1. Introduction

In recent years, chiral metamaterials (CMMs) have intrigued much interest and have been extensively investigated in terms of linear [1–6] or circular polarizers [7–13], asymmetric transmissions [14–19], and negative refractive index [20,21] for their unique electromagnetic (EM) characteristics such as elliptical or circular dichroism, bianisotropy, and magneto-electric coupling. Giant chirality and optical activity are also observed on the basis of these original physics [22,23]. Due to the anisotropic EM properties (i.e., the lack of mirror symmetry), CMMs exist considerable cross-coupling between electric and magnetic fields at resonance and provide strong chirality around resonance, leading to the breaking of the degeneracy between the right-handed circularly polarized (RCP) and left-handed circularly polarized (LCP) EM waves and different transmission coefficients. Under this condition, CMMs also have the capability of tailoring the propagation state of EM waves [24,25]. The constitutive relations in a chiral medium associated with the strength of the coupling, namely chirality parameter κ, can be written as [20]

A survey of recent literature indicates that CMMs have also been found in the applications of antenna [26] and polarization spectra filter [27], etc. For the circular polarizers, three-dimensional (3D) gold helical metamaterials have been employed to obtain broadband chirality and thus broadband circular polarizer exceeding one octave [7]. However, only one kind of circularly polarized waves can be transformed by such helix CMM, and the realization and fabrication in the visible wavelength range is hindered by the minimum realizable features imposed by the diffraction limit. Most recently, Alu et al. proposed an alternative avenue of stacked nanorod arrays with a tailored rotational twist to obtain the analogous broadband 3D chiral effects and to simultaneously relax the requirement on fabrication [10]. U-shaped split ring resonator with only one ring (SRR) [21,23] is commonly arranged in four-fold rotational symmetry (C4) and thus the optical activity is insensitive to the plane polarization angle of the incident linearly polarized wave. Meanwhile, based on the same concept, Ye et al. proposed a wheel pattern composed of a cross and a four-gap split ring that exhibits C4 symmetry to obtain circular polarizer with arbitrary polarizations [9]. However, the CMMs in [21] and [23] have relatively low polarization extinction ratio and polarization-conversion efficiency. To address these issues over dual or multiple operation bands, asymmetric U-shaped SRR pair [11], planar spiral structure [12], and twisted split ring resonators (SRRs) with double or more concentric rings [13] have afterwards been continuously reported. However, to the authors’ best knowledge, most meta-atoms utilized in previous articles are not compact enough except for that proposed in [13]. Moreover, it seems that the twisted SRR is rarely reported in the circular-polarization transformation although it has been reported to convert the linear polarization to its cross polarization. In the scenario of electrically large elements, the resulting metamaterial slab or bulk medium is more influenced by the parasitic diffraction effects than that made of compact elements, and the collective electromagnetic response cannot be described appropriately by the macroscopic constitutive material parameters defined under the effective medium approximation. In this frame, the elements in deep subwavelength would benefit considerably the design to some degrees.

2. Fundamentals and design

In this work, we present a type of alternative compact CMM that is capable of emitting circularly polarized waves from linearly polarized incident wave with high transformation coefficients over dual microwave bands under the normal incidence. The chiral behavior is enhanced by the fractal perturbation and thus renders the twisted SRR the capability of linear-to-circular conversion. Figures 1(a) and 1(b) show the schematic and geometrical parameters of the proposed meta-atom that is utilized as the basic building block of the CMM circular polarizer. The meta-atom is composed of two metallic SRR with Hilbert fractal curve of second iteration order on both sides of a dielectric substrate twisted by 90° with respect to each other. The metallic pattern is the dual part of the complementary structure proposed in [28], which is a variation of conventional Hilbert curve. The plane wave with linear polarization along x or y-direction normally impinges on the sample along z-direction. Due to the strong space-filling property of fractals, the electrical current path along the zig-zag boundary is significantly extended and thus considerable reduction of resonant frequencies, namely compactness of the meta-atom can be envisioned. Most importantly, the fractal perturbation can readily introduce geometrical asymmetry due to the asymmetric fractal boundary. Therefore, the circular dichroism and optical activity can be achieved since there is no mirror symmetry, and the EM response of the normally incident wave with x polarization distinguishes from that with y polarization due to the lack of C4 symmetry.

 

Fig. 1 (a) Schematic and (b) parameter illustration of the proposed CMM unit cell. (c) Photograph of the fabricated sample, and (d) its orientation with the incident wave. The geometrical parameters are designed as p = 6.6 mm, a_x = 1.08 mm, a_y = 0.78 mm, b_x = 4.44 mm, b_y = 5.04 mm, h = 1 mm, and d = g = 0.24 mm.

