IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 10, OCTOBER 2003 2641

Averaged Transition Conditions for ElectromagneticFields at a Metafilm

Edward F. Kuester, Fellow, IEEE,, Mohamed A. Mohamed, Melinda Piket-May, andChristopher L. Holloway, Member, IEEE

Abstract—This paper derives generalized sheet transition con-ditions (GSTCs) for the average electromagnetic fields across asurface distribution of electrically small scatterers characterizedby electric and magnetic polarization densities. We call such anarrangement of scatterers ametafilm—the two-dimensional (2-D)equivalent of a metamaterial. The derivation is based on a replace-ment of the discrete distribution of scatterers by a continuous one,resulting in a continuous distribution of electric and magnetic po-larization densities in the surface. This is done in a manner analo-gous to the Clausius–Mossotti–Lorenz–Lorentz procedure for de-termining the dielectric constant of a volume distribution of smallscatterers. The result contains as special cases many particularones found throughout the literature. The GSTCs are expected tohave wide application to the design and analysis of antennas, re-flectors, and other devices where controllable scatterers are usedto form a “smart” surface.

Index Terms—Metafilm, metamaterial, polarizability den-sity, smart controllable surface, transition boundary condition(GSTC).

I. INTRODUCTION

I N THIS PAPER, we develop an equivalent transition(boundary) condition for the specular interaction of electro-

magnetic waves with a surface of electrically small scatterers.These scatterers are assumed to be characterized completely bytheir electric and magnetic polarizabilities and their density ofdistribution in the surface. Such a surface constitutes a sort oftwo-dimensional (2-D) metamaterial that we will call ametafilm,which, when properly designed, can have certain desired re-flection and transmission properties (e.g., total reflection ortotal transmission). One of the key advantages of the transitioncondition obtained in this paper is that it allows one to relatethe functional dependence of the polarizability densities tothe reflection and transmission properties of the surface. Thiscould in principle enable us to realize controllable surfaces, ablefor instance to switch electronically between reflecting andtransmitting states.

Historically, the problem of the interaction between electro-magnetic waves and a metafilm has been extensively researchedin many different contexts, but always with some specific restric-tions limiting the generality of the result. In this paper, we will

Manuscript received September 30, 2002; revised February 4, 2003.Publication of the U.S. Government, not subject to U.S. copyright.E. F. Kuester, M. A. Mohamed, and M. Piket-May are with the Department

of Electrical and Computer Engineering, University of Colorado, Boulder, CO80309 USA (e-mail: kuester@schof.colorado.edu).

C. L. Holloway is with the National Institute of Standards and Technology,RF Technology Division, U.S. Department of Commerce, Boulder Laboratories,Boulder, CO 80305 USA.

Digital Object Identifier 10.1109/TAP.2003.817560

derive what we believe to be the most flexible form of a gener-alized sheet transition condition (GSTC) applicable to metafilmslocated in a homogeneous medium. The case of a metafilm ona dielectric interface will be considered in a separate paper.

A. History

We give here a brief outline of previous work on this problem.Although we make no claim to completeness, we believe theseto be the most important literature to date on this subject. Stra-chan [1] was perhaps the first to conduct research on reflectionand transmission of an incident optical wave from a surface witha distribution of polarizable scatterers. In his work, Strachan con-sidered the case when a dielectric interface was present. How-ever, we believe his treatment of this effect is not correct as faras the components of the electric fieldand surface polarizationdensity perpendicular to the interface are concerned. Addi-tionally, he has only used the electric polarization while ignoringthe magnetic polarization density. Strachan also ignored possiblefielddependenceononeof the tangential coordinates, anddidnotconsider the interaction effect between the dipoles. A step for-ward was made by Sivukhin, who considered interaction effectsbetween dipoles in a square array [2]–[4]. Again, only electricsurface polarization effects were considered, and the consider-ation of a dielectric interface leaves something to be desired.

Wait [5] and [6] computed the waves reflected from a perfectlyconducting surface which has a uniform distribution of hemi-spherical bosses whose electrical constants are arbitrary. Theresulting boundary condition is one-sided, rather than of a tran-sition nature, and connects fields to Hertz potentials rather thanto the fields themselves. Wait limited consideration to sphericalscatterers in his work. Vainshtein [7] considered a periodic arrayof 2-D “particles” (wires) of special cross-sectional shape. Vain-shtein does not express the boundary condition in terms of theelectric and magnetic polarizability densities, but rather in termsof other parameters related to a conformal mapping. Hollowayand Kuester ([8]–[10]) have studied the effect of surface rough-ness for a conductor surface in the two dimensions, extendingand synthesizing the work of Wait, Vainshtein, and others. Theyobtained one-sided generalized impedance boundary conditions(GIBCs), using the relationship between both the electric andmagnetic polarization densities and the average fields. This workwas only done for a periodic distribution of surface roughness.

