as metafilms,have also been heavily analyzed.These structures are typically formed by the 2-D periodic repetition of non-intersecting,properly-engineered scatterers or meta-atoms.Initially,their theoretical treatment has been,in essence,identical to that of their 3-D counterparts,thus leading to the assignment of bulk effective constitutive parametersεeffandµeff.Later,the authors of[Holloway,Dienstfrey,Kuester,O’Hara,Azad and Taylor(2009)]have proven the inconsistency of such procedures,as for realistic metafilms(when the thickness of the structure along the direction normal to the periodicity isd≪λ),the values ofεeffandµeffdepend on thicknessd.Sincedcannot be uniquely defined,due to the lack of periodicity along this direction,the corresponding bulk effective constitutive parameters do not have their usual physical meaning and cannot be considered as characteristic parameters of the structure under study.

A viable alternative for the electromagnetic characterization of metafilms is the extraction of appropriate boundary conditions,which efficiently correlate the electromagnetic fields on the two sides of the metafilm to a macroscopic average of its actual micro-structure.These macroscopic parameters are called effective surface susceptibilities and represent the surface equivalent ofεeffandµeffbulk constitutive parameters.Nonetheless,contrary to them,surface susceptibilities can be unambiguously attributed to a metafilm and hence,they constitute a sufficient set of parameters for its description.In fact,once these parameters are known or can be calculated,an effective-medium representation of the metafilm is achieved and boundary conditions are only needed to compute the reflection and transmission coefficients of the metafilm.

Such generalized boundary conditions have been first developed in[Kuester,Mohamed,Piket-May and Holloway(2003)],based on the dipole approximation of the individual meta-atoms of the metafilm.Apart from this important contribution,the authors of this work have also presented a simple procedure for the determination of the surface susceptibility matrix for arbitrary non-bianisotropic scatterers.This process,which can be regarded as the 2-D translation of the well-known Clausius-Mossotti formulas of the classical mixing theory[Sihvola(1999)],assumes only quasistatic electromagnetic interactions between meta-atoms.Two years later,analyticalexpressionsforthe reflection and transmission coefficientsofsuch metafilms have been derived in[Holloway,Mohamed,Kuester and Dienstfrey(2005)],thus completing the first general-purpose surface susceptibility model.Furthermore,the extension of this model to bianisotropic scatterers has been presented in[Belokopytov,Zhuravlev and Terekhov(2011)],in which the cross-polarized reflection and transmission coefficients of a metafilm have also been evaluated.Similar results to the latter paper have been published in[Koledintseva,Huang,Drewniak,DuBroff and Archambeault(2012)],where the more general form of the metafilm’s boundary conditions has also been introduced.

An alternative approach has been suggested in[Holloway,Dienstfrey,Kuester,O’Hara,Azad and Taylor(2009)],according to which the desired surface susceptibilities can be directly retrieved from the simulated values of the reflection and transmission coefficients of the metafilm.To this aim,the previously reported analytical expressions of the scattering coefficients[Holloway,Mohamed,Kuester and Dienstfrey(2005)]have been properly inverted.This extraction technique,which is-to some extent-similar to the well-known Nicolson-Ross-Weir retrieval algorithms for bulk metamaterials,has been proven very reliable and also applicable to realistic scatterer arrays,imprinted on a substrate material.Moreover,it has been successfully employed for the electromagnetic characterization of two parallel metafilms with shperical nanoparticles,located on the opposite sides of a dielectric substrate[Morits and Simovski(2010)].However,the main shortcoming of this algorithm is that it has been rigorously developed only for structures with non-bianisotropic particles,thus restricting its general applicability.

