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Isotropic Backward Waves Supported by a Spiral Array Metasurface

A planar metallic metasurface formed of spiral elements is shown to support an isotropic backward wave over a narrow band of microwave frequencies. The magnetic field of this left-handed mode is mapped experimentally using a near-field scanning technique, allowing the anti-parallel group and phase velocities to be directly visualised. The corresponding dispersion relation and isofrequency contours are obtained through Fourier transformation of the field images.

wave 18 . The dispersion curve of this mode shows that it has negative group velocity and positive phase velocity. Other studies have reported similar effects, all incorporating current-loop elements in a planar arrangement [20][21][22][23][24] .
Alternatively, such negative mode index behaviour can be explained in terms of an array of coupled magnetic and/or electric dipoles. Dipoles oriented longitudinally (transversely) to the direction of propagation result in a positive (negative) gradient in the dispersion curve 25 . It is the transverse case that is of interest to backward wave studies. The dispersion of the modes supported by an array of split-ring resonators has been theoretically analysed by simply describing the system as an array of out-of-plane magnetic dipoles and in-plane electric dipoles 26 . The analysis of magnetic dipoles in isolation demonstrated a negative gradient in their dispersion relation, however the realisation of such a system is difficult and limited to studies of high-index dielectric spheres 27 .
In this study, out-of-plane magnetic dipoles are created by using arrays of planar metallic spirals. In this geometry, current loops around the spiral produce a magnetic field through the centre that couples, albeit weakly, to neighbouring elements. By placing three-armed spirals in a hexagonal lattice, backward waves are supported in all in-plane directions to a high degree of isotropy. The structure can be viewed as a two-dimensional analogy of the negative index metamaterial 6 , and future studies using this metasurface design could demonstrate new opportunities for imaging and manipulation of surface waves, such as perfect lensing and negative refraction.

Experiment
The proposed metasurface design (shown in Fig. 1) consists of a hexagonal array (pitch, a = 4.95 mm) of spiral-shaped copper elements, supported on a non-magnetic dielectric substrate (Mylar: ε = 2.8(1 + 0.03i)). Each spiral inclusion has three arms each with five turns. The arm width, w, and spacing between them, g, are both chosen to be 0.25 mm. The thicknesses of the metal and dielectric are 18 and 25 μm respectively. The eigenmodes and corresponding electromagnetic fields of the structure were simulated using COMSOL Multiphysics, where an infinite structure is modelled using the unit cell geometry (Fig. 1 inset) and periodic boundary conditions. A sample was subsequently fabricated using a 'print and etch' 28 method with overall size 400 × 280 mm.
Bound microwave surface waves were excited on the structure via coupling to the near-field of a loop antenna (shown in Fig. 1). The magnetic field, formed through the centre of the loop as a current is driven around it, couples to the out-of-plane component of the magnetic field of the surface mode when the antenna is placed adjacent to the surface (<1 mm). On the opposite surface a second loop antenna raster scans the surface and the transmission between the two antennas (mediated by the surface wave) is measured as a function of position. A Vector Network Analyser (VNA) and integral microwave generator sweeps through 'continuous-wave' frequencies in the range 5 to 15 GHz driving the source antenna. A Fourier transform of the resultant field maps for each frequency reveal the reciprocal space isofrequency contours from which the isotropy of propagation can be inferred (circular contours correspond to isotropic propagation). Since power must flow away from the source antenna, the group velocity of any waves must be in the positive radial direction, whilst the sign of the mode-index can be found by studying the propagation of the wave's phase fronts. These will be away from the source for forward modes and towards the source for backward modes. For a given direction of propagation along the surface, a frequency-wavevector diagram of Fourier amplitude reveals the dispersion of the mode. A backward wave will manifest itself as a region of positive gradient (positive group velocity) for negative wavevectors (negative phase velocity), or vice-versa within the first Brillouin zone.

Discussion
The experimentally obtained dispersion curves are shown in Fig. 2, with results from numerical simulations overlaid. This dispersion covers the 'irreducible Brillouin zone' , defined as a path through reciprocal space connecting the Γ, Κ, and Μ symmetry points. In both directions (ΓX and ΓM) there exists a band of frequencies for which the third-order mode has two solutions, one a forward wave, the other a backward wave.   This isotropy is more apparent in the isofrequency contours presented in Fig. 3, which show both forward and backward modes as approximately circular contours. The first panel shows the predictions from the FEM model of the contours as a function of increasing frequency; the behaviour of the positive and negative index modes is clearly evident. The experimental Fourier amplitude data is illustrated in the remaining panels, where the light cone is marked by the black circle. These results demonstrate a significant improvement in isotropy compared to previous studies of backward surface waves using Sievenpiper mushroom structures 12 .
In this report on mushroom structures, numerical modelling demonstrated that the time-averaged power flow was negative beneath the metallic patches, and positive above them. The direction of group velocity of the mode was given by the direction of net power flow through both of these competing regions. Similarly in the present case, despite the geometry being planar, two regions of power flow are observed. In Fig. 4 we plot the time-averaged power flow (integrated over all space in the y-direction) near the sample surface for the third Eigenmode at a value of |k| along ΓK at the turning point in its dispersion (blue square in Fig. 2). While the group velocity of the mode is zero, there exists an envelope of negative power flow in the immediate vicinity of the sample (located at z = 0), which is completely balanced by a surrounding region of positive power flow (|z| > 0.5 mm). The zero-group-velocity point occurs at the mode's maximum frequency (Fig. 3; 9.16 GHz), and with decreasing frequency, we observe the evolution into backward-(high k) and forward-(low k) mode solutions. It is important to note that both solutions originate from the centre of the first Brillouin Zone |k| = 0, as is clear from Fig. 3, and are therefore not a consequence of diffractive coupling. In order to observe the magnetic field distribution associated with the backward wave alone, the recorded field data needs to be filtered in wavevector since a positive index mode exists at the same frequency. An inverse Fourier transform is performed on the isofrequency contours after a windowing function is applied. This window reduces the Fourier amplitude within a given radius in k-space to zero, effectively removing the forward wave. The resulting real space magnetic field map is shown in Fig. 5. To observe the direction of phase front propagation the complex field is advanced in phase by some step, Δφ such that the field after the nth step is given by n 0 Fig. 5 shows the real part of this complex field in the vicinity of the source position for values of Δφ between 0° and 135°. With advancing phase, the phase fronts move towards the source indicating a backward wave, whilst the circular wavefronts demonstrate the isotropy.

Conclusion
In this paper, the near-field excitation and field-mapping of a bound surface wave on a spiral metasurface has demonstrated that, across a narrow band of frequencies, both a forward and backward wave are supported. The backward wave represents a 2D analogy of waves within a negative index medium, with group and phase velocities in opposite directions. The array of spirals realises an approximation to an array of out-of-plane magnetic dipoles 25 , supporting a transverse magnetic mode that disperses with a negative gradient.
In recent years the manipulation of surface waves supported by metasurfaces has been a topic of extensive research resulting in graded index lenses and cloaking devices at microwave frequencies. The work presented here expands on this topic and adds isotropic negative mode index to the list of available properties when designing metasurface devices. In particular this research could enable negative refraction of surface waves and subsequently the development of a perfect lens for surface waves. 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. Figure 5. Measured instantaneous magnetic field (H z , arbitrary scale) at a height of 0.5 mm above the metasurface. The surface mode is excited at the centre of the image. The complex field is advanced in phase from 0 to 135° and the phase fronts (e.g. the null in electric field marked by the black circle) propagate towards the source (negative phase velocity) whilst, for our measurement system, power must flow away from the source (positive group velocity).