Bedauxet al. have considered this class of problems exten-sively. A comprehensive view of their work can be found inthe book [11]. To cite some of their work specifically, Bedeaux[12] considered only electric polarization density, but did con-sider the interaction between neighboring dipoles, and the dis-continuity of the normal component of the electric field at the

0018-926X/03$17.00 © 2003 IEEE

2642 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 10, OCTOBER 2003

Fig. 1. Metafilm in a cermet topology.

dielectric interface. As with the work of Strachan, we believethis treatment of the interface needs to be studied carefully tojudge its validity. In [13]–[16], a statistical theory is used to de-termine surface dielectric susceptibilities of a thin island film.There, interactions between islands are accounted for by usingthe pair distribution function describing the random locationsof the scatterers. This paper also considers an interface betweentwo media, but studied the effect of electric dipoles only, ig-noring the effect of the magnetic dipole distribution. Windet al.(1984) [17] have included both electric and magnetic polariza-tion effects, but in their model the distribution of the dipoles isassumed continuous from the start. This is due to the fact thatthey used a continuous layer of thin film rather than using sep-arate islands, and this prevents obtaining a connection of theirresult to the “microscopic” structure of the problem. In all thiswork, the surface is characterized by surface dielectric suscep-tibilities or permittivities rather than by the equivalent general-ized sheet transition conditions (GSTCs).

Dignamet al. [18] and [19], have also handled the case whenthe scatterers are randomly distributed in the surface (rather thanthe periodic array assumed by most workers). Interaction be-tween neighboring dipoles was taken into account. However,they did not include in the study an explicit boundary condi-tion, nor were effects of magnetic dipoles included. Twersky[20] has made an extensive study of the behavior of an acousticmetafilm, including the effects of random distribution and non-specularly scattered fields, but his formulas are quite compli-cated and one does not easily extract tractable information abouttransition conditions from them. Perssonet al.[21] have consid-ered only electric polarization density in their work, and embedthe surface distribution in a homogeneous medium. Within cer-tain limitations, they consider random distributions of the scat-terers in the surface, but obtain only the plane wave reflectionand transmission coefficients, not a more general transition con-dition. Barreraet al. [22] have also examined the randomly dis-tributed case, for electric dipoles only but including the effectof a dielectric substrate.

Langreth [23] identified an effective boundary condition, butconsidered only electric dipoles and did not account for the in-teraction between neighboring dipoles. Daweset al.[24] did notobtain a boundary condition, but only calculated the reflection

and transmission coefficients of plane waves for some specificperiodic arrays of planar scatterers.

Rumyantsevet al. [25] obtain a result which in some sensecomes closest to the results we will obtain in this paper. Theytook into account both magnetic and electric polarization den-sities while also taking into considerations the interaction be-tween the dipoles. They considered the material on both sidesof the metafilm to be the same. However, they only studied thecase of a plane incident wave, so their transition condition iseffectively expressed only in the spatial Fourier transform do-main. Maslovskiet al. [26] came up with the accurate interac-tion field for the case of a square periodic array of scatterers.However, they did not pursue the derivation to the point of ob-taining a boundary condition. Tretyakovet al. [27] produce re-sults for an electric dipole array, but only mention that the mag-netic dipole case can be treated by using duality. They obtainan average boundary condition which is also close to what wewill derive here; their results can actually can be viewed as aspecial case of ours. Yatsenkoet al. [28] studied the case whenthere are two parallel metafilms. They have taken the interac-tion field into consideration, but have not obtained an equivalentboundary condition. Simovskiet al.[29] and Grahamet al.[30]have studied the effect of higher-order multipole distributions,but only for the case of electric multipoles. The former interprettheir result in terms of an effective permittivity of a thin layer ofdielectric, rather than giving an effective boundary condition.

In this paper, we aim to give a systematic derivation of a gen-eral form of a boundary condition which relates the fields oneither side of a metafilm. It will contain the foregoing results asspecial cases, at least for the case when the media on both sidesof the metafilm are identical. Generalization of our work to thecase of a metafilm lying on a magnetodielectric substrate willbe done in a future paper.

B. Outline of This Paper

The purpose of this paper is to obtain a boundary condi-tion linking the average electric and magnetic fields on oppositesides of a metafilm consisting of a distribution of isolated elec-trically small scatterers (the so-calledcermettopology as shownin Fig. 1). Such conditions have been called GSTCs by Senior

KUESTERet al.: AVERAGED TRANSITION CONDITIONS FOR ELECTROMAGNETIC FIELDS 2643

and Volakis [31] (see also [32] and [33]), who have given exam-ples of simple structures which can be approximately describedby them. Our result will be a GSTC of the second order, similarto those presented in [31, ch. 5], or [32, eqns. (43) and (44)].We begin by regarding the distribution of scatterers as dense bycomparison with any large-scale dimension of the structure inquestion (perhaps a wavelength, or perhaps some other macro-scopic length), but sparse as measured in terms of the sizes ofthe scatterers themselves. The first condition means that the av-erage field varies slowly enough that its discontinuities acrossthe surface can be regarded as due tocontinuoussurface distri-butions of electric polarization and magnetization . Thesecond condition means that the field acting on one of the scat-terers can be calculated by assuming all other scatterers are re-placed by continuous polarization and magnetization densities,in a sense to be made more precise below.

