Encrypted metasurfaces with inherent asymmetric-like digitized keys under decoupled near-field parameters

With the increasing development of information technology, the issue of information security has attracted widespread attention across diverse domains. In the field of optics, the properties of optical parameters are utilized to protect information and communication security and various solutions have been devised, including quantum key distribution and one-time pads. As integrated devices evolve, encryptions based on metasurfaces with the convenience of modulating optical parameters^{1,2,3,4,5,6,7,8} have gained considerable interest. Due to the inherent confidentiality of polarization, polarization encryption^{9,10,11,12,13} became the initial preference. But due to its simple form and easy cracking, now it only serves as the foundation for the transmission and encryption of multidimensional information. With the expansion of far-field degrees of freedom, holographic encryption^{14,15,16,17,18} based on Pancharatnam-Berry (PB) phase has emerged as an effective approach. The utilization of far-field parameters increases the number of channels and expands the workspace capacity for encrypted metasurfaces. Due to the combination of encrypted metasurfaces and PB phase, library construction becomes more complex. Therefore, adopting the phase parameter to ensure the logical correctness of the near field is a feasible method. Whether in the near-field or far-field, the manipulation of electric fields by metasurfaces can be adopted as a method of multi grayscale display. Consequently, researchers have devised steganography algorithms of images^19,20 to bolster the confidentiality of metasurfaces, which can be divided in two types. The first involves hiding plaintext within an image using algorithms, and achieving information transmission through the metasurfaces’ light field modulation capability. The second type involves converting the image into a static QR code and loading it onto the metasurface. Since the metasurface only needs to process binary data, this approach ensures the accuracy of the information and reduces crosstalk. However, a single encryption method is easily cracked. To ensure security, researchers proposed multiple^21,22,23, and progressive^24,25,26 encryption methods, allowing different algorithms loaded on or creating fake text for confusing thieves. Guo et al.²⁷ achieved data transmission within various polarization channels based on four meta-atoms via spatial multiplexing, laying the foundation for the application of different algorithms on the same metasurface. Similarly, Zheng et al.²⁸ combining graphics, hided the plaintext and the cypher code in the form of images within different channels based on adjusting the arrangement of meta-atoms, realizing the combination of space reuse and image encryption. Li et al.²⁹ introduced additional calculations among information channels within spatially multiplexed structures, hiding the real nanoprinting with ciphertext behind two images. Above, we have elaborated on the progress of encrypted metasurfaces from the perspective of the underlying logic. Accompanying with the development of underlying logic, various research efforts have highlighted new functionalities, such as cascading metasurfaces³⁰, reprogrammable metasurfaces in encryption^31,32,33, continuous spectral response³⁴, and orbital angular momentum metasurfaces³⁵. These metasurfaces have ingeniously expanded encryption functions, significantly increasing the capacity of encryption channels. For example, cascaded metasurfaces allow multiple metasurfaces to work together, expanding the degrees of freedom and increasing the upper limit of encryption channels. Reprogrammable metasurfaces, by repeatedly altering the structure of nanopillars, can erase plaintext or keys, enabling device reuse. Furthermore, encrypted metasurfaces with continuous broad spectral response operate across multiple wavelengths but have stringent requirements for the optical responses of the nanopillars. As for orbital angular momentum metasurfaces, they can reuse information by modifying far-field parameters, but controlling crosstalk becomes more challenging.

Recently, Zhang et al.³⁶ searched for a method based on wavelength multiplexing, achieving image transmission using two wavelengths to establish multiple channels to hide the real channels, balancing design complexity and transmission performance. While these studies have contributed greatly to the development of metasurface encryption, it is important to note that in all methods mentioned above, the selected meta-atoms need to be capable of generating specific responses to multiple wavelengths or arranged in a spatially multiplexed form. As a result, this leads to the complexity of structure optimization and insufficient space utilization. Crucially, the existing loading theory and the matching of different light field and information are no longer adequate to meet the demands of the rapidly evolving field of encrypted metasurfaces. Therefore, we categorize the methods based on saving encrypted information through structures as structural encryptions, and attempt to find a new perspective to break this current impasse which is a new challenge.

