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Abstract
In the past few years, orbital angular momentum (OAM) has aroused great interest in the scientific communities, because it shows great potential for enhancing capacities of radio and optical communication systems. Here, we propose anisotropic metasurfaces to generate multiple OAM vortex beams at microwave frequencies. A phase compensation theory is presented, in order to determine the phase distributions on metasurfaces, This theory enables independent control of beam numbers, polarizations, orientations, and topological charges of OAM vortex beams, respectively. The metasurface is composed of anisotropic elements, whose reflection phases can be engineered separately in different polarization directions. The scheme is validated by both simulation and experimental results and shows great potential for the polarization division multiplexing in OAM communication systems.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Waves with angular momentum can rotate when they travel in space. There are two primary momentums in physics: one is spin angular momentum (SAM) associated with the rotation of polarization vector, and the other is orbital angular momentum (OAM) arising from a helical phase structure. SAM has two possible quantized values of ± ћ depending on the handedness of circular polarization, where ћ is the Planck constant [1,2]; while OAM of lћ can be described by a spiral phase of exp(ilφ), in which l is the topological charge or mode, manifesting the repeating rate of 360° phase shift azimuthally along the beam cross section [3]. The striking difference between SAM and OAM is the range of allowed value. OAM can be many times greater than the SAM, of which the topological charges are unlimited and mutually orthogonal to each other. Therefore, OAM can be regarded as a new set of communication channels without increasing the frequency bandwidth.
Recently, OAM and related realms have become a research hotspot owing to their excellent performance and wide applications, including optical tweezers [4,5], multiplexing communication [6], and quantum memories and metrology [7]. Toward the diverse applications of OAM, how to efficiently generate, manipulate and receive OAM beams are three key issues to be deal with. To date, many methods have been proposed to create OAM beams such as spiral phase plate [8], holographic diffraction grating [9], spiral reflectors [10] and antenna arrays [11]. For example, the spiral phase plate (SPP) has attracted tremendous interest for the structural simplicity and easy implementation. However, the SPPs are realized by the accumulation of phase delays during wave propagation, but suffer from the large thicknesses and fabrication inaccuracy in practical applications [12]. The antenna array can create OAM beams by additional phase shifters in each channels, but it is limited by the complicated architecture and high cost in the realization [13,14].
Nevertheless, recent development of metasurface demonstrates the unique ability of arbitrary wavefront control, giving rise to a host of optical functionalities with flat and ultra-thin profile [15–23]. The abrupt phase discontinuities from the internal subwavelength resonators provide a different route to bend the beams in a predefined manner, and allow for in situ control of electromagnetic (EM) waves at subwavelength scale. By regulating the phase and amplitude responses of the meta-atoms, a vortex beam can be easily realized with arbitrary topological charge, as confirmed by a series of experiments from microwave to optical regime [24–27]. In addition, the generation of multiple OAM beams [28,29] have been reported by the isotropic metasurface, and even at second harmonic frequency with nonlinear metasurface [30]
However, it remains a significant challenge to control the polarization of each beam freely, which is critical for expanding the channel capacity in the wireless communications. Vortex beams with different polarizations have been realized by pancharatnam-berry coding metasurfaces [31], but it is hard to control the topological charges of the beams independently. Instead, anisotropic metasurfaces [32–34] have greatly extended our abilities to design planar devices with polarization-selective functionalities, because of the varying geometries in two dimensions that respond distinctly to different polarization states of the external stimulus. On this basis, a strategy of polarization control for single OAM beam is proposed by controlling the phase difference of the meta-atom in two orthogonal directions [35].
In this paper, we investigate the feasibility of generating multiple OAM beams with customized polarization states and topological charges based on the anisotropic metasurface, which is composed of homogenous, orthogonal I-shaped elements. A general scheme is proposed to synthesize the phase patterns of the metasurface under dual polarizations, as can be easily implemented by tuning the arm lengths of the meta-atoms in two direction. This method is verified by both simulations and experiments with good agreement. Our theory suggests an attractive way for diverse applications in polarization multiplexing of OAM communication systems.
