Analytical theory of wave propagation through stacked fishnet metamaterials

Analytical theory of wave propagation through stacked fishnet metamaterials R. Marqu´ es,1∗L. Jelinek,1F. Mesa,2and F. Medina1 1Departamento de Electr´ onica y Electromagnetismo. Universidad de Sevilla Avd. Reina Mercedes, 41012 Sevilla (Spain) 2Departamento de F´ ısica Aplicada 1, ETSII, Universidad de Sevilla Avd. Reina Mercedes, 41012 Sevilla (Spain) ∗mar[email protected] Abstract: This work analyzes the electromagnetic wave propagation through periodically stacked fishnets from zero frequency to the first Wood’s anomaly. It is shown that, apart from Fabry-Perot resonances, these structures support two transmission bands that can be backward under the appropriate conditions. The first band starts at Wood’s anomaly and is closely related to the well-known phenomena of extraordinary transmission through a single fishnet. The second band is related to the resonances of the fishnet holes. In both cases, the in-plane periodicity of the fishnet cannot be made electrically small, which prevents any attempt of homogenization of the structure along the fishnet planes. However, along the normal direction, even with very small periodicity transmission is still possible. An homogenization procedure can then be applied along this direction, thus making that the structure can behave as a backward-wave transmission line for such transmission bands. Closed-form design formulas will be provided by the analytical formulation here presented. These formulas have been carefully validated by intensive numerical computations. © 2009 Optical Society of America OCIS codes: (160.3918) Metamaterials; (050.2065) Effective medium theory; (160.1245) Artificially engineered materials; (160.5298) Photonic crystals References and links 1. S. Zhang, W. Fan, K. J. Malloy, and S. R. J. Brueck, “Near-infrared double negative metamaterials,” Opt. Express 13, 4922-4930 (2005). 2. S. Zhang, W. Fan, N. C. 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Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature (London) 455, 299-300 (2008). 8. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32 53-55 (2007). 9. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature (London) 391, 667-669 (1998). #110974 - $15.00 USD Received 5 May 2009; revised 17 Jun 2009; accepted 17 Jun 2009; published 25 Jun 2009 (C) 2009 OSA 6 July 2009 / Vol. 17, No. 14 / OPTICS EXPRESS 11582 10. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000). 11. A. Mary, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, “Theory of negative-refractive-index response of double-fishnet structures,” Phys. Rev. Lett. 101, 103902 (2008). 12. F. Medina, F. Mesa, and R. Marqu´ es, “Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective,” IEEE Trans. Microwave Theory Tech. 56, 3108–3120 (2008). 13. R. Marqu´ es, F. Mesa, L. Jelinek, and F. Medina “Analytical theory of extraordinary transmission through metallic diffraction screens perforated by small holes,” Opt. Express 17, 5571-5579 (2009) 14. J. D. Jackson, Classical Electrodynamics, (Edt. Wiley, New York 1999), 3rd Ed. 15. R. Gordon, “Bethe’s aperture theory for arrays,” Phys. Rev. A 76, 053806 (2007). 16. R. F. Collin Field Theory of Guided Waves, (Edt. IEEE Press, New York 1991), 2nd Ed., 17. S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, (Edt. Wiley, New York 1994), 3rd Ed., pp.263-264. 18. G. V. Eleftheriades, O.Siddiqui, and A. K. Iyer, “Transmission line models for negative refractive index media and associated implementations without excess resonators,” IEEE Microwave and Wirel. Compon. Lett. 13, 51- 53 (2003). 