## Abstract

Orbital angular momentum (OAM) vortex waves generated by conventional spiral phase plates and metasurfaces have been widely discussed. In this work, we propose an innovative OAM generation method based on transformation optics (TO). By solving Laplace’s equation with specific boundary conditions, an oblate cylindrical shaped physical domain is designed to imitate a gradient shaped virtual domain which is able to generate a vortex beam upon reflection. As a proof-of-concept demonstration, a broadband all-dielectric microwave lens for vortex beam generation is presented with a topological charge of + 1. The corresponding far-field patterns as well as near-field helical phase and doughnut-shaped amplitude distributions of the lens, obtained from numerical simulations, are reported along with a wide operational bandwidth spanning from 8 to 16 GHz. As a transformation method, the proposed TO technique provides an effective way to realize a conversion from plane waves to vortex waves, which can greatly facilitate the potential implementation of OAM waves in microwave wireless communication systems.

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

## 1. Introduction

Vortex waves carrying orbital angular momentum (OAM) have attracted considerable attention in both the optical and radio frequency regimes, due to their unique properties that can significantly improve the channel capacity of a communication system [1–3]. In sharp contrast to common wireless communication systems that typically employ schemes based on the linear momentum of an electromagnetic (EM) field, vortex waves facilitate a new angular momentum based form of signal modulation. The signal space consisting of orthogonal OAM eigenstates provides wireless communication systems with a new dimension of physical characteristics by exploiting the rotational degree of freedom. An EM wave carrying OAM has a helical transverse phase structure quantified by exp (i*lϕ*), in which *ϕ* is the transverse azimuthal angle and *l* is an unbounded integer (the OAM state number). OAM beams with different *l* values are mutually orthogonal, allowing them to be multiplexed together along the same beam axis and de-multiplexed with low crosstalk. Such vortex waves offer a solution to the problem of radio-band congestion by allowing the development of new techniques for achieving high spectral density in wireless communications.

As a consequence, generation techniques for vortex waves carrying OAM have recently been explored in depth. Spiral phase plates (SPPs) [4–10] and helicoidal parabolic antennas [1], as conventional OAM generation methods, tailor the phase of the transmitting wave in proportion to the azimuthal angle *φ* around the center. Antenna array synthesis techniques [11-12] represent another effective way to generate OAM vortex beams. However, they generally require an intricate feeding system to achieve radiating beams with the desired helical phase fronts, which dramatically increases the complexity and cost of the corresponding systems. Recently, metasurfaces operating in reflection and transmission modes have been designed to generate vortex beams due to their considerable potential for manipulation of electromagnetic waves [13–16]. However, most of them suffer from narrow-band response and/or moderate efficiency. Nevertheless, OAM generation methods based on spatial transformations, which can modify the phase distribution by compressing the field, are rarely discussed.

Transformation optics has been demonstrated as a powerful design tool for the manipulation of EM fields in unprecedented ways through the use of judiciously engineered materials with parameters that vary spatially [17–19]. An equivalence has been proposed by Hu *et al.* [20] between coordinate transformation and spatial deformation by using Laplace’s equation to determine the deformation of coordinate grids during the transformation. The transformed material parameters were determined by the solution of Laplace’s equation under proper Dirichlet-Neumann boundary conditions. This concept has led to the widely publicized carpet cloak design [21], but also to the development of many other conceptual as well as functional devices such as novel waveguiding structures [22–27], microwave lens antennas [28–37], illusion devices [38–43] and so on.

In this work, we propose a spatial transformation optics based OAM generation method. An oblate cylindrical shaped physical domain is designed to mimic a gradient shaped virtual domain whose bottom surface acts as a helicoidal parabolic reflective surface. In order to establish the quasi-conformal mapping between the physical domain and the virtual domain, Neumann and Dirichlet sliding boundary conditions are set at the edges of the transformation zones. The Jacobian matrix of the coordinates in the physical and virtual domains is calculated by solving the Laplace’s equation under the specific boundary conditions. The transformed medium with quasi-isotropic material permittivity distribution is calculated from the Jacobian matrix and assigned to the physical domain to achieve the spiral phase creation. To further demonstrate the proposed TO based OAM generation method, an all-dielectric lens which can be easily illuminated by a standard feeding source and which can operate over a wide frequency band is designed. It is shown that the concentric circular phase distribution and Gaussian amplitude distribution on the cross-section of a plane wave is tailored into helical shaped and doughnut-shaped distributions respectively. Full wave finite element method (FEM) numerical simulations are performed to validate the broadband performances of the proposed device due to its non-resonant properties.