Surface electric polarization and magnetization density dis-tributions cause discontinuities in the macroscopic electric andmagnetic fields (defined as the incident fields plus the fields ofthe entire sheet), according to [34] and [35]. However,and

are themselves a consequence of averaging a distributionof discrete electric and magnetic dipolesand , where theinteger denotes a particular scatterer. Each dipole moment isproportional to the field acting on the scatterer, the proportion-ality factor involving the electric and magnetic polarizabilitiesof the th scatterer [36]. The calculation then requires that webe able to identify the electric and the magnetic fields actingon the scatterer at the position, where is a position vector.This acting field cannot be the average or macroscopic field re-ferred to earlier, since this field is discontinuous at the scattererlocation, whereas the acting or microscopic field must be con-tinuous and well defined. The acting electric and magnetic fieldsat this point are thus taken to be equal to the sum of the incidentfields plus the field created by the sheet of (continuous) electricand magnetic polarization density distributions, from which thepolarization and magnetization in a small disk surroundinghave been removed (this idea is used, e.g., in [26]). To calculatethe fields of this punctured sheet, we have to calculate the fieldscaused by the small disk alone, and subtract this field from thefields of the entire sheet (without the hole removed). The actingfield is thus the macroscopic field minus the field of the diskalone.

The radius of the disk is chosen in such a way that the field ofthe punctured continuous distribution of electric and magneticpolarization density is, to a leading order approximation, equalto the field caused by the distribution of discrete scatterers (ex-cepting the one at , of course). This calculation has been doneby previous researchers for the case of regular square arraysof scatterers and for random distributions, and we make use oftheir work in this paper to complete our result. We should notethat SI units are used throughout the paper, and the fields andsources are assumed time harmonic, with an assumed complextime factor . These are important issues when comparing toresults by other researchers, primarily from physics and chem-istry backgrounds, who may have used Gaussian units and/or adifferent convention for time dependence.

II. ELECTRIC AND MAGNETIC FIELDS ACTING ON A SCATTERER

AT

Our goal is to obtain a boundary condition for the macro-scopic fields. The macroscopic field is defined as the incidentfield plus the field of the whole sheet, averaged so that rapidvariations of the field over distances on the order of typical par-ticle separations in the sheet are eliminated

(1)

(2)

In an attempt to simplify notation where possible, the macro-scopic field will be denoted with no superscripts. This field isassociated with continuous electric and magnetic polarizationdensities, possibly nonuniformly distributed, in the plane of thescatterers. The macroscopic field has discontinuities at all pointsof the sheet (from the classical boundary conditions). These dis-continuities can be expressed in the form [34], [35]

(3)

(4)

(5)

(6)

where is the surface electric polarization density, and isthe surface magnetization (magnetic polarization) density. Here

is the direction perpendicular to the surface containing thescatterers, taken to be the plane , and the subscript de-notes the two directions (and ) tangential to that surface.

The polarization and magnetization densities are found fromthe densities of electric and magnetic dipole moments in theplane . These dipole moments in turn depend on the fieldacting on each of the scatterers, along with the polarizabilitiesof the scatterers. The acting field is defined as the incidentfield, plus the field from the entire sheet of electric and mag-netic polarization density, but excluding the contribution of asmall circular disk of radius centered at the position of thescatterer under consideration. In the following section, we willcalculate the acting field for a possibly nonuniform electric andmagnetic polarization density distributions, and relate them tothe macroscopic fields. Specifically, the field acting on thescatterer at can be obtained by subtracting the field due tothe disk of radius from the macroscopic field. This resultingfield must be continuous at . The acting electric field canbe expressed as

(7)

2644 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 10, OCTOBER 2003

We now define the average field (for either the macroscopicfield or the field of the disk) at the surface as

(8)

so that the acting field at the surface becomes

(9)

A similar argument can be made for the magnetic field, and asa result

(10)

III. ELECTRIC AND MAGNETIC FIELDS FOR THEDISK

In this section, the objective is to calculate the magnetic andelectric fields and respectively due to a small diskwith radius lying in the - plane, and at . By “small”we mean that the surface magnetization and polarization den-sities may be assumed approximately constant on the disk. Tofind the electric and magnetic fields for the disk with uniformsurface polarization and magnetization distribution, electric andmagnetic vector potentials will be used. In the case when boththe electric polarization surface current and the magnetic polar-ization surface current exist at the same time, the fields can bewritten as a superposition of electric and magnetic potentials

(11)

and

(12)

The electric and magnetic vector potentials,and respec-tively, can be obtained by solving the Helmholtz equation witha source term containing the surface currents at the disk. TheHelmholtz equation can be written as follows for a disk of elec-tric polarization:

(13)

where stands for the component, , or . Likewise

(14)

