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Abstract
We demonstrate the synthesis of split dual circularly polarized (CP) beams with an additional linearly polarized (LP) radiation simultaneously from dual superposed birefringent meta-surface lenses formed of zebra crossing shaped meta-gratings. More specifically, each of the meta-surface lens is capable of generating birefringent CP refractions with different helicities and will readily achieve triple beams of different polarizations when a couple of left-hand CP (LHCP) and right-hand CP (RHCP) lobes radiating in the same direction. Polarization states of such triple-beam radiations can be specifically defined, and the vibrating electric field of each beam would thus be prescribed to form the LHCP and RHCP waves as well as a co- /cross-LP wave. Especially, we show that such a spatial aliasing of dual superposed birefringent meta-surface lenses will achieve promising aperture efficiencies for the multiple-beam generation on the basis of the complementary arrangement of different meta-gratings, and should thus pave the way for more advanced spatial-phase engineering with specific radiating characteristics using meta-surfaces.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Meta-surfaces have demonstrated the great capacity of synthesizing perfect dual beams in the birefringence regime recently, where both reflecting and transmissive meta-surfaces are proved to be qualified candidates to fulfil the beam splitting with predefined characteristics [1–13]. In particular, the present literatures have also witnessed the synthesis of multiple beams using the shared aperture radiations [14–27]. Xu et al. presented a flexible reflective meta-surface strategy to control the four-beams with the measured aperture efficiency of 32.3% through the alternating projection method with closed-loop iterative optimizations [24]. Shao et al. proposed a multi-channel meta-surface for versatile wavefront and polarization manipulation with cross-distribution of four different meta-atom arrays [25]. Yang et al. demonstrated the Cassegrain system with polarization-twisting meta-surface for synthesis of four-beams through the superposition of aperture fields, and had the measured aperture efficiency of 25.1% [26]. Zhang et al. proposed a transmissive meta-surface based cavities for generating dual-beams with the measured aperture efficiency of 31.3% through splitting an LP electromagnetic wave into dual CP beams [27]. These meta-surfaces, usually consisting of a dense layout of subwavelength gradient meta-cells with much less than half-wavelength element spacing, have been widely adopted for being capable of generating multiple electromagnetic fields. However, compared with the reflective meta-surface designs, the aperture efficiency in the transmission regime experiences slight degradations as it is not trivial to obtain the desired refractions and the transmittances of meta-surface lenses are usually sensitive with the variation of transmitting phases. In addition, it would thus be more challenging to achieve the functionalities of controlling polarization states of multiple beams, while efficiently calibrating the refractions for different radiating directions.
Normally, the spacing between the meta-atoms needs to be sub-wavelength to form a dense array or at least to be less than half wavelength to reduce aliasing effects at the operating frequency. Such a requirement often limits the design of the shared physical aperture by separating specific physical regions for the synthesis of specific radiating characteristics, thus lead to the limited aperture efficiency as the energy received from the source will be reduced with different radiating area. On the other hand, the spatial aliasing effects have paved another way to manipulate the electromagnetic fields by complementary arranging different meta-array together within a common radiating aperture [28–30]. One approach to fully utilize the radiation aperture is to employ the zebra crossing shaped meta-gratings, and the spatial aliasing for such a sparse arrangement of the meta-gratings can also introduce an additional beam with the complementary setting of different meta-array. Based on these considerations, we demonstrate the generation of triple beams with different polarizations from dual superposed birefringent meta-surface lenses formed of a zebra crossing shaped meta-grating. Each of meta-surface lens is capable of generating birefringent CP refractions with different helicities on the basis of the complementary arrangement of the meta-grating and will thus readily achieve triple beams of different polarizations when a couple of LHCP and RHCP lobes radiating in the same direction. In addition, polarization states of such triple-beam radiations can be specifically defined, and the vibrating electric fields would thus be prescribed to form LHCP / RHCP waves as well as co- / cross-linearly polarized (Co- / Cro-LP) wave. Especially, we show that such a spatial aliasing of dual superposed birefringent meta-surface lenses will have promising aperture efficiencies for the multiple-beam generation and should pave the way for more advanced spatial-phase engineering with specific radiating characteristics using meta-surfaces.
