Vortex electromagnetic (EM) waves hold promise for their ability to significantly increase the transmission capacity of wireless communication systems via the torsion resistance defined by different topological charges associated with the orbital angular momentum (OAM). However, the application of vortex waves in remote distance transmission is limited by its characteristic of divergence. In this paper, a lens based on a phase-modulation metasurface (MS) is proposed that enables vortex EM waves to converge, thereby improving their propagation performance at microwave frequencies. A phase-shift distribution on the plane of the MS is obtained based on the concept of the optical converging axicon, which can convert a Laguerre-Gaussian (LG) beam to a Bessel beam based on changing the propagation direction. Simulation results verify the ability of the MS lens to achieve OAM beam focusing, which is advantageous for enhancing the propagation directivity and increasing the gain in the main lobes of vortex waves. This is of particular importance in microwave wireless communication applications.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Vortex waves carrying orbital angular momentum (OAM) have received a considerable amount of attention both in the optical and radio frequency domains due to their potential for increasing the spectral efficiency and channel capacity of wireless communications . In 2007, the first attempt was made to demonstrate OAM in the microwave frequency regime, which proved that antenna arrays can be used to generate vortex EM waves with similar characteristics to Laguerre-Gauss vortex beams in the near-axis direction . In 2011, Professor Bo Thidé of the Swedish Institute of Space Physics and his team of Italian colleagues conducted a seminal experiment in the lagoon of Venice . The experiment used a Yagi and a spiral parabolic antenna to send uniform plane waves and vortex EM waves, respectively, realizing that the two signals were transmitted simultaneously in the same frequency band [4, 5]. By exploiting the orthogonality between different OAM modes, vortex EM waves can transfer more information than general electromagnetic waves without transmission bandwidth. Consequently, OAM vortex waves have become an important research focus in wireless communications [6, 7]. However, OAM based EM waves are hollow and divergent. In addition, the degree of the divergence increases as the order of the OAM-mode and the propagation distance increases. This shows that OAM based EM waves are not well-suited for long distance communication. Therefore, it is an important topic of research is to investigate effective methods of creating converging vortex EM waves.
In optics, non-diffractive Bessel beams have become the subject of intense research due to their unique properties including small center spots, unchanged intensity distribution during transmission, highly focused light intensity and self-reconstruction . Because of their advantages of structural simplicity, high conversion efficiency, and high light damage threshold, axicon devices are widely used to experimentally generate Bessel beams. Furthermore, a new method for transformation of a Laguerre-Gaussian beam to a vortex, non-divergent Bessel beam by a helical axicon was proposed in [9,10]. Recently, a lens with a gradient-index was proposed to focus microwave radiation [11–13], which has a subwavelength thickness and a high focusing strength. In addition, the results in  showed that a flat MS with a parabolic reflection-phase distribution can focus an impinging plane EM wave to a point image in the reflection geometry. However, the limitation of this lens is that it is only capable of focusing an incident plane wave to a point image relatively close to the lens rather than at a long distance away from it.
A metasurface is a type of composite material composed of artificial subwavelength structures that are typically periodic, which can realize nearly arbitrary control of EM waves (e.g., phase, polarization, propagation) according to its unit cell structure . In optics, a high efficiency all-silicon MS is proposed in , which presents good transmission coefficient over a wide bandwidth. An all-metallic MS with high reflection coefficient has also been proposed to control optical beams over a wide bandwidth in . In recent years, MSs have been widely used in conjunction with antennas to increase their performance. In 2011, a V-type antenna structure was proposed to achieve phase control for realizing anomalous transmission and reflection of EM waves . Based on the electromagnetic resonance mechanism, artificial EM materials can, in principle, acquire almost any equivalent permittivity and permeability, which far exceeds the range of parameters available in natural materials. Therefore, MSs are utilized to achieve unprecedented control of EM waves (both radiation intensity and phase distributions). Currently, MSs have important potential applications in stealth technology, antenna technology, microwave and terahertz devices, optoelectronic devices and many other fields [19,20].
In this paper, inspired by the phase-modulation of incident waves on an optical axicon lens , we propose a MS lens which is able to converge vortex EM waves. On the basis of the EM wave vector direction, the transmissive-type MS corresponds to the mirrored axicon, modulating the phase across a specific plane instead of a circular conical surface. The formula for determining the required phase-compensation is derived and then applied to design the MS. Ultimately, the numerical simulation results show that the MS converging lens is effective for minimizing the angle of the main lobe while improving the gain of the vortex beams without changing the original spiral phase wavefront.
