## Abstract

The manipulation of the wave-front with versatile vectorial polarization channels has intrigued huge attention in many fields, including encryption, detection, and vectorial hologram. However, there still lacks an efficient method to adequately achieve vectorial beams at will. As an exotic phenomenon, circular dichroism(CD) becomes an alternative strategy to overcome this academic challenge. Here, a scheme based on CD enantiomers to customize arbitrary vectorial beams in K-space is proposed. The strategy is illustrated with analytical calculations on the checkerboard-type arrangement to establish a concise relationship between the Full-Poincaré Polarization (FPP) and CD-based orientation enantiomers. On this basis, to expand its potential in practical applications, here we combine the genetic algorithm(GA) with a gradient optimization algorithm. Four beams with left-handed-circular, right-handed-circular, linear, and 5-axial-ratio left-handed-elliptical polarizations with the energy ratio of 7:8:9:10 are achieved as the proof of principle. Hence, this proposed paradigm could manipulate arbitrary vectorial beams and has great significance in multi-polarized distribution radio communications, encryption, and vector-holographic imaging.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

As a natural characteristic of electromagnetic waves, the polarization state, which consists of spin, axial ratio (AR), and polarization azimuth, illustrates the electric field vector's progression in the time domain [1–6]. Hence, which is essential in structure sensing [7–9], information encryption [10], remote detection [11], hologram [12–17], and especially wireless communication [18]. Fortunately, the emergence of metasurface provides a powerful electromagnetic control method. According to Stokes parameters, FPP manipulation can be realized via modulating two orthogonal linear components of electromagnetic waves. Hence, the birefringence medium and anisotropic metasurfaces are feasible [14,19–28]. However, most of them are based on the resonant phase [25], propagation phase [26]. Hence, the sophisticated searching for geometric parameters is inevitable. Meanwhile, maintaining broadband performance is also intricate. However, as for a pair of orthogonal circular polarization, Pancharatnam-Berry (PB) [6,13,29] phase provides a flexible and broadband platform for phase modulation. For example, Song et al. realized a vectorial holography via geometric-phase metasurfaces, which is realized via decomposing the linear polarization into two orthogonal circular polarizations in opposite direction by setting opposite phase gradients [17]. Hence, a row of units for phase gradients is inevitable. As a natural mechanism, CD which could realize different absorption between the LCP and RCP [28–40] is a optional tool to tackle this shortcoming. Hence, a pair of CD enantiomers is selected in our work to manipulate the vectorial beam in K-space. The comparisons of our work with some published works on vectorial wave shaping are summarized in Table 1.

In this paper, the vectorial beam customization in K-space basing on CD enantiomers is proposed (schematic diagram of the functions of CD-enantiomers-based metasurface is shown in Fig. 1). A pair of wideband orthogonal circular polarization enantiomers is designed via embedding a lumped resistor in the chiral metal structure and symmetrically copying the whole structure [15]. Taking the checkerboard type arrangement of this pair of enantiomers as an example, the concise relationship between the FPP modulation in specific directions and the orientation direction of enantiomers is established. Full-wave simulation and experiments prove its excellent FPP modulation performance. Further, simultaneously optimizing the arrangement and orientation direction of CD enantiomers with the combination of the genetic optimization algorithm and the gradient optimization algorithms, the customization of vectorial beams is realized. As verification, four vectorial beams with a specific energy ratio are customized. The electromagnetic simulation results are in good agreement with expectations. Basing on the synergy of this pair of CD enantiomers and using optimization algorithms, this scheme paves a way for vectorial beams modulation, which has significant application prospects in radio communications, especially in the application scenarios subdivision and the multi-polarization channels.

