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Abstract
In this paper, a novel method to generate multiple orbital angular momentum (OAM) vortex beams is proposed using a 1-bit metasurface. Through carefully adjusting the phase shift of each element, mirror-symmetrical OAM vortex beams are generated in targeted directions by a 1-bit metasurface under plane-wave illumination. Moreover, the topological charges of the generated vortex beams are opposite. Based on this phenomenon, dual-beam, four-beam, and full-space OAM vortex waves are respectively studied and generated by conducting full-wave simulations. Finally, a prototype of the proposed metasurface is fabricated and measured in an anechoic chamber. The measurement results show that multiple OAM vortex beams are successfully generated and detected, verifying the effectiveness of the proposed method.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Orbital angular momentum (OAM) has attracted tremendous interests since being discovered in 1992 [1]. It has been experimentally proved that OAM vortex beam can be utilized to improve communication capacity without increasing the bandwidth [2]. Considering the scarce spectrum resource, the research of OAM vortex beam is of great significance. The generation of OAM vortex beam has been studied extensively covering the optical domain [3–5] and microwave bands [6–9]. The circular phased array antenna [10–13] is one of the most frequently used devices in OAM vortex beam generation within microwave bands. Quite recently, a novel method was proposed based on the reflectarray theory by utilizing a reflective metasurface [14]. Due to the powerful manipulation ability on electromagnetic fields, metasurface owns huge potential in research of OAM vortex beam.
In general, metasurface acts as a reflector or a lens in generation of OAM vortex beam [15–21]. By adjusting the size or rotation angle of the elements constituting the metasurface, the reflection/transmission phase of the incident wave can be flexibly tailored to satisfy the phase condition of OAM vortex beam. However, the phase changes of the incident wave caused by these metasurfaces are usually rather wide, which will increase the difficulty in element designs especially for transmission-type element. By contrast, the 1-bit metasurfaces possess unique advantages, of which the elements have only two discrete reflection/transmission phase states. Thus, the design of metasurface can be simplified greatly. It is also beneficial to reconfigurable metasurface designs. Hence, the 1-bit metasurface has been studied a lot [22,23]. Nevertheless, to the best of our knowledge, the research is still not sufficient for OAM vortex beam by using 1-bit metasurfaces.
In this paper, we propose a novel method to generate multiple OAM vortex beams based on 1-bit metasurface. The proposed metasurface is illuminated by normally incident plane wave and redirects the incident energy to targeted directions. According to the mathematical modeling and full-wave simulations, the scattering patterns of 1-bit metasurface are mirror symmetrical. Quite interestingly, the symmetrical main lobes both carry OAM but the topological charges are opposite. Inspired by this phenomenon, multiple OAM vortex beams are further studied and generated under different situations.
2. Mathematical modeling
Consider a metasurface composed of M × N elements. It is illuminated by plane wave and the incident angle is θinc as shown in Fig. 1. By carefully adjusting the reflection phase of each element, the incident wave can be scattered to the desired direction of (θ, φ) and the scattered field is given by
where is used to fit the element pattern and qe is 1 unless otherwise specified. |Гmn| represents the reflection magnitude of the mnth element and is assumed to be 1. The meaning of other parameters can be found in [24]. Thus, in order to generate an OAM vortex beam in direction, the phase delay φmn is supposed to satisfyin which Фmn is the azimuthal angle of the elements in the metasurface and l is the topological charge of OAM vortex beam. Based on the above mathematical model, we can rapidly predict the field distributions.Figure 2(a) depicts the distribution of phase delay φmn for generating OAM vortex beam with topological charge l = 1 in normal direction at 8.8GHz. Note that the incident angle of plane wave is 0°. The metasurface consists of 26 × 26 elements and the element size is 12mm × 12mm. Figures 2(b) and (c) are the corresponding scattering pattern and phase distribution in uv-plane (u = sinθ·cosφ, v = sinθ·sinφ), respectively. The doughnut-like main lobe with an energy null in the center is clearly observed. Combining with the phase distribution map, it can be seen that OAM vortex beam with topological charge l = 1 is successfully generated. This is the conventional method to generate OAM vortex beam. For the sake of differentiation, we call this device as conventional metasurface in this paper.
