Simultaneous Generation of Linear- and Circular-Polarization Vortex Based on Symmetrical Metallic Quasi-Metasurface

Abstract

Through comprehensive research on the working principles of the generalized Snell’s law, this paper presents a quasi-metasurface based on the theory of wavefront difference. The proposed structure successfully demonstrates the vortex manipulation of both linearly polarized waves and circularly polarized waves. The symmetrical metallic column design modulates the phase of one period at 18 GHz with a reflection loss of less than −0.5 dB. By decomposing circularly polarized waves into a set of mutually orthogonal linearly polarized waves with a phase difference of π/2, a simple synthesis method is proposed to convert existing linearly polarized waves into circularly polarized waves, thereby demonstrating vortex wave effects. The quasi-metasurface exhibits excellent performance, and the vortex characteristics of both linearly and circularly polarized waves are verified through simulations.

1. Introduction

The phase-gradient metasurface model based on the generalized Snell’s law has been well researched [1,2,3]. These studies on fundamental electromagnetic theories have enabled phase-gradient metasurfaces to play a significant role in various intelligent communication fields, such as signal reconfiguration, signal enhancement, and signal tracking [4,5,6]. Furthermore, advancements in phase-gradient metasurfaces with orbital angular momentum (OAM) technology have expanded signal capacity in the microwave frequency range, opening new avenues for the exploration, validation, and application of vortex wave technology [7,8,9,10].

As an extension of electromagnetic waves, vortex electromagnetic waves follow the same classification and fundamental principles as wave superposition and decomposition [11]. In addition to metasurfaces, various approaches for generating vortex waves have been explored [12,13,14]. However, most metasurfaces are designed to manipulate either linearly or circularly polarized waves, with significant success achieved in both areas [15,16]. For instance, a programmable metasurface for the generation of multimode vortex waves was recently proposed to control linearly polarized waves [17,18], enabling independent and fast communications. Research on controlling circularly polarized vortex waves using metasurfaces is less comprehensive due to the complexity of the manipulation theory and the need for independently designed control meta-atoms based on specific requirements [19,20,21]. For high-performance OAM, both LP and CP waves must adhere to the generalized Snell’s law, although the principles of phase-modulation meta-atoms differ. For linearly polarized vortex waves, control is achieved by identifying and arranging meta-atoms with different discontinuous and periodic reflecting phases, a result validated since the earliest proposal of the generalized Snell’s law [7,22]. In contrast, circularly polarized waves are controlled based on a combination of the generalized Snell’s law and the Pancharatnam–Berry (PB) phase principle [23,24], which has been used to manipulate the behavior of circularly polarized vortex waves on metasurfaces.

This study leverages the fundamental principles of the generalized Snell’s law, as shown in Figure 1a, to propose a quasi-metasurface structure using symmetrical cylindrical metals, depicted in Figure 1b. This structure facilitates both linear- and circular-polarization wavefront differences through phase-to-height mapping, as demonstrated in Figure 1c. Within the high-frequency structure simulator (HFSS), each metallic column is designed with dimensions of 10 mm × 10 mm × height, where the height variation modulates the reflection phase of the electromagnetic (EM) wave. A master–slave boundary condition was implemented for the periodic model calculations. By systematically sweeping variables and analyzing the results, the quasi-metasurface structure successfully manipulates the phases of both linearly and circularly polarized waves at a frequency of 18 GHz. Additionally, a circular observation surface with a radius of 200 mm was positioned 1000 mm from the quasi-metasurface to capture and plot the amplitude and phase distributions at that location. Through the strategic design of symmetrical metallic columns, two quasi-metasurface models were developed, excited by active spherical waves. These models achieve precise control over the electromagnetic wave phase, enabling the simultaneous generation of first- and second-order linearly and circularly polarized vortex waves.

2. Fundamental Theory

The generalized Snell’s law can be employed to calculate the wavefront difference between the feed source and the target plane. Subsequently, the resonance principle and the PB phase principle can be utilized to compensate for the wavefront differences under linearly and circularly polarized conditions. The quasi-metasurface model employs symmetrical metallic columns with nearly full reflection to directly compensate for the wavefront difference (height), achieving simultaneous manipulation of both linearly and circularly polarized waves [25,26].

2.1. Generalized Snell’s Law (GSL) and Orbital Angular Momentum (OAM)

The proposition and refinement of the generalized Snell’s law have provided a solid scientific basis for using metasurfaces to manipulate the phase factor (e^−jφ) of electromagnetic waves. Figure 2a illustrates the specific mathematical principles for using metasurfaces to manipulate spherical waves and generate plane-wave beams carrying OAM.

