Three-Dimensional Simultaneous Arbitrary-Way Orbital Angular Momentum Generator Based on Transformation Optics

In wireless communications, people utilize the technology of diversity against multipath fading, so as to improve the reliability of communication equipment. One of the long-standing problems in diversity antennas is the limited number of diversity in a certain space. In this paper, we provide a solution to this issue by a three-dimensional (3D) simultaneous arbitrary-way orbital angular momentum (OAM) generator (3D SAWOG) based on transformation optics. The proposed 3D SAWOG consists of a metamaterial block and a group of transformation cylinders, by which arbitrary-way planar wavefronts can be converted to helical wavefronts with various topological charges simultaneously. The 2D four-way OAM generator and the 3D SAWOG are analyzed, designed, and simulated. The simulation results validate the performance of a 3D SAWOG successfully, indicating that the proposed model possess a high mode purity and expansibility. The SAWOG can be used as a novel diversity antenna array due to the orthogonal property among different modes, which could provide more degrees of freedom than traditional dual-polarization antennas, further improving the reliability of the communication systems.

array, which improves the number of the antenna diversity compared with traditional antenna arrays. The use of OAM antennas can improve the isolation due to its orthogonal property of different modes. It facilitates significantly shorting the distance of adjacent receiving antennas and reducing the overall size of the receiving device. Therefore, the study of the SAOG is meaningful and valuable for practical applications.
In this paper, based on the theory of transformation optics, a 3D simultaneous arbitrary-way orbital angular momentum generator is proposed for the first time to achieve OAM-carrying beams with various topological charges at the same time, as shown in Fig. 1. Firstly, a 2D four-way OAM generator with a variable emitting source is proposed, which can be split into arbitrary paths by employing coordinate transformations. Then, a 3D OAM generator with different topological charges on corresponding paths has been designed. The simulations of electric field and phase in different situations are presented to validate the generator's performance. The proposed generator shows competitive advantages, including arbitrary-way generation of OAM, simultaneous different topological charges, and simple design procedure.

Results
Theoretical design. For a Gaussian-enveloped beam carrying an orbital angular momentum, its field is an eigenmode of the paraxial Helmholtz equation [1][2][3]22 , where E 0 is the amplitude of electric field, ω 0 is the waist radius. In addition, ω ω = 2 / , and λ refers to the beam width, Rayleigh length, wavenumber and wavelength in vacuum, respectively. The phase can be expressed as following 22 is the wavefront radius of curvature. It is the phase for LG l 0 beam. In particular, this beam will degenerate into the Gaussian beam when l = 0. In order to generate OAM, it is important to manipulate phase and generate a helical wavefront e −ilφ .
The key to generating OAM by transformation optics is to accurately set up the coordinate transformation of phase. The Jacobian matrix A depicts mapping relations between virtual space and physical: The relative permittivity ε′ ij and permeability μ′ ij tensors of transformation media are defined as 23 : Figure 1. Diagram of the 3D simultaneous arbitrary-way OAM generator. In the central position, there is a square filled with metamaterials, which can split the Gaussian beam source into arbitrary ways of beams, so as to produce the input for each metamaterial cylinder. In addition, transformation cylinders with specific topological charges are placed outside the source. On the different paths, the input Gaussian beams will be converted into the LG beams with corresponding helical wave fronts and topological charges.
Scientific RepoRts | 6:38667 | DOI: 10.1038/srep38667 According to equations (3)-(4), constructive parameters of the metamaterial cylinders could be calculated. The parameters obtained could effectively manipulate the wavefront in the transformation cylinder so as to generate a beam with OAM. Through this transformation cylinder, Gaussian beam propagating along axis x will be converted to a helical Laguerre-Gaussian beam with a phase e −ilφ , as shown in Fig. 2.

