Average Transition Conditions for Electromagnetic Fields at a Metascreen of Vanishing Thickness
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Average Transition Conditions for ElectromagneticFields at a Metascreen of Vanishing Thickness
Edward F. Kuester,
Life Fellow, IEEE , Enbo Liu and Nick J. Krull
Abstract —Using a dipole interaction model, we derive gener-alized sheet transition conditions (GSTCs) for electromagneticfields at the surface of a metascreen consisting of an array ofarbitrarily shaped apertures in a perfectly conducting screenof zero thickness. Use of the GSTCs permits modeling ofstructures containing perforated surfaces much more rapidlythan is possible with full-wave numerical simulations. Theseconditions require that the period of the array be smaller thanabout a third of a wavelength in the surrounding media, andgeneralize many results previously presented in the literature.They are validated by comparison with results of finite-elementmodeling, and show excellent agreement when conditions of theirderivation are satisfied.
Index Terms —boundary conditions, generalized sheettransition conditions (GSTC), metamaterials, metasurfaces,metascreens
I. I
NTRODUCTION R ECENT years have seen a large body of research into thefield of electromagnetic metamaterials, and in particularinto metasurfaces, which are two-dimensional (surface) ver-sions of engineered artificial magnetodielectric materials (see,e. g., [1]-[7]). Not only do metasurfaces offer the possibilityof devices with lower loss than those based on bulk metamate-rials, but they have found application in controlling reflection,transmission and polarization of waves; as filters, absorbers,sensors and beamformers; and performing the functions offocusing and field transformation.In [8], equivalent boundary conditions were derived fora metafilm —a particular kind of metasurface consisting ofa planar array of isolated scatterers characterized by theirpolarizabilities. These equivalent boundary conditions are aspecial case of what are called generalized sheet transitionconditions or GSTCs [9]. Subsequent work has found manyapplications of this theory, summarized for example in [1].A complementary structure, to which we will give thename metascreen , consists of an array of electrically smallapertures in a planar conducting screen, as shown in Fig. 1.Sakurai [10] first obtained a form of GSTC for the caseof a mesh of wires whose radius is small compared to themesh openings. Later, Kontorovich and his colleagues [11]-[18] independently derived the same result and carried out
Manuscript received May 16, 2019.E. F. Kuester and N. J. Krull are with the Department of Electrical,Computer and Energy Engineering, University of Colorado, Boulder, CO80309 USA (email: [email protected]). Enbo Liu is with theDepartment of Mechatronics, University of Electronic Science and Technologyof China (UESTC), Chengdu, China. d d y x
PEC (cid:1) α M α E Fig. 1. Metascreen consisting of a square array of identical apertures in aplanar conducting screen. extensive developments of that theory. Other approaches to theproblem have considered the problem of plane-wave incidenceonto the metascreen, representing it as an equivalent shuntreactance, calculated with a more or less approximated mode-matching technique [19]-[35] or finite-difference method [36]-[37]. For wire-mesh screens, a Bloch-Floquet-expansion-basednumerical technique has been applied [38]-[39]. Some ofthese results are restricted to the case of a normally incidentplane wave, of identical media on both sides of the screen orto particular geometric forms of the metascreen. Of course,modern software modeling tools allow full-wave numericalsolutions of the fields to be obtained, but these can requireextensive computing time and resources, making the designof such surfaces inconvenient by comparison with the use ofanalytical models [36]-[39].In the present paper, we will derive GSTCs that relate theaverage or macroscopic fields on a metascreen of fairly generalform, restricted only by the requirements that the screen be aperfect conductor of zero thickness, and that the apertures arearranged in a square lattice whose period is small comparedto a wavelength. In [40], GSTCs for a metascreen of a fairlyarbitrary geometry were derived by the method of multiple-scale homogenization, but computation of the coefficientsappearing in the conditions obtained in this way requires thesolution of several static boundary problems (which must becarried out numerically in general). Our approach here willbe similar in some ways to that of [45]-[46] for an analogousacoustical problem, and to that of [47]-[48] for the problemof diffusion through a porous membrane. Latham [49] andCasey [50] have partially carried out such an analysis forthe electromagnetic problem, but have not obtained averagedboundary conditions for the screen. Delogne [51] has outlinedan approach that would lead to averaged boundary conditions,but presents explicit results only considering magnetic polar-