1. Introduction Stacked metallic plates perforated by periodic hole arrays –the so-called fishnet structures– were first proposed to obtain low-loss negative refractive index metamaterials at near-infrared frequencies [1, 2], and subsequently demonstrated at microwave and terahertz frequencies (where metals are almost perfect conductors) [3–5], in the far-infrared frequency range [6,7], and in the visible range [8]. This behavior was first observed in double fishnet structures [1,2,6,8], and later in multilayer structures [3,4] and prisms [5,7], where negative refraction at the output interface of the prism has been reported. Although the relation between negative refractive index in fishnet structures and extraordinary optical transmission through metallic plates perforated by periodic hole arrays [9] was clear from the beginning [2], the first attempts to explain the behavior of fishnet structures followed the standard interpretation of negative refraction in wire and split ring metamaterials [10]. Thus, the straight conducting strips of the fishnets were supposed to act as a “wire medium” providing a negative ε , whereas the coupled metallic strips along the orthogonal direction were supposed to form an LC resonator providing a negative μ [1,2,6–8]. This interpretation, however, could not explain why the transverse periodicity of all fishnet metamaterials was close to a wavelength in all reported experiments and electromagnetic simulations. For this reason, other interpretations linking the behavior of stacked fishnet metamaterials to the presence of extraordinary optical transmission through a single layer fishnet have been proposed [3,5,11]. The main aim of this letter is to develop an analytical theory of wave propagation through stacked fishnet structures from zero frequency to the first Wood’s anomaly. The analysis will start with a canonical structure (a periodic arrangement of single fishnets of square periodicity and negligible thickness) showing the most important properties of these stacked fishnets. This analysis generalizes our previous analysis of extraordinary transmission through metallic plates perforated by periodic hole arrays [12,13], and shows that extraordinary transmission in singlelayer fishnets and extraordinary propagation in stacked fishnets can both be explained within the same theoretical framework. From this analysis, closed-form expressions for the dispersion relation of an infinite periodic array of single-layer fishnets will be derived. After careful testing of these expressions by appropriate numerical simulations, they will be discussed and the main physical mechanisms behind wave propagation through stacked fishnet metamaterials will be identified. Finally, the analysis will be generalized to structures of higher complexity. #110974 - $15.00 USD Received 5 May 2009; revised 17 Jun 2009; accepted 17 Jun 2009; published 25 Jun 2009 (C) 2009 OSA 6 July 2009 / Vol. 17, No. 14 / OPTICS EXPRESS 11583 2. Analysis For the sake of simplicity we begin our analysis with the structure shown in Fig.1 assuming negligible thickness and square holes perforated in a periodic array of perfectly conducting metallic screens. Propagation along the normal direction (i.e. along zaxis) will be considered, with a time dependence ∝exp(−i ω t). The unit cell of this structure is also shown in Fig.1(c) and (d). Fig. 1. Periodic array of perfect conductor screens perforated with square holes: (a) front view and (b) lateral view. (c) Front and (d) lateral views of the structure unit cell or equivalent waveguide. For propagation along the zaxis the structure and the field are both invariant by reflections at the x=±a/2 and y=±a/2 planes. Owing to the specific properties of electric and magnetic fields under mirror symmetry [14, pp. 267-273], these planes must be virtual perfect electric or perfect magnetic conducting (PEC or PMC) walls. Choosing x-planes as PMCs and y-planes as PECs we obtain the unit cell depicted in Fig.1(c) and (d), which implies an y-polarized mode. It is also possible to make the opposite choice for the x- and yplanes, which leads to the orthogonal propagating mode (which, for the considered square periodicity, has the same dispersion equation). Choosing x- and y-planes both as PECs (or PMCs) leads - below Wood’s anomaly - to non propagative modes, which will not be considered here. Thus, for the considered propagation, the analyzed structure is equivalent to the TEM square waveguide periodically loaded by square irises shown in Fig.1(c) and (d). Following our previous analysis [13], the electric field, E0 y, produced by the currents flowing on the metallic