## 2. CD meta-enantiomers

According to spatial symmetry analysis, CD absorption based on metasurface would happen only when there is no mirror symmetry and the rotation symmetry below C2 [36]. 2D chirality, which refers to a system that lacks mirror and reverses symmetries, satisfies this condition exactly. Via combining these 2D chiral structures with lumped resistors, photosensitive semiconductor germanium (Ge), and some lossy medium such as polyimide, a great deal of work has explored the CD enhancement mechanism. [15,39] Based on the aforementioned principle, a circular dichroic meta-enantiomer is designed as shown in Fig. 2(a). An 0.018 mm Archimedean spiral metal structure (perfect electric conductor, PEC) is etched on an F4B dielectric substrate (*ɛ _{r}*=2.65 and loss tangent tan

*δ*=0.001). The back of the substrate is a metal plate, and the period of meta-enantiomer is

*p*. The polar coordinate formula of the Archimedean metallic structure is

*ρ*=

*at*. In which, the Archimedean spiral coefficient

*a*is 0.2 mm/rad,

*ρ*is the distance between the origin coordinate and corresponding coordinate on Archimedean spiral; the total rad of t is 12. The line width of Archimedean metallic structure is

*w*. A lumped resistor is inserted into the Archimedean spiral, and the center of the resistor is located at

*f*=11.2 rad of Archimedean spiral’ course. The resistant value is optimized as 241 Ω.

The reflection coefficients of the meta-enantiomers plotted in Figs. 2(b) and (c) are simulated in Commercial Software CST Microwave Studio with periodic boundary conditions. Figure 2(b) shows that this meta-enantiomer has a robust circular polarization differential absorption from 12.5GHz to 16.5GHz. This Archimedean spiral structure absorbing RCP and converting LCP to RCP is called an R-enantiomer. Further, we monitored the distribution of surface currents in different incident waves. (Note S1, Supporting Information) The results show that when LCP incident, the surface current is relatively strong and the ohmic loss is negligible; when RCP is incident, the surface current is concentrated on both ends of the resistor, and the energy is lost by resistor. According to the Jones matrix's symmetry analysis, the mirror symmetry replication can change its eigenvector from left to right spin [15]. The L-enantiomer could be obtained by mirror symmetry copying of the R-enantiomer. This pair of high extinction ratio units can provide a couple of orthogonal circular polarization bases for any FPP wave modulation. Figure 2(c) indicates that these enantiomers strictly abide by the PB phase principle, which is beneficial for modulating the energy and polarization in the K-space via rotating these mixed enantiomers.

## 3. Checkerboard-type enantiomers FPP manipulation

Take a pair of orthogonal circular polarization bases as an example, and denote their amplitude and phase are *E _{R}*,

*E*, and

_{L}*α*,

_{L}*α*, respectively. Converting these to the coordinate (S

_{R}_{1}, S

_{2}, S

_{3}) in the Stokes-Cartesian coordinate system, it can be deduced:

AR's absolute value is the axis ratio of synthetic polarization; The sign of AR indicates the handle of polarization(the positive sign is left- handle). This analysis indicates the concise FPP manipulation by the synthesis of the right and left circular polarization. Here, this pair of enantiomers are arranging in the form of the checkerboard serving as a particular case. In Fig. 2(e), the yellow dotted frame shows the smallest array. The blue part represents the R-enantiomer, and the pink part represents the L-enantiomer. Each color block comprises 3×3 identical atoms. Figure 2(f) indicates the rotation angle rot* _{i}* of the enantiomer in the i-th quadrant. When a

*y*-pol wave illuminates this array, the normalized superimposed Farfield can be expressed as

*n*-

*th*quadrant; ${{\boldsymbol r}_n} = \sin \theta \cos \varphi \cdot {{\boldsymbol X}_n} + \sin \theta \sin \varphi \cdot {{\boldsymbol Y}_n}$;

*φ*and

*θ*are the azimuth angle and the elevation angle of the spatial scattering direction, respectively. The center coordinates of four quadrants are X

*=[1.5*

_{n}*p*, -1.5

*p*, -1.5

*p*, 1.5

*p*] and Y

*=[1.5*

_{n}*p*, 1.5

*p*, -1.5

*p*, -1.5

*p*]. For enantiomers in the first and third quadrants, $|{{R_{RR({LL} )}}} |= 0$, $|{{R_{RL}}} |= 0$, and for enantiomers in the second and fourth quadrants, $|{{R_{RR({LL} )}}} |= 0$, $|{{R_{LR}}} |= 0$ approximately. Assuming that ${\delta _1} = \frac{{\pi a}}{\lambda }\sin \theta (\cos \varphi + \sin \varphi )$ and ${\delta _2} = \frac{{\pi a}}{\lambda }\sin \theta (\cos \varphi - \sin \varphi )$, formula 2 can be simplified as

