We performed theoretical and experimental investigations of the magnetic properties of metamaterials based on asymmetric double-wire structures. Using the multipole model for the description of metamaterials, we investigated the influence of the geometrical asymmetry of the structure on the macroscopic effective parameters. The results show that the larger wire in the system dominates the dynamics of the structure and defines the orientation and the strength of the microscopic currents. As a result the magnetization of the structure can be significantly enhanced for certain asymmetric configurations of the double-wire structure.
© 2011 OSA
In the last decade, nanostructured metamaterials (MMs) exhibiting a magnetic response in the optical domain were of great interest [1, 2]. A magnetic response of MMs is realized by means of a special design of its unit cell, which commonly consists of metal nano-structures embedded in a dielectric host. These nano-structures support plasmonic modes, which can be excited by the electric field of an illuminating wave . The modes with antisymmetric current profile are responsible for a magnetic response of MMs. Excitation of those antisymmetric modes requires either a spatially non-symmetric illuminating field distribution or an asymmetric geometry of the unit cell. It is intuitively clear, that fulfillment of both requirements should potentially lead to an enhancement of the magnetic response.
Numerical and experimental investigations on the tailoring of the magnetic properties of MMs by breaking the symmetry of their constitutive elements have been undertaken in microwave [4–7] and optical frequency ranges [8–10]. Different types of structures, in particular: split ring resonators [5–8] and double-wire structures  were investigated. Experiments on numerous configurations and combinations of the elements have shown that the asymmetry of the structure indeed is a powerful tool for the engineering of the magnetic response of MMs, providing broad tunable frequency range and access to the additional modes. However there is a lack of simple theoretical models providing qualitative understanding of the mechanisms behind the observed effects and, importantly, enabling to predict them. To the best of our knowledge, only in a few works such theoretical models were used, for the split ring resonators  and for nano-sandwiches .
In our work we performed theoretical and experimental investigations of the magnetic properties of MMs consisting on asymmetric double-wire structures. The double-wire system  is one of the favorite configurations of MM’s unit cells owing to its rather simple geometry, which is an advantage for the experimental realization. In our investigations, we considered the general case of double-wires, which are infinitely extended in the transverse direction (Fig. 1). Such two dimensional configuration is efficient for the numerical simulations and obtained results can be easily transfered to the case of wires of finite length. The theoretical analysis of the asymmetric double-wire structures was done using the model presented in  extended to the asymmetric case. The analytical approach  describing the effective properties of a MM is based on a standard homogenization procedure accepted in the theory of electrodynamics of continuous media . The material equations for MMs are written taking into account contributions of the electric dipole, electric quadrupole, and magnetic dipole moments . These moments are expressed in terms of the carrier dynamics in a double-wire structure considered as a metaatom. It has been found that the complex carrier dynamics in a metaatom can be adequately described by a model of coupled linear harmonic oscillators. The parameters of the oscillators (eigenfrequencies, damping constants, and amplitudes) and the coupling between the oscillators are determined by the geometry of the unit cell and are supposed to be found from comparison with numerical simulations.
MMs with a similar asymmetry of the unit cell have been considered in Ref.  where the magnetic properties of the gold nano-discs of different diameters were investigated experimentally and theoretically using a point-dipole model and numerical simulations. Even though the systems considered in Ref.  and in our work are different, the influence of asymmetry on the magnetic response of the structures is similar in both cases. In contrast to Ref.  implementation of the theoretical model , which facilitated understanding of the system dynamics, allowed us to investigate not only microscopic properties but also effective material properties of the asymmetric MMs.
2. Theoretical approach for the description of metamaterials with an asymmetrical unit cell
The structure under consideration, was a double-wire structure infinite in z direction - see Fig. 1. To control the asymmetry of the structure, we chose the length of wires L1,2 in x direction (Fig. 1(a)). We use the notation length (L) for this dimension, because it corresponds to the length of the plasmonic resonator in our consideration. The parameter ΔL = L1 – L2 characterized the asymmetry quantitatively. The length of the first wire ranged from to , while the length of the second one ranged from to providing the conservation of the amount of metal in the structure (L1 + L2 = constant). The wires were placed in air with a separation distance of 40 nm between them. This was a simplification compare to the common experimental arrangement, where structures are placed on a substrate and are separated with a dielectric. The presence of a substrate breaks the symmetry of the structure even for wires of the same length. The consideration of the system in a symmetric environment allowed us to concentrate on effects caused by the asymmetry of the wires only. Nevertheless, the simplified model qualitatively describes the effects observed in the experimental system as it is shown in section 3.5.
