## Abstract

We theoretically analyze the properties of metamaterials which have been designed by taking advantage of Babinet’s principle. It is shown that the complementary structure exhibits both a complementary spectral response and field distribution of the respective eigenmodes. For complementary split-ring resonators, we show that the spectral resonance features can be explained from two different perspectives. On one hand they can be explained as plasmon polariton resonances in dielectric nanostructures surrounded by metal, on the other hand they can be understood as guided mode resonances with vanishing propagation constant. The physical origin of these modes and differences to the conventional split-ring geometry are discussed.

©2008 Optical Society of America

## 1. Introduction

Metamaterials (MMs) are media that permit control over the propagation characteristics of light [1, 2]. They derive their properties primarily from their geometry rather than from intrinsic material properties of the constituents. MMs usually consist of periodically arranged unit cells so that the light propagation is entirely governed by the dispersion relation of the respective Bloch modes [3]. Frequently, effective material parameters for MMs can be derived if a single Bloch mode dominates the propagation characteristics of the wave field [4]. If metals are incorporated into the unit cell, this dominating Bloch mode has a significantly smaller imaginary part of its complex wave vector when compared to other Bloch modes [5, 6]. Additionally, strong dispersion of the effective material parameters can be encountered if resonances are employed. These resonances must be independent from the period of the structure and are determined by the response of the unit cell. Such an approach resembles the concept of a polarizable atom where an effective polarization can be determined upon spatial averaging [7]. In designing a MM, one should therefore aim for a nanostructure that shows a resonantly enhanced scattering response similar to an optical nanoantenna [8]. If the scattering response is dominated by an electric/magnetic dipole, the effective permittivity/permeability is altered. Presently, two resonance mechanisms are employed to control the MMs response. The first is based on Mie resonances in dielectric spheres [9, 10] and the second on plasmon polariton resonances in appropriately shaped metallic nanoparticles [11].

By applying Babinet’s principle to the latter class of MMs, a resonant scattering response has been encountered in the microwave domain [12, 13]. Basically, Babinet’s principle states that a complementary structure illuminated by a complementary wave field (for a plane wave this implies that the polarization is rotated by 90° in the normal direction) causes a complementary scattering response with reflectance and transmittance being interchanged [14]. Babinet’s principle in its original formulation requires the scattering structures to be made of an infinitely thin perfect conductor.

Babinet’s principle is a fundamental principle in optics with a variety of applications. In the past, it was exploited to understand light propagation in conductive and inductive wire grids [15, 16]. Recently, experiments with the complementary split-ring resonator (c-SRR) have probed the limits of the applicability of Babinet’s principle to design MMs with resonance frequencies in the optical domain [17]. It was shown that the principle remains valid, although the strict requirements, as imposed in its original formulation, are violated to a certain extent. For c-SRR structures at normal incidence the complementary behavior comprises the scattering response, the classification of observable resonances, and the effective material parameters. As an open question, it remains to clarify to which extent the physical origin of the resonances in complementary MMs is likewise complementary to the excitation of plasmon polaritons in MMs. The understanding of the resonance mechanism will certainly lead to more efficient design approaches for this MM type, which is mandatory for the transfer of the concept into practical devices.

With the present work, we aim at bridging this gap with a detailed analysis of the origin of resonances in c-SRRs. However, this physical analysis applies to nanoaperture MMs in general, in particular to apertures having a rectangular or circular shape [18]. Our observations are connected to the work related to enhanced light transmission through sub-wavelength apertures evoked by vertical resonances [19, 20, 21] which are essentially Fabry-Pérot resonances in the nanoapertures. Additionally, there are also horizontal resonances causing an enhanced transmission by the excitation of surface-plasmon-polaritons (SPPs) on the periodically patterned metal film surface[22]. A comprehensive review about the scattering of light at particles and hole arrays can be found in Ref. 23. Enhanced transmission evoked by the latter effect depends sensitively on the nanoaperture period and can be disregarded in the present work where the resonance spectrum is independent of the period.