This formula represents the electric field superposition of LCP and RCP waves in the whole space. In which *α*_{n} is the phase of the *n-th* quadrant and *α*_{mn} represents the phase difference of the *m-th* and the *n-th* quadrant (*α*_{m}-*α*_{n}). According to formula 3, the phase of a synthetic polarization beam could be regulated by *α*_{1}, and the phase and amplitude of LCP and RCP of the synthetic polarization are shown as:

Based on the combining formulas 1, 4, and 5, we can indicate that the axial ratio and spin direction of the polarization is jointly determined by rot_{31} and rot_{42}, and the polarization azimuth is manipulated by rot_{21}, as Fig. 2(d) shows. For showing the feasibility of this strategy and the performance of the proposed enantiomers, ten polarization are selected to verify our FPP manipulation in the normal direction. They are four linear polarization(with polarization azimuth of 0°, 45°, 90°, and 135°), 3-axial-ratio Left-spin-elliptic polarization(with polarization azimuth of 0°, 45°, 90°, and 135°), and two orthogonal circular polarization. And the corresponding rotation angles of the enantiomers in each quadrant are calculated. (Note S2, Supporting Information) A checkerboard type metasurface with 6×6 arrays (aperture size is 108×108 mm^{2}) is constructed. A Full-wave simulation is performed. Setting a *y*-polarization wave to illuminate it and a Fairfield monitor to record the far-field electrical field distribution at 13GHz, the simulation results of axis ratio and polarization azimuth are shown in Fig. 3, where the blue band is the half-power angular domain.

Figure 3(a), (b), (c), and (d) show the linear polarizations with *β* = 0°, 45°, 90°, and 135°, respectively. (The corresponding energy distribution, amplitude, and phase simulation results are shown in S3, Supporting Information) Fig. 3(e), (f), (g), and (h) show 3-axial-ratio left-spin-elliptic polarizations with *β* = 0°, 45°, 90°, and 135°, respectively. (The corresponding energy distribution, amplitude, and phase simulation results are shown in S4, Supporting Information) These results demonstrate the feasibility of this customized FPP scheme. To quantitatively analyze the effectiveness of the FPP modulation proposed, it is necessary to take the modulation of both *β* and AR into consideration. Here, the error angle (EA) on Poincaré Sphere (showed in Fig. 2(d)) is used to characterize manipulation error. The calculated EAs of above four linear polarizations are 2.94°, 4.58°, 6.71°, and 7.03°; 4.10°, 5.21°, 7.24°, and 8.26° are EAs of four 3-AR left-spin-elliptic polarization, respectively. These calculation results quantitatively indicate accurate FPP customization. The manipulations of LCP(*α*_{13} = 180°, *α*_{24} = 0°) and RCP (*α*_{13} = 0°, *α*_{24} = 180°) also verify the feasibility of the scheme(Note S5, Supporting Information).

When phases difference *α*_{13} and *α*_{24} satisfy *α*_{13}=*α*_{24}=180°, the energy is mainly distributed in the four first-level side lobes, as Figs. 4(a)-(d) show. (Their radiation directions are *φ*=±90°, *θ*=±40°, respectively) According to formula 5, the polarization azimuth *β* of each lobe can be customized via changing *α*_{21} from 0 to 180°. For verification, making rot_{21}=0° and setting a *y*-pol plane wave to illuminate the metasurface, the lobes at *φ*=0°, *θ*=±40° are *x*-polarization, and the lobes at *φ*=90°, *θ*=±40° are *y*-polarization. Figure 4 a shows the polarization azimuth of four sidelobes and the energy scattering distribution in the far-field, and Fig. 4(e) and (i) are the simulation results of *β* and AR at the plane of *φ *= 0° and *φ* = 90° respectively. Then, setting rot_{21} = 45°, 90°, and 135° respectively, the corresponding simulation results are shown in Fig. 4. With the increase of rot_{21}, the linear polarization of each sidelobe can achieve 360° rotation. Hence, arbitrary modulation of *β* of the first sidelobe is achieved.