The symmetric double-wire system has been considered in detail within the framework of the multipole approach in . Here we will not repeat the whole calculations leading to the dispersion relation of light in a MM, instead we only emphasize peculiarities related to an asymmetric structure. In this work, we restricted our consideration to the case of normal incidence with the polarization along the x axis.
As it was shown in , the system of double-wires can be described by a set of two coupled oscillators, each of which is associated with the fundamental dipole mode of a wire:
The solution of the system (1) can be easily found in the Fourier domain under assumption that x(t) = x̃(ω)exp(−iωt):14] and found the dispersion relations with the standard retrieval algorithm . Then the parameters of the oscillators have been found from the fitting of the numerically and analytically calculated dispersion relations, as it has been done in Ref. .
3. Results and discussion
3.1. Eigenmodes of asymmetric double-wire structures
A symmetric double-wire structure possesses two modes with frequencies separated due to the interaction between the wires [11, 16, 17]. One of the modes has a symmetric field profile and the other an antisymmetric one. Though, in a system of two different wires the field profiles of the corresponding modes are no more purely symmetric or antisymmetric, we will retain the notation of ’symmetric’ and ’antisymmetric’ modes for asymmetric structures, for the sake of conciseness.
According to the coupled mode theory, the stronger the wires couple, the larger is the frequency splitting of the modes. Obviously, the coupling between wires is maximal if the wires have the same lengths and becomes weaker if the difference of the lengths increases. The dependence of the frequency splitting on the length difference ΔL can be extracted from the spectra presented in Fig. 2(a)–(c), where the resonances corresponding to the excitation of the two modes can be easily identified. Both resonances appear in the spectra as a minimum of the transmission - see Fig. 2(a). The antisymmetric mode is associated with a carrier distribution having a pronounced quadrupole moment. Therefore, excitation of the antisymmetric mode is accompanied by an increase of absorption, which is enhanced by its non-radiating character - see Fig. 2(c). The symmetric mode, in contrast, has a radiating property of a dipole and appears as high reflection in the spectra - see Fig. 2(b). Thus, we observe that the resonances in Fig. 2 indeed have the largest spectral separation in the symmetric case (ΔL = 0 nm) and move towards the eigenfrequencies of the uncoupled wires if the difference of their lengths increases. In this work, we use spatial frequencies 1/λ for the presentation of the spectral information to facilitate considerations of the modes spitting.
3.2. Parameters of harmonic oscillators
For the analytical description of the double-wire structure the parameters of the oscillators (ω0i, γi, ai) and the coupling constant (σ) corresponding to the systems under consideration have to be defined. This problem is reduced to finding the connection between ω0i, γi, ai, σ and the length L of a wire. This dependency is the same for the two wires. The parameters of the oscillators can be found from the simulations of the single-wire (Fig. 1(b)) and the double-wire (Fig. 1(d)) structures. To demonstrate the accuracy of the fitting procedure, we show in Fig. 3 the wave vectors obtained from the fitting procedure (black curves) and from numerical calculations (red curves) for double-wire structures. The wave vectors from the numerical simulations were calculated with the retrieval algorithm , using complex transmission and reflection coefficients from the FMM simulations.
The parameters providing the best fit of the analytically calculated dispersions to the numerically calculated ones for the single-wire structures are shown in Fig. 4(a)–(c) with blue triangle marks. The eigenfrequency of the oscillator (Fig. 4(a)) is inversely proportional to the length of the wire, the amplitude (Fig. 4(c)) is proportional to the length, while the damping constant (Fig. 4(b)) in a first approximation does not dependent on the length of the wire.
The same set of parameters extended by σ was calculated for the double-wire configuration in a similar manner from the fitting of the theoretical dispersion relations to the numerical ones. The parameters obtained from the double-wire system (shown in Fig. 4, red diamond marks) have the same dependence on the length of the wire, however the eigenfrequencies and amplitudes are shifted relative to the values from the single-wire configuration. This deviation appears due to the interaction of the wires with the neighbors in the lateral direction, which is different for the single- and double-wire configurations. This interaction is not included in the theoretical model, however unavoidably appears in the FMM simulations and in the experiment, where the wires have a periodical arrangement with a small period. Hence, the results obtained from the simulations of single-wire structures can be used only for a qualitative estimation of the parameters. Nevertheless, the effect of the interaction can be taken into account under the assumption that the oscillators have a larger effective mass in the double-wire arrangement in comparison with the single-wire configuration.
