## Abstract

Radio waves carrying orbital angular momentum (OAM) may potentially increase spectrum efficiency and channel capacity based on their extra rotational degree of freedom. However, due to their divergence characteristics, vortex waves are not suitable to transmit over a long distance in the radio frequency (RF) and microwave domains. In this paper, a transformation optics (TO) based all-dielectric converging lens is proposed. The beam divergence angle of the vortex wave passing through the lens can be decreased from 25° to 9°. The transformed material parameters of the converging lens are determined by solving Laplace’s equation subject to specific boundary conditions. Far-field antenna radiation patterns as well as near-field helical phase and electric field amplitude distributions obtained from numerical simulations are reported, demonstrating the broadband characteristics of the proposed microwave lens. Moreover, the all-dielectric compact lens design comprised by a graded permittivity profile can be fabricated by additive manufacturing technology, which greatly facilitates the potential development and application of vortex wave based wireless communications.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Wireless communication is facing serious challenges in terms of scarce spectrum resources and limited polarization modes due to the increasing demands for higher data rates and service quality. In 1992, Allen et al. observed that electromagnetic waves with helical phase fronts could carry orbital angular momentum (OAM). Since then, considerable research has been devoted to investigating their applications in the telecommunications domain for the substantial enhancement in information storage capacity. The reason for this is that OAM, a property of electromagnetic waves associated with the beam vorticity and phase singularity, possess unlimited eigenstates which are orthogonal to each other [1]. Thus, EM waves with different eigenstates are able to offer multiple channels in order to increase transmission capacity and spectral efficiency. Electromagnetic fields possess angular momentum, which consists of both spin angular momentum (SAM) [2] and orbital angular momentum (OAM) related to the phase front of spiral distributions [3, 4]. Owing to this OAM feature, the use of orthogonal spatially overlapping and co-propagating spatial OAM modes can be treated as an approach of mode-division multiplexing. However, the vortex wave has a doughnut-shaped field intensity profile due to phase singularity in the beam center. In particularly, the radius of the beam carrying an OAM mode increases with the transmission distance due to the divergence characteristics of RF vortex waves. In addition, simulations and experiments have revealed that the beam divergence angle becomes larger with an increase in the OAM mode [5]. Hence, there is a limitation on the propagation of OAM radio waves in free space. In order to use a traditional space domain reception method (similar to what is currently done in fiber-based systems), the size of the receiving antenna needs to be extremely large, making the reception of OAM radio signals difficult to achieve in practice, especially for long-distance transmission [6]. Therefore, the need to generate converging vortex waves carrying OAM is of paramount importance for their successful integration into wireless telecommunication systems.

Both metamaterials [7–9] and transformation optics (TO) [10–12] are possible technologies that can be used to manipulate electromagnetic waves in a user-defined manner and have been applied to many engineering designs in the realm of geometrical optics and electromagnetics [13–15]. Transformation optics has enabled the propagation path of electromagnetic waves to be controlled by spatially varying the effective material parameters of an electromagnetic device (e.g., a TO lens). The physical basis of transformation optics is the invariance of Maxwell’s equations under coordinate transformations. This concept involves the mapping between a physical space, which is the transformed medium, and a virtual space, which is free space. In order to establish an equivalence in the fields between the physical and virtual spaces, Neumann and Dirichlet sliding boundary conditions are set at the limits of the transformation zones. The transformed material parameters are determined by the solution of Laplace’s equation under the specific boundary conditions. The complex distribution of electromagnetic parameters is usually achieved using metamaterials with unique electromagnetic properties that can be tailored to alter the normal principles of light and electromagnetic wave propagation [16–18]. As such, the TO concept has motivated a series of studies on conceptual and functional devices including waveguiding structures [19–23], lens antennas [24–28] and directive antennas [29–31], multi-beam radiating structures [32–34], illusion systems [35–37], and isotropic emitters [38,39]. We have previously investigated a TO based OAM generation lens [40], the convergence to be solved. Inspired by the conventional optical axicon converging lens, TO method is able to provide a flat lens solution with gradient permittivity distribution for OAM beam convergence.

In this work, we propose a beam converging method based on a space transformation that can significantly enhance the directivity of a vortex wave. A cylindrical shaped converging lens is designed and numerically demonstrated in the microwave domain. Laplace’s equation is applied to calculate the constitutive electromagnetic parameters of the transformation medium under Neumann and Dirichlet sliding boundary conditions set at the edges of the transformation zones. Full wave simulations based on the finite element method are employed to validate the design method such that a broadband all-dielectric lens is illuminated by an antenna array generating vortex waves. Near-field phase and amplitude distributions as well as far-field radiation patterns are presented to demonstrate the converging characteristics and low-loss performances of the proposed lens over a wide frequency range, which spans from 6 GHz to 14 GHz.