2. Theory and design
Figure. 1 depicts the anisotropic reflective metasurface comprising orthogonal I-shaped copper elements periodically arranged in two dimensions, where the left-handed, right-handed and linear polarized OAM beams can be generated simultaneously. To seek the polarization-dependent EM characteristics, the arm lengths of the meta-atom are deliberately varied in x and y directions, which leads to the disparate amplitude and phase responses owing to the difference of resonance positions. The substrate is F4B with the relative permittivity εr = 2.65 × (1-j0.001). A metallic ground layer is attached to the bottom of the substrate to block the EM penetration. Other detailed dimensions in the inset of Fig. 1 are p = 6.0 mm, w = 0.2 mm, a = 1.8 mm and h = 2.0 mm. Assume the metasurface is excited by a u-polarized point source in Fig. 1(a), where. If we want to create single OAM beam along the normal direction, the reflection phases of the element (m, n) are given by [35]
where the superscript x and y stand for the polarization directions. Here and are initial phases for x- and y-polarized components. k0 is the wavenumber of free space. and refer to the position vectors of the element (m, n) and the source. lφmn corresponds to the OAM phase, in which l is the topological charge and φmn is the rotation angle between and the unit vector. It has been revealed that the polarization of the OAM beam is determined by the phase difference of the first items in Eqs. (1) and (2) [35]. The second items are responsible for the twisted wavefront of the beam, while the third ones aim to compensate the phase variance from the source to the elements at different locations.
Fig. 1 (a) Schematic of multiple OAM beams generation based on an anisotropic metasurface fed by a point source. Each reflected beam carries the OAM with customized topological charge, orientation and polarization state. (b-c) Dependence of the reflection amplitude and phase on the arm length lx of the meta-atom, where the blue and red lines represent the EM responses under x- and y-polarized waves respectively.
Although the phase requirements from Eqs. (1) and (2) are only valid for single OAM beam generation, it provides a promising avenue for multi-beams manipulation by controlling the polarization phasesand. Assume that there are k beams reflected from the metasurface. For each output beam, the phase components along x and y directions for the element (m,n) can be expressed as
where i indicates the beam number. (xi, yi) is the position of the element center. lx(i) and ly(i) are the topological charges for x- and y-polarized waves respectively. The supplementary phase results from the beam deflection along the direction, which demands additional phase lags for the meta-atoms. Therefore, for all the beams yielded from the metasurface with predefined topological charge, polarization state and orientation angle, the reflection phase of the element (m,n) should satisfyIt is worth noting that, for isotropic meta-atom with = 0, the required element phase has identical form to that in Ref [24], which focused on the multiple OAM beams generation without polarization control. Actually, the introduction of the term suggests new possibility of polarization control for each beam individually, thereby adds a new dimension to the information carried by the OAM waves. Specifically, when lx(i) = ly(i), we can realize the linear or cross polarization for the ith beam by setting = 0 or π, or left- /right-handed circular polarization (LCP/RCP) by setting = π/2 or -π/2.
Actually, more interesting functionalities can be achieved based on the proposed metasurface, apart from the control of beam polarization, orientation and topological charge. For example, if lx(i)≠ly(i), the x- and y-polarized waves will carry OAM with different topological charges from Eqs. (5) and (6), which may reflect more information during the transmission. Furthermore, the beam orientation vector in Eqs. (5) and (6) can also be replaced by different vectors and, implying that the x- and y-polarized waves are directed at various angles in the upper space. On the whole, the anisotropic metasurface greatly enriches our ability to manipulate the features of the OAM, and provides more degrees of freedom in the modulation of baseband signals for further enhancing the total channel capacity.
Figures 1(b) and1(c) demonstrate the dependence of the simulated reflection amplitude and phase on the arm length lx of the meta-atom at 15 GHz, as retrieved from the commercial full-wave solver, CST Microwave Studio 2012. Here ly is kept at 5.0 mm in the simulation. The unit-cell boundaries are applied to x and y directions to consider the mutual coupling among neighboring elements, and the Floquet ports are applied to + z and −z directions in Fig. 1(a). The amplitude curve indicates the low loss nature of the meta-atom, with the reflection amplitude approaching unity for both polarizations. Meanwhile, a moderate phase range from −206°to 66° is exhibited for x polarization, when lx is swept from 2.0 mm to 5.0 mm, but the reflection phase for y polarization stays unchanged. This feature enables independent phase adjustment with respect to the incidence polarization. Due to the symmetry of the unit cell, the same variation trend is also applied to the reflection trace as a function of length ly (not shown here). To simplify the design progress, eight discrete values of the arm length lx (ly) 5.0 mm, 4.3 mm, 3.8 mm, 3.5 mm, 3.2 mm, 2.9 mm, 2.3 mm and 2.0 mm are selected for the meta-atoms, with the corresponding phases −206°, −180°,-141°, −97°, −43°, 0°, 43° and 66°, as indicated by the circles in Figs. 1(b) and 1(c). In consequence, there are 64 basic elements in total to constitute the whole metasurface.