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION izabilities and neglecting the interaction of the apertures. Thesame comment applies to the work of Gordon [52]. A briefpreliminary version of the present paper was presented in [53],after which the paper [54] appeared. The latter has a certaindegree of overlap with the present paper, but our work relies ona derivation independent of Babinet’s principle and allows fordifferent media on either side of the metascreen. The methodof this paper is also extensible to a metascreen of nonzerothickness [55]. These and other differences would seem tojustify publication of this work.II. D
ERIVATION OF THE
GSTC S Consider the metascreen of Fig. 1. The screen is a perfectelectric conductor (PEC) of zero thickness, in which a periodicsquare array of apertures is cut, whose lattice constant is d .Each aperture of the array in isolation has an electric polar-izability α E and a (dyadic) magnetic polarizability α ↔ M , inaccordance with Bethe’s small aperture theory. The definitionof these polarizabilities is given in Appendix A. When in thepresence of a field, the effect of the apertures is to producean additional field approximately equal to that produced byarrays of normal electric and tangential magnetic dipoles p ± = u z p z ± and m t ± located slightly above and below (at z = ± δ ) a PEC screen with no holes (here u v denotes a unitvector in the direction v = x, y or z of a cartesian coordinatesystem). These dipole arrays are in turn approximated by con-tinuous distributions of surface polarization and magnetizationdensities: P ± Sz = N p z ± ; M ± St = N m t ± (1)where N = 1 /d is the density of apertures per unit area.The small distance δ will be allowed to approach zero later inour derivation. The resulting situation is shown in Fig. 2. Thesurface dipole densities on the bottom side of the metascreenare in opposite directions from those on the top side becauseof formulas (49)-(52). z x z = 0 z = (cid:0) z = (cid:2)(cid:3) M + P + M (cid:4) P (cid:5) StSt S z S z (cid:6) , (cid:7) (cid:8) , (cid:9) Fig. 2. Side view of a metascreen showing equivalent surface polarizationand magnetization densities.
Now, between z = 0 and z = δ , and between z = 0 and z = − δ (the shaded regions in Fig. 2), the tangential electricfield will be zero because of the PEC at z = 0 . Because ofthe jump condition on tangential E across a sheet containing P ± Sz and M ± St ([8], [56]), we have E | z = δ + × u z = jωµ M + St − ∇ t (cid:18) P + Sz ǫ (cid:19) × u z (2)and E | z = − δ − × u z = − jωµ M − St + ∇ t (cid:18) P − Sz ǫ (cid:19) × u z (3) Using (1) and (49)-(52) from Appendix A, then taking thelimit as δ → , we obtain that tangential E is continuous atthe metascreen: E | z =0 + × u z = E | z =0 − × u z ≡ E | z =0 × u z (4)We emphasize that the electric field in equation (4) is notthe actual field on the PEC between the two sheets of dipoledensities, but the fields external to these dipole sheets ( z > δ or z < − δ ), extrapolated to z = 0 . Tangential E also obeysthe jump condition E | z =0 × u z = jωµ av N α ↔ M · [ H sc t ] + z =0 − + 1 ǫ av ∇ t n N α E [ D sc z ] + z =0 − o × u z (5)where ǫ av = ǫ + ǫ µ av = 2 µ µ µ + µ (6)and as detailed in Appendix A, the short-circuit fields D sc , B sc are the fields acting on one of the apertures when thataperture is metalized. These fields are those produced bysources located on both sides of the screen, including thoseproduced by all the other apertures.We must now obtain suitable expressions for the short-circuit fields. Following the procedure used in [8], these fieldsat z = 0 ± are equal to the macroscopic field D z or H t at themetascreen, minus the fields of a disk of radius R = 2 πd P ′ m,n ( m + n ) / ≃ . d (7)cut out of the surface polarization and magnetization sheetslocated at z = ± δ respectively [the prime in (7) denotesthat the term with m = n = 0 is to be omitted from thesummation]. The situation is illustrated in Figure 3 for thearray of magnetic dipoles; a similar configuration holds forthe electric dipoles. In this procedure, the fields of all dipoles d d y x R M s Fig. 3. Field of magnetic dipole array approximated by that of a puncturedsheet of magnetization density M S . except those at the center of the exclusion circle have beenapproximated by the fields of the continuous distribution ofsurface magnetization density or surface polarization densitywith the disks removed. The fields of these disks are computedwith the disks acting in the presence of the PEC screen at z = 0 that has no apertures . For example, the disk at z = δ acts above the PEC and produces a field that is the same asif the PEC were removed and its effect replaced with that ofan image disk radiating together with the original disk in an RANSITION CONDITIONS FOR METASCREEN 3 unbounded region with material parameters µ and ǫ . In otherwords, the