We show that Babinet’s principle is an excellent means of applying a unified point of view on various fields of modern optics, e.g., light propagation in MMs and through sub-wavelength apertures as well as localized plasmons in metallic nanoparticles and optical nanoantennas. Although not all implications can be revealed in their entirety in the present work, we shall put forward some essential ideas.

## 2. Properties of the resonant modes

First, we compare the scattering response of a split-ring resonator (SRR) with that of its complementary structure (c-SRRs) and summarize the essential features. Most of these properties have already been reported in the literature [12, 13, 17].

For our investigations, the geometry of both structures (SRR and c-SRR) is characterized by an arm length of *l* = 300 nm, a width of *w* = 40 nm, and a height of *h* = 15 nm. The gap is completely open on one side of the ring. They are assumed to be arranged in a periodic lattice with the period being Λ = 400 nm. In the simulations, the SRR is assumed to be made of gold and the c-SRR is assumed to be the respective air nanoaperture in a thin gold film. The dielectric function of gold has been taken from Ref. 24. The effect of an experimentally necessary substrate is omitted in our work as it has solely a weak quantitative impact on the optical response and, strictly speaking, Babinet’s principle to be applied requires a symmetric cladding [25].

Figure 1 shows the normalized reflected and transmitted intensity for normal incidence and the two polarization directions. All spectral calculations have been performed by using the Fourier modal method [26]. In the simulation 31 × 31 plane waves were retained, sufficiently enough to ensure convergence of all quantities in the order of a percent. We observe resonances appearing as reflection/transmission peaks for the SRR/c-SRR at approximately the same wavenumbers, if the polarization of the illuminating wave field is interchanged. The resonances are labeled in ascending order of the wavenumber. For SRR resonances, that number equals the number of amplitude nodes of the electric field component normal to the SRR surface [27]. The origin of these resonances is the excitation of plasmonic eigenmodes in the gold nanostructures [28]. The larger line width of the c-SRR resonances in Fig. 1 can be potentially attributed to the larger part of the electric mode volume in gold, which increases the absorption losses. Furthermore, the spectral shift of the higher-order resonances is attributed to a partial violation of the validity conditions of Babinet’s principle. Particularly, the conductivity of gold degrades at optical frequencies and the ratio of structure height to wavelength becomes larger - contradicting the assumption of nanoapertures in an infinitely thin and perfectly conducting film. It has to be mentioned that the skin depth of gold in the spectral region of interest takes approximately 24 nm, which is already comparable to the thickness of the metal film.

The close relation between resonances of the SRR and the c-SRR can not only be deduced from the spectral response, but also from the field distribution of the eigenmodes. Babinet’s principle states that **E**-*c***B*** ^{c}* =

**E**

_{Inc}, therefore at resonance, where the strength of the incident field is negligible (

**E**

_{Inc}≪

**E**,

**B**

*), the total electric field of the SRR eigenmode must resemble the total magnetic field of the c-SRR eigenmode [17]. Here,*

^{c}**E**is the total electric field behind the structure,

**B**

*is the total magnetic field behind the complementary structure, and*

^{c}**E**

_{Inc}is the illuminating electric field component. This analogy can be easily perceived from Fig. 2. There, the amplitude of the magnetic field component, normal to and shortly (20 nm) behind the surface of the c-SRR, is shown for the three lowest-order resonance frequencies of Fig. 1(c,d). The fields were calculated by using the finite-difference time-domain (FDTD) method [29]. As already outlined above, a striking similarity with the eigenmodes of a SRR is observed (see Ref. [27]). The c-SRR higher order modes have an increasing number of nodes in the magnetic field component normal to the c-SRR plane. This similarity of field shapes is not restricted to the normal component, but holds for the other components too.