2. Design and numerical results
Figure 1(a) demonstrates the spatial aliasing based complementary meta-gratings with dual superposed birefringent meta-surfaces in a zebra crossing shape to generate triple beams with different polarizations. Every meta-column with three groups of meta-arrays is spaced out from each other to form the birefringent meta-surface lens and transform the LP waves from the source into the well converged CP waves. In this way, the spacing between every column of meta-grating should be more than half the wavelength and thus generate an additional grating lobe. When we set the LHCP and RHCP grating lobes from each meta-surface lens radiating in the same direction to form an LP radiation, such spatial aliasing based complementary meta-gratings would readily achieve triple beams with different polarizations.
Fig. 1. The spatial aliasing based complementary meta-gratings. Every meta-column is equally separated with three groups of meta-arrays. The meta-grating has the whole aperture size of 180 $\times$ 180 mm$^{2}$ with the thickness of 2 mm and the focal from source to meta-grating is 100 mm. (a) Schematic illustration of the spatial aliasing based complementary meta-gratings to generate triple beams with different polarizations simultaneously from an LP source. (b) The physical parameters of the meta-atom. (c) The polarization manipulations of meta-atom array under the Floquet mode analysis.
More specifics, the meta-atom is composed of outer CP meta-atom, intermediate metal ground with a defect hole at the center, and the inner LP meta-atom. The inner and outer layers are interconnected by the metallic vias across the central hole in the metal ground, as shown in Fig. 1(b). The physical parameters of the meta-atom are $a = 2.69$ mm, $b = 2.13$ mm, $c = 6$ mm, $d_{1} = 0.2$ mm, $d_{2} = 1.2$ mm, $r_{1} = 0.4$ mm, $r_{2} = 0.8$ mm, $r_{3} = 1.1$ mm, $r_{4} = 2.5$ mm. The F4B ($\varepsilon _{r}=3.5$, tan$\delta =0.001$) is chosen as the dielectric substrate. The LP meta-surface consisting of periodic C-slots etched over the circular patch array would solely receive the LP fields from source with the E-fields parallel to the orientation of the C-slots. On the other hand, two kinds of the zebra crossing shaped meta-gratings will transform the LP incidence into the LHCP and RHCP waves respectively, as shown in Fig. 1(c). The CP meta-surface consisting of periodic C-slots etched over the elliptical patch array would generate LHCP beams when $\alpha =-45^{\circ }$ and RHCP beams when $\alpha =45^{\circ }$, where $\alpha$ refers to the included angle between the orientation of the C-slot and the minor axis of the elliptical patch. The gradient orientated elliptical patch array with rotating angle of $\beta$ ranging from $0^{\circ }$ to $360^{\circ }$ will create sufficient phase coverage to modulate the illumination.
Figure 2(a) and (b) demonstrate the electric field distribution of the LHCP-atom array under the illuminations of the $x$-polarized and the $y$-polarized incidences respectively at 15 GHz. We can observe that the $x$-polarized field can hardly reach the elliptical patches through the metallic vias as the C-slots etched over the circular patches are polarization selective. On the other hand, the C-slots will give the passport to the $y$-polarized incidence and this can be observed from the strong resonant filed distributions over the elliptical patches. Figure 2(c-h) demonstrate the CP transmitting phase, transmission coefficient and conversion efficiency of the meta-atom array at 15 GHz with structural parameters $\beta$ varying from $0^{\circ }$ to $360^{\circ }$ under the illuminations of $y$-polarized electromagnetic waves from different incident angles of $[0^{\circ },40^{\circ }]$. We use the $\eta _{CP-LP}=\mid \frac {\mid E_{L} \mid ^{2} - \mid E_{R} \mid ^{2} }{{\mid E_{L}\mid }^{2}+{\mid E_{R}\mid }^{2}}\mid$ to describe the conversion efficiency from the LP wave into CP field, where $E_{L}$ and $E_{R}$ refer to the amplitudes of LHCP and RHCP components respectively. We can observe that the transmitting phases of the meta-atom array would be capable of covering $360^{\circ }$. In addition, the transmission coefficient and conversion efficiency $\eta _{LHCP-YLP}/\eta _{RHCP-YLP}$ can always keep a high level with $\alpha =\pm 45^{\circ }$ for the generations of both LHCP and RHCP waves.
