Vortex beams (VBs) are optical beams characterized by a helical phase structure that enables them to carry orbital angular momentum (OAM). These beams exhibit a phase distribution of the form eilθ, where l represents the topological charge and θ represents the azimuthal angle. A key feature of VBs is the presence of a central phase singularity, which generates a core with no intensity and a surrounding ring-shaped intensity distribution [1–6]. This unique characteristic has spurred interest in VBs for various applications, such as optical manipulation, quantum communication, and high-resolution imaging [7–11]. The ability of VBs to carry multiple OAM modes allows for increased information multiplexing, thereby enhancing the data transmission capacity of optical systems [12–14]. In imaging, VBs play a crucial role in super-resolution microscopy [15–17] and optical trapping [18,19], offering improved spatial resolution and precision. Furthermore, VBs have shown considerable potential for information encryption [20–23], wherein variations in OAM modes facilitate secure and efficient communication. However, standard approaches for generating vortex beams, including spiral optical elements and devices for spatial light modulation [24,25], often suffer from issues such as bulky setups and low efficiency, which limits their practical use.

Metasurfaces are artificial two-dimensional materials comprising subwavelength building blocks with ultrathin structural features. By arranging these elements in a specific manner, they can generate uniquely tailored electromagnetic wave modulation capabilities that enable various applications. Compared with conventional devices, metasurfaces benefit from their subwavelength size avoiding phase accumulation dependent on wave propagation distances comparable to or even greater than the operating wavelength. In addition, metasurfaces can tune the wavefront of electromagnetic waves by introducing abrupt phase changes through ultrathin slices of subwavelength resonators. The first such metasurface was proposed by Yu and Capasso [26], and subsequent designs have been demonstrated in the terahertz [27], microwave [28], near-infrared [29], and visible ranges [30,31]. These metasurfaces have found use in various applications such as optical cloaking [32,33], generation of the photonic spin Hall effect [34], generation of diffraction-free beams [35], and preparation of digitally encoded metasurfaces [36,37]. In the field of optical information encryption, metasurface technology provides innovative ideas for realizing high-dimensional and high-security encryption schemes. Liu et al. [38] designed a metasurface encoding multidimensional SAM and OAM states based on an all-dielectric metasurface platform, realizing optical encryption in the visible broadband. Song et al. [39] developed a surface-relief plasmonic metasurface consisting of shallow nanoapertures, which enables optical decryption through specific combinations of incident and reflected polarized light by incorporating four different orientations of nanoapertures. The remarkable optical properties of the metasurfaces have driven the continuous development of ultrathin optical devices. Although metasurfaces provide their unique freedom in modulating the wavefront, most metasurface-based devices usually suffer from inefficiency, which somewhat hinders the precise modulation of electromagnetic waves by transmissive metasurfaces. Therefore, improving the transmittance of metasurface devices is imperative. The geometric phase [40,41], often referred to as the Pancharatnam–Berry phase, represents a fundamental optical phase effect, whereby light undergoes spatial or geometric transformations that result in phase shifts due to changes in its polarization state. Unlike conventional phase changes, the formation of the geometric phase depends on the geometry of the object and the angle of incidence, but not on the light propagation path. This effect plays a crucial role in the precise manipulation of polarization, phase, and OAM generation. In recent years, vortex beam generation in the microwave band has been investigated using various modulation techniques [42,43]. However, studies on multilayer metasurfaces in the microwave band for imaging and information encryption remain limited. Consequently, developing more efficient metasurfaces to generate VBs and facilitate information encoding and encryption is crucial.