## 2. Theoretical design of the converging lens

The TO based converging method is designed to transform a prescribed zone of the virtual space with a triangular-shaped region on the top into a rectangular physical space, as illustrated in Fig. 1. By transmitting through the calculated physical space, a vortex wave emitted from a circular microstrip patch array source is transformed into a converged beam while keeping the same topological charge of +1. In other words, we design a flattened axicon lens where the incident Laguerre-Gaussian (LG) beam is transformed into a high-order Bessel beam after passing through the lens.

To determine the transformation from the virtual space to the physical space, we propose a design based on the TO concept, as illustrated by the schematics shown in Figs. 1(a) and 1(b). The physical and virtual space coordinates are respectively denoted by (*x’*, *y’*) and (*x*, *y*). The coordinates *A* and *A’*, *B* and *B’*, *C* and *C’*, *D* and *D’* share respectively the same locations such that the segments *BC* and *B’C’* are equal to *W*. Segment *AB* and segment *CD* share the same length which is assumed to be *H*. *BC* is perpendicular to *AB* and *CD*. The angle between *AE* and *AD* is considered as *θ*. Therefore, the segment *AE* and *DE* have a length of *W*/(2*cosθ*). The segment *AB* is transformed to segment *A’B’*, while the segment *BC* is transformed to segment *B’C’*. Similarly, the segment *CD* is transformed to segment *C’D’*. Therefore, the rectangular physical space defined as *A’B’C’D’* is mapped from the air filled virtual space *ABCDE*.

In order to establish the mapping between the two spaces, Neumann and Dirichlet sliding boundary conditions are set at the edges of the transformation zones:

## 3. Demonstration of the convergence lens model

In order to demonstrate the proposed beam converging method, a lens comprised by the inhomogeneous medium calculated from the spatial transformation is designed and analyzed in the microwave domain. Continuous permittivity distributions are calculated by employing the commercial partial differential equation (PDE) solver Comsol Multiphysics [41]. The continuous distribution of calculated permittivity (*ε _{zz}*) in the

*x’*-

*z’*plane of the physical domain is presented in Fig. 2. The range of

*ε*is related to the deformation between the physical space and the virtual space. In the proposed design case, the permittivity distribution ranges from 1 to 2.8. To refrain from using resonant metamaterials and to favor a potential all-dielectric realization process, we assign

_{zz}*ε*values below 1 to be unity.

_{zz}It is worth noting that the two-dimensional (2D) design can easily be extended into a 3D model due to its rotationally symmetric feature. A three-dimensional (3D) discrete lens model is designed for further full-wave numerical simulations, as shown in Fig. 3. The 3D lens is discretized into 216 values of permittivity, which represents a realistic practical implementation. The feeding source, which consists of a microstrip antenna array with eight radiating elements arranged in a circular configuration, is selected for its ability to generate vortex waves. Vortex waves are formed by shifting the phase of the signal emitted from each radiating element. The microwave lens is comprised of 12 layers, where each layer consists of 18 circular rings. As a result, the entire lens is composed of 216 different circular rings corresponding to the discretized permittivity values. The distance between the feeding array antenna and the converging lens is 4 cm. The proposed discrete lens model has an oblate cylindrical shape with a height of 4.8 cm and a radius of 7.2 cm.

The phase distribution on the cross section perpendicular to the direction of propagation plays a decisive role in the channel transmission of vortex waves carrying OAM. The key feature of the OAM beam is that its electric field has a rotating phase wavefront structure. According to existing theoretical verifications, the spatial phase front of EM beams carrying OAM has a spiral profile along the forward direction and hence in the cross-section perpendicular to the beam axis, one can observe a helical phase distribution. Meanwhile, the order of the corkscrew is determined by the eigenmode number *l*.

The electromagnetic field amplitude distributions are numerically calculated at several frequencies in the 6 GHz-14 GHz frequency range. As presented in Fig. 4, that displays the amplitude distribution of the electric field plotted in the *x*-*z* plane for different configurations [Figs. 4(a)–4(e) and Figs. 4(k)–4(o)], a hollow beam, which is the main characteristic of vortex waves, can be clearly observed. It is obvious that the divergence of the field wavefronts of vortex wave is efficiently reduced with almost no leakage outside the lens region. The intensity of the field is increased and the surface of the hollow annular ring is reduced along the direction of wave propagation. Moreover, the electric field distribution plotted in the *x*-*y* plane [Figs. 4(f)–4(j) and Figs. 4(p)–4(t)] at a distance of 5.4*λ* away from the feeding source clearly shows a hollow beam along the propagation direction of the vortex wave. The angle of beam divergence of the vortex wave passing through the lens is greatly reduced in the presence of the lens. Compared to wavefronts emitted by the circular patch array alone (without lens), the distribution of the wavefronts of the vortex waves passing through the lens are much closer to each other. The intensity of the field is increased and the surface of the hollow annular ring is reduced along the direction of the wave propagation, implying concentration of energy.