Fig. 2. The performance of the mate-atom array. The electric field distribution of the LHCP-atom array under the $x$-polarized (a) and the $y$-polarized (b) incidences respectively. The transmitting phases varied with different oriented CP elliptical patches when $\alpha =-45^{\circ }$ for LHCP-atom (c) and $\alpha =45^{\circ }$ for RHCP-atom (d) under the illumination from $[0^{\circ },40^{\circ }]$. The transmission coefficient varied for LHCP-atom (e) and RHCP-atom (f) under the illumination from $[0^{\circ },40^{\circ }]$. The conversion efficiency varied for LHCP-atom (g) and RHCP-atom (h) under the illumination from $[0^{\circ },40^{\circ }]$.
Figure 3 demonstrates the detailed phase decomposition process of the triple beam synthesis from spatial aliasing based complementary meta-gratings. More specifically, the phase distributions of the meta-grating $\Phi$ can be given as
Fig. 3. Phase decomposition process of the triple beam synthesis from spatial aliasing based complementary meta-gratings. The required phase distributions of complementary meta-gratings with (a) $\Phi_{0}$ for the single beam, (b) $\Phi_L$ for the dual LHCP beams and (c) $\Phi_R$ for the dual RHCP beams.
Through summing up $\Phi _{0}$ and $\Phi _{L/R}$, the required phase distribution over the meta-grating can be represented as shown in Fig. 4(a), where an additional $x$-LP radiation would appear in $0^{\circ }$ through combing the LHCP and the RHCP waves if we have $\Phi _{La}=0$ and $\Phi _{Ra}=0$. On the other hand, such an LP radiation will be $y$-LP if we set $\Phi _{La}=-90^{\circ }$ and $\Phi _{Ra}=90^{\circ }$ with the phase distribution as shown in Fig. 4(b). The full-wave simulations (CST Microwave Studio) are performed to verify the triple beam generations with different polarizations from the spatial aliasing based complementary meta-grating as shown in Fig. 4(c-d). We can observe in Fig. 4(e-f) that both the LHCP and RHCP meta-gratings would perfectly create dual beams with the main beam and an additional grating lobe. In the meanwhile, the grating LHCP and RHCP lobes in $\theta =0^{\circ }$ re-create an $x$-LP wave forming the triple beam radiation with the gains of 21.10 dBic, 21.47 dBi, 21.28 dBic in the direction of $\theta =[-34^{\circ },0^{\circ },34^{\circ }]$. Similarly, the additional radiation in the middle would be $y$-LP if we set $\Phi _{La}=-90^{\circ }$ and $\Phi _{Ra}=90^{\circ }$. The corresponding gains are 21.21 dBic, 21.51 dBi, 21.21 dBic in the direction of $\theta =[-34^{\circ },0^{\circ },34^{\circ }]$. The aperture efficiencies can be calculated by $\eta _{ae}=\frac {\lambda ^{2}\sum _{i=1}^{3}G_{i}}{4\pi D^{2}}$, where $G_{i}$ refers to the gains of beam-$\#i$, $D$ refers to the aperture of the meta-surface grating. As a result, we can have the aperture efficiencies of the split dual CP beams with an additional $x$- / $y$-LP as $39.3\%$ / $39.8\%$ in the radiation direction of $\theta =[-34^{\circ },0^{\circ },34^{\circ }]$. One can observe the polarization status of the corresponding radiations over the Poincare sphere in Fig. 4(g-h), where the normalized Stokes parameters $S_{1}=\frac {{\mid E_{x}\mid }^{2}-{\mid E_{y}\mid }^{2}}{{\mid E_{x}\mid }^{2}+{\mid E_{y}\mid }^{2}}$ are 0.99 and −0.99 of beam-$\#2$ and beam-$\#5$ demonstrating the good quality of the $x$-LP and $y$-LP radiation states. In addition, the values of $S_{3}$ are −0.98 for beam-$\#1$, 0.99 for beam-$\#3$, −0.99 for beam-$\#4$ and 0.99 for beam-$\#6$ illustrating nearly perfect LHCP and RHCP radiating states.
