# fundamental modal properties of surface waves on metamaterial grounded slabs

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 4, APRIL 2005 1431

Fundamental Modal Properties of Surface Waveson Metamaterial Grounded Slabs

Paolo Baccarelli, Member, IEEE, Paolo Burghignoli, Member, IEEE, Fabrizio Frezza, Senior Member, IEEE,Alessandro Galli, Member, IEEE, Paolo Lampariello, Fellow, IEEE, Giampiero Lovat, Student Member, IEEE, and

Simone Paulotto, Student Member, IEEE

Abstract—This paper deals with the analysis of surface wavessupported by a metamaterial layer on a ground plane, and inves-tigates the potentiality of these grounded slabs as substrates forplanar antennas. Both double- and single-negative media, eitherepsilon- or mu-negative, are considered. As is known, such struc-tures may support two kinds of surface waves, i.e., ordinary (trans-versely attenuating only in air) and evanescent (transversely at-tenuating also inside the slab) surface waves. A graphical anal-ysis is performed for proper real solutions of the dispersion equa-tion for TE and TM modes, and conditions are presented that en-sure the suppression of a guided-wave regime for both polariza-tions and kinds of wave. In order to demonstrate the feasibility ofsubstrates with such desirable properties, numerical simulationsbased on experimentally tested dispersion models for the permit-tivity and permeability of the considered metamaterial media arereported. Moreover, the effects of slab truncation on the field radi-ated by a dipole source are illustrated by comparing the radiationpatterns at different frequencies both in the presence and in the ab-sence of surface waves. The reported results make the consideredstructures promising candidates as substrates for planar antennasand arrays with reduced edge-diffraction effects and mutual cou-pling between elements.

Index Terms—Grounded slab, guided waves, metamaterialmedia, surface-wave suppression.

I. INTRODUCTION

FABRICATION and experimental verification of materialswith negative values of magnetic permeability and/or di-

electric permittivity have been demonstrated in recent years byseveral research groups (see, e.g., [1]–[8]). The proposed de-signs are based on one-, two-, or three-dimensional periodicstructures that may be modeled, at least in certain frequencyranges, as homogeneous materials (metamaterials) exhibitingscalar negative and dispersive constitutive parameters.

Various peculiar features of electromagnetic-wave propaga-tion exist when such metamaterials are involved, some of whichwere examined by Veselago in his seminal paper published inthe late 1960s [9], [10]. In this connection, during the last fewyears, different fundamental issues have been addressed. Prop-erties of plane waves in a double-negative (DNG) medium havebeen considered, e.g., in [11] and [12], where the “backward” or“left-handed” nature of these waves and the negative refractionindex of the medium are discussed. Reflection and refraction of

Manuscript received May 31, 2004; revised September 13, 2004.The authors are with the Department of Electronic Engineering,

“La Sapienza” University of Rome, 00184 Rome, Italy (e-mail:[email protected]).

Digital Object Identifier 10.1109/TMTT.2005.845208

plane waves at planar interfaces are of special interest in viewof the proposed application of planar configurations to obtainfocusing effects at optical or microwave frequencies [13], [14].

Guided-wave propagation in structures containing metama-terial media has also been considered (see, e.g., [15]–[19]). Inparticular, modes supported by DNG slabs surrounded by an or-dinary medium are studied in [17]–[19], where the existence ofboth ordinary (transversely attenuating only in air) and evanes-cent (transversely attenuating also inside the slab) surface wavesis pointed out, and a number of modal features differing fromthose found in double-positive (DPS) slabs are reported, in-cluding dispersion and energy-flux properties.

In this paper, we aim at studying the modal properties ofa basic open structure for microwave circuits and antennas,namely, a slab placed on a perfectly conducting ground plane,when the medium of the slab is assumed to be a metamaterial,either DNG or single-negative (SNG). On the basis of simplegraphical discussions of the involved dispersion equations, con-ditions are derived for the existence of real modes either ordinaryor evanescent. As a result of our analysis, sufficient conditionsare obtained, which ensure the absence of proper surface wavesof both TE and TM polarizations in the considered structure(some preliminary results of this kind have recently beenpublished in [20] and [21] for the case of a DNG groundedslab). These conditions of surface-wave suppression are veryattractive in view of a possible employ of a metamaterial slabas a substrate for planar antennas and arrays with reducededge-diffraction effects and mutual coupling between elements.

This paper is organized as follows. In Section II, we describethe considered structure and we derive the dispersion equationsfor both TE and TM modes, either ordinary or evanescent,via transverse resonance. In Section III, we present results forDNG grounded slabs, discussing in particular the possibility ofachieving surface-wave suppression. In Section IV, a similaranalysis is presented for SNG grounded slabs, both mu-negative(MNG) and epsilon-negative (ENG). In Section V, numericalresults are presented for the dispersion properties of modessupported by DNG slabs, on the basis of simple experimentallytested dispersion models for the permittivity and permeabilityof the considered metamaterial media, demonstrating the pos-sibility to obtain surface-wave suppression in certain frequencyranges. In Section VI, the effects of surface-wave suppressionon the radiated field excited by a magnetic dipole placed on theground plane of truncated DNG slabs are illustrated. Finally,in Section VII, conclusions are drawn on the various results ofthis study.

0018-9480/$20.00 © 2005 IEEE

1432 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 4, APRIL 2005

Fig. 1. (a) Reference metamaterial grounded-slab structure with the relevantphysical and geometrical parameters and coordinate system. (b) Transverseequivalent network for TE and TM modes of the grounded slab.