screen at z=0 (see Fig.1) is expanded as the following series of the TEM, TE2n,2m, and TM2n,2mmodes of the infinite TEM waveguide with appropriate coefficients: E0 y(x,y)=A0+ N ∑ n=1 ATE n0fn0(x,y)+ M ∑ m=1 ATM 0mf0m(x,y)+ N,M ∑ n,m=1 (ATE nm +ATM nm )fnm(x,y)(1) where fnm(x,y)=cos(2n π x/a)cos(2m π y/a)(2) #110974 - $15.00 USD Received 5 May 2009; revised 17 Jun 2009; accepted 17 Jun 2009; published 25 Jun 2009 (C) 2009 OSA 6 July 2009 / Vol. 17, No. 14 / OPTICS EXPRESS 11584 gives the the dependence on x,yof the different TE2n,2mand TM2n,2mmodes of the waveguide. In (1) only even modes have been considered due to the symmetry of the structure, which includes two additional symmetry planes at x=0 and at y=0, respectively. From (1), using standard waveguide theory, the electric field component E0 xproduced by these currents is written as E0 x(x,y)= ∞ ∑ n,m=1m nATE nm −n mATM nm gnm(x,y)(3) where gnm(x,y)=sin(2n π x/a)sin(2m π y/a). Let us now assume that a Bloch wave of phase dependence ei β zis propagating along the structure. The total field at z=0 is given by the addition of (1) and the incident field, Einc y, created by the currents on all the remaining screens at z=qp, where pis the periodicity along the z-axis and q=±1,±2,.... Near Wood’s anomaly, where the first higher order modes TE20 and TM02 in the expansion (1) are at cutoff, the attenuation constant of such modes α =k0 λ a2 −1; k0= ω √ ε 0 μ 0(4) is very small and these modes contribute to the incident field at z=0 regardless of the periodicity of the structure. Thus, for not very small periodicities, where additional higher order modes may be present, the incident field at z=0 can be expressed as Einc y=ETEM y+ETE y+ETM y(5) where ETEM y=∑ q=0 A0ei β qpeik0|q|p=A0cos( β p)−eik0p cos(k0p)−cos( β p)(6) ETE y=∑ q=0 ATE 10 f10 ei β qpe− α |q|p=ATE 10 f10 cos( β p)−e− α p cosh( α p)−cos( β p)(7) ETM y=∑ q=0 ATM 01 f01 ei β qe− α |q|p=ATM 01 f01 cos( β p)−e− α p cosh( α p)−cos( β p).(8) The presence of the terms ETE yand ETM yin the incident field at z=0 makes our analysis different from the standard analysis of periodic structures [16, Sec. 9], which usually only considers the coupling through the fundamental TEM mode of the waveguide. As it will be shown in the following, it is the coupling through the TM02 mode what creates a transmission band starting at Wood’s anomaly which, depending on the structural parameters, can provide a backward wave propagating along the stacked fishnet. After this point the analysis follows the same procedure as in [13]. First, the coefficients of (1) are obtained in the frame of the small hole approximation. In this approximation the xcomponent of the electric field (3) is Ex(x,y)0≈0, which follows from the boundary conditions at the metallic screen, and from the specific behavior of gnm(x,y)for x,y≈0. This approximation gives the relation m nATE nm −nn mATM nm ≈0; n,m=0.(9) In order to complete the determination of the coefficents, we make use of the approximation [15]: wEyfnm dxdy=hEyfnm dxdy≈hEydxdy=wEydxdy(10) #110974 - $15.00 USD Received 5 May 2009; revised 17 Jun 2009; accepted 17 Jun 2009; published 25 Jun 2009 (C) 2009 OSA 6 July 2009 / Vol. 17, No. 14 / OPTICS EXPRESS 11585 where Ey=E0 y+ETEM y+ETE y+ETM yis the total field at z=0, and subindex wand hstand for the waveguide and hole sections respectively. This condition provides the equation wEyfnm dxdy=a2A01+cos( β p)−eik0p cos(k0p)−cos( β p)(11) which, after expanding Eyin a series similar to (1) and using the orthogonality properties of fnm, suffices for the determination of all the coefficients as a function of the first coefficient A0. Once all coeffcients in (1) have been determined, the tangential magnetic field created by the currents at z=0, H0 x, is obtained from (1) using waveguide theory [16, Sec. 5]. Finally, the phase constant β of the Bloch wave is obtained by imposing the condition H0 x=0 at the aperture [14, p. 486]. After some cumbersome but straightforward calculations, the dispersion equation for the Bloch wave can be written in the following way: cos( β p)=1+Yp Ys(12) where Yp=−2iY0sin2(k0p/2) sin(k0p)+4YTE 20 +YTM 02 sinh2( α