High-dimensional Poincaré beams generated through cascaded metasurfaces for high-security optical encryption

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Introduction
With the rapid development of information technology, information security is becoming of vital significance and so do the derived encryption techniques.Optical encryption possesses the advantages of parallel processing, large capacity and low power consumption, and offers an ideal platform for information protection owing to abundant degrees of freedom (DoFs) of the photon [1][2][3][4][5].However, conventional optical encryption system commonly involves in a series of complicated optical elements and results in a bulky volume.In addition, due to the limited optical field manipulation capability of traditional optical elements, the multi-dimensional DoFs of light have not been fully exploited.
As an ultra-thin planar optical element composed of subwavelength nanostructures, metasurface has shown powerful capability of multi-dimensional optical field manipulation and emerges as a promising candidate for optical encryption in a compact form [6][7][8][9][10][11][12][13].For metasurface-empowered optical encryption system, the key space could be defined as the total number of keys and attributed to both incident and output states.
While the encoding space refers to the information capacity of the output images or beams through metasurfaces.So far, various DoFs of light, including incident angle [14][15][16], wavelength [17], polarization [18,19], orbital angular momentum (OAM) mode [20][21][22][23] and so on [24,25], have been explored to construct key space for optical information security.The combination of multiple DoFs has been demonstrated the capability to enhance the complexity of keys [26][27][28][29][30]. Besides, through encoding information into optical images in the form of nano-printing or holography, encoding space has also been established and depends on the number of multiplexing channels of metasurface [31,32].To further expand the key and encoding spaces, encryption algorithms and dynamic encryption schemes were aroused for high security and large capacity performance [33][34][35][36][37][38].Nevertheless, the above two strategies usually suffer from timeconsuming decryption process and the lack of pixelated real-time modulation.Featured with spatial variant polarization states and phase distributions, Poincaré beams gain a great deal of attentions in many applications [39][40][41][42][43][44], especially in optical encryption and encoding owing to versatile independent DoFs [45][46][47][48][49].In the recent works, optical information was encoded into the polarization order and ellipticity of perfect hybridorder Poincaré beams (HyOPBs) [45].It presents a new scheme of optical encryption to enlarge the encoding space, which is distinguished from conventional nanoprinting or holography encryption.However, such Poincaré beams encryption strategy has constraints due to limited key space (with only one incident state in essence) and stays a low security level.With the advantages of sub-wavelength modulation capability and inherent ultra-thin size, metasurface also allows spatial multiplexing frameworks such as arrayed metasurface [45] and cascaded metasurfaces [25,[50][51][52].Therefore, the combination of cascaded and arrayed metasurfaces boasts great potentials for enhancement of key and encoding spaces towards high-security and large-capacity features.
In this work, we propose a high-security and large-capacity optical encryption strategy based on multi-channel perfect high-dimensional Poincaré beams (HDPBs) generated by cascaded metasurfaces.Benefiting from the infinite modes of HDPBs and the scalability of arrayed metasurface, both key space and encoding space have been considerably expanded.Accordingly, by cascading two arrayed metasurfaces composed of 16 × 10 cells, an optical encryption platform is implemented by encoding information into five independent parameters of HDPBs.Our work provides a promising solution to extensively enlarge the key and encoding spaces and may open an avenue towards advanced optical encryption systems.

Concept and principle of high-dimensional Poincaré beams encryption
Figure 1 schematically illustrates the HDPBs optical encryption scheme using cascaded metasurfaces, which is capable of generating multi-channel HDPBs with diverse DoFs to encode optical information.In principle, HyOPBs can be described as a linear superposition of right-handed circular polarized (RCP) and left-handed circular polarized (LCP) bases with different topological charges [39].Here, we generalize a more complex vector vortex beam superposed by two HyOPBs as HDPBs, which can be represented by high-dimensional Poincaré sphere indicated by the magenta or blue sphere in Fig. 1.Taking the blue sphere for instance, the north ). Arbitrary state |ψ� on the surface of high-dimensional Poin- caré sphere can take the form as follows where Here, sin( χ 2 + π 4 ) and cos( χ 2 + π 4 ) denote the relative amplitude while Φ is the phase difference of two poles.Also, l 1 , m 1 , n 1 , l 2 , m 2 and n 2 are the topological charges and the corresponding numbers 1 and 2 refer to the contributions by two cascaded metasurfaces.R and L are abbreviations of RCP and LCP states.êx and êy represent the unit vectors along x and y axis and φ is the azimuth coordinate.For instance, � ψ n 1 +m 2 L corresponds to the LCP state carried with topological charge of n 1 + m 2 and phase of Φ.To acquire such HDPBs, two metasurfaces made up of planar chiral meta-atoms are required to be aligned to generate incident HyOPBs and perfect HDPBs.With the spin-decoupled modulation channels, the generated perfect HDPBs can lie on two different high-dimensional Poincaré spheres (that is the magenta and blue ones) according to the circular polarization states of incident light.In order to encrypt information with higher security and larger capacity, two arrayed metasurfaces are established so as to generate a variety of perfect HDPBs.Once the authentic coordinates (x 1 , y 1 , x 2 , y 2 ) of two arrayed metasurfaces are obtained, the desired states of HDPBs could be achieved through alignment of two specific metasurfaces.The plaintext can be subsequently decoded by characterizing the properties of HDPBs based on the annular intensity distributions through RCP/ LCP analyzers for RCP/LCP incidences.Otherwise, the misalignment of coordinates (a counterfeit key) would lead to generation of wrong HDPBs and make the plaintext unavailable.It should be mentioned that the scale of the arrayed metasurfaces, could be extremely expanded and contributes to a high security level.In addition, the encoding space with dependence on the states of HDPBs can also be enlarged through plenty of combinations of two cascaded metasurfaces. (1a)

