Growing Evanescent Envelopes and Anomalous Tunneling in Cascaded Sets of Frequency-Selective Surfaces in Their Stop Bands

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1 Growing Evanesent Envelopes and Anomalous Tunneling in Casaded Sets of Frequeny-Seletive Surfaes in Their Stop ands Andrea Alù Dept. of Applied Eletronis, University of Roma Tre, Rome, Italy. Nader Engheta Dept. of Eletrial and Systems Engineering, University of Pennsylvania, Philadelphia, PA, USA. Abstrat: The presene of wave tunneling and the growing evanesent envelope for field distributions in suitably designed, periodially layered staks of frequeny seletive surfaes (FSS) is disussed in this paper. It is known that a slab of double-negative (DNG) or left-handed metamaterial sandwihed between two vauum half-spaes may lead to growing evanesent plane waves, whih may be used to restore sub-wavelength information in Pendry s perfet lens [Phys. Rev. Lett., 85, 3966, (000)]. In this paper, it is shown that a ompletely different setup allows an analogous buildup of evanesently modulated waves. In partiular, it is shown how an interfae resonane phenomenon similar to the one present at the interfae between metamaterials with oppositely signed onstitutive parameters may be indued by a proper hoie of the periodiities of the FSS staks and the geometrial properties of these surfaes. The analysis is performed through an equivalent transmission-line approah, and some physial insights into this phenomenon are presented. Salient features, suh as the omplete wave tunneling through the pair of asaded FSS, eah operating at its bandgap, are presented and disussed. PACS numbers: 4.0.Jb, 78.0.Ci, 4.5.s, 4.30.Wb INTRODUCTION In his paper desribing the idea of perfet lens using the slab of properly seleted double-negative (DNG) media, i.e., a medium in whih the real permittivity and permeability are simultaneously negative (also known as left-handed or negative index media), Pendry theoretially showed that inside suh a metamaterial slab the evanesent wave has a growing exponential, whih ontributes to the higher resolution of the image []. This idea has generated a lot of interests in the field of metamaterials, and many groups worldwide have been reently studying this issue of reonstrution of the evanesent waves and its sub-wavelength fousing. A natural question may arise: May the growing exponential happen in other lasses of media or in properly designed strutures, perhaps arguably more easily realizable than DNG media? Some ideas have been already reported in the literature [], [3]. We have been interested to explore the answers to this question and to find other alternatives. Here, we show that a somewhat analogous phenomenon in the form of growth of evanesently modulated waves may our inside suitably designed periodi arrangements of staks of frequeny-seletive surfaes (FSS) within onventional (i.e., double-positive (DPS)) media. Frequeny Seletive Surfaes (FSS), whih are onstruted as planar -dimensional periodi arrays of metalli elements with speifi geometrial shapes, or as periodi apertures in a metalli sreen, have been widely studied and known for many years [4]. The transmission and refletion oeffiients for these surfaes are dependent on the frequeny of operation (hene their names) and may also depend on the polarization and angle of inidene. In partiular, the possibility of having frequeny bands at whih a given FSS is ompletely opaque (stop-bands) and others at whih the same surfae allows wave transmission has attrated the attention of sientists and designers. In this work, we show how by periodially staking two sets of FSS with dual behaviors (e.g., one stak of FSS with holes in the metalli plate, and the other a stak of FSS with metalli pathes over the surfae of a very thin dieletri holder) one may ahieve growing field distributions towards the interfae between these two staks, and may have omplete tunneling of wave through this pair, even though eah stak alone would be below ut-off (i.e., in its stop band), similar to what we have obtained for an inident plane wave tunneling through a pair of properly hosen single-negative (SNG) bilayers [5]. To whom orrespondene should be addressed, engheta@ee.upenn.edu

