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Abstract
We propose a compact cascaded meta-surface system (CCMS) to produce well converged orbital angular momentum (OAM) vortex waves with tailored spin angular momentum (SAM) by integrating a meta-surface lens (ML) with an assistant meta-mirror (AM). Specifically, the co-linearly polarized (LP) waves from the feed would be reflected by the ML firstly and then twisted into the cross-LP counterparts by the AM to penetrate the ML for the perfect synthesis of the OAM vortex beams while performing the linear-to-circular polarization conversion. Especially, the CCMS can pack the ML and the AM closely together with a quarter of the ML focal length when we apply proper phase distributions on the AM. In addition, the proposed CCMS can readily be extended to the generation of multiple circularly polarized OAM vortex waves with different modes. Our design should thus pave the way for building up more efficient wireless communication systems with expanded channel capacity.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Vortex waves carrying spin angular momentum (SAM) and orbital angular momentum (OAM) simultaneously have received increasing attentions in developing wireless communication systems due to the great potential to improve the spectral efficiency and channel capacity [1–6]. Meta-surface, as an intelligent artificial material for flexible manipulation of EM wave fronts, has demonstrated the great ability to combine such OAM vortex waves with prescribed SAM of EM fields [7–11]. Especially, functional metasurfaces have been proposed to generate the OAM vortex waves with integer and fractional modes [12], and synthesize the spin-decoupled multi-beams [13] and conical beams [14] carrying desired OAMs. In addition, quadruplex polarization channels of vortex waves have been achieved by proposing the chirality-assisted geometric-phase meta-surfaces [15].
These designs, possessing the great multiplexing capacity in increasing the spectral efficiency of EM waves, are highly desirable and essential for the modern wireless communications. Generally, the spin and orbital helicities of the released circularly polarized (CP) fields can be controlled by regulating either the physical dimensions of meta-atoms [16,17] or the orientations of Pancharatnam-Berry (PB) elements [18–35]. Recently, the literatures have also demonstrated the generation of OAM vortex waves using meta-surfaces with linear-to-circular polarization conversions at the same time [16–19]. For example, reflectarrays generating CP vortex waves were successfully implemented by properly adjusting the linear excitations for equal amplitude elements in two orthogonal directions and 90${^\circ }$ phase difference with vortex wave fronts [16,17]. The PB phase meta-mirrors have also fulfilled the perfect conversion of a LP waves into multiple CP vortex beams with flexible beam directions and polarization states [18,19]. However, most literatures prefer to solely employ the phase modulations to synthesize the OAM vortex waves, while simultaneously regulating the polarizations states. Such operations would inevitably couple the SAM and OAM generations, thus may degrade the aperture efficiency and purity of the vortex waves during the simultaneous control of the SAM and OAM of the released EM fields. With these considerations, we propose a compact cascaded meta-surface system (CCMS) to create well converged OAM vortex waves with tailored SAM by integrating a meta-surface lens (ML) with an assistant meta-mirror (AM). Such an ML can directly achieve SAM modulation with the inherent structural characteristics of the meta-surface performing the linear-to-circular polarization conversion. Meanwhile, it also shows the great ability to achieve OAM beam modulation with prescribed meta-atoms of different orientations. When integrated with the AM to construct the CCMS, the co-LP waves from the feed would be reflected by the ML in the first place and then twisted into the cross-LP counterparts by the AM to penetrate the ML for the perfect synthesis of the OAM vortex beams while performing the linear-to-circular polarization conversion. We will show the CCMS can pack the ML and the AM closely together with a quarter of the ML focal length when we apply proper phase distributions over the AM. And such a design can also be extended to the generation of multiple CP OAM vortex waves having different modes.
