Strong beaming of microwave surface waves with complementary split-ring-resonator arrays

A thin copper sheet, populated by an array of complementary split ring resonators, presents strong surface wave beaming in orthogonal directions at two distinct frequencies. This simple array is significantly thinner than existing single frequency beaming surfaces. The observed beaming frequencies are associated with the two lowest resonance modes of the split rings, and the beams are subwavelength in width and approximately non-diverging. The beaming is analysed through comparison of near-field scans of the surface-normal electric fields with numerical simulations.

subwavelength unit cell size, and their broader bandwidth were listed as the key advantages of the CSRRs over the alternative structures.
By exploiting the advantages of CSRRs, this current study demonstrates that the self-collimation of surface waves, reported in previous work, may be achieved with structures of far more compact dimensions. By associating the beaming frequencies with the resonant eigenmodes of the underlying CSRR elements, a full explanation of the phenomenon is achieved. Significantly, the CSRRs display orthogonal beaming of surface waves at two completely separate frequencies, a phenomenon as yet unreported in the existing literature. Furthermore, both observed beams are highly subwavelength in width and approximately non-diverging in nature. Finally, the simplicity of the CSRR structures enables faster and easier fabrication than before. It is believed that these much thinner and simpler structures could prove useful for a number of engineering applications in the microwave regime, such as in surface based communications and device-to-device links where directionality matters and on devices where radar energy needs to be collected and channelled to reduce electromagnetic cross-section, such as wind turbine blades. Figure 1 shows the typical geometry of the CSRRs used in our experiments. The CSRRs are etched onto the copper sample in a square lattice of pitch d = 2 mm. Samples consist of 18 μm thick copper sheets backed by 50 μm thick polyethylene terephthalate (plastic) sheet.
The CSRRs were etched into the copper surface to expose the dielectric backing beneath. Experimentally, stripped-end coaxial cables of outer diameter 2.1 mm and exposed wire diameter 0.5 mm, with exposed wire length of 1 mm, connected to a vector network analyser (VNA) were used as near-field probes to both excite and detect microwave surface waves on the sample surface. It is important to understand that our antenna probes are operating at frequencies below their resonance in order to minimise perturbation of the modes of interest. While we are not attempting to quantitatively measure the field strength of any particular field component, our probes will be most sensitive to the z-component of the electric field. In the discussion that follows, we make a comparison of the field distribution of the transverse-magnetic modes that propagate across the surface. The samples were placed in an xyz translation stage, with microwave absorber placed along all edges to minimize the effects of reflections. On one side of the surface, an excitation probe was used, being swept through frequency from 5 GHz to 70 GHz. On the other side of the sample, a similar probe was used to scan across the surface in the xy plane with step size of 0.25 mm in both directions measuring both the magnitude of the normal component of the electric field, E z , and the phase relative to an internal reference signal in the VNA. Both probes were arranged normal to the sample, at a distance of approximately 500 μm from the surface. A finite element method (FEM) model (COMSOL 18 ), with the assumption of perfect conducting rings and lossless dielectric, was used to determine the eigenmodes of the system, and to model the expected resonant fields.

Results
Maps of the fields were obtained directly from the S12 measurements from the VNA at frequencies up to 70 GHz with a frequency step size of 0.05 GHz. Two frequency regions display strong surface wave beaming. The lower frequency beaming, shown in Fig. 2, occurs at close to 20 GHz and demonstrates self-collimation in a direction orthogonal to the CSRR split axis, along the x-axis of the sample. Note that this beaming is very tight with a width of order 8 mm compared to the wavelength of 15 mm. The higher frequency beaming, shown in Fig. 3, occurs around 49 GHz and here the surface waves are tightly collimated along the CSRR split axis, or the y-axis of the sample, with now a width of order the wavelength or about 2 unit cells.
A structured metal plate is known to support two families of surface modes, one with symmetric external magnetic fields parallel to the surface (H // ) and one with antisymmetric external H // fields. In the CSRR structures, we see the effects of the first and second order antisymmetric modes experimentally as the bright beams in By combining the measured time-averaged electric field magnitude and phase, we obtain the complex electric field. From this a 2D Fourier transform produces the k-space image of the excited eigenmodes of the system. This process has been used previously both in the optical regime (see supplementary information of 19 ) and in the microwave regime 15 . Inspection of these k-space iso-frequency contours reveals information about the propagating modes supported in the system.
These iso-frequency contour maps, shown in Figs 2b and 3b, demonstrate the highly anisotropic response of the surface. Again we observe the different mode symmetries, with the first and second order antisymmetric mode contours elongating from circles into straight lines, which leads to the strong collimation of power or beaming. The symmetric mode remains largely close to the light line, appearing as the circle in the iso-frequency contours of Figs 2b and 3b.
Power propagation is given by the group velocity: So for iso-frequency contour plots, the group velocity is normal to the k-vector contour lines 20 : Bearing this in mind, it is clear that straight regions of the iso-frequency contour lines will set all normals for the straight region parallel to each other, forcing the power (group velocity) to travel in a straight and non-diverging path. Thus, the appearance of flat contour lines in the experimental iso-frequency data in Figs 2b and 3b accords fully with the beaming of the surface wave power. This strong power beaming anisotropy for this surface observed in experimental results is further verified by plotting model dispersion curves obtained from FEM 18 simulations by solving for the eigenmodes of an infinite periodic array of these CSRRs, which are treated as perfect conductors. Results from such modelling is shown in