Fig. 4. Triple beams with different polarizations from the spatial aliasing based complementary meta-grating. The required phase distributions for generating triple beams with an x-LP wave (a) and y-LP wave (b) radiating in the middle at 15 GHz. The 3D patterns of the split dual CP beams with an additional x-LP (c) and y-LP (d) radiation. The embedded pictures refer to the corresponding phase patterns of the radiation beams. The 2D patterns in $\varphi=0^{\circ}$ of the split dual CP beams with an additional x-LP (e) and y-LP (f) radiation. The polarization states of the Poincare sphere for the triple beams #1~3 (g) and #4~6 (h).
Figure 5 continues to demonstrate the triple beams in the direction of $\theta =[-40^{\circ },0^{\circ },40^{\circ }]$ and $\theta =[-45^{\circ },0^{\circ },45^{\circ }]$ from the spatial aliasing based complementary meta-gratings. Different from the equal meta-column in the previous design, the different sized meta-columns offer an alternative way to devise the radiation directions of multiple beams under the complementary arrangement meta-arrays. More specifically, such coordinate values $A(x_{pq})$ of the meta-atom over the meta-surface lenses can be given as
Fig. 5. Triple beams with different polarizations from the spatial aliasing based complementary meta-grating with unequal meta-column. The required phase distributions for generating triple beams in the direction of $\theta =[-40^{\circ },0^{\circ },40^{\circ }]$ (a) and $\theta =[-45^{\circ },0^{\circ },45^{\circ }]$ (b) at 15 GHz. The 3D patterns of the triple beams in the direction of $\theta =[-40^{\circ },0^{\circ },40^{\circ }]$ (c) and $\theta =[-45^{\circ },0^{\circ },45^{\circ }]$ (d). The 2D patterns in $\varphi =0^{\circ }$ of the triple beams in the direction of $\theta =[-40^{\circ },0^{\circ },40^{\circ }]$ (e) and $\theta =[-45^{\circ },0^{\circ },45^{\circ }]$ (f). The polarization states of the Poincare sphere for the triple beams $\#7\sim 9$ (g) and $\#10\sim 12$ (h).
The bandwidth characteristics of the proposed spatial aliasing based complementary meta-grating are demonstrated in Fig. 6. We can observe that the transmission coefficient and conversion efficiency of the LHCP-atom and RHCP-atom array can always keep greater than 0.9 at the frequency range of 14.4$\sim$15.7 GHz. Within such a bandwidth, the spatial aliasing based complementary meta-gratings can perfectly generate dual CP waves together with an additional LP beam. The triple beams possess the gains of 20.27 dBic, 20.29 dBi, 20.27 dBic in the direction of $\theta =[-35^{\circ },0^{\circ },35^{\circ }]$ at 14.4 GHz with the aperture efficiency of 34.1%, and 21.27 dBic, 20.27 dBi, 21.27 dBic in the direction of $\theta =[-32^{\circ },0^{\circ },32^{\circ }]$ at 15.7 GHz with the aperture efficiency of 33.6%. The radiating angles are getting smaller as the operating frequency goes higher. We can also observe the polarization statues of the corresponding radiations over the Poincare sphere and the normalized values of $S_{1}$ are −0.99 and −0.99 of beam-$\#14$ and beam-$\#17$ demonstrating the good quality of the $y$-LP radiation states. In addition, the values of $S_{3}$ are −0.99 for beam-$\#13$, 0.99 for beam-$\#15$, −0.99 for beam-$\#16$ and 0.99 for beam-$\#18$ illustrating nearly perfect LHCP and RHCP radiating states.