The electric surface current is , and the magneticsurface current is , where and are thesurface polarization and magnetization densities respectively.Since the disk only occupies , we have

(15)

and

(16)

where is the Heaviside unit step function

(17)

At this stage, we use the double Fourier transformation whichis defined as follows:

(18)

(19)

Substituting (18) into (15), defining and using, we obtain the ordinary

differential equation

(20)

Using , , and gives

(21)

We make the transformations andand define the variable . Then (21) becomes

(22)From [44], we have that

(23)

We substitute (23) into (22) and obtain

(24)

Using the identity , and letting, we have

(25)

KUESTERet al.: AVERAGED TRANSITION CONDITIONS FOR ELECTROMAGNETIC FIELDS 2645

Applying (25) to (24) leads to

(26)

Similar steps applied to (16) lead to

(27)

Equation (26) has solutions of the form

(28)

and since is continuous at , we set .Applying the jump condition at leads to

(29)

and thus

(30)

Substituting from (30) into (28) gives

(31)

We now apply the inverse Fourier transform to to get

(32)

Using , , andas before, we transform (32) into

(33)

Again, let and , so that the magneticpotential vector takes the form

(34)Using (23), the magnetic potential can be written in the form

(35)

Similarly, the electric potential can be written in the form

(36)

Equations (35) and (36) for the electric and magnetic poten-tials will next be used with (11) and (12) to calculate the electricand the magnetic fields and due to the disk with a smallradius as and as .

A. Electric Field Components for a Uniformly Polarized andMagnetized Disk

Since we have identified the electric and magnetic potentialvectors, we can substitute these vectors into (11) and (12) to findthe electric and magnetic fields of the disk. We expand (11) as

(37)

We observe that , , , andoperating on , , , , and all tend to zero

as goes to zero (see Appendix A). The components of theelectric field as are then

(38)

(39)

and

(40)

To find the -component of the electric field , we sub-stitute for both the electric and magnetic vector potentialsand from (35) and (36), respectively, into (11):

(41)

2646 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 10, OCTOBER 2003

The operator affects only and results in(see Ap-

pendix A). As a result (41) can be written in the form

(42)

Letting

(43)

where is the signum (algebraic sign) function

(44)

In a similar way, one can find

(45)

To find the -component of the electric field, we first find

(46)

and then let

(47)

Likewise, the magnetic field components can be calculated

(48)

(49)

and

(50)

B. Average Fields at the Center of the Disk

The electric and magnetic field components given in (43),(45), and (47)–(50) will be used later to calculate the electricand magnetic fields acting on a scatterer in the plane . Tothis end, we need [see (9) and (10)] the average field of the disk,which we obtain as follows:

(51)

so that

(52)

In a similar way

(53)

while the component is continuous

(54)

In a similar way the average magnetic field components are

(55)

(56)

and again, the component is continuous

(57)

IV. I NDUCED DIPOLE MOMENT AND THE

BOUNDARY CONDITIONS

The induced dipole moments for a scatterer with an electricand magnetic polarization density distribution can be found inthe form

(58)

(59)

where and are the electric and magnetic dipole moments,respectively, while and are the electric and magneticpolarizability dyadics for that scatterer (in using a minus signon the right-hand side (RHS) of (59), we have followed the con-vention used in [8], among others, in order that the magneticpolarizability has positive values for simple scatterers). The re-lation between the the electric and magnetic polarization densi-ties and the electric and magnetic dipole moments can be givenas follows:

(60)

where is the number of scatterers per unit area and the symboldenotes an average over the scatterers in the vicinity of the

KUESTERet al.: AVERAGED TRANSITION CONDITIONS FOR ELECTROMAGNETIC FIELDS 2647

point where and are defined. Substituting forand ,we obtain

(61)

We will assume for simplicity that the polarizability dyadicsare diagonal. Then thecomponent of the electric polarizationdensity from (9) and (10) and (52)–(57) is

(62)

Then solving for we get

(63)Now, the term in square brackets in the denominator is approx-imately one for small

(64)

However, we have already committed an error of at leastin replacing the field due to all scatterers except the

one at by that of the continuous distributions andminus the field of auniformly polarized and magnetized disk.Thus, we will replace the square brackets in (63) by one, andobtain

(65)

where

(66)

In a similar manner, we get thecomponent of the polarizationto be

(67)

where

(68)

The component of the electric polarization density obeys

(69)so that

(70)

Once again, we observe that

(71)

and, thus, to our order of approximation

(72)

where

(73)

In similar fashion, the component of the magnetic polariza-tion density is

(74)

where

(75)

the component is

(76)

where

(77)

and the component is

(78)

where

(79)

By substituting from (65), (67), (72), (74), (76), and (78) into(3)–(6) one can obtain the GSTCs

(80)

(81)

(82)

(83)

where and are the diagonal dyadics assembled from(66), (68), and (73), or (75), (77), and (79), respectively

(84)

(85)

These can be interpreted as effective electric and magnetic po-larizability densities per unit area, respectively (the actual den-sities multiplied by correction factors).