In this paper, we introduce an innovative approach to constructing an encrypted metasurface, referred to as asymmetric-like encryptions. This approach can utilize various encodings and logical judgment conditions in design to generate diverse mathematical keys, thus achieving comprehensive digitization of encryption and decryption processes. Through layered-control, wherein dichroism and polarization conversion are individually manipulated in two distinct layers, we have effectively decoupled the anti-diagonal entries of the corresponding Jones matrix. This breakthrough allows the metasurface to flexibly modulate both cross-polarization components independently for the first time. Using an orthogonal polarization basis, we systematically integrated potential spatial light field distributions and devised working and output states, directly and indirectly elucidating the modulation characteristics of meta-molecules, respectively. Building upon this foundation, we designed logical states to encode composite information. We encoded three predefined plaintexts into three-bit data strings according to channel order, then encrypted and embedded them within the four co- and cross-polarization states of spatial light field distributions, termed intermediate states. The correlation between channels (information transmitted) and intermediate states provides a basis for constructing mathematical keys which are in form of logical expressions.

Jones Matrix Derivation for the Layered-Control Metasurface

The proposal includes a layered-control theory that decouples the cross-polarization components of the near-field and an overall architecture of asymmetric encryption platform. The predesigned messages should be converted into the form of QR codes, with each pixel of the QR code corresponding to the nanopillars within a 3×3 area of the metasurface, as shown in Fig. 1. To prioritize the display of metasurface structures within the Fig. 1, we used a one-to-one pixel-to-nanopillar representation. The predesigned messages are arranged as ordered 3-bit data strings, where each string corresponds to a logical state containing 4-bit Boolean values. Each logical state can be matched with the metasurface structure according to established logic. When a pair of orthogonal polarized light passes through the metasurface, the co- and cross polarization components can be detected and defined as four light field distributions in the near field, which we call intermediate states, with the transmitted information hidden within them. It is worth noting that we have taken into account the characteristics of visual secret in our metasurface design, ensuring that all ciphertexts cannot be directly decoded. The design of the metasurface guarantees that the ciphertext of each channel must be decrypted using the corresponding key. In the decryption process of the three channels, a progressive encryption method is adopted. The difficulty of decrypting information from different channels varies to ensure the security of the channel with the highest confidentiality, which requires obtaining all the light field distributions to match the corresponding key.

The spatial positions of the predesigned information correspond one-to-one with those on the metasurface. The highlighted information in color is conveyed through nanopillars marked with the same color on the metasurface, as shown in the blue and yellow regions. By using a pair of orthogonally polarized light incident on the metasurface, four encrypted intermediate states can be generated in the near field. The encrypted information can then be extracted by decoding the intermediate states using mathematical keys.

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The proposed double-layer metasurface manipulate transmittance and polarization conversion through two functional layers independently. This approach allows us to decouple the anti-diagonal entries of the corresponding Jones matrix, constructing logical states for encoding with selected modulation combinations. The electric field passing through the metasurface can be expressed as

where ({J}_{u}) and ({J}_{l}) represent the Jones matrices of the upper and lower layers, respectively. The lower layer controls transmittance as a dichroic element, while the upper layer controls polarization conversion efficiency with ideal perfect transmittance. Thereby, the two properties, transmittance and polarization, can be independently modulated by corresponding functional layer according to transmitted lights. The corresponding Jones matrix ({J}_{T}) of the metasurface can be described as

where (psi) represents the rotation angle of the upper meta-atoms, (Rleft(psi right)) is the rotation matrix, ({t}_{x}) and ({t}_{y}) denote transmittances of the lower layer, ({phi }_{{x}_{1}}) and ({phi }_{y1}) represent the phase delay of the fast axis and the slow axis in the upper layer, while ({phi }_{{x}_{2}}) and ({phi }_{{y}_{2}}) represent the phase delay of the fast axis and the slow axis in the lower layer. The subscript ({x}_{1}) and ({y}_{1}) represent the parameters of the upper layer and the subscript ({x}_{2}) and ({y}_{2}) represent the parameters of the lower layer. By substituting Eq. (2) into Eq. (1) and squaring (|{E}_{out}rangle), while (|{E}_{in}rangle) polarized along (hat{x}) or (hat{y}), the intensity of two orthogonally polarized lights can be formulated as

