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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 14 — Jul. 6, 2009
  • pp: 11582–11593
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Analytical theory of wave propagation through stacked fishnet metamaterials

R. Marqués, L. Jelinek, F. Mesa, and F. Medina  »View Author Affiliations


Optics Express, Vol. 17, Issue 14, pp. 11582-11593 (2009)
http://dx.doi.org/10.1364/OE.17.011582


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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

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

1. S. Zhang, W. Fan, K. J. Malloy, and S. R. J. Brueck, “Near-infrared double negative metamaterials,” Opt. Express 13, 4922–4930 (2005). [CrossRef] [PubMed]

, 2

2. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental Demonstration of Near-Infrared Negative-Index Metamaterials,” Phys. Rev. Lett. 95, 137404 (2005). [CrossRef] [PubMed]

], and subsequently demonstrated at microwave and terahertz frequencies (where metals are almost perfect conductors) [3

3. M. Beruete, M. Sorolla, and I. Campillo “Left-handed extraordinary optical transmission through a photonic crystal of subwavelength hole arrays,” Opt. Express 14, 5445–5455 (2006). [CrossRef] [PubMed]

5

5. M. Navarro-Cía, M. Beruete, M. Sorolla, and I. Campillo, “Negative refraction in a prism made of stacked subwavelength hole arrays,” Opt. Express 16, 560–566 (2008). [CrossRef] [PubMed]

], in the far-infrared frequency range [6

6. G. Dolling, Ch. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 311800–1802 (2005). [CrossRef]

, 7

7. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila1, D. A. Genov1, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature (London) 455, 299–300 (2008). [CrossRef]

], and in the visible range [8

8. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 3253–55 (2007). [CrossRef]

]. This behavior was first observed in double fishnet structures [1

1. S. Zhang, W. Fan, K. J. Malloy, and S. R. J. Brueck, “Near-infrared double negative metamaterials,” Opt. Express 13, 4922–4930 (2005). [CrossRef] [PubMed]

, 2

2. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental Demonstration of Near-Infrared Negative-Index Metamaterials,” Phys. Rev. Lett. 95, 137404 (2005). [CrossRef] [PubMed]

, 6

6. G. Dolling, Ch. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 311800–1802 (2005). [CrossRef]

, 8

8. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 3253–55 (2007). [CrossRef]

], and later in multilayer structures [3

3. M. Beruete, M. Sorolla, and I. Campillo “Left-handed extraordinary optical transmission through a photonic crystal of subwavelength hole arrays,” Opt. Express 14, 5445–5455 (2006). [CrossRef] [PubMed]

, 4

4. M. Beruete, I. Campillo, M. Navarro-Cía, F. Falcone, and M. Sorolla “Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays,” IEEE Trans. Microwave Theory Tech. 55, 1514–1521 (2007).

] and prisms [5

5. M. Navarro-Cía, M. Beruete, M. Sorolla, and I. Campillo, “Negative refraction in a prism made of stacked subwavelength hole arrays,” Opt. Express 16, 560–566 (2008). [CrossRef] [PubMed]

, 7

7. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila1, D. A. Genov1, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature (London) 455, 299–300 (2008). [CrossRef]

], 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

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). [CrossRef]

] was clear from the beginning [2

2. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental Demonstration of Near-Infrared Negative-Index Metamaterials,” Phys. Rev. Lett. 95, 137404 (2005). [CrossRef] [PubMed]

], the first attempts to explain the behavior of fishnet structures followed the standard interpretation of negative refraction in wire and split ring metamaterials [10

10. D. R. Smith, W. J. Vier, D. C. Padilla,, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). [CrossRef] [PubMed]

]. 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

1. S. Zhang, W. Fan, K. J. Malloy, and S. R. J. Brueck, “Near-infrared double negative metamaterials,” Opt. Express 13, 4922–4930 (2005). [CrossRef] [PubMed]

, 2

2. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental Demonstration of Near-Infrared Negative-Index Metamaterials,” Phys. Rev. Lett. 95, 137404 (2005). [CrossRef] [PubMed]

, 6

6. G. Dolling, Ch. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 311800–1802 (2005). [CrossRef]

8

8. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 3253–55 (2007). [CrossRef]

]. 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

3. M. Beruete, M. Sorolla, and I. Campillo “Left-handed extraordinary optical transmission through a photonic crystal of subwavelength hole arrays,” Opt. Express 14, 5445–5455 (2006). [CrossRef] [PubMed]

, 5

5. M. Navarro-Cía, M. Beruete, M. Sorolla, and I. Campillo, “Negative refraction in a prism made of stacked subwavelength hole arrays,” Opt. Express 16, 560–566 (2008). [CrossRef] [PubMed]

, 11

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). [CrossRef] [PubMed]

].