II. PROBLEM STATEMENT AND BACKGROUND

The reference structure considered here is a grounded slabof height made of an ideal linear stationary isotropic homo-geneous lossless metamaterial medium with permeability

and permittivity , both of which may be nega-tive (see Fig. 1(a), where the relevant coordinate system is alsoshown).

The problem addressed here is the study of guided modessupported by this structure, propagating along the longitudinal

-direction with a real propagation constant . Atime–harmonic dependence is assumed and suppressedthroughout. No variation of the electromagnetic field is as-sumed along the -direction, thus, the two-dimensional natureof the problem allows us to separately study TE and TM modes.

As is well known, a transverse equivalent network in the -di-rection can be associated to each TE or TM mode, as shown inFig. 1(b). The expressions of the relevant characteristic imped-ances for the two polarizations in the air and slab regions (sub-scripts 0 and , respectively) are as follows:

(1)

where

(2)

with and .The dispersion equation for TE and TM modes can be ob-

tained by enforcing the transverse-resonance condition, e.g., at, on the relevant equivalent network. The result is [22]

(3)

This equation that is, of course, the same as the ones reportedin [17]–[19] and [22] will be studied in Sections III–VI for

Fig. 2. Transverse profile of the absolute value of the E (TE case) and E(TM case) component of the electric field of ordinary and evanescent surfacewaves propagating along the grounded slab of Fig. 1.

TE and TM modes by considering DNG and SNG slabs sep-arately. Since the grounded slab is a transversely open struc-ture, its modes can be either proper, i.e., attenuating at infinityin the transverse -direction , or improper, i.e.,diverging at infinity [23]. In particular, surfacewaves have a purely imaginary transverse wavenumber in airwith for proper surface waves andfor improper surface waves .

In addition to this, when one (or both) of the constitutive pa-rameters is negative, two kinds of real solutions, correspondingto surface waves supported by the structure, have to be con-sidered, i.e., ordinary surface waves with a real transversewavenumber inside the slab, and evanescent surfacewaves with an imaginary wavenumber inside theslab. A sketch of the transverse profile of the modal field forordinary and evanescent waves on a grounded metamaterialslab is depicted in Fig. 2: the field of ordinary modes is trans-versely attenuating only in air, whereas the field of evanescentmodes is transversely attenuating also inside the slab. It isknown that ordinary surface waves cannot exist in SNG slabs,while evanescent surface waves cannot exist in a DPS isotropicgrounded slab, although they are known to be present in SNGand DNG slabs [18], [19], [24], and in other specific structures,e.g., ferrite slabs [25], [26].

The issue of the location of surface-wave poles on the Rie-mann surface of the spectral Green’s function for the groundedslab will not be addressed here for the sake of brevity. How-ever, we would emphasize that, in the case of a grounded slab,a guided mode is constrained to have a longitudinal phase con-stant greater than : this implies that the relevant pole inthe complex plane is always captured in a steepest descentdeformation of the integral path. On the other hand, the graph-ical method presented in Section III will be applied to functionsthat give rise to proper modes only (i.e., modes corresponding topoles located on the top Riemann sheet); improper modes (i.e.,modes corresponding to poles located on the bottom Riemannsheet) could be found only by considering different functions.

III. MODAL PROPERTIES OF DNG GROUNDED SLABS

Since in a DNG medium both the permeability and permit-tivity are negative, the squared wavenumber is positive.Therefore, surface waves may be either ordinary (if ) orevanescent (if ). A graphical analysis of the dispersionequation for surface waves is provided in Section III-A for TE

BACCARELLI et al.: FUNDAMENTAL MODAL PROPERTIES OF SURFACE WAVES ON METAMATERIAL GROUNDED SLABS 1433

waves and in Section III-B for TM waves. A discussion on thepossibility to achieve proper surface-wave suppression in DNGslabs is provided in Section III-C.

A. TE Surface Waves

1) Ordinary TE Surface Waves: In this case, the dispersionequation (3) for proper real modes can be written as

(4)

By taking into account that , (4) becomes

(5)

The condition implies that ordinary TE surfacewaves cannot exist if (which would imply ).Therefore, only the case will be studied, for which (5)can be written as

(6)

in terms of the adimensional variables

(7)

Since we aim at studying surface waves, positive real solu-tions for of (6) are sought. These can be obtained graphicallyby finding the intersections between the tangent function at theleft-hand side of (6) and the function at the right-handside of the same equation. A straightforward analysis shows thatthe function is defined for , is concave and mono-tonically increasing in the interval , is zero at , andtends to infinity for (see Fig. 3).

We seek conditions that inhibit propagation of guided modes:this corresponds to determine the conditions under which nointersection occurs between and . By observingFig. 3, it can be seen that sufficient conditions to avoid suchintersection are either: 1) there exists a straight linesuch that lies below it and lies above it, for

with [see Fig. 3(a)] or 2) there exists a straightline such that lies above it and liesbelow it, for with [see Fig. 3(b)].

Since the derivative of is monotonically increasing,we can choose equal to the value of such derivative at ,i.e., equal to : in order to satisfy condition 1), the deriva-tive of has to be less than the derivative of at

and has to be less than . After some algebra,this can analytically be expressed as

(8)

On the other hand, since the derivative of is mono-tonically increasing in , we can choose equal to thevalue of such derivative at , i.e., equal to 1: in order tosatisfy condition 2), has to be less than with

Fig. 3. Graphical representation of the functions occurring in (6) forordinary proper TE surface waves. Two cases of surface-wave suppression,corresponding to the curves labeled f (�), are shown in (a) and (b).

. After some algebra, this can analytically be ex-pressed as

(9)

2) Evanescent TE Surface Waves: In this case, andthe dispersion equation (3) for proper real modes can be writtenas

(10)Both the cases now have to be considered.

