We explain the origin of the electric and particular the magnetic polarizabiltiy of metamaterials employing a fully electromagnetic plasmonic picture. As example we study an U-shaped split-ring resonator based metamaterial at optical frequencies. The relevance of the split-ring resonator orientation relative to the illuminating field for obtaining a strong magnetic response is outlined. We reveal higher-order magnetic resonances and explain their origin on the basis of higher-order plasmonic eigenmodes caused by an appropriate current flow in the split-ring resonator. Finally, the conditions required for obtaining a negative index at optical frequencies in a metamaterial consisting of split-ring resonators and wires are investigated.
©2007 Optical Society of America
Metamaterials (MMs) are media which permit control of the material response upon incident electromagnetic fields through artificial structuring . If they consist of periodically arranged unit cells, their optical properties can be derived from the dispersion relation of the spatially periodic Bloch modes. For periods substantially less than the wavelength, only the lowest-order eigenmode plays a role and effective material parameters can be introduced. These effective material parameters mimic a homogeneous medium with the equivalent optical response of the nanostructured MM. Care must be taken by interpreting the effective material parameters if higher-order Bloch eigenmodes are excited. Specifically, if these modes exhibit larger amplitudes than the lowest-order eigenmode, the structure might be yet sub-wavelength but effective material parameters cease to be valid . For a dielectric material, the effective material parameters equal in most cases the averaged bulk properties of its constituents. However, they may strongly deviate from this average if resonances are excited in the unit cells. Mie type resonances in densely packed spheres, made of a polaritonic (enhanced ϵ) material are a typical example for a MM with strongly dispersive effective permittivity ϵ eff and permeability μ eff . For this purpose one usually relies on polaritonic materials or semiconductors with strong ex-citonic resonances [4, 5, 6]. Alternatively, unit elements made of metallic nanostructures with plasmonic resonances were proposed for designing MMs [7, 8].
A representative example is the split-ring resonator (SRR), originally proposed for the microwave domain. First variants consisted of two (or more) interleaved thin metallic rings with small gaps on opposite sides . An electromagnetic field with a magnetic component perpendicular to these rings induces currents in the conductors. Typically, spacing between the two rings is small compared to their sizes and thus the rings act as a capacitance, hence only a small current flows between the two interleaved sheets. The entire element may be regarded as an LC-oscillator that shows a dipolar response to an external field. Consequently, a medium made of closely packed SRRs alters the effective permeability, causing a Lorentz-type resonance. Position, width, and strength of the resonance are dictated by both the geometrical and intrinsic material parameters. Such a MM offers new perspectives for designing materials with unprecedented properties. In particular, a negative real part for the effective permeability is feasible. In conjunction with an element providing a negative effective permittivity, a negative refractive index was proposed and successfully demonstrated .
Soon after the initial experiments in the microwave domain, first attempts were undertaken to adopt the approach in the optical domain . The quest for miniaturization resulted in various modifications of the original design. Thus, the second ring was dropped for its negligible effect [12, 13]. Furthermore, the gap was significantly enlarged to open eventually one side of the SRR completely . The influence of such geometrical modifications on the spectral position of the resonances, which causes a change in the effective permeability was investigated . Hence, the design changed from a ring with a small gap to a U-shaped SRR. Furthermore, the influence of the orientation of the illuminating field on the resonances was investigated experimentally .
In addition, the properties of the SRR materials changed from those with a large imaginary permittivity and no noticeable influence of its real part at microwave frequencies  to those with a small negative real part of the permittivity where the remaining imaginary part usually contributes to losses in the visible. However despite of these modifications, the explanation of the origin of the SRR resonances at optical frequencies remained largely unaltered in many publications. The theoretical basis of this explanation is Maxwell’s theory in the limit of slowly varying fields, i.e., a quasi-static LC-circuit theory. For the stated reasons, a reinterpretation of the optical response of nanostructured metamaterials in terms of the full Maxwell theory is in order and will be given in this paper. As a first step in this direction an improved non-quasi-static Lorentz-model for the response of SRRs in the optical domain was presented, recently . Within such a non-quasi-static model, the influence of adjacent SRRs, e.g., by radiative coupling, can be considered, especially when the spacing becomes comparable to the wavelength of the light.