2. Design principle
Figure 1 schematically demonstrates the proposed CCMS consisting of the ML and the AM for the synthesis of OMA vortex beams while transforming the LP waves into CP waves. The ML consists of the top-layer CP meta-surface, metallic middle-layer ground with a defect hole at the center of each unit, and bottom-layer LP meta-surface, where the meta-cells on the top and bottom layers are interconnected by the metallic probes across the central holes of the metallic middle-layer ground. In such a design, the bottom-layer LP meta-surface would function as the receiving component of the LP illumination, while the top-layer CP meta-surface would thus operate as the transmitting component for the CP radiations. At the same time, the metallic middle-layer ground with defect holes and the metallic probes couple the EM waves from the receiving component into the transmitting component to form an antenna-filter-antenna array with bandpass filtering transmissions and linear-to-circular polarization conversion. Specifically, the $x$-polarized waves received by bottom-layer LP meta-surface can be coupled to the top-layer CP meta-surface through the metallic probes at certain resonant frequencies. In the meanwhile, the gradient CP meta-cells over the top-layer would offer enough phase distributions to produce the convergent vortex waves. Moreover, the periodic LP meta-cells over bottom-layer would receive solely $x$-polarized waves and reflect the $y$-polarized waves at the same time. On the other hand, the AM with gradient skew double-headed arrow resonators (DHARs) on the grounded substrate can convert the $y$-polarized waves into its cross counterpart to penetrate the ML, while offering the proper phase gradients to achieve more compact structure. According to the ray tracing, $O_{1}$ is defined as the focal point of the ML and the compensation path of the ML is thus determined by the physical length $l_{O_{1}D}$ when we employ the focal length $f_{1}$ to generate the OAM vortex wave. The conventional folded transmittarray satisfies the expressions $l_{O_{1}D} = l_{AB}+l_{BC}+l_{CD}$ and $l_{AB}=l_{BC}=l_{CD}$. Therefore, the distance between AM and ML can be reduced to 1/3 focal length $f_{1}$. For the compact cascaded meta-surface system, if we place the AM at the quarter of the focal length $f_{1}$ from the ML, the compensation path of the system will become $l_{O_{2}D} = l_{AB}+l_{BC}+l_{CD}$, where $O_{2}$ is the focal point of the CCMS and the extra compensation path of $\Delta l = l_{O_{1}D} - l_{O_{2}D}$ should be imposed on the AM to ensure the compensation path of the ML as the physical length $l_{O_{1}D}$. The proposed scheme should be applicable to adjusting the separating distance between the AM and the ML freely so as to achieve the compact configuration. The phase distributions provided by the AM and ML can be calculated as
where $k$ refers to the wave number in the free space, $\varphi$ refers to the additional phase to produce OAM vortex waves, $\Phi _{0}$ refers to an arbitrary phase constant that can adjust the phase distributions freely. Specifically, the phase compensations would follow where $f_{1}$ refers to the focal length of the ML, $l$ is the mode number of vortex beam. Given the profile height of $f_{1}/4$, we can have the corresponding phase compensations on the AM and the ML. The halo shaped phase pattern on the AM offers the required phase distribution to achieve more compact configuration and transform the $y$-polarized EM waves into $x$-polarized EM waves. The ML possesses the ability to synthesize the vortex beam, while transforming the $x$-polarized EM waves into left-hand CP (LHCP) waves. For the proposed CCMS, the aperture size of the ML is 156 $\times$ 156 mm$^{2}$ with 26 $\times$ 26 meta-atoms of 6 $\times$ 6 mm$^{2}$ and the focal length $f_{1}$ of 136 mm for the synthesis of CP vortex beams, while the aperture size of the AM is 104 $\times$ 104 mm$^{2}$ also with 26 $\times$ 26 meta-atoms of 4 $\times$ 4 mm$^{2}$. In this way, each unit over the AM and ML can have one-to-one correspondence in the ray tracing to perform the phase compensation. The illuminations casting over the ML and AM would have the incident angles of $\theta _{1}$ = [0$^\circ$, 40$^\circ$] and $\theta _{2}$ = [0$^\circ$, 48$^\circ$].