1434 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 4, APRIL 2005

Fig. 4. Graphical representation of the functions occurring in (11) and (14) forevanescent proper TE surface waves. Two cases of surface-wave suppressionwhen � " > 1 are shown, corresponding to the curves labeled f (�). Thecase of surface-wave suppression when � " < 1 corresponds to the curvelabeled g (�).

If , (10) can be written as

(11)

in terms of the adimensional variables

(12)

The function at the right-hand side of (11) is defined for, is concave and monotonically decreasing in ( , ),

tends to infinity for , and tends to for(see Fig. 4). Therefore, by observing Fig. 4, it can easily beseen that a necessary and sufficient condition to avoid intersec-tion between and is that the horizontal asymp-tote of lies below the horizontal asymptote of .The necessary and sufficient condition to avoid propagation ofproper evanescent TE surface waves when is then

(13)

If , (10) can be written as

(14)

The function at the right-hand side of (14) is defined forevery , is convex and monotonically increasing in the interval

, is zero at , and tends to for (seeFig. 4).

By observing Fig. 4, it can be seen that sufficient conditionsto avoid intersection between and are that:1) the horizontal asymptote of lies above the hori-zontal asymptote of , and there exists a straight line

such that lies below it and liesabove it, as far as lies below the horizontal asymptote of

; alternatively: 2) the horizontal asymptote oflies below the horizontal asymptote of , and thereexists a straight line such that lies belowit, and lies above it, as far as lies below thehorizontal asymptote of . By taking into account thatthe horizontal asymptote of is equal to and thehorizontal asymptote of is equal to 1, the first part of

Fig. 5. Graphical representation of the functions occurring in (18) for ordinaryproper TM surface waves.

condition 1) requires . Moreover, since the deriva-tive of is monotonically decreasing, we can choose

equal to the value of such derivative at , i.e., equal to1. After some algebra, condition 1) can then analytically beexpressed as

(15)

On the other hand, the first part of condition 2) requires that. Moreover, since the derivative of is mono-

tonically decreasing, we can choose equal to the value of suchderivative at , i.e., equal to . After some algebra,condition 2) can then analytically be expressed as

(16)

B. TM Surface Waves

1) Ordinary TM Surface Waves: In this case, the dispersionequation (3) for proper real modes can be written as

(17)As for the TE case, the condition implies thatordinary surface waves cannot exist if . The case

will then be examined, and (17) can be written as

(18)

The function on the right-hand side of (18) is definedand monotonically decreasing in , tends to plus infinity for

, and is zero at (see Fig. 5).By observing Fig. 5, it can easily be seen that a sufficient

condition to avoid intersection between andis

(19)

BACCARELLI et al.: FUNDAMENTAL MODAL PROPERTIES OF SURFACE WAVES ON METAMATERIAL GROUNDED SLABS 1435

Fig. 6. Graphical representation of the functions occurring in (21) and (23) forevanescent proper TM surface waves. The curve labeled f (�) correspondsto � " > 1, while the curve labeled g (�) corresponds to � " < 1.

2) Evanescent TM Surface Waves: In this case, andthe dispersion equation (3) for proper real modes can be writtenas

(20)Both the cases have to be considered.

If , (20) can be expressed as

(21)

The function on the right-hand side of (21) is definedfor , is convex and monotonically increasing in ,is zero at , and tends to for (see Fig. 6).Therefore, by observing Fig. 6, a sufficient condition to avoidintersection between and is that the horizontalasymptote of lies below the horizontal asymptote of

and that the point lies on the right of the point. A sufficient condition to avoid propagation

of proper evanescent TM surface waves when is then

(22)

Finally, if , (20) can be written as

(23)

The function on the right-hand side of (23) is definedfor , is concave and monotonically decreasing in the in-terval , tends to infinity for , and tends tofor (see Fig. 6). Therefore, by observing Fig. 6, anecessary and sufficient condition to avoid intersection between

and is that the horizontal asymptote oflies above the horizontal asymptote of . The necessaryand sufficient condition to avoid propagation of proper evanes-cent TM surface waves when is then

(24)

TABLE ISUMMARY OF CONDITIONS FOR SUPPRESSION OF PROPER SURFACE WAVES OF

DIFFERENT KINDS ON DNG GROUNDED SLABS WITH � " < 1. THE

CONDITION FOR EVANESCENT TE WAVES IS NECESSARY AND SUFFICIENT,THE CONDITION FOR EVANESCENT TM WAVES IS ONLY SUFFICIENT.

ORDINARY WAVES CANNOT EXIST IN THIS CASE

C. Discussion on Surface-Wave Suppression in DNGGrounded Slabs

The analysis in Sections III-A and III-B has shown that, foreach kind of proper surface wave supported by a DNG groundedslab, conditions may be found that inhibit its propagation. How-ever, it is necessary to ascertain if simultaneous suppression ofall kinds of such waves can be obtained. In this connection, thecases of will be considered separately.

Sufficient conditions to avoid proper surface-wave propa-gation in the case are summarized in Table I. Inparticular, the condition reported for TE modes is also a neces-sary condition, whereas that for TM modes is only a sufficientone.

By examining Table I, it can be concluded that, in order toinhibit the propagation of every kind of surface wave (both or-dinary and evanescent) when , a sufficient condition isthat the following set of inequalities is satisfied:

(25)

provided that

(26)

In particular, since , the last inequality canbe satisfied, at a fixed frequency , by choosing the slab height

sufficiently large; in fact, the condition on in (26) can alsobe expressed as

(27)

where is the speed of light in a vacuum.Sufficient conditions to avoid proper surface-wave propaga-

tion in the case are collected in Table II. In particular,the condition reported for TM evanescent waves is also a nec-essary condition, while the other conditions are only sufficientones. By examining the alternative conditions for TE-mode sup-pression, it can be deduced that the only consistent pairs are(o1)–(e1) or (o2)–(e2). However, it can easily be seen that theonly pair compatible with the reported conditions for TM-modesuppression is the (o1)–(e1) pair.