The phase of the electromagnetic wave excited by the feed source is indicated by the dashed red line in Figure 2a. It can be observed that the wavefront phase of the electromagnetic wave is a spherical surface, denoted as $k_{0}d$, where $k_0$ is the propagation coefficient 2π/λ, related to the frequency of the electromagnetic wave, and d is the distance between the feed source and wavefront. When the beam propagates from the feed source to the blue pillar shown in Figure 2a, the distance is $d_1,$ and the phase is $k_0{d}_1$. When the beam propagates from the feed source to the red metal pillar shown in Figure 2a, the distance is $d_2,$ and the phase is $k_0{d}_2$. Based on this principle, vortex electromagnetic waves carrying OAM can be generated by compensating and manipulating these phases using a designed metasurface.

In the spatial domain, the wavefront phase of a spherical wave forms a two-dimensional surface. Therefore, considering practical situations, the phase compensation formula for a two-dimensional metasurface based on the generalized Snell’s law is given by Equation (1) [27]:

$$\Phi (x, y)=-k_{0}d(x, y)=-k_0\sqrt{x^2+y^2+Z^2}$$

(1)

In Equation (1), the center of the metasurface is the origin of the Cartesian coordinate system, and d (x, y) represents the distance function between the (|x| + |y|)th unit on the metasurface and the feed source. The constant Z denotes the vertical distance between the metasurface and feed source.

Using the phase compensation expressed in Equation (1), a spherical wave originally excited by the feed source transforms into a plane wave with high-gain characteristics. To enable the beam to carry OAM, a phase modulation formula for vortex waves is described in Equation (2) [10]:

$${\Phi }_{oam}(x, y)=-k_{0}d(x, y)+l {\tan }^{-1}({\displaystyle \frac{y}{x}})\begin{array}{cc}& (x\ne 0)\end{array}$$

(2)

where l represents the order of the vortex wave carrying the OAM. When l = 0, it corresponds to a high-gain plane wave without any OAM.

Once the theoretical method for compensating and modulating the generation of vortex electromagnetic waves based on the generalized Snell’s law is determined, the next step is to design unit structures that can control the phase characteristics of electromagnetic waves.

2.2. Synthesis of Circular-Polarization Wave from Linear-Polarization Wave

Figure 2b illustrates the meta-atom devised in this study, which is capable of phase modulation of electromagnetic waves by manipulating the height of the pillar across different frequencies. Analysis of the reflection amplitude and phase demonstrated that this structure achieved a whole-period phase shift at 18 GHz. Finally, the phase and height were calculated using Equation (3):

$$H_{mn}=p_1\times {\Phi }_{mn}+p_2$$

(3)

It can be observed that the height ($H_{mn}$, in mm) exhibited a linear relationship with the phase shift. After fitting, parameters $p_1$ and $p_2$ were obtained as 0.02313 and 3.006, respectively.

The above result was based on the assumption of linearly polarized waves and the fact that circularly polarized waves can be considered as a composition of orthogonally polarized linear waves with a phase difference of π/2. Therefore, in practical measurement processes, it is possible to measure a set of mutually orthogonal linearly polarized waves using Equations (4) and (5) to synthesize right-hand circular polarization (RHCP) or left-hand circular polarization (LHCP):

$${\mathrm{e}}_{\mathrm{lhcp}}=A{\mathrm{e}}_x^{-j\phi }+jB{\mathrm{e}}_y^{-j\phi }$$

(4)

$${\mathrm{e}}_{\mathrm{rhcp}}=A{\mathrm{e}}_x^{-j\phi }-jB{\mathrm{e}}_y^{-j\phi }$$

(5)

where the imaginary component “j” in the formula indicates a phase shift of π/2. In these equations, A equals B. The variables $e_x$ and $e_y$ represent linearly polarized waves in the x- and y-polarized directions, respectively.

Figure 3a,b show the amplitude and phase of the linearly polarized waves in the x- and y-direction obtained using the simulation software (HFSS15). The intensity and phase of the leading and lagging π/2 electric fields obtained using Equations (4) and (5) are shown in Figure 3c,d, respectively. Finally, the co- and cross-polarization of the circularly polarized waves obtained using the above equations are shown in Figure 3e,f, respectively. It can also be observed that within the same range, the co-polarization wave (LHCP) had a stronger amplitude and clearer phase.