Generation of multi-channel perfect hybrid-order Poincaré beams
As a precondition for HDPBs encryption, generation of multi-channel perfect HyOPBs should be realized at the first stage.To this end, a single-layer metasurface based on planar chiral meta-atoms is employed for spin-decoupled phase manipulation through breaking the mirror symmetry and higher-order rotational symmetry [53,54].As depicted in Fig. 2(a), the transmitted co-polarized φ co and two cross-polarized channels φ RL and φ LR enable three independent phase modulation channels under RCP and LCP incidences (Supplementary Note 1).By incorporating chiral meta-atoms with geometric phase, the designed metasurface can function as a combination of four optical elements, namely a quarter-wave plate, a spiral phase plate, an axicon and a Fourier lens, for generation of multi-channel perfect vortex beams with controllable topological charges of l 0 , m 0 and n 0 (see Supplementary Note 2 for detailed phase distributions of chiral metaatoms).Therefore, through coherent superposition of the co-polarized and cross-polarized perfect vortex beams, two different perfect HyOPBs ( ψ ) can be accessed according to the incident circular polarization states (Supplementary Note 3).
For experimental demonstration, a metasurface made of silicon nitride chiral metaatoms with height of 1.2 μm on fused silica substrate was prepared (See Methods for fabrication process).Figure 2(b) shows the optical microscope image of the fabricated metasurface with radius of 50 μm and topological charges l 0 , m 0 and n 0 of -3, 4 and 1, respectively.Through illuminating 470 nm light, the transmitted beams through metasurface were collected as exhibited in Fig. 2 Methods).The first to fourth columns list the generated beams under RCP, LCP, horizontal linearly polarized (H-polarized) and vertical linearly polarized (V-polarized) incidences.It is clear that the intensities of the generated beams are split into seven and four bright lobes after passing through linear polarization analyzer under RCP and LCP incidences, which correspond to the polarization orders of ψ 4

(c) (optical characterization can be referred to
Meanwhile, the annular intensity distributions present almost identical radiuses regardless of the number of topological charges.The above experimental results are coincident with the theoretically calculated ones listed in Fig. S2(b), validating the generation of multi-channel perfect HyOPBs.Additionally, these two HyOPBs can be further linearly superposed into HDPBs under linear-polarized incidences, as shown in the third and fourth column in Fig. 2(c).By changing the incident states from H-polarization to V-polarization, the generated perfect HDPBs evolves from point I to II on the equator of high-dimensional Poincaré sphere displayed in Fig. 2(d).Characterized by non-uniformly distributed polarization states with space-variant phase distribution, HDPBs can serve as information carriers to establish an enormous key and encoding spaces for optical encryption.