Fig. 1. The CCMS for OAM vortex beam synthesis with linear-to-circular polarization conversion at 15 GHz. (a) The configuration of the CCMS. The embedded pictures refer to the ray tracing of the system and the AM unit. (b) Structural information of the ML. The physical parameters are $a_{1}$ = 4 mm, $h_{1}$= 3 mm, $r$ = 3.5$\sqrt {2}$ mm, $\alpha$ = -45${^\circ }$, $\beta$= +45${^\circ }$, $\varepsilon _{rAM}$ = 2.2, $w_{1}$ = 0.2 mm, $r_{1}$ = 2.5 mm, $r_{2}$ = 1.3 mm, $r_{3}$ = 0.4 mm, $r_{4}$ = 0.8 mm, $a_{2}$ = 6 mm, $d_{1}$ = 1.2 mm, $d_{2}$ = 1.9 mm, $d_{3}$ = 0.61 mm, $d_{4}$ = 1.2 mm, $h_{2}$ = 1 mm, $\varepsilon _{rML}$ = 3.5. (c) The required phase distributions over the AM and the ML.
Figure 2 shows the transmission and reflection properties of the ML and the AM with different structural parameters and different incident angles. For the ML, the transmission amplitude $|T_{LHCP,x}|$ is very close to 1 when the rotation angle $\delta$ varies from 0$^\circ$ to 360$^\circ$ under the illuminations from 0$^\circ$ to 40$^\circ$. In the meanwhile, the transmission amplitude $|T_{RHCP,x}|$ would always be near 0 when $\delta$ and $\theta _{1}$ varies the same. The ML is thus capable of converting the $x$-polarized waves fully into LHCP waves. The polarization conversion efficiency $\eta _{CP}$ is defined as $\eta _{CP} = ( |T_{LHCP,x}|^{2}-|T_{RHCP,x}|^{2})/(|T_{LHCP,x}|^{2}+|T_{RHCP,x}|^{2})$, where $|T_{LHCP,x}|$ and $|T_{RHCP,x}|$ are the transmission amplitudes of the LHCP and RHCP components. For the $x$-polarized waves, the periodic ML units displays a high-level of more than 90% polarization conversion efficiency $\eta _{CP}$ when the rotation angle $\delta$ varies from 0$^\circ$ to 360$^\circ$ under the illuminations from 0$^\circ$ to 40$^\circ$. And the corresponding transmission phases also fulfill 360$^\circ$ phase coverage when $\delta$ and $\theta _{1}$ change the same. Such transmission amplitudes and phases are plenty to create vortex wave radiations with prefect linear-to-circular polarization conversion as we expected. Meanwhile, the periodic ML unit would have nearly perfect reflections for the $y$-polarized waves, and the reflection phases of $R_{y,y}$ can keep almost as a constant regardless the variations of $\delta$ and $\theta _{1}$. For the AM, the polarization conversion efficiency $\eta _{LP}$ is defined as $\eta _{LP} = |R_{x,y}|^{2}/(|R_{y,y}|^{2}+|R_{x,y}|^{2})$, where $|R_{y,y}|$ and $|R_{x,y}|$ are the reflection amplitudes of the co- and cross-polarized components. The polarization conversion efficiency $\eta _{LP}$ from the -45$^\circ$ AM unit keeps a high-level and the reflection phases change from -139$^\circ$ to 54$^\circ$ when the $d_{1}$ varies from 0.2 mm to 3.4 mm under the illuminations from 0$^\circ$ to 48$^\circ$. Meanwhile, the +45$^\circ$ AM unit has the same quality polarization conversion efficiency, and the reflection phases can cover [41$^\circ$, 180$^\circ$] and [-180$^\circ$, -125$^\circ$] when the $d_{1}$ varies the same. Therefore, combining -45$^\circ$ and +45$^\circ$ AM units would provide the enough phase distributions to achieve the compact design and twist the $y$-polarized EM waves into $x$-polarized EM waves simultaneously.