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 z axis) will be considered, with a time dependence ∝ exp(-iωt). The unit cell of this structure is also shown in Fig.1(c) and (d). For propagation along the z axis the structure and the field are both invariant by reflections at the xa/2 and ya/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 y- planes, 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).

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.

Ey0(x,y)=A0+n=1NAn0TEfn0(x,y)+m=1MA0mTMf0m(x,y)+n,m=1N,m(AnmTE+AnmTM)fnm(x,y)
(1)

where

fnm(x,y)=cos(2nπxa)cos(2mπya)
(2)

gives the the dependence on x,y of the different TE2n,2m and TM2n,2m modes 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 E0x produced by these currents is written as

Ex0(x,y)=n,m=1(mnAnmTEnmAnmTM)gnm(x,y)
(3)

where gnm(x,y)=sin(2nπx/a) sin(2mπy/a).

Let us now assume that a Bloch wave of phase dependence eiβz is propagating along the structure. The total field at z=0 is given by the addition of (1) and the incident field, E inc y, created by the currents on all the remaining screens at z=qp, where p is 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

a=k0(λa)21;k0=ωε0μ0
(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

Eyinc=EyTEM+EyTE+EyTM
(5)

where

EyTEM=q0A0eiβqpeik0qp=A0cos(βp)eik0pcos(k0p)cos(βp)
(6)
EyTE=q0A10TEf10eiβqpeαqp=A10TEf10cos(βp)eαpcosh(αp)cos(βp)
(7)
EyTM=q0A01TMf01eiβqeαqp=A01TMf01cos(βp)eαpcosh(αp)cos(βp).
(8)

The presence of the terms E TE y and E TM y in the incident field at z=0 makes our analysis different from the standard analysis of periodic structures [16

16. R. F. CollinField Theory of Guided Waves, (Edt. IEEE Press, New York1991), 2nd Ed.,

, Sec. 9

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). [CrossRef]

], 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

13. R. Marqués, 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) [CrossRef] [PubMed]

]. First, the coefficients of (1) are obtained in the frame of the small hole approximation. In this approximation the x component 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

mnAnmTEnnmAnmTM0;n,m0.
(9)

In order to complete the determination of the coefficents, we make use of the approximation [15

15. R. Gordon, “Bethe’s aperture theory for arrays,” Phys. Rev. A 76, 053806 (2007). [CrossRef]

]:

wEyfnmdxdy=hEyfnmdxdyhEydxdy=wEydxdy
(10)

where Ey=E 0 y+E TEM y+E TE y+E TM y is the total field at z=0, and subindex w and h stand for the waveguide and hole sections respectively. This condition provides the equation

wEyfnmdxdy=a2A0(1+cos(βp)eik0pcos(k0p)cos(βp))
(11)

which, after expanding Ey in 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 A 0. Once all coeffcients in (1) have been determined, the tangential magnetic field created by the currents at z=0, H 0 x, is obtained from (1) using waveguide theory [16

16. R. F. CollinField Theory of Guided Waves, (Edt. IEEE Press, New York1991), 2nd Ed.,

, Sec. 5]. Finally, the phase constant β of the Bloch wave is obtained by imposing the condition H 0 x=0 at the aperture [14