field will be that produced by M + St and P + Sz ,calculated as in section III of [8]. From (47)-(49) of [8], we have D disk z (cid:12)(cid:12) z = δ − = 2 P + Sz F ( R ); H disk t (cid:12)(cid:12) z = δ − = 2 M + St G ( R ) (8)where F ( R ) = 12 R e − jk R (1 + jk R )= 12 R (cid:2) O ( k R ) (cid:3) ( k R ≪ (9) G ( R ) = − R (cid:2) e − jk R (1 − jk R ) + 2 jk R (cid:3) = − R (cid:2) O ( k R ) (cid:3) ( k R ≪ (10)where k = ω √ µ ǫ . In a similar way, D disk z (cid:12)(cid:12) z = − δ + = 2 P − Sz F ( R ); H disk t (cid:12)(cid:12) z = − δ + = 2 M − St G ( R ) (11)where F and G are obtained from F and G by replacingthe subscript 1 with 2. Therefore, letting δ − → + and − δ + → − , and assuming that k , R ≪ , we have finally [ D sc z ] + z =0 − = [ D z ] + z =0 − − D disk z (cid:12)(cid:12) z =0 + + D disk z (cid:12)(cid:12) z =0 − = [ D z ] + z =0 − − R (cid:0) P + Sz − P − Sz (cid:1) (12)and [ H sc t ] + z =0 − = [ H t ] + z =0 − − H disk t (cid:12)(cid:12) z =0 + + H disk t (cid:12)(cid:12) z =0 − = [ H t ] + z =0 − + 12 R (cid:0) M + St − M − St (cid:1) (13)Now, from (1), (49) and (51) we have P + Sz − P − Sz = − N α E [ D sc z ] + z =0 − (14)which gives [ D sc z ] + z =0 − = [ D z ] + z =0 − + 2 NR α E [ D sc z ] + z =0 − (15)or [ D sc z ] + z =0 − = 11 − NR α E [ D z ] + z =0 − (16)Similarly, from (1), (50) and (52) we have M + St − M − St = 2 N α ↔ M · [ H sc t ] + z =0 − (17)so that [ H sc t ] + z =0 − = [ H t ] + z =0 − + NR α ↔ M · [ H sc t ] + z =0 − (18)or [ H sc t ] + z =0 − = (cid:20) ↔ t − NR α ↔ M (cid:21) − · [ H t ] + z =0 − (19) Note that eqn. (48) of [8] contains a typographical error; the last term onthe right side should be + j ω P sy z ) but since in the present paper P sx = P sy = 0 , this error has no effect here. where ↔ t = u x u x + u y u y is the tangential identity dyadic. Inthe case where α ↔ M is diagonal, (19) reduces to [ H sc x ] + z =0 − = 11 − NR α xxM [ H x ] + z =0 − (20)and (cid:2) H sc y (cid:3) + z =0 − = 11 − NR α yyM [ H y ] + z =0 − (21)Substituting (16) and (19) into (5), we obtain as our GSTCthat E t is continuous at z = 0 , and: E | z =0 × u z = jωµ av π ↔ tMS · [ H t ] + z =0 − − ǫ av ∇ t n π zzES [ D z ] + z =0 − o × u z (22)where we have defined electric and magnetic surface porosities of the metascreen as π zzES = − N α E − NR α E (23)and π ↔ tMS = N α ↔ M · (cid:20) ↔ t − NR α ↔ M (cid:21) − (24)If the magnetic polarizability dyadic is diagonal, we cansimplify (24) to π ↔ tMS = u x u x N α xxM − NR α xxM + u y u y N α yyM − NR α yyM (25)The minus sign in (23) is chosen to achieve a certain duality inthe form of (22) when compared to the GSTCs of a metafilm.A consequence of this sign choice is that π zzES will be negative.The fact that π ↔ tMS has only tangential components, while π zzES has only normal components is a consequence of the fact thattangential electric dipoles and normal magnetic dipoles placedon a perfectly conducting surface produce no external fields.Equation (22) has the same form as eqn. (3) of [1], andconforms with that of [57] for a wire grating. It can be shownthat (4) and (22) are also special cases of the GSTCs derivedin [40] if the screen thickness is set equal to zero. It mustbe emphasized that this form of the GSTCs applicable to ametascreen differs in a fundamental way from those that havebeen derived for a metafilm in [8], and used extensively sincethen (see, e. g., [41]-[44]). A metafilm consists of an array ofunconnected scatterers that can be modeled as polarization andmagnetization sheets that cause discontinuities in both tangen-tial E and tangential H that are proportional to those surfacepolarizations. Equation (22) on the other hand expresses thetangential electric field itself in terms of the jumps of themacroscopic fields (along with the derivative of one of them),and is not expressible in the form of a GSTC for a metafilm.Thus, even though our derivation of (22) has been basedon the dipole-interaction model (analogous to the Clausius-Mossotti-Lorentz-Lorenz model of dielectric permittivity), wesee that (22) will hold even without that restriction, but that ex-pressions (23)-(25) for the surface porosities will no longer betrue in general. Indeed, (22) has the same form as the boundarycondition obtained by Sakurai [10], though his results applyto the case of a mesh of thin wires for which (23)-(25) cannotbe expected to hold. We obtain expressions for π zzES and π ↔ tMS IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION for some specific metascreen geometries in Appendix B, validin both the dipole-interaction approximation, as well as in thelimit of a thin-wire mesh.It should be emphasized that we should not expect theGSTC (22) to be accurate if the lattice constant d is toolarge. Not only have we approximated the interaction betweenapertures by invoking the quasistatic approximations (9)-(10),but the very form of the GSTC itself precludes the presence ofpropagating higher-order