Therefore, in order to inhibit the propagation of every kindof surface wave (both ordinary and evanescent) when

1436 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 4, APRIL 2005

TABLE IISUMMARY OF CONDITIONS FOR SUPPRESSION OF PROPER SURFACE WAVES OF

DIFFERENT KINDS ON DNG GROUNDED SLABS WITH � " > 1. THE

CONDITION FOR EVANESCENT TM WAVES IS NECESSARY AND SUFFICIENT,THE OTHER CONDITIONS ARE ONLY SUFFICIENT

, a sufficient condition is given by the following set ofinequalities:

(28)

that can also be written as

(29)

provided that

(30)

In this case, since , the condition on can beachieved, at a fixed frequency , by choosing the slab heightsufficiently small; in fact, the inequality in (30) can be expressedas

(31)

IV. MODAL PROPERTIES OF SNG GROUNDED SLABS

For the sake of completeness, propagation of surface waveson SNG grounded slabs is briefly addressed here. In SNGgrounded slabs, the squared wavenumber inside the slab isnegative: therefore, for surface waves with a real propagationconstant , the transverse wavenumber inside theslab is always purely imaginary, i.e., only evanescent surfacewaves may exist.

In the following, we separately consider proper real waveson MNG and ENG metamaterials in Sections IV-A and IV-B,respectively; a discussion on surface-wave suppression on SNGgrounded slabs is provided in Section IV-C.

A. MNG Grounded Slabs

In the case of MNG grounded slabs, the dispersion equationfor proper real TE waves is from (3)

(32)

in terms of the adimensional variables and of (12). Since (32)is formally equal to (11) for the case of evanescent TE properreal waves on a DNG grounded slab with , on the basisof the analysis carried out in Section III-A, it can be concludedthat a necessary and sufficient condition to avoid proper TE sur-face waves on an MNG grounded slab is that .

The dispersion equation for proper real TM waves on a MNGgrounded slab is from (3)

(33)

Since the left- and right-hand sides of (33) have opposite signs,it can be concluded that proper real TM waves cannot exist onan MNG grounded slab.

B. ENG Grounded Slabs

In the case of ENG grounded slabs, the dispersion equationfor proper real TE waves is from (3)

(34)

Since the left- and right-hand sides of (34) have opposite signs,it can be concluded that proper real TE waves cannot exist onan ENG grounded slab.

The dispersion equation for proper real TM waves on an ENGgrounded slab is from (3)

(35)

Since (35) is formally equal to (21) for the case of evanescentTM proper real waves on a DNG grounded slab with ,on the basis of the analysis carried out in Section III-B, it can beconcluded that a sufficient condition to avoid proper TM surfacewaves on a MNG grounded slab is

(36)

C. Discussion on Surface-Wave Suppression in SNG GroundedSlabs

On the basis of the results of Sections IV-A and IV-B, condi-tions for surface-wave suppression on SNG grounded slabs canbe summarized as shown in Table III.

For the case of MNG grounded slabs, surface-wave suppres-sion is achieved if and only if without any additionalcondition on the slab height. For the case of ENG grounded

BACCARELLI et al.: FUNDAMENTAL MODAL PROPERTIES OF SURFACE WAVES ON METAMATERIAL GROUNDED SLABS 1437

TABLE IIISUMMARY OF CONDITIONS FOR SUPPRESSION OF PROPER SURFACE

WAVES ON MNG AND ENG GROUNDED SLABS

slabs, a sufficient condition to achieve surface-wave suppres-sion is that provided that the slab height satisfies theinequality

(37)

V. NUMERICAL RESULTS

In order to verify the possibility to achieve proper surface-wave suppression in DNG grounded slabs, numerical resultsare presented here for the dispersion properties of TE and TMmodes of different kinds supported by two different structures,indicated in the following as Case 1 and Case 2. Results on im-proper and complex modes will also be shown in the dispersiondiagrams for the sake of completeness.

Case 1

The first considered case consists of a DNG grounded slabmade of a metamaterial medium modeled as in [2] and [3] withslab height and constitutive parameters chosen as in [19], i.e.,

mm and relative permeability and permittivity given by

(38)

where , GHz, and GHz. Theregion of simultaneously negative permeability and permittivityin this case ranges from GHz to GHz.

In Fig. 7, the values of the relative permeability (solid linewith circles) and permittivity (solid line with diamonds) are re-ported in a frequency range from GHz to GHz,together with the values of the product (light-gray solidline). For the considered medium model, when , theconditions expressed in (25) are never satisfied, while when

the conditions in (29) hold in the range of frequenciesfrom approximately GHz to GHz, representedas a shaded area in this figure. In order to have surface-wavesuppression inside this range of frequencies, the additional con-dition in (31), which fixes an upper limit for the slab height,has to be satisfied. Such a limit is also reported in Fig. 7 as afunction of frequency (dashed line). Since the slab height hasbeen chosen as mm, by inspection of this figure, it canbe concluded that the range of surface-wave suppression is ap-proximately from GHz to GHz, as delimited by thevertical dashed lines (more exactly, between 5.103–5.2 GHz).