Furthermore we proved, if the coupling constant σ can be elaborated using a model of two interacting point dipoles. The potential energy of two point dipoles, modeling the wires, with dipole moments arranged parallel to the x-axis:
Then the coupling constant σ can be easily introduced  as:Fig. 4(c), we obtain:
Defining α from the symmetric case (ΔL = 0) we calculate the dependence of σ on ΔL - see Fig. 4(d) (red line). Comparison of σ from Eq. (15) with the coupling constant from the fitting procedure (Fig. 4(d), green triangles) shows that the model of two coupled dipoles describes qualitatively the interaction between the wires.
A more realistic model for the interaction between the wires is the model where each wire is associated with a chain of point dipoles. The potential energy of the interaction for the whole system can be written as a double integral over all single dipoles in the chains. To account for the geometrical factor, the integration should be performed over the lengths of the wires. However, our simulations show that for the systems under consideration this model provides the same accuracy of approximation of the coupling constant as the model of two coupled point dipoles. The more detailed comparison of the models, for example the issue of the distance dependency, will be done elsewhere.
3.3. Dynamics of currents in an asymmetric double-wire system
Here we will consider in detail the dynamics of the system in two limited cases ΔL = 100 nm and ΔL = −100 nm. Data corresponding to the system with ΔL = 100 nm are presented in Fig. 5(a)–(g), where the phase information for the amplitudes x1,2 (Fig. 5(b)) and the currents j1 = q1ẋ1 and j2 = q2ẋ2 (Fig. 5(d)) are given with respect to the phase of the electric field in the center of the unit cell. It is seen, that at frequencies around the eigenfrequency of the antisymmetric mode (0.9 μm−1 – 1.1 μm−1, see Fig. 5(a)), the first oscillator x1 has a larger amplitude than the second one x2. The possible reasons are the larger size and the top position of the first wire, which is affected by the incoming wave first. In this case, according to Eq. (1), the first largest oscillator dominates over the second one, whereas the influence of x2 on the dynamics of x1 is negligible. Indeed, out of the resonance, at frequencies lower than the eigenfrequency of the antisymmetric mode (1.05 μm−1) the first oscillator follows the electric field almost in phase (Fig. 5(b)). The second oscillator is mostly driven by the first one and acquires a phase shift of about π with respect to x1 at frequencies close to the eigenfrequency of the antisymmetric mode. In the resonance both oscillators undergo a phase jump close to π; as a result for frequencies higher than the resonance frequency, x1 oscillates out of phase and x2 in phase with the electric field. The currents j1 and j2 associated with these oscillations are presented in Figs. 5(c) and (d).
For the structure with ΔL = −100 nm (see Fig. 5(h)–(n)) the situation should be inverse, as the larger oscillator is x2, which is below the first one. However, excitation conditions for x2 are not optimal, due to the first oscillator, which absorbs a part of the energy of the electric field. As a result, the amplitudes of oscillators x1 and x2 are of the same order and smaller than in the case ΔL = 100 nm. Nevertheless, the phase of the larger oscillator x2 is closer to the phase of the electric field than the phase of the smaller one x1, however, it deviates from the phase of the electric field, due to the stronger influence of the oscillator x2. As a result, at the frequencies below the frequency of the magnetic resonance, the loop current orientation is reversed to that of the structure with ΔL = 100 nm. An additional minimum in the amplitudes at frequencies around the eigenfrequency of the antisymmetric mode can be explained as a complex answer of the system on the excitation.