Another analytical method has been proposed in our latest works[Dimitriadis,Sounas,Kantartzis,Caloz and Tsiboukis(2012);Dimitriadis,Kantartzis and T-siboukis(2013)].Contrary to the other models,this technique results in a set of non-local effective susceptibilities,namely parameters which depend on the incident wavevector.Despite the fact that these parameters do not represent meaningful physical entities,characteristic of the specific structure,they have been proven very efficient in the prediction of the reflection and transmission properties of metafilms.The key difference is the availability of a large number of off-diagonal matrix components,which can flexibly model the weak spatial dispersion phenomena,usually associated with such devices.Nevertheless,this approach has been previously presented only for specific metafilms illuminated by a TE-polarized plane wave.

In this paper,we review the most important aspects of the three general surface susceptibility models mentioned above,which are exclusively founded on the dipole approximation of the constituting meta-atoms.The structure of the paper is as follows:In Section 2,we provide a brief introduction to the dipole approximation technique and the properties of the corresponding particle polarizabilities.The differences between the two existing types of polarizabilities,i.e.the quasistatic and the dynamic ones,are carefully addressed,by providing,also,the basic principles that apply to each category.In Section 3,the three prescribed models are more elaborately present,highlighting possible implementation issues and other important traits.The third approach(the“proposed"technique)is adequately generalized compared to its structure-specific form in our previous publications.Furthermore,in Section 4,the three algorithms are implemented and extensively compared for various structures with lossless and lossy particles of increased practical interest.Their outcomes are also certified via numerical simulations,acquired by means of a commercialcomputationalsuite.In the process,a setofnew analyticalformulas for the scattering coefficients of metafilms,comprising magneto-dielectric spheres and Ω-shaped bianisotropic particles,are derived and many novel physical insights on the phenomena under study are provided.Finally,in Section 5,we briefly summarize the main conclusions deduced during our investigation.Note that throughout the following analysis anejωttime dependence is assumed and suppressed.

2 Polarizabilities of electrically-small scatterers

Prior to introducing the different surface susceptibility models for the electromagnetic characterization of a metafilm,we should,first,focus on the building blocks of such periodic structures,namely the meta-atoms.The properties of the individual particles are indeed very important,since they significantly affect the behavior of the overall device.The periodicity in metafilms is not as crucial a factor as in the traditional frequency selective surfaces(FSSs),because it only affects the strength of the inter-particle interactions on the lattice.In principle,periodicity is not even necessary for the operation of a metafilm,although it is usually preferable,in order to facilitate the analysis and fabrication procedures.

For the scatterers themselves,the most critical parameters are their shape and the electric/magnetic properties of the materials from which they are made.However,these parameters are not convenient for a unified description of such periodic arrays and hence,the latter are typically modeled via the classical multipole theory[Raab and De Lange(2005)].According to this approach,charges and currents,induced on an isolated scatterer by externally-applied electromagnetic fields,can be expressed as the superposition of various polarization terms with increasing order of complexity(dipole,quadrupole,octopole etc.).The proportionality factors between these polarization moments and the externally applied electromagnetic fields are called polarizabilities and are tensors of increasing rank(second-rank for dipole polarizabilities,third-rank for quadrupole polarizabilities and so on).

In this paper,we examine only metafilm models which have been developed under the dipole approximation of the constitutive meta-atoms.The main parameters and notations of this approach are briefly introduced in the following subsections.

2.1　Dipole approximation

If a meta-atom is sufficiently smaller than the free-space wavelength of the impinging radiation(typicallyD≤λ/4,whereDis the largest dimension of the particle),its electromagnetic response to any external excitation can be modeled via the point-dipole approximation[Collin(1991)].Specifically,each scatterer of the array may be substituted by an electric dipole moment,p,and a magnetic dipole momen-t,m,which are placed on its geometrical center.According to this approximation,these dipole moments can be related to the local field acting at the center of every scatterer1This local field is the superposition of the external excitation and the sum of the scattered fields microscopically produced by every single scatterer of the array.as in[Tretyakov(2003)]

are the normalized dipole moments and local fields six-vectors,respectively.