p/2) sinh( α p)sinc π b a+Y(13) Ys=iY0csc(k0p)+2YTE 20 +YTM 02 csch( α p)sinc π b a(14) with sinc(x)=sin(x)/x, and Y=2N ∑ n=2 YTE 2n,0sincn π b a+2M ∑ m=2 YTM 0,2msincm π b a +4N,M ∑ n,m=1YTE 2n,2mn2 n2+m2+YTM 2n,2mm2 n2+m2sincn π b asincm π b a(15) where Y0= ε 0/ μ 0is the wave admittance of free space, and YTE 2n,2mand YTM 2n,2mare the admittances of the TE and TM waveguide modes given by YTE 2n,2m=iY0n λ a2 +m λ a2 −1 (16) YTM 2n,2m=−iY0/n λ a2 +m λ a2 −1 (17) and the upper limits of the summations in (15) are taken as N,M=round(a/b)[13]. It should be emphasized that, for practical values of aand b, the value of N,Mis usually very low (no more than five or six), which means that very few terms should be summed up in the series in (15). 3. Discussion and numerical results The dispersion equation (12) formally corresponds to the periodic equivalent circuit whose unit cell is shown in Fig.2(a) [16, Sec. 9], which takes the well known form β p=−2Yp Ys(18) #110974 - $15.00 USD Received 5 May 2009; revised 17 Jun 2009; accepted 17 Jun 2009; published 25 Jun 2009 (C) 2009 OSA 6 July 2009 / Vol. 17, No. 14 / OPTICS EXPRESS 11586 Fig. 2. Unit cells of (a) the periodic transmission line corresponding to the dispersion equation (12) for a stacked fishnet witht→0, (b) the backward-wave transmission line reported and analyzed in [17, pp. 263-264] and [18], (c) the equivalent circuit proposed in Sec. 4 for thick stacked fishnet. in the limit of electrically small periodicities ( β p1). This last equation corresponds to an infinite transmission line of periodicity p, parallel admittance Yp, and series admittance 2Ys. The interpretation of the different admittances in (13)–(15) is straightforward. The parallel admittance (13) has three terms. The third termYis the contribution of all the waveguide modes localized at the discontinuities, i.e., of all the TE2n,2m, and TM2n,2mmodes with n+m>1. The first and the second terms in (13) correspond to the contributions of the non-localized modes, i.e., the TEM, TE20, and TM02 modes. As expected, only these non localized modes contribute to the series admittance (14). The main difference between these equations and the standard equations for periodically loaded transmission lines and waveguides [16, Sec. 9] is the presence of the non-localized TE20 and TM02 modes in the series admittance Ys.Asitwas already mentioned, this presence is imposed by the small values of the attenuation constant (4) near Wood’s anomaly, and it provides the most relevant physical effects in stacked-fishnet structures. In particular, just at Wood’s anomaly, the attenuation constant (4) vanishes and the corresponding TM02 mode admittance (17) becomes infinite. Also the series admittanceYs(14) becomes infinite and the phase constant β vanishes. Hence, there is always a transmission band starting at Wood’s anomaly with zero phase and group velocity. Just below Wood’s anomaly, the series admittance is dominated by the TM02 admittance, which is capacitive (ℑ(YTM 02 )< 0). Thus, if the parallel admittance is inductive (ℑ(Yp)>0), according to (18) there must be a backward-wave transmission band [17, pp. 263-264] just below Wood’s anomaly. Now we will examine under which conditions the parallel admittance is inductive. For small holes, the contribution of the localized modes (15) is always inductive [12,16]. For small values of the longitudinal periodicity p, the contribution of the two first terms in (13) is negligibly small (of the order ofY0k0p). Therefore, for small holes and periodicities, a backward-wave transmission band is always expected just below Wood’s anomaly. After the discussion of some of the qualitative predictions of our analytical model, some numerical results will be presented next. Thus, Fig.3 and Fig.4 show some dispersion diagrams for the stacked-fishnet structure shown in Fig.1 for different structural parameters. For comparison purposes, results obtained using the commercial electromagnetic solver CST Microwave Studio are also shown. These plots qualitatively confirm the abovementioned backward-wave behav- #110974 - $15.00 USD Received 5 May 2009; revised 17 Jun 