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 [16]. Hence, which is essential in structure sensing [79], information encryption [10], remote detection [11], hologram [1217], 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 [14,1928]. 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 [2840] 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.

Tables Icon

Table 1. Reported works on vectorial wave shaping.

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 (ɛr=2.65 and loss tangent tanδ=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.

 figure: Fig. 1.

Fig. 1. Schematic diagram of the functions of CD-enantiomers-based metasurface. The CD enantiomers: L-atom and R-atom are in the upper right corner of the figure. CD-enantiomers are rotated and coded into the corresponding position according to the requirements of vector beam customization to constitute metasurface. When the linear polarization strikes the metasurface, four beams with left-handed, right-handed, linear, and 5-axial-ratio left-handed-elliptic polarizations with the energy ratio of 7:8:9:10 are achieved.

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 figure: Fig. 2.

Fig. 2. Meta-enantiomer structure, simulation results, and checkerboard type arrangement diagram. (a) Illustration of R-enantiomer, where period p=6mm, width w=0.78mm, thickness h=1.5mm. (b) S-parameters curves of R-enantiomer from 10 GHz to 18GHz. (c) At 12GHz, the amplitude and phase response of cross-polarization of L-enantiomer (box) and R-enantiomer (triangle) under different rotate angles. (d) The FPP manipulation by checkerboard enantiomers. Polarization azimuth is modulated by rot31 and rot42, and the axial ratio is modulated by rot21. EA is the central angle between the simulation and theoretical polarization on the Poincaré Sphere. (e) Schematic of checkerboard type arrangement of the enantiomers. (f) rot1 and rot3 are the rotation angles from the x-axis's negative direction to the spiral's tail. Counterclockwise rotation is positive, and vice versa. rot2 and rot4 are the rotation angles from the x-axis's positive direction to the spiral's tail, clockwise rotation is positive and negative, and vice versa.

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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 ER, EL, and αL, αR, respectively. Converting these to the coordinate (S1, S2, S3) in the Stokes-Cartesian coordinate system, it can be deduced:

$$\beta = \frac{{{\alpha _R} - {\alpha _L}}}{2}, AR = \tan (0.5\ast \textrm{arcsin(}\frac{{{E_R}^2 - {E_L}^2}}{{E_R^2 + E_L^2}})),$$

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 roti of the enantiomer in the i-th quadrant. When a y-pol wave illuminates this array, the normalized superimposed Farfield can be expressed as

$${{\boldsymbol E}_r} = \left[ {\begin{array}{cc} {\sum\nolimits_{n = 1}^4 {R_{LL}^n{e^{ - i{\boldsymbol k} \cdot {{\boldsymbol r}_n}}}} }&{\sum\nolimits_{n = 1}^4 {R_{LR}^n{e^{ - i{\boldsymbol k} \cdot {{\boldsymbol r}_n}}}} }\\ {\sum\nolimits_{n = 1}^4 {R_{RL}^n{e^{ - i{\boldsymbol k} \cdot {{\boldsymbol r}_n}}}} }&{\sum\nolimits_{n = 1}^4 {R_{RR}^n{e^{ - i{\boldsymbol k} \cdot {{\boldsymbol r}_n}}}} } \end{array}} \right]\left[ {\begin{array}{c} {{\boldsymbol E}_i^L}\\ {{\boldsymbol E}_i^R} \end{array}} \right], $$
where, $R_{LL({RR} )}^n\; $ and $R_{LR({LR} )}^n$ are the co- and cross-polarization reflection coefficients of the 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 Xn=[1.5p, -1.5p, -1.5p, 1.5p] and Yn=[1.5p, 1.5p, -1.5p, -1.5p]. 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
$${{\boldsymbol E}_r} = {e^{i{\alpha _1}}}\left[ {\begin{array}{cc} {\frac{{i{\boldsymbol x} + {\boldsymbol y}}}{{\sqrt 2 }}}&{\frac{{ - i{\boldsymbol x} + {\boldsymbol y}}}{{\sqrt 2 }}} \end{array}} \right]\left[ {\begin{array}{c} {{e^{ - i{\delta_1}}} + {e^{i({\delta_1} + {\alpha_{31}})}}}\\ {{e^{i({\delta_2} + {\alpha_{21}})}} + {e^{i( - {\delta_2} + {\alpha_{42}} + {\alpha_{21}})}}} \end{array}} \right], $$

This formula represents the electric field superposition of LCP and RCP waves in the whole space. In which αn is the phase of the n-th quadrant and αmn represents the phase difference of the m-th and the n-th quadrant (αm-αn). According to formula 3, the phase of a synthetic polarization beam could be regulated by α1, and the phase and amplitude of LCP and RCP of the synthetic polarization are shown as:

$${\alpha _L} - {\alpha _R} = \frac{{2{\alpha _{21}} + {\alpha _{42}} - {\alpha _{31}}}}{\textrm{2}}, $$
$$\frac{{{E_L}}}{{{E_R}}} = \frac{{\sqrt {{{[{1 + \cos (2{\delta_1} + {\alpha_{31}})} ]}^2} + {{\sin }^2}(2{\delta _1} + {\alpha _{31}})} }}{{\sqrt {{{[{1 + \cos (2{\delta_2} + {\alpha_{42}})} ]}^2} + {{\sin }^2}(2{\delta _2} + {\alpha _{42}})} }}, $$

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 rot31 and rot42, and the polarization azimuth is manipulated by rot21, 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 mm2) 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: Fig. 3.

Fig. 3. The β and AR simulation results of customized FPP at φ=0°, -15°<θ<15°. The purple axis is the y-axis, and the green axis is the x-axis. (a), (b), (c), and (d) the linear polarization with β = 0°, 45°, 90°, and 135° respectively. (e), (f), (g), and (h) the simulation results of 3-AR left-spin-elliptic polarization with β = 0°, 45°, 90°, and 135° respectively.

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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(α13 = 180°, α24 = 0°) and RCP (α13 = 0°, α24 = 180°) also verify the feasibility of the scheme(Note S5, Supporting Information).

When phases difference α13 and α24 satisfy α13=α24=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 α21 from 0 to 180°. For verification, making rot21=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 rot21 = 45°, 90°, and 135° respectively, the corresponding simulation results are shown in Fig. 4. With the increase of rot21, the linear polarization of each sidelobe can achieve 360° rotation. Hence, arbitrary modulation of β of the first sidelobe is achieved.

 figure: Fig. 4.

Fig. 4. The simulation results of polarization azimuth of four side lobes (φ = 0° or 90°, θ = ± 40°). The purple axis is the y-axis, and the green axis is the x-axis. Figure 4. (a-d) show energy is concentratively distributed in four sidelobes. Figure 4. (e-h) are the β and AR simulation results at φ = 0°. Figure 4(i)-(l)) are the β and AR simulation results at φ = 90°.

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Taking the case in Fig. 4(b) as a testing prototype to verify the scheme, a 108×108 mm2 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.

 figure: Fig. 5.

Fig. 5. The configuration for measurements and the experimental results. (a) Photograph of 108×108 mm2 enantiomers metasurface. (b) Photographs of the measure configuration. The transmitter emits the y-pol wave; the revolving stage can change θ; the receiver with an oblique angle receives a corresponding polarization wave. (c) Measured normalized two-dimensional scattering patterns in the plane of φ = 0°. (d) Measured normalized two-dimensional scattering patterns in the plane of φ = 90°.

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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 [4043]. 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 [4648]. 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:

$${\boldsymbol E}(\theta ,\varphi )\textrm{ = }\sum\limits_{m = 1}^M {\sum\limits_{n = 1}^N {[{{\boldsymbol x} + i{{(\textrm{ - 1})}^{{\nu_{n,m}}}}y} ]} } {e^{i[\frac{{\pi {\sigma _{n,m}}}}{4} - k(\sin \theta \cos \varphi {X_{n,m}} + \sin \theta \sin \varphi {Y_{n,m}})]}}, $$
where νn,m, and σn,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):
$$\varepsilon = \sum\limits_{q = 1}^Q {\left\{ {({E_{xq}^2 + E_{yq}^2} ){{\sin }^2}({\gamma_q} - \gamma_q^c) + {{\left[ {\sqrt {(E_{xq}^2 + E_{yq}^2)} \cos ({\gamma_q} - \gamma_q^c) - E_q^c} \right]}^2}} \right\}}, $$
where Q is the total number of the sampling point, and q is the number of sampling points. First, through the annealing algorithm, we evenly distribute the sampling points in the half-space to facilitate the optimization algorithm to approach the optimal solution. Next, we can set an appropriate fitness function and enter a series of initial values, and find the global approximate optimal solution. When evolution is slow, the approximate optimal solution is obtained, which contains the arrangement of the R- and L-enantiomers and the phase profile. In general, there are often optimal solutions near the optimized coding-array. Hence, gradient-based optimization algorithms are necessary. (The rationality of this optimization algorithm combination is analyzed in the Supplement 1) Here, the chirality 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 νn,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.

 figure: Fig. 6.

