Polarization-Encoded Structured Light Generation Based on Holographic Metasurface

In view of large information capacity of holographic metasurface and wide applications of structured light, the polarization-encoded structured light generation is proposed based on holographic metasurface. The polarization-encoded structured light generators consist of rotated L-shaped nanoholes, and they work under linearly polarized light illumination. The rotated L-shaped nanoholes simultaneously manipulate the intensity, phase, and polarization of light field and generate the structured light composed by multiple perfect vortices with polarization-encoded orbital angular momentum (OAM) modes. The generated polarization-encoded structured lights also behave the incident polarization dependence. The polarization multiplexing performance of holographic metasurface is embodied in both incident and output fields, and it is also verified by the numerical simulations. The designed method can be expanded to construct any transmission or reflection holographic metasurfaces, and the generated structured lights provide more switchable OAM channels. The polarization-encoded structured light and the compact holographic metasurface are benefit to broadening the wide applications of structured lights.


Introduction
Structured light points to the light field with high customized amplitude, phase, and polarization distribution [1][2][3]. Vector beam possessing spatial varying polarization state and vortex beam carrying spiral phase are two typical structured lights. Since the vortex beams with different topological charges are orthogonal and they provide the orbital angular momentum (OAM) for the interaction of light and matter [4], optical vortices have been applied in many fields including optical communication [5], particles trapping [6], optical encryption [7], and quantum memory [8]. Because of the special focusing effect, high-resolution imaging, and polarization sensitivity, vector beams have also been applied in many fields [9][10][11]. Till now, several methods have been proposed to generate the structured beams. Spiral phase plate [12], forked grating [13], interferometer [14], and metasurface [15,16] are utilized to generate vortex beams. Sub-wavelength grating [17], q-plate [18], spatial light modulator [19], and metasurface [20,21] are used to generate vector beams. Certainly, the superposition of vortex fields with different OAM modes can generate vector beam [22]. With comparison to traditional methods depending on the spatial accumulation effect along an optical path, metasurface realizes the light field manipulation with the help of the interaction of light field and nanometer scatterers [23][24][25][26][27][28][29]; therefore, it has advantages of ultra-thin and compact structure, ease to integration, and convenience to operate.
Simple metasurface can directly modulate the amplitude, phase, or polarization of light field using the local light modulation of naonounits, and holographic metasurface makes the light field manipulation ability of metasurface fully exert [30,31]. Shan et al. proposed the holographic metasurface consisting of nanorods and realized the generation of wavelength-encrypted characters with circularly polarized light illumination [32]. Liu et al. designed the holographic metasurface consisting of nanopillars and generated different patterns in near-field and far-field with the control of incident circularly polarized light [33]. Wan et al. proposed the multiplexing vectorial metaholograms through varying 1 3 phase difference of two orthogonal circularly polarized lights and obtained the polarization-encrypted images [34]. Our team designed the holographic metasurface consisting with rectangular nanoholes and obtained the structured vortex fields with the control of incident circularly polarized light [35]. These above works exhibit the powerful light control of holographic metasurface, and the obtained diversely encrypted images show us the multiplexing function of holographic metasurface. Combining the incident polarization state, the more output polarization states may bring more channels for the applications of light field.
In this paper, we propose polarization-encoded structured light (PESL) generation based on holographic metasurface. The output structured lights are encoded by the different linear polarization states, and they can be adjusted with the incident linear polarization state. The proposed polarization-encoded structured light generator (PESLG) consists of L-shaped nanoholes etched on sliver film, and the rotated nanoholes under the linearly polarized light illumination simultaneously modify the amplitude, phase, and polarization of light field. The PESLs are generated by the designed holographic metasurface, and the obtained vortex fields with different OAM modes are encoded by different linear polarization states and they also change with the incident polarization. The theoretical analysis provides the design idea of holographic metasurface. The numerical simulations verify the feasibility of the designed holographic metasurface for the generation of PESLs. The proposed structured light generator realizes the multiple-parameter control of light field. The combination of the polarization-encoded OAM modes and the polarization-dependent output provides more mode choices for the applications of the structured light. The intensity setting of PESL equivalent to the perfect vortex are benefit to the optical encryption, polarization multiplexing, and light coupling. We believe that the proposed PESL will be helpful for expanding the applications of structured light in particle manipulation, optical encryption, optical communication, quantum information processing, and other areas.