The radius of the disk remains to be determined. It is chosenin such a way that the field of the punctured continuous distri-bution of electric and magnetic polarization density is equal tothe field caused by the distribution of all the discrete scatterersexcept the one at , in the static limit (note that we again

2648 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 10, OCTOBER 2003

commit an error of at least in doing so). This calcu-lation has been done by Maslovskiet al. [26], who obtain

(86)

for a periodic distribution of scatterers in a square array of period(see also [37] and [38]). In (86), the prime indicates the sum is

to be taken over all integer values ofand from to ,except for the combination . Essentially equivalentresults can also be found in [18], [39] and many others (keepingin mind the need to convert from Gaussian units used inthesepapers to the SI units used here: ).

For a random distribution, Dignam and Moskovits [18] havegiven a formula which can be used to obtainin terms of the2-D radial pair distribution function1 [40]–[43] describingthe statistical distribution of the scatterers in the surface (similarexpressions are given in [13]–[16], [21] and [22]). Its meaningis that, for a collection of scatterers lying in a region in aplane, the probability that there is a scatterer lying at a distancebetween and from some given scatterer is

(87)

where also denotes the area occupied by the scatterers (whichcan later be allowed to approach infinity). Converting the resultsof [18] to SI units, we find that

(88)

Since can be determined from electron micrographs orfrom diffraction measurements, it should be straightforward tocompute in the case of random scatterer distribution.

Equations (80) and (81) are of the same form as the GSTCspresented in [31, eqns. (5.17) and (5.19)], or [32, eqns. (43)and (44)], though these only applied to thin magnetodielectricplanar sheets or to wire grids. We can see that they are also ofessentially the same form as those given in [25, [eqns. (4)–(6)and (10)]], but with a different value for the interaction constant(that is, a different value of is used than we have used here)and with assumed plane wave dependence onand rather thanthe general variation we permit.

In the form presented here, we emphasize the interpretationof the coefficients of the conditions as polarizability densities, inmuch the same way as are the generalized impedance boundaryconditions which we derived for periodically rough surfaces in[8] and [9]. We expect that this will facilitate further generaliza-tions of these conditions as well as the practical computation ofthe coefficients in particular cases. Only dipole interactions havebeen taken into account in our model, meaning that quadrupoleand higher interaction terms such as accounted for in [29] and[30] are absent. This will mean that the accuracy of (80)–(83)will deteriorate if the scatterers occupy too large a fraction ofthe surface, and this limitation must be quantitatively assessedby comparison with accurate numerical simulations.

1Some authors employ the related pair correlation functionh (r) =g (r) � 1. See [40] or [41].

V. CONCLUSION

In this paper, we have obtained GSTCs for the average fieldsat a metafilm in terms of its electric and magnetic polarizabilitydensities. The interaction between the scatterers, which may beperiodically or randomly distributed, is taken into account tothe first (dipole) order of approximation. Our GSTCs contain asspecial cases a number of results previously given in the litera-ture. A more thorough assessment of the range of their validitymust still be done, including limits on the size of the scatterersas well as their separation (in particular, how large maybe?).

The GSTCs have several important potential applications.When dealing with numerical modeling of fields, using any ofthe standard techniques such as FDTD, FEM, or the momentmethod, an important consideration in reducing computationtime and storage requirements is how finely the spatial regionmust be subdivided into cells or grids in order to achieveadequate computational accuracy. When the structure beinganalyzed contains very small features, the spatial grid mustbe correspondingly fine, resulting in far more unknownsto determine and, hence, slower and more resource-hungrycomputation. If a metafilm can be replaced by the GSTCsderived in this paper, the spatial resolution necessary will bemuch coarser (if we are willing to forgo precise knowledge ofthe field in or near the metafilm). Hence a substantial saving incomputer memory and simulation time can be achieved.

In a paper to be published separately, we will present an appli-cation of these GSTCs to the calculation of plane wave reflec-tion and transmission coefficients from a metafilm. Examina-tion of the GSTCs shows that the properties of the metafilm arecontrollable through the effective polarizability densities (i.e.,through the polarizabilities of the individual scatterers, their sur-face density and the geometry of their distribution in the planeas expressed by the parameter). For example, the polarizabil-ities could be controlled by applying a bias field to scatterersmade of ferroelectric material. As a result, the plane-wave re-flection and transmission coefficients could in principle be con-trolled, so that total reflection or total transmission could be ob-tained through electronic means. In this way, we design a digitalswitch. Practical aspects such as the switching time need to beconsidered in order to evaluate the feasibility of such a devicefor applications.