Based on the theoretical analysis and our numerical verifications, we can observe that the phase distributions of the electromagnetic field components in all subplots of Fig. 5 possess a helical phase profile with a topological charge *l* = +1. As a basis for comparison, the system without the lens is also simulated and presented in Figs. 5(a)–5(e) at several different frequencies. As it can be clearly observed, the converging properties of the microwave lens do not have any influence on the phase profile of the incident vortex waves.

The characteristics of the 2D and 3D far-field radiation patterns for two different configurations (with and without the TO lens) are presented in Fig. 6. The 3D far-field radiation patterns of the source alone are shown in Figs. 6(a), 6(d), 6(g), 6(j), and 6(m), where a wide main beam can be observed. However, in Figs. 6(b), 6(e), 6(h), 6(k), and 6(n), the far-field radiation along the direction of the wave propagation is significantly concentrated into a narrow hollow beam for the microwave lens system. As it can also be clearly observed from the 2D far-field radiation patterns shown in Figs. 6(c), 6(f), 6(i), 6(l), and 6(o), it is obvious that the main lobe of the vortex beam emitted by the source is narrowed in the presence of the TO lens. The angle of beam divergence of the vortex wave passing through the lens is reduced to roughly 9 degrees in the targeted frequency range. In addition, it can also be seen that the average gain increase of the main lobe is about 5 dB.

## 4. Influence of the lens on higher order OAM models

The lens proposed in this study has been optimized for a specific beam divergence angle, namely that of the *l* = +1 topological charge. However, the beam divergence angle increases for higher order OAM modes. To investigate the influence of the lens on other higher order modes, the near-field and far-field simulation results when a vortex beam with a topological charge of +2 and +3 is used as the source are shown at 10 GHz in Fig. 7. As illustrated in Figs. 7(a), 7(d), 7(h) and 7(k), the intensity of the field increases and the surface of the null in the hollow beam is reduced along the direction of wave propagation, which means that the angle of beam divergence of the vortex wave passing through the lens is decreased. A reduction of the beam divergence angle of 30° and 25° is noted for the *l* = +2 and *l* = +3 modes, respectively. Such an effect is further confirmed by the electric field distributions in the *x*-*z* plane plotted in Figs. 7(c), 7(f), 7(g), and 7(m). From Figs. 7(b), 7(e), 7(i), and 7(l), we can observe that the phase profile of the incident vortex wave with a topological charge of +2 and +3 remains unchanged in the presence of the lens.

The characteristics of the antenna radiation patterns for the two higher order modes are presented in Fig. 8. The main lobe along the direction of the wave propagation is significantly concentrated when passing through the microwave lens, such that the radiated beam becomes more directive. Numerical simulation results presented Figs. 7 and 8 reveal that the lens can be used to converge a vortex beam with any topological charge.

The simulation results show that the proposed all-dielectric lens can be used to effectively reduce the angle of beam divergence of vortex waves with any topological charge over a wide frequency range. The engineering of the lens can be achieved with gradient distributed permittivity using all-dielectric materials. When vortex waves pass through the lens, the intensity of the electric field increases and the surface of the beam hollow decreases along the direction of wave propagation, which implies concentration of electric field energy. By utilizing the proposed field converging method, it becomes feasible to increase the transmission distance of vortex waves with different mode numbers over a broad operating frequency bandwidth. The broadband performance together with the low-cost dielectric realization make the device very attractive. Theoretically, the achieved bandwidth can be very broad since all-dielectric non-resonant materials are employed in the design.

## 5. Conclusion

In summary, using transformation optics, we have presented the theoretical design and numerical validation of a compact all-dielectric converging lens operating in the microwave regime over a wide frequency range. The proposed all-dielectric converging lens illuminated by an antenna array generating vortex beams was able to significantly reduce the angle of beam divergence of the vortex wave. Such a manipulation in beam divergence is achieved by employing a transformation optics concept based on solving Laplace’s equation with specific boundary conditions. The simulated electric field amplitude and phase distributions have confirmed the concentration of electric field energy along the direction of wave propagation. Moreover, the lens has no influence on the phase profile of the incident vortex wave. In order to further demonstrate the proposed lens, far-field antenna radiation patterns have been simulated to show the reduction in the beam divergence angle of the main lobe. The lens has been numerically validated over a wide operating frequency band extending from 6 GHz to 14 GHz due to its non-resonant properties. The proposed design is easy to fabricate, low-cost and represents a potential pathway toward the implementation of practical microwave communications systems based on vortex waves that carry OAM.

## Funding

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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