Fig. 6. Bandwidth characteristics of the proposed spatial aliasing based complementary meta-grating. The transmission coefficient and conversion efficiency of the LHCP-atom (a) and RHCP-atom (b) arrays. The 3D patterns of the triple beams at 14.4 GHz (c) and 15.7 GHz (d). The 2D patterns in $\varphi =0^{\circ }$ of the triple beams at 14.4 GHz (e) and 15.7 GHz (f). The polarization states over the Poincare sphere for the triple beams $\#13\sim 15$ at 14.4 GHz (g) and $\#16\sim 18$ at 15.7 GHz (h).
The direction of LP beam can readily be re-directed by introducing additional phases over the original of complementary meta-gratings, as shown in Fig. 7. We can observe that the triple beams are reset in the direction of $\theta =[-35^{\circ },5^{\circ },45^{\circ }]$ with the gains of 20.45 dBic, 21.61 dBi, 20.11 dBic. The normalized value of $S_{1}$ is 0.99 for beam-$\#20$ demonstrating the good quality of the $x$-LP and the values of $S_{3}$ are −0.96 for beam-$\#21$, 0.97 for beam-$\#22$ illustrating nearly perfect LHCP and RHCP radiating states.
Fig. 7. Triple beams with different polarizations from the spatial aliasing based complementary meta-grating with non-normal direction of the LP beam. (a) The additional phases imposed over the original of complementary meta-gratings in Fig. 5. (b) The 3D patterns of the triple beams in the direction of $\theta =[-35^{\circ },5^{\circ },45^{\circ }]$. (c) The 2D patterns in $\varphi =0^{\circ }$ of the triple beams in direction of $\theta =[-35^{\circ },5^{\circ },45^{\circ }]$. (d) The polarization states of the triple beams $\#19\sim 21$ over the Poincare sphere.
3. Fabrication and experimental results
Finally, we fabricate the proposed spatial aliasing based complementary meta-gratings with equal meta-column and experientially test the radiation performances as demonstrated in Fig. 8. In the experiment, the F4B circuit board ($\varepsilon _{r}=3.5$, tan$\delta =0.001$) has the size of 180 $\times$ 230 mm$^{2}$ with additional 50 mm as the supporting frame. The thickness of the dielectric substrate is 2 mm. The physical parameters of the meta-atom in the fabrications are the same as the simulations with $a = 2.69$ mm, $b = 2.13$ mm, $d_{1} = 0.2$ mm, $d_{2} = 1.2$ mm, $r_{3} = 1.1$ mm, $r_{4} = 2.5$ mm. Meta-atom arrays with $\alpha =\pm 45^{\circ }$ will generate the RHCP and LHCP radiations respectively. The different rotating angles of $\beta$ ranging from $0^{\circ }$ to $360^{\circ }$ are employed to create the desired phase coverage to modulate the illumination. We employ the horn as the excitation operating at 15 GHz and place it at the focal with 100 mm from the meta-grating. We can observe that the triple beam radiations are achieved in the direction of $\theta =[-32^{\circ },0^{\circ },32^{\circ }]$ with 17.24 dBi, −4.99 dBi, 17.26 dBi $x$-polarized components and 17.04 dBi, 20.66 dBi, 16.92 dBi $y$-polarized components respectively. The measured aperture efficiency is thus 31.7%. The normalized $S_{1}$ is −0.99 of beam-$\#5$ demonstrating the good quality of the $y$-LP states. In the meanwhile, $S_{3}$ are −0.99 of beam-$\#4$ and 0.99 of beam-$\#6$ illustrating the nearly perfect LHCP and RHCP states respectively. Compared with the simulation results, the experimental outcomes have a little bit shifts in the radiating directions and about 1 dB gain loss. These are mainly attributed to the fabrication tolerance and the material losses of the meta-surface. However, the overall radiation performances are still satisfactory with triple beams of different polarizations as we devised.