In this paper, we analyze numerically the origin of the electric and magnetic polarizability of MMs at optical frequencies at the example of the SRR by using rigorous diffraction theory. We show that effective material resonances are solely related to plasmonic eigenmodes in the relevant structure. Although in a dynamic simulation electric and magnetic fields are inseparable, we prove that the polarization of the electric field is of primary importance for the excitation of eigenmodes in the entire SRR. The plasmonic eigenmodes, which are driven by the external field, are the primary source of the observed magnetic and/or electric polarizability. By using this excitation mechanism, we reveal the existence of higher-order SRR resonances which in particular cause a magnetic polarizability. We have chosen the common split-ring geometry for our discussion. Remarkably, our approach is also valid for different metamaterial structures such as cut-wire pairs or fishnet structures.
We shall show that the eigenmodes, which can be excited at normal incidence with the electric field pointing parallel to the gap, will cause resonances in the permeability for appropriate excitation configurations. On the other hand, modes which can be excited at normal incidence with the electric field perpendicular to the gap, will only cause a resonance in the permittivity function. Furthermore, we investigate the possibility to obtain a negative refractive index at optical frequencies by incorporating metallic wires into the MM unit cell. Because the filling fraction of the unit cell with wires is usually quite small, the effective plasma frequency of this artificial medium decreases and becomes smaller than the magnetic resonance frequency of the SRR, rendering a negative index impossible. This leads to the conclusion that metallic plates rather than wires have to be used, because they exhibit an effective plasma frequency only slightly smaller than that of the bulk material .
2. Resonances at normal incidence
For investigation of the resonance properties of SRRs at optical frequencies, the illumination direction is usually chosen perpendicular to the SRR plane . Recently, it has been shown  that this configuration allows to determine the spectral positions of the SRR eigenmodes, but it does not provide a useful resonance of the effective permeability. A typical spectral response for both polarizations is shown in Fig. 1.
In our numerical investigations, we assume that the SRRs are made of gold (data from Ref. 18) and entirely surrounded by air. A potential substrate alters only slightly the spectral position of the resonances, and no qualitative differences can be observed for experimentally available substrates. Thus we neglected it in our calculations. If not otherwise stated, we assume periodically arranged SRRs with a period of Λ = 400 nm. The lengths of the SRRs parallel and perpendicular to the gap are equally set to be lII = l ⊥ = 300 nm. The in-plane width of the wires forming the SRR is w = 40 nm and their height is h = 15 nm. The simulation of the spectral response was performed by using the Fourier Modal Method (FMM) . 31 diffraction orders were retained in the calculation for both the x- and the y-direction. This ensures sufficient numerical convergence for all relevant quantities deduced from the calculations.
The FMM is a well established method to analyze diffraction at periodic structures. It is based on an appropriately chosen modal expansion of the electro-magnetic field in the incident, the transmitted, and the grating region. Inside the grating region the field is expanded into a superposition of Bloch waves that are solutions to the wave equation. Outside the grating the field is expanded into plane waves, which correspond to the various diffraction orders. By matching subsequently the amplitude of each mode such that the boundary conditions for the electro-magnetic field components are fulfilled between the three respective regions, the interaction problem is solved and diffraction efficiencies can be calculated. The method is particularly well suited for the present task, as it allows to take the correct value of the dielectric constant for each frequency into account .
In the chosen spectral domain, resonances appear at three frequencies (we used the respective wave numbers ) 1 = 3100 cm-1, 2 = 6200 cm-1, and 3 = 8100 cm-1. The first and third resonance are excited by an electric field polarized parallel to the gap [Fig. 1(a)]. The second resonance is excited by a polarization perpendicular to the gap [Fig. 1(c)]. All resonances may be related to plasmonic eigenmodes of the SRR.  As usual the modes can be labeled according to the number of amplitude nodes of the electric field component perpendicular to the SRR. The higher the frequency required for their excitation the larger is the number of nodes. Symmetry considerations imply that odd/even modes are excited with the electric field polarization parallel/perpendicular to the gap.
3. Resonances for parallel incidence
In order to obtain a strong magnetic response from the medium made of SRRs, the illumination direction has to be in plane with the SRRs. Additionally, the orientation of the electric field relative to the SRR has to be appropriately chosen.