Bloch-Floquet modes on the lattice,which means that we should restrict the lattice constant to beless than half a wavelength.We may convert our GSTC into a somewhat different formby defining the surface current density J S = u z × [ H t ] + z =0 − (26)and using the result D z = − jω ∇ t · ( u z × H t ) (27)that follows from Amp`ere’s law, so that (22) can be expressedas E t | z =0 = j X ↔ ms · J S − j ∇ t (cid:20) π zzES ωǫ av ∇ t · J S (cid:21) (28)where X ↔ ms = ωµ av ( u x u x π yyMS (29) − u x u y π yxMS − u y u x π xyMS + u y u y π xxMS ) is the dyadic surface reactance of the metascreen. Equation(28) has the form of the boundary condition obtained byKontorovich and his colleagues [11]-[18] for a thin-wire mesh,again with different expressions for the surface porosities than(23)-(25).A boundary condition on the normal component of B canbe obtained by taking the tangential divergence of E × u z andusing a vector identity together with Faraday’s law: ∇ t · ( E × u z ) = u z · ∇ t × E = − jωB z (30)From this, we see that B z must be continuous at themetascreen, while applying the tangential divergence to bothsides of (22) and employing a further vector identity gives: u z · ∇ t × E | z =0 = jωµ av ∇ t · n π ↔ tMS · [ H t ] + z =0 − o (31)Applying (30) to (31) and dividing the result by jω if thefrequency is not zero, we obtain B z | z =0 = − µ av ∇ t · n π ↔ tMS · [ H t ] + z =0 − o (32)which has the same form as eqn. (12) of [58].To summarize the main results of this paper, themacroscopic electromagnetic field at a perfectly conductingmetascreen of zero thickness must obey the following condi-tions: •
1. The tangential components of E and the normalcomponent of B are continuous across the metascreen. •
2. The field must obey (22) and (32) at the metascreen. III. T HE S TATIC L IMIT
In the electrostatic limit, we let ω → in (22) to obtain E | z =0 × u z = − ǫ av ∇ t n π zzES [ D z ] + z =0 − o × u z (33)A static electric field can be expressed in terms of a scalarpotential, E = −∇ Φ , and the condition that tangential E must be continuous at z = 0 means that Φ( x, y, z = 0 + ) − Φ( x, y, z = 0 − ) must be a constant, which we can chooseto be zero with no loss of generality so that Φ is continuousat z = 0 , and equal to a function Φ ( x, y ) there. Providedthat Φ and D z are not independent of x and y , we have theelectrostatic form of the GSTC: Φ = π zzES ǫ av [ D z ] + z =0 − (34)where an arbitrary additive constant in the potential has beenchosen to give the indicated value at the metascreen. If Φ and D z are constant, equation (33) simply states that and nothing further can be deduced from it, but if we take thelimit of (34) as Φ approaches a constant function, we canregard it as applying to this case as well. Equation (34) hasthe same form as equation (25) of [58].Suppose now that we place the metascreen between twoconducting plates at z = d and z = − d as shown in Fig. 4.In < z < d , the electric field is z x z = 0 z = d z = Fig. 4. Metascreen between two conducting plates. E z = E = 1 d (Φ − Φ ) (35)where Φ is the potential at the upper plate, while in − d
IV. E
QUIVALENT C IRCUIT
Some researchers prefer to describe metasurfaces usingsurface impedances (see, e. g., [60]). A simple impedanceequivalent circuit for the metascreen of zero thickness can beobtained in certain special cases. Suppose that the field hasno variation in the y -direction ( ∂/∂y ≡ ) and that all fieldsvary with x as e − jk x x . Then, as is well known, the field canbe written as the superposition of a TE part (consisting ofthe field components E y , H x and H z only) and a TM part(consisting of the field components H y , E x and E z only). Ifwe suppose in addition that the magnetic porosity dyadic isdiagonal: π ↔ tMS = u x u x π xxMS + u y u y π yyMS , the metascreen willproduce no conversion between these two polarizations, andthey may be modeled independently of each other.For the TE field, let E y → V and H x → − I . Since E y iscontinuous at z = 0 , the GSTC (22) can be interpreted as ashunt reactance X TE at z = 0 , where X TE = ωµ av π xxMS (39)Likewise, for the TM field let E x → V and H y → I . ByAmp`ere’s law and the assumptions above about the x - and y -dependences of the field, we have E z = − ( k x /ωǫ ) H y . Thusthe GSTC (22) in this case is equivalent to a shunt reactanceat z = 0 of X TM = ωµ av π yyMS + k x π zzES ωǫ av (40)We should note that jX TE and jX TM are often called thetransfer impedance when describing braided shields of, forinstance, coaxial cables (see [61]-[63] and [51]).V. P LANE W AVE R EFLECTION AND T RANSMISSION
In this section, we will apply the GSTCs obtained aboveto the determination of the reflection and transmission coef-ficients of a plane wave incident on a metascreen. As in theprevious section, the magnetic porosity dyadic will be assumedto be diagonal. θ z x metascreen @ z =0 ε , μ ε , μ (cid:10) θ (cid:11) θ Fig. 5. Plane wave incident at a metascreen.