2. Metasurface lens design
The transformation of a divergent LG beam to a convergent Bessel beam via a mirrored axicon lens is schematically shown in Fig. 1(a). The transformation function of the axicon lens is given in , which can be described by Eq. (1):
To achieve the desired convergence property of vortex EM waves, a flat phase-modulated MS lens, which is equivalent to a mirrored axicon lens, is proposed here for use in the microwave domain as shown in Fig. 1(b). By propagating through the MS lens, the vortex wave emitted from a patch antenna array source experiences a non-uniform (inhomogeneous) phase-shift that locally transforms the propagation direction of the incident wave to realize the desired convergent effect.
Suppose we consider the transmission MS lens, which is formed from an M×N array of elements. The transmission function of the MS, which provides the required inhomogeneous phase delay, is given by:
Theoretically, the distribution of compensative phase shown in Fig. 2(a) can be calculated by using the non-relativity approximation equation for the wave number. The effect of mutual coupling between elements can be ignored in the application of the superposition principle for electromagnetic fields during the evaluation procedure. The desired phase-shift on the MS lens can be calculated in terms of a quasi-continuous phase change based on the sub-wavelength element design. The discrete phase distribution on a MS lens consisting of 16×16 elements is displayed in Fig. 2(b).
Next, the unit cell structure of the MS shown in Fig. 3(b) is designed in order to provide the desired phase distribution. Each element, consisting of four layers of uniform square metallic patches with a cross slotted central metallic patch, is simulated at 10 GHz using the HFSS software. The periodicity P = 0.3λ = 9 mm, and the distance between each layer is t = 0.05λ = 1.5 mm. The slot length and width is 6.5 mm and 0.5 mm respectively. The length of square patches varies from 1 to 7 mm. The metallic parts composing the metasurface are printed on a low loss Arlon AD350ATM substrate having dielectric constant, εr = 3.5 and tangential losses, tan δ = 0.003. The HFSS simulation results of the unit cell shown in Fig. 3(d) indicate that the unit cell can realize a full transmission phase range of 406°, while maintaining the transmission coefficient to be better than −3 dB. Based on the square and cross-slot patch elements, the transmission MS lens with 16×16 elements was designed to focus vortex EM waves at 10 GHz. The final lens design is square with dimension of 14.4×14.4 cm2, as shown in Fig. 3(a).
From Fig. 3(d), it can be observed that the absorption of the material gradually increases as the incident angle increases. When the incidence angle reaches 80°, the curve of absorption rate changes drastically with geometric parameter S. As shown in Fig. 3(e), the transmission performance of the cell structure also deteriorates as the incident angle increases. When comparing absorptivity and transmission performance as incidence angle changes from 0 to 80°, we observe that the losses of unit cell mainly stem from the low diffraction efficiency at high incidence angles.
3. Numerical verification of proposed converging lens system
In order to verify the proposed MS converging lens, we use the HFSS software to simulate its performance by employing the finite element method. The converging lens system is excited by a circular patch antenna array source consisting of 8 patches with the same phase shift between each of the neighboring elements. The proposed converging lens is placed at a distance 2λ above the antenna source.
Theoretically, efficient convergence can be achieved as long as the radiated beams are not diffracted by the lens. As shown in the Fig. 4, side-lobes level appears for D = 30 mm, 45 mm, 60 mm, and little diffraction also appears on the edges of the MS lens due to small size. However, according to the angle between main lobe and z axis, we can observe that more efficient convergence is obtained with D = 60 mm above the antenna source. So, we chose D = 60 mm to place the MS lens. The performance of the antenna and the converging lens system are both verified over a frequency band varying from 9.3 GHz to 10.3 GHz.
The amplitude profiles at three different frequencies in the x–z plane perpendicular to the beam axis for the antenna and MS converging lens are presented in Fig. 5. The ability of the antenna to generate a vortex EM wave with topology charge +1 is simulated and presented in Figs. 5(a)–5(c) at 9.3 GHz, 10 GHz, and 10.3 GHz, respectively, which display the vortex EM field amplitude distribution for ϕ = 0 (ϕ is the azimuthal angle of the spherical coordinate system). By further comparing the electric field amplitude distribution of the converging lens to the antenna only, the divergence angle of the main lobe is obviously minimized as shown in Figs. 5(d)–5(f), which also demonstrates that the MS lens operates over a frequency band ranging from 9.3 GHz to 10.3 GHz. Theoretically, the achieved bandwidth is relatively narrow since resonant materials are employed in the design.
The EM field expression indicates that the intensity distribution in the cross-section of the vortex has a donut shape. In Fig. 6, we present the E-field distribution of the source and converging MS lens antenna in the x–z plane at z = 90 mm, 120 mm, and 150 mm with a fixed 2λ distance between the source and MS lens antenna system. A hollow characteristic of the vortex beam results and the divergence angle increases rapidly as the wave propagates further from the source, which can be observed in Figs. 6(a)–6(c). For the MS converging lens, the donut shape of the E-field magnitude distribution at the three different positions are all similar as shown in Figs. 6(d)–6(f), which demonstrates the superior convergence properties of the lens.