14. J. D. Jackson, Classical Electrodynamics, (Edt. Wiley, New York1999), 3rd Ed.

, p. 486]. After some cumbersome but straightforward calculations, the dispersion equation for the Bloch wave can be written in the following way:

cos(βp)=1+YpYs
(12)

where

Yp=2iY0sin2(k0p2)sin(k0p)+4(Y20TE+Y02TM)sinh2(αp2)sinh(αp)sinc(πba)+Y
(13)
Y3=iY0csc(k0p)+2(Y20TE+Y02TM)csch(αp)sinc(πba)
(14)

with sinc(x)=sin(x)/x, and

Y=2n=2NY2n,0TEsinc(nπba)+2m=2MY0,2mTMsinc(mπba)
+4n,m=1N,M(Y2n,2mTEn2n2+m2+Y2n,2mTMm2n2+n2)sinc(nπba)sinc(mπba)
(15)

where Y0=ε0μ0 is the wave admittance of free space, and Y TE 2n,2m and Y TM 2n,2m are the admittances of the TE and TM waveguide modes given by

Y2n,2mTE=iY0(nλa)2+(mλa)21
(16)
Y2n,2mTM=iY0(nλa)2+(mλa)21
(17)

and the upper limits of the summations in (15) are taken as N,M=round(a/b) [13

13. R. Marqués, 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) [CrossRef] [PubMed]

]. It should be emphasized that, for practical values of a and b, the value of N,M is 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

16. R. F. CollinField Theory of Guided Waves, (Edt. IEEE Press, New York1991), 2nd Ed.,

, Sec. 9], which takes the well known form

βp=2Y2Ys
(18)

in the limit of electrically small periodicities (βp≪1). This last equation corresponds to an infinite transmission line of periodicity p, parallel admittance Yp, and series admittance 2Ys.

Fig. 2. Unit cells of (a) the periodic transmission line corresponding to the dispersion equation (12) for a stacked fishnet with t →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.

The interpretation of the different admittances in (13)–(15) is straightforward. The parallel admittance (13) has three terms. The third term Y is the contribution of all the waveguide modes localized at the discontinuities, i.e., of all the TE2n,2m, and TM2n,2m modes 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. As it was 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 admittance Ys (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(Y TM 02 <0). Thus, if the parallel admittance is inductive (ℑ(Yp)>0), according to (18) there must be a backward-wave transmission band [17

17. S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, (Edt. Wiley, New York1994), 3rd Ed., pp.263–264.

, 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

12. F. Medina, F. Mesa, and R. Marqués, “Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective,” IEEE Trans. Microwave Theory Tech. 56, 3108–3120 (2008). [CrossRef]

, 16

16. R. F. CollinField Theory of Guided Waves, (Edt. IEEE Press, New York1991), 2nd Ed.,

]. For small values of the longitudinal periodicity p, the contribution of the two first terms in (13) is negligibly small (of the order of Y 0 k 0 p). 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 behavior, 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

15. R. Gordon, “Bethe’s aperture theory for arrays,” Phys. Rev. A 76, 053806 (2007). [CrossRef]

]. 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.

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.
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.

ε=1ω2Lpp,μ=2ω2Csp
(19)

where Cs and Ls come from Ys=-iωCs and 1/Yp=-iωL.

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 x and y axes, ax and ay, and for rectangular holes of dimensions bx and by along these axes, the generalization of (12)–(15) is straightforward. Moreover, if the periodicity along the longitudinal direction p is small, coupling between the metallic screens by higher order evanescent modes may appear. Specifically, this coupling will appear when the product αnmp is small, where αnm=k0(nλax)2+(mλay)21 is the attenuation constant of the TE2n,2m or TM2n,2m modes. For the above cases, the analysis leads to the following general expressions for Yp and Ys in (12):

Yp=2iaxayY0sin2(k0p2)sin(k0p)
+4axayn=1NY2n,0TEsinh2(αn0p2)sinh(αn0p)sinc(nπbxax)+4axaym=1mY0,2mTMsinh2(α0mp2)sinh(α0mp)sinc(mπbyay)
+8axayn,m=1N,M(Y2n,2mTEn2n2+m2+Y2n,2mTMm2n2+m2)sinh2(αnmp2)sinh(αnmp)sinc(nπbxax)sinc(mπbyay)
(20)
Ys=iaxayY0csc(k0p)
+2axayn=1NY2n,0TEcsch(αn0p)sinc(nπbxax)+2axaym=1MY0,2mTMcsch(α0mp)sinc(mπbyay)
+4axayn,m=1N,M(Y2n,2mTEn2n2+m2+Y2n,2mTMm2n2+m2)csch(αnmp)sinc(nπbxax)sinc(mπbyay).
(21)

The factor ax/ay in (20) and (21) does not play any role in the dispersion equation. However, the introduction of this factor is convenient in order to make the admittances consistent with its circuit definition, namely, the ratio between the average current and the average voltage through the waveguide. Note that (20) and (21) reduces to (13) and (14) as αnmp increases up to a value that there is no coupling through the corresponding mode.