The condition of surface-wave suppression will now be il-lustrated by means of the dispersion diagrams of the TE andTM modes supported by the considered structure. In Fig. 8, the

Fig. 7. Relative permeability � (solid line with circles) and relativepermittivity " (solid line with diamonds) as a function of frequency f for themedium model described by (38). The product � " is shown with a light-graysolid line. The frequency range in which the conditions in (29) are satisfied isrepresented as a shaded area. The maximum slab height h to have surface-wavesuppression, calculated according to (31), is reported with a dashed line.Adimensional units (a.u.) are reported on the left vertical axis.

Fig. 8. Dispersion curves of the TE and TE modes supported by agrounded metamaterial slab with slab medium as in Fig. 7 and slab heighth = 20 mm. The shaded area represents the predicted range of surface-wavesuppression for both TE and TM modes. Legend: normalized phase constants� =k : Solid lines: proper real ordinary waves, dotted lines: improper realordinary waves, light-gray solid line: proper real evanescent wave, blackdashed–dotted lines: complex waves. Normalized attenuation constants � =k .Gray dashed lines: complex waves. Thin solid line: � = k .

dispersion curves for the two TE modes with cutoff frequencynearest to the predicted range of surface-wave suppression (rep-resented as a shaded area) are reported. The modes are conven-tionally labeled and without any reference to the stan-dard mode labeling of DPS slabs; moreover, we will show fre-quency ranges in which the reported modes have a leaky regimewith a complex propagation constant . Withreference to normalized phase constants , proper real or-dinary waves are represented with solid lines, improper realordinary waves are represented with dotted lines, proper realevanescent waves are represented with a light-gray solid line,and complex waves are represented with dashed–dotted lines.Normalized attenuation constants are represented withgray dashed lines. The thin solid line represents the line where

, and divides the region corresponding to evanescentsurface waves above it from the region of ordinary surface wavesbelow it. For both the reported modes, at the cutoff frequency,

1438 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 4, APRIL 2005

Fig. 9. Dispersion curves of the TM , TM , and TM modes supportedby a grounded metamaterial slab, as in Fig. 8. The shaded area representsthe predicted range of surface-wave suppression for both TE and TM modes.Legend: as in Fig. 8.

where , one improper real solution turns into a realproper ordinary one. By increasing the frequency, this mergeswith another proper real ordinary solution of backward type ata splitting-point frequency, giving rise to a proper complex so-lution [27], [28]. It can be observed that the upper real properbranch for the mode is evanescent at lower frequencies andordinary at higher frequencies.

Turning now to TM modes, in Fig. 9, three modes are re-ported, conventionally labeled , , and (withoutany reference to the standard mode labeling of DPS slabs) withcutoff frequencies inside the same frequency range, as in Fig. 8;the predicted range of surface-wave suppression is representedagain as a shaded area. The and modes behave as the

mode in Fig. 8, their real proper branches being always ofthe ordinary kind. The mode instead behaves differently,being proper, real, and evanescent above cutoff, and improper,real, and ordinary below cutoff.

It can be observed that the predicted region of surface-wavesuppression starts at GHz, i.e., slightly above thesplitting-point frequency for the mode, and ends exactly atthe cutoff frequency of the mode. Inside such region, noproper real waves exist, as expected. Instead, proper complexbranches exist (for instance, of the , , , andmodes), together with improper real branches (for instance, the

mode). It can also be observed that the attenuation con-stants of the complex modes inside the predicted range of sur-face-wave suppression are very high, thus, they are not expectedto give rise to directive radiation if excited by a finite source, aswill be shown in Section VI.

It can also be pointed out that, on the basis of the dispersionresults presented in Figs. 8 and 9, the actual frequency rangewhere no proper surface waves exist is from GHzto GHz, thus, it contains, but is larger than, the predictedrange of surface-wave suppression. This is consistent with thefact that the latter was obtained by using a set of sufficient condi-tions for surface-wave suppression. In particular, with referenceto Table II, the condition for TE-wave suppression is only a suf-ficient one, whereas that for TM evanescent-wave suppressionis also necessary. Therefore, the reason why the region of sur-face-wave suppression is exactly predicted at its upper bound,

Fig. 10. Relative permeability � (solid line with circles) and relativepermittivity " (solid line with diamonds) as a function of frequency f forthe medium model described by (39). The product � " is shown with a graysolid line. The frequency range in which the conditions in (25) are satisfied isrepresented as a shaded area. The minimum slab height h to have surface-wavesuppression in the DNG and ENG ranges, calculated according to (27) and(37), respectively, is reported with a dashed line. Adimensional units (a.u.) arereported on the left vertical axis.

whereas it is not exactly predicted at its lower bound, is that itis limited on the left by the splitting-point frequency of a TEmode, whereas on the right, it is limited by the cutoff frequencyof an evanescent TM mode.

Case 2

The second considered case consists of a DNG grounded slabwith height mm and a metamaterial medium modeled asin [4] with relative permeabilities and permittivities chosen asin [29] and given by

(39)

where GHz, GHz, (i.e.,losses have been neglected), and GHz. The regionof simultaneously negative permeability and permittivity in thiscase ranges from GHz to GHz.

In Fig. 10, the values of the relative permeability (solid linewith circles) and permittivity (solid line with diamonds) are re-ported in a frequency range from GHz to GHz,together with the values of the product (light-gray solidline). For the considered medium model, when , theconditions expressed in (29) are never satisfied, while when

, the conditions in (25) hold in the range of frequen-cies from GHz to GHz, represented as a shadedarea in the figure. In order to have surface-wave suppression in-side this range of frequencies, the additional condition in (27),which fixes a lower limit for the slab height, has to be satis-fied. Such a limit is also reported in Fig. 10 as a function offrequency (dashed line). Since the slab height has been chosenas mm, by inspection of Fig. 10, it can be concluded that,when the medium is DNG, the range of surface-wave suppres-sion is the whole interval from GHz to GHz.