Fig. 2 (a) The phase delay distribution across the metasurface. (b) The scattering pattern in uv-plane. (c) Scattering phase distribution in uv-plane. Black circles in (b) and (c) denote the main lobe and the insets at lower right corner are the megascopic main lobe.
Once getting the phase delay of conventional metasurface φmn, the phase delay of 1-bit metasurface φmn|1-bit then can be obtained. The numerical relationship between φmn|1-bit and φmn is [24]
Figure 3(a) depicts the distribution of φmn|1-bit. It can be seen that φmn|1-bit has only two values and their difference is 180° so that phase cancellation occurs, resulting in the split scattering pattern as shown in Fig. 3(b). This pattern is in consistence with AMC-PEC (artificial magnetic conductor-perfect electric conductor) structure [25,26] but apparently different from the doughnut-like main lobe of OAM vortex beam. Besides, the scattering phase distribution shown in Fig. 3(c) does not agree with the typical characteristic of OAM vortex beam, either.Fig. 3 (a) 1-bit phase delay distribution. (b) and (c) are the normalized scattering pattern and phase distribution in uv-plane, respectively.
Quite interestingly, the observed phenomenon will become very different when the main lobe is deflected to non-normal directions. Figure 4 shows the phase delay distribution, normalized pattern, and scattering phase distribution for conventional metasurface and 1-bit metasurface, respectively. The main lobe is set to point to θ = 30°. For conventional metasurface, a single main lobe appears for θ = 30° and carries OAM with topological charge l = 1 as expected. While, two main lobes are simultaneously observed in the pattern of 1-bit metasurface for θ = 30° and θ = −30°, respectively. Moreover, both main lobes carry OAM but the topological charges are opposite, which is 1 for θ = 30° and −1 for θ = −30°. Comparing the phase delay distributions shown in Figs. 4(a) and (d), it is found that the 1-bit phase delay distribution is axisymmetric so that the main lobe always appears in pair and exhibits mirror symmetry. The rotation directions of the scattering phase are opposite against each other because the image and preimage are opposite in a mirror-symmetrical system. The above observed phenomenon demonstrates that multiple OAM vortex beams can be simultaneously generated by using 1-bit metasurface. Nevertheless, it must be pointed out that the lobe level will be decreased by using 1-bit metasurface due to the phase quantization error [24]. As shown in Fig. 5, the maximum lobe level of 1-bit metasurface is 1.73dB lower than that of conventional metasurface, which will reduce the aperture efficiency of the metasurface.
Fig. 4 (a) The phase delay distribution, (b) the normalized scattering pattern in uv-plane and (c) the scattering phase distribution in uv-plane for conventional metasurface. The main lobe is deflected to θ = 30°. (d), (e) and (f) are the corresponding maps for 1-bit metasurface.
The above results are all obtained under condition of θinc = 0. It is worth mentioning that multiple OAM vortex beams may still be generated under oblique incidence. As shown in Fig. 6, the incident angle of 1-bit metasurface is θinc = 10° and the main beam is set to point to θ = 5°. By computing the scattering field using Eq. (1), it is found that one more beam appears at θ = −26°. From the scattering phase distribution it can be seen that the topological charges of the two beams are also opposite. The numerical relationship between the incident angle and the direction of two OAM vortex beams is worthy of researching.
Fig. 6 (a) The phase delay distribution, (b) the normalized scattering pattern in uv-plane and (c) the scattering phase distribution in uv-plane for 1-bit metasurface. The incident angle is θinc = 10° and one beam is set to point to θ = 5°. The other beam appears at θ = −26°.
3. Full-wave simulations
In this section, full-wave simulation is conducted to further demonstrate the observed phenomenon. Three different occasions are respectively studied as follows.