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In this design, the substrate is chosen as the commonly available inexpensive F4B board with dielectric constant of ε_r = 2.65, thickness of h = 1 mm, and loss tangent of 0.001, while the metallic layers on both sides are copper sheet with thickness 36 μm. By periodically arranging the well-designed meta-atom with period constant p, a chiral slab with appealing circular behavior at desired frequencies can be engineered. To study the behavior of the chiral structure, we conduct numerical simulations using the Ansoft HFSS, a commercial full-wave finite-element-method (FEM) EM-field solver. In the simulation setup shown in Fig. 1(a), the periodical boundary condition (PBC) are adopted in the transverse xoy plane while two floquet ports are assigned on the top and bottom boundary along z direction. Figures 1(c) and 1(d) portray the photograph of the finally fabricated sample and its orientation with the incident wave. Under the Cartesian coordinate system, the intensity of transmitted waves $E_x^t$ and $E_y^t$ associates with that of incident waves $E_x^i$and $E_y^i$ through four linear transmission components (the complex Jones matrix).

In the circular polarization basis, the RCP ($E_{\text{+}}^t$) and the LCP ($E_{\text{-}}^t$) transmitted components associate with the incident linear components as [13]

In the experiment, we fabricate the designed chiral structure composed of 35 × 35 unit cells and occupied an overall area of 231 × 231 mm² through conventional printed circuit board technology, and measure the transmission coefficients in microwave anechoic chamber. A pair of double-ridged broadband horn antennas with VSWR<2 over a wide frequency range from 1 to 18 GHz are used and they are distributed by a distance of 1.2 m to eliminate the near-field effects. The chiral slab is placed in the middle between the antennas and the transmission coefficients are recorded through Agilent N5230C vector network analyzer. The time-domain gating strategy was used to eliminate the undesirable reflections. By changing the orientation of the two horn antennas to generate and receive EM waves with different linear polarizations, all four different linear transmission coefficients are measured and the four circular transmission coefficients can be easily transformed from them according to Eq. (3).

3. Results and discussions

Figure 2 shows the simulated transmission spectra of the chiral slab for propagation in the forward ( + z) and backward (−z) directions from 7 to 13 GHz, while Fig. 3 compares the simulated and measured transmission coefficients only for the backward propagation. As is expected from Fig. 2, the cross-polarization components t_yx and t_xy are obviously different in the case of backward and forward propagations, which originates from the different polarization conversion of the chiral structure. Moreover, t_yx and t_xy not only interchange their magnitudes but also get an 180° phase deviation when the propagation direction is reversed, which is an necessity to maintain the reciprocity of the chiral system [15,18]. From Fig. 3, we observe that the simulation and measurement results are in good agreement in the entire frequency range. The slight frequency shift is due to the inherent tolerances and random errors induced in fabrications and measurements. For the x-polarized normally incident wave, the co-polarization and cross-polarization transmitted waves present almost the same amplitudes and −90° ( + 90°) phase difference at f₂ = 9.77 and f₃ = 11.84 GHz, respectively, indicating a pure circular polarization wave. At f₁ = 8.72 GHz, although the phase difference is on the order of + 90°, t_xx reaches its minimum value and is much lower than t_yx, and thus an elliptic polarization is expected. As to the y polarization, the phase difference is observed as nearly + 90°, −90° and + 90° at f₁ = 8.84, f₂ = 10.08 and f₃ = 11.78 GHz, respectively. However, pure circular polarization can be effectively formed only at f₃, whereas at f₂ the transmission difference between t_yy and t_xy is quite large and hence the conversion efficiency of a circular wave is low. The performance at f₁ resembles that in the x-polarization case despite of a slight improvement. The slight difference of f₁, f₂, and f₃ between the x and y polarizations are observed due to the lack of C4 symmetry [12].

 

Fig. 2 Simulated transmission spectra of the four matrix components for (a) backward and (b) forward propagations.

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Fig. 3 Simulated and measured four linear transmission coefficients for the backward propagation. (a) x polarization; (b) y polarization. The top row is the magnitude while the bottom row is the phase.