Manipulation of perfect high-dimensional Poincaré beams
By virtue of the multi-channel phase modulation of planar chiral metasurface, generation of HDPBs has been demonstrated above.Nonetheless, the states of HDPBs are limited within a single metasurface because of the identical co-polarized phase modulation channels.In terms of HDPBs encryption, to achieve larger key and encoding spaces, another metasurface (meta_1) is adopted to generate multi-channel HyOPBs as incidences for the following metasurface (meta_2) (see Supplementary Note 4 for detailed phase modulation design).As illustrated in Fig. 3(a), the incident RCP and LCP light are transformed into two HyOPBs ψ l 1 R + � 11 ψ m 1 L and ψ l 1 L + � 12 ψ n 1 R through meta_1.Cascaded by meta_2, the HyOPBs are subsequently modulated into multi-channel perfect HDPBs.In this work, HDPBs are designed to lie on the equator of two high-dimensional Poincaré spheres and can be expressed as (referred to equations S9a-b) and for RCP and LCP incidences.Φ 12 and Φ 11 (Φ 22 and Φ 21 ) denote the phase differences between co-polarized and cross-polarized channels of meta_1 (meta_2), respectively.After passing through RCP/LCP analyzers, the output beams turn into for RCP and LCP incidences and appear as annularly distributed bright lobes.Hence, polarization orders of two HDPBs, defined as P R =(l 1 + l 2 -m 1 -n 2 )/2 and P L =(l 1 + l 2 -n 1 -m 2 )/2, can be flexibly manipulated by designing six topological charges and emerged as the number of bright lobes.Figure 3(b) and (c) display the top and side view of the scanning electron (2) microscope images of the fabricated chiral metasurface, respectively.The experimental setup for generating HDPBs through alignment of two cascaded metasurfaces is illustrated in Supplementary Note 5. Endowed with topological charges of l 1 = 7, m 1 =-5, n 1 = 7, l 2 = 8, m 2 = 1 and n 2 = 4 (l 1 = 7, m 1 = 4, n 1 =-5, l 2 = 8, m 2 = 6 and n 2 = 8), the generated HDPBs with independent polarization orders P R and P L of 8 and 7/2 (3/2 and 7) demonstrates 16 and 7 (3 and 14) bright lobes through RCP and LCP analyzers, as shown ).The rotation angles θ R and θ L of annular intensity profiles are related to the azimuth angles Φ R = Φ 11 + Φ 22 and Φ L = Φ 12 + Φ 21 in the left (right) part of Fig. 3(d).Besides, due to the axicon and Fourier lens phase modulation provided by meta_2, HDPBs possess almost identical radiuses, exhibiting perfect characteristics.In this regard, the outline of the HDPBs could also be adjustable according to the designed ellipticities γ (defined as the ratio of horizontal and vertical radiuses).Figure 3(e) shows the intensity profiles of the perfect HDPBs with ellipticities of 0.6, 0.8, 1.0 and 1.2 (with topological charges of l 1 = 7, m 1 = 7, n 1 = 7, l 2 = 8, m 2 = 8 and n 2 = 8).The normalized intensities along horizontal and vertical directions indicated by white solid and dashed lines are plotted in Fig. S4.The corresponding ellipticities are measured to be 0.62, 0.77, 0.99 and 1.24, which match well with the theoretical values and demonstrate the capability of shaping the outlines of perfect HDPBs (Supplementary Note 6).By the way, the alignment accuracy of two cascaded metasurfaces is required to be about 5 μm (see Supplementary Note 7), which can be well accessible in experiments.
In addition, the states of HDPBs on the equators of high-dimensional Poincaré spheres shown in Fig. 3(f ), which are represented by azimuth angles Φ R = Φ 11 + Φ 22 and Φ L = Φ 12 + Φ 21 and reflect in the rotation angles θ L and θ R of bright lobes, is accessible to be manipulated by adjusting Φ 12 , Φ 11 , Φ 22 and Φ 21 .It is worth noting that the influence of Gouy phase on azimuth angles should be taken into careful consideration (see Supplementary Note 8 for detailed analysis).The rotation angles θ L and θ R of bright lobes (indicated by the white dashed lines in Fig. 3(f )), which are defined as the minimum positive angle between the center of bright lobes and the x-coordinate axis, can be employed to retrieve the azimuth angles Φ R and Φ L of HDPBs (see details in Supplementary equations S9e-f and Supplementary Note 9).As depicted in the four corners of Fig. 3 , are characterized by RCP and LCP analyzers.For both RCP and LCP incidences, the generated beams via four sets of cascaded metasurfaces all present 31 bright lobes determined by polarization orders P R and P L of 31/2 but with different rotation angles θ R and θ L , corresponding to the states marked by Greek numerals I, II, III and IV featured with azimuth angles Φ R of 270°, 0°, 90°, 180° and Φ L of 0°, 90°, 180°, 270°.Therefore, flexible and independent manipulation of polarization orders P L and P R , ellipticities γ along with azimuth angles Φ R and Φ L of perfect HDPBs have been demonstrated in virtue of two cascaded multi-channel chiral metasurfaces.