In Fig. 10, the shown range from GHz to GHzis a part of the frequency interval where the medium is ENG.According to the results reported in Table III, in order to achievesurface-wave suppression, a sufficient condition is that

, provided that the slab height is higher than the lower limit

BACCARELLI et al.: FUNDAMENTAL MODAL PROPERTIES OF SURFACE WAVES ON METAMATERIAL GROUNDED SLABS 1439

Fig. 11. Dispersion curves of the TE , TE , and TE modes supported bya grounded metamaterial slab with slab medium as in Fig. 10 and slab heighth = 5 mm. The shaded area represents the predicted range of surface-wavesuppression for both TE and TM modes. Legend: normalized phase constants� =ko: Gray dashed–dotted lines: improper complex waves. Normalizedattenuation constants � =ko: Black dashed lines: improper complex waves.Other lines: as in Fig. 8.

expressed in (37). Such a lower limit has also been reported inFig. 10, and it is seen to be the continuation of the lower limitvalid for the DNG case. Therefore, since mm and therelative permittivity is less than one in absolute value in all theENG range (although it is not completely shown in Fig. 10), itcan be concluded that no surface waves may also exist in theENG range.

As in Case 1, the condition of surface-wave suppression willbe illustrated by means of the dispersion diagrams of the rel-evant TE and TM modes. In Fig. 11, the dispersion curves ofthree TE modes, conventionally labeled , , and ,are reported in a frequency range between GHz and

GHz, together with the line ; the shaded areaagain represents the predicted range of surface-wave suppres-sion for the DNG range. Modes labeled and behaveas mode in Fig. 8, while mode is improper real belowcutoff and proper real above cutoff. Moreover, the proper realbranch of the mode is ordinary at lower frequencies andevanescent at upper frequencies; its normalized phase constanttends to infinity at GHz since there becomesgreater than one and, thus, the curve does not intersectthe curve any more (see Fig. 4).

It is interesting to note that, as the structure becomes ENG atGHz, the phase constants of the proper complex

and modes vanish, and these modes turn into impropercomplex modes (whose normalized phase constant is rep-resented with gray dashed–dotted lines and whose normalizedattenuation constant is represented with black dashedlines) [27], [28].

In Fig. 12, the dispersion curves of three TM modes, conven-tionally labeled , , and are reported in the samefrequency range as in Fig. 11, again, with the line andthe shaded area of predicted surface-wave suppression in theDNG range. The three modes have a similar behavior, the onlydifference being that the proper real branch of the modeis evanescent, while the proper real branches of the and

modes are ordinary; in this case, the complex branches ofthe three modes also remain proper in the shown ENG rangeabove GHz [27], [28].

Fig. 12. Dispersion curves of the TM , TM , and TM modes supportedby a grounded metamaterial slab as in Fig. 11. The shaded area representsthe predicted range of surface-wave suppression for both TE and TM modes.Legend: as in Fig. 8.

Fig. 13. Truncated metamaterial grounded slab with relevant physical andgeometrical parameters, as in Fig. 11. The finite-size slab is assumed to becircular with radius R = 10� .

As already observed in Case 1, also in Case 2, inside thepredicted range of surface-wave suppression, no proper surfacewaves, but only proper complex or improper real waves, exist.Moreover, also in this case, the attenuation constants of the com-plex modes are very high in almost the entire suppression range.

Finally, it can be observed that, in Case 2, the region of sur-face-wave suppression, in the DNG range, is exactly predicted,being limited on the left by the frequency GHz atwhich the phase constant of one evanescent TE mode tends toinfinity, whereas on the right being limited by the frequency

GHz, at which the material ceases to be DNG, be-coming SNG. In fact, with reference to Table I, the conditionfor TE proper evanescent surface-wave suppression is a neces-sary and sufficient one.

VI. RADIATION PATTERNS IN THE PRESENCE OF

TRUNCATED DNG SUBSTRATES

Here, the far field radiated by a dipole source in the pres-ence of a finite-size DNG slab is considered in order to verifysurface-wave suppression and to show its effects on radiationpatterns. A magnetic dipole source is assumed to be placed onthe ground plane of a DNG slab along the -axis to model ra-diation from a short and narrow slot; the finite-size slab is as-sumed to be circular with radius (see Fig. 13). Comparisonswill be presented between the far fields radiated in the presenceof an infinite and a finite-size DNG slab. In the infinite case, theradiated field can easily be calculated by means of the dyadicGreen’s function of the infinite grounded slab, which is known

1440 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 4, APRIL 2005

Fig. 14. Normalized radiation patterns (in decibels) of a magnetic dipoleplaced on the ground plane of a DNG slab along the x-axis in the elevationplanes: (a) � = 0 and (b) � = 90 . Physical parameters: as in Fig. 11 atf = 22:5 GHz. Legend: infinite structure: black line with diamonds; circularfinite structure (radius R = 10� ): gray line.

in a simple closed form [22]. In the finite case, a perfect ab-sorber is assumed to be present beyond the edges of the slab sothat the radiated field may be calculated through a physical-op-tics approximation of the aperture field on the air–slab interface;in particular, the convenient expressions reported in [30] havebeen used.