Fig. 8. The manufactured photos and the measurement results. (a-e) Experimental setup and manufactured photos of the spatial aliasing based complementary meta-gratings. (f) The measured radiation patterns of the split dual CP beams with an additional $y$-LP radiation in $\varphi =0^{\circ }$. (g) The polarization states of the triple-beams over the Poincare sphere.
The detailed comparisons with the previous publications concerning meta-surface designs for multiple beams [24–27] are demonstrated in Table 1. Reference [24] proposed an anisotropic reflective meta-surface to realize $y$-LP four-beams in direction of $\theta =40^{\circ }$ at 10 GHz with the measured aperture efficiency of $32.3\%$. Reference [25] proposed a single-layer anisotropic reflective meta-surface with cross-arrangement meta-atom arrays for realizing independent wave-front manipulations and polarization-multiplexing. Compared with these multiple beams generated from the planar reflecting meta-surface designs, our proposal can generate triple beams with different polarizations in the transmission regime based on dual superposed birefringent meta-surface lenses. In addition, the separations of the adjacent meta-atoms for the above papers are always less than half wavelength to reduce aliasing effects at the operating frequency, while our demonstration of the zebra crossing shaped meta-gratings is intentionally set sparsely with larger than half wavelength cell spacing to fully utilize the grating lobes for the synthesis of an additional LP radiation beam. On the other hand, Ref. [26] employed the Cassegrain system for synthesis of LP four-beams in direction $\theta =15^{\circ }$ at 15 GHz and achieved the measured aperture efficiency of $25.1\%$. Reference [27] proposed a cavities transmission meta-surface for two symmetrical CP beams with opposite helicities in direction of $\theta =18^{\circ }$ at 15 GHz possessing the measured aperture efficiency of $31.3\%$. Compared with these radiations from the transmissive meta-surfaces, our design using spatial aliasing based complementary meta-gratings has realized better aperture efficiency and also achieved much larger elevation angles for the synthesis of triple beams with an LP radiation together with LHCP and RHCP beams simultaneously.
Table 1. Performance Comparisons Between Multiple Beam Meta-Surfaces
4. Conclusions
In conclusion, we have proposed the spatial aliasing based complementary meta-gratings with dual superposed birefringent meta-surface lenses for generating triple beams with different polarizations. On the basis of the sparse arrangement of the meta-grating, we have shown that the LHCP and RHCP grating lobes from each meta-surface lens radiating in the same direction can efficiently merge into an additional LP radiation for the different polarized triple beams. However, the radiating directions of LHCP and RHCP waves are always symmetrical for all the samples in the demonstrations. When comes to the designs of asymmetric diffraction angles of LHCP and RHCP waves, three beams with different polarizations can be also realized as long as we have the dual CP grating lobes radiating in the same directions. However, the present designs in this investigation are based on same periodic sized meta-atoms, and the asymmetric setting of dual CP waves will lead to the overlap of the LHCP and RHCP meta-columns, thus should degrade the over performances of the radiations in the aperture efficiency and polarization conversion. In this way, one can further build up different sized LHCP-atoms and RHCP-atoms for the spatial aliasing based complementary meta-gratings to generate different polarized multiple beams. We expect the present design scheme would pave the way for more advanced spatial-phase engineering applications with specific radiating characteristics using meta-surfaces.
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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