Fig. 6. 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 three different rectangular hole sizes. Solid lines were calculated using formulas (12) and (20),(21). Circles correspond to numerical simulations obtained by CST Microwave Studio.

When the dispersion diagrams of Fig. 4 are computed using (20) and (21) instead of (13) and (14), the plotted results are the red dashed curves also shown in Fig. 4 As it can be seen, for small periodicities p, the introduction of this additional coupling substantially improves the quantitative agreement with numerical simulations. New dispersion diagrams for a stacked fishnet structure with rectangular holes have also been computed. The results are shown in Fig.6. This Figure shows that there is again a satisfactory qualitative and quantitative agreement with numerical simulations.

Finally, the analysis can also be generalized to the case of thick screens using the equivalent circuit proposed in [12

12. F. Medina, F. Mesa, and R. Marqués, “Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective,” IEEE Trans. Microwave Theory Tech. 56, 3108–3120 (2008). [CrossRef]

] and [13

13. R. Marqués, 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) [CrossRef] [PubMed]

] for the analysis of extraordinary transmission through thick screens. This equivalent circuit is shown in Fig.2(c), where the transformer turns ratio is n=ax/bx and the transmission line admittance, defined as the ratio between the average current and the average voltage through the rectangular waveguide of length t, is Yh=YTE10 (by/bx), where Y TE10 is the wave admittance of the fundamental TE10 mode excited in the waveguide formed by the hole in the screen. After solving the circuit of Fig.2(c), it turns out that this circuit can still be represented by the elemental circuit of Fig.2(a), provided the original series and parallel admittances Ys and Yp are substituted by the following new series and parallel admittances:

Ys={1Ys+iYpsin(kht)+2Yh(cos(kht)1)(Yp2+Yh)2eikht(Yp2Yh)2eikht}1
(22)
Yp=(Yp2+Yh)2eikht(Yp2Yh)2eikht2Yh
(23)

5. Conclusions

Stacked fishnets have been previously proposed as an alternative way to negative refractive index metamaterials. Along this paper we have developed an analytical theory of electromagnetic wave propagation through stacked fishnet stuctures made of perfect conducting screens. Since the same phenomena observed at optical frequencies were also observed in the microwave range, where metals are almost perfect conductors, we feel that the main results of our analysis can be extended to stacked fishnets operating in the optical range, at least qualitatively.

Fig. 7. Dispersion diagrams for a periodic array of stacked fishnets as that shown in Fig.1 with periodicity p=a/3 and hole size b=a/6. Two different screen thicknesses have been considered. Solid lines correspond to our analytical formulas. Circles correspond to numerical simulations obtained by CST Microwave Studio.

Acknowledgments

This work has been supported by the Spanish Ministerio de Educación y Ciencia and European Union FEDER funds (projects TEC2007-65376, TEC2007-68013-C02-01, and CSD2008-00066), and by Junta de Andalucía (project TIC-253). L. Jelinek also thanks for the support of the Czech Grant Agency (project no. 102/08/0314).

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). [CrossRef] [PubMed]

2.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental Demonstration of Near-Infrared Negative-Index Metamaterials,” Phys. Rev. Lett. 95, 137404 (2005). [CrossRef] [PubMed]

3.

M. Beruete, M. Sorolla, and I. Campillo “Left-handed extraordinary optical transmission through a photonic crystal of subwavelength hole arrays,” Opt. Express 14, 5445–5455 (2006). [CrossRef] [PubMed]

4.

M. Beruete, I. Campillo, M. Navarro-Cía, F. Falcone, and M. Sorolla “Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays,” IEEE Trans. Microwave Theory Tech. 55, 1514–1521 (2007).

5.

M. Navarro-Cía, M. Beruete, M. Sorolla, and I. Campillo, “Negative refraction in a prism made of stacked subwavelength hole arrays,” Opt. Express 16, 560–566 (2008). [CrossRef] [PubMed]

6.