No pronounced resonances are observed for an incident electric field polarized perpendicular to the SRR plane. Here, only weak plasmonic resonances are excited at high frequencies which contribute to small dispersion of the effective permittivity in the visible. Their spectral position is determined by the ratio of the SRR wire height h to its width w. Similar resonances appear for single wires with the same size and aspect ratio. These resonances are of no importance in the present context and will be neglected in the further discussion.
For an electric field polarization parallel to the SRR plane, two possibilities exist. First, the electric field may be polarized parallel to the gap. This allows for a realization where the gap is either at the front or the back of SRR relative to the propagation direction of the incident light (see Fig. 2). Second, the electric field can be polarized perpendicularly to the gap. Hence, the gap is parallel to the propagation direction of the incident light (see Fig. 3).
Reflectance and transmittance for the first option with the gap being either at the front or the back of the SRR are shown in Fig. 2(a). The subscript 1/2 labels the case with the gap at the front/back of the SRR, as shown on top of Fig. 2. No differences have be observed in the reflectance/transmittance spectra for both orientations. Two resonances appear at frequencies almost identical to the resonance frequencies of the plasmonic eigenmodes, which were excited at normal incidence with the electric field polarized parallel to the gap [see Fig. 1(a)]. The inspection of the near-field amplitude of the electric field revealed equal mode patterns when compared to the eigenmodes at normal incidence. Especially the amplitude of the field component perpendicular to the SRR plane, formerly used for labeling the eigenmodes , shows one and three nodes at the two relevant frequencies, respectively. Because of their symmetry, even numbered nodes cannot be excited in this geometry. The electric field amplitude parallel to the gap is antisymmetric with respect to the D 1 symmetry axis for these modes, whereas the illuminating plane wave provides only a symmetric field.
Besides the question which mode will be excited in a particular configuration, it is of major relevance to reveal which effective material parameter will be altered by the excitation of a particular mode. The present configuration (see Fig. 2) is asymmetric with respect to the inversion of the propagation direction. Hence the parameter retrieval algorithm will lead to effective material parameters which depend on the orientation of the MM . This results from the fact that the effective refractive indices are identical [see Fig. 2(b)] but the impedances differ. However, it was shown that at least a unique effective refractive index can be derived for such asymmetric structures, whereas the effective impedance differs for the two orientations. The reason for this behavior is the different phase behavior of the reflected field . Although an unambiguous definition of unique effective material parameters is impossible, their computation provides valuable insight to conclude that the observed resonances primarily affect the effective permittivity.
The different effective material parameters (n eff, ϵ eff and μ eff) are shown in Fig. 2(b–d), and the asymmetry of the structure was fully taken into account. For retrieving the effective material parameters we have applied the procedure as described in . Focusing first on the effective permittivity ϵ eff, a well pronounced Lorentz-type resonance can be observed for both SRR orientations at 1 = 3100 cm-1. The field of the eigenmodes in the two side arms of the SRR has no particular influence on the effective properties. The response is dominated by an electric dipole radiating from the bottom of the U-shaped SRR parallel to the gap. The eigenmode excitation is identical for both cases shown on top of Fig. 2. Hence, due to the electron oscillation in the bottom wire of the SRR the electric dipole alters mainly the electric polarizability of the medium irrespective of the SRR orientation. Essentially the same consideration holds for the higher-order resonance at 3 = 8000 cm-1. Although it is not being as well defined, the increase of the imaginary part of ϵ eff and the general shape of the real part are well in accordance with a Lorentz resonance.
The effective permeability shows resonances at both frequencies. For SRRs with the gap at the front, the real part exceeds unity whereas for the opposite orientation of the gap the real part becomes less than unity. However, both resonances exhibit a similar shape. These properties of the dispersion relation hold for both resonances. The imaginary parts of the retrieved effective permeability change their signs at resonance frequencies. In addition, to a first approximation they differ by sign over the entire spectral domain. In general, the resonances of permeability are fairly weak. Overall, the extension of such a medium with continuous metallic wires placed parallel to the electric field inside the unit cell will therefore not lead to a negative refractive index which holds true for either orientation. In the simulations, we found positive values close to zero over an extended range, as expected for an effective metallic medium. This response is caused by the additional metallic wires. In the spectral region of the SRR resonances, the real part of the refractive index is symmetrically increased due to the superimposed resonance of the effective permittivity as provided by the SRR.