Fig. 2. Transmission and reflection properties of the ML and the AM at 15 GHz. The relationships between the rotation angle $\delta$ of the periodic ML units and transmission amplitude $|T_{LHCP,x}|$ (a), transmission amplitude $|T_{RHCP,x}|$ (b), polarization conversion efficiency $\eta _{CP}$ (c) and transmission phases arg($T_{LHCP,x}$) (e) under the illumination of $x$-polarized EM waves from different incident angles of $\theta _{1}$ = [0$^\circ$, 40$^\circ$]. The relationships between the rotation angle $\delta$ of the periodic ML units and reflection amplitudes $|R_{y,y}|$ (d) and reflection phases arg($R_{y,y}$) (f) under the illumination of $y$-polarized EM waves from different incident angles of $\theta _{1}$ = [0$^\circ$, 40$^\circ$]. The relationships between the dimension $d_{1}$ of the periodic AM units and polarization conversion efficiency $\eta _{LP}$ (g) and reflection phases arg($R_{x,y}$) (h) when $\alpha$ = -45$^\circ$ under the illumination of $y$-polarized EM waves from different incident angles of $\theta _{2}$ = [0$^\circ$, 48$^\circ$]. The relationships between the dimension $d_{1}$ of the periodic AM units and polarization conversion efficiency $\eta _{LP}$ (i) and reflection phases arg($R_{x,y}$) (j) when $\beta$ = +45$^\circ$ under the illumination of $y$-polarized EM waves from different incident angles of $\theta _{2}$ = [0$^\circ$, 48$^\circ$].
3. Numerical results and experimental verifications
The full wave simulations (CST Microwave Studio) are then performed to verify the radiation performance of the CCMS as shown in Fig. 3. The CCMS produces a perfect LHCP vortex beam with radiation depressions in 0$^\circ$ and circular radiation peaks in the propagating direction, and the maximum gain of the vortex radiations is 19.1 dBic. The magnitude pattern and phase pattern of the released LHCP OAM vortex beam are demonstrated at 2750 mm away from the ML with a scanning range of 700 $\times$ 700 mm$^{2}$. The amplitude distribution shows a circular distribution with central defect singularity, while the phase patterns show a 360$^\circ$ phase change around center of the radiation aperture as we expected. These results show the proposed CCMS effectively radiates a CP vortex wave with mode number of $l$ = 1. Meanwhile, the maximum E-plane gains are 18.0 dBic at $\theta$ = -5$^\circ$ and 18.1 dBic at $\theta$ = 6$^\circ$, and the maximum H-plane gains are 18.1 dBic at $\theta$ = -6$^\circ$ and 18.4 dBic at $\theta$ = 6$^\circ$. The axial ratios in E-plane are 0.5 dB at $\theta$ = -5$^\circ$ and 0.3 dB at $\theta$ = 6$^\circ$, and the axial ratios in H-plane are 0.2 dB at $\theta$ = -6$^\circ$ and 0.7 dB at $\theta$ = 6$^\circ$, thereby further validating the synthesis of CP vortex waves. Mode spectra are calculated to quantify the mode purity of CP vortex waves with $P_{L} = E_{L}^{2}/\sum _{n=1}^{N}E_{n}^{2}$ [10], where $N$ refers to the total number of OAM modes, $E_{L}$ and $E_{n}$ refer to the electric field amplitude of the $L$th and $n$th order vortex beam. We can observe the CP vortex beam with $l$ = 1 has high mode purity of 97% at 15 GHz.
Fig. 3. The radiation performances of the CCMS at 15 GHz. (a) LHCP OAM vortex beam. (b) The amplitude pattern and phase pattern of LHCP OAM vortex beam. (c) E-plane and (d) H-plane radiations patterns. (e) The axial ratio of the E-plane and H-plane. (f) The mode purity of the LHCP OAM vortex waves.