In Fig. 14, a DNG slab with physical parameters, as in Fig. 11,is considered at the frequency GHz; the radius of thefinite structure, as in Fig. 13, is , where is thefree-space wavelength. From Figs. 11 and 12, it can be seen that,at this frequency, one TE surface wave of evanescent type andno TM surface waves are present. In Fig. 14(a), the radiationpatterns of the infinite (black line with diamonds) and of the fi-nite (gray line) structures are presented in the elevation plane

; in Fig. 14(b), the same are shown in the elevation plane. The effect of TE surface-wave diffraction at the edges

of the finite structure is clearly evident in the plane,where the far field is mostly due to TE waves. On the other hand,the pattern in the plane is mainly due to TM waves,thus, the TE-wave diffraction deteriorates the pattern only closeto broadside. The broad maxima at are due to the exci-tation of the mode in a physical leaky regime with a nor-malized attenuation constant (see Fig. 12, wherethe and modes are seen to have a much higher atten-uation constant, thus giving a negligible contribution to the ra-diation pattern); however it may be noticed that, due to its rapiddecay, this leaky mode does not contribute appreciably toedge diffraction.

In Fig. 15, the same structure is considered at GHz.From Figs. 11 and 12, it can be seen that now neither TE, norTM surface waves (of any kind) are present. As a consequence,no diffraction effects are found in the radiation patterns, neither

Fig. 15. Same as in Fig. 14 at f = 23:5 GHz.

in the plane [see Fig. 15(a)], nor in the plane[see Fig. 15(b)]. In the plane, an almost circular patterncan be observed since the TE leaky waves have an extremelyhigh attenuation constant that does not give rise to any visiblebeam. Instead, in the plane, the maxima at areagain due to the excitation of the leaky mode, now with ahigher attenuation constant than at GHz, which resultsin an even broader beam.

VII. CONCLUSION

In this paper, an investigation has been done on modal prop-erties of metamaterial grounded slabs with particular referenceto surface waves. By means of a simple graphical approach tothe dispersion equations of the involved TE and TM modes onDNG grounded slabs, it has been shown that conditions may befound that ensure the absence (suppression) of any proper sur-face wave propagating along the considered structures. Theseconditions involve the constitutive parameters of the medium,slab height, and operating frequency. SNG media have also beenconsidered by deriving analogous conditions for proper surface-wave suppression both in the MNG and ENG cases.

Two different specific structures have been simulated with ex-perimentally tested models for the dispersion behavior of therelevant constitutive parameters. The results on the dispersionproperties of the TE and TM modes supported by the consid-ered structures have fully validated the predictions made on thebasis of the theoretical analysis concerning such novel modalfeatures. Moreover, the radiation patterns of a magnetic dipoleplaced on the ground plane of both infinite and truncated meta-material slabs have been presented in order to demonstrate thereduction of edge-diffraction effects when surface waves areabsent.

Even though the considered structures are idealized, and therealization of practical metamaterials with predictable constitu-tive relations may be difficult, the results of this study show that

BACCARELLI et al.: FUNDAMENTAL MODAL PROPERTIES OF SURFACE WAVES ON METAMATERIAL GROUNDED SLABS 1441

metamaterial grounded slabs have a potential as enhanced sub-strates for planar antennas and arrays with reduced edge-diffrac-tion and mutual coupling between elements.

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[13] J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.,vol. 85, pp. 3966–3969, Oct. 2000.

[14] G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative re-fractive index media using periodically L–C loaded transmission lines,”IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2702–2712, Dec.2002.

[15] I. S. Nefedov and S. A. Tretyakov, “Waveguide containing a backward-wave slab,” Radio Sci., vol. 38, pp. 1101–1109, Dec. 2003.

[16] A. Alú and N. Engheta, “Guided modes in a waveguide filled with apair of single-negative (SNG), double-negative (DNG), and/or double-positive (DPS) layers,” IEEE Trans. Microw. Theory Tech., vol. 52, no.1, pp. 199–210, Jan. 2004.

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[26] P. Baccarelli, C. Di Nallo, F. Frezza, A. Galli, and P. Lampariello, “Novelbehaviors of guided and leaky waves in microwave ferrite devices,” inProc. 8th Mediterranean Electrotechnical Conf., vol. 1, Bari, Italy, May13–16, 1996, pp. 587–590.

[27] P. Baccarelli, P. Burghignoli, F. Frezza, A. Galli, P. Lampariello, G.Lovat, and S. Paulotto, “The nature of radiation from leaky waves onsingle- and double-negative metamaterial grounded slabs,” in IEEEMTT-S Int. Microwave Symp. Dig., vol. 1, Fort Worth, TX, Jun. 8–13,2004, pp. 309–312.

[28] , “Effects of leaky-wave propagation in metamaterial groundedslabs excited by a dipole source,” IEEE Trans. Microw. Theory Tech.,vol. 53, no. 1, pp. 32–44, Jan. 2005.

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[30] H. Ostner, E. Schmidhammer, J. Detlefsen, and D. R. Jackson, “Radia-tion from dielectric leaky-wave antennas with circular and rectangularapertures,” Electromagnetics, vol. 17, pp. 505–535, May 1997.

Paolo Baccarelli (S’96–M’01) received the Laureadegree in electronic engineering and Ph.D. degree inapplied electromagnetics from “La Sapienza” Uni-versity of Rome, Rome, Italy, in 1996 and 2000, re-spectively.

In 1996, he joined the Department of ElectronicEngineering, “La Sapienza” University of Rome,where he is a Contract Researcher since 2000. FromApril 1999 to October 1999, he was a Visiting Scholarwith the University of Houston, Houston, TX. Hisresearch interests concern analysis and design of

planar leaky-wave (LW) antennas, numerical methods, periodic structures, andpropagation and radiation in metamaterials and anisotropic media.

Paolo Burghignoli (S’97–M’01) was born in Rome,Italy, on February 18, 1973. He received the Laureadegree (cum laude) in electronic engineering andPh.D. degree in applied electromagnetics from “LaSapienza” University of Rome, Rome, Italy, in 1997and 2001, respectively.