We have to make an additional remark regarding the meaning of the effective material parameter if the amplitude of the strongest higher-order Bloch eigenmode is slightly larger than the amplitude of the lowest-order Bloch eigenmode in a narrow spectral domain close to both resonances. This is true for the present case, where ratios as large as 1.3 are observed directly in the resonance. Such a situation was recently shown to invalidate the assignment of an effective material parameter because the medium must not be considered homogenous. By contrast the properties are affected by the periodicity . Therefore, although the character of the resonances remains deducible, the absolute values of the effective material parameter should be merely regarded as an indication.
Based on these results we can argue that for a pronounced magnetic polarizability the induced currents have to flow perpendicularly to the propagation direction of the incident light in at least two different wire pieces and with a phase difference of π. In other words, magnetic polarizability requires currents related to plasmonic eigenmodes which must flow in opposite directions in both side arms. In such a configuration, the scattered field resembles that of an electric quadrupole. As this field resembles that of a magnetic dipole, the effective permeability of the medium exhibits a Lorentz-type resonance. Although the quadrupole is driven by the electric field it is the cause of the magnetic polarizability. Similar arguments have been used to explain transmission peaks, caused by a negative refractive index in the spectra of two parallel metallic wire pieces separated by large distances .
For obtaining a resonance in the permeability, the arms of such U-shaped SRRs have to be oriented perpendicularly to the propagation direction and the electric field has to be polarized perpendicularly to the gap. Figure 3 shows the transmittance and reflectance (a), the illumination geometry of the structure (b), and the retrieved effective permittivity (c) and permeability (d). The effective parameters remain unchanged when inverting the propagation direction and thus can be unambiguously assigned to the metamaterial. For this and all the following structures that show inversion symmetry, we have applied for the parameter retrieval the procedure as described in . It bases on an inversion of the thin film reflectance and transmittance coefficient. Resonances in the spectra appear at nearly the same frequencies as for the normal incidence (Fig. 1). To clarify the character of the plasmonic eigenmodes, the amplitude of the electric field component perpendicular to the SRR plane is shown in Fig. 4 detected 22.5 nm above the surface of the SRR. Calculations have been performed with a Finite-Difference-Time-Domain method . The dispersion of gold was modeled by a Drude oscillator where the parameters of the model were adjusted to match the permittivity of the material as reported in literature .
The plasmonic eigenmode will exhibit more nodes at higher resonance frequencies, similar to the findings at normal incidence . This similarity holds for all field components. The eigen-modes with an odd number of nodes cause a Lorentz-type resonance in the effective permeability. This is particularly true for the third-order resonance, usually denoted as an ‘electrical resonance’ or the ‘particle-plasmon resonance’ of the SRR . As stated above the conditio sine qua non for a magnetic resonance to appear is a mode profile that induces currents having opposite directions in both arms. In Fig. 4 the direction of the currents is indicated by arrows. The direction of all currents were deduced from FDTD simulations. The first-order resonance at 1 = 3900 cm-1 (Fig. 4-left) and the third-order resonance at 3 = 8100 cm-1(Fig. 4-right) meet this condition. It is evident that the base length affects the frequencies of the plasmonic eigenmodes and thus the spectral position of the effective material resonance . In the present geometry, the resonance is too weak to cause a negative permeability (Fig. 3), but this can be easily achieved by increasing the height h of the SRR as will be shown later. The effective permittivity is only slightly affected in the spectral domain of the permeability resonances.
For the second resonance at 2 = 6400 cm-1 the currents in the two side arms of the SRR point in the same direction as indicated in Fig. 4-center. This eigenmode corresponds to the lowest-order resonance, which can be excited at normal incidence with the electric field polarization perpendicular to the gap. The scattered field is mainly generated by an electrical dipole rather than a quadrupole. Hence, only the effective permittivity is altered at that frequency. Anti-resonances are observed in the effective permeability, characterized by a negative imaginary part and an inverted line shape for the real part. Their presence can be attributed to the excitation of higher Bloch eigenmodes with non-negligible amplitude . In particular, for higher-order resonances their amplitude is larger than that of the lowest-order Bloch eigen-mode, which makes the assignment of an effective material parameter to a certain extent invalid (the medium does not show the required homogeneity) . From such an effective material parameter, we can deduce the character of the resonance and understand how it affects the light propagation, but the absolute values of the strength of the effective material parameters should be evaluated with care.