Note that,even in this first-order approximation,the complete description of a specific scatterer requires the knowledge of 36 complex polarizabilities in(1).However,for the vast majority of particles,most of these parameters are negligibly small.Furthermore,the reciprocity theorem enforces several limitations to the polarizability tensors,since the following symmetries and anti-symmetries must apply[Serdyukov,Semchenko,Tretyakov and Sihvola(2001)]

These formulas are known in the literature as the Onsager-Casimir principle and can be easily derived from the corresponding symmetries of the dyadic Green functions[Serši´c,Tuambilangana,Kampfrath and Koenderink(2011)].As a result,the total number of independent polarizability terms may be reduced from 36 to 21 complex parameters,in the more general case.For lossless particles as well as for specific geometries,like the planar metallic particles,further simplifications of the polarizability matrix are possible.

2.2　Quasistatic and dynamic polarizabilities

We should,now,distinguish between two different types of polarizabilities that we will encounter.The first category includes the polarizabilities which are developed via the use of equivalent circuit models,like theRLCequivalent circuits of diverse split-ring resonators[Marqués,Martín and Sorolla(2008)].Despite the fact that these models can predict the location of the first particle resonance with decent accuracy,they are limited by the assumptionc0=∞(or,equallyk0=0)of lumped circuit models.This leads to inaccurate results when retardation effects significantly influence the performance of the structure,as is usually the case in various metamaterial devices.For this reason,these polarizabilities are not the most adequate choice for the study of the electrodynamic behavior of metafilms.In the rest of our work,we will refer to such polarizabilities as quasistatic,a term coined in[Serši´c,Tuambilangana,Kampfrath and Koenderink(2011)].

Conversely,to study metafilms where the retardation effects should be properly taken into account,a set of dynamic polarizabilities is required.These parameters should-by definition-match some important criteria:(1)involve the speed of light(or,equivalently,the wavenumber,k0)as a parameter,(2)satisfy the reciprocity theorem,in the form of the Onsager-Casimir principle,and(3)satisfy the energy conservation theorem.In the case of a lossless particle and to comply with(3),the polarizability tensors should be related to each other through the so-called Sipe-Kranendonk conditions[Belov,Maslovski,Simovski and Tretyakov(2003)]

Finally,we should mention two important techniques that will be systematically employed in the rest of the paper.The first one,proposed in[Serši´c,Tuambilangana,Kampfrath and Koenderink(2011)],concerns the transformation of the qua sistatic polarizabilities of a lossless scatterer into the corresponding dynamic ones.

This can be done by properly adding the radiation damping term to the quasistatic polarizability matrix,which can be mathematically described as

Such a procedure is applicable in any case,provided that the polarizability matrix can be inverted.Since,for many realistic scatterers,most of the 36 elements of[α]are zero,the latter frequently reduces to a square matrix of lower order(for example 4×4 or 3×3).In these cases,the reduced matrix needs to be invertible,as the full 6×6 one is obviously singular and cannot be inverted.

The second technique,introduced in[Yatsenko,Maslovski,Tretyakov,Prosvirnin and Zouhdi(2003)],refers to the inverse procedure,namely the determination of the quasistatic polarizabilites of a lossless particle,given its dynamic polarizability matrix.Nevertheless,contrary to the previous case,this technique can only be applied in some special cases,as it requires the solution of a system of equations,which is not always invertible.

3 Effective surface susceptibility models

with Jsand Ksthe electric and magnetic surface currents,respectively,induced on the boundary surface.For a metafilm under the dipole approximation,followed herein,these surface currents can be related to the(electric)surface polarization,Ps,and magnetization,Ms,vectors induced on its surface.Thus,letting alsod→0,(7)can be written as[Idemen(1988)]

Figure 1:Geometry of an arbitrary periodic metafilm located on the z=0 plane.

where the indextrefers to the tangential components of the surface polarizations/magnetizations or differential operators.