where (delta ={phi }_{{y}_{1}}-{phi }_{{x}_{1}}), (P(delta )=1-cos delta), (D(psi )={sin }^{2}psi {cos }^{2}psi) and the subscript ({xy}) denotes (hat{y}) polarized component that has been switched by (hat{x}) polarized incident light. Detailed derivation can be found in Supplementary Note 1, Supplementary Materials. With a layered design approach, the complex transmission matrix of the upper layer simplifies into a diagonal matrix, comprising perfect transmission coefficients multiplied by phase delays. Since the upper layer is no longer influenced by additional transmission coefficients, the phase difference and rotation angle of the upper layer can collectively impact the polarization conversion efficiency through a product of newly defined functions. However, due to the removal of rotational degrees of freedom in the lower layer, the corresponding Jones matrix of meta-atoms transitions from a conventional four-element square matrix to a diagonal matrix containing only complex transmission coefficients. Consequently, the transmittance of lower layer contributes to the four elements of the Jones matrix through the form of product terms. Besides, the prerequisite of light output is that the lower layer provides non-zero transmittance. By controlling the polarization conversion efficiency in the upper layer, the phase difference (P(delta )) and the rotation angle (D(psi )) can modulate the output light intensity. A comparison between Eq. 3b and Eq. 3c reveals two light intensity expressions affected by the different transmittance provided by the lower layer and the polarization conversion efficiency by the upper layer, indicating decoupling of the anti-diagonal elements of Jones matrix. Therefore, in the subsequent logic coding design, the Boolean values represented by these two parameters can be designed differently. This decoupling allows for a flexible modulation of all elements within the Jones matrix, facilitating the construction of logical states for message encryption in subsequent logic coding designs.

Screening and Construction of Logical States

Because both parameters, ({I}_{{xx}}) and ({I}_{{yx}}) represent responses in the x direction, while ({I}_{{xy}}) and ({I}_{{yy}}) correspond to the y direction responses, we combine ({I}_{{xx}}) & ({I}_{{yx}}) into ({I}_{x}^{{out}}), and ({I}_{{xy}}) & ({I}_{{yy}}) into ({I}_{y}^{{out}}) to simplify the compositions of the logical states. So, now these four parameters, ({t}_{x}), ({t}_{y}), ({I}_{x}^{{out}}), ({I}_{y}^{{out}}) serve as logical states’ parameters in the form of Boolean value, configuring independent encrypted channels. Note that ({I}_{x}^{{out}}) here contains the co-polarization component when x-polarized light is incident, as well as the cross-polarization component when y-polarized light is incident. Similarly, ({I}_{y}^{{out}}) contains the cross-polarization component when x-polarized light is incident, as well as the co-polarized component when y-polarized light is incident. When only one direction of polarized light is allowed to pass through the lower layer, the first two bits of the logical state represent the polarized light allowed to pass, and the last two bits represent the polarization of output light. While the phase difference (delta) equals 0, the output light contains only co-polarization component. While the phase difference (delta) equals to (pi /2), only the cross-polarization component exists in output light. Besides, co- and cross- polarization components will coexist when the phase difference equals (pi /4). In situation that both (hat{x}) and (hat{y}) are concurrently permitted to pass through the lower layer, the relation between an individual logical state and the component of output light is no longer an injective function. Thereby, the logical state necessitates a supplementary definition in this scenario: the last two digits signify the polarization conversion efficiency under (hat{x})-polarized light passing through.

In order to filter all permutations and combinations, the following guidelines should be observed: (1) The presence of input necessitates an output, and vice versa. (2) Polarization conversion efficiency remains unaltered under varying polarized light conditions in a specific meta-molecule. Ten feasible scenarios following the guidelines strictly have been selected and enumerated in Table 1. To construct encrypted channels, at least two logical states should be adopted in each channel for recognition. In the case of Channel 1, it utilizes ({t}_{x}):1, and ({t}_{x}):0, signifying that when (x) polarized light is permitted to transmit, the pixel is recorded as “TRUE”. Similarly, Channel 2 chooses ({I}_{y}^{{out}}):1 and ({I}_{y}^{{out}}):0 as the configuration. With the commonality between these logical states, they can be identified as “x11x” and “xxxx”. So, the Boolean value “TRUE” of Channel 3 can be constructed based on the condition “x11x”, where x represents the undefined bit. For more details about the process followed screening mechanisms, please refer to Supplementary Table 1–6 in Supplementary Note 2, Supplementary Materials.