## 3. Origin of resonances in complementary metamaterials

After having outlined the basic properties of c-SRR resonances, we will proceed in clarifying their physical origin. In the optical domain SRR resonances are associated with plasmonic eigenmodes [27, 30]. For c-SRRs the mechanism is very similar but even richer in some details. The origin of these resonances can be explained by using two different pictures, namely on the one hand plasmon polariton resonances in a dielectric (air) nanostructure embedded into metal, and on the other hand Fabry-Pérot resonances of guided waves propagating perpendicular to the structure.

Figure 3 shows the transmittance of the c-SRR as a function of the metal film thickness for the two possible polarizations. The spectral positions of the resonances at
${\overline{v}}_{{1}^{c}}=4000\text{\hspace{0.17em}}{\mathrm{cm}}^{-1}$
,
${\overline{v}}_{{2}^{c}}=8500\text{\hspace{0.17em}}{\mathrm{cm}}^{-1}$
, and
${\overline{v}}_{{3}^{c}}=10000\text{\hspace{0.17em}}{\mathrm{cm}}^{-1}$
are almost independent of the film thickness, although a minor shift to lower frequencies is observed for decreasing thickness, particularly for the higher-order resonances. An additional weak resonance (4^{th} order eigenmode) is observed at
${\overline{v}}_{{4}^{c}}=15000\text{\hspace{0.17em}}{\mathrm{cm}}^{-1}$
in Fig. 3(b) with a resonance frequency already in the spectral domain where the conductivity of gold is abated. However, this eigenmode behaves similarly as the 2^{nd} order eigenmode which was discussed in detail. Most notably, we further observe that for all resonances the transmittance degrades only weakly with increasing thickness.

In addition to these thickness independent transmission resonances, a further set of (weaker) resonances appears in the spectrum. With increasing thickness their spectral positions tend asymptotically towards the corresponding fundamental transmission resonance. This response resembles that observed for enhanced transmission through nanoapertures in the presence of vertically propagating guided modes [19].

The transmission spectrum can be qualitatively understood in terms of the plane wave response of a symmetric Fabry-Pérot resonator, given by

which also holds in first approximation for guided modes. Here, *T* is the amplitude transmission coefficient, *t* and *r* are the complex amplitude transmission and reflection coefficients of the guided mode, *k _{z}* is their propagation constant, and

*h*is the height of the structure. If a transmission resonance is independent of the height of the structure (see Fig. 3), the guided mode has to be close to its cut-off frequency (all modes in a dielectric waveguide embedded into metal have a zero cut-off propagation constant). Thus the real part of the wave vector has to be close to zero. Hence,

*k*can be approximated by ${k}_{z}\approx \frac{\sqrt{{\omega}^{2}-{\omega}_{c}^{2}}}{c}$ , where

_{z}*ω*is the cut-off frequency. Higher-order transmission peaks at larger height occur because the guided mode is strongly dispersive (see Fig. 5). For a given height of the structure, there exist wavenumbers which minimize the denominator in Eqn. 1, thereby enhancing transmission. Therefore, these resonances may be understood as higher order Fabry-Pérot resonances of the guided eigenmodes.

_{c}For the very small thickness (*h* = 15 nm) considered here, these higher-order transmission resonances are therefore not excited. For larger thicknesses, they move towards the mode cut-off frequency, as observed in Fig. 3.

It is now clear that for very small thicknesses, the system adopts to the very first resonance at (ℜ*k _{z}h* = 0) which entails (ℜ

*k*= 0), hence c-SRR resonances will appear at the cut-off frequencies of the modes.