2 THEORETICAL ANALYSIS It is well known that an equivalent iruit model representation may be used for plane wave propagation and i t for its interation with FSS [4]. In this transmission-line analogy, under a time-harmoni e ω exitation, an FSS may be represented as a shunt load, either indutive or apaitive depending on its geometrial parameters, in a transmission line (TL) segment representing the surrounding spae. A asade of FSS with dieletri materials filling the spae between them (shown in Fig., top part) would therefore be modeled as a series of TL ells periodially loaded by shunt elements, as in Fig., bottom part. The distane between every FSS would orrespond to the distane between the loads, and the host TL parameters Z and k TL depend on the angle of inidene, the frequeny and the polarization of the inident plane wave. The values of the shunt admittanes representing the FSS depend diretly on its geometrial parameters, in addition to the frequeny, polarization, and angle of inidene of the inoming wave. Suh parameters have been extensively studied and reported in the literature (see e.g., [4]) for several types of FSS. Obviously, in the limit of no loss, the values of the shunt admittanes of FSS are assumed to be purely imaginary. Referring to Fig., therefore, the length of every single ell d equals the distane between the two neighboring FSS and every TL unit ell has been hosen to have its shunt load in the middle (at d /) to provide symmetry for the ease of analysis. The shunt admittane Y = i represents the lossless FSS. Fig. A asade of equally spaed FSS (top) and its iruit model representation (bottom) The problem of plane wave interation with suh a system, therefore, is equivalent to finding the solution for the voltage and urrent propagation along suh a periodially loaded TL. The solutions an be determined from the well known loh equation (see e.g., [6]): Z oshγ d = os ktld sin ktld, () where γ is the loh wave number, whih desribes the e γ z -exponential dependene of the propagation/attenuation along z. A purely imaginary γ orresponds to a propagating loh mode, whereas a positive real γ orresponds to a deaying evanesent wave, representing the stop-band of this periodi struture. In the present work, we show how two different asades of FSS (one with indutive FSS and the other with apaitive FSS), when properly designed, may exhibit growing evanesently modulated field distributions and omplete tunneling of inident wave through this pair, although eah asade alone operates in its stop-band. In our previous work [5] for a pair of only-epsilon-negative (ENG) and only-mu-negative (MNG) layers (or for a pair of DPS-DNG layers), we have derived the onditions for the transpareny, resonane and omplete tunneling of a wave inident on the pair, whih depended on the harateristi impedanes of the two metamaterials and their eletrial lengths. In the ase here, analogous onditions may be derived for wave transpareny and interfae resonane for the above-mentioned pair of asades of dual FSS. The loh impedane, whih represents the ratio of voltage and urrent at the input setion of every ell for an infinite periodially and homogeneously loaded TL, may be written as:

3 Z Z os ktld + sin ktld Z =± iz. () Z os ktld sin ktld Not surprisingly, by omparing () and () it an be verified that a real-valued Z is possible only in the pass-bands, and onversely Z an be imaginary in the stop-bands, when no real power is transferred. The sign should be positive when we are in the pass-bands (and Z is real), whereas it may be positive or negative when Z is imaginary, depending on the apaitive or indutive behavior of the evanesent deay along the TL. Let us onsider now the ase of two different periodially loaded TL segments, with loh impedanes Z and Z and eletrial loh lengths g = N γ d and g = N γ d, where N and N are the numbers of unit ells in the first and seond asades, staked one after the other as in Figure a. We are not limiting ourselves to be only in the stop-bands or only in the pass-bands, i.e., the loh parameters may be in general omplex. The input impedane at the onnetion between the two segments looking to the right may be found by the onventional TL formula: Z + Ztanh g ( z = Nd) = Z. (3) Z + Z tanh g The input impedane at the input of the whole system beomes: ( z = Nd) + Ztanh g ( z = 0) = Z. (4) Z + Z z = N d tanh g in( ) Requiring that the system is totally transparent to the inident wave implies that ( 0) z = = Z. y inspetion, the neessary and suffiient onditions to this equation are given by the two alternative solutions: Z = Z Z = Z, (5) g = g g = g whih are formally equivalent to those derived in [5] for metamaterial bilayers. For passive loads, the ondition on the right learly orresponds to loh segments in their pass-bands (with imaginary γ and positive real Z ), whereas the ondition on the left requires the two segments to be both in the stop-band (with positive γ and onjugate imaginary Z ). When the left ondition is applied, if one FSS asade in its stop-band has an indutive behavior (i.e., Z = iz ) the other should have a apaitive behavior ( Z = iz ). An interfae resonane in this ase may our, analogous to what we have shown for the ENG-MNG bilayers [5], leading to the ondition for having growing exponential modulated waves. Proper seletion of eletrial lengths for the two asades satisfying the ondition N γ d = N γ d, moreover, leads to a omplete tunneling of the evanesent loh waves through the overall system. In other words, if eah of the two FSS asades alone is illuminated by a plane wave, most of the inident energy is refleted bak beause it is designed to be at its band gap. However, by pairing the two FSS asades that satisfy the left ondition given in (), an inident wave may tunnel through ompletely, resulting in no refletion from this pair. We also observe a growing evanesent envelope for the field distribution inside this pair. It is interesting to note that: a) the ondition does not depend on the outer medium, i.e., the system is transparent independently on what surrounds it as long as the outside media on both sides are the same, and b) there is potentially no limitation on the total length of the whole system, but the ondition is on the ratio between the two lengths. Of ourse, when realisti losses are onsidered a higher tunneling is expeted for shorter segments, due to absorbing phenomena and to the fat that the ondition on the harateristi impedanes an no longer be satisfied exatly, however the phenomenon still remains oneptually unhanged. oth these peuliar harateristis had been shown in [5] for the ENG-MNG bilayer. Fig. presents the voltage distribution along the pair of FSS asades shown in Fig. a. The TL representation of the system has also been depited. Fig. b learly shows that the voltage at the entrane fae is ompletely reovered at the exit fae, thus it has tunneled through, and at every ell of the first