We continue to display the generation of CP dual vortex beams and triple vortex beams from the CCMS, as shown in Fig. 4. The synthesized dual- and triple- vortex beams are directed in $\begin {bmatrix} \theta _{i}\\\varphi _{i}\\l_{i} \end {bmatrix}_{i = 1\sim 2}$ = $\begin {bmatrix} 20^\circ & 20^\circ \\90^\circ & 270^\circ \\ -1 & 1\end {bmatrix}$ and $\begin {bmatrix} \theta _{i}\\\varphi _{i}\\l_{i} \end {bmatrix}_{i = 1\sim 3}$ = $\begin {bmatrix} 20^\circ & 20^\circ & 20^\circ \\60^\circ & 180^\circ & 300^\circ \\ -1 & -2 & 1\end {bmatrix}$. The superpositions of aperture fields are employed to calculate the corresponding phase distributions of the MLs. Such designs can redirect the released fields into specific angles as we expected with well converged CP vortex beams. Then the phase distribution of the AM still keeps the same as the single vortex beam generation. For the dual vortex beam radiation, the maximum gains of the LHCP vortex beams are 15.8 dBic and 15.3 dBic. The magnitude patterns and phase patterns of the released LHCP vortex beams are also demonstrated at 2750 mm away from the ML with a scanning range of 700 $\times$ 700 mm$^{2}$. The amplitude distributions show a circular distribution with central singularity and the phase distributions are all agree with our devised OAM beams having mode numbers of $l$ = -1 and $l$ = 1. Meanwhile, the generated LHCP vortex beams with $l$ = -1 and 1 have the mode purities of 83 % and 86 % at 15 GHz. For the triple vortex beam radiation, the maximum gains of the LHCP vortex beams are 14.9 dBic, 12.3 dBic, and 13.9 dBic. The amplitude distributions have a circular distribution with central singularity and the phase distributions demonstrate the mode numbers of $l$ = -1, $l$ = -2 and $l$ = 1. Meanwhile, the generated CP vortex beams with $l$ = -1, -2 and 1 have the mode purities of 77 %, 80 % and 77 % at 15 GHz.
Fig. 4. Multiple vortex beam radiations from the CCMS at 15 GHz. The phase distributions of the ML for (a) dual vortex beams and (b) triple vortex beams. 3D far-field patterns for (c) LHCP dual vortex beams and (d) LHCP triple vortex beams. The amplitude patterns and phase patterns for (e) LHCP dual vortex beams and (f) LHCP triple vortex beams. The mode purities for (g) LHCP dual vortex beams and (h) LHCP triple vortex beams.
Figure 5 shows the bandwidth properties of the CCMS. We can see the VSWRs of all these designs can keep less than 2 from 14.7 GHz to 15.5 GHz. The working bandwidths with -0.5 dB gain degradations from the peak radiations are 14.8 GHz - 15.4 GHz for single vortex beam radiation, 14.8 GHz - 15.3 GHz for dual vortex beam radiation and 14.8 GHz - 15.4 GHz for triple vortex beam radiation. The proposed CCMS obtains the peak radiations of 19.4 dBic at 15.1 GHz, 15.7 dBic at 14.9 GHz and 13.9 dBic at 14.9 GHz for the three cases. The corresponding aperture efficiencies can be calculated by $\eta =\lambda ^{2}G_{s}/4\pi D^{2}$, where $G_{S}$ refers to the overall gains of multiple vortex beams following the definition of $G_{S}=10lg(i\times 10^{G_{ave}/10})$ with $i$ standing for the number of the multiple vortex beams and $G_{ave}$ as the average gain of the multiple vortex beams. The proposed CCMS obtains the maximum aperture efficiencies of 11.4% at 15.1 GHz, 9.6% at 14.9 GHz and 9.7% at 14.9 GHz for the three cases from single vortex beam to triple vortex beams respectively. The aperture efficiencies of dual and triple vortex beams are comparable. But they are lower than that of the single vortex beam due to the beam deflection in $\theta$ = 20$^\circ$ for both cases. The mode purities have some fluctuations when the working frequency varies, where the maximum values of the mode purities are 97% at 15.0 GHz for the single vortex beam radiation, 86% at 14.8 GHz and 86% at 15.0 GHz for dual vortex beam radiation, 84% at 14.8 GHz, 80% at 15.0 GHz and 84% at 15.1 GHz for triple vortex beam radiation. Meanwhile, the axial ratios from the preferred working frequency band are less than 3 dB, thereby further validating the generations of the CP vortex beams.