Construction of Intermediate States and Derivation of Decryption Keys

Based on properties of incident and output light, we introduce an additional logic set, intermediate states, to realize analog-to-digital conversion. The angle of the receiver can be adjusted to match that of the incident light polarizations, or it could remain perpendicular. Thereby, four polarization components can be detected when the metasurface is subjected to orthogonally linearly polarized light, corresponding to the combination of “({I}_{{xx}}), ({I}_{{xy}}), ({I}_{{yx}}), ({I}_{{yy}})” which are spatial light field distributions defined as intermediate states. For the convenience of recording, (A,{B},{C},{D}) are chosen as new symbols correspondingly. It facilitates the specification of the outputs that hiding the messages, and establish connections to logical states.

A maximum of three independent channels can be designed, due to the limitation of finite distinguishable parameters. Thus, each meta-molecule can be considered to carry a three-bit length data string. The overall relationship between the data string and the logical state is shown in Table 2. By looking up the table, the accurate logical state of each pixel on the metasurface can be obtained. Once the data string is determined, the combination of nanopillars in each layer will be selected accordingly.

The complexity of the key generation process is determined by the confidentiality of channels. In a relatively simple case, the key construction for channel 1 begins by screening the intermediate states where the high-order (leftmost) bit of the data string is “1”. The relevant data strings are [100], [110], [101], and [111], with their corresponding intermediate states being (A), (B), ({AD}) and ({ABCD}), respectively. The next step involves taking the union of these sets. Since both ({AD}) and ({ABCD}) are subsets of (A), the key for channel 1 can be expressed as: ({Ch}1=A+B+{AD}+{{{rm{A}}}}{BCD}=A+B).

While the relationship between the data string and its logical state is well-defined and unique, the relationship between logical states and intermediate states necessitates further considerations due to the presence of dual-input light. As a result, the intermediate state ({AD}) does not contribute to channel 2, as confirmed by the corresponding data string [101]. However, its superset, (D) (which corresponds to the data string [010]), takes precedence in the calculation. Therefore, it is essential to first remove the intermediate state ({AD}) and subsequently supplement its subset, ({ABCD}) to ensure that the calculation proceeds correctly.

For channel 3, the key is derived from the union of all intermediate states where the lower bits (rightmost) of the corresponding data strings are “1”. To sum up, the logical expressions of all three channels can be derived as

Traditional asymmetric encryption is characterized by the use of algorithms to generate a pair of mathematically related keys, where the public key is used for encryption and the private key is used for decryption. In our proposal, data is converted into corresponding physical entities. The encryption part takes the form of a set of logical states, while the decryption part is akin to a traditional private key, since the light field can be mathematically expressed.

The proposed transmission model provides more adjustable parameters for independent control of the anti-diagonal entries, which means cross-polarization components of light field could be precisely modulated. From the perspective of information encoding, the asymmetric encryption can achieve more accurate recognition of the light field. By constructing and subdividing two logical states, the description of non-reciprocal devices has been achieved. Our driving force is to create new platforms for complex light field modulation and creative functions loading.

To overcome this limitation that only partial near-field parameters could be independently modulated under a pair of orthogonal linear polarization bases, we employed layered control for dual parameter modulation and conducted theoretical derivations, laying the groundwork for asymmetric-like encryptions to obtain more parameters used in designing digital encoding.

Traditional asymmetric encryption and decryption involves converting a string into another string, while in our proposal, plaintext will be translated into Boolean values in two-dimensional coordinates at first and converted into the modulation capability of nanopillars for light. The output form of ciphertext is the spatial light field distribution, and decryption keys convert it back into Boolean values in two-dimensional coordinates afterwards.

By decoupling the anti-diagonal entries of Jones matrices, we obtained parameter combinations with identifiable modulation capacities of meta-molecules, serving as logical states for the construction of channels. The co- and cross-components of outgoing light retain distinct portions of the encrypted messages for decryption analysis. Through the mathematical key (logical expressions), encrypted messages can be derived and reproduced. This work activates the self-confidentiality of metasurfaces, realizing asymmetric-like encryption and a comprehensive process of digitized encryption and decryption.

Previously, channel design relied on only three parameters, necessitating eight types of meta-molecules with fixed properties to achieve logical encoding. However, with the introduction of an additional free control parameter in the near-field, structural possibilities have expanded. We have the capability to customize the correspondence between the structure and encoding, implying that the encoding is not unique, although adherence to certain rules is still necessary. In our work, this advancement achieves cross-storage of information, enabling real visual secret. Unlike traditional methods, our encryption process incorporates an intermediate state for cross-storage of information. This introduces an encryption protection from the metasurface. Moreover, this encryption can be characterized using mathematical expressions, offering heightened confidentiality compared to previous approaches.

