Our study is divided in two steps. First, in

Section 2.1, a general study of twist-symmetric complementary split ring resonators (CSRRs) is conducted. In this initial study, the effects of locally adding an approximate twist symmetry to an array of CSRRs is highlighted. Secondly, in

Section 2.2, a lens design consisting of an array of tailored CSRRs is conducted based on the results obtained in

Section 2.1.

#### 2.1. Study of Twist-Symmetric Complementary Split Ring Resonators

In this work, four different structures with local twist symmetry are studied. The structures are illustrated in

Figure 1. In this initial study, the structures are 3D periodic. The different unit cells consisted of different numbers of layers of perforated metallic sheets of a thickness

t, separated by an air gap of thickness

h. The perforation in the metal consisted of two concentric semi-circular slots, also called CSRRs, of radius,

R. The slots had a width

$sw$, and a metallic bridge of width

g connected the metal on each side of the slot. The studied structures were 1-, 3-, 4-, and 6-fold twist-symmetric with 1, 3, 4, and 6 metallic sheets per unit cell, respectively. The twist symmetry was local since only the unit cell was twist-symmetric, and not the full array of CSRRs. Furthermore, the twist symmetry was approximate since only the slots were rotated between adjacent sub-unit cells, and not the entire sub-unit cell. Moreover, since the sub-unit cell exhibited mirror symmetry, the rotation between two adjacent sub-unit cells in the smallest geometrical full unit cell was halved. This means that the rotation between two adjacent sub-unit cells was not

$2\pi /m$, but rather

$\pi /m$, where

m is the order of the twist symmetry. Consequently, the rotation between adjacent sub-unit cells was 180

${}^{\circ}$, 60

${}^{\circ}$, 45

${}^{\circ}$, and 30

${}^{\circ}$ in the 1-, 3-, 4-, and 6-fold twist-symmetric structures, respectively.

In

Figure 2a, the propagation constants for the four structures are represented. In all structures, the lateral periodicity,

$uw$, was 20 mm, the slots were placed at a radius

$R=7$ mm; the slot width,

$sw$, was 3 mm, and the metallic bridge separating the slots had a width of

$g=1.5$ mm. The separation between the metallic sheets,

h, was 4 mm, and the thickness of the metallic sheets,

t, was 1 mm. The final implementation (i.e., the lens) will operate with the second mode of the CSRR array. Therefore, only this mode is plotted in

Figure 2a. The electric field profile of the second mode is illustrated in the inset of

Figure 2a. The mode was odd with respect to the center of the unit cell, in contrast to the first mode, which was even. The odd field distribution was consistent with the excitation employed in the final implementation of the lens. Notably, the electric field pattern in the slot resembled that of a TE

${}_{10}$-mode in a rectangular waveguide. In fact, the dispersion characteristics of the mode were similar to the ones in a periodically-loaded rectangular waveguide.

From the results of

Figure 2a, we can conclude that the phase delay can be controlled by changing the order of the symmetry. In previous works, it has been demonstrated that by increasing the order of twist symmetry, an increased density can be obtained [

16,

17,

18,

24]. However, in

Figure 2a, the highest density is not necessarily achieved in the structure with the highest order of the symmetry. The reason for this discrepancy with previously-reported results is that, in this study, the sub-unit cell period was kept constant, in contrast to previous studies where the full unit cell periodicity was kept constant. Moreover, the lowest order of twist symmetry (i.e., the three-fold structure) was the most different from the purely-periodic structure. There were two reasons for this. First, the cut-off frequency of the CSRR was shifted upwards in the twist-symmetric structures compared to the purely-periodic structure. The CSRRs in different layers of the structures effectively formed a waveguide with a cut-off frequency. When adjacent layers were rotated, the effective width of the waveguide was reduced, leading to an up-shift in the cut-off frequency. The largest rotation between the two sub-unit cells was obtained in the three-fold structure, and hence, the shift was more severe in this configuration, as the effective width of the waveguide was the smallest. Secondly, the slope of the dispersion curve was more gradual in the twist-symmetric structures compared to the purely-periodic structure. The more gradual slope was caused by the reduced coupling between the CSRRs in different layers for the twist-symmetric structures, compared to the purely-periodic structure. The reduced coupling resulted in a narrower pass band for the second mode [

30].