Next,if we define the effective surface susceptibilities as

However,note that(9)are valid only for metafilms comprising biaxially anisotropic meta-atoms.For the more general case of bianisotropic metafilms,4 surface susceptibility tensors are required,which can be defined from[Dimitriadis,Sounas,Kantartzis,Caloz and Tsiboukis(2012)]are the normalized surface polarization/magnetization and average field six-vectors,respectively.Plugging the prior expressions into(8),we finally obtain[Koledintseva,Huang,Drewniak,DuBroff and Archambeault(2012)]

3.1　Quasistatic interaction model

Consider a metafilm,formed by the periodic repetition of biaxially anisotropic meta-atoms which are described by the polarizability matrix

Following an analytical procedure for the calculation of the local field,acting on a single scatterer of the lattice,the authors of[Kuester,Mohamed,Piket-May and Holloway(2003)]have proven that the elements of the surface susceptibility matrix[χ]that describes the metafilm may be computed via

wherei=(e,m),N=(ab)-1is the number of scatterers per unit surface,andRthe radius of a circular disk,whose center is located on the scatterer where the local field is calculated.This radius depends on the periodicities of the structure and,for the special case of a square lattice(a=b),it takes the valueR=0.6956a[Collin(1991)].It is worth mentioning that,for the derivation of(15),the quasistatic approximationk0R≪1 has been made,thus justifying the name usually attributed to this technique.Moreover,the analogy of(15)to the well-known Clausius-Mossotti mixing rule[Sihvola(1999)]is evident,as the only difference lies on the values of the depolarization tensor of the circular disk(¯L=diag{1/4R,1/4R,-1/2R}),compared to the depolarization tensor of the sphere.

The extension of the above technique to the more general case of bianisotropic scatterers has been performed in[Belokopytov,Zhuravlev and Terekhov(2011)].

Selecting a similar methodology,its authors reached the matrix formula of

with the elements of[β]matrix defined as

Observe that,for the computation of surface susceptibility matrix[χ]through this quasistatic approach,it is necessary to know the quasistatic polarizability matrix[α]′of the constituting meta-atom and the lattice parameters of the metafilm.It should be stressed that,in the case of lossless metafilms,if the dynamic polarizability matrix[α]of the scatterer is utilized instead of its quasistatic counterpart,the parameters derived from(15)and(16)do not satisfy the energy conservation law,and lead to an incorrect modeling of the structure.When this pitfall is avoided,the resulting parameters of the model should comply with the locality conditions and may be treated as characteristic parameters of the metafilm,since they are independent on its excitation[Simovski and Tretyakov(2007)].In what follows,this technique will be referred to as the “K-B method”,from the initial letters of the first authors in the aforementioned publications.

3.2　S-parameter retrieval algorithm

Another approach for the calculation of the surface susceptibility matrix[χ]of a metafilm,that contains biaxially anisotropic scatterers,has been proposed in[Holloway,Dienstfrey,Kuester,O’Hara,Azad and Taylor(2009)].Essentially,it is based on the inversion of the analytical expressions for the reflection and transmission coefficients.The latter can be obtained by inserting the field expressions for the incident,reflected,and transmitted waves into the boundary conditions(10),as explained in[Holloway,Mohamed,Kuester and Dienstfrey(2005)].Then,by solving the resulting systems of equations for the perpendicular(Fig.2(a))and the parallel(Fig.2(b))polarization,we obtain

Figure 2:Incident,reflected,and transmitted waves for an arbitrary periodic metafilm.(a)Perpendicular and(b)parallel polarization.

whereθis the incidence angle on theyz-plane.If these coefficients are determined for a normal incidence(θ=0°)and for another arbitrary angle of incidenceθ,(18)and(19)can be inverted,and the surface susceptibilities read

It is mentioned that,for the computation of[χ]tangential components,only the scattering coefficients of normally incident waves are required,while for the normal components it is necessary to use the coefficients of an obliquely incident wave sitive to the numerical noise of the scattering coefficients,as later discussed.