2009; accepted 17 Jun 2009; published 25 Jun 2009 (C) 2009 OSA 6 July 2009 / Vol. 17, No. 14 / OPTICS EXPRESS 11587 Fig. 3. Dispersion diagrams for a periodic array of stacked fishnets as that shown in Fig.1 with negligible thickness t→0, periodicity p=a/3, and different values of the hole size b. Solid lines are calculated using formulas (12)–(15). Circles correspond to numerical simulations by CST Microwave Studio. Fig. 4. Dispersion diagrams for a periodic array of stacked fishnets as that shown in Fig.1 with negligible thickness t→0, hole size b=a/6, and different values of the periodicity p. Blue solid lines (Analytical 1) are calculated using formulas (12)–(15). Red dashed lines (Analytical 2) are obtained using (20)–(21) for the parallel and series admittances. Circles correspond to numerical simulations by CST Microwave Studio. ior, and also show a good quantitative agreement in most cases. The disagreement between the analytical and intensive numerical computations for the biggest hole sizes shown in Fig.3 can be attributed to the approximations made in our model. In fact, approximation (10) is expected to work properly only for small holes [15]. In Fig.4 there is also some disagreement between the numerical values obtained from formulas (12)–(15) [the curve denoted as “Analytical 1”] and the CST numerical computations, specifically for small periodicities. This disagreement is explained by the extra coupling between consecutive screens by higher order modes. When this extra coupling is taken into account (see Section 4) these discrepancies disappear [as shown by the satisfactory agreement between the so-called “Analytical 2” curve and the CST results]. Figs.3 and 4 also show that the bandwidth of the backward transmission band increases as the hole size increases but it decreases as the periodicity grows. #110974 - $15.00 USD Received 5 May 2009; revised 17 Jun 2009; accepted 17 Jun 2009; published 25 Jun 2009 (C) 2009 OSA 6 July 2009 / Vol. 17, No. 14 / OPTICS EXPRESS 11588 Fig. 5. Dispersion diagrams for a periodic array of stacked fishnets as that shown in Fig.1 with negligible thickness t→0, hole size b=a/2, and periodicity p=a/3. Solid lines were calculated using formulas (12)–(15). Circles correspond to numerical simulations by CST Microwave Studio. As is well known, the holes become resonant when their size approaches to a halfwavelength. Below Wood’s anomaly, this situation only occurs if ba/2. In that case, there appears an additional transmission band associated with the resonance of the holes. In our model, the appearance of this additional transmission band can be linked to a zero value of the parallel admittance Ypin (12). Since the presence of this zero causes a change of sign in the parallel admittance, the transmission band starting at Wood’s anomaly will be forward. However the nature of the transmission band related to the resonance of the hole cannot be predicted in advance as it will depend on the sign of the series admittance. This last behavior is shown in Fig.5, where two forward transmission bands (one of them starting at Wood’s anomaly) can be seen. The agreement between our analytical results and the numerical simulations is still quite good in Fig. 5, in spite of the fact that now the approximation (10) can be hardly justified. This agreement is surprisingly good for the upper band as β p2. When the periodicity pbecomes a multiple of a half-wavelength, our analytical model still predicts the expected additional narrow transmission bands linked to Fabry-Perot resonances (their nature can be forward/backward depending on the longitudinal periodicity). However, the study of this kind of resonances is beyond the scope of the present work. In summary, there can be three independent mechanisms for electromagnetic wave propagation through stacked-fishnet structures operating below or near Wood’s anomaly. One of them is associated with the pole that appears in the first evanescent TM mode admittance just at Wood’s anomaly, and it is always present regardless of the hole size and periodicity p. The second one is related to the resonances of the holes, and appears as the