Extensions of the work reported in this paper will be under-taken in the future. One important problem for applications isthat of modeling a perforated conductor (the “inductive grid”)which is in some sense the dual of the problem treated here. Itshould be possible to treat it using aperture polarizability den-sities and obtain GSTCs complementary to (80)–(83). Anotherinteresting problem that deserves future study is the extensionof the present work to the case where the metafilm is locatedat the surface of a dielectric substrate. There are more subtlefactors which come into play in the analysis in this case. Forexample, an induced dipole perpendicular to the interface pro-duces a field which is ambiguously defined, depending on its“microscopic” location relative to the substrate surface (i. e., isit completely above, partially inside, or under the dielectric sur-face?). Moreover, the field produced by this layer of scatterershas more complicated discontinuities than when embedded in

KUESTERet al.: AVERAGED TRANSITION CONDITIONS FOR ELECTROMAGNETIC FIELDS 2649

a homogeneous medium. In particular, the acting field is nowdiscontinuous, so the question arises as to precisely how polar-izability is to be defined when a scatterer is at an interface. Theseissues must be properly resolved in order to extend these ideasto the more realistic situation when the metafilm supported bya substrate surface. They will be the subject of a future paper.

APPENDIX AEVALUATIONS OF SOME DIFFERENTIATIONS

A. Evaluation of

From the chain rule, we can evaluate

(89)

where we have used , from which followand . Using Bessel’s

differential equation

(90)

Equation (89) can be written in the form

(91)Letting , then

(92)

In a similar way

(93)

B. Evaluation of

We have

(94)

Since and , we have that

(95)

Using (90) in (95)

(96)so as , . As a result

as .It is clear from (89) that the first derivatives and

both approach 0 as .

APPENDIX BEVALUATION OF SOME INTEGRALS

A. Evaluation the Integral

Using [45, formula 6.637.1], with imaginary, we put, , , and

. Let have a little loss (that is, it has a small negativeimaginary part which will be allowed to approach zero in thefinal result: , and ) so that has a positivereal part. We then have

(97)

since

(98)

B. Calculating [ and ]

(99)

where

and

defines the signum (algebraic sign) function.

2650 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 10, OCTOBER 2003

C. Calculating

(100)

Letting

(101)

(102)

(103)

REFERENCES

[1] C. Strachan, “The reflection of light at a surface covered by a monmolec-ular film,” in Proc. Camb. Phil. Soc., vol. 29, 1933, pp. 116–130.

[2] D. V. Sivukhin, “Contribution to the molecular theory of light re-flection,” Comptes Rendus (Doklady) Acad. Sci. URSS, vol. 36, pp.231–234, 1942.

[3] , “Molecular theory of the reflection and refraction of light” (inRussian),Zh. Eksper. Teor. Fiz., vol. 18, pp. 976–994, 1948.

[4] , “Elleiptical polarization of light reflected from roughness” (inRussian),Zh. Eksper. Teor. Fiz., vol. 21, pp. 367–376, 1951.

[5] J. R. Wait, “Guiding of electromagnetic waves by uniformally roughsurfaces,”IRE Trans. Antennas Propagat., vol. 7, pp. S154–S162, 1959.

[6] , “Guiding of electromagnetic waves by uniformally rough sur-faces,”IRE Trans. Antennas Propagat., vol. 7, pp. S163–S168, 1959.

[7] L. A. Vainshtein (Wainstein), “On the electrodynamic theory of grids,”in Elektronika Bol’shikh Moshchnostei, P. L. Kapitsa and L. A. Vain-shtein, Eds. Moscow: Nauka, 1963, vol. 2, pp. 26–74.

[8] C. L. Holloway and E. F. Kuester, “Equivalent boundary conditions for aperfectly conducting periodic surface with cover layer,”Radio Sci., vol.35, pp. 661–681, 2000.

[9] , “Impedance-type boundary conditions for a periodic interface be-tween a dielectric and a highly conduction medium,”IEEE Trans. An-tennas Propagat., vol. 48, pp. 1660–1672, 2000.

[10] , “Power loss associated with conducting and superconductingrough interfaces,”IEEE Trans. Antennas Propagat., vol. 48, pp.1601–1610, 2000.

[11] D. Bedeaux and J. Vlieger,Optical Properties of Surfaces. London,U.K.: Imperial College Press, 2002.

[12] , “A phenomenological theory of the dielectric properties of thinfilms,” Physica, vol. 67, pp. 55–73, 1973.

[13] , “A statistical theory of the dielectric properties of thin island films.I. The surface material coefficients,”Physica, vol. 73, pp. 287–311,1974.

[14] J. Vlieger and D. Bedeaux, “A statistical theory for the dielectric prop-erties of thin island films,”Thin Solid Films, vol. 69, pp. 107–130, 1980.

[15] D. Bedeaux and J. Vlieger, “A statistical theory for the dielectric proper-ties of thin island films: Application and comparison with experimentalresults,”Thin Solid Films, vol. 102, pp. 265–281, 1983.

[16] M. M. Wind, P. A. Bobbert, and J. Vlieger, “Optical properties of2D-systems of small particles on a substrate,”Physica A, vol. 157, pp.269–278, 1989.

[17] M. M. Wind and J. Vlieger, “Optical properties of thin films on roughsurfaces,”Physica, vol. 125A, pp. 75–104, 1984.

