1. Introduction

A metamaterial (MM) is a man-made structure with an appropriate design intended to strongly influence the propagation characteristic of electromagnetic fields, and in particular, of light. In the broader sense, a photonic crystal (PhC) may be considered a typical example of a MM. It allows for engineering the control of the propagation characteristics such as dispersion, diffraction, or group velocity [1], where the characteristic period of the structure is on the order of the wavelength. Frequently, the term metamaterials is only used for structures with feature sizes and periods much less than the wavelength. Thus, the optical properties of such MMs are independent of the period and even on deviations from periodicity. Light propagating inside such a MM experiences solely effective material properties, namely an effective complex permittivity and permeability [2]. These artificial materials allow to overcome a long standing dogma, stating that spectral domains with resonances in the dispersion of permittivity and permeability do not coincide [3]. In particular, to date no natural material is known that exhibits an enhanced response to the magnetic field in the optical domain. A material with both permittivity and permeability being resonant can cause a negative refractive index [4], provided that ε ₁ µ ₂+ε ₂ µ ₁<0 where the subscripts 1 and 2 denote the real and imaginary part of permittivity and permeability, respectively. In such a medium a multitude of counterintuitive physical effects may take place and very appealing applications can be viewed, with the perfect lens being the most prominent example [5].

The idea to compose a material of unit cells which influence either the effective permittivity [6], the effective permeability [7, 8], or both simultaneously allows to envisage an artificial material with a negative effective refractive index. Various designs for such specific unit cells were proposed, such as spheres with a large dielectric constant [9], a pair of small parallel metallic wires [10, 11], or its inverse structure [12]. The most prominent example is a combination of thin metallic wires and metallic split-ring-resonators (SRRs), having the shape of a ‘U’. While the wire provides a negative effective permittivity, the SRR accounts for a negative effective permeability [13]. The working principle of a SRR may be understood in terms of electrical engineering. The gap of the SRR is regarded as a capacitance, and the ‘U’ as a single winding of an inductance loop. Both together constitute an LC-circuit, coupled to an external light field and driven into resonance at appropriate frequencies. Originally the idea was proposed and experimentally verified in the microwave region [13] and initiated many efforts aimed at translating the concept into the optical domain [14].

A resonance of the effective permeability requires the magnetic field to oscillate perpendicularly to the SRR plane and the light to propagate parallel to this plane. To date, such a configuration has not been accomplished in the optical domain, as it challenges nano-fabrication technology. Nevertheless, for probing the spectral position of this resonance the SRR can be illuminated at normal incidence. If the electric field oscillates parallel to the gap (x-direction in Fig. 1b), it can couple to the so-called LC-resonance of the SSR [15, 16]. Additionally, a strong resonance at higher frequencies is usually encountered. This resonance is heuristically associated with a plasmon excitation in the piece of wire that opposes the gap (base wire), hence called the plasmonic resonance of the SRR [14]. This conclusion was supported by observations of a strong resonance at nearly the same frequency for perpendicular polarization. However, recent investigations showed that for SRRs with a maximum gap width this plasmon resonance is blue shifted in comparison to the plasmon frequency of the base wire [17]. This cannot be explained by assuming that the origin of this resonance is a plasmonic excitation in this base wire. The finding did not fit into the established interpretation and raised new questions on the actual nature of those higher-frequency resonances.

Based on a detailed analysis of the spectral position of the resonances as a function of SRR geometry and by simulating the near-field, we show that these high-frequency resonances are not solely associated with charge oscillations along a single wire piece of the ‘U’-structure. They rather constitute higher order plasmon modes excited in the entire ‘U’-structure. These higher-order resonances are observed for both polarizations. The effective material parameter that is altered in this excitation configuration is the effective permittivity, based on plasmonic excitations in the structure. Additionally, we show that excitation of the LC-resonance does not necessarily require a gap. The resonance remains present in the limit of a vanishing gap where the SRR consists of a single wire piece that previously formed the bottom.

2. Analyzed structures and numerical tools

A typical periodically arranged structure along with a sketch of a single unit cell and all relevant geometrical parameters are shown in Fig. 1. The parameters in the present work were chosen to match closely those of technologically feasible physical systems. The SRRs are made of gold and assumed to be deposited on a glass substrate with a refractive index of n _S =1.5. The height of the structure is h=20 nm, the period is Λ=500 nm, the thickness of the wire is w=60 nm and the length of the SRR parallel to the gap is l _‖=400 nm. One parameter of the SRRs that was changed in the simulations is the length of the SRRs perpendicular to the gap l _⊥. For simplifying the description of the SRRs in the text, we use throughout the manuscript the phrase “a bottom of the ‘U’-structure” for the base wire piece of the SRR, and “legs of the ‘U’-structure” for the two wire pieces of the SRR perpendicular to the gap.