Fig. 6. Schematic diagram of the encoding, spatial sampling, the optimal process, objective function construction, and the combination of GA and quasi-Newton algorithm.

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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 mm2. 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 mm2 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.

 figure: Fig. 7.

Fig. 7. Simulation results of polarization and normalized amplitude in K-space of GA & L-BFGS-B based enantiomers metasurface. (a), (b), and (c) K-space normalized amplitude distribution of the RCP, the LCP, and the synthetic polarization. (d) Axial ratio distribution of synthetic polarization in K-space.

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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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11. F. Yue, V. Aglieri, R. Piccoli, R. Macaluso, A. Toma, R. Morandotti, and L. Razzari, “Highly Sensitive Polarization Rotation Measurement through a High-Order Vector Beam Generated by a Metasurface,” Adv. Mater. Technol. 5, 1901008 (2020). [CrossRef]  

12. Z. L. Deng, J. Deng, X. Zhuang, S. Wang, K. Li, Y. Wang, Y. Chi, X. Ye, J. Xu, G. P. Wang, R. Zhao, X. Wang, Y. Cao, X. Cheng, G. Li, and X. Li, “Diatomic Metasurface for Vectorial Holography,” Nano Lett. 18(5), 2885–2892 (2018). [CrossRef]  

13. Z. L. Deng, M. Jin, X. Ye, S. Wang, T. Shi, J. Deng, N. Mao, Y. Cao, B. O. Guan, A. Alù, G. Li, and X. Li, “Full-Color Complex-Amplitude Vectorial Holograms Based on Multi-Freedom Metasurfaces,” Adv. Funct. Mater. 30(21), 1910610 (2020). [CrossRef]  

14. J. W. Wu, Z. X. Wang, Z. Q. Fang, J. C. Liang, X. Fu, J. F. Liu, H. T. Wu, D. Bao, L. Miao, X. Y. Zhou, Q. Cheng, and T. J. Cui, “Full-State Synthesis of Electromagnetic Fields using High Efficiency Phase-Only Metasurfaces,” Adv. Funct. Mater. 30(39), 2004144 (2020). [CrossRef]  

15. Q. Wang, E. Plum, Q. Yang, X. Zhang, Q. Xu, Y. Xu, J. G. Han, and W. L. Zhang, “Reflective chiral meta-holography: multiplexing holograms for circularly polarized waves,” Light Sci Appl 7(1), 25 (2018). [CrossRef]  

16. D. Wen, F. Yue, G. Li, G. Zheng, K. Chan, S. Chen, M. Chen, K. Li, K. W. Cheah, E. Y. B. Pun, S. Zhang, and X. Chen, “Helicity multiplexed broadband metasurface holograms,” Nat. Commun 6(1), 8241 (2015). [CrossRef]  

17. Q. Song, A. Baroni, R. Sawant, P. Ni, V. Brandli, S. Chenot, S. Vézian, B. Damilano, P. Mierry, S. Khadir, P. Ferrand, and P. Genevet, “Ptychography retrieval of fully polarized holograms from geometric-phase metasurfaces,” Nat Commun 11(1), 2651 (2020). [CrossRef]  

18. R. C. Devlin, A. Ambrosio, D. Wintz, S. L. Oscurato, A. Y. Zhu, M. Khorasaninejad, J. Oh, Pasqualino. Maddalena, and Federico. Capasso, “Spin-to-orbital angular momentum conversion in dielectric metasurfaces,” Opt. Express 25(1), 377–379 (2017). [CrossRef]  

19. Y. Huang, N. Rubin, A. Ambrosio, Z. Shi, R. C. Devlin, C. Qiu, and F. Capasso, “Versatile total angular momentum generation using cascaded J-plates,” Opt. Express 27(5), 7469–7484 (2019). [CrossRef]  

20. N. A. Rubin, G. D’Aversa, P. Chevalier, Z. Shi, W. Chen, and F. Capasso, “Matrix Fourier optics enables a compact full-Stokes polarization camera,” Science 365(6448), 1839 (2019). [CrossRef]  

21. N. A. Rubin, A. Zaidi, M. Juhl, R. P. Li, J. P. B. Mueller, R. C. Devlin, K. Leósson, and F. Capasso, “Polarization state generation and measurement with a single metasurface,” Opt. Express 26(17), 21455–21478 (2018). [CrossRef]  

22. L. L. Huang, X. Z. Chen, H. Mühlenbernd, Z. Hao, S. Chen, B. Bai, Q. F. Tan, G. F. Jin, KW. Cheah, C. Qiu, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nat. Commun. 4(1), 2808 (2013). [CrossRef]  

23. S. M. Kamali, E. Arbabi, A. Arbabi, and A. Faraon, “A review of dielectric optical metasurfaces for wavefront control,” Nanophotonics. 7(6), 1041–1068 (2018). [CrossRef]  

24. D. Wang, F. Liu, T. Liu, S. L. Sun, and L. Zhou, “Efficient generation of complex vectorial optical fields with metasurfaces,” Light Sci Appl 10(1), 67 (2021). [CrossRef]  

25. S. L. Sun, Q. He, S. Y. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012). [CrossRef]  

26. A. Arbabi, Y. Horie, A. Ball, M. Bagheri, and A. Faraon, “Subwavelength-thick lenses with high numerical apertures and large efficiency based on high-contrast transmitarrays,” Nat Commun 6(1), 7069 (2015). [CrossRef]  

27. K. Zhang, Y. Wang, S. N. Burokur, and Q. Wu, “Generating Dual-Polarized Vortex Beam by Detour Phase: From Phase Gradient Metasurfaces to Metagratings,” Transactions on Microwave Theory and Techniques. (2021). [CrossRef]  

28. K Zhang, Y. Y. Yuan, X. M. Ding, H. Y. Li, B. Ratni, Q. Wu, J. Liu, S. N. Burokur, and J. B. Tan, “Polarization-Engineered Noninterleaved Metasurface for Integer and Fractional Orbital Angular Momentum Multiplexing,” Laser Photonics Rev. 15(1), 2000351 (2021). [CrossRef]  

29. B. H. Li, X. W. Li, R. Z. Zhao, G. C. Wang, W. N. Han, B. Q. Zhao, L. L. Huang, Y. Zhang, Y. F. Lu, and L. Jiang, “Polarization Multiplexing Terahertz Metasurfaces through Spatial Femtosecond Laser-Shaping Fabrication,” Adv. Opt. Mater. 8(12), 2000136 (2020). [CrossRef]  

30. S. Beychok, “Circular dichroism of biological macromolecules,” Science 154(3754), 1288–1299 (1966). [CrossRef]  

31. A. L. Rucker and T. P. Creamer, “Polyproline II helical structure in protein unfolded states: lysine peptides revisited,” Protein science. 11, 980–985 (2002). [CrossRef]  

32. Z. Liu, Y. Xu, C.-Y. Ji, S. Chen, X. Li, X. Zhang, Y. Yao, and J. Li, “Fano-Enhanced Circular Dichroism in Deformable Stereo Metasurfaces,” Adv. Mater. 32(8), 1907077 (2020). [CrossRef]  

33. M. Qiu, L. Zhang, Z. Tang, W. Jin, C. W. Qiu, and D. Y. Lei, “3D Metaphotonic Nanostructures with Intrinsic Chirality,” Adv. Funct. Mater. 28(45), 1803147 (2018). [CrossRef]  

34. J. T. Collins, C. Kuppe, D. C. Hooper, C. Sibilia, M. Centini, and V. K. Valev, “Chirality and Chiroptical Effects in Metal Nanostructures: Fundamentals and Current Trends,” Advanced Optical Materials 5(16), 1700182 (2017). [CrossRef]  

35. H. Kazemi, M. Albooyeh, and F. Capolino, “Simultaneous Perfect Bending and Polarization Rotation of Electromagnetic Wavefront Using Chiral Gradient Metasurfaces,” Phys. Rev. Applied 13(2), 024078 (2020). [CrossRef]  

36. C. Menzel, C. Rockstuhl, and F. Lederer, “Advanced Jones calculus for the classification of periodic metamaterials,” Phys. Rev. A 82(5), 053811 (2010). [CrossRef]  

37. J. Li, Y. T. Zhang, J. N. Li, X. Yan, L. J. Liang, Z. Zhang, J. Huang, J. H. Li, Y. Yang, and J. Q. Yao, “Amplitude modulation of anomalously reflected terahertz beams using all-optical active Pancharatnam-Berry coding metasurfaces,” Nanoscale 11(12), 5746–5753 (2019). [CrossRef]  