To validate the theoretical model, two typical phase patterns of the metasurface are designed for the purpose of multiple OAM beams generation with different characteristics. All of them are composed of 35 × 35 elements. Figure 2 details the design flow of synthesizing double OAM beams situated symmetrically around the surface normal, with the topological charge l = −1 and + 1 respectively. The two beams numbered 1 and 2, are designed to point toward φ1 = 45°, θ1 = 35°and φ2 = 225°, θ2 = 35°. The excitation antenna is located at a distance of 240 mm above the metasurface center. The corresponding phase patterns of andare calculated from Eqs. (3) and (4), in which their phase differences remain -π/2 and π/2 to give RCP and LCP for the reflected waves. The first two rows in Figs. 2(a) and 2(b) illustrate the phase distributions of x and y components for the two OAM beams, where the white arrows indicate the excitation polarizations. The three-dimensional (3D) radiation patterns and two dimensional (2D) radiation patterns at a cut-plane with θ = 35° are listed in the third and fourth rows of Figs. 2(a) and 2(b), showing the same pointing angles as designed for beam 1# and beam 2# respectively. Based on the superposition principle in Eqs. (5) and (6), we can get the synthesized phase pattern of the metasurface in Fig. 2(c), in which the above mentioned two polarized beams are generated simultaneously. Figures 2(d)-2(g) show the wavefronts perpendicular to the axes of beam 1# and beam 2# in Fig. 2(c). It can be seen that the topological charges of the two beams are l = −1 and + 1 as desired.
Fig. 2 Detailed procedure to synthesize a metasurface with two OAM beams, in which the polarizations, orientations and topological charges are controlled independently. (a-b) Calculated phase patterns of x and y components of the metasurface for single beam generation, with the corresponding 3D radiation patterns and 2D radiation patterns at a cut-plane with θ = 35°. The OAM beam is directed at the angle φ1 = 45°, θ1 = 35° with RCP, l = −1 (a), and φ2 = 225°, θ2 = 35° with LCP, l =+ 1 (b). (c) Calculated synthesized phase patterns of x and y components for the metasurface for double beams generation, with the corresponding 3D and 2D radiation patterns at a cut-plane with θ = 35°. The RCP and LCP OAM beams are directed at the angles φ1 = 45°, θ1 = 35° and φ2 = 225°, θ2 = 35°, with l = −1and l =+ 1 respectively. (d-e) Simulated phase distributions of the RCP and LCP electric fields at the cut-planes perpendicular to the axes of the two beams respectively. (f-g) Simulated amplitude distributions of the RCP and LCP electric fields at the cut-planes perpendicular to the axes of the two beams respectively.
To further verify our proposed approach, another metasurface is proposed to generate triple OAM beams (named beam 1#, 2# and 3#) with the topological charge l = −1, + 1 and + 1 respectively. The feeding source lies at the same position as that in the former case. Here beam 1# is tilted at the direction (φ1 = 0°, θ1 = 30°) with linear and cross polarization (v-polarization in Fig. 1(a)). In the meanwhile, beam 2# and beam 3# are deflected at the angle of (φ2 = 120°, θ2 = 30°) and (φ3 = 240°, θ3 = 30°) with RCP and LCP. The phase distributions of the metasurface along x and y directions for beams 1#-3# are illustrated in the first two rows of Figs. 3(a)- 3(c). All the beams are generated at the specified angles, as proved by the 3D radiation pattern and 2D radiation pattern at a cut-plane with θ = 30° in Fig. 3. The superposition phase pattern can be found in Fig. 3(d), which gives rise to three beams simultaneously at the same directions. Different OAM features can be observed from the spiral phase distributions at the cut-planes perpendicular to the v-polarization, RCP and LCP beam axes in Figs. 3(e)- 3(g), with the topological charges l = −1, l = + 1 and l = + 1 as expected.
Fig. 3 Detailed procedure to synthesize a metasurface with three OAM beams, in which the polarizations, orientations and topological charges are controlled independently. (a-c) Calculated phase patterns of x and y components of the metasurface for single beam generation, with the corresponding 3D radiation patterns and 2D radiation patterns at a cut-plane with θ = 30°. The OAM beam is directed at the angle φ1 = 0°, θ1 = 30° with linear polarization (v-polarization), l = −1 (a), φ2 = 120°, θ2 = 30° with RCP, l =+ 1 (b) and φ2 = 240°, θ2 = 30° with LCP, l =+ 1 (c). (d) Calculated phase patterns of x and y components of the metasurface for triple beams generation, with the corresponding 3D radiation pattern and 2D radiation pattern at a cut-plane with θ = 30°. The v-polarization, RCP and LCP OAM beams are directed at the angles (φ1 = 0°, θ1 = 30°), (φ2 = 120°, θ2 = 30°), and (φ3 = 240°, θ3 = 30°), with l1 = −1, l2 =+ 1 and l3 =+ 1 respectively. (e-g) Simulated phase distributions of the v-polarization, RCP and LCP electric fields at the cut-planes perpendicular to the axes of the three beams respectively. (h-j) Simulated amplitude distributions of the v-polarization, RCP and LCP electric fields at the cut-planes perpendicular to the axes of the three beams respectively.