Optical encryption based on cascaded metasurfaces
After investigating the manipulation capability of both the outline and states of HDPBs via cascaded metasurfaces, five explored independent parameters, namely γ, Φ R , Φ L , P R and P L , can be employed for optical information encoding.Furthermore, in order to construct an optical encryption system on the basis of HDPBs, two arrayed metasurfaces are implemented to enable a high-security platform with enlarged encoding capacity, as presented in Fig. 4(a).The first arrayed metasurfaces (denoted as arrayed meta_1) is composed of 16 × 10 meta_1 cells for generating multi-channel HyOPBs as incidences for the second arrayed metasurfaces (denoted as arrayed meta_2) consisting of 16 × 10 meta_2 cells (see Tables S2 and S3 in Supplementary Note 10).Through combination of different cells in two arrayed metasurfaces, an abundant number of perfect HDPBs featured with different parameters are available.As a demonstration, the plaintext messages of four Chinese words "南京", " 金陵", "建邺" and "秦淮" (four names of Nanjing in Chinese), are encrypted into two metasurfaces (ciphertext) through the Unicode encoding in the form of five parameters of perfect HDPBs, accompanied with the customized keys in the form of coordinates (x 1 , y 1 , x 2 , y 2 ).Once the ciphertext and customized keys are transferred into four users, the decryption process begins with the alignment of two arrayed metasurfaces.In accordance with the received coordinates (x 1 , y 1 , x 2 , y 2 ), the target cells are subsequently illuminated by 470 nm RCP and LCP light for generation of perfect HDPBs, respectively.As a consequence, each user who owns two keys could obtain two sets of HDPBs, depicted in Fig. 4(b).Then, the parameters of HDPBs, including γ, Φ R and Φ L , P R and P L , are retrieved and transformed into digital forms, as listed in Fig. 4(c).With reference to the self-defined look-up table in Supplementary Note 11 each user could translate the obtained messages into two four-bit hexadecimal digits in Fig. 4(d).As indicated in Fig. 4(e), the ciphertext would eventually be decoded through Unicode encoding as "南京", "金陵", "建邺" and "秦淮", respectively.So far, the optical encryption platform has been established via two cascaded metasurfaces based on multi-channel perfect HDPBs.Then, an electron beam resist (PMMA-A4) with the thickness of 200 nm was spincoated on top of the SiN x layer.The patterns of chiral meta-atoms were subsequently defined on PMMA resist by Electron-beam lithography (EBL) system (ELS-F125, Elionix).After development, the generated pattern was transferring to an electron beam evaporated chromium layer, which served as a hard mask for the following dry etching process of SiN x layer in a mixture of CHF 3 and SF 6 plasma (Oxford Instruments, PlasmaPro100 Cobra300).At last, the remaining chromium mask was removed by a solution of ammonium cerium nitrate.

Optical characterization
A 470 nm laser was utilized for optical measurement and focused onto the fabricated metasurfaces through an objective (20×, NA = 0.40).The polarization states of the incident light were controlled by the combination of linear polarizer and quarter-wave plate.
After illuminating the metasurface/cascaded metasurface, the transmitted light was subsequently collected by an objective (20×, NA = 0.40) and analyzed by polarizers.Subsequently, the intensity profiles of the generated Poincaré beams were recorded using a CMOS camera.The alignment of two cascaded metasurfaces is detailed in Supplementary Note 5.

Fig. 1
Fig. 1 Schematic illustration of cascaded metasurfaces based perfect HDPBs for optical encryption.The plaintext is encrypted into two cascaded metasurfaces with multiple cells.According to the customized keys in the form of coordinates (x 1 , y

Fig. 2
Fig. 2 Generation of multi-channel perfect HyOPBs.a Planar chiral meta-atom for phase modulation of co-polarized and cross-polarized channels (φ co , φ RL and φ LR ) under RCP and LCP incidences.The single-layer chiral metasurface functions as the combination of quarter-wave plate (QWP), spiral phase plate (SPP), Axcion and Fourier lens.b The optical microscope image of the fabricated chiral metasurface.Scale bar: 20 μm.c The measured intensity profiles of perfect HyOPBs and HDPBs with topological charges l 0 , m 0 and n 0 of -3, 4 and 1 at 470 nm under RCP, LCP, H-and V-polarized incidences.The white arrow in the left corner of each image corresponds to the input polarization state while that in the right corner depicts the polarization state of analyzers.The first row represents the output intensity without polarization analyzers.d The high-dimensional Poincaré sphere that describes the states of the generated Poincaré beams under linearly polarized incidences

Fig. 4
Fig. 4 Optical encryption demonstration based on cascaded metasurfaces.a The optical microscope images of two arrayed metasurfaces both including 16 × 10 cells which serve as ciphertext.Scale bar: 200 μm.b Information decryption by generating HDPBs through alignment of two metasurfaces according to the customized keys in the form of coordinates (x 1 , y 1 , x 2 , y 2 ) for four users (each user possesses two keys).c The obtained messages about five parameters of the generated HDPBs and (d) the corresponding retrieved 4-bit Unicode characters based on look-up table.e The decrypted plaintext of four Chinese characters through Unicode encoding by four users