38. M. Amin, O. Siddiqui, and M. Farhat, “Linear and Circular Dichroism in Graphene-Based Reflectors for Polarization Control,” Phys. Rev. Applied 13(2), 024046 (2020). [CrossRef]  

39. X. J. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nat Commun 4(1), 2807 (2013). [CrossRef]  

40. Q. Wang, X. Zhang, Y. Xu, J. Gu, Y. Li, Z. Tian, R. Singh, S. Zhang, J. G. Han, and W. L. Zhang, “Broadband metasurface holograms: toward complete phase and amplitude engineering,” Sci. Rep. 6(1), 32867 (2016). [CrossRef]  

41. G. Y. Lee, G. Yoon, S. Y. Lee, H. Yun, J. Cho, K. Lee, H. Kim, J. Rho, and B. Lee, “Complete amplitude and phase control of light using broadband holographic metasurfaces,” Nanoscale 10(9), 4237–4245 (2018). [CrossRef]  

42. A. C. Overvig, S. Shrestha, S. C. Malek, M. Lu, A. Stein, C. Zheng, and N. F. Yu, “Dielectric metasurfaces for complete and independent control of the optical amplitude and phase,” Light: Sci. Appl. 8(1), 92 (2019). [CrossRef]  

43. D. Gies and Y. Rahmat-Samii, “Particle swarm optimization for reconfigurable phase-differentiated array design,” Microw. Opt. Technol. Lett. 38(3), 168–175 (2003). [CrossRef]  

44. A. Konak, D. W. Coit, and A. E. Smith, “Multi-objective optimization using genetic algorithms: A tutorial. Reliability Engineering and System Safety,” Reliability Engineering & System Safety 91(9), 992–1007 (2006). [CrossRef]  

45. D. Whitley, “A genetic algorithm tutorial,” Stat Comput 4(2), 65–85 (1994). [CrossRef]  

46. R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A limited-memory algorithm for bound constrained optimization,” SIAM J. Sci. Comput. 16(5), 1190–1208 (1995). [CrossRef]  

47. C. Y. Zhu, R. Byrd, P. H. Lu, and J. C. Nocedal, “Algorithm 778: L-BFGS-B: Fortran Subroutines for Large-Scale Bound-Constrained Optimization,” ACM Trans. Math. Softw. 23(4), 550–560 (1997). [CrossRef]  

48. J. Arora and A. S. Jan, “Practical Mathematical Optimization: An introduction to basic optimization theory and classical and new gradient-based algorithms,” Struct Multidisc Optim 31(3), 249 (2006). [CrossRef]  

References

  • View by:

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  2. A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Simultaneous and Complete Control of Light Polarization and Phase using High Contrast Transmitarrays,” Nat. Nanotechnol. 10(11), 937–943 (2015).
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  11. F. Yue, V. Aglieri, R. Piccoli, R. Macaluso, A. Toma, R. Morandotti, and L. Razzari, “Highly Sensitive Polarization Rotation Measurement through a High-Order Vector Beam Generated by a Metasurface,” Adv. Mater. Technol. 5, 1901008 (2020).
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  15. Q. Wang, E. Plum, Q. Yang, X. Zhang, Q. Xu, Y. Xu, J. G. Han, and W. L. Zhang, “Reflective chiral meta-holography: multiplexing holograms for circularly polarized waves,” Light Sci Appl 7(1), 25 (2018).
    [Crossref]
  16. D. Wen, F. Yue, G. Li, G. Zheng, K. Chan, S. Chen, M. Chen, K. Li, K. W. Cheah, E. Y. B. Pun, S. Zhang, and X. Chen, “Helicity multiplexed broadband metasurface holograms,” Nat. Commun 6(1), 8241 (2015).
    [Crossref]
  17. Q. Song, A. Baroni, R. Sawant, P. Ni, V. Brandli, S. Chenot, S. Vézian, B. Damilano, P. Mierry, S. Khadir, P. Ferrand, and P. Genevet, “Ptychography retrieval of fully polarized holograms from geometric-phase metasurfaces,” Nat Commun 11(1), 2651 (2020).
    [Crossref]
  18. R. C. Devlin, A. Ambrosio, D. Wintz, S. L. Oscurato, A. Y. Zhu, M. Khorasaninejad, J. Oh, Pasqualino. Maddalena, and Federico. Capasso, “Spin-to-orbital angular momentum conversion in dielectric metasurfaces,” Opt. Express 25(1), 377–379 (2017).
    [Crossref]
  19. Y. Huang, N. Rubin, A. Ambrosio, Z. Shi, R. C. Devlin, C. Qiu, and F. Capasso, “Versatile total angular momentum generation using cascaded J-plates,” Opt. Express 27(5), 7469–7484 (2019).
    [Crossref]
  20. N. A. Rubin, G. D’Aversa, P. Chevalier, Z. Shi, W. Chen, and F. Capasso, “Matrix Fourier optics enables a compact full-Stokes polarization camera,” Science 365(6448), 1839 (2019).
    [Crossref]
  21. N. A. Rubin, A. Zaidi, M. Juhl, R. P. Li, J. P. B. Mueller, R. C. Devlin, K. Leósson, and F. Capasso, “Polarization state generation and measurement with a single metasurface,” Opt. Express 26(17), 21455–21478 (2018).
    [Crossref]
  22. L. L. Huang, X. Z. Chen, H. Mühlenbernd, Z. Hao, S. Chen, B. Bai, Q. F. Tan, G. F. Jin, KW. Cheah, C. Qiu, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nat. Commun. 4(1), 2808 (2013).
    [Crossref]
  23. S. M. Kamali, E. Arbabi, A. Arbabi, and A. Faraon, “A review of dielectric optical metasurfaces for wavefront control,” Nanophotonics. 7(6), 1041–1068 (2018).
    [Crossref]
  24. D. Wang, F. Liu, T. Liu, S. L. Sun, and L. Zhou, “Efficient generation of complex vectorial optical fields with metasurfaces,” Light Sci Appl 10(1), 67 (2021).
    [Crossref]
  25. S. L. Sun, Q. He, S. Y. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
    [Crossref]
  26. A. Arbabi, Y. Horie, A. Ball, M. Bagheri, and A. Faraon, “Subwavelength-thick lenses with high numerical apertures and large efficiency based on high-contrast transmitarrays,” Nat Commun 6(1), 7069 (2015).
    [Crossref]
  27. K. Zhang, Y. Wang, S. N. Burokur, and Q. Wu, “Generating Dual-Polarized Vortex Beam by Detour Phase: From Phase Gradient Metasurfaces to Metagratings,” Transactions on Microwave Theory and Techniques. (2021).
    [Crossref]
  28. K Zhang, Y. Y. Yuan, X. M. Ding, H. Y. Li, B. Ratni, Q. Wu, J. Liu, S. N. Burokur, and J. B. Tan, “Polarization-Engineered Noninterleaved Metasurface for Integer and Fractional Orbital Angular Momentum Multiplexing,” Laser Photonics Rev. 15(1), 2000351 (2021).
    [Crossref]
  29. B. H. Li, X. W. Li, R. Z. Zhao, G. C. Wang, W. N. Han, B. Q. Zhao, L. L. Huang, Y. Zhang, Y. F. Lu, and L. Jiang, “Polarization Multiplexing Terahertz Metasurfaces through Spatial Femtosecond Laser-Shaping Fabrication,” Adv. Opt. Mater. 8(12), 2000136 (2020).
    [Crossref]
  30. S. Beychok, “Circular dichroism of biological macromolecules,” Science 154(3754), 1288–1299 (1966).
    [Crossref]
  31. A. L. Rucker and T. P. Creamer, “Polyproline II helical structure in protein unfolded states: lysine peptides revisited,” Protein science. 11, 980–985 (2002).
    [Crossref]
  32. Z. Liu, Y. Xu, C.-Y. Ji, S. Chen, X. Li, X. Zhang, Y. Yao, and J. Li, “Fano-Enhanced Circular Dichroism in Deformable Stereo Metasurfaces,” Adv. Mater. 32(8), 1907077 (2020).
    [Crossref]
  33. M. Qiu, L. Zhang, Z. Tang, W. Jin, C. W. Qiu, and D. Y. Lei, “3D Metaphotonic Nanostructures with Intrinsic Chirality,” Adv. Funct. Mater. 28(45), 1803147 (2018).
    [Crossref]
  34. J. T. Collins, C. Kuppe, D. C. Hooper, C. Sibilia, M. Centini, and V. K. Valev, “Chirality and Chiroptical Effects in Metal Nanostructures: Fundamentals and Current Trends,” Advanced Optical Materials 5(16), 1700182 (2017).
    [Crossref]
  35. H. Kazemi, M. Albooyeh, and F. Capolino, “Simultaneous Perfect Bending and Polarization Rotation of Electromagnetic Wavefront Using Chiral Gradient Metasurfaces,” Phys. Rev. Applied 13(2), 024078 (2020).
    [Crossref]
  36. C. Menzel, C. Rockstuhl, and F. Lederer, “Advanced Jones calculus for the classification of periodic metamaterials,” Phys. Rev. A 82(5), 053811 (2010).
    [Crossref]
  37. J. Li, Y. T. Zhang, J. N. Li, X. Yan, L. J. Liang, Z. Zhang, J. Huang, J. H. Li, Y. Yang, and J. Q. Yao, “Amplitude modulation of anomalously reflected terahertz beams using all-optical active Pancharatnam-Berry coding metasurfaces,” Nanoscale 11(12), 5746–5753 (2019).
    [Crossref]
  38. M. Amin, O. Siddiqui, and M. Farhat, “Linear and Circular Dichroism in Graphene-Based Reflectors for Polarization Control,” Phys. Rev. Applied 13(2), 024046 (2020).
    [Crossref]
  39. X. J. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nat Commun 4(1), 2807 (2013).
    [Crossref]
  40. Q. Wang, X. Zhang, Y. Xu, J. Gu, Y. Li, Z. Tian, R. Singh, S. Zhang, J. G. Han, and W. L. Zhang, “Broadband metasurface holograms: toward complete phase and amplitude engineering,” Sci. Rep. 6(1), 32867 (2016).
    [Crossref]
  41. G. Y. Lee, G. Yoon, S. Y. Lee, H. Yun, J. Cho, K. Lee, H. Kim, J. Rho, and B. Lee, “Complete amplitude and phase control of light using broadband holographic metasurfaces,” Nanoscale 10(9), 4237–4245 (2018).
    [Crossref]
  42. A. C. Overvig, S. Shrestha, S. C. Malek, M. Lu, A. Stein, C. Zheng, and N. F. Yu, “Dielectric metasurfaces for complete and independent control of the optical amplitude and phase,” Light: Sci. Appl. 8(1), 92 (2019).
    [Crossref]
  43. D. Gies and Y. Rahmat-Samii, “Particle swarm optimization for reconfigurable phase-differentiated array design,” Microw. Opt. Technol. Lett. 38(3), 168–175 (2003).
    [Crossref]
  44. A. Konak, D. W. Coit, and A. E. Smith, “Multi-objective optimization using genetic algorithms: A tutorial. Reliability Engineering and System Safety,” Reliability Engineering & System Safety 91(9), 992–1007 (2006).
    [Crossref]
  45. D. Whitley, “A genetic algorithm tutorial,” Stat Comput 4(2), 65–85 (1994).
    [Crossref]
  46. R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A limited-memory algorithm for bound constrained optimization,” SIAM J. Sci. Comput. 16(5), 1190–1208 (1995).
    [Crossref]
  47. C. Y. Zhu, R. Byrd, P. H. Lu, and J. C. Nocedal, “Algorithm 778: L-BFGS-B: Fortran Subroutines for Large-Scale Bound-Constrained Optimization,” ACM Trans. Math. Softw. 23(4), 550–560 (1997).
    [Crossref]
  48. J. Arora and A. S. Jan, “Practical Mathematical Optimization: An introduction to basic optimization theory and classical and new gradient-based algorithms,” Struct Multidisc Optim 31(3), 249 (2006).
    [Crossref]