Design Principle
Here, we design the holographic metasurface in terms of the Fresnel holography technique to generate the structured light field. Suppose the target light field is U 0 (x 0 , y 0 ), the diffraction light field at the propagation distance of d away from the target field can be expressed by Fresnel integration of U(x,y) = ∫∫U 0 (x 0 , y 0 )K(x − x 0 , y − y 0 )dx 0 dy 0 , where the point spread function of K(x − x 0 , y − y 0 ) satisfies K(x − x 0 , y − y 0 ) = [exp(jkd)/(jλd)]exp{jk[(x − x 0 ) 2 + (y − y 0 ) 2 ]/(2d)} with k = 2π/λ denoting the wave number of the incident field and λ representing the wavelength [35]. Since the scatterers of metasurface are distributed discretely, the diffraction field should be discretized, and it can be written as where p, q, m, and n are integers, δ x and δ y represent the sampling intervals along the x and y axes of the diffraction field, and δ x0 and δ y0 represent the sampling intervals along the x 0 and y 0 axes of the target field. While the diffraction field performs the reverse Fresnel diffraction, the obtained field is U H (x 0 ′,y 0 ′) = U 0 (x 0 ,y 0 )/(λd) 2 and the discretized form is U H (mδ x ,nδ y ) = U 0 (mδ x ,nδ y )(λd) 2 . One can see that the diffraction field is the same as the target field except for the constant term. This means that the target field can be reproduced at the defaulted distance, and it is also the working principle of metasurface holography. When the target field is a radially structured light including several concentric vortices encoded by different polarization states, one of polarization mode U 0i (mδ x0 ,nδ y0 ) can be expressed as, where C i represents the amplitude of vortex, r i is the radius of vortex, w i denotes the waist radius of vortex, and l i is the topological charge. For different polarization mode, these parameters may take the same or different values. The amplitude and phase information of the holographic metasurface can be obtained by inserting Eqs. (2) into (1). For simplification, we set the amplitude threshold to construct the metasurface hologram. Here, the phase distribution of radially structured light is realized with the help of the phase delays introduced through rotating the L-shaped nanoholes. As we know, the phase delay introduced by an anisotropic nanohole equals to twice of the rotation angle of nanohole and it is always carried by the cross circularly polarized light [31]. Therefore, one spiral phase with the circularly polarized state can be easily generated through rotating the nanoholes.
As we know, one linear polarization state of cosγe x + sinγe y with the polarization angle of γ can be expressed as the superposition of two orthogonal circularly polarized states, namely, cosγe x + sinγe y = 2 0.5 [exp(jγ)e R + exp(−jγ)e L ]/2, where e x = (1, 0), e y = (0, 1), e R = 2 0.5 (1, −j)/2, and e L = 2 0.5 (1, j)/2 represent the unit vectors for x-polarization, y-polarization, and right-and left-handed circular polarization. Thus, two suits of nanoholes with opposite rotation angles can consist of one metasurface with linear polarization output. Figure 1A shows the sketch map of one structured light generator with x-polarization output. The compound metasurface CM is formed through combining the complementary structures of R1 and R2, whose nanoholes are randomly chosen from two metasurfaces of M1 and M2 corresponding to the right-and left-handed circular polarization. This compound metasurface takes effect under the x-polarized light illumination. Similarly, for the structured light encoded by two orthogonal linearly polarized states, one can construct the compound PESLG in terms of above principle. Suppose one OAM mode of exp(jl 1 φ) takes the x-polarization and the other OAM mode of exp(jl 2 φ) takes the y-polarization, the target field with two orthogonal linearly polarized OAM modes can be written as, while w 1 = w 2 and r 1 = r 2 , the above equation can be simplified into exp(−r 2 /w 2 )exp(jpφ) [exp(jqφ)e x + exp(−jqφ)e y ] with p = (l 1 + l 2 )/2 and q = ( l 1 − l 2 )/2 and it is the so-called vector vortex beam [28]. While p = 0, it changes into the pure vector beam. Figure 1B shows the schematic diagram for the generation of PESL by the constructed holographic metasurface. This holographic metasurface is composed of ]exp(jl 2 )e y four sets of rotated nanoholes, where two sets of nanoholes take effect under the right-handed circularly polarized light illumination and the other two sets of nanoholes take effect under the left-handed circularly polarized light illumination. The rotation angle θ of nanohole at any position equals to l 1 φ/2 for the first set, −l 1 φ/2 for the second set, l 2 φ/2 + π/4 for the third set, and −l 2 φ/2 + π/4 for the fourth one.
In order to ensure the quality of the generated PESL, we should optimize the structure parameters of metasurface. The structure parameters include the thickness of silver film, the length and width of L-shaped nanohole, and the separation of two adjacent nanoholes, which are labeled in the magnified nanohole inserted in the lower right corner of Fig. 1A. Here, the working wavelength of λ is set at 632.8 nm and the propagation distance of d is set at 15 μm. Through the optimization, the separations of nanoholes of δ x and δ y along two orthogonal directions takes 200 nm, which satisfies Nyquist sampling theorem δ 2 N ≤ λd with the sampling number of N taking 200. Two lengths of l a and l b take 120 nm and two width of w a and w b take 35 nm. The thickness of silver film takes h = 150 nm. The optimization process is finished with the help of finite-difference timedomain technique [36].