If a TE (perpendicular) polarized plane wave is incident atan angle θ to the z -axis as shown in Figure 5, the electricfield E = u y E y is given by E y = e − jk x sin θ (cid:2) e − jk z cos θ + Γ TE e jk z cos θ (cid:3) ( z < ) = e − jk x sin θ T TE e − jk z cos θ ( z > ) (41) where k , = ω √ µ , ǫ , , Γ TE is the reflection coefficient, T TE is the transmission coefficient, and the transmitted angle θ is related to the incident angle by Snell’s law: k sin θ = k sin θ = k x (42)The magnetic field is obtained from Faraday’s law ∇ × E = − jωµ H . Enforcing continuity of E y and the GSTC (22) at z = 0 in the usual way leads to the following formulas for thereflection and transmission coefficients: Γ TE = − − jX TE (cid:16) cos θ ζ − cos θ ζ (cid:17) jX TE (cid:16) cos θ ζ + cos θ ζ (cid:17) (43)and T TE = 2 jX TE cos θ ζ jX TE (cid:16) cos θ ζ + cos θ ζ (cid:17) (44)where X TE is given by (39) and ζ , = p µ , /ǫ , are thewave impedances of the upper and lower half-spaces. Thisresult could of course have also been obtained by using theequivalent shunt reactance (39) connected across the junc-tion of two transmission lines with characteristic impedances ζ / cos θ and ζ / cos θ . It will be readily observed that when π xxMS → , we obtain Γ TE = − and T TE = 0 consistent withan unperforated PEC screen, while for π xxMS → ∞ (meaningthat the metalization is removed), we retrieve the Fresnelcoefficients for perpendicular polarization.The reflection and transmission coefficients for a TM (par-allel) polarized incident wave are derived in a similar manner,with somewhat more complicated expressions arising due tothe presence of a normal component of the electric field. Weobtain: Γ TM = − − jX TM (cid:16) ζ cos θ − ζ cos θ (cid:17) jX TM (cid:16) ζ cos θ + ζ cos θ (cid:17) (45)and T TM = jX TM ζ cos θ jX TM (cid:16) ζ cos θ + ζ cos θ (cid:17) (46)where X TM given by (40) with k x is given by (42). Again,this result could have been obtained by using the equivalentshunt reactance (40) connected across the junction of twotransmission lines, this time with characteristic impedances ζ cos θ and ζ cos θ . We obtain the appropriate limits ifwe allow the porosities to approach zero or infinity, with theexception of the special angle of incidence θ = arcsin s − π yyMS π zzES µ av ǫ av µ ǫ ! (47)for which case we have Γ TM = − .Equations (43)-(46) agree with various results previouslygiven in the literature if certain approximated forms of X TE , TM are used—for normal incidence at a screen in anonmagnetic material interface [29], [30]; and for oblique in-cidence at a wire mesh in free space [11], [15]. The expressiongiven in [52] for the normal-incidence transmission coefficient IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION differs from our result by a factor of 2 in the magnetic surfaceporosity; we believe our formula to be correct, as validated bythe numerical results in the next section. The case of obliqueincidence at a wire mesh at a height h above a material half-space has been treated in [14], [17], [18, sect. 3.3], but theseresults cannot be reduced to our situation by letting h → ,because the near-field interaction of the mesh with the half-space is not accounted for in these works.VI. S OME N UMERICAL R ESULTS
Aside from checking limiting cases and special cases in-vestigated by other authors, the accuracy of (43)-(46) using(39) and (40) can be assessed by comparing them to theresults of full-wave numerical simulation. Our model will bea metascreen with square apertures as shown in Fig. 6. In d a w Fig. 6. Metascreen with square apertures.