In

Figure 2b, the effective refractive index at 11 GHz is presented. The order of symmetry and slot width were swept. The remaining parameters were:

$R=7.5$ mm,

$uw=20$ mm,

$g=2$ mm,

$h=4$ mm, and

$t=1$ mm. Again, the density of a periodic structure can be controlled by varying the order of the symmetry. The increased density was related to the increased path traveled by the mode, which was forced to revolve in a helical manner around the periodicity axis. The smallest pitch of the helix, and consequently the highest density, was obtained in the three-fold structure. In fact, for increasing order of twist symmetry, we approached the purely-periodic structure as the rotation between subsequent sub-unit cells was decreasing. Additionally, if another geometrical parameter was allowed to vary simultaneously (the slot width,

$sw$, in this case), a large continuous range of effective indices can be obtained. This control of the phase delay has been previously used to design compact phase shifters [

19]. Here, we employed this effect for the design of a flat lens.

#### 2.2. Lens Design Using CSRRs

The operation of the lens was conceptually similar to the one of transmit arrays [

31,

32,

33,

34]. The phase delay experienced by a wave propagating through the lens was controlled throughout the aperture. The full lens structure is presented in

Figure 3. The lens consisted of 13 perforated metallic sheets with a thickness of 1 mm. The sheets were made of aluminum and were separated by 4 mm of air. These sheets were thick enough to remain flat in a practical realization of the lens. However, due to the manufacturing process employed here (laser cutting), the sheets deformed significantly. Therefore, a layer a 4 mm-thick Rohacell 51HF (

${\epsilon}_{r}=1.065$) was added for structural support in between the metallic layers in the realized prototype. The manufactured lens (in the measurement setup) is illustrated in

Figure 3c.

The perforation in the metallic sheets consisted of an array of CSRRs. The CSRRs were tailored throughout the aperture so the lens provided the required phase correction necessary to transform a spherical wave emanating from the focal point into a plane wave at the opposite side. A similar configuration has been employed to obtain a wideband linear-to-linear polarization transformation with very low insertion losses [

30]. However, in that work, a normally incident plane wave was assumed, and the CSRRs remained unchanged throughout the aperture. Moreover, the first and last layers were different in order to produce a polarization transformation.

Thirteen metallic layers were employed here to produce 3-, 4-, and 6-fold twist-symmetric unit cells. In this way, each twist configuration can fit an integer number of periods into 12 metallic layers. One additional layer, identical to the first, was inserted at the end to ensure that the lens was symmetric and no polarization transformation was performed. If desired, polarization transformation can be integrated into the structure if the first and last layers are different. The phase delay throughout the lens was tailored so that a spherical wave emanating from the focal point, $fp$, arrived in phase at the opposite side of the lens. To obtain this, $\psi \left(r\right)=l\left(r\right)+T\xb7{n}_{\mathrm{eff}}\left(r\right)$ must remain constant throughout the aperture (up to an integer addition of free-space wavelengths), where r is the radial coordinate in the aperture, $l\left(r\right)=fp/cos\left[{tan}^{-1}(r/fp)\right]$ is the total optical path from the focal point through the lens, and ${n}_{\mathrm{eff}}\left(r\right)$ is the effective refractive index of the lens at the position r. The focal point in the designed lens was 130 mm, and the width of the lens, w, was 220 mm. The total thickness of the lens, T, was 61 mm.

The simulated electric field for the lens, at 11 GHz, excited by a half-wavelength dipole, is presented in

Figure 4a,b for the E-plane and H-plane. The lens successfully transformed the spherical wave into a plane wave. To estimate how successful the transformation was, the normalized radiation pattern at 11 GHz, both for the lens fed with a rectangular waveguide (WR90) and for the isolated waveguide (normalized to the maximum realized gain of the full lens antenna), is plotted in

Figure 4c,d for the E-plane and H-plane. A clear improvement in the gain of roughly 6 dB was achieved with the lens. The measured H-plane radiation pattern is included in

Figure 4d, and the measurement corroborated the simulation. The measured E-plane radiation pattern was distorted by the struts necessary for mounting the antenna in the anechoic chamber. Therefore, the E-plane cut is not presented here. The losses in the metallic sheets were below 1% of the stimulated power in simulations.