## 2. Theoretical design of the vortex beam generation

A helicoidal parabolic antenna, which is commonly fabricated by joining together several discrete surfaces, is used to generate a vortex wave with a feeding source located along the *y*-axis. The helicoidal parabolic system is composed of discrete parabolic surfaces and an associated feeding port, as illustrated in Fig. 1(a). The plane wave generated by the feeding source is transformed into a vortex wave through the reflection from the discrete parabolic surfaces. The parabolic antenna is composed of 8 discrete surfaces with an average gap of *K*/7 along the *y*-axis and has an overall diameter *N*. The vertices of the top surface are located in the *xoz* plane. The feeding source of the discrete parabolic antenna is positioned at a distance *T* from the origin. The general formula for each discrete parabolic surface in the *x-y* plane is given by

*DC*is

*W*

^{2}/(4

*H*), such that

*W*and

*H*are constants, and

*L*is a variable.

The TO based generation method is designed to transform a certain zone of virtual space above the helicoidal parabolic antenna into an oblate cylindrical physical space, as presented in Fig. 1(b). By transmitting through the calculated physical space, a plane wave emitted from the feed is transformed into a vortex beam after its reflection from the grounded planar surface rather than from a set of discrete helicoidal parabolic surfaces.

To determine the transformation from the helicoidal paraboloid to the planar surface, we propose a design based on the concept of spatial transformations, as illustrated by the schematics shown in Figs. 2(a) and 2(b). The physical and virtual space coordinates are respectively denoted by (*x*, *y*) and (*x’*, *y’*). The physical space is placed over a ground plane (metallic surface) and is illuminated by a plane wave generated by a feeding source as represented in Fig. 2. The distance between the wave port and the origin is identical in both the virtual and physical spaces and is set to a value of *T* = 90 mm, found after a parametric study is performed to evaluate the optimal distance *T*. As illustrated in Fig. 2(a), the orange colored arc *CD* represents the side view of one section of the discrete helicoidal parabolic surface. Both the arc *CD* and the straight line segment *C’D’* represent perfect electric conductor (PEC) surfaces. Due to the transformation from arc *CD* to the straight line segment *C’D’*, the OAM beam can be tailored by a planar surface rather than a curved one. To obtain the desired mapping between the virtual space, which is free space, and the physical space, which is the transformed medium, the commercial partial differential equation (PDE) solver Comsol Multiphysics [44] is employed to evaluate Laplace’s equation subject to predefined boundary conditions. The virtual space represented in yellow and the physical space represented by a gradient color scheme are presented in Figs. 2(a) and 2(b), respectively.

The coordinates *A* and *A’* as well as *B* and *B’* share the same locations such that the segments *AB* and *A’B’* are equal to *W*. The length of the segment *BC* is taken to be a variable *L*, while the segment *DE* is assumed to be *H*, where *CE* is perpendicular to *DA*. The segment *DA* is transformed to segment *D’A’* whose length is *M*, while the segment *BC* is transformed to segment *B’C’*. Similarly, the arc *CD* is transformed to the horizontal line segment *C’D’*. Therefore, the rectangle defined as *A’B’C’D’* is mapped from the quadrilateral *ABCD*.

The designed model is based on a spatial transformation and achieved by solving Laplace’s equation. In order to establish an equivalence in the fields at the outer boundaries with the virtual space, Neumann and Dirichlet sliding boundary conditions are set at the edges of the microwave lens. These boundary conditions are:

## 3. Demonstrated microwave lens model

In order to demonstrate the generation method introduced in the previous section, a lens assigned by the inhomogeneous medium calculated from the space transformation is built and analyzed at microwave frequencies. Continuous permittivity distributions for the different sectors are calculated by employing the PDE solver according to the corresponding values of surfaces *m*. Next, each of the sectors is discretized into 147 values of permittivity. The calculated permittivity distributions corresponding to the 8 discrete surfaces are shown in Fig. 3, which share similar parameter variations but have different ranges (minimum and maximum values).

A 3D discrete lens model is designed for further realistic full-wave numerical simulations, as shown in Fig. 4. The microwave lens is comprised of 8 sectors (the cross section of sector 1 is shown illustrated in different colors), which correspond to the original 8 discrete helicoidal parabolic surfaces. The variable *L* is equal to *md*, where *m* is a serial number ranging from 1 to 8. Different values of *m* correspond to different surfaces, with the top surface indicated by *m* = 1. The gap between any two neighboring elements is *d* = *K*/7. The physical space presented in Fig. 2(b) represents the side view of one of the sectors. We further discretize each sector into 7 layers along the *y*-axis, where each layer consists of 21 unit cells. We assume that each unit cell has dimensions 5mm × 5mm × 5mm, small enough with regard to the operating wavelength of 3 cm (frequency of 10 GHz) so that an effective medium can be considered. As a result, the entire lens is composed of 1176 different circular rings containing a total of 9352 cubical unit cells. The proposed discrete lens model has an oblate cylindrical shape with a height of 3.5 cm and a bottom radius of 10.5 cm.

According to existing theory, radio waves that carry OAM possess the particularity of having spiral shaped spatial phase fronts of the EM wave component along the propagation direction. The order of the OAM mode is determined by the eigenmode number *l* representing the topological charge, such that the helical phase distribution can be observed in a cross-section perpendicular to the beam axis.