Taking the case in Fig. 4(b) as a testing prototype to verify the scheme, a 108×108 mm^{2} sample of the proposed enantiomers metasurface is fabricated, as shown in Fig. 5(a). The experiment is carried out in a microwave anechoic chamber, as Fig. 5(b) shows. At the frequency of 13 GHz, the normalized measured results are given in Fig. 5(c) and (d). These results indicate that the energy is evenly distributed in the four side lobes. At *φ* = 0°, the enantiomers metasurfaces radiate 45° polarized wave in the direction of *θ* = ±40°; and at *φ* = 90°, the 135° polarized wave are obtained in the direction of *θ* = ±40°. In terms of the radiation energy and the synthesized polarization, the measured results are consistent with numerical ones, verifying the feasibility of the scheme and meta-enantiomer.

## 4. Vectorial beam customization via mixed enantiomers

The above discusses the FPP manipulation of specific directions basing on theoretical derivation, and achieving multi-vector-beams independent modulation in arbitrary directions is more meaningful for practical applications. Based on the principle of backward propagation, the amplitude and phase of the target field can be realized by the meta-enantiomers with both phase and amplitude modulations [40–43]. However, if the electromagnetic wave's modulation dimension is only polarization and phase profile, the backward propagation no longer works, and utilizing the optimized algorithm is a feasible way to control the polarization of K-space [14]. The optimization algorithms for the issue of optimal solution can be divided into Gradient free optimization algorithm and Gradient-based optimization algorithm. The typical representatives are the genetic algorithm (GA) [44,45] and L-BFGS-B [46–48]. In this paper, enantiomers-based multi-vector-beams independent modulation would be realized via combining GA and L-BFGS-B.

Here, each element on the metasurface is assigned a four-digit binary number. From the highest bit to the lowest, for the first digit, 0 means L-enantiomer, and 1 means R-enantiomer. The last three(L-3) digits indicate that the phase takes an arithmetic sequence in 360°. The table in Fig. 6 shows the specific encoding. According to the far-field superposition formula, the far-field pattern can be expressed as:

*ν*

_{n,m,}and

*σ*

_{n,m}are the first and the L-3 bits of the binary number of the element(n,m) respectively; ${X_{n,m}}$ and ${Y_{n,m}}$ are the x coordinate and y coordinate value of the center of each element;

*k*is wave number. In this way, distributions of the energy and the polarization in the K-space are obtained. Normalizing the energy, the normalized meta-surface-modulation ${{\boldsymbol E}_x}$ and ${{\boldsymbol E}_y}$ in the whole space are obtained. A polar coordinate system that can characterize energy and the AR is established, shown on the right side of Fig. 6. The polar diameter is $\sqrt {{\boldsymbol E}_x^2 + {\boldsymbol E}_y^2} $, and the polar angle is (The Latitude of the polarization on PS). ${{\boldsymbol E}^c}$ and ${{\boldsymbol \gamma }^c}$ are the polar diameter normalized and ${\boldsymbol \gamma } = 2atan\frac{1}{{{\boldsymbol AR}}}$ polar angle of the customized. The Far-field pattern of the meta-surface and customization can be characterized in the polar coordinate. In this way, the square of Euclidean distance between these two vectors can be used to consist of the figure of merit (FOM):

*ν*of each element is determined by GA, and the phase profile optimized by GA is set as the initial values of the gradient-based optimization algorithms to solve the Local optimization problem. The FOM is Eq. (7). Simultaneous, the ν

_{n,m}in the construction field (Eq. (6)) is no longer a variable, and the phase profile becomes the only variable that needs to be optimized. Through iterations, this method can quickly converge to a nearby extremum. Figure 6 shows the whole optimization process. Overall, combining GA and L-BFGS-B and based on CD enantiomers’ synergy, the customization of energy and polarization in K-space is realized.