Fig. 5. Radiation performances of the CCMS over the operating bandwidth from 14.5 GHz to 15.5 GHz. (a) The VSWRs. (b) The average gains and the aperture efficiencies. (c) and (d) The mode purities and the axial ratios.
Finally, we fabricate the CCMS and experientially test the radiation performances as shown in Fig. 6. Both MLs with single vortex beam radiation and triple vortex beam radiation are fabricated for the experiments to verify the radiation performances of the proposed design. In the fabrication, metal wires are used to replace each metal probe to ensure the tight connection between the top-layer CP meta-surfaces and bottom-layer LP meta-surfaces. For the measurements of the single vortex beam, the maximum gains of the LHCP vortex beams in E-plane are 17.1 dBic at $\theta$ = -6$^\circ$ and 17.6 dBic at $\theta$ = 5$^\circ$, and the corresponding axial ratios are 0.9 dB and 1.1 dB. The magnitude patterns and phase patterns of the released LHCP vortex beams are demonstrated at 2750 mm away from the ML with a scanning range of 700 $\times$ 700 mm$^{2}$. The amplitude distributions show a circular distribution with central singularity and the phase distributions are all agree with our devised OAM beams having mode numbers of $l$ = 1. Meanwhile, the generated CP vortex beam with $l$ = 1 has the mode purity of 86 % at 15 GHz. For the measurements of the triple vortex beams, we rotate the CCMS to calibrate each vortex beam to the receiving horn to obtain the triple vortex radiations from the CCMS. The maximum gains of the LHCP vortex beams are 11.0 dBic at $\theta$ = -6$^\circ$ and 10.3 dBic at $\theta$ = 6$^\circ$ for $l$ = -1, 8.2 dBic at $\theta$ = -9$^\circ$ and 9.3 dBic at $\theta$ = 10$^\circ$ for $l$ = -2, 10.6 dBic at $\theta$ = -6$^\circ$ and 10.7 dBic at $\theta$ = 6$^\circ$ for $l$ = 1. The corresponding axial ratios are 2.6 dB and 2.3 dB for $l$ = -1, 2.9 dB and 2.2 dB for $l$ = -2, 1.3 dB and 1.6 dB for $l$ = 1. The amplitude distributions have a circular distribution with central singularity and the phase distributions demonstrate the mode numbers of $l$ = -1, $l$ = -2 and $l$ = 1. Meanwhile, the generated CP vortex beam with $l$ = -1, -2 and 1 have the mode purities of 64 %, 50 % and 68 % at 15 GHz. The LHCP vortex radiations from the measurements experience a little bit shifts in the radiation directions and slight gain degradations compared with the simulations. These are mainly attributed to the fabrication tolerance and material losses of the meta-surfaces, also the measurement error of the vortex waves in the experiments. However, the overall radiation performances are still satisfactory with LHCP vortex beams as we devised.
Fig. 6. The comparisons between the measurement and the simulation results of the CCMS at 15 GHz. (a) Experimental setup and manufactured photos. (b) Far-field patterns and axial ratios for single vortex beam. (c) The amplitude patterns and phase patterns for single vortex beam. (d) The mode purities for single vortex beam. Far-field patterns and axial ratios for triple vortex beams with (e) $l$ = -1, (f) $l$ = -2 and (g) $l$ = 1. (h) The amplitude patterns and phase patterns for triple vortex beams. (i) The mode purities for triple vortex beams.
4. Conclusions
In conclusion, we have demonstrated simultaneously controlling the spin and orbital angular momentums of EM fields from the CCMS. Especially, our proposed design has achieved compact profile with only a quarter of the ML focal length when we impose proper phase distributions over the AM, and can readily be extended to the generation of multiple CP OAM vortex waves possessing different modes. The design scheme, through cascading functional meta-surfaces for more advanced controlling EM fields, should pave the way for building up contemporary compact wireless communication systems with tangible applications to greatly expand channel capacity.
Funding
National Natural Science Foundation of China (61301072, 61671344).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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