Finally,the authors of[Morits and Simovski(2010)]have successfully applied this method in the electromagnetic characterization of a bi-layered metafilm,i.e.a structure comprising two closely-spaced arrays of magneto-dielectric spheres.Nonetheless,to the best of our knowledge,this approach has not yet been applied to metafilms with bianisotropic scatterers,since the corresponding analytical expressions for the reflection and transmission coefficients are too complicated to be inverted.In what follows,we will use the abbreviation “H-M method”,when referring to the technique described in this subsection.

3.3　Dynamic non-local approach

Recently,we have developed an efficient algorithm for the electromagnetic characterization of metafilms consisting of biaxially anisotropic[Dimitriadis,Sounas,Kantartzis,Caloz and Tsiboukis(2012)]and planarbianisotropic[Dimitriadis,Kantartzis and Tsiboukis(2013)]scatterers.This method is based on the combination of an analytical microscopic modeling approach,which accurately accounts for the dynamic dipolar interactions between the meta-atoms in the lattice,with a rigorous macroscopic averaging procedure,in order to obtain the desired surface susceptibility matrix.However,the analysis in these works has been performed only for TE-polarized incident waves and for some special cases of constituting particles,thus limiting the general applicability of the overall formulation.Here,we settle this issue by presenting generalized expressions,valid for any periodic metafilm and plane wave excitation of arbitrary polarization.

To this objective,the surface susceptibility matrix is evaluated from

where[α]is the dynamic polarizability matrix,[D]is the jump condition matrix

and[C]is the intraplanar interaction coefficient matrix,defined as

andg0(Rmn)corresponds to the scalar Green function of free space

where Rmnis the vector pointing from(m,n)to(0,0).From the above expressions,it follows that only half of the elements of[C]are non-zero,namely

which lead to the corresponding relations between the elements of[C]in(26)

Hence,the matrix formula(21)indicates that[χ]can be obtained as the superposition of three terms with distinct physical meaning:[α],representing the microscopic properties of every individual scatterer,[C],accounting for the dynamic intraplanar interactions between the meta-atoms,and[D],expressing the field discontinuities across the metafilm in relation to the surface polarization and magnetization six-vector.Note that both[C]and[D]depend on the wavevector of the incident radiation.Therefore,contrary to the previous methods,the parameters of this model are non-local and cannot be treated as meaningful physical parameters of the structure.Nevertheless,they are more flexible and very instructive for the correct prediction of the reflection and transmission properties of various metafilms.

For the very important practical case of lossless metafilms,it can be proven that[χ]is an Hermitian matrix3This property,proven in the appendix of[Dimitriadis,Sounas,Kantartzis,Caloz and Tsiboukis(2012)],is valid for any surface susceptibility model.and(21)can be simplified into

as follows from the Sipe-Kranendonk conditions of(5).

Lastly,it is interesting to stress that,for metafilms formed by planar scatterers,the jump condition matrix reduces to the 3×3 form

and,similarly,the general expression for[χ]is written

In the next section,we will verify the validity of this generalized approach(henceforth called “proposed”method),in comparison to the aforementioned techniques.

4 Numerical results

In this section,we will extensively compare the aforementioned techniques for various cases of infinite lossless and lossy metafilms.Their outcomes will also be compared with the numerical results obtained from the commercial simulation package[CST MWS™(2012)],which are considered as the reference solutions.In all simulations,a single unit-cell of the metafilm under study has been analyzed,by placing periodic boundary conditions(PBCs)at thex=±a/2 andy=±b/2 planes(see Fig.1).This approach,which is fully equivalent to the study of an infinite periodic metafilm,stems from the Floquet-Bloch theory[Tretyakov(2003)]and is generally considered as an efficient approximation for the analysis of finite periodic structures as well,provided that the latter have minimum dimensions of about 2λalong the axes of the periodicity[Bhattacharyya(2014)].