 

Fig. 1. Typical arrangement of the structure in a) and schematic sketch of the pertinent SRR along with a definition of the geometrical parameters in b).

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Fig. 2. Reflection from SRRs with a fully opened gap. The electric field is polarized in a) parallel to the gap and in b) perpendicular to it. The resonances are labeled in the figure with an arrow and an appropriate number.

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The spectral response of the SRRs was calculated by a grating algorithm based on the Fourier Modal Method (FMM) [18, 19]. The advantage of the method is its reliability and accuracy. As the computation takes place in the spectral domain, it is possible to include experimentally available data for the dielectric functions. In the present work the dielectric functions of gold as published by Johnson and Christy were used [20]. The near fields at the resonances were calculated by the finite difference time domain (FDTD) method [21]. Within this method, electrical currents inside the gold structure were simulated assuming a Drude polarizability. The parameters of the Drude model were adjusted to match the real dielectric function at the pertinent frequencies. The FDTD method was chosen for this part of the calculations as the necessary truncation of the Fourier-expansion in the FMM makes it cumbersome to simulate precisely all fine details of the near-field amplitudes. Bothmethods were cross-checked to verify their accuracy, and it turned out that they give identical results.

3. Results

3.1. Near-field distributions of the plasmon modes

The closed SRR in the absence of a gap shows primarily a single, well-pronounced resonance [17]. By opening the gap, two strong resonances appear if the polarization of the electric field is parallel to the gap. In Fig. 2 a) these two resonances can be seen at approximately ν̄=2,350 cm^-1 and ν̄=6,250 cm^-1 for a structure with l _‖=l _⊥=400 nm. The figure shows the reflection as a function of the wave number. A third peak, not mentioned so far, appears at a frequency of ν̄=10,000 cm^-1. This is no artifact of the numerical simulation but represents a genuine third-order mode excited in the SRR. The mode is experimentally difficult to detect, as its excitation strength depends strongly on the quality of the fabricated structures. Slight deviations of the geometry from unit cell to unit cell and particularly residual surface roughness of the SRR can cause a strong inhomogeneous broadening and damping of this resonance. Additionally, slight oscillations denoted as a fourth resonance appear at ν̄≈13,000 cm^-1. They correspond to the frequency at which the first diffraction order into the glass substrate changes from an evanescent to a propagating one. This type of oscillation is well-known and called Rayleigh or Wood anomaly [22]. Its frequency is independent of the geometrical structure but depends solely on the period of the SRR unit cell and the refractive index of the medium on either side of the SRR. A fifth resonance is excited around a frequency of ν̄=17,500 cm^-1. It is unambiguously related to a plasmon excitation perpendicular (oscillation in x-direction) to the leg wires. Like all plasmon resonances for structures with the present dimensions, their resonance frequency depends on the ratio of wire height to wire width. The resonance can be observed at the same spectral position in the single wire with the same height h and width w. Simulations show that its frequency depends negligibly on other SRR parameters.

 

Fig. 3. Magnitude of the electric field distribution for the first three plasmon modes. The illuminating electric field $E_y^i$ is chosen to be perpendicular to the gap (y-polarized). The fields are normalized to the illuminating electric field.

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A similar response in the reflectance spectra is observed if the electric field polarization is chosen perpendicular to the gap. As shown in Fig. 2b), the first resonance at ν̄=5,000 cm^-1 is broad and rather strong. In addition, two higher-order resonances are observed at ν̄=9,000 cm^-1 and ν̄=11,400 cm^-1. Also the same residual spectral features appear at higher frequencies. The Wood anomaly in reflection is visible once more at ν̄=13,000 cm^-1. The plasmonic peak (fifth resonance) at around ν̄=17,500 cm^-1 is associated with a charge density oscillation perpendicular to the bottom of the ‘U’-structure (oscillation in y-direction).

 

Fig. 4. Magnitude of the electric field distribution for the first three plasmon modes. The illuminating electric field $E_x^i$ is chosen to be polarized parallel to the gap (x-polarized). The fields are normalized to the illuminating electric field.

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For visualizing the excited plasmons, the near-field amplitudes of the first three resonances have been calculated for the present structure. Results for the magnitude of the electric field are shown in Fig. 3 and Fig. 4 for the two polarization directions. The phase distribution of the modes is omitted, as no additional information can be deduced. In general a phase jump of π occurs along each node of the magnitude of the electric field. The phases are rather constant in the spatial domain between the nodes as expected for a resonance. The fields are referenced to the total field at z=20 nm above the structure in reflection. The illuminating field is a linearly polarized plane wave at normal incidence (positive z-direction). Please note that the amplitude of the field component in the polarization direction is a superposition of the incident and the reflected field, whereas the other two components are due to reflected fields only.