3. Performance Results

The proposed concept was experimentally analyzed in the electromagnetic simulation software HFSS (Version 15). The amplitude, phase, and gain of the first- and second-order vortex waves of the linearly polarized waves, as well as their main polarization and cross-polarization, are shown in Figure 4 when the values of l in Equation (2) were set to 1 and 2.

The amplitudes and phases of the first- and second-order vortex waves in Figure 4a,b, respectively, show significant phase vortex phenomena. Additionally, the amplitudes of the first- and second-order vortex waves formed one and two (merged) central nulls, respectively. Furthermore, the co- and cross-polarization gains of vortex waves of different orders show that with phase compensation and manipulation, the maximum gain could be higher than 20 dBi. However, as the order of the vortex waves increased, the co-polarization gain gradually decreased, whereas the cross-polarization gain remained relatively low. The aforementioned results for linearly polarized waves also provide valuable references for subsequent studies on circularly polarized vortex waves.

Figure 5 shows the results obtained by compensating and manipulating the circularly polarized vortex waves using the same quasi-metasurface based on Equation (2) and calculating the results using Equation (4). The similarity between the results shown in the figure and the analysis of the results of the linearly polarized waves indicates that the quasi-metasurface can compensate for and manipulate the phase of circularly polarized vortex waves.

To present the vortex modulation ability of the linearly and circularly polarized vortex waves of the quasi-metasurface more intuitively, the spectrum purity (SP) Equations (6)–(8) of the electromagnetic waves were used [28,29,30]. Using the SP equations, the collected linearly polarized vortex waves were substituted with the calculated circularly polarized vortex waves. The OAM modes considered ranged from $l=-5$ to $l=5$. The calculated spectrum purity for each vortex wave order is shown in Figure 6.

$$A_l=\frac{1}{2\pi }{\displaystyle {\int }_0^{2\pi }\psi \left(\phi \right)e^{-jl\phi }d\phi }$$

(6)

$$\psi \left(\phi \right)={\displaystyle \sum _l{A}_l{e}^{jl\phi }}$$

(7)

$$spectrum-purity\left(sp\right)=A_l/{\displaystyle \sum _{l^{\prime }=-5}^5{A}_{l^{\prime }}}$$

(8)

where $A_l$ indicates the Fourier transform result and $\psi (\phi )$ is a function of the sampled field along the circumference of the z-axis.

As shown in Figure 6a, the SP of the first-order vortex wave generated by a first-order quasi-metasurface was 82.5%, which was higher than those of the other orders. Based on the results generated by the four quasi-metasurfaces shown in Figure 6b, the following conclusions were obtained: (1) the purity of the first-order vortex wave was higher than that of the other vortex wave orders, regardless of whether the waves were linearly or circularly polarized, and (2) the vortex wave modulation of the prepared metasurface model is feasible. Therefore, the quasi-metasurface model can modulate both linearly and circularly polarized vortex waves at a frequency of 18 GHz.

Based on the results presented in

Table 1. Comparison between results reported in the literature and our results.

Ref.	Frequency (GHz)	Polarization	Gain (dBi)	Principle
[24]	7.8	Circular	>15	PB
[23]	28	Circular	23.8 and 28.7	PB
[7]	10	Linear	17.6	Resonant
[22]	18	Linear	18.45	Resonant
[21]	15	Linear	Null	Resonant
Our work	18	Linear and circular	16–20	wavefront difference

on the manipulation of linearly and circularly polarized waves using supersurfaces, most supersurfaces focus exclusively on the control of either linearly polarized waves [7,22] or circularly polarized waves [23,24]. By contrast, the proposed quasi-metasurface could simultaneously control both linearly and circularly polarized waves, achieving a minimum gain greater than 16 dBi in the Ku-band. This result demonstrates that the designed supersurface can ensure multipolarization control without sacrificing efficiency or gain.

4. Conclusions

To effectively manipulate beam deflection in terms of individual cells based on the generalized Snell’s law, this study developed a quasi-metasurface model. A quasi-metasurface model capable of simultaneously controlling linearly and circularly polarized waves at a frequency of 18 GHz was designed by incorporating a novel vortex wave technology. The proposed model not only validated the beam control principle grounded in generalized Snell’s law from a theoretical standpoint but also demonstrated exceptional compatibility in practical engineering applications, including aerospace communications and anti-interference information transmission, resulting in substantial savings in processing and manufacturing costs.