Furthermore,the appropriate excitation ports have been placed at thez=±ℓ=±3aplanes,also considered as the reference planes for the phase of the reflection and transmission coefficients.Note,also,that our computational domain is terminated by applying perfectly-matched layers(PMLs),just after the excitation ports.The distanceℓ=3ahas been selected,in order to make sure that any evanescent mode,possibly radiated by the structure,is drastically attenuated before reaching the excitation ports and that the PMLs are properly functioning.Finally,for the implementation of the H-M method,aθ=45°angle has been chosen,in all cases,for the derivation of the parameters in(20c)and(20d).

4.1　Magneto-dielectric spheres

4.1.1Lossless case

In first place,we investigate a metafilm comprising spherical magneto-dielectric meta-atoms.These particular scatterers have lately received an increasing scientific interest,since they can be successfully employed for the implementation of isotropic DNG materials[Holloway,Kuester,Baker-Jarvis and Kabos(2003);Shore and Yaghjian(2007)].Moreover,they usually exhibit lower losses within the resonance band,compared to most of the-commonly used-metallic scatterers,and may be fabricated by utilizing ferrimagnetic materials with externally controllable properties,like the yttrium-iron garnet[Holloway,Kabos,Mohamed,Kuester,Gordon,Janezic and Baker-Jarvis(2010)].Moreover,due to their canonical geometrical shape and the existence of analytical expressions for the calculation of their electric and magnetic polarizabilities,they constitute a convenient choice for testing the validity of various effective-medium theories[Alù(2011)].

So,let us presume a doubly-periodic repetition of the unit cell of Fig.3(a)along thexandydirections.Initially,we consider a rather sparse distribution of lossless scatterers,consisting of a material withεr=13.8 andµr=11.0,while the filling ratio of the unit cellisγ=r/a=0.15,forrthe radiusofthe spheres.The quasistatic polarizabilities of this particular meta-atom in free-space can be evaluated by the analytical formulas[Holloway,Mohamed,Kuester and Dienstfrey(2005)]

Figure 3:(a)Unit-cell of the metafilm under study with a=6 mm and r=0.9 mm,(b)quasistatic,and(c)dynamic polarizabilities of a magneto-dielectric sphere made up of a material with εr=13.8 and µr=11.0.

Figure 4:Surface susceptibilities for a lossless metafilm with magneto-dielectric spheres for(a),(b)the K-B method,(c),(d)the H-M method,and(e),(f)the proposed method for θ =75°.

To highlight the importance of these off-diagonal terms,we will,now,compare the efficiency of the three aforementioned models in the prediction of the reflection and transmission coefficients of the metafilm.Therefore,taking into account the form of[χ]in(34)and inserting it into(13),the reflection and transmission coefficients for the two possible polarizations are determined by

Figure 5:Comparisons of the scattering coefficients predicted from the various models.(a),(b)Magnitude and phase of the transmission coefficients T⊥and(c),(d)magnitude and phase of the reflection coefficients R‖ for θ =75°.

Next,by considering aθ=75°incidence and inserting the surface susceptibilities of Fig.4 into(35b),we acquire the magnitude(Fig.5(a))and phase(Fig.5(b))of the transmission coefficient,T⊥,around the first resonance frequency.We first note that,due to the rather large electrical length of the unit cell of the metafilm at this frequency band,the K-B method can approximate only the shape of the simulated scattering coefficients,as a result of its quasistatic approximations.Moreover,fo-cusing on the specific frequencya/λ=0.357 and the susceptibility terms of(35b),we promptly detect that both the K-B and H-M models predict non-zero value only approach,compared to the K-B and H-M methods,justifies its superior predictive performance in this particular frequency band.

4.1.2Lossy case

Let us now assume that the magneto-dielectric spheres are made up of a material with constitutive parametersεr=13.8(1-j0.002)andµr=11.0(1-j0.002),while all other geometric parameters of the metafilm remain the same as in the previous subsection.The presence of material losses is expected to lead to the occurrence of imaginary parts in the surface susceptibilities for all the models.