3.1 Dual OAM vortex beams generation
Figure 7 illustrates the generation of dual OAM vortex beams. The incident wave in normal direction is an x-polarized regular plane wave, of which the topological charge can be seen as 0. By arranging elements satisfying the phase delay distribution shown in Fig. 4(d), OAM vortex beams pointing to θ = ± 30° with opposite topological charges can be generated. In addition, the polarization of electromagnetic wave is not changed during this process. The elements are selected as metallic square patch etched on substrate with εr = 2.2. The periodicity of the elements is 12mm and the thickness of substrate is 3mm. The patch lengths for the two kinds of elements are 6.86mm and 12mm, respectively. Figure 8(a) depicts the reflection phase of the elements. It is observed that the phase difference of the two elements exactly equals to 180° at 8.8GHz, satisfying the condition of 1-bit phase quantization. The scattering pattern of the designed metasurface is plotted in Fig. 8 (b). Two lobes pointing to θ = ± 30° are clearly identified. To observe the phase distribution, two sampling planes, of which the size is 0.8m × 0.8m, are adopted 3m away from the center of the metasurface. Obviously, the helical phase distributions with opposite rotational orientations are successfully generated. The scattering pattern and phase distribution are mirror symmetrical with respect to xoz-plane, validating the correctness of mathematical modeling.
Fig. 8 (a) The reflection phase against frequency of the two kinds of elements. (b) The 3D pattern and phase distribution in sampling plane. Point O is the center of the metasurface.
3.2 Four OAM vortex beams generation
The angular momentum of electromagnetic waves contains not only OAM but also SAM (Spin Angular Momentum), which is related with the polarization state of EM waves. The manipulation of polarization state gives an extra degree of freedom to apply the proposed method. Cross dipole is a classical structure to manipulate EM waves with different polarizations. We combine cross dipole with 1-bit phase quantization to design metasurface elements for generating four OAM vortex beams simultaneously.
The lengths of the two arms of the cross dipole are 5mm and 10.5mm, respectively and the widths are both 1.5mm. The other parameters are the same with those of Section 3.1. Table 1 lists the elements structures and their co-polarization reflection phases under y-polarized incidence. In the table, dm-n denotes the reflection phase difference of Element m and Element n. It can be seen that for a fixed arm in y direction, the change of arm in x direction has very small influence on reflection phase (see d1-3 and d2-4). Moreover, the phase difference of elements with different arms in y direction is very close to 180° (see d1-2, d1-4, d2-3, and d3-4), which meets the condition of 1-bit quantization quite well. The same conclusion can be drawn for x-polarized incidence due to the symmetry of the element structure. Thus, we can apply the proposed theory in both x and y polarization directions. As a result, four OAM vortex beams can be generated when the metasurface is illuminated by 45°-polarized (angular bisector direction of the angle between x and y axes) incident wave since it can be decomposed to x-polarized incidence and y-polarized incidence.
Table 1. Elements Structure and Co-polarization Reflection Phase
By properly adjusting the phase delay distributions, the beam pointing direction can be controlled feasibly. Figures 9(a) and (b) illustrate the two typical occasions to generate four OAM vortex beams. The directional vectors of four beams in Fig. 9 (a) are all in yoz-plane. While in Fig. 9(b), OAM vortex beams with l = ± 1 are in yoz-plane and those with l = ± 2 are in xoz-plane. In both occasions, OAM vortex beams with l = ± 1 are x-polarized and those with l = ± 2 are y-polarized. It is worth mentioning that interference may occur when the beams are too close in space.
Fig. 9 Schematic illustration for four OAM vortex beams generation. (a) The four beams are all in yoz-plane. OAM vortex beams with l = ± 1 are x-polarized while those with l = ± 2 are y-polarized. (b) OAM vortex beams with l = ± 1 are in yoz-plane and are x-polarized, while those with l = ± 2 are in xoz-plane and are y-polarized.