4. How to obtain a negative index?
After having identified electrical quadrupoles, evoked by plasmonic resonances in the U-shaped SRRs, as the source of resonances in the magnetic response, the natural question arises how a negative refractive index can be achieved at optical frequencies. It was frequently stated that the inclusion of a metallic wire laid out parallel to the electric field polarization allows for a negative refractive index. To date, to the best of the author’s knowledge, this was not experimentally verified at optical frequencies in the SRR geometry up to now. Evidently, an ensemble of metallic wires exhibits a smaller plasma frequency compared to the corresponding bulk material . Because the resonance frequencies of the plasmonic SRR eigenmodes are close to the bulk plasma frequency, the downshift caused by the material ‘dilution’ can potentially result in a positive permittivity, rendering a negative refractive index impossible regardless of the negative permeability.
Figure 5 shows the transmittance/reflectance spectra and all relevant effective material parameters for a unit cell that contains either a SRR, a metallic wire element, or both elements, as indicated at top. In calculating the effective parameters of the wire medium, its spatial extension in longitudinal direction was set to the length l ∥ = 300 nm of the SRR. The SRR height h was assumed to be 40 nm to ensure a strong magnetic resonance with slightly negative values within a narrow spectral domain. We concentrate here on the first-order eigenmode of the SRR. The metallic wires are infinitely extended into the direction parallel to the polarization of the incident electric field. They have a width of 100 nm in the opposite transversal direction and a height of 80 nm in the longitudinal direction. They are placed such that they have both equal distances to adjacent SRRs and to both arms of a SSR. The period of the structure remained 400 nm. This new unit cell hosts two elements hosts, the SRR and the metallic wire. To counteract the decrease of the plasma frequency by ‘dilution’ the wires are quite massive and resemble a metallic plate.
As shown in Fig. 5(c,d), the wires primarily affect the effective permittivity of the SRR-wire medium. This material parameter shows an effective Drude-type behavior with the plasma frequency shifted towards lower frequencies when compared to bulk gold. Particularly for the present example the effective plasma frequency remains well above the frequency of the lowest-order plasmonic eigenmode of the SRR at 1 = 4100 cm-1. For smaller frequencies, the effective permittivity of the medium, which comprises both elements, is dominated by the wire, whereas for frequencies larger than the plasma frequency the effective permittivity is dominated by the resonance of the second-order (dipolar) plasmonic eigenmode at 2 = 7900 cm-1. This holds for both the real and the imaginary part. Between these two frequency domains a smooth transition takes place [see Fig. 5(c,d)].
The effective permeability of the SRR-wire medium [Fig. 5(e,f)] follows essentially the permeability of the SRRs almost throughout the entire spectral domain except for > 9000 cm-1. This deviation can be attributed to the excitation of higher-order Bloch modes which are propagating in the structure due to a pronounced coupling among both elements. For a ratio of period/ resonance wavelength close to unity, the parameter retrieval algorithm yields artifacts related to the excitation of these higher-order Bloch modes. In the present example, this ratio is 0.36 for a frequency at = 9000 cm-1. Nevertheless, the effective material parameters of the SRR-wire medium can be generally regarded to be valid in the entire spectral domain, as we verified that their amplitudes never exceed that of the lowest-order Bloch eigenmode, although these higher-order Bloch eigenmodes are present.
Although the combination of the two elements causes a decrease of the effective plasma frequency, the permittivity remains sufficiently negative at the resonance frequency of the SRR and a negative real part for the refractive index is observed [Fig. 5(g)]. Similar observations on the interplay between wire and SRR causing a decrease of the effective plasma frequency were reported for unit elements working in the microwave domain .
In the optical domain, it is of particular importance to choose metallic wires of appropriate thickness to limit the decrease of the effective plasma frequency. This is demonstrated in Fig. 6. The results regarding the effective parameters are shown for the same SRR as before, but for three different heights of the wire h. We consider in the figure only the transmittance and the real part of the effective material parameters. Due to the stronger ‘dilution’ of the metal, the effective plasma frequency experiences a larger downshift. Above the frequency of the second-order plasmonic eigenmode, the permittivity is dominated by the effective permittivity provided by the SRRs, hence it is rather independent of the wire height. The effective permeability is likewise independent of the wire height. The increase of the permittivity in the spectral domain of the magnetic resonance for thinner wires causes a less negative refractive index. In the present example a negative refractive index requires a wire height of at least 40 nm. A detailed investigation of the strength of the resonance and the figure-of-merit for the material as a function of the detailed geometrical parameters (SRR and wire size and shape, as well as their relative position in the unit cell) is beyond the scope of this paper and will appear elsewhere. This figure-of-merit has been defined as the ratio of real to imaginary part of the refractive index . Here, we emphasize that particularly at optical frequencies the unit cell of MMs has to contain sufficient amounts of metal to limit the downshift of the plasma frequency compared to its bulk value. This is necessary to observe a negative refractive index at optical frequencies. The problem becomes more severe if the resonance frequencies of the SRR are closer to the visible, which will finally limit the use of SRRs in metamaterials.