In this study, a vortex beam generator that utilizes a U-shaped symmetric multilayer metallic transmissive element surface is presented. This device exhibits high transmission efficiency at an operating frequency of approximately 10.67 GHz and meets geometric phase modulation conditions. The designed U-shaped structure affords considerable design freedom, and calculations of the vortex effect at different topological charges confirm its scalability. Next, we explore the reduction of the array size to produce high-quality vortex effects while maintaining high purity. Moreover, we confirm the meta-quaternion array [44] as the smallest unit, which is used as a pixel-based metasurface to construct subsequent target images. A digitally patterned array plate comprising meta-quaternion arrays is prepared and experimentally characterized using right-handed circularly polarized (RCP) light to evaluate their performance in constructing target images. Experimental results agree well with theoretical simulations, indicating that the geometric phase-based meta-quaternion arrays can be used as vortex pixels for constructing arbitrary numerical and alphabetic patterns, a finding further confirmed in subsequent simulations. In addition, we constructed metasurfaces featuring various numerical and alphabetic patterns comprising these meta-quaternion arrays and achieved multilayer encryption of target characters by combining the classical Luoshu nine-grid with a weighted superposition computation technique. Our study presents (i) new methods for manipulating vortex fields using metasurfaces and (ii) their applications in encryption, thereby offering promising prospects for encryption applications in wireless communications.

Figure 1 shows the layout of the transmissive metasurface designed to produce VBs along the light propagation direction. A 20×20 metasurface array is constructed to generate VBs with distinct topological charges. Additionally, a 2×2pixel metasurface array is employed to form the pixel elements of the target digital pattern. When an RCP beam is incident, VBs and the corresponding target digital patterns are observed at an observation plane located at a specified distance behind the metasurface. The arrangement of pixel metasurfaces corresponding to various digital patterns can be designed to generate the required target pattern. Furthermore, by incorporating the classical Luoshu nine-grid and weighted superposition calculations, multilayer encryption of the target plaintext is achieved, thereby implementing an encryption function.

The geometric phase principle underlying the constructed metasurface can be conveniently explained using the Jones matrix formalism [45]. The conversion of circular polarization is achieved through a spatially varying phase shift. Specifically, a wave plate with a spatially varying fast axis is described by the Jones matrix: T(r,φ)=cos(δ2)(1001)+isin(δ2)(cos(2θ)sin(2θ)sin(2θ)−cos(2θ)),(1)where δ is the phase delay of the wave plate and θ is the angle between the fast axis and the x-axis. For incident light with left-handed circular polarization, the output beam can be expressed as $$\begin{gathered} Eout(r,φ)=T(r,φ)Ein(r,φ) \\ =E0cos(δ2)(1iσ)+iE0sin(δ2)ei2σθ(1−iσ). \end{gathered}$$(2)

In this context, σ=±1 corresponds to the polarization states of left-handed and right-handed circular polarization. Equation (2) indicates that the output beam comprises two components: the first is the co-polarized transmitted light, which preserves the polarization characteristics of the incident light, and the second is the cross-polarized transmitted light, which exhibits a phase modulation of 2σθ and is oppositely polarized to the incident light. By adjusting the angle θ between the fast axis of the wave plate element and the x-axis, it can be varied between 0 and π phase shifts over the entire range from 0 to 2π, while preserving the same transmission amplitude. The amplitude of the cross-polarized transmitted light depends on the phase retardation δ. Theoretically, a phase retardation of π allows for 100% polarization conversion efficiency; under such conditions, the metasurface functions equivalently to a half-wave plate (HWP), achieving perfect polarization conversion.

In this study, we employ a geometric phase-based metasurface to manipulate RCP light, thereby generating VBs with distinct topological charges. The geometric phase is introduced through the structural design of metasurface elements, such as rotated metallic units, which modulate the light’s propagation path and consequently alter its phase distribution. The geometric phase can be described mathematically by ΦPB=2σθ,(3)where σ=±1 denotes the handedness of the incident polarization and θ represents the rotation angle of the metasurface unit. By precisely adjusting θ, the phase of the incident light is modulated, enabling the generation of VBs with varying topological charges.

For the generation of VBs, we employ geometric phase-based metasurfaces, specifically using a spiral phase plate (SPP) approach to directly modulate the incident light’s phase. The phase of the vortex beam is manipulated using the SPP, which can be expressed by the following formula: Φ(x,y)=l·arctan(yx).(4)

Here, Φ(x,y) represents the phase pattern at the spatial point (x, y), and l is the topological charge that determines the beam’s helicity. The function arctan (y/x) generates the spiral phase, causing the light to acquire a helical structure in the spatial domain and resulting in the formation of OAM.