In this context,the parameters of the K-B method are shown in Figs 6(a)and 6(b),while those of the H-M method are presented in Figs 6(c)and 6(d)4Parameters χeyey and χmyym are not included in these figures,since they are again equal to χexex and χmxxm,respectively,as in the previous case..Evidently,apart from the occurrence of the imaginary parts,the addition of losses leads to more wide and less sharp resonances of surface susceptibilities,compared to the ity condition around thea/λ=0.356 frequency,since its imaginary part becomes positive.This implies that the H-M model is not local at this frequency range and its parameters cannot be treated as characteristic parameters of the structure.On the contrary,the surface susceptibilities of the proposed method forθ=45°are illustrated in Figs 7(a)-7(c),where only the real parts of the off-diagonal terms take positive values,as anticipated from the non-local nature of the extracted parameters.As a consequence,the latter parameters represent more accurately the physics of the particular problem.Finally,by substituting the surface susceptibilities into(35a),we obtain the magnitude of the reflection coefficientR⊥of Fig.7(d).The proposed method is in very good agreement with the CST MWS™outcomes,apart from a narrow band arounda/λ=0.3555,where a small fluctuation ofχexexleads to a subsequent deviation from the simulation results.This is the region where the H-M method also loses its accuracy,due to the positive values of Im{χezez},while the K-B method deviates from the other approaches around the resonance band.

Figure 6:Surface susceptibilities for a lossy metafilm with magneto-dielectric spheres for(a),(b)the K-B method and(c),(d)the H-M method.

Figure 7:(a),(b),(c)Surface susceptibilities for a lossy metafilm with magneto dielectric spheres for the proposed method for θ =45°,and(d)comparison of the magnitude of the reflection coefficients R‖ of the various models for θ =45°.

4.2　Microstrip Ω-shaped resonator

4.2.1Lossless case

Planar metallic scatterers are another attractive solution for the design of practical metafilms,since they can be easily fabricated via standard photolithographic techniques.Here,we will investigate metafilms consisting of the microstrip Ω-shaped resonator of Fig.8(a).This specific meta-atom has been utilized in the implementation of various realistic devices,like reciprocal microwave phase shifters[Saadoun and Engheta(1994)],DNG materials with low losses[Ran,Huangfu,Chen,Li,Zhang,Chen and Kong(2004);Lheurette,Houzet,Carbonell,Zhang,Vanbesien and Lippens(2008)],waveguide power splitters[Di Palma,Bilotti,Toscano and Vegni(2012)]and antenna radomes[Basiry,Abiri and Yahaghi(2011)].This par-ticle can be modeled via an electric dipole moment,px,and a magnetic dipole moment,mz,which are induced when it is excited from anx-directed electric field and/or az-directed magnetic field.Furthermore,electric charges can also be accumulated along they-direction,when ay-directed electric field is externally applied.However,the latter polarization is not coupled to the previous ones.To sum up,the polarizability matrix[α]of the Ω particle can be written as

Figure 8:(a)Ω-shaped resonator with dimensions:l=3.5 mm,r=1.2 mm,w=0.3 mm,and g=0.2 mm,(b)real,and(c)imaginary part of the dynamic particle polarizabilities derived via[Karamanos,Dimitriadis and Kantartzis(2012)].

where,due to the Onsager-Casimir principle,αexmz=-αmzxealso holds.

Figure 9:(a)Quasistatic susceptibilities of the lossless Ω-resonator with the dimensions of Fig.8(a).Surface susceptibilities of the(b)K-M method,(c)H-M method,and(d)proposed method for θ =75°.(e)Magnitude and(f)phase of the transmission coefficients T⊥ predicted from the various models for θ =75°.