We conclude from the near-field amplitudes that all excited resonances can be attributed to plasmonic modes of the entire SRR. With increasing order of the mode the resonance frequency becomes larger. By labeling the appearing modes according to the number of their nodes in the magnitude of the E _z -component, we can recognize modes up to an order of six. The character of the various plasmon modes is best explained looking at the E _z -component. Modes with an odd number of nodes are excited if the incident field is polarized parallel to the gap, whereas modes with an even number of nodes are excited for polarization perpendicular to the gap. This is a result of the D₁ symmetry group for the SRR and the illuminating light field. For the polarization perpendicular to the gap (Fig. 3), the internal field has to be in phase at both legs of the SRR. The SRR has mirror symmetry with respect to this polarization. Hence, for preserving this symmetry, the reflected fields have to have an equal phase along the legs of the ‘U’-structure. As a zero order mode was not found (e.g., a mode with a constant phase across the SRR), all modes must have a non-vanishing even number of nodes (2,4,6, ..) to meet the required symmetry. The strength of the highest order mode at ν̄=11,364 cm^-1 is low but its symmetry can be inferred from the figure (e.g., there are seven maxima in the E _z -component and six nodes).

For the polarization with the electric field parallel to the gap, the illuminating light-field causes internal fields in the two legs of the SRR that have opposite flow direction. Therefore, the field in the two legs has a phase difference of π. Consequently, only plasmon modes with an odd number of nodes (1,3,5, ..) can be excited (see Fig. 4).

Based on the near-field patterns it is comprehensible, why the first, the second, and the third resonance for the polarization perpendicular to the gap appear at slightly higher energies than their counterparts for parallel polarization. The additional field nodes require higher energies of the light field to excite the particular plasmonmodes. The process is identical to the excitation of a quadrupole or a hexapole resonance in metallic nanoparticles, where the excitation of those higher order modes takes likewise place at higher frequencies when compared to the dipolar resonance. Therefore, it is not justified to identify the origin of the second resonance for parallel polarization to be equal to the origin of the first resonance for perpendicular polarization.

To sum up, the response resembles results recently obtained in elongated nanowires, which form plasmonic resonators [23]. In experiments using a scanning near-field optical microscope, standing waves along the nanowires were found with short oscillation periods. More specifically, the modes found in the SRRs correspond to the lowest-order modes of such a plasmonic resonator.

Also explained on the basis of the E _z -component of the field, the same line of argumentation on the exact mode character applies to all the other field components. With an increasing resonance frequency the nodes in the amplitude in the respective modes increases.

3.2. Spectral response as a function of the geometry

In order to obtain further insight into the character of the resonances, a number of geometrical parameters were altered. Reducing the leg length shifts the plasmon resonances associated with the entire SRR (resonances labeled 1–3 in Fig. 2) continuously and simultaneously to higher frequencies (Fig. 5). Note that the sample with the shortest length of l _⊥=60 nm=w in Fig. 5 corresponds to a single nanowire. For the polarization of the electric field parallel to the gap as shown in Fig. 5a), the strength of the first resonance remains nearly constant whereas the second and third resonance become significantly weaker with decreasing l _⊥. The strength of these higher-order resonances becomes weaker with a decreasing geometrical size of the structure. Particularly the third-order resonance is no longer visible for the structure with l _⊥=273 nm, but remains visible for structures with a slightly larger leg length. The second resonance appears with negligible strength at a frequency of ν̄=12,500 cm^-1 for the structure with l _⊥=103 nm. In the limit of a vanishing leg length of the SRR where the length l _⊥ is equal to the width w, only a dipolar mode can be excited.

In this limit, the first resonance converges to the plasmon resonance of the single wire. In electrical engineering terms, this resonance corresponds to the so-called LC-resonance, where oscillating charges in the entire structure are excited due to coupling at the gap [15]. From a plasmonic point of view, this resonance is characterized by a charge oscillation along the entire wire that forms the ‘U’-structure. It corresponds to the fundamental plasmon mode of the structure which naturally coincides with the plasmon mode of the single wire in the limit of a vanishing leg length. This is supported by the computed near-field distribution of the SRR as shown in Fig. 4. The shift in the resonance wavelength with decreasing l _⊥ can be understood by unfolding conceptually the ‘U’ into an extended single wire piece that supports the electron oscillations. Its resonance frequency depends dominantly on the ratio of the entire ‘U’- structure length to its height. An increasing ratio results in a stronger resonance shift towards lower frequencies. Therefore, reducing the length of the unfolded ‘U’-structure as in the present simulation shifts the resonance to higher frequencies. The third geometrical parameter, namely the width, has minor influence.