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Figure 4 depicts the magnitude and polarization extinct ratio (defined by 20log₁₀ |T₊|/|T_-|) of the linear-circular transmission coefficients. As is discussed for the linear transmission coefficients, there is also obviously three transmission dips at three frequencies f₁, f₂, and f₃, indicating three distinct resonances. For x polarization, numerical (experimental) results indicate that the transmitted RCP wave reaches its minimum of −25.2 dB (−21.6 dB) at 9.77 GHz (9.89 GHz) while the LCP wave reaches its minimum of −30.6 dB (−24.5 dB) at 11.84 GHz (11.99 GHz). At both frequencies, the transmission coefficients of the LCP and RCP wave are observed as −1.28 dB (−1.42 dB) and −2.98 dB (−3.59 dB), respectively. Consequently, the calculated (measured) polarization extinction ratios are observed as −23.9 (−20.2 dB) and 27.6 dB (20.9 dB) at the above two frequencies, suggesting a prominent LCP wave at f₂ while a prominent RCP wave at f₃. The measured large transmissions indicate high conversion efficiency of the chiral structure. The extinction ratio of 6.24 dB (4.8 dB) at 8.72 GHz (8.87 GHz) indicates a degenerated right handed elliptical at f₁. For the y-polarized incident wave, the simulated (measured) difference between the two transmitted CP components reaches a level of 15.5 dB (8.3 dB) at 8.84 GHz (8.98 GHz), −7.27 dB (−7.84 dB) at 9.87 GHz (9.89 GHz), and 29.5 dB (21.2 dB) at 11.78 GHz (11.86 GHz). Therefore, the degenerated right handed elliptical, left handed elliptical and pure RCP wave are observed at f₁, f₂ and f₃, respectively. The most intriguing characteristic of CMM should be the almost completely converted RCP waves from x- and y-polarized linear waves in the vicinity of 11.8 GHz, where an approximately polarization-independent operation regime in a C4 asymmetric structure is suggested. The induced similar distribution of localized currents in both cases gives rise to the abnormal function of polarization conversion.

 

Fig. 4 Simulated and measured four linear-circular transmission coefficients for the backward propagation. (a) x polarization; (b) y polarization. The top row is the magnitude while the bottom row is the polarization extinct ratio.

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The emitted wave can be described by the polarization azimuth rotation angle θ and ellipticity η defined by the following equations:

The corresponding numerical and experimental results for both x and y polarization are portrayed in Fig. 5. Again, the simulation and measurement results agree well with each other in both cases. From Fig. 5(a), we notice that the calculated (measured) value of η is 19° (13.1°), −41.4° (−39.4°), and 42.6° (39.85°) at 8.72 GHz (8.87 GHz), 9.77 GHz (9.89 GHz) and 11.84 GHz (11.99 GHz), respectively for the x-polarized incident wave, further confirming a nearly pure LCP wave and RCP wave at f₂ and f₃, respectively. Note that η = 45° corresponds to the pure circularly polarized wave whereas η = 0° to the pure linearly polarized wave, and the wave is right handed elliptical when η>0°, otherwise it is left handed elliptical. Moreover, the simulated (measured) polarization azimuth rotation angles are found to be θ = −55.9° (−59.4°) and θ = 47.7° (50.5°) with respect to the incident linearly polarized wave when η = 0° at 9.06 GHz (9.15 GHz) and 11.29 GHz (11.48 GHz), respectively, revealing a giant optical activity. This also suggests that the transmitted wave is still linear polarization with rotation angle θ relative to the incident linearly polarized wave. In the case of y polarization, the value of η is 35.4° (23.9°), −21.6° (−22.94°) and 43.1° (40°) at 8.84 GHz (8.98 GHz), 9.87 GHz (9.89 GHz) and 11.78 GHz (11.86 GHz), respectively, and the rotation angles are observed as θ = −33° (−32.5°) and θ = 38.1° (30.9°) when η = 0° at 9.33 GHz (9.4 GHz) and 11.27 GHz (11.32 GHz), respectively.

 

Fig. 5 Simulated and measured polarization azimuth rotation angle and ellipticity of the CMM for both (a) x polarization and (b) y polarization.