In order to validate the proposed all-dielectric lens, full-wave simulations have been performed by employing the finite element method based HFSS software from ANSYS [45]. The simulated phase and amplitude profiles in the lateral *xoz* plane perpendicular to the beam axis for three different configurations are presented in Fig. 5. A regular wave port placed in the center of the top boundary of the cylindrically-shaped simulation domain is utilized as the feeding source. As a basis for comparison, a planar reflective surface and a helicoidal parabolic reflective surface are simulated and presented in Figs. 5(a)–5(d). A small circular region is omitted from the phase distribution cartography due to the presence of the feed structure. However, the major features of the spatial phase distribution of a beam carrying an OAM mode can be clearly observed from the proposed transformed lens (Fig. 5(e)), which is in good agreement with that of the helicoidal parabolic reflector shown in Fig. 5(c). The phase distributions shown in Figs. 5(c) and 5(e) present only one twist, corresponding to a topological charge of 1.

The generation methods for OAM-carrying radio waves are derived from the study of OAM light beams, mainly those based on Laguerre-Gaussian (LG) beams. The EM field expression formula indicates that the intensity distribution in the cross-section of the LG light beam has a doughnut shape. In Fig. 5, we present the simulated EM field intensity distribution in the cross-section parallel to *xoz* plane for the three different configurations. Based on further comparisons between the amplitude distributions of the three different configurations, it is found that the simulated OAM spatial field distribution and the theoretical predictions are in very good agreement. It should be noted that a stronger radiation intensity suggests that better reception of the radio beams can be achieved at a fixed distance away from the transmitter. The EM field amplitude distribution of the transformed lens shows a doughnut shaped pattern as for the helicoidal parabolic reflector.

The 3D far-field radiation patterns for the three different configurations at 12 GHz are presented in Fig. 6. So as to highlight the influence of the lens, the radiation pattern obtained for the planar reflector is shown in Fig. 6(a), where a broad main beam can be observed. The radiation along *y*-axis is significantly reduced by the lens which produces a hollow-shaped main lobe. Although the lens imparts a slight dispersive effect on the two main lobes compared to the pattern of the helicoidal parabolic reflector, it can be clearly observed that OAM vortex waves can indeed be generated by the proposed all-dielectric lens. It should be noted that when solving the Laplace’s equation with both Dirichlet and Neumann boundaries, a material parameter distribution with non-diagonal terms is obtained, thus leading to challenges for a physical realization of the device. Therefore the anisotropy is ignored for a quasi-isotropic material implementation of the lens model. However, such approximation causes some degradation in the functionality of the lens [32-33], leading to slightly different radiation patterns of the proposed lens compared to those of the parabolic reflector.

The all-dielectric lens operates over a wide frequency band ranging from 8 GHz to 16 GHz. Numerical simulation results for different operating frequencies are depicted in Fig. 7. The characteristics of the far-field patterns obtained from the lens are further compared in Fig. 7 at 8 GHz, 10 GHz, 14 GHz, and 16 GHz, which demonstrates a null in the amplitude at the center of each beam. Theoretically, the achieved bandwidth can be very broad since all-dielectric non-resonant materials are employed in the design.

The simulation results show that the proposed all-dielectric lens can be indeed used to generate vortex waves with orbital angular momentum in an effective manner over a wide frequency range. The vortex phase wavefronts can be generated with gradient distributed permittivity, due to the configuration of the sub-wavelength metamaterial elements. As a result, OAM vortex waves with different mode numbers can be feasibly realized. By utilizing the proposed configuration, it becomes much easier to produce vortex radio waves with different mode numbers over a broad operating bandwidth. Moreover, the proposed flat lens design provides a relatively simple way to generate OAM vortex waves for radio and microwave wireless communication applications.

## 4. Conclusion

In summary, a new generation method of vortex beam with orbital angular momentum has been proposed. Such a manipulation in phase variation was achieved by employing a spatial transformation optics concept by solving Laplace’s equation with specific boundary conditions. A flat surface below the oblate physical space, which is assigned by the designed inhomogeneous (*i.e.* gradient) permittivity profile, is able to transform a plane wave into a vortex wave. In order to further demonstrate the proposed design method, a microwave lens in the shape of the physical space was introduced as a vortex wave generator with a topological charge of + 1 for full-wave analysis. Analogous phase front and hollow wave front amplitude distributions represent the vortex wave generation functionality of the lens. Furthermore, far-field radiation patterns show good agreement with the performance of the conventional helicoidal parabolic reflector antenna counterpart. The lens has been numerically simulated and has shown to possess a wide operating frequency band extending from 8 GHz to 16 GHz due to the non-resonant material properties. Such transformation optics based design method, is not only an efficient way to realize the transformation between the plane wave and the vortex wave, but also can serve to significantly facilitate the potential implementation of OAM in microwave wireless communication systems by achieving the benefits of simple excitation, wide bandwidth, ease of fabrication and integration.

## 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 of China (No. 2017JQ6025).

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