Simulation Generation of PESL
In order to testify the reliability of metasurface PESLG, we construct one holographic metasurface that can generate two orthogonal linearly polarized vortex fields with the same radii of 6 μm and the topological charges taking l 1 = 1 and l 2 = 3. The silver film with the thickness of 150 nm is deposited on the glass substrate, and the L-shaped nanoholes with the optimized dimensions are fabricated in the silver film. The nanoholes rotate in terms of the phase distributions of holographic metasurface. During the simulation process, the boundary condition of perfect matching layer is used to prevent non-physical scattering at the boundary. The minimum mesh step is set at 2 nm, the calculation region is set at 42 μm × 42 μm × 20 μm; the dielectric constant of silver is taken from the value given by Palik [37]. The observation plane is set at the propagation distance of 15 μm above the silver film. Figure 2A shows the structure diagram of compound PESLG, and the magnified part at the right reflects the relative rotation of four adjacent nanoholes. Figure 2B, C give the simulated intensity and phase distributions with the incident polarization along the horizontal direction, where the results in Fig. 2B are the intensity and phase distributions for the x-component of transmission field, and the results in Fig. 2C are the intensity and phase distributions for the y-component of transmission field. Figure 2D, E give the simulated intensity and phase distributions with the incident polarization  Fig. 2B, one can see that the intensity distribution takes on the annular shape and the phase uniformly increases 2π along the clockwise direction. It means the x-polarized vortex with the topological charge taking 1 generates. From the results in Fig. 2C, one can see that the intensity distribution also takes on the annular shape and the phase uniformly increases thrice of 2π along the clockwise direction. It means the y-polarized vortex with the topological charge taking 3 generates. Although the annular intensity distributions for two cases are the same, the different phase distributions indicate two channels with x-and y-polarization have no crosstalk.
From the results in Fig. 2D, one can see that the intensity distribution still takes on the annular shape and the phase uniformly increases thrice of 2π along the clockwise direction. It means the x-polarized vortex with the topological charge taking 3 generates. The results in Fig. 2E show that the intensity distribution also takes on the annular shape and the phase uniformly increases 2π along the clockwise direction. It means the y-polarized vortex with the topological charge taking 1 generates. Two regular phase distributions indicate two channels with x-and y-polarization have no crosstalk. Moreover, with comparison to the results in Fig. 2B, C, two orthogonal polarization-encoded outputs are just exchanged. This is because the change of incident polarization direction makes the left-handed circularly polarized light add the additional phase of π with respect to the right-handed circularly polarized light, and it leads the linear polarization direction to rotate π/2.