3.4. Effective magnetic response of double-wire structures
For the analysis of the magnetic response of the asymmetric structures, it is reasonable to consider the magnetization with respect to the magnetic field of an incoming wave. Therefore we introduced a complex coefficient χ (ω) = χ′ (ω) + iχ″ (ω) which defines the relation between the magnetic field of the wave and the magnetization of the MM:Eq. (7) taking into account the relation between the electric and the magnetic field in a plane wave: Figs. 5(e) and (f). In the resonance the phase of χ (ω) is π (see Fig. 5(f)), i.e. the equivalent microscopic loop currents (Figs. 5(c) and (d)) produce the magnetization oscillating out of phase with the magnetic field. The phase relations between the fields, the magnetization and the currents at the frequencies below (0.97 μm−1), at (1.05 μm−1) and above (1.15 μm−1) the resonance are illustrated in Fig. 6, where normalized real parts of these quantities are presented. In Figs. 6(c),(f), and (i) the phase relations between the fields, the magnetization, and the currents are illustrated in phase diagrams. The absolute value and the phase of χ (ω) for the structure ΔL = −100 nm is shown in Figs. 5(i) and (m). One can see that the microscopic currents of the structure ΔL = −100 nm give a magnetization which is almost in phase with the magnetic field (−π/2 < arg(χ (ω)) < π/2), and only at frequencies above the resonant frequency the phase shift between the magnetization and the magnetic field becomes larger than π/2. The phase relations between the fields, the magnetization and the currents at the frequencies below (0.97 μm−1), at (1.05 μm−1) and above (1.15 μm−1) the resonance are shown in Fig. 7. To describe effective magnetic properties of MMs, in some cases it is useful to introduce the effective magnetic permeability μeff(ω), for instance, for the consideration of boundary conditions. Definition of an effective magnetic permeability in a common way as Eq. (7) magnetizations using the parameters from Fig. 4 are shown in Fig. 8; the phase of the magnetization is given relative to the phase of the electric field of an incoming wave. In Fig. 9, the effective magnetic permeability calculated according Eq. (20) (black curves) together with the results from numerical simulations (red curves) are shown. It is seen that the larger the wire on the top, the higher the absolute value of the effective permeability. That may appear contradictory to the behavior of the magnetization shown in Fig. 8. However, a thorough consideration of the relation between μeff(ω) and the magnetization clears up this seeming discrepancy. According to Eq. (17) and Eq. (20) the absolute value of the permeability can be written as
Indeed, when the magnetization is in phase with the magnetic field (χ′ (ω) > 0) and grows, the absolute value of μeff(ω) increases, as the denominator in Eq. (21) decreases. This is the case for structures with a negative ΔL, for which the phases of χ (ω) lie between −π/2 and π/2 (Fig. 5(m)). Further, if the magnetization increases and is out of phase with the magnetic field (χ′(ω) < 0), μeff(ω) decreases, as the denominator in Eq. (21) increases. This is the case for structures with positive ΔL, for which the phases of χ (ω) lie between π/2 and −π/2 (Fig. 5(f)).
The real part of μeff(ω) becomes negative if χ′(ω) exceeds 1/μ0, which means that the magnetization oscillates in phase with the magnetic field and is strong enough. Furthermore, if the magnetization is out of phase with the magnetic field the real part of μeff(ω) is always positive.
3.5. Experimental investigations of the asymmetric double-wire structures
For experimental investigations of the asymmetric double-wire structures, a set of samples was produced using an electron-beam lithography technique. A SEM image of one of the samples is shown in Fig. 10(a). The wires were made of gold (dAu = 20 nm) and separated by a MgO dielectric layer (dMgO = 30 nm) with n = 1.72 (see Fig. 10(b)). The period of the structure was 500 nm. In the experiment only the length L1 of the wire on the top was varied in the range from L1 = 160 nm to L1 = 260 nm, whereas the length of the wire L2 on the bottom for all samples was kept at L2 = 260 nm. It should be reminded that, in accordance with our earlier notation, we use the term ’length’ to describe the dimension of the wire related to the considered resonances.
As the geometrical parameters of the structures can slightly vary along a wafer (due to technological reasons), we produced a set of ten samples on a compact area of 1 x 0.5 mm2, where the size of each sample was 1 mm x 50 μm. Such a compact arrangement of the samples assured that the observed effects were caused by the variation of the wire length only. The measurements of transmission and reflection spectra were performed using a Bruker Vertex 80 spectrometer combined with a Hyperion 2000 microscope. The light was focused on the set of the samples with an objective with NA = 0.4 and a certain sample was selected by an appropriate aperture setting.