The phase delay distributions and corresponding arms configurations with different topological charges and beam directions (θ, φ) are shown in Fig. 10. By assembling dipole arms with different polarization orientations into a whole metasurface, four OAM vortex beams can be simultaneously generated. For example, the metasurface shown in Fig. 9(a) is the assembling of arms configurations shown in Figs. 10 (e) and (f) while the metasurface shown in Fig. 9 (b) is the assembling of arms configurations shown in Figs. 10 (g) and (h). When the metasurface is illuminated by 45°-polarized incident plane wave, the orthogonal arms will be both excited to generate four OAM vortex beams.
Fig. 10 (a), (b), (c), and (d) are phase delays and (e), (f), (g), and (h) are the corresponding arms distributions. (a) and (e) are for l = 1, θ = 15°, φ = 90°. (b) and (f) are for l = 2, θ = 40°, φ = 90°. (c) and (g) are for l = 1, θ = 30°, φ = 90°. (d) and (h) are for l = 2, θ = 30°, φ = 0°. (a), (e), (c), and (g) are x-pol. (b), (f), (d), and (h) are y-pol.
The simulated scattering patterns of metasurface shown in Figs. 9 (a) and (b) are depicted in Figs. 11(a) and (b), respectively. It is observed in Fig. 11 (a) that the four beams are all in yoz-plane as expected. The OAM vortex beams with l = ± 1 are x-polarized and point to θ = ± 15° while those with l = ± 2 are y-polarized and point to θ = ± 40°. The sampling planes are 3m away from the metasurface, of which the sizes are 0.8m × 0.8m and 1.2m × 1.2m, respectively. The sampling plane for l = ± 2 are larger than that of l = ± 1, which is attributed to the larger divergence angle of OAM vortex beams with l = ± 2. The helical phase distributions are clearly observed in the sampling planes and the rotation orientations are opposite in the symmetrical planes. On the other hand, OAM vortex beams with l = ± 1 in Fig. 11(b) are x-polarized and in yoz-plane. Meanwhile, OAM vortex beams with l = ± 2 are generated in xoz-plane and are y-polarized. These four beams are all steered by 30° as expected. The helical phase distributions are once again clearly observed in sampling planes. It is worth mentioning that the normal reflection field shown in the figure may be strengthened or weakened with the variation of the deflection angle of the OAM vortex beams.
Fig. 11 The patterns of OAM vortex beams. The four OAM beams in (a) are all in yoz-plane. In (b), OAM beams with l = ± 1 are in yoz-plane and OAM beams with l = ± 2 are in xoz-plane.
3.3 Full-space OAM vortex beams generation
In the above simulations, the designed metasurfaces only work in pure reflection mode, leaving half space separated by metasurfaces unutilized. In this section, full-space metasurface is proposed to generate four OAM vortex beams. The schematic illustration is shown in Fig. 12. Reflection or transmission will occur according to the polarization orientation of the incident wave. By tailoring the reflection/transmission phase, OAM vortex beams can be generated in half-space of the metasurface. When the incident wave is 45°-polarized, OAM vortex beams will be generated in full-space.
Fig. 12 Schematic illustration of full-space OAM vortex beams generation. (a) OAM vortex beams in full-space are all in yoz-plane. (b) OAM vortex beams in upper half-space are in yoz-plane while those in lower half-space are in xoz-plane.
The element consists of three-layer metallic patches with two-layer substrates (εr = 2.2, tanδ = 0.0009) sandwiched between the patch layers as shown in Fig. 13. In the second and third patch layers, the patches in x direction are the same in length with the element period P. Thus, these patches constitute a two-layer polarization grid, which can reflect the x-polarized incident wave but let y-polarized incidence pass through. Meanwhile, by adjusting the arm lengths of a and b, the phase of reflected x-polarized wave and transmitted y-polarized wave can be tailored, respectively.