Discussion
When attempting to describe this highly directional propagation of microwave surface wave power over the CSRR samples, it is helpful to understand the origin of the surface mode. 'Spoof ' SPPs are a result of the collective excitation of resonances within the surface of a structured metal. The dispersion of the surface waves is dictated by the anticrossing of a grazing photon with a localised, in this case split ring, resonance of the system in question. Any periodic collection of resonant elements (often referred to as 'meta atoms') will produce such an effect. The branch of the dispersion curve beyond the light line (grazing photon) corresponds to the localised surface mode, 'spoof ' surface plasmon. Whilst there are 'spoof ' SPPs associated with many higher order resonances associated with the CSRRs, the beaming observed in the surface-normal electric field data shown in Figs 2 and 3 arises from just the first and second order resonances. Consequently in the following simulated results fields related to these two modes only are shown. Figure 5 shows the electric fields calculated by the FEM model for the eigenmodes of the CSSR. The lower frequency eigenmode is characterised by a π phase change in the electric field around the perimeter of the split ring, with a zero in electric field intensity at the split. This is the fundamental (half-wavelength) mode of the element. By contrast the higher mode fits a full wavelength around the ring and has two field zeros, with one sitting at the centre of the ring length. The length of the conductor is about 6.2 mm, which would naively give a fundamental resonance of 24.2 GHz with the harmonic at 48.4 GHz. Both simple estimates are close to the computed values of 23.2 GHz and 41.6 GHz.
The fields along the left and right sides of the CSRR for the lower mode of the model in Fig. 5a are equal and opposite, leaving one remaining field component along the y-axis of the ring. Taken in isolation, one can compare this component to the fields that would occupy a rectangular hole of the same length along this lower side, oriented along the x-axis. Similarly, the two high field regions of the higher mode in Fig. 5b could be compared to two rectangular holes oriented along the y-axis. The dispersion surfaces arising from arrays of rectangular holes have already been shown to be distorted 21 , with elongation of the iso-frequency contours occurring along the short axis of the rectangles. Thus, by representing the CSRR structure in this way, the switch in orientation of surface wave beaming between the two modes may be explained.   To test this analogy, a further sample was created with an extra split introduced into the lower ring section where the main field component for the lower frequency mode resides. The addition of this new split was designed to remove this high field region, thus effectively removing the part of the ring acting as the single rectangular hole for the lower mode and hence destroying the beaming previously seen along the x-axis.
First, a simulation of this sample was carried out to verify the effect of the extra split on the supported eigenmodes of the structure. This is shown in Fig. 6. The lower mode still exists as an eigenmode of the structure; however it is significantly changed in frequency and the fields now display only the equal and opposite components on the left and right sides of the ring. The component contributing to the lower mode and its associated beaming has been removed. As expected, the higher order mode remains.
The doubly split sample was then fabricated and scanned in the same way as the original CSRRs. The fields from the exciting antenna no longer couple to the lower mode responsible for beaming along the x-axis while the remaining mode displays beaming along the y-axis (see Fig. 7), similar to the 49.15 GHz mode of the original CSRR structure. The removal of the lower mode supports the analogy presented above, and explains why the singly split CSRRs display perpendicularly oriented beaming of the surface wave power for the two modes.
The beaming frequency of the remaining mode for the doubly split CSRR has, however, been reduced to 38 GHz somewhat below the calculated resonant frequency of 42.2 GHz as compared to the original beaming at 49.15 GHz, significantly above the original resonant frequency of 41.6 GHz. This change arises from the strength of diffractive scattering and the resulting band-gap at the edge of the first Brillouin zone.
In summary, complementary split ring resonator arrays display propagating surface modes, and for the lower two of these the power in the surface wave is at selected frequencies very strongly collimated, being confined to a width somewhat less than the wavelength. This behaviour can be explained through analogy to rectangular holes, which display similarly anisotropic dispersion. In contrast to previous work demonstrating surface wave beaming, the planar CSRR array presents a far thinner structure capable of supporting this phenomenon, and the simplicity of these thin structures allows for easier sample fabrication. Additionally, the arrays also display beaming at two separate frequencies, in orthogonal directions. Such structures could therefore prove useful for engineering applications in the microwave frequency regime. Data Availability All data created during this research are openly available from the University of Exeter's institutional repository at https://ore.exeter.ac.uk/.