Numerical calculation.
In the simulation, the Gaussian beam is employed to propagate along the symmetrical axis x through the metamaterial cylinder whose central axis is also parallel to the axis x. Parameters a, r, and l denotes the thickness, radius, and the topological charge, respectively. A proportional coefficient n must satisfy the following relation: This relation is the prerequisite of Gaussian beam's transformation in the range of cylinder. In next part, some simulations performed by the multi-physics simulation tool (COMSOL) will be presented to reveal the physical process and confirm the generation of OAM.
Based on the principle of transformation optics, a transformation from physical space (x, y, z) to the virtual space (x′ , y′ , z′ ) is defined as following: x x y z x n c y x y z y z x y z z where n is the proportional coefficient, and c is a constant related to initial point coordinate so as to avoid the singularity in the calculation. In addition, θ = − z y tan ( / ) 1 is the azimuthal angle in the yoz plane of the transformation cylinder and range of θ is from 0 to 2π. Then, the Jacobian matrix A 24 according to equation (3) can be calculated: where ′ = ′ + ′ r y z 2 2 . Generally, relative permittivity and permeability ε ij and μ ij both equal to 1 in free space. Therefore, the relative permittivity and permeability of transformation media 24 can be acquired using equation (4): From the above, the wavefront transformation is ndθ/2π and the longitudinal phase shift is ndθ/λ at the position with azimuthal angle θ. Therefore, the OAM L generated in the x-direction is where h is the Planck's constant. Considering the property of OAM, the phase difference is lλθ/2π and the helical wavefront is e −ilφ .
2D four-way OAM generator (FWOG). In our proposed scheme, the central metamaterial square can split a source into arbitrary beams. Some methods have been proposed to support the realization of this scheme. A general method 25 is employed here, as Fig. 3. The virtual space OA′ B′ (x, y, z) is mapped to physical space OAB (x′ , y′ , z′ ) with transformation function as: and  For a rectangle, width is 2λ and the length is 4λ. The Gaussian beam located at the center is divided into four directions as the input beams. Each rectangle contains two sections, the azimuthal angle of one section is set as 0 and another is π/2. Constitutive tensors of material in the transformation region can be obtained:  where r is a radius of the circle, and l is the vertical distance from the source to a side of the polygon, and the θ is the central angle of a circle. The radiation direction varies with the relative position of points (A, A′ , B, B′ ). For the TE incident wave, we can simplify the parameters as following: This method results in generally arbitrary emission direction for an individual beam. In addition, this design can be fabricated with copper strips and dielectric substrates by current techniques 25 . Therefore, the scheme that splits one beam into arbitrary beams is viable in the practical applications.
In order to verify the results of the 2D FWOG, simulations are carried on based on the method of finite elements (i.e., Comsol). Considering the performance of computer and complexity, a 2D FWOG is firstly presented in Fig. 4. The constructive parameters can be obtained by equation (8) under the condition of z = 0. The Gaussian beam located at the center is divided into four directions to perform as the input for each transformation cylinder. Four rectangles represent the cylinders' profiles with different topological charges, and the value of each cylinder increases from l = 1 to l = 4 anticlockwise.
In order to observe the change of the electric field in a 2D model, each rectangle is divided into two sections, the azimuthal angle of one section is set as 0 and another is π/2. As shown in Fig. 4, there is a significant difference between the electric field and power flow, confirming that the wavefronts of output beams have shifted at different azimuthal angles, which indicates the wavefront from an output surface would spirally propagate along the symmetrical axis x. The proposed generator can be further extended to an arbitrary-way generator by changing the angle θ , which is related to the number of splitting beams. Compared with the method to overlap more than one grating, the proposed method performs high convenience and flexibility, and can be put into practice based on transformation optics [26][27][28][29][30] . 3D multiway OAM generator (MWOG). Compared with the 2D OAM simulations, a 3D MWOG provides a better way to show the characteristics of the helical electric field and phase, for which reason, a 3D transformation cylinder is built according to equation (8), as depicted in Fig. 5(a). In general, the intensity of an output beam is zero at the central position, while the intensity around center does not equal to zero because the output beam propagates along axis x in a spiral way. The simulated result observed from the yoz plane is shown in Fig. 5(b), i.e., a typical distribution for Laguerre-Gaussian beam when its topological charge is one.
Several reasons cause the lack of symmetry in the intensity maps. Firstly, the intensity of the orbital angular momentum that we observe is the average intensity over a period. In the simulation, the result that we get is the distribution at a certain time. Therefore, a diagram of power density may not be a standard ring shape. Secondly, the degree of mesh and proportional coefficient c both will influence final results. However, the former is the main reason.
In order to clearly display the output field and phases with different topological charges, results of the transverse section will be presented. The constructive parameters can be calculated by equations (7)- (9). The z-component electric field and phases of the exit surface in the situation l = 1 to l = 4 are depicted in Figs 6 and 7. Each transformation cylinder is excited by a Gaussian beam with the planar wavefront, and resultant electric fields and phases are observed at the position of 3λ away from the exit surface. In one hand, for the case of l = 1 in Fig. 6(a), two helical arms circling clockwise. Furthermore, the number of helical arms will also increase an additional two along with the topological charges gradually increasing. In the other hand, it clearly indicates that the phase gradually changes from − π (blue color) to π (red color) in Fig. 7(a), which experience a phase variation of 2π from anticlockwise direction. This result is consistent with equation (1) and proves that the wavefront of the output beam carries OAM in situation l = 1. The rest results of phase also verify this thought. In terms of relative knowledge of LG beam, a conclusion can be drawn that each output beam has successfully possessed an OAM of L = lh′ .

Discussion
In this paper, a 3D SAWOG has been proposed for the first time, by which, arbitrary beams with a variety of modes can be obtained simultaneously. The simulation results have verified the performance of a 3D SAWOG successfully, indicating that the proposed model possess a high mode purity and expansibility. The 3D SAWOG can be used as a novel diversity antenna array due to the orthogonal property among different modes, which would provide more degrees of freedom than traditional dual-polarization antennas, further improving the reliability of the communication systems. Furthermore, compared with traditional polarization diversity antennas, the proposed SAWOG can significantly reduce the distance between adjacent antennas. Due to the decreased distance between adjacent antennas, the SAWOG facilitates the miniaturization of the antenna arrays, which is very promising for future applications in the next generation wireless communication system [31][32][33][34][35][36] .