2021 (2)

D. Wang, F. Liu, T. Liu, S. L. Sun, and L. Zhou, “Efficient generation of complex vectorial optical fields with metasurfaces,” Light Sci Appl 10(1), 67 (2021).
[Crossref]

K Zhang, Y. Y. Yuan, X. M. Ding, H. Y. Li, B. Ratni, Q. Wu, J. Liu, S. N. Burokur, and J. B. Tan, “Polarization-Engineered Noninterleaved Metasurface for Integer and Fractional Orbital Angular Momentum Multiplexing,” Laser Photonics Rev. 15(1), 2000351 (2021).
[Crossref]

2020 (11)

B. H. Li, X. W. Li, R. Z. Zhao, G. C. Wang, W. N. Han, B. Q. Zhao, L. L. Huang, Y. Zhang, Y. F. Lu, and L. Jiang, “Polarization Multiplexing Terahertz Metasurfaces through Spatial Femtosecond Laser-Shaping Fabrication,” Adv. Opt. Mater. 8(12), 2000136 (2020).
[Crossref]

Z. Liu, Y. Xu, C.-Y. Ji, S. Chen, X. Li, X. Zhang, Y. Yao, and J. Li, “Fano-Enhanced Circular Dichroism in Deformable Stereo Metasurfaces,” Adv. Mater. 32(8), 1907077 (2020).
[Crossref]

H. Kazemi, M. Albooyeh, and F. Capolino, “Simultaneous Perfect Bending and Polarization Rotation of Electromagnetic Wavefront Using Chiral Gradient Metasurfaces,” Phys. Rev. Applied 13(2), 024078 (2020).
[Crossref]

Y. Y. Yuan, K. Zhang, B. Ratni, Q. H. Song, X. M. Ding, and Q. Wu, “Independent phase modulation for quadruplex polarization channels enabled by chirality-assisted geometric-phase metasurfaces,” Nat. Commun. 11(1), 4186 (2020).
[Crossref]

Z. J. Shi, Alexander. Y. Zhu, Z. Y. Li, Y. W. Huang, W. T. Chen, C. W. Qiu, and F. Capasso, “Continuous angle-tunable birefringence with freeform metasurfaces for arbitrary polarization conversion,” Sci. Adv. 6, 23 (2020).
[Crossref]

Z. L. Deng, M. Jin, X. Ye, S. Wang, T. Shi, J. Deng, N. Mao, Y. Cao, B. O. Guan, A. Alù, G. Li, and X. Li, “Full-Color Complex-Amplitude Vectorial Holograms Based on Multi-Freedom Metasurfaces,” Adv. Funct. Mater. 30(21), 1910610 (2020).
[Crossref]

J. W. Wu, Z. X. Wang, Z. Q. Fang, J. C. Liang, X. Fu, J. F. Liu, H. T. Wu, D. Bao, L. Miao, X. Y. Zhou, Q. Cheng, and T. J. Cui, “Full-State Synthesis of Electromagnetic Fields using High Efficiency Phase-Only Metasurfaces,” Adv. Funct. Mater. 30(39), 2004144 (2020).
[Crossref]

H. Q. Zhou, B. Sain, Y. T. Wang, C. Schlickriede, R. Z. Zhao, X. Zhang, Q. S. Wei, X. W. Li, L. L. Huang, and T. Zentgraf, “Polarization-Encrypted Orbital Angular Momentum Multiplexed Metasurface Holography,” ACS Nano 14(5), 5553–5559 (2020).
[Crossref]

F. Yue, V. Aglieri, R. Piccoli, R. Macaluso, A. Toma, R. Morandotti, and L. Razzari, “Highly Sensitive Polarization Rotation Measurement through a High-Order Vector Beam Generated by a Metasurface,” Adv. Mater. Technol. 5, 1901008 (2020).
[Crossref]

Q. Song, A. Baroni, R. Sawant, P. Ni, V. Brandli, S. Chenot, S. Vézian, B. Damilano, P. Mierry, S. Khadir, P. Ferrand, and P. Genevet, “Ptychography retrieval of fully polarized holograms from geometric-phase metasurfaces,” Nat Commun 11(1), 2651 (2020).
[Crossref]

M. Amin, O. Siddiqui, and M. Farhat, “Linear and Circular Dichroism in Graphene-Based Reflectors for Polarization Control,” Phys. Rev. Applied 13(2), 024046 (2020).
[Crossref]

2019 (5)

A. C. Overvig, S. Shrestha, S. C. Malek, M. Lu, A. Stein, C. Zheng, and N. F. Yu, “Dielectric metasurfaces for complete and independent control of the optical amplitude and phase,” Light: Sci. Appl. 8(1), 92 (2019).
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H. X. Xu, G. W. Hu, L. Han, M. H. Jiang, Y. J. Huang, Y. Li, X. Yang, X. H. Ling, L. Z. Chen, J. L. Zhao, and C. W. Qiu, “Chirality-Assisted High-Efficiency Metasurfaces with Independent Control of Phase, Amplitude, and Polarization,” Adv. Opt. Mater. 4(7), 1801479 (2019).
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J. Li, Y. T. Zhang, J. N. Li, X. Yan, L. J. Liang, Z. Zhang, J. Huang, J. H. Li, Y. Yang, and J. Q. Yao, “Amplitude modulation of anomalously reflected terahertz beams using all-optical active Pancharatnam-Berry coding metasurfaces,” Nanoscale 11(12), 5746–5753 (2019).
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Y. Huang, N. Rubin, A. Ambrosio, Z. Shi, R. C. Devlin, C. Qiu, and F. Capasso, “Versatile total angular momentum generation using cascaded J-plates,” Opt. Express 27(5), 7469–7484 (2019).
[Crossref]