The phase distribution of the vortex beam is presented in Fig. 7. As a basis for comparison, the theoretical and simulated phase distributions of the antenna and MS converging lens antenna at z = 70 mm are shown in Figs. 7(a) and 7(b), and Figs. 7(c) and 7(d) respectively. The consistency between simulation and theory strongly indicates that the MS converging lens does not change the vortex characteristics of the electromagnetic waves.
The 3D simulations for the source and MS converging lens antenna at 9.3 GHz, 10 GHz, and 10.3 GHz are presented in Fig. 8. By comparing of the radiation patterns obtained for the source and converging lens antenna system, it can be observed that the angle between the main lobe and the propagation axis is significantly reduced from about 50° to 18°, and the gain of the main lobe is also considerably increased from 9 dBi to 15 dBi. This is also confirmed by the 2D gain curves in Figs. 8(c), 8(f) and 8(i).
The convergence functionality of the lens is further tested for other topological charges at 10 GHz. In Fig. 9, the E-field distribution of vortex waves with topological charges of +2 and +3 are verified. One can easily observe that, through the lens, the divergence angle of the antenna is significantly minimized and the energy in the main lobe is considerable increased, which verifies the excellent convergence efficiency of the MS lens for multiple mode vortex waves.
The transmission losses between lossless MS lens and lossy MS lens from 9.9 GHz to 10.2 GHz are shown in the Fig. 10(a). The inherent losses of the selected substrate is 0.003, and the transmission loss ranges from 0.36 dB to 0.48 dB, indicating that the losses of the MS lens are acceptable. The transmitted efficiency of the MS lens antenna is calculated as follows:
The transmission losses resulting from the use of a low-loss substrate from 9.9 GHz to 10.25 GHz are shown in Fig. 10(a). Moreover, the transmitted efficiency of the MS lens has been calculated via Eq. (4) and shown from 9.9 GHz to 10.25 GHz in Fig. 10(b). We observe that the the transmission efficiency of the MS lens is more than 75 %, and it is even 88.14 % at 10 GHz. This is consistent with the simulation results of the transmission efficiency of the unit cell.
The corresponding power distributions of source and MS lens antenna at different OAM modes are analyzed based on the purity calculation of OAM modes . The calculation result is shown in Fig. 11. The mode purity of the +1 OAM order vortex wave generated by the source is 0.874. When the vortex wave passes through the MS lens, the mode purity changes to 0.865. It can be observed that the purity of the convergence lens antenna system for the predominant OAM mode is reduced by only 1 % compared to the source alone. The proposed method of convergence has little effect on the purity of vortex wave.
The compared simulations of the axicon lens antenna and MS lens antenna are shown in Fig. 12. From Figs. 12(a) and 12(d), it can be clearly observed that the conventional axicon lens and the MS lens can both converge the vortex wave with excellent effect of divergence angle reduction. Simultaneously, by comparing the radiation patterns obtained for the axicon lens and MS lens shown in Figs. 12(b) and 12(e), it can be observed that the angle between the main lobe and the z axis is significantly reduced to 20° and 18°, respectively. This is also confirmed by the 2D gain curves presented in Fig. 12(c), where the gain of the MS lens antenna is found to be slightly higher than the mirrored axicon lens antenna. In addition, Fig. 12(f) presents the reflection coefficient (S11) of the source antenna, axicon lens antenna and MS lens antenna. We can observe that the matching bandwidth are quite similar for the three configurations. Moreover, the MS antenna presents the advantage of being flat, and can therefore be easily integrated in communication systems.
In summary, a flat MS lens which is capable of converging vortex electromagnetic waves has been proposed. The structure of the MS lens can provide a phase variation from 0 to 2π. As numerically verified, the electric field amplitude distribution has been shown to produce a significant convergence effect. Along the propagation direction, the dark spot associated with the EM field intensity increases much slower when the MS converging lens is applied. Both 2D and 3D radiation patterns demonstrate that the proposed lens enhances the directivity of a circular OAM generating array antenna by converging and increasing the level of the main lobe. A numerical study was performed to prove that the lens is able to operate from 9.3 GHz to 10 GHz. Although the lens design was optimized for OAM waves with topology charge +1, it has also shown promising performance for OAM waves with topology charges of +2 and +3. The high performance and practicality of the lens make it a prime candidate for improving the medium-long range distance performance of microwave communication systems based on vortex electromagnetic waves.
National Natural Science Foundation of China (NSFC) (No. 61601345); Fundamental Research Funds for the Central Universities (No. XJS16046, JB160109); Natural Science Foundation of Shaanxi Province, China (No. 2017JQ6025); The Pennsylvania State University John L. and Genevieve H. McCain Endowed Chair Professorship.
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