_{z}The relationship between c-SRR and nanoaperture cavity resonances can be understood by looking at the field distribution and the dispersion relation of the lowest-order guided modes for a nanoaperture of infinite thickness. For this purpose, the wave equation has been solved with the ansatz
$\mathbf{E}(x,y,z)={\mathbf{E}}_{0}(x,y){e}^{i{k}_{z}z}$
using a finite-difference scheme [31]. The amplitudes of the *x*- and *y*-component of the electric field for the two lowest-order guided modes (1^{c} and 2^{c}) are shown in Fig. 4. They are almost identical (by visual inspection) to the corresponding amplitudes of the electric field of a c-SRR with a finite height at normal light incidence, when the frequency of the illumination corresponds to the first- or second-order plasmon resonance, respectively (see Fig. 1). The *x*-components for both guided modes are nearly identical [Fig. 4(a) and (c)], whereas the *y*-components differ in the number of nodes. We conclude that the observed spectral resonance originates in a coupling of the incident field to a guided eigenmode inside the nanoaperture.

As the guided modes have a continuous spectrum, it is not *a priori* clear at which frequency the transmission resonance will appear. However, this question can be addressed in analyzing the entire dispersion relation of the nanoaperture modes.

Figure 5 shows the effective refractive index *n*_{eff} of the first three eigenmodes of the infinitely thick c-SRR as a function of the frequency. The refractive index was derived from the calculated propagation constant *k _{z}* by

*n*

_{eff}=

*k*/

_{z}*k*

_{0}with

*k*

_{0}= 2

*π*/

*λ*being the wavenumber at a particular wavelength. A large imaginary part of

*n*

_{eff}implies that the guided mode is strongly attenuated upon propagation, therefore being nearly evanescent. The mode changes its character at a certain frequency (cut-off frequency). There, the imaginary part of the mode tends towards zero. As it remains close to zero beyond this frequency, the guided mode propagates in the nanoaperture with low losses. In contrast to the imaginary part of

*n*

_{eff}, the real part remains close to zero below the cut-off frequency. At frequencies above the cut-off frequency, the modes have a propagating character with a strong dispersion in the real part of

*n*

_{eff}.

The cut-off frequencies for the three guided modes displayed in Fig. 5 coincide precisely with the height-independent transmission resonances of the c-SRR (shown in Fig. 3). At cut-off frequency, the calculated propagation constant shows exactly the properties required for a transmission resonance which is independent of the thickness of the structure. Both, the real and the imaginary parts of the propagation constant are close to zero.

The slight increase in the imaginary part of *n*_{eff} for all modes at frequencies beyond 15000 cm^{-1} is associated with the interband absorption of gold in this spectral domain.

## 4. Eigenmodes at parallel light incidence

An essential issue for SRR MMs is the comparison of resonant modes excited for normal and in-plane incidence [33]. In this section, we will address this issue for c-SRRs.

In the following, we restrict ourselves to the case where the electric field is also in-plane. Then two c-SRR orientations have to be discriminated. The c-SRR gap can be situated either normal or parallel with respect to the illuminating wave.

Figure 6(a) shows the reflected and transmitted intensity for in-plane illumination of normal oriented c-SRRs where it does not matter whether the gap is at the front or the back side except a phase difference for asymmetric structures [32]. The single resonance observed at
${\overline{\nu}}_{{2}^{c}}=7000\text{\hspace{0.17em}}{\text{cm}}^{-1}$
is close to the resonance of the second-order guided mode. To verify that this guided mode is indeed excited, the *z*-component of the magnetic field 20 nm above the c-SRR plane is shown in Fig. 6(b). Furthermore, the electric field in the central plane of the c-SRR, split into its *x*- and *y*-component, is shown in (c) and (d), respectively. Note that the height of the c-SRR is again only 20 nm. Indeed, the field distribution corresponds to that of the secondorder mode. The magnetic field component perpendicular to the c-SRR exhibits two amplitude minima which are not exactly zero because this field is a superposition of the incident and the mode field. Nevertheless, the expected amplitude modulation of the field along the c-SRR is clearly seen. Furthermore, the electric field equals the second-order guided mode as calculated for the infinite extended system and for normal incidence (see Fig. 4). In contrast to a SRR, which supports in this configuration odd modes, only even modes are excited in the c-SRR. This response can be explained by the symmetry constraints. The illuminating magnetic field has an even symmetry, hence it can only couple to eigenmodes with the same symmetry. Therefore, the first- and the third-order modes are not excited. This adds another complementary aspect, as for SRRs, only odd modes are excitable in this configuration [30].