Measured transmission, reflection, and absorption spectra are shown in Fig. 11. As it was discussed above, symmetric and antisymmetric modes can be identified in the spectra by the maximum of reflection and absorption, respectively. In the transmission the resonance corresponding to the excitation of the antisymmetric mode appears as a minimum at about 0.8 μm−1 and the symmetric mode as a minimum at about 1.1 μm−1. The difference in the absolute values for the resonance positions in the numerical simulations and in the experiment is due to the different refractive index of the spacer and slightly different lengths of the wires - see Fig. 1 and Fig. 10. In addition, in the presence of the substrate the wire on the bottom becomes effectively longer. As a result, the spectral separation of the resonances for ΔL = 0 nm is smaller for the experimental system with a substrate as for the system without a substrate considered earlier. Nevertheless, the splitting of the symmetric and antisymmetric modes becomes weaker when ΔL tends to −100 nm. It is interesting to note that in both experiment and simulation the absorption of the material at the antisymmetric resonance is maximal (Fig. 2(c) and Fig. 11(c)) for negative ΔL and decreases when ΔL becomes positive, whereas the reflection of the material (Fig. 2(b) and Fig. 11(b)) increases. This behavior can be explained by the increase of the electric dipole moment of the double-wire structure as ΔL becomes positive (the length of the wire on the top increases), which was also confirmed by numerical calculations (not shown here).
To access the magnetic properties of the MM’s layers, we performed FMM simulations of the produced structures. The simulated transmittances and reflectances have shown good agreement with the experimental ones. This allowed us to define the effective parameters from the simulated complex transmission and reflection coefficients. Using Eq. (17) and Eq. (19), the coefficients χ (ω) characterizing the magnetic moment of the structures can be expressed as a function of the effective magnetic permeability. The absolute values and phases of the parameters χ (ω) for the structures with ΔL = 0 nm and ΔL = −100 nm are presented in Fig.12. As the amount of metal in the experimental systems was not conserved (L2=constant for all structures), an interpretation of the absolute values of χ (ω) can be misleading. The phase of the coefficients χ (ω) is a more appropriate parameter in this case. It is seen that for the structure with ΔL = −100 nm the magnetic moment oscillates almost in phase with the magnetic field (arg[χ (ω)]≈ 0), at the eigenfrequency of the antisymmetric mode (0.85 μm−1).The phase delay increases when the length of the wire on the top becomes larger. For the structure with ΔL = 0 nm the phase delay is about π/4 at the frequency 0.65 μm−1. Thus, the orientation of the magnetic moment relative to the magnetic field in the experimental system corresponds to the predictions of the analytical model.
We presented results on the investigation of optical properties of asymmetric double-wire structures. The investigations were performed using the multipole model , FMM simulations, and experimental measurements. The analytical description of the structures required the extension of the multipole model presented in Ref.  to the asymmetric case. The connection between the parameters of the oscillators in the analytical model and the real geometry was found through the fitting of the analytically obtained dispersions to the numerically calculated ones. Comparison of the parameters obtained from the simulations of the single-wire and double-wire structures reveals an interaction between the neighboring wires in the lateral direction. This interaction is not included in the model. Nevertheless, it can be taken into account by increasing the effective mass of the oscillators in the double-wire configuration in comparison to that for the single-wire configuration. We also have shown that the interaction between the wires in a double-wire structure can be qualitatively described as the interaction of two coupled dipoles.
The investigation verified that the asymmetry of the structure influences the magnetization significantly. In the system of the double-wires the asymmetry was introduced by tuning of the wire lengths. The magnitude of the magnetization and the phase shift relative to the magnetic field depends strongly on the configuration of the system. In general, the dynamics of the system is dominated by the larger wire, where plasmon oscillations follow the external electric field. This defines the orientation of the effective micro currents in the double-wire structure. For a MM with a unit cell, where the wire on the top is longer than the wire on the bottom, the magnetization is strong and has a phase shift of π with respect to the magnetic field of the illuminating wave. This leads to a decreasing of the effective magnetic permeability. In the configuration where the wire on top is shorter than the one on the bottom, the magnetization is weaker and oscillates in phase with the magnetic field; the corresponding effective magnetic permeability grows. The obtained results correlate with the ones presented in Ref. , where a set of asymmetric nano-disks was considered. However, the implementation of the analytical model in our work allowed us to connect the geometrical asymmetry of the structure with the macroscopic effective parameters.
The experimental investigations have shown that the results obtained in the model based on the double-wires structures in a symmetrical environment qualitatively describes dynamics of a double-wire system on a substrate.
The authors gratefully acknowledge financial support from the German Federal Ministry of Education and Research (PhoNa and MetaMat), the State of Thüringian State Government (MeMa), and the German Research Foundation (SPP 1391).
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