In 1997, he joined the Electronic EngineeringDepartment, “La Sapienza” University of Rome,where he is currently a Contract Researcher. FromJanuary 2004 to July 2004, he was a Visiting Re-search Assistant Professor with the University of

Houston, Houston, TX. His scientific interests include analysis and designof planar leaky-wave (LW) antennas, numerical methods for the analysis ofpassive guiding and radiating microwave structures, periodic structures, andpropagation and radiation in metamaterials.

Dr. Burghignoli was the recipient of the 2003 Graduate Fellowship Awardpresented by the IEEE Microwave Theory and Techniques Society (IEEEMTT-S).

Fabrizio Frezza (S’87–M’90–SM’95) received theLaurea degree (cum laude) in electronic engineeringand Doctorate degree in applied electromagneticsfrom the “La Sapienza” University of Rome, Rome,Italy, in 1986 and 1991, respectively.

In 1986, he joined the Electronic EngineeringDepartment, “La Sapienza” University of Rome,where, from 1990 to 1998, he was a Researcher,from 1994 to 1998, a Temporary Professor ofelectromagnetics, and since 1998, an AssociateProfessor of electromagnetics. His main research

activity concerns guiding structures, antennas and resonators for microwavesand millimeter waves, numerical methods, scattering, optical propagation,plasma heating, and anisotropic media.

Dr. Frezza is a member of Sigma Xi, the Associazione Elettrotecnica ed Elet-tronica Italiana (AEI), the Italian Society of Optics and Photonics (SIOF), theItalian Society for Industrial and Applied Mathematics (SIMAI), and the ItalianSociety of Aeronautics and Astronautics (AIDAA).

1442 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 4, APRIL 2005

Alessandro Galli (S’91–M’96) received the Laureadegree in electronic engineering and Ph.D. degreein applied electromagnetics from the “La Sapienza”University of Rome, Rome, Italy, in 1990 and 1994,respectively.

In 1990, he joined the Electronic EngineeringDepartment, “La Sapienza” University of Rome.In 2000, he became and Assistant Professor and, in2002, an Associate Professor with “La Sapienza”University of Rome, where he currently teacheselectromagnetic fields for telecommunications en-

gineering. His scientific interests mainly involve electromagnetic theory andapplications, particularly regarding analysis and design of passive devices andantennas (dielectric and anisotropic waveguides and resonators, leaky-wave(LW) antennas, etc.) for microwaves and millimeter waves. He is also active inbioelectromagnetics (modeling of interaction mechanisms with living matter,health safety problems for low-frequency applications and mobile communi-cations, etc.).

Dr. Galli was the recipient of the 1994 and 1995 Quality Presentation Recog-nition Award presented by the IEEE Microwave Theory and Techniques Society(IEEE MTT-S).

Paolo Lampariello (M’73–SM’82–F’96) was bornin Rome, Italy, on May 17, 1944. He received theLaurea degree (cum laude) in electronic engineeringfrom the University of Rome, Rome, Italy, in 1971.

In 1971, he joined the Institute of Electronics,University of Rome, where he was an AssistantProfessor of electromagnetic fields. Since 1976,he has been engaged in educational activities in-volving electromagnetic-field theory. In 1986, hebecame a Professor of electromagnetic fields. FromNovember 1988 to October 1994, he was Head of

the Department of Electronic Engineering, “La Sapienza” University of Rome,Rome, Italy. Since November 1993, he has been the President of the Councilfor Electronic Engineering Curriculum of “La Sapienza” University of Rome.Since September 1995 he has been the President of the Center Interdepartmentalfor Scientific Computing, “La Sapienza” University of Rome. From September1980 to August 1981, he was a North Atlantic Treaty Organization (NATO) Post-Doctoral Research Fellow with the Polytechnic Institute of New York, Brooklyn.He has been engaged in research in a wide variety of topics in the microwavefield including electromagnetic and elastic wave propagation in anisotropicmedia, thermal effects of electromagnetic waves, network representations of mi-crowave structures, guided-wave theory with stress on surface waves and leakywaves, traveling-wave antennas, phased arrays, and, more recently, guiding andradiating structures for the millimeter- and near-millimeter-wave ranges.

Prof. Lampariello is a member of the Associazione Elettrotecnica ed Elet-tronica Italiana (AEI). He is a past chairman of the Central and South ItalySection of the IEEE and a past chairman of the IEEE Microwave Theory andTechniques (MTT)/Antennas Propagation (AP) Societies Joint Chapter of thesame section. He is currently the President of the Specialist Group “Electro-magnetism” of the AEI.

Giampiero Lovat (S’02) was born in Rome, Italy,on May 31, 1975. He received the Laurea degree(cum laude) in electronic engineering from “LaSapienza” University of Rome, Rome, Italy, in2001, and is currently working toward the Ph.D.degree in applied electromagnetics at “La Sapienza”University of Rome.

From January 2004 to July 2004, he was a VisitingScholar with the University of Houston, Houston,TX. His scientific interests include theoretical andnumerical studies on leakage phenomena in planar

structures and guidance and radiation in metamaterials and general periodicstructures.

Simone Paulotto (S’97) received the Laurea degree(cum laude and honorable mention) in electronic en-gineering from “La Sapienza” University of Rome,Rome, Italy, in 2002, and is currently working towardthe Ph.D. degree in applied electromagnetics at “LaSapienza” University of Rome.

In 2002, he joined the Electronic EngineeringDepartment, “La Sapienza” University of Rome. Hisscientific interests focus on electromagnetic propa-gation/radiation in planar structures and scatteringtheory.