3. Experiments and discussions
In the experiment, we fabricate the sample based on the standard Printed Circuit Board (PCB) technology, and measure it in the home-built near field scanning system as illustrated in Fig. 4(a). The sample has identical phase pattern to that in Fig. 2(c) to generate a pair of OAM beams with RCP, l = −1 and LCP, l = + 1 respectively. A wideband waveguide antenna in Fig. 4(c) is employed as the excitation at the distance of 240 mm above the center of the metasurfaces. It is reasonable to neglect its impact on the OAM modes due to the small size (20 mm) and low profile. The sample is attached to a plastic holder, and laid on a rotary stage so that it can be rotated in both azimuth and elevation directions independently. A linearly-polarized waveguide probe is fixed on the scanning stage to detect the electric field around the sample. To measure the circularly-polarized waves, two rounds of the measurements should be performed when the probe polarization is aligned with two orthogonal axes, so that we can calculate the RCP and LCP patterns manually. In order to get full view of the tilted OAM beams, the sample should be rotated accordingly to stay parallel to the scanning surface at a distance of 780 mm. Here the rotation angles of sample are set as (φ1 = 45°, θ1 = 35°) and (φ2 = 225°, θ2 = 35°) as predicted from the simulation. A small scanning step 10 mm (λ/2 at 15 GHz) is selected to acquire phase distributions at the sampling plane.
Fig. 4 (a) Experimental setup to measure the near fields of the fabricated metasurface, where the sample is attached to the plastic holder on a rotary stage, and a waveguide probe is employed as the detector on the left. (b) Zoomed-in image of the sample. (c) The waveguide antenna as the excitation of the metasurface.
The measured amplitude and phase distributions for the RCP and LCP beams are provided in Figs. 5(a) -5(d), where the doughnut-shaped intensities and spiral phases can be clearly recognized. Opposite phase rotations can also be found from Figs. 5(b) and 5(d), from which we can determine the topological charges l = −1 and + 1 respectively.
Fig. 5 (a-b) Measured amplitude and phase distributions at the cut-plane perpendicular to the RCP beam (beam 1#) axis. (c-d) Measured amplitude and phase distributions at the cut-plane perpendicular to the LCP beam (beam 2#) axis.
To gain further insight of the performance for the LCP and RCP beams, we have also measured their radiation patterns in the microwave chamber. Owing to the polarization difference of the beams, a pair of RCP and LCP horn antennas are used to record the far field characteristics of the metasurface. Both the measured (orange lines) and simulated (blue lines) results are shown in Figs. 6(a) and 6(b), which are in excellent accordance with the deflection angles emerging at 35° and −35° as expected. The amplitude nulls can be clearly observed at the beam centers, complying with the features of the OAM mode. The slight discrepancy between the curves in Figs. 6(a) and 6(b) is likely due to the measurement error and the slight deviation of the phase center.
Fig. 6 2D radiation patterns of the RCP (a) and LCP (b) beams respectively at a cut-plane with φ = 45°.
4. Conclusions
In summary, we have theoretically and experimentally demonstrated a new scheme to realize multiple polarization-controllable OAM beams by means of an anisotropic coding metasurface. The metasurface, consisting of periodic orthogonal I-shaped structures, shows powerful ability to control the polarization, orientation and topological charge of multiple OAM beams independently by tailoring the phase distributions along two polarization directions. Although the design is implemented at microwave frequencies, it can also be applied in terahertz and visible regime. Thanks to design simplicity and ultrathin profile, the metasurface is especially advantageous for constructing low-cost, multi-functional OAM devices in future wireless communication systems.
Funding
National Key Research and Development Program of China (2017YFA0700201, 2017YFA0700202, 2017YFA0700203); National Natural Science Foundation of China (61631007, 61138001, 61371035, 11227904, 61731010, 61571117, 61501112, 61501117, 61522106, 61722106, 61701107, 61701108); the 111 Project (111-2-05); the Natural Science Foundation of Jiangsu Province (BK20150020).
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