N. A. Rubin, G. D’Aversa, P. Chevalier, Z. Shi, W. Chen, and F. Capasso, “Matrix Fourier optics enables a compact full-Stokes polarization camera,” Science 365(6448), 1839 (2019).
[Crossref]

2018 (7)

N. A. Rubin, A. Zaidi, M. Juhl, R. P. Li, J. P. B. Mueller, R. C. Devlin, K. Leósson, and F. Capasso, “Polarization state generation and measurement with a single metasurface,” Opt. Express 26(17), 21455–21478 (2018).
[Crossref]

S. M. Kamali, E. Arbabi, A. Arbabi, and A. Faraon, “A review of dielectric optical metasurfaces for wavefront control,” Nanophotonics. 7(6), 1041–1068 (2018).
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M. Qiu, L. Zhang, Z. Tang, W. Jin, C. W. Qiu, and D. Y. Lei, “3D Metaphotonic Nanostructures with Intrinsic Chirality,” Adv. Funct. Mater. 28(45), 1803147 (2018).
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W. Lin, T. Jin, and G. Zheng, “Controlling phase of arbitrary polarizations using both the geometric phase and the propagation phase,” Phys. Rev. B. 97(24), 245426 (2018).
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Z. L. Deng, J. Deng, X. Zhuang, S. Wang, K. Li, Y. Wang, Y. Chi, X. Ye, J. Xu, G. P. Wang, R. Zhao, X. Wang, Y. Cao, X. Cheng, G. Li, and X. Li, “Diatomic Metasurface for Vectorial Holography,” Nano Lett. 18(5), 2885–2892 (2018).
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Q. Wang, E. Plum, Q. Yang, X. Zhang, Q. Xu, Y. Xu, J. G. Han, and W. L. Zhang, “Reflective chiral meta-holography: multiplexing holograms for circularly polarized waves,” Light Sci Appl 7(1), 25 (2018).
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G. Y. Lee, G. Yoon, S. Y. Lee, H. Yun, J. Cho, K. Lee, H. Kim, J. Rho, and B. Lee, “Complete amplitude and phase control of light using broadband holographic metasurfaces,” Nanoscale 10(9), 4237–4245 (2018).
[Crossref]

2017 (3)

R. C. Devlin, A. Ambrosio, D. Wintz, S. L. Oscurato, A. Y. Zhu, M. Khorasaninejad, J. Oh, Pasqualino. Maddalena, and Federico. Capasso, “Spin-to-orbital angular momentum conversion in dielectric metasurfaces,” Opt. Express 25(1), 377–379 (2017).
[Crossref]

J. P. Balthasar Mueller, Noah A. Rubin, Robert C. Devlin, B. Groever, and F. Capasso, “Independent Phase Control of Arbitrary Orthogonal States of Polarization,” Phys. Rev. Lett. 118(11), 113901 (2017).
[Crossref]

J. T. Collins, C. Kuppe, D. C. Hooper, C. Sibilia, M. Centini, and V. K. Valev, “Chirality and Chiroptical Effects in Metal Nanostructures: Fundamentals and Current Trends,” Advanced Optical Materials 5(16), 1700182 (2017).
[Crossref]

2016 (3)

X. D. Qiu, L. G. Xie, X. Liu, L. Luo, Z. Y. Zhang, and J. L. Du, “Estimation of optical rotation of chiral molecules with weak measurements,” Opt. Lett. 41(17), 4032 (2016).
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Y. C. Zhang, J. Q. Liu, D. Li, X. Dai, F. H. Yan, Xavier A. Conlan, R. H. Zhou, Colin J. Barrow, J. He, X. Wang, and W. R. Yang, “Self-Assembled Core–Satellite Gold Nanoparticle Networks for Ultrasensitive Detection of Chiral Molecules by Recognition Tunneling Current,” ACS Nano 10(5), 5096–5103 (2016).
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Q. Wang, X. Zhang, Y. Xu, J. Gu, Y. Li, Z. Tian, R. Singh, S. Zhang, J. G. Han, and W. L. Zhang, “Broadband metasurface holograms: toward complete phase and amplitude engineering,” Sci. Rep. 6(1), 32867 (2016).
[Crossref]

2015 (3)

A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Simultaneous and Complete Control of Light Polarization and Phase using High Contrast Transmitarrays,” Nat. Nanotechnol. 10(11), 937–943 (2015).
[Crossref]

D. Wen, F. Yue, G. Li, G. Zheng, K. Chan, S. Chen, M. Chen, K. Li, K. W. Cheah, E. Y. B. Pun, S. Zhang, and X. Chen, “Helicity multiplexed broadband metasurface holograms,” Nat. Commun 6(1), 8241 (2015).
[Crossref]

A. Arbabi, Y. Horie, A. Ball, M. Bagheri, and A. Faraon, “Subwavelength-thick lenses with high numerical apertures and large efficiency based on high-contrast transmitarrays,” Nat Commun 6(1), 7069 (2015).
[Crossref]

2013 (2)

L. L. Huang, X. Z. Chen, H. Mühlenbernd, Z. Hao, S. Chen, B. Bai, Q. F. Tan, G. F. Jin, KW. Cheah, C. Qiu, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nat. Commun. 4(1), 2808 (2013).
[Crossref]

X. J. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nat Commun 4(1), 2807 (2013).
[Crossref]

2012 (1)

S. L. Sun, Q. He, S. Y. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
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2010 (1)

C. Menzel, C. Rockstuhl, and F. Lederer, “Advanced Jones calculus for the classification of periodic metamaterials,” Phys. Rev. A 82(5), 053811 (2010).
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2006 (3)

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45(22), 5453–5469 (2006).
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A. Konak, D. W. Coit, and A. E. Smith, “Multi-objective optimization using genetic algorithms: A tutorial. Reliability Engineering and System Safety,” Reliability Engineering & System Safety 91(9), 992–1007 (2006).
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J. Arora and A. S. Jan, “Practical Mathematical Optimization: An introduction to basic optimization theory and classical and new gradient-based algorithms,” Struct Multidisc Optim 31(3), 249 (2006).
[Crossref]

2003 (1)

D. Gies and Y. Rahmat-Samii, “Particle swarm optimization for reconfigurable phase-differentiated array design,” Microw. Opt. Technol. Lett. 38(3), 168–175 (2003).
[Crossref]

2002 (1)

A. L. Rucker and T. P. Creamer, “Polyproline II helical structure in protein unfolded states: lysine peptides revisited,” Protein science. 11, 980–985 (2002).
[Crossref]

1997 (1)

C. Y. Zhu, R. Byrd, P. H. Lu, and J. C. Nocedal, “Algorithm 778: L-BFGS-B: Fortran Subroutines for Large-Scale Bound-Constrained Optimization,” ACM Trans. Math. Softw. 23(4), 550–560 (1997).
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1995 (1)

R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A limited-memory algorithm for bound constrained optimization,” SIAM J. Sci. Comput. 16(5), 1190–1208 (1995).
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1994 (1)

D. Whitley, “A genetic algorithm tutorial,” Stat Comput 4(2), 65–85 (1994).
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1966 (1)

S. Beychok, “Circular dichroism of biological macromolecules,” Science 154(3754), 1288–1299 (1966).
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Aglieri, V.

F. Yue, V. Aglieri, R. Piccoli, R. Macaluso, A. Toma, R. Morandotti, and L. Razzari, “Highly Sensitive Polarization Rotation Measurement through a High-Order Vector Beam Generated by a Metasurface,” Adv. Mater. Technol. 5, 1901008 (2020).
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Albooyeh, M.

H. Kazemi, M. Albooyeh, and F. Capolino, “Simultaneous Perfect Bending and Polarization Rotation of Electromagnetic Wavefront Using Chiral Gradient Metasurfaces,” Phys. Rev. Applied 13(2), 024078 (2020).
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Alù, A.

Z. L. Deng, M. Jin, X. Ye, S. Wang, T. Shi, J. Deng, N. Mao, Y. Cao, B. O. Guan, A. Alù, G. Li, and X. Li, “Full-Color Complex-Amplitude Vectorial Holograms Based on Multi-Freedom Metasurfaces,” Adv. Funct. Mater. 30(21), 1910610 (2020).
[Crossref]

Ambrosio, A.

Amin, M.

M. Amin, O. Siddiqui, and M. Farhat, “Linear and Circular Dichroism in Graphene-Based Reflectors for Polarization Control,” Phys. Rev. Applied 13(2), 024046 (2020).
[Crossref]

Arbabi, A.