This work was conducted under a pair of orthogonal linear polarization bases, where the angular degree of freedom provided by nanopillars contributes to the near-field. Thus, in theory, it is not possible to attach far-field phase information. We predict that at least one more channel without self-confidentiality can be constructed on a circularly polarized basis to approach the realization of a full-space digital metasurface. However, the theoretical derivation of the corresponding near-field parameters will be more challenging.

To characterize each channel and verify the theory, simulations of the predesigned metasurface are carried out by FDTD Solutions (Lumerical Inc., Vancouver, BC, Canada). By sweeping elaborately, the structure of each meta-molecule can be obtained to realize predesigned functions. In the lower layer, four types of nanopillars are engineered to either permit the passage of x-polarized, y-polarized, orthogonally linearly polarized light, or induce complete reflection. In the upper layer, three types of nanopillars are configured to control polarization conversion efficiency at levels of 0%, 50%, 100%, respectively. Structural details regarding meta-molecules and their correspondence to logical states are provided in Table 3. Detailed field and phase information could be found in Supplementary Fig. 1–7 and Supplementary Table 7–9, Supplementary Note 3, Supplementary Materials.

In our proposal, the working wavelength is 532 nm and each meta-molecule comprises dielectric nanopillars made of silicon. The bottom substrate, with a height of 200 nm, and the filling medium between the two layers, with a height of 400 nm, are composed of SiO2. Previous work^13,36,37,38 has demonstrated that this structure can be fabricated using Silicon-On-Insulator technology, including the processes of deposition, patterning, lift off and etching. Additionally, the structural parameters of the meta-atoms in each layer are illustrated in Fig. 2. The period of the meta-molecule is 300 nm, and the height of the meta-atoms is fixed at 500 nm.

All parameters used in the sweep are indicated in the corresponding positions.

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In actual QR code loading, we used (63times 63) pixels, which correspond to (189times 189) meta-molecules. When orthogonal linearly polarized light is incident, the outgoing light field can extract co-polarized and cross-polarized components, with these spatial light field distributions corresponding to intermediate states. We provided a detailed example to illustrate the information loading process, as depicted in Fig. 3. The section marked with a white box corresponds to a fixed position in the preset information. Since all three preset information at the same position are “1”, the data string is [1 1 1]. Based on the correspondence of Table 2 and Table 3, we set the meta-molecular structure at the same position on the metasurface using the lookup table method. Each binary information at the corresponding position is arranged sequentially into a data string and translated into a logical state using Table 2. Each logical state represents a nanopillar with a specific structure, thereby confirming the entire metasurface design.

The proposed architectural model consists of two layers of metasurfaces, with the upper layer controlling polarization conversion efficiency and the lower layer controlling transmittance. The data string is composed of Boolean values extracted from the same positions in the predesigned information, such as the highlighted white area, which corresponds to a unique logical state for the nanopillar structure design.

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By extracting ({E}_{x}^{{out}}) and ({E}_{y}^{{out}}) from the analyzer under 532 nm incident light polarized along (x)-axis and (y)-axis, four original images of intermediate states are obtained, as shown in Fig. 4(a). We optimize four original images for the obtainment of reproduced messages, including normalizing, filtering, threshold denoising, and binarization. The threshold for binarization is determined based on the count of bright points in pixels marked with red diagonals because these pixels consistently convey the same Boolean value, as depicted in Fig. 4(b). Consequently, the minimum count of bright points within pixels at specific fixed positions have been established as the threshold for binarization. Finally, the predesigned information can be restored by utilizing the mathematical key (Eqs. 4a–c), which is illustrated in Fig. 4(c). Error rate analysis could be found in Supplementary Fig. 8–10, Supplementary Note 4, Supplementary Materials.

a Direct outputs from the analyzer in the optical path. b The patterns after preliminary optimizations including normalizing, filtering and threshold denoising. The minimum value among all pixels marked with a red diagonal is adopted as the threshold for binarization in the single image. c After binarization, the four digital images can be converted into three channel outputs through logical expressions, reproducing the encrypted information.

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