[18] M. J. Dignam and M. Moskovits, “Optical properties of sub-monolayermolecular films,”J. Chem. Soc. Faraday Trans. II, vol. 69, pp. 56–64,1973.

[19] M. J. Dignam and J. Fedyk, “Microscopic model for the optical prop-erties of thin films,”J. Physique Colloque, vol. 38, no. C5, pp. C5–57,1977.

[20] V. Twersky, “Multiple scattering of sound by correlated monolayers,”J.Acoust. Soc. Amer., vol. 73, pp. 68–84, 1983.

[21] B. N. J. Persson and A. Liebsch, “Optical properties of two-dimensionalsystems of randomly distributed particles,”Phys. Rev. B, vol. 28, pp.4247–4254, 1983.

[22] R. G. Barrera, M. de Castillo-Mussot, G. Monsivais, P. Villaseñor, andW. L. Mochán, “Optical properties of two-dimensional disordered sys-tems on a substrate,”Phys. Rev. B, vol. 43, pp. 13 819–13 826, 1991.

[23] D. C. Langreth, “Macroscopic approach to the theory of reflectivity,”Phys. Rev. B, vol. 39, pp. 10 020–10 027, 1989.

[24] D. H. Dawes, R. C. McPhedran, and L. B. Whitbourn, “Thin capacitivemeshes on dielectric boundary: Theory and experiment,”Appl. Opt., vol.28, pp. 3498–3509, 1989.

[25] V. V. Rumyantsev and V. T. Shunyakov, “Propagation of electromagneticexcitations in layered crystalline media,”Kristallografiya, vol. 36, pp.535–540, 1991.

[26] S. I. Maslovski and S. A. Tretyakov, “Full-wave interaction field in two-dimensional arrays of dipole scatterers,”Int. J. Electron. Comun., Arch.Elek.Übertragungstech. (AEÜ), vol. 53, pp. 135–139, 1999.

[27] S. A. Tretyakov, A. J. Viitanen, S. I. Maslovski, and I. E. Saarela,“Impedance Boundary Conditions for Regular Dense Arrays of DipoleScatterers,” Helsinki Univ. Technol., Finland, 304, Electromagn. Lab.Rep. Ser., 1999.

[28] V. Yatsenko, S. Moslovski, and S. Tretyakov, “Electromagnetic inter-action of parallel arrays of dipole scatterers,”Progress in Electromagn.Res. (PIER), vol. 25, pp. 285–307, 2000.

[29] C. R. Simovski, M. Popov, and S. He, “Dielectric properties of thin filmconsisting of a few layer of molecules or particles,”Phys. Rev. B, vol.62, pp. 13 718–13 730, 2000.

[30] E. B. Graham and R. E. Raab, “The role of the macroscopic surfacedensity of bound electric dipole moment in reflection,” inProc. R. Soc.Lond. A, vol. 457, 2001, pp. 471–484.

[31] T. B. A. Senior and J. L. Volakis,Approximate Boundary Conditions inElectromagnetics. London, U.K.: Inst. Elect. Eng., 1995.

[32] , “Sheet simulation of a thin dielectric layer,”Radio Sci., vol. 22,pp. 1261–1272, 1987.

[33] M. A. Ricoy and J. L. Volakis, “Derivation of generalized transi-tion/boundary conditions for planar multiple-layer structures,”RadioSci., vol. 25, pp. 391–405, 1990.

[34] M. Idemen, “Straightforward derivation of boundary conditions onsheet simulating an anisotropic thin layer,”Electron. Lett., vol. 24, pp.663–665, 1988.

[35] , “Universal boundary relations of the electromagnetic field,”J.Phys. Soc. Jpn., vol. 59, pp. 71–80, 1990.

[36] L. D. Landau and E. M. Lifshitz,Electrodynamics of ContinuousMedia. Oxford, U.K.: Pergamon, 1960, p. 7 and 192.

[37] J. Topping, “On the mutual potential energy of a plane network of dou-blets,” inProc. Roy. Soc. London A, vol. 114, 1927, pp. 67–72.

[38] R. E. Collin,Field Theory of Guided Waves. New York: IEEE, 1991,pp. 760–761.

[39] J. Vlieger, “Reflection and transmission of light by a square nonpolarlattice,” Physica, vol. 64, pp. 63–81, 1973.

[40] I. Z. Fisher,Statistical Theory of Liquids. Chicago, IL: Univ. ChicagoPress, 1964, ch. 2.

[41] J. P. Hansen and I. R. McDonald,Theory of Simple Liquids. London:Academic Press, 1976, sec. 2.6 and 5.1.

KUESTERet al.: AVERAGED TRANSITION CONDITIONS FOR ELECTROMAGNETIC FIELDS 2651

[42] A. L. Fetter, “Lectures on correlation functions,” inCritical Phenomena,F. J. W. Hahne, Ed. Berlin: Springer-Verlag, 1983, vol. 186, LectureNotes in Physics, pp. 301–353.