Figure 14 illustrates the reflection/transmission phase and magnitude with different values of a and b. The operation frequency is set to be 8.5GHz. According to the reflection property of x-polarized incident wave shown in Fig. 14(a), the reflection phase difference of a = 7.5mm and a = 11.4mm at 8.5GHz is 180° no matter what value b is, satisfying the condition of 1-bit phase quantization. In addition, the reflection magnitude is higher than −1dB, indicating that the x-polarized wave is almost totally reflected. On the other hand, for transmission property of y-polarized incident wave shown in Fig. 14(b), the transmission phase difference and magnitude also satisfies the condition of 1-bit phase quantization at 8.5GHz.
Fig. 14 The phase and magnitude of (a) reflected x-polarized incident wave and (b) transmitted y-polarized incident wave.
The simulated patterns of metasurface shown in Figs. 12 (a) and (b) are depicted in Figs. 15(a) and (b), respectively. It can be seen that OAM vortex beams are generated in full-space. In Fig. 15 (a), the four OAM beams are all in yoz-plane. While in Fig. 15 (b), OAM vortex beams with l = ± 1 are in xoz-plane and those with l = ± 2 are in yoz-plane. In both of the situations, the x-polarized component of the incident wave is reflected to the upper half-space and the y-polarized component is transmitted to the lower half-space. The size and distance of the sampling planes are the same with that in Section 3.2. The helical phase distributions can be clearly observed in the sampling planes, further verifying the successful generation of full-space OAM vortex beams.
Fig. 15 The patterns of full-space OAM vortex beams. The four OAM beams in (a) are all in yoz-plane. In (b), OAM beams with l = ± 2 are in yoz-plane and OAM beams with l = ± 1 are in xoz-plane. In both (a) and (b), OAM beams in upper half-space are x-polarized and OAM beams in lower half-space are y-polarized.
4. Experimental verification
In order to verify the mathematical modeling and full-wave simulations, a prototype of the metasurface for generation of four OAM vortex beams (corresponding to Fig. 9(b)) was fabricated using printed-circuit-board (PCB) technology and measured in an anechoic chamber as shown in Fig. 16(a). A horn antenna was placed 500mm (about 15λ8.8GHz) away from the metasurface and acted as the illumination source. The angle between the horn aperture and horizontal plane was 45°so that the orthogonal element arms of the metasurface were excited simultaneously. In the experiment, a standard probe was used to detect the reflected wave of the metasurface as shown in Fig. 16(b). Note that the probe was vertically polarized and we respectively put x-axis and y-axis of the metasurface shown in Fig. 9(b) along the vertical direction to match the polarization state of the probe. The angle between the probe and metasurface was 30° and the probe scanning in the sampling plane is for recording the phase information of each OAM beam. The distance between two adjacent scanning points was 20mm.
The measured results are depicted in Fig. 17. The helical phase distributions can be obviously observed and the rotation orientations of l = ± 1 and l = ± 2 are opposite. Meanwhile, the measurement error exists mainly due to two reasons. One is that the incident wave in the experiment is not ideal plane wave compared with simulation. The other one is the position error of the devices. Nevertheless, the measurement results can sufficiently verify the proposed method for multiple OAM vortex beams generation.
5. Conclusion
In summary, we propose a novel method to generate multiple OAM vortex beams based on 1-bit metasurface. It is found that mirror-symmetrical OAM vortex beams can be generated in targeted directions when the metasurface is illuminated by plane wave. Moreover, the symmetrical main lobes simultaneously carry OAM with opposite topological charges. This study can be applied to many occasions. As a basic application, dual OAM vortex beams are firstly generated. Then, by employing cross dipole structure, the x- and y-polarized components of the incident wave are simultaneously converted to OAM vortex beams so that four OAM vortex beams are generated. Furthermore, a multilayer meta-device is carefully designed, which can reflect or transmit the incident wave according to the polarization directions. By applying 1-bit design, full-space OAM vortex beams are also generated. At last, a prototype of the metasurface for generation of four OAM vortex beams is fabricated and measured. The measurement results validate the effectiveness of the proposed method.
Funding
National Natural Science Foundation of China (NSFC) (61671464, 61701523, 61471389).
Acknowledgments
The authors thank Haipeng Li and Yuzhou Ran for their assistance in experiment.
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