 

Fig. 5. Reflection from SRRs as a function of the leg. The electric field is polarized in a) parallel to the gap and in b) perpendicular to it. The geometries of the pertinent structures are shown in the upper part of the figure in colors according to the lines.

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The same resonance shift applies to the two higher-order modes. This continuous shift is more evident from the extracted resonance position of the first two lowest-order modes as shown in Fig. 6. The higher order plasmonic modes can be excited, because the spatial extension of the structure is not negligible in comparison to the wavelength. Retardation will take its toll. If the second resonance was a plasmon excited in the bottom of the ‘U’-structure, no change in its spectral position could be expected. Nevertheless, we observe exactly the same tendency for its shift in resonance position as compared to the first resonance.

It is worthwhile to note that in the limit of a vanishing leg length and hence in the limit of a vanishing gap, the first-order resonance remains excited. Therefore, not only a coupling to this resonance via the gap takes place, but coupling must be also be accomplished at the remaining wire piece that forms the bottom of the ‘U’-structure.

The resonance associated with theWood-anomaly appears as expected at ν̄=13,000 cm^-1 independently of the exact SRR geometry in Fig. 5. The fifth resonance at ν̄=17,500 cm^-1 appears nearly at the same frequency but is decreased in strength for a reduced length of l _⊥. This can be easily understood, as the resonance is associated with a charge density oscillation perpendicular to the wire pieces that forms the legs of the ‘U’-structure. If these particular piece of wires vanishes, the plasmon is no longer excitable.

For the polarization with the electric field perpendicular to the gap, a comparable behavior is observed, as shown in Fig. 5 b). By reducing l _⊥, the resonances associated with the plasmon excitation in the entire SRR shift simultaneously to higher frequencies. The extracted resonance positions for the first two modes are also shown in Fig. 6. The strength of the first resonance gets significantly weaker than for the second resonance. The third resonance is weakly seen for the structure with a length of l _⊥=273 nm but vanishes at a further reduction of l _⊥.

 

Fig. 6. Resonance frequency of the first two modes as a function of the length of of the ‘U’-structure perpendicular to the gap for the two possible polarizations.

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We conclude from this observation that the first resonance in this polarization is dominantly associated with a plasmon excitation in the two legs of the ‘U’-structure. The bottom of the ‘U’-structure mediates a coupling among the wires and shifts the resonance to significantly lower frequencies. The remaining higher-order resonances are associated with charge oscillations in accordance with their eigenmodes in the structure. The fifth resonance is nearly independent in spectral position and strength when l _⊥ is changed. The part of the ‘U’ that causes the resonance, an oscillation of the charge density perpendicular to the bottom of the SRR along the width of this wire piece, is not affected. This observation is similar to the response with the polarization parallel to the gap.

4. Conclusion

In conclusion, we have theoretically analyzed the character of the resonances in SRRs at normal incidence. A consistent framework of arguments can explain the respective resonances in terms of plasmon excitations in the entire structure. There is no particular need for employing electrical engineering terms, such as L-C-resonances. Especially, the second-order mode previously considered a plasmon mode excited in base line of the ‘U’-structure, was shown to be a higher-order mode of the entire ‘U’-structure. Adopting a plasmonic point of view, it has been shown that the only difference for the two polarizations consists in the symmetry of the excited modes. For the polarization of the electric field parallel to the gap, the modes have an odd symmetry, whereas the modes possess even symmetry for the other polarization. The difference in the resonance frequencies of differently polarized modes can be explained by the mode orders. In addition, it was shown that coupling to the fundamental resonance at normal incidence can be accomplished via the base line wire and not exclusively via the gap of the SRR.

For the polarization parallel to the gap, preliminary results for SRRs with further elongated legs showed the possibility to excite modes with an even higher order. By closing the resonator, all odd modes vanish. In previous studies, this was understood as a sign that this mode is associated with a resonance in the permeability if the polarization of the incident light was chosen such that the magnetic field is perpendicular to the plane. The question on whether similar conclusions are valid for higher-order modes is presently under investigation. As those higher modes appear at significantly larger frequencies, it seems to be an attractive concept to employ higher-order resonances of SRRs for obtaining a metamaterial with a negative permeability. Although the frequencies of higher-order modes will suffer from the same limitations as their first-order counterparts, they could potentially pave the way to obtain a negative index metamaterial at higher frequencies as compared to SRR concepts that rely on the fundamental mode which are up to now limited by the technologically obtainable structure size. This seems to be advantageous for the fabrication of SRRs on a substrate with the plane of the SRR being perpendicular to the substrate. With such a medium, a negative refractive index could be obtained at normal incidence, as outlined in the introduction.