[43] L. Tsang, J. A. Kong, and K.-H. Ding,Scattering of ElectromagneticWaves: Theories and Applications. New York: Wiley, 2000, ch. 4, sec.5.2.

[44] A. Jeffrey,Handbook of Mathematical Formulas and Integrals. SanDiego, CA: Academic, 1995, p. 277.

[45] I. S. Gradshteyn and I. M. Ryzhik,Table of Integrals, Series and Prod-ucts. New York: Academic, 1980.

Edward F. Kuester (S’73–M’76–SM’95–F’98) wasborn in St. Louis, MO, on June 21, 1950. He receivedthe B.S. degree from Michigan State University, EastLansing, in 1971, and the M.S. and Ph.D. degreesfrom the University of Colorado, Boulder, in 1974and 1976, respectively, all in electrical engineering.

Since 1976, he has been with the Department ofElectrical and Computer Engineering, University ofColorado, Boulder, where he is currently a Professor.In 1979, he was a Summer Faculty Fellow with theJet Propulsion Laboratory, Pasadena, CA. During

1981–1982, he was a Visiting Professor at the Technische Hogeschool, Delft,The Netherlands. During 1992–1993, he wasprofesseur invitéat the ÉcolePolytechnique Fédérale de Lausanne, Switzerland. In 2002, he was a VisitingScientist at the National Institute of Standards and Technology (NIST), Boulder.His research interests include the modeling of electromagnetic phenomena ofguiding and radiating structures, applied mathematics, and applied physics.

Dr. Kuester is a member of the Society for Industrial and Applied Mathe-matics and Commissions B and D of the International Union of Radio Science(URSI).

Mohamed A. Mohamedwas born in Sohag, Egypt,on February 3, 1965. He received the B.Sc. degreein physics and the M.Sc. degree with a thesisentitled “Some properties of radio frequency plasmanitriding of stainless steel” from South ValleyUniversity, Sohag, in 1987 and 1991, respectiely.He is currently working toward the M.S. degree inmathematics and the Ph.D. in electrical engineeringat the University of Colorado, Boulder.

Melinda Piket-May received the B.S.E.E. degreefrom the University of Illinois, Champaign, in 1988and the M.S.E.E. and Ph.D. degrees in electrical en-gineering from Northwestern University, Evanston,IL, in 1990 and 1993, respectively.

Her work experience includes positions at FermiNational Accelerator Lab, Naval Research Lab, andCray Research. In 1993, she joined the ECE Depart-ment, University of Colorado, Boulder, where sheis currently an associate professor. She has an activeresearch program in computational electromagnetics.

Her work includes the development of general methods to extract industriallyrelevant information from numerical computer simulations. She serves on manyprofessional committees. She is also very active in engineering education;working on undergraduate engineering design issues and incorporating researchinto the classroom in an interactive and meaningful way.

Professor Piket-May received a 1996 URSI Young Scientist Award and wasnamed a Sloan New Faculty Fellow in 1997. She was awarded an NSF CA-REER award in 1997 for her research and teaching activities, was awarded the2000 College of Engineering Peebles Award for teaching, and was selected as an2001/2001 Carnegie CASTLE fellow. She also received the University of Col-orado 2002 Elizabeth Gee Award for outstanding scholarship in teaching andresearch.

Christopher L. Holloway (S’86–M’92) was born inChattanooga, TN, on March 26, 1962. He receivedthe B.S. degree from the University of Tennessee,Chattanooga, in 1986, and the M.S. and Ph.D. degreesfrom the University of Colorado, Boulder, in 1988and 1992, respectively, both in electrical engineering.

During 1992 he was a Research Scientist withElectro Magnetic Applications, Incorporated, Lake-wood, CO. His responsibilities included theoreticalanalysis and finite-difference time-domain modelingof various electromagnetic problems. From the fall

of 1992 to 1994, he was with the National Center for Atmospheric Research(NCAR), Boulder. While at NCAR, his duties included wave propagationmodeling, signal processing studies, and radar systems design. From 1994 to2000, he was with the Institute for Telecommunication Sciences (ITS), U.S.Department of Commerce, Boulder, where he was involved in wave propaga-tion studies. Since 2000, he has been with the National Institute of Standardsand Technology (NIST), Boulder, where he works on electromagnetic theory.He is also on the Graduate Faculty of the University of Colorado, Boulder.

Dr. Holloway was awarded the 1999 Department of Commerce SilverMedal for his work in electromangetic theory and the 1998 Department ofCommerce Bronze Medal for his work on printed curciut boards. His researchinterests include electromagnetic field theory, wave propagation, guided wavestructures, remote sensing, numerical methods, and EMC/EMI issues. He is aMember of Commission A of the International Union of Radio Science (URSI)and is an Associate Editor for the IEEE TRANSACTIONS ONELECTROMAGNETIC

COMPATIBILITY . He is the chairman for the Technical Committee on Compu-tational Electromagnetics (TC-9) of the IEEE Electromagnetic CompatibilitySociety.