S. M. Kamali, E. Arbabi, A. Arbabi, and A. Faraon, “A review of dielectric optical metasurfaces for wavefront control,” Nanophotonics. 7(6), 1041–1068 (2018).
[Crossref]

A. Arbabi, Y. Horie, A. Ball, M. Bagheri, and A. Faraon, “Subwavelength-thick lenses with high numerical apertures and large efficiency based on high-contrast transmitarrays,” Nat Commun 6(1), 7069 (2015).
[Crossref]

A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Simultaneous and Complete Control of Light Polarization and Phase using High Contrast Transmitarrays,” Nat. Nanotechnol. 10(11), 937–943 (2015).
[Crossref]

Arbabi, E.

S. M. Kamali, E. Arbabi, A. Arbabi, and A. Faraon, “A review of dielectric optical metasurfaces for wavefront control,” Nanophotonics. 7(6), 1041–1068 (2018).
[Crossref]

Arora, J.

J. Arora and A. S. Jan, “Practical Mathematical Optimization: An introduction to basic optimization theory and classical and new gradient-based algorithms,” Struct Multidisc Optim 31(3), 249 (2006).
[Crossref]

Bagheri, M.

A. Arbabi, Y. Horie, A. Ball, M. Bagheri, and A. Faraon, “Subwavelength-thick lenses with high numerical apertures and large efficiency based on high-contrast transmitarrays,” Nat Commun 6(1), 7069 (2015).
[Crossref]

A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Simultaneous and Complete Control of Light Polarization and Phase using High Contrast Transmitarrays,” Nat. Nanotechnol. 10(11), 937–943 (2015).
[Crossref]

Bai, B.

L. L. Huang, X. Z. Chen, H. Mühlenbernd, Z. Hao, S. Chen, B. Bai, Q. F. Tan, G. F. Jin, KW. Cheah, C. Qiu, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nat. Commun. 4(1), 2808 (2013).
[Crossref]

Ball, A.

A. Arbabi, Y. Horie, A. Ball, M. Bagheri, and A. Faraon, “Subwavelength-thick lenses with high numerical apertures and large efficiency based on high-contrast transmitarrays,” Nat Commun 6(1), 7069 (2015).
[Crossref]

Balthasar Mueller, J. P.

J. P. Balthasar Mueller, Noah A. Rubin, Robert C. Devlin, B. Groever, and F. Capasso, “Independent Phase Control of Arbitrary Orthogonal States of Polarization,” Phys. Rev. Lett. 118(11), 113901 (2017).
[Crossref]

Bao, D.

J. W. Wu, Z. X. Wang, Z. Q. Fang, J. C. Liang, X. Fu, J. F. Liu, H. T. Wu, D. Bao, L. Miao, X. Y. Zhou, Q. Cheng, and T. J. Cui, “Full-State Synthesis of Electromagnetic Fields using High Efficiency Phase-Only Metasurfaces,” Adv. Funct. Mater. 30(39), 2004144 (2020).
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Baroni, A.

Q. Song, A. Baroni, R. Sawant, P. Ni, V. Brandli, S. Chenot, S. Vézian, B. Damilano, P. Mierry, S. Khadir, P. Ferrand, and P. Genevet, “Ptychography retrieval of fully polarized holograms from geometric-phase metasurfaces,” Nat Commun 11(1), 2651 (2020).
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Barrow, Colin J.

Y. C. Zhang, J. Q. Liu, D. Li, X. Dai, F. H. Yan, Xavier A. Conlan, R. H. Zhou, Colin J. Barrow, J. He, X. Wang, and W. R. Yang, “Self-Assembled Core–Satellite Gold Nanoparticle Networks for Ultrasensitive Detection of Chiral Molecules by Recognition Tunneling Current,” ACS Nano 10(5), 5096–5103 (2016).
[Crossref]

Beychok, S.

S. Beychok, “Circular dichroism of biological macromolecules,” Science 154(3754), 1288–1299 (1966).
[Crossref]

Brandli, V.

Q. Song, A. Baroni, R. Sawant, P. Ni, V. Brandli, S. Chenot, S. Vézian, B. Damilano, P. Mierry, S. Khadir, P. Ferrand, and P. Genevet, “Ptychography retrieval of fully polarized holograms from geometric-phase metasurfaces,” Nat Commun 11(1), 2651 (2020).
[Crossref]

Burokur, S. N.

K Zhang, Y. Y. Yuan, X. M. Ding, H. Y. Li, B. Ratni, Q. Wu, J. Liu, S. N. Burokur, and J. B. Tan, “Polarization-Engineered Noninterleaved Metasurface for Integer and Fractional Orbital Angular Momentum Multiplexing,” Laser Photonics Rev. 15(1), 2000351 (2021).
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K. Zhang, Y. Wang, S. N. Burokur, and Q. Wu, “Generating Dual-Polarized Vortex Beam by Detour Phase: From Phase Gradient Metasurfaces to Metagratings,” Transactions on Microwave Theory and Techniques. (2021).
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Byrd, R.

C. Y. Zhu, R. Byrd, P. H. Lu, and J. C. Nocedal, “Algorithm 778: L-BFGS-B: Fortran Subroutines for Large-Scale Bound-Constrained Optimization,” ACM Trans. Math. Softw. 23(4), 550–560 (1997).
[Crossref]

Byrd, R. H.

R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A limited-memory algorithm for bound constrained optimization,” SIAM J. Sci. Comput. 16(5), 1190–1208 (1995).
[Crossref]

Cao, Y.

Z. L. Deng, M. Jin, X. Ye, S. Wang, T. Shi, J. Deng, N. Mao, Y. Cao, B. O. Guan, A. Alù, G. Li, and X. Li, “Full-Color Complex-Amplitude Vectorial Holograms Based on Multi-Freedom Metasurfaces,” Adv. Funct. Mater. 30(21), 1910610 (2020).
[Crossref]

Z. L. Deng, J. Deng, X. Zhuang, S. Wang, K. Li, Y. Wang, Y. Chi, X. Ye, J. Xu, G. P. Wang, R. Zhao, X. Wang, Y. Cao, X. Cheng, G. Li, and X. Li, “Diatomic Metasurface for Vectorial Holography,” Nano Lett. 18(5), 2885–2892 (2018).
[Crossref]

Capasso, F.

Z. J. Shi, Alexander. Y. Zhu, Z. Y. Li, Y. W. Huang, W. T. Chen, C. W. Qiu, and F. Capasso, “Continuous angle-tunable birefringence with freeform metasurfaces for arbitrary polarization conversion,” Sci. Adv. 6, 23 (2020).
[Crossref]

Y. Huang, N. Rubin, A. Ambrosio, Z. Shi, R. C. Devlin, C. Qiu, and F. Capasso, “Versatile total angular momentum generation using cascaded J-plates,” Opt. Express 27(5), 7469–7484 (2019).
[Crossref]

N. A. Rubin, G. D’Aversa, P. Chevalier, Z. Shi, W. Chen, and F. Capasso, “Matrix Fourier optics enables a compact full-Stokes polarization camera,” Science 365(6448), 1839 (2019).
[Crossref]

N. A. Rubin, A. Zaidi, M. Juhl, R. P. Li, J. P. B. Mueller, R. C. Devlin, K. Leósson, and F. Capasso, “Polarization state generation and measurement with a single metasurface,” Opt. Express 26(17), 21455–21478 (2018).
[Crossref]

J. P. Balthasar Mueller, Noah A. Rubin, Robert C. Devlin, B. Groever, and F. Capasso, “Independent Phase Control of Arbitrary Orthogonal States of Polarization,” Phys. Rev. Lett. 118(11), 113901 (2017).
[Crossref]

Capasso, Federico.

Capolino, F.

H. Kazemi, M. Albooyeh, and F. Capolino, “Simultaneous Perfect Bending and Polarization Rotation of Electromagnetic Wavefront Using Chiral Gradient Metasurfaces,” Phys. Rev. Applied 13(2), 024078 (2020).
[Crossref]

Centini, M.

J. T. Collins, C. Kuppe, D. C. Hooper, C. Sibilia, M. Centini, and V. K. Valev, “Chirality and Chiroptical Effects in Metal Nanostructures: Fundamentals and Current Trends,” Advanced Optical Materials 5(16), 1700182 (2017).
[Crossref]

Chan, K.

D. Wen, F. Yue, G. Li, G. Zheng, K. Chan, S. Chen, M. Chen, K. Li, K. W. Cheah, E. Y. B. Pun, S. Zhang, and X. Chen, “Helicity multiplexed broadband metasurface holograms,” Nat. Commun 6(1), 8241 (2015).
[Crossref]

Cheah, K. W.