The spectral response of a c-SRR for parallel gap orientation is shown in Fig. 7(a). Two pronounced resonances are observed at frequencies corresponding to even modes for excitation normal to the c-SRR plane (see Fig. 1). The calculation of the magnetic field component *B _{z}* perpendicular to the c-SRR plane for the resonance frequencies of ${\overline{\nu}}_{{1}^{c}}=3500\text{\hspace{0.17em}}{\text{cm}}^{-1}$
and ${\overline{\nu}}_{{3}^{c}}=8000\text{\hspace{0.17em}}{\text{cm}}^{-1}$
reveals that the resonances correspond indeed to the first- and third-order guided modes [Fig. 7(b) and (c)]. The second-order guided mode may also be excited because no restrictions on the symmetry are imposed by the structure. It appears as an extremely weak dip in reflectance at
${\overline{\nu}}_{{2}^{c}}=6000\text{\hspace{0.17em}}{\text{cm}}^{-1}$
. The weakness of this excited resonance can be explained by analyzing the

*y*-component of the electric mode field. This component is provided by the incident field. The electric field is displayed for the first two lowest modes in Fig. 4. The

*y*-component of the first-order (and likewise for the third-order) modes shows no node along the bottom of the c-SRR (opposite to the gap). Therefore, an illuminating plane wave - showing a phase variation much larger than the c-SRR size - can excite these modes easily. In contrast, the amplitude has a single node for the second-order resonance, hence reducing significantly a potential coupling.

## 5. Conclusion

In conclusion, we have analyzed the resonance mechanisms which govern the spectral response of complementary MMs. The geometry of such MMs is derived by applying Babinet’s principle to geometries of MMs that employ localized plasmon polaritons as resonance mechanisms. For complementaryMMs the resonances have been shown to be Fabry-Pérot resonances guided modes in nanocavities. These guided modes are resonances of the infinitely extended nanostructure. It has been shown that for complementary MMs with a near zero thickness, resonances appear only at spectral positions where the eigenvalue disappears (cut-off frequencies). This makes the spectral positions of these resonances independent of the height of the MM. The spatial distribution of the field of the resonances in c-SRRs as compared to the field of resonances in SRRs is shown in the conceptual sketch in Fig. 8. The field distributions are extracted from rigorous simulations using FDTD. It can be seen that the resonances in the SRR are associated with plasmons, with an electric current inside the metallic nanoparticle that oscillates parallel to the structure. The resonances in the c-SRR are associated with the field that oscillates perpendicular to the structure. With increasing resonance number, an additional node appears in both amplitudes and a phase jump of *π* occurs.

At increased height, further resonances appear in the spectra for the c-SRRs, corresponding to higher order Fabry-Pérot transmission resonances of the guided modes in the nanocavity. These resonances are similarly excitable for an in-plane illumination relative to the complementary structure.

Although we have analyzed in the present work the example of the complementary split-ring resonator, the origin of the resonance mechanism applies equally to all the other possible shapes of complementary MMs, such as rectangular or annular apertures. By employing Babinet’s principle in the design of MMs, we expect to see in the near future a constant flow of ideas among the different fields of nanophotonics.

## Acknowledgments

Parts of the computations were done on the IBM p690 cluster (JUMP) of the Forschungs-Zentrum in Jülich, Germany. T.M. and T.Z. thank the Alexander von Humboldt Foundation and the Landesstiftung Baden-Württemberg, respectively. Our work was supported by the German Federal Ministry of Education and Research (Grant No. 13N9155).

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