Appendix B, we have obtained the uniformly valid (for anyratio of a/d ) expressions (71) and (72) for the electric andmagnetic surface porosities of this structure. These will beused in (39)-(40) and (43)-(46) to obtain GSTC-based resultsfor reflection and transmission.In Fig. 7, we present a comparison of the predictions ofthe present paper to finite-element simulations of normal-incidence plane wave reflection and transmission coefficientsfor the case when the medium on each side of the screenis free space. In this case, Γ TE = Γ TM = S and T TE = T TM = S . The finite-element simulations were carried outin ANSYS HFSS, using pairs of PEC and PMC side walls atthe boundaries of the period cell to force the normally incidentplane waves at the metascreen. We have chosen the latticeconstant to be d = 20 mm, which is equal to a half wavelengthat f λ/ = 7 . GHz. We can see that the magnitudes andphases of the transmission coefficient show good agreementbetween the GSTC prediction and numerical results up to f λ/ ,and the phases of the reflection and transmission coefficientsare quite accurate well above that frequency. Of course, atnormal incidence, the higher-order Floquet mode that beginsto propagate above f λ/ is not excited due to symmetry, andit is to be expected that good agreement will be obtaineduntil we near the next Floquet-mode threshold at f λ/ = 15 GHz. The finite-element solution was carried out with asfine a mesh as reasonably possible on a PC with 8 GB ofmemory; this typically resulted in more than 300,000 first-order elements, and small (less than 0.05%) variations incomputed S -parameters as the mesh was refined to its finalsize, so high accuracy can be attributed to these results, f , GHzGSTC HFSS | S | : a = 8 mm| S | : a = 18 mm| S | : a = 8 mm| S | : a = 18 mm | S | , d B (a) f , GHz GSTC HFSS S : a = 8 mm S : a = 18 mm S : a = 8 mm S : a = 18 mm S , r a d i a n s (b) Fig. 7. Normal incidence reflection and transmission coefficients from themetascreen of Fig. 6 with d = 20 mm, ǫ = ǫ = ǫ and µ = µ = µ :(a) magnitude, (b) phase. although when | S | or | S | becomes numerically small,the small discretization errors in the finite-element simulationseem to magnify the discrepancies observed on the dB scalein Figure 7(a), especially as the frequency begins to approach15 GHz.For oblique incidence, Floquet ports were used in the HFSSsimulations. In Fig. 8, we show comparisons between thetransmission coefficients obtained numerically and those basedon the GSTCs for the metascreen of Fig. 6, with variouspolarizations, angles of incidence and material constants ofthe second medium. Results are displayed as S , where S = T (TEorTM) s ζ cos θ ζ cos θ (48)At oblique incidence, the onset of the next higher-order Flo- RANSITION CONDITIONS FOR METASCREEN 7 f , (cid:12)(cid:13)(cid:14) G(cid:15)(cid:16)(cid:17) (cid:18)
FSS TE: (cid:19) (cid:20)(cid:21)(cid:22) (cid:23) (cid:24) = (cid:25) | S (cid:26)(cid:27)(cid:28)(cid:29) TE: (cid:30) (cid:31) !" = 4 & T’( ) * +-. / 0 = 4
345 6 = 4 = (a) >?@ ABC DEF HIJ f K LMN
OPQR S
FSS TE: U V WXY Z [ = \ TE: ] ^ _‘a b c = 4 d efg h i jkl m n = 4 o pqr s t uvw x y = 4 z S , r a d i a n s (b) Fig. 8. Oblique incidence transmission coefficients from the metascreen ofFig. 6 with a = 18 mm, d = 20 mm, ǫ = ǫ and µ = µ = µ : (a)magnitude, (b) phase. quet mode (also known as the grating frequency or Rayleighfrequency [64]) occurs at f R = cd (cid:0) √ ǫ r + sin θ (cid:1) where ǫ r = ǫ /ǫ . For θ = 45 ◦ and ǫ = ǫ = ǫ , we have f R = 8 . GHz; for θ = 45 ◦ and ǫ = ǫ = 4 ǫ , we have f R = 5 . GHz; and for θ = 75 ◦ and ǫ = ǫ = 4 ǫ , wehave f R = 5 . GHz. It is observed that for those cases theagreement is very good below f R / , but as f R is approached,significant discrepancies arise (the phase especially deviateswildly above about . f R ). Note that in the oblique incidencesimulations, the finite-element solution mesh had to be muchsmaller, typically around 20,000 or 30,000 first-order elements,and larger variations in computed S -parameters (on the orderof a few tenths of a percent) could often be observed at thefinal mesh refinement, so accuracy better than this for thefinite-element simulations should not be assumed.VII. C ONCLUSION