D. Wen, F. Yue, G. Li, G. Zheng, K. Chan, S. Chen, M. Chen, K. Li, K. W. Cheah, E. Y. B. Pun, S. Zhang, and X. Chen, “Helicity multiplexed broadband metasurface holograms,” Nat. Commun 6(1), 8241 (2015).
[Crossref]

Cheah, KW.

L. L. Huang, X. Z. Chen, H. Mühlenbernd, Z. Hao, S. Chen, B. Bai, Q. F. Tan, G. F. Jin, KW. Cheah, C. Qiu, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nat. Commun. 4(1), 2808 (2013).
[Crossref]

Chen, L. Z.

H. X. Xu, G. W. Hu, L. Han, M. H. Jiang, Y. J. Huang, Y. Li, X. Yang, X. H. Ling, L. Z. Chen, J. L. Zhao, and C. W. Qiu, “Chirality-Assisted High-Efficiency Metasurfaces with Independent Control of Phase, Amplitude, and Polarization,” Adv. Opt. Mater. 4(7), 1801479 (2019).
[Crossref]

Chen, M.

D. Wen, F. Yue, G. Li, G. Zheng, K. Chan, S. Chen, M. Chen, K. Li, K. W. Cheah, E. Y. B. Pun, S. Zhang, and X. Chen, “Helicity multiplexed broadband metasurface holograms,” Nat. Commun 6(1), 8241 (2015).
[Crossref]

Chen, S.

Z. Liu, Y. Xu, C.-Y. Ji, S. Chen, X. Li, X. Zhang, Y. Yao, and J. Li, “Fano-Enhanced Circular Dichroism in Deformable Stereo Metasurfaces,” Adv. Mater. 32(8), 1907077 (2020).
[Crossref]

D. Wen, F. Yue, G. Li, G. Zheng, K. Chan, S. Chen, M. Chen, K. Li, K. W. Cheah, E. Y. B. Pun, S. Zhang, and X. Chen, “Helicity multiplexed broadband metasurface holograms,” Nat. Commun 6(1), 8241 (2015).
[Crossref]

L. L. Huang, X. Z. Chen, H. Mühlenbernd, Z. Hao, S. Chen, B. Bai, Q. F. Tan, G. F. Jin, KW. Cheah, C. Qiu, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nat. Commun. 4(1), 2808 (2013).
[Crossref]

Chen, W.

N. A. Rubin, G. D’Aversa, P. Chevalier, Z. Shi, W. Chen, and F. Capasso, “Matrix Fourier optics enables a compact full-Stokes polarization camera,” Science 365(6448), 1839 (2019).
[Crossref]

Chen, W. T.

Z. J. Shi, Alexander. Y. Zhu, Z. Y. Li, Y. W. Huang, W. T. Chen, C. W. Qiu, and F. Capasso, “Continuous angle-tunable birefringence with freeform metasurfaces for arbitrary polarization conversion,” Sci. Adv. 6, 23 (2020).
[Crossref]

Chen, X.

D. Wen, F. Yue, G. Li, G. Zheng, K. Chan, S. Chen, M. Chen, K. Li, K. W. Cheah, E. Y. B. Pun, S. Zhang, and X. Chen, “Helicity multiplexed broadband metasurface holograms,” Nat. Commun 6(1), 8241 (2015).
[Crossref]

Chen, X. Z.

L. L. Huang, X. Z. Chen, H. Mühlenbernd, Z. Hao, S. Chen, B. Bai, Q. F. Tan, G. F. Jin, KW. Cheah, C. Qiu, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nat. Commun. 4(1), 2808 (2013).
[Crossref]

Chenault, D. B.

Cheng, Q.

J. W. Wu, Z. X. Wang, Z. Q. Fang, J. C. Liang, X. Fu, J. F. Liu, H. T. Wu, D. Bao, L. Miao, X. Y. Zhou, Q. Cheng, and T. J. Cui, “Full-State Synthesis of Electromagnetic Fields using High Efficiency Phase-Only Metasurfaces,” Adv. Funct. Mater. 30(39), 2004144 (2020).
[Crossref]

Cheng, X.

Z. L. Deng, J. Deng, X. Zhuang, S. Wang, K. Li, Y. Wang, Y. Chi, X. Ye, J. Xu, G. P. Wang, R. Zhao, X. Wang, Y. Cao, X. Cheng, G. Li, and X. Li, “Diatomic Metasurface for Vectorial Holography,” Nano Lett. 18(5), 2885–2892 (2018).
[Crossref]

Chenot, S.

Q. Song, A. Baroni, R. Sawant, P. Ni, V. Brandli, S. Chenot, S. Vézian, B. Damilano, P. Mierry, S. Khadir, P. Ferrand, and P. Genevet, “Ptychography retrieval of fully polarized holograms from geometric-phase metasurfaces,” Nat Commun 11(1), 2651 (2020).
[Crossref]

Chevalier, P.

N. A. Rubin, G. D’Aversa, P. Chevalier, Z. Shi, W. Chen, and F. Capasso, “Matrix Fourier optics enables a compact full-Stokes polarization camera,” Science 365(6448), 1839 (2019).
[Crossref]

Chi, Y.

Z. L. Deng, J. Deng, X. Zhuang, S. Wang, K. Li, Y. Wang, Y. Chi, X. Ye, J. Xu, G. P. Wang, R. Zhao, X. Wang, Y. Cao, X. Cheng, G. Li, and X. Li, “Diatomic Metasurface for Vectorial Holography,” Nano Lett. 18(5), 2885–2892 (2018).
[Crossref]

Cho, J.

G. Y. Lee, G. Yoon, S. Y. Lee, H. Yun, J. Cho, K. Lee, H. Kim, J. Rho, and B. Lee, “Complete amplitude and phase control of light using broadband holographic metasurfaces,” Nanoscale 10(9), 4237–4245 (2018).
[Crossref]

Coit, D. W.

A. Konak, D. W. Coit, and A. E. Smith, “Multi-objective optimization using genetic algorithms: A tutorial. Reliability Engineering and System Safety,” Reliability Engineering & System Safety 91(9), 992–1007 (2006).
[Crossref]

Collins, J. T.

J. T. Collins, C. Kuppe, D. C. Hooper, C. Sibilia, M. Centini, and V. K. Valev, “Chirality and Chiroptical Effects in Metal Nanostructures: Fundamentals and Current Trends,” Advanced Optical Materials 5(16), 1700182 (2017).
[Crossref]

Conlan, Xavier A.

Y. C. Zhang, J. Q. Liu, D. Li, X. Dai, F. H. Yan, Xavier A. Conlan, R. H. Zhou, Colin J. Barrow, J. He, X. Wang, and W. R. Yang, “Self-Assembled Core–Satellite Gold Nanoparticle Networks for Ultrasensitive Detection of Chiral Molecules by Recognition Tunneling Current,” ACS Nano 10(5), 5096–5103 (2016).
[Crossref]

Creamer, T. P.

A. L. Rucker and T. P. Creamer, “Polyproline II helical structure in protein unfolded states: lysine peptides revisited,” Protein science. 11, 980–985 (2002).
[Crossref]

Cui, T. J.

J. W. Wu, Z. X. Wang, Z. Q. Fang, J. C. Liang, X. Fu, J. F. Liu, H. T. Wu, D. Bao, L. Miao, X. Y. Zhou, Q. Cheng, and T. J. Cui, “Full-State Synthesis of Electromagnetic Fields using High Efficiency Phase-Only Metasurfaces,” Adv. Funct. Mater. 30(39), 2004144 (2020).
[Crossref]

D’Aversa, G.

N. A. Rubin, G. D’Aversa, P. Chevalier, Z. Shi, W. Chen, and F. Capasso, “Matrix Fourier optics enables a compact full-Stokes polarization camera,” Science 365(6448), 1839 (2019).
[Crossref]

Dai, X.

Y. C. Zhang, J. Q. Liu, D. Li, X. Dai, F. H. Yan, Xavier A. Conlan, R. H. Zhou, Colin J. Barrow, J. He, X. Wang, and W. R. Yang, “Self-Assembled Core–Satellite Gold Nanoparticle Networks for Ultrasensitive Detection of Chiral Molecules by Recognition Tunneling Current,” ACS Nano 10(5), 5096–5103 (2016).
[Crossref]

Damilano, B.