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The physical origin of the triple-band resonances for the proposed CMM structure can be understood by examining the current and field distributions as shown in Fig. 6, where the phase of axial component of the local magnetic field is plotted in two planes cutting through the top and bottom Hilbert layer, and arrows denote the current directions. The longitudinal magnetic dipole to magnetic dipole coupling [11,12,23] formed in different localized regions accounts for the LCP and RCP resonances while the length of the effective current path is responsible for the operation frequency. As can be seen, there are obviously four small SRR with different orientations formed in the Hilbert structure. At 8.72 GHz, the currents in the upper two SRR of the top layer (enclosed by black dashed) are parallel (coherent) while in the bottom layer the induced currents are parallel in more than three quarters of region including the upper two SRR, enabling six magnetic dipoles with different strength of magnetic moments. The magnetic dipoles in lower two SRR of the bottom layer are not effectively excited due to the disordered (antiparallel) currents, and the weak magnetic moments in the lower two SRR give rise to the low conversion efficiency of the magnetic dipole to magnetic dipole coupling. At 9.77 GHz, seven magnetic dipoles are effectively shaped in corresponding SRR each with coherent current direction, and the currents in the right two SRR (enclosed by black dashed) of either top layer or bottom layer are parallel. The magnetic dipoles in these regions of top layer couple to those in the bottom layer. Moreover, the induced approximate half-region current path in the superposition region of the bilayer interprets the higher resonant frequency with respect to the fundamental resonance. The enhanced intensity of magnetic moments accounts for the moderate polarization transformation. At 11.84 GHz, two electric dipoles form in the diagonal of the top and bottom layer, respectively while six magnetic dipoles are excited in the residual region. The magnetic dipole to magnetic dipole coupling formed in the off-diagonal of the top and bottom layer (see the black dashed) dominates the residual magnetic (electric) dipole to electric (magnetic) dipole coupling and mainly contributes to the polarization conversion at the third frequency. The strongest intensity of the magnetic moments and the coherent current path on the order of three eighths of the entire Hilbert resonator in the superposition region of the bilayer gives rise to the best transformation efficiency and the highest operation frequency. The unique mechanism for the LCP and RCP wave due to the changed orientation of SRR distinguishes current design from any previously reported one.

 

Fig. 6 Axial component of the local magnetic field and surface current distribution of the Hilbert chiral structure in the case of x-polarized incident wave for the (a) right handed elliptical wave at 8.72 GHz, (b) LCP wave at 9.77GHz and (c) RCP wave at 11.84 GHz. Note that * denotes the position where the current direction changes. The top row is snapshot of the top pattern while the bottom row is that of the bottom pattern.

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4. Controllability over resonances

Finally, we perform two complementary parametric analyses for independent control over the resonances and chiral characteristics. Figure 7(a) depicts the effects of gap position on the polarization extinct ratio while Fig. 7(b) the effects of periodicity which correspond to the effects of element spacing. In both cases the residual geometrical parameters are kept constant as those shown in the caption of Fig. 1 and the results are obtained under the excitation of x-polarized EM wave. From Fig. 7(a), we learn that f₂ undergoes continuously blue shifts when the gap of the top Hilbert SRR moves along the x axis from the end to the center in steps of 0.3mm, whereas f₁ suffers slight blue shifts, and f₃ is almost fixed. Therefore, the gap position along the x axis affords dependent control over f₂. On the other hand when the gap alters to the center of the structure along the y axis, f₁, f₂ and f₃ suffer from significant red shifts which can be employed for more compact designs. Despite this, the polarization purity deteriorates at f₁ and f₂ to some degrees. Most importantly, the transmitted RCP wave at f₃ degenerates to a left-handed elliptical, and asymmetric transmission for linear polarization (t_xy≠t_yx, t_xx = t_yy) is obtained due to the breaking of the mirror symmetry of Hilbert SRR along the y axis. From Fig. 7 (b), the coupling between adjacent elements is significant when the Hilbert SRR is closely spaced and is negligible when the element spacing is sufficiently large. Therefore, all three resonant frequencies go upwards when p increases from 5.64 to 6.36 mm and then are almost constant no matter how p increases. A further inspection also indicates that the polarization extinction ratio deteriorates at f₂, while improves to some degree at f₃ as p goes up. As a consequence, a tradeoff between the LCP wave at f₂ and RCP wave at f₃ should be cautiously considered in the selection of the lattice constant.

 

Fig. 7 Control of the resonances and chiral characteristics through (a) the gap position and (b) element periodicity under the case of x-polarized illuminated EM wave.

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5. Conclusions

In summary, we have demonstrated numerically and experimentally that the twisted Hilbert SRR exhibits triple-band resonances and chiral characteristics. The formed localized magnetic or electric dipoles give rise to the distinguished resonances and polarization conversions. A properly designed dual-band circular polarizer has been demonstrated with high conversion efficiency and large polarization extinction ratio. An extra-noticeable advantage of the proposed structure is its planar compact size evaluated as λ0/6.83 × λ0/6.83 × λ0/34.4 at 8.72 GHz, which shows remarkable advantage in terms of miniaturization with respect to λ0/3.92 in [11] and λ0/1.83 in [12]. Moreover, the complicated element consists of only one resonator but possesses three resonant bands with respect to that incorporating two resonators with different geometrical parameters in [11] for dual-band operation. The element size would be more compact provided a substrate board with a higher dielectric constant is used, which should find promising applications when the polarizer needs to be integrated with other compact devices. Another advantage is the realization of the multifunctional CMMs which show high stability, reliability and integrity. Most importantly, the CMM structure is not restricted to the low-frequency operation and can be scaled up to higher spectrum due to the geometrical scalability.