Integrated PESLs
From the above theoretical analysis, we know as the rotation angles of two sets of nanoholes satisfy θ 1 = −θ 2 , the output beam is the horizontal linear polarization, and as the rotation angles of two sets of nanoholes satisfy θ 1 = −θ 2 + π/2, the output beam is the vertical linear polarization. Similarly, as the rotation angles of two sets of nanoholes satisfy θ 1 = −θ 2 + π/4, the output linear polarization is along the diagonal direction, and as the rotation angles of two sets of nanoholes satisfy θ 1 = −θ 2 − π/4, the output linear polarization is along the anti-diagonal direction. Thus, we can also design the integrated PESLG to modulate the amplitude, phase, and polarization of light field based on holographic metasurface. For clearness, we arrange the integrated PESLs at the different quadrants. Meanwhile, in order to fully show the intensity, phase, and polarization manipulation of holographic metasurface, we customize four structured lights with the same radii, different topological charges, and different polarization states. Figure 3A, B give the target intensity, phase and polarization distributions of four integrated PESLs with the horizontal linearly polarized light illumination. The integrated PESLs have the same radii of 5 μm and their topological charge taking 2, −1, 4, and −3. The green arrows inserted in Fig. 3A denote the polarization states of the integrated PESLs. Figure 3C shows the simulated total intensity distribution of the integrated PESG under horizontal linearly polarized light illumination, and Fig. 3D, E show the phase distributions of x-and y-components of the integrated PESLs. The intensity distributions of x-and y-components of the integrated PESLs are also inserted in the free corners of the phase patterns. The inserted blue arrows denote the incident polarization directions and the red arrows denote the detection polarization directions The results in Fig. 3D, E show that three vortices appear among the phase distributions of x-and y-components of the integrated PESLs and the topological charges for the corresponding vortices equal to 2, 4, and −3, which are consistent with the defaulted ones. The annular intensity distributions with the same radii have different brightness. For the vortex with the polarization along the detection direction, the intensity is the brightest. As the polarization of vortex has one cross angle with the detection direction, the intensity of vortex is lower than the former. And the vortex with the polarization orthogonal to the detection direction does not appear. All these results verify the effectiveness of the designed holographic metasurface for the generation of integrated PESLs. Figure 3F, G show the phase distributions of x-and y-components of the integrated PESLs with the incident polarization along the vertical direction. Just like the analysis in above section, the rotated incident polarization direction brings the additional phase of π to the lefthanded circularly polarized light and it leads the output linear polarization direction to rotate π/2. One can see that as the detection polarization is along the horizontal direction, the output results are the same as the ones of Fig. 3E, and as the detection polarization is along the vertical direction, the output results are the same as the ones of Fig. 3D. It indicates that the integrated PESLs can be adjusted by the incident polarization. The results in Fig. 3H, I further verify the polarization multiplexing characteristic of the integrated PESLG, where the incident polarization is along diagonal direction. Figure 3H shows that three vortices appear as the detection polarization is along the horizontal direction except that the one at the lower right corner is invisible. Three vortices appear in Fig. 3I as the detection polarization is along the vertical direction except the one at the upper left corner is invisible. This is because the rotated incident polarization direction brings the additional phase of π/2 to the lefthanded circularly polarized light and it leads the output linear polarization direction to rotate π/4. Just because the polarization directions of four vortices rotate π/4, the intensity of the vortex at the upper left corner is the largest as the detection direction is along the horizontal direction, and the vortex at the lower right corner does not appear. Similarly, as the detection direction is along the vertical direction, the intensity of the vortex at the lower right corner is the largest and the vortex at the upper left corner disappears The results also indicate that the output PESLs may be controlled by the incident polarization. The integrated PESLG provides more mode combinations with comparison to the traditional polarization multiplexing metasurface with two channels controlled by left-and right-handed circularly polarized light or the horizontal and vertical linearly polarized light. The output perfect vortices with the changeless intensity rings are convenient for the coupling and detecting for different OAM modes. Therefore, this kind of polarization-controllable structured lights are favorable for the coupling, switching and manipulation of parallel OAM modes. The design of compact holographic metasurface is benefit to the integration and miniaturization.
In this work, we mainly carry out theoretical analysis and numerical simulations. The preparation for the practical metasurface samples can be divided into two steps, like the former work [15,35,36]. The first one is to deposit a silver film with the certain thickness on the glass substrate by using the magnetron sputtering method. The second step is to fabricate the PESLGs by means of the focused ion beam etching method. The diffraction intensity distribution of the integrated PESLs may be magnified using the microscope objective and detected by one charge coupled device target.

Conclusions
This paper constructs holographic metasurfaces based on Fresnel diffraction to generate the PESLs. The proposed holographic metasurfaces consisting of L-shaped nanoholes simultaneously modulate the amplitude, phase and polarization of light field. The polarization-encoded vortex beams with different OAM modes are obtained the proposed holographic metasurfaces. The output of PESLs also depend on the incident state of polarization. Therefore, the designed PESLGs possess the dual polarization multiplexing function and their performances of the generated PESLs are verified by the numerical simulations. The compact structure of the designed multifunctional metasurface is benefit to the integration. The more polarization modes generated by the proposed PESLG helps to increase the capacity of information propagation and the level of information encryption. The polarization-encoded perfect vortices bring more convenience for the coupling, switching and sensing of OAM modes. These achievements predict that the generated PESL based on holographic metasurface may be applied in optical encryption, large-capacity optical communication and parallel processing of optical information. The design principle of our proposed holography metasurface has universality, and it can be expanded into geometric nanounits with any shapes, any transmission or reflection situation, active control, and digital coding metasurface. The more performances of PES-LGs like the broadband characteristic, high efficiency, and speed coding can be improved.
Author Contribution Writing-original draft preparation, C. Z.; data curation, P. L., J. X., and Y. Z.; supervision, project administration, funding acquisition, S. T. All authors have read and agreed to the published version of the manuscript.

Funding
The authors received funding from the National Natural Science Foundation of China (10874105) and Shandong Provincial Natural Science Foundation of China (ZR2020KA009).

Availability of Data and Materials
The data that support the findings of this study are available from the corresponding author upon reasonable request.