Q. Song, A. Baroni, R. Sawant, P. Ni, V. Brandli, S. Chenot, S. Vézian, B. Damilano, P. Mierry, S. Khadir, P. Ferrand, and P. Genevet, “Ptychography retrieval of fully polarized holograms from geometric-phase metasurfaces,” Nat Commun 11(1), 2651 (2020).
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S. L. Sun, Q. He, S. Y. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
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Q. Wang, E. Plum, Q. Yang, X. Zhang, Q. Xu, Y. Xu, J. G. Han, and W. L. Zhang, “Reflective chiral meta-holography: multiplexing holograms for circularly polarized waves,” Light Sci Appl 7(1), 25 (2018).
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S. L. Sun, Q. He, S. Y. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[Crossref]

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Z. Liu, Y. Xu, C.-Y. Ji, S. Chen, X. Li, X. Zhang, Y. Yao, and J. Li, “Fano-Enhanced Circular Dichroism in Deformable Stereo Metasurfaces,” Adv. Mater. 32(8), 1907077 (2020).
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Q. Wang, E. Plum, Q. Yang, X. Zhang, Q. Xu, Y. Xu, J. G. Han, and W. L. Zhang, “Reflective chiral meta-holography: multiplexing holograms for circularly polarized waves,” Light Sci Appl 7(1), 25 (2018).
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Q. Wang, X. Zhang, Y. Xu, J. Gu, Y. Li, Z. Tian, R. Singh, S. Zhang, J. G. Han, and W. L. Zhang, “Broadband metasurface holograms: toward complete phase and amplitude engineering,” Sci. Rep. 6(1), 32867 (2016).
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Y. C. Zhang, J. Q. Liu, D. Li, X. Dai, F. H. Yan, Xavier A. Conlan, R. H. Zhou, Colin J. Barrow, J. He, X. Wang, and W. R. Yang, “Self-Assembled Core–Satellite Gold Nanoparticle Networks for Ultrasensitive Detection of Chiral Molecules by Recognition Tunneling Current,” ACS Nano 10(5), 5096–5103 (2016).
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J. Li, Y. T. Zhang, J. N. Li, X. Yan, L. J. Liang, Z. Zhang, J. Huang, J. H. Li, Y. Yang, and J. Q. Yao, “Amplitude modulation of anomalously reflected terahertz beams using all-optical active Pancharatnam-Berry coding metasurfaces,” Nanoscale 11(12), 5746–5753 (2019).
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Q. Wang, E. Plum, Q. Yang, X. Zhang, Q. Xu, Y. Xu, J. G. Han, and W. L. Zhang, “Reflective chiral meta-holography: multiplexing holograms for circularly polarized waves,” Light Sci Appl 7(1), 25 (2018).
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Y. C. Zhang, J. Q. Liu, D. Li, X. Dai, F. H. Yan, Xavier A. Conlan, R. H. Zhou, Colin J. Barrow, J. He, X. Wang, and W. R. Yang, “Self-Assembled Core–Satellite Gold Nanoparticle Networks for Ultrasensitive Detection of Chiral Molecules by Recognition Tunneling Current,” ACS Nano 10(5), 5096–5103 (2016).
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H. X. Xu, G. W. Hu, L. Han, M. H. Jiang, Y. J. Huang, Y. Li, X. Yang, X. H. Ling, L. Z. Chen, J. L. Zhao, and C. W. Qiu, “Chirality-Assisted High-Efficiency Metasurfaces with Independent Control of Phase, Amplitude, and Polarization,” Adv. Opt. Mater. 4(7), 1801479 (2019).
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ACM Trans. Math. Softw. (1)

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Y. C. Zhang, J. Q. Liu, D. Li, X. Dai, F. H. Yan, Xavier A. Conlan, R. H. Zhou, Colin J. Barrow, J. He, X. Wang, and W. R. Yang, “Self-Assembled Core–Satellite Gold Nanoparticle Networks for Ultrasensitive Detection of Chiral Molecules by Recognition Tunneling Current,” ACS Nano 10(5), 5096–5103 (2016).
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Z. Liu, Y. Xu, C.-Y. Ji, S. Chen, X. Li, X. Zhang, Y. Yao, and J. Li, “Fano-Enhanced Circular Dichroism in Deformable Stereo Metasurfaces,” Adv. Mater. 32(8), 1907077 (2020).
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Supplementary Material (1)

NameDescription
Supplement 1       Supplemental Document

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.

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Figures (7)

Fig. 1.
Fig. 1. Schematic diagram of the functions of CD-enantiomers-based metasurface. The CD enantiomers: L-atom and R-atom are in the upper right corner of the figure. CD-enantiomers are rotated and coded into the corresponding position according to the requirements of vector beam customization to constitute metasurface. When the linear polarization strikes the metasurface, four beams with left-handed, right-handed, linear, and 5-axial-ratio left-handed-elliptic polarizations with the energy ratio of 7:8:9:10 are achieved.
Fig. 2.
Fig. 2. Meta-enantiomer structure, simulation results, and checkerboard type arrangement diagram. (a) Illustration of R-enantiomer, where period p=6mm, width w=0.78mm, thickness h=1.5mm. (b) S-parameters curves of R-enantiomer from 10 GHz to 18GHz. (c) At 12GHz, the amplitude and phase response of cross-polarization of L-enantiomer (box) and R-enantiomer (triangle) under different rotate angles. (d) The FPP manipulation by checkerboard enantiomers. Polarization azimuth is modulated by rot31 and rot42, and the axial ratio is modulated by rot21. EA is the central angle between the simulation and theoretical polarization on the Poincaré Sphere. (e) Schematic of checkerboard type arrangement of the enantiomers. (f) rot1 and rot3 are the rotation angles from the x-axis's negative direction to the spiral's tail. Counterclockwise rotation is positive, and vice versa. rot2 and rot4 are the rotation angles from the x-axis's positive direction to the spiral's tail, clockwise rotation is positive and negative, and vice versa.
Fig. 3.
Fig. 3. The β and AR simulation results of customized FPP at φ=0°, -15°<θ<15°. The purple axis is the y-axis, and the green axis is the x-axis. (a), (b), (c), and (d) the linear polarization with β = 0°, 45°, 90°, and 135° respectively. (e), (f), (g), and (h) the simulation results of 3-AR left-spin-elliptic polarization with β = 0°, 45°, 90°, and 135° respectively.
Fig. 4.
Fig. 4. The simulation results of polarization azimuth of four side lobes (φ = 0° or 90°, θ = ± 40°). The purple axis is the y-axis, and the green axis is the x-axis. Figure 4. (a-d) show energy is concentratively distributed in four sidelobes. Figure 4. (e-h) are the β and AR simulation results at φ = 0°. Figure 4(i)-(l)) are the β and AR simulation results at φ = 90°.
Fig. 5.
Fig. 5. The configuration for measurements and the experimental results. (a) Photograph of 108×108 mm2 enantiomers metasurface. (b) Photographs of the measure configuration. The transmitter emits the y-pol wave; the revolving stage can change θ; the receiver with an oblique angle receives a corresponding polarization wave. (c) Measured normalized two-dimensional scattering patterns in the plane of φ = 0°. (d) Measured normalized two-dimensional scattering patterns in the plane of φ = 90°.
Fig. 6.
Fig. 6. Schematic diagram of the encoding, spatial sampling, the optimal process, objective function construction, and the combination of GA and quasi-Newton algorithm.
Fig. 7.
Fig. 7. Simulation results of polarization and normalized amplitude in K-space of GA & L-BFGS-B based enantiomers metasurface. (a), (b), and (c) K-space normalized amplitude distribution of the RCP, the LCP, and the synthetic polarization. (d) Axial ratio distribution of synthetic polarization in K-space.

Tables (1)

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Table 1. Reported works on vectorial wave shaping.

Equations (7)

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β = α R α L 2 , A R = tan ( 0.5 arcsin( E R 2 E L 2 E R 2 + E L 2 ) ) ,
E r = [ n = 1 4 R L L n e i k r n n = 1 4 R L R n e i k r n n = 1 4 R R L n e i k r n n = 1 4 R R R n e i k r n ] [ E i L E i R ] ,
E r = e i α 1 [ i x + y 2 i x + y 2 ] [ e i δ 1 + e i ( δ 1 + α 31 ) e i ( δ 2 + α 21 ) + e i ( δ 2 + α 42 + α 21 ) ] ,
α L α R = 2 α 21 + α 42 α 31 2 ,
E L E R = [ 1 + cos ( 2 δ 1 + α 31 ) ] 2 + sin 2 ( 2 δ 1 + α 31 ) [ 1 + cos ( 2 δ 2 + α 42 ) ] 2 + sin 2 ( 2 δ 2 + α 42 ) ,
E ( θ , φ )  =  m = 1 M n = 1 N [ x + i (  - 1 ) ν n , m y ] e i [ π σ n , m 4 k ( sin θ cos φ X n , m + sin θ sin φ Y n , m ) ] ,
ε = q = 1 Q { ( E x q 2 + E y q 2 ) sin 2 ( γ q γ q c ) + [ ( E x q 2 + E y q 2 ) cos ( γ q γ q c ) E q c ] 2 } ,

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