In this paper, we have derived GSTCs for a metascreenconsisting of a square array of arbitrarily shaped apertures ina perfectly conducting planar screen of vanishing thickness.From these, equivalent shunt reactances have also been ob-tained. These conditions contain as special cases many suchresults previously presented in the literature. The boundaryconditions contain parameters known as surface porosities thatdepend only on the shape and spacing of the apertures, and wehave presented some formulas for these porosities for severalmetascreen geometries. The GSTCs enable us easily to deriveformulas for reflection and transmission coefficients of planewaves at a metascreen, and their accuracy has been demon-strated by comparison with results of full-wave simulations.We claim that the GSTC model derived in this paper (andextended to metascreens of nonzero thickness in a separatepaper [55]) holds for apertures of arbitrary shape, providedthe general conditions set forth in this paper are satisfied.Validation has been carried out for an array of asymmetricapertures in [65], where retrieval methods for the surfaceparameters of the GSTCs are also derived..This work could readily be extended to other array geome-tries such as rectangular or hexagonal, and it seems likelythat frequency-dependent surface porosities could be foundthat would provide more accuracy when the lattice constantbecomes comparable to or larger than half a wavelength, asseen for example in [28], [32] and [66]. These are subjects forfuture research. A
PPENDIX AD EFINITION OF A PERTURE P OLARIZABILITIES
Since various authors have defined the polarizabilities of anaperture in different ways, we present here the definition tobe used in this paper, modified slightly from the presentationin [67]. Let a PEC screen of zero thickness lie in the plane z = 0 , a medium with material parameters ǫ , µ occupyingthe region z > and one with ǫ , µ occupying the region z < . With no aperture in the screen, a set of sources locatedon both sides of the screen is said to produce the short-circuit fields E sc , H sc (and the corresponding D sc , B sc ). When anelectrically small aperture is cut into the screen, the total fieldis equal to the short-circuit field plus an additional field E ap , H ap due to the aperture. Sufficiently far from the aperture,this field is equal to that produced by a set of electric andmagnetic dipoles, acting in the presence of the screen, with IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION the aperture closed off (metalized) . The dipoles acting in theupper half space z > are located at the center of the apertureat z = 0 + and have the values p + = − u z ǫ ǫ + ǫ α E [ D sc z ] + z =0 − , (49) m + = 2 µ µ + µ α ↔ M · [ H sc ] + z =0 − (50)while the dipoles acting in the lower half space z < arelocated at the center of the aperture at z = 0 − and have thevalues p − = u z ǫ ǫ + ǫ α E [ D sc z ] + z =0 − , (51) m − = − µ µ + µ α ↔ M · [ H sc ] + z =0 − (52)These equations define the polarizabilities α E and α ↔ M ; themagnetic polarizability dyadic has only components in thetangential ( x and y ) directions. Note that the polarizabilitydefinitions used here have excluded factors dependent on thematerial properties on either side of the screen, in contrast, forexample, with the definition used in [68].A PPENDIX BA NALYTICAL E XPRESSIONS FOR THE S URFACE P OROSITIES
All metascreens considered in this appendix have sufficientsymmetry that π xyMS = π yxMS = 0 and that π xxMS = π yyMS ≡ π tMS . A. Small aperture limit
When the size of the aperture is significantly smaller thanthe lattice constant d , we may use (23) and (24) to obtainanalytical expressions for the surface porosities in certaincases. For a circular aperture of radius r , it is well knownthat α E = 23 r ; α ↔ M = 43 r ( u x u x + u y u y ) (53)Thus, π zzES = − r d − r Rd (54) π tMS = 4 r d − r Rd (55)valid if r ≪ d . By duality, equations (54)-(55) are containedimplicitly in the results of Eggimann and Collin [69]-[70];formula (54) has been given in [59]. If the denominator in (55)is eliminated, we retrieve a result obtained in, e. g., [71], [62]and [51], for which interaction between neighboring aperturesis neglected.For square apertures of side a , Fabrikant [72]-[73] has giventhe accurate analytical approximations α E = 16 √ a ; α ↔ M = 29 ln(1 + √ a ( u x u x + u y u y ) (56) and therefore, the metascreen shown in Fig. 6 has the surfaceporosities π zzES = − a √ d − a √ Rd (57) π tMS = 2 a √ d − a √ Rd (58)valid for a ≪ d . Fabrikant has also given expressions forthe polarizabilities of a number of other shapes, from whichfurther expressions for surface porosities may be obtained(e. g., for an array of cross-shaped apertures [74]). B. Larger apertures