hole size approaches to a half of the transverse periodicity a. Finally, when the longitudinal periodicity pbecomes a multiple of a half-wavelength, narrow transmission bands linked to Fabry-Perot resonances appear. Note that in all the above mechanisms one or more of the periodicities are not electrically small. Therefore, stacked-fishnet structures cannot be homogenized, at least when “homogenization” is taken in its usual meaning. However, the two first abovementioned mechanisms of wave propagation are compatible with an electrically small longitudinal periodicity p. In this case, the dispersion equation can be approximated by (18). Under this approximation, and assuming that the resulting transmission band is backward, the stacked-fishnet structure behaves like the backward-wave transmission line whose unit cell is sketched in Fig.2(b). These transmission lines are well known by electrical engineers, and have been analyzed in many textbooks [17, pp. #110974 - $15.00 USD Received 5 May 2009; revised 17 Jun 2009; accepted 17 Jun 2009; published 25 Jun 2009 (C) 2009 OSA 6 July 2009 / Vol. 17, No. 14 / OPTICS EXPRESS 11589 263-264]. More recently, they have been proposed as an alternative way to obtain left-handed metamaterials (see [18] and references therein) operating at microwave frequencies. Thus, in this specific case and in this specific sense, the stacked-fishnet structure can be considered as a 1D left-handed metamaterial with effective permittivity, ε , and permeability, μ , given by [18] ε =−1 ω 2Lpp, μ =−2 ω 2Csp(19) where Csand Lscome from Ys=−i ω Csand 1/Yp=−i ω L. An additional conclusion of our analysis is that backward-wave propagation in stacked fishnets can be dealt within the same theoretical framework as extraordinary transmission through single-layer fishnets (see [12] and [13]). It also shows that the physical mechanism behind this behavior is not linked to the presence of any kind of LC resonant circuits, as it was proposed in many previous works [1, 2, 6–8]. On the contrary, backward-wave propagation in stacked fishnets closely resembles backward-wave propagation in transmission line metamaterials [18], as it was previously suggested in [3]. 4. Generalization of the analysis The analysis of the canonical structure studied in Section 2 has shown the most salient features of stacked fishnets, and has provided qualitative results that can easily be generalized to more specific structures. In this section, this analysis will be extended to more general configurations. These generalizations include thick screens, arbitrary periodicities in the transverse plane, rectangular holes, and very small longitudinal periodicities. In all these generalization, the metallic screens will still be considered as perfect conductors. Since the same phenomena observed in stacked fishnets at infrared and optical frequencies were also observed at RF and microwave frequencies, where metallic screens are almost perfect conductors, we feel that the main conclusions extracted from our analysis can also be generalized to the optical range, at least qualitatively. For arbitrary periodicities along the xand yaxes, axand ay, and for rectangular holes of dimensions bxand byalong these axes, the generalization of (12)–(15) is straightforward. Moreover, if the periodicity along the longitudinal direction pis small, coupling between the metallic screens by higher order evanescent modes may appear. Specifically, this coupling will appear when the product α nmpis small, where α nm =k0(n λ /ax)2+(m λ /ay)2−1 is the attenuation constant of the TE2n,2mor TM2n,2mmodes. For the above cases, the analysis leads to the following general expressions for Ypand Ysin (12): Yp=−2i ax ayY0sin2(k0p/2) sin(k0p) +4ax ay N ∑ n=1 YTE 2n,0sinh2( α n0p/2) sinh( α n0p)sincn π bx ax+4ax ay M ∑ m=1 YTM 0,2msinh2( α 0mp/2) sinh( α 0mp)sincm π by ay +8ax ay N,M ∑ n,m=1YTE 2n,2mn2 n2+m2+YTM 2n,2mm2 n2+m2sinh2( α nmp/2) sinh( α nmp)sincn π bx axsincm π by ay (20) #110974 - $15.00 USD Received 5 May 2009; revised 17 Jun 2009; accepted 17 Jun 2009; published 25 Jun 2009 (C) 2009 OSA 6 July 2009 / Vol. 17, No. 14 / OPTICS EXPRESS 11590