G. Dolling, Ch. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 311800–1802 (2005). [CrossRef]

7.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila1, D. A. Genov1, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature (London) 455, 299–300 (2008). [CrossRef]

8.

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 3253–55 (2007). [CrossRef]

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). [CrossRef]

10.

D. R. Smith, W. J. Vier, D. C. Padilla,, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). [CrossRef] [PubMed]

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). [CrossRef] [PubMed]

12.

F. Medina, F. Mesa, and R. Marqués, “Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective,” IEEE Trans. Microwave Theory Tech. 56, 3108–3120 (2008). [CrossRef]

13.

R. Marqués, 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) [CrossRef] [PubMed]

14.

J. D. Jackson, Classical Electrodynamics, (Edt. Wiley, New York1999), 3rd Ed.

15.

R. Gordon, “Bethe’s aperture theory for arrays,” Phys. Rev. A 76, 053806 (2007). [CrossRef]

16.

R. F. CollinField Theory of Guided Waves, (Edt. IEEE Press, New York1991), 2nd Ed.,

17.

S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, (Edt. Wiley, New York1994), 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). [CrossRef]

OCIS Codes
(160.1245) Materials : Artificially engineered materials
(050.2065) Diffraction and gratings : Effective medium theory
(160.3918) Materials : Metamaterials
(160.5298) Materials : Photonic crystals

ToC Category:
Metamaterials

History
Original Manuscript: May 5, 2009
Revised Manuscript: June 17, 2009
Manuscript Accepted: June 17, 2009
Published: June 25, 2009

Citation
R. Marqués, L. Jelinek, F. Mesa, and F. Medina, "Analytical theory of wave propagation through stacked fishnet metamaterials," Opt. Express 17, 11582-11593 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-11582


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References

  1. S. Zhang,W. Fan, K. J. Malloy, and S. R. J. Brueck, "Near-infrared double negative metamaterials," Opt. Express 13, 4922-4930 (2005). [CrossRef] [PubMed]
  2. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental Demonstration of Near-Infrared Negative-Index Metamaterials," Phys. Rev. Lett. 95, 137404 (2005). [CrossRef] [PubMed]
  3. M. Beruete, M. Sorolla, and I. Campillo "Left-handed extraordinary optical transmission through a photonic crystal of subwavelength hole arrays," Opt. Express 14, 5445-5455 (2006). [CrossRef] [PubMed]
  4. M. Beruete, I. Campillo, M. Navarro-Cıa, F. Falcone, and M. Sorolla "Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays," IEEE Trans. Microwave Theory Tech. 55, 1514-1521 (2007).
  5. M. Navarro-Cıa, M. Beruete, M. Sorolla, and I. Campillo, "Negative refraction in a prism made of stacked subwavelength hole arrays," Opt. Express 16, 560-566 (2008). [CrossRef] [PubMed]
  6. G. Dolling, Ch. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden "Low-loss negative-index metamaterial at telecommunication wavelengths," Opt. Lett. 31, 1800-1802 (2005). [CrossRef]
  7. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, "Three-dimensional optical metamaterial with a negative refractive index," Nature (London) 455, 299-300 (2008). [CrossRef]
  8. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, "Negative-index metamaterial at 780 nm wavelength," Opt. Lett. 32, 53-55 (2007). [CrossRef]
  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). [CrossRef]
  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). [CrossRef] [PubMed]
  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). [CrossRef] [PubMed]
  12. F. Medina, F. Mesa, and R. Marques, "Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective," IEEE Trans. Microwave Theory Tech. 56, 3108-3120 (2008). [CrossRef]
  13. R. Marques, 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) [CrossRef] [PubMed]
  14. J. D. Jackson, Classical Electrodynamics, (Wiley, New York 1999), 3rd Ed.
  15. R. Gordon, "Bethe’s aperture theory for arrays," Phys. Rev. A 76, 053806 (2007). [CrossRef]
  16. R. F. CollinField Theory of Guided Waves, (IEEE Press, New York 1991), 2nd Ed.,
  17. S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics, (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 Wirel. Compon. Lett. 13, 51-53 (2003). [CrossRef]

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