Equations (23) and (24) can only be expected to be validwhen multipole interactions of order higher than dipoles canbe neglected—in other words, when d is significantly largerthan the dimensions of the apertures. There are some resultsin the literature that apply when this condition does not hold.For square apertures whose side length is nearly equal to thelattice constant ( w = d − a ≪ d ), the results of Sakurai [10],and of Kontorovich and his colleagues [11]-[18], give: π zzES = − d π ln 2 dπw (59) π tMS = d π ln 2 dπw (60)Related results for the surface electric porosity were obtainedearlier in [75]-[78] in connection with studies of grids invacuum tubes.For an array of circular apertures whose diameter can bea larger fraction of the lattice constant, a different formulathan (55) for the magnetic porosity can be inferred from acomparison of the formula for normal-incidence transmissioncoefficient given in [27, equation (11)] to our formulas (44)and (39), in which we put k = k = k and ζ = ζ = ζ .In the limit when k d ≪ , this formula becomes π tMS = d π J ′ ( πr d )1 − (cid:16) πr j ′ d (cid:17) + √ J ′ ( πr d √ − (cid:16) πr j ′ d (cid:17) (61)where J is the Bessel function of order 1 and j ′ = 1 . . . . is the smallest positive root of J ′ . The formula in [27] isasserted to be accurate only for . d < r < . d , and indeedit does not give π tMS → as r → as would be expectedon physical grounds. Several semi-empirical formulas for anequivalent sheet impedance have been derived by Ramaccia et al. [7, sect. 4.4] (see also [79]-[80]) for the case whenthe diameter r is nearly equal to the lattice constant ( w = d − r ≪ d ). By comparing the resulting formulas for thereflection coefficient of a normally incident plane wave, wecan infer from [79] the following expression for the magneticsurface porosity: π tMS = d π ln sec π r d (62) RANSITION CONDITIONS FOR METASCREEN 9
Values for π tMS can be inferred from full-wave simulationresults for the normal incidence transmission coefficient in thecase when ǫ = ǫ = ǫ and µ = µ = µ , using (39) and(44) to obtain π tMS = T TE jk (1 − T TE ) as ω → (63)A comparison of formulas (55), (61) and (62) with results from(63) is given in Figure 9, and shows, perhaps surprisingly, that(55) is quite accurate over the entire range ≤ r /d ≤ . ,despite being based on a small-hole approximation. In fact, r / d (55)(62)(61)HFSS { mst d Fig. 9. Comparison of three formulas for the magnetic surface porosity of asquare array of circular holes. we see that (55) is the best of the three expressions, losingaccuracy only when r → . d , in which case we shouldexpect a percolation threshold π tMS → ∞ on physical groundsas the boundary condition changes from the form (22) to thatfor a metafilm (the first of equations (1) in [1]). No suchbehavior is exhibited by any of the three formulas. We havenot found any closed-form analytical approximations for π zzES in the literature other than (54). C. Uniform expressions for square lattice of square apertures
Following an idea of Grosser and Schulz [59] (see also[81], eqn. (60)), we can construct interpolated formulas for themetascreen of Fig. 6 that are uniformly valid for any value of a/d . We propose expressions of the form π zzES = − f (cid:16) ad (cid:17) d π ln sec πa d (64) π tMS = f (cid:16) ad (cid:17) d π ln sec πa d (65)where f and f are functions to be determined. If f , → as a/d → , then (64) and (65) will approach the limits givenin (59)-(60) above, because sec πa d = 1sin πw d ≃ dπw (66) Expression (65) with f ≡ has been given in [82], and is the low-frequency limit of the result obtained in [83]. if w/d ≪ . We now want to choose f , in forms as simpleas possible so that the limits in (57) and (58) are achievedwhen a/d ≪ . Put f ( x ) = C x + (1 − C ) x (67)where x = a/d and C is a constant to be determined. Thisfunction obeys f (1) = 1 as required, while f ( x ) ≃ C x for x ≪ . Since ln sec πx ≃ π x for x ≪ (68)we have that π zzES d ≃ − πC x for x ≪ (69)and equating this to x / √ from (57), we get C = 8 √ π = 1 . . . . (70)and thus π zzES d = − ln sec πa d π (cid:20) C ad + (1 − C ) a d (cid:21) (71)is an approximation for π zzES valid uniformly for < a < d .In a similar manner, π tMS d = ln sec πa d π C ad + (1 − C ) a d + sin (cid:16) π a d (cid:17) (72)is a uniform approximation for the magnetic porosity, where C = 329 π ln(1 + √
2) = 1 . . . . (73)We included the extra term sin (cid:16) π a d (cid:17) / in order to improvethe accuracy of (72), which was verified against full-wave nu-merical simulation results using (63). Both (71) and (72) havebeen compared with results of full-wave numerical simulations(Figs. 7 and 8 are examples of this) and shown to be accurateto within a few percent for any value of a/d between 0 and 1.Note that by complementarity, the result of [81] correspondsto the choice f ( x ) = sin ( πx/ in (65), which was foundto give discrepancies with full-wave simulation results about3 times larger than does (72), so the latter is to be preferred.A CKNOWLEDGMENT
The authors are grateful to Dr. Chris Holloway of theNational Institute of Standards and Technology for a numberof useful discussions about this paper.R
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