4 asade the loh mode experienes an exponentially growing envelope for the field distribution. In Fig., the first asade is effetively removed and the only the seond asade is present. As an be seen, the voltage distribution in the remaining asade exhibits a deaying envelope, and the refleted wave is non zero. A similar effet may be observed if we remove the seond asade and keep the first one only. y omparing Figs. b and, we see the effet of pairing the two FSS asades that results in the growing evanesent envelope at the interfae. The seond ondition in (5) orresponds to the propagating loh modes (i.e., when the FSS are both operating at their pass-band) with opposite behaviors; one having a forward propagation, while the other having a bakward propagation. Also in this ase, somehow analogous to the pair of DPS-DNG slabs [7], [8], a total transpareny may be ahieved with a proper design of the line parameters. The phenomenon underlying this anomalous behavior, as also in the ase of Pendry s lens, relies on the interfae resonane present at the juntion of the two FSS asades. We should remember, in fat, that this analysis is valid in the steady-state regime, when the multiple refletions at the resonant interfae have already built up. A detailed explanation of the iruit equivalene of suh growing evanesent fields for Pendry s lens, has been given by us in [9]. a) b) ) Figure a) Staked FSS asades with dual behavior and their iruit model, e.g. one stak of FSS with holes in the metalli plate, and the other a stak of FSS with metalli pathes over the surfae of a very thin dieletri holder; b) Voltage distribution for two staks satisfying the first of (), kd = 7π / 4, kd = π, N = 6, N = 0, Z = 099., Z = ; ) The same, but Z = 0. CONCLUDING REMARKS Here we have shown how even a simple olletion of FSS with onjugate (dual) behavior, whih is indeed easily onstrutible, may support the growing exponential modes that have attrated so muh attention in reent years. In partiular, it has been shown how a resonane phenomenon similar to the one present at the interfae between ENG and MNG or DPS and DNG media may be indued by a proper hoie of the spaing and the geometrial properties of the frequeny seletive surfaes. Interesting features, suh as the presene

5 of growing exponential envelopes and omplete tunneling through the pair of staked FSS, while eah alone operating at its band gap, are presented. These results are onsidered a first step towards sub-wavelength imaging systems employing FSS strutures. ACKNOWLEDGEMENTS This work is supported in part by the Fields and Waves Laboratory, Department of Eletrial and Systems Engineering, University of Pennsylvania. Andrea Alù was supported by the sholarship Isabella Sassi onadonna from the Italian Eletrial Assoiation (AEI). Portions of these results were presented at the 004 URSI International Symposium on Eletromagneti Theory, Pisa, Italy, May 4-7, 004. REFERENCES [] J.. Pendry, Phys. Rev. Lett., 85, 3966, (000). [] C. Luo, S. G. Johnson, J. D. Joannopoulos, and J.. Pendry, Phys. Rev., 68, 0455, (003). [3] S. Maslovski and S. Tretyakov, arxiv:ond-mat/0303 v, (003). [4]. A. Munk, Frequeny Seletive Surfaes: Theory and Design, (Wiley, New York, 000). [5] A. Alù, and N. Engheta, IEEE Trans. Antennas Propag., 5, 558, (003). [6] R. E. Collin, Field Theory of Guided Waves, (IEEE Press, 99). [7] A. Alù, and N. Engheta, IEEE Trans. Mirowave Theory Tehn., 5, 99, (004). [8] A. Alù, and N. Engheta, in Pro. of ICEAA 03, Turin, Italy, Sept. 8-, 003, pp [9] A. Alù, and N. Engheta, submitted to IEEE Trans. Ant. Prop., under review.