As an example, letting a *y*-pol plane wave illuminate the meta-surface, we customize the far-field energy and polarization at 13 GHz. We customize an LCP beam at *θ* = 40°, *φ* = 90°, an RCP beam at *θ* = 30°, *φ* = 10°, a linear polarization beam at *θ* = 0°, and a 5-axial-ratio left-spin-elliptical beam at *θ* = 30°, *φ* = 180°. And set their energy ratio to 7:8:9:10. We selected 15×15 elements (each one contains 2×2 identical enantiomers) with an area of 180×180 mm^{2}. This aperture size can guarantee sufficient variables in the subsequent optimization and restrict the full-wave simulation and fabrication costs. Q=60 sampling points in the space, including the above four directions, are selected to characterize the customized far-field. Here, the annealing algorithm distributes the remaining 56 sampling points in the half-space evenly, guaranteeing the objective function converges successfully. There are 500 variables (225 chiral variables and 225 phase variables) for the GA. Randomly setting 2000 sets of initial values, we get the optimized chiral unit distribution and phase profile. Then, keep the arrangement of enantiomers unchanged and assigned the phase profile to the quasi-Newton genetic algorithm as initial values for iterative optimization. The number of optimized variables reduces to 100, and the algorithm will quickly find the optimal solution. Each iteration requires the values and gradients of FOM simultaneously and approximates the second derivatives by a limited memory matrix. Next, it defines a quadratic model of the objective function and computes the search direction by approximating the minimizer of the quadratic model.

According to the above, meta-surface with 180×180 mm^{2} diameter is modeled in the CST microwave studio. Setting the Open add space boundary conditions and a *y*-pol plane wave. The simulated energy distribution of RCP and LCP components of the synthetic field in K-space at 13 GHz are shown in Fig. 7 a and b. The synthetic field energy is shown in Fig. 7(c), and their energy ratio also meets the customization requirements (the energy ratio between is 7:8:9:10). The energy of some other direction is suppressed heavily. Simultaneously, the AR of each beam is consistent with the customized as shown in Fig. 7(d). Besides, we have also simulated the results which without using the L-BFGS-B algorithm. (Note S7, Supporting Information) From the results, that the beam energy is not concentrated and the clutter is obvious, which illustrates the necessity of the combination of GA and L-BFGS-B. Overall, this method has realized the high-quality customization of energy and polarization in K-space.

## 5. Conclusion

This paper presents a customization scheme of energy and polarization in K-space based on CD enantiomers. Compared with anisotropic metasurfaces, our scheme can modulate arbitrary vectorial beams in K-space without sophisticated structure searching of massive geometric parameters. Via embedding a lumped resistor in the chiral metal structure and symmetrically copying this whole structure, a pair of CD enantiomers is obtained. The customized FPP modulation in specific directions is realized via the checkerboard arrangement of the enantiomers. The simulation results and experimental tests verify the correctness of the scheme. Combines GA and gradient optimization algorithm to optimize the chirality distribution and phase profile on the entire meta-surface. The co-aperture independent modulation of multi-vector beams is realized, which has been successfully verified by Full-wave simulation in the CST. It has reference significance for the tunable metasurface to control far-field radiation flexibly and has broad application prospects in many fields such as the multi-polarized distribution of wireless communications, polarization encryption, information storage, and so on.

**Experimental section:** The sample was fabricated with a standard printed-circuit-board (PCB) and surface mount device (SMD) resistance soldering technology. In our Far-field experiments, the fabricated sample is placed at the center of the revolving stage. A linear polarized horn antenna serves as the transmitter is placed at the front of the sample with a distance of 1.5 m, rotating with the revolving stage. The *θ* can be adjusted by rotating the transmission horn and metasurface. Anther linear polarized horn antenna serves as the receiver is fixed on the other side of the microwave anechoic chamber 7 m away from the sample to receive the 45° or 135° polarized wave (*φ *= 45° or 135°). The transmission and the receiving antennas are connected to an Agilent vector network analyzer (5230C).

## Funding

the National Science Foundation for Post-doctoral Scientists of China (2019M651644); National Natural Science Foundation of China (61971435, 61971437).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

## Supplemental document

See Supplement 1 for supporting content.

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