Next,letusconcentrate on a metafilm consisting ofthe doubly periodic repetition of Ω resonators,with the dimensions provided in the caption of Fig.8(a).Assuming a square unit cell witha=b=7.5 mm,the analysis is performed for frequencies 8

In order to apply the K-B method,it is necessary to determine the quasistatic polarizabilities of the scatterer.The latter can be directly obtained via the procedure described in[Yatsenko,Maslovski,Tretyakov,Prosvirnin and Zouhdi(2003)]and are given in Fig.9(a).We detect that by “removing”the radiation losses from the dynamic polarizabilities,the resonances in the corresponding quasistatic terms become narrower and sharper.Then,the[χ]matrix of the K-B method,which has a similar form to the[α]matrix of(37),can be calculated.These susceptibilities,obtained via(16),are shown in Fig.9(b)and are similar in shape with the corresponding quasistatic polarizability terms.In contrast,the parameters of the H-M,which are depicted in Fig.9(c),lead to some very interesting conclusions.Specif ically,since this model includes only diagonal terms of the[χ]matrix,the magnetothe structure.

For a direct comparison of the methods,under study,we insert(32)into(13)and,similarly to the previous section,closed-form expressions are determined for the reflection and transmission coefficients of the two linear eigen-polarizations

4.2.2Lossy case

To examine the performance of a lossy structure with planar metallic scatterers,we simply downscale the dimensions of the unit cell and the resonators by two orders of magnitude(i.e.l=35µm,r=12µm,w=3µm,g=2µm,anda=b=75µm,with reference to Fig.8(a)).The frequency range of our study is,similarly,upscaled and becomes 0.8

At this point,it is particularly instructive to study the extracted surface susceptibilities of the H-M method,presented in Figs 11(a)and 11(b).One observes that,

Figure 10:(a)Real and(b)imaginary parts of the dynamic polarizabilities for the lossy(downscaled)Ω-resonator.Surface susceptibilities of(c),(d)the K-B method and(e),(f)the proposed method for θ =75°.

Figure 11:Surface susceptibilities of the H-M method derived from(20)(a),(b)for θ =45° and(c),(d)real and imaginary parts of χmzzm for various values of angle θ.

Figure 12:(a)Magnitude,(b)phase of the transmission coefficient T⊥,and(c)total scattered power|R⊥|2+|T⊥|2 for θ =75°,as predicted from the various models.

5 Conclusions

In this paper,we have comprehensively examined the three main general-purpose surface susceptiblity models existing in the literature,which have been developed withing the realm of the dipole approximation technique.Via exhaustive comparisons,both for lossless and lossy metafilms of magneto-dielectric spheres and microstrip Ω-shaped resonator,we have managed to trace the main assets and limitations of these methods.Specifically,it has been found that the K-B method,based on the assumption of quasistatic particle interactions,is the least accurate approach.This can be attributed to the size of the typical meta-atoms as well as to their dense packing in realistic metafilms.Hence,the interactions between the scatterers are usually strong and depend on the excitation method,while the resulting weak spatial dispersion phenomena,potentially important for the proper prediction of the metafilm’s scattering properties,are totally ignored from this approach.On the other hand,the H-M method is proven very reliable and accurate,despite some defects that frequently occur,due to the sensitivity of its retrieval formulas on the noise of its input parameters(simulated reflection and transmission coefficients).However,its extracted parameters may lose their physical meaning in some cases,even in the absence of bianisotropic effects at the particle level.Finally,the proposed non-local procedure is found to be the most accurate in all cases,but it comes with a cost of a higher number of surface susceptibilities and,thus,with a higher implementation complexity.However,this drawback is counterbalanced from the proper incorporation of the spatial dispersion phenomena of the metafilms.

Acknowledgement:This research has been co-financed by the European Union(European Social Fund-ESF)and Greek national funds through the Operational Program “Education and Lifelong Learning”of the National Strategic Reference Framework(NSRF)-Research Funding Program:Aristeia.Investing in knowledge society through the European Social Fund.

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