Finally, we wanted to estimate the influence of a surrounding media and a possible substrate on the properties of such a MM at optical frequencies. Figure 7 shows the real part of the effective refractive index for the same structure as analyzed in Fig. 6 with a height for the wire of h = 80 nm. First, the structure is placed on a substrate with a refractive index of n = 1.5. Such a substrate has only a minor impact on the dispersion of the refractive index. A more realistic structure will be potentially embedded in a dielectric medium. Figure 7 shows likewise the real part of the effective refractive index if the structure is embedded in a dielectric host medium with a refractive index of 1.5 and finally when this structure is placed on a substrate with the same refractive index. Again, the presence of the substrate does alter the dispersion, but the surrounding media strongly shifts all resonances towards smaller wavenumbers. This is in complete agreement with the shift of the spectral position of the plasmon resonances in small metallic nanoparticles if they are embedded in a dielectric host media, except this shift of the dispersion remains qualitatively the same.
5. Conclusion and outlook
In this paper, we explained the origin of the electric and the magnetic polarizability in MMs at optical frequencies employing an electrodynamic approach. Exemplarily, the investigation was done employing SRRs that support plasmonic eigenmodes. The impact of such resonances for various configurations of the incident field to the SRR on the effective material parameters was investigated. We concentrated on the propagation direction parallel to the SRR. Identical modes can be excited for different illumination and polarization directions, as long as their symmetry coincides with the symmetry of the illuminating wave-field. For observing a magnetic polarizability in the MM, the field radiated by the SRR has to be dominantly an electrical quadrupole field, i.e., consisting in the simplest case of two electric dipoles oscillating π out of phase. The corresponding currents which can be excited at the frequencies of the SRR plasmonic eigenmodes are the origin of this field. Because the currents are driven by the electric field component of the incident wave, the orientation of the electric field vector relative to the SRR is an important parameter.
We found that for the analyzed U-shaped SRR the electric field has to be polarized perpendicular to the gap and the propagation has to be aligned in the SRR plane. For such a configuration, a magnetic polarizability was found even in higher-order resonances of the SRR eigenmodes. It turned out that the resonances observed at normal incidence with the electric field parallel to the gap are due to the same eigenmodes. All these eigenmodes cause a strong magnetic response for the in-plane propagation. The resonances excited at normal incidence with the electric field being polarized perpendicular to the gap, cause a strong electric response at the aforementioned in-plane propagation direction.
In addition, we have shown that extension of the SRR unit cell with a metallic wire medium will cause potentially a negative refractive index, if the unit cell contains a sufficient amount of metal. We have concentrated in the present work on the lowest-order eigenmode to observe in its spectral vicinity a negative refractive index. It remains an open question, whether a magnetic polarizability at higher-order resonances can be employed to observe a negative refractive index, as the geometrical size of the SRR becomes comparable to its resonance wavelength. However, this violates the key assumption that unit cells must be much smaller than the wavelength which is associated with the medium effective material parameters.
This work provides a comprehensive explanation of resonances in SRRs based on the full Maxwell theory. Moreover, the consequences of these resonances for the dispersion of the effective optical parameters have been revealed. Our results open the avenue for new design approaches of metamaterials that support magnetic resonances in the optical domain.
Parts of the computations were done on the IBM p690 cluster (JUMP) of the Forschungs-Zentrum in Jülich, Germany. The authors acknowledge support by the Federal Ministry of Education and Research (Unternehmen Regio, ZIK ultra optics, and 13N9155) and the Deutsche Forschungsgemeinschaft (FOR 557 and FOR 730), (Research Unit 532). T.Z. is indebted to the Landesstiftung Baden-Württemberg for facilitating the analysis entailed in this paper.
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