Based on these considerations, we design a multilayer metallic structure, as illustrated in Fig. 2. The unit cell consists of top and bottom U-shaped metallic resonators and an intermediate metallic plate of the same structure. These three layers are separated by two dielectric layers, each composed of a 2-mm-thick F4B high-frequency board (dielectric constant εr=2.65; dielectric loss tanδ=0.001). The interlayer coupling induces effective magnetic currents within the structure, and the U-shaped design provides sufficient anisotropic freedom for electromagnetic responses. By precisely tuning the structural parameters, we achieve the final design of the unit cell design. As shown in Fig. 2(a), the unit cell configuration is fixed: the period p=12.8mm, metallic layer thickness hm=0.065mm, dielectric layer thickness hD=2mm, l=7.5mm, w1=3.5mm, w2=4mm, w3=2mm, and R=5.8mm. We use CST software to verify the performance of the designed metasurfaces by computing the transmission coefficients and phase profiles for x- and y-polarized components over an operating frequency range of 8–12 GHz, as shown in Figs. 2(b) and 2(c). According to the simulation results, at the operational frequency of 10.67 GHz (indicated by the red dashed line), the transmission coefficients for the x- and y-polarized components are 0.99 and 0.86, respectively, with a transmission phase difference of approximately 180°. Thus, the designed unit structure functions as an HWP near 10.67 GHz, satisfying the geometric phase modulation conditions.

To verify the capability of the proposed metasurfaces to manipulate vortex light fields via geometric phase modulation, we conducted simulations using CST software. Three metasurfaces, each with an array size of 20×20, were modeled. By varying the topological charge, VBs with topological charges of 1, 2, and 3 were generated. The near-field simulation outcomes for these VBs are shown in Fig. 3. The results demonstrate that the designed metasurfaces successfully manipulate the geometric phase of the incident beam to generate the corresponding VBs. Furthermore, the VBs exhibit high mode purity for topological charges 1, 2, and 3, rendering the metasurfaces suitable for applications in vortex beam imaging and encryption. Additionally, simulations were extended to explore phase modulation for topological charges ranging from 1 to 9 in Fig. 4. In our previous investigation, we examined the vortex generation effects of arrays with topological charge values of 1, 2, and 3. To further explore the scalability of the proposed structure, we extend our analysis by considering six additional cases, increasing the topological charge up to 9. The results demonstrate that a series of high-purity vortex beams can still be effectively generated, confirming the scalability of the designed metasurface. Moreover, it is noteworthy that as the topological charge increases, the central region of the metasurface exhibits deviations in both electric field intensity and phase from the theoretical distribution of vortex beams. This discrepancy arises from the low phase sampling rate, leading to reduced resolution. Overall, these findings indicate that the proposed array configuration can reliably generate vortex beams with high mode purity corresponding to the assigned topological charge values.

Next, based on the arrangement of the aforementioned arrays, it is observed that the generation of a well-defined vortex beam relies on relatively large arrays. This raises the question of whether smaller vortex plate structures can produce high-quality VBs. As shown in Figs. 5(d)–5(f), a smaller 4×4 array structure is designed and simulated using CST. With incident light at the same frequency used for the larger arrays, the VBs maintain good performance in both the near-field and far-field regions. However, this configuration may not represent the smallest achievable array size. To explore this further, the array is further reduced to a 2×2 configuration, as illustrated in Figs. 5(g)–5(i). Following the same simulation procedure, both the amplitude and phase of the VBs exhibit satisfactory results in the near-field region. The designed 2×2 array is treated as a single sub-array unit, and its arrangement is determined based on the contour information of the target pattern. The spacing between sub-arrays is set to 0.4×p. As shown in Fig. 6, a 12×12 sub-array configuration is arranged, demonstrating satisfactory amplitude performance in the near field. The designed array can be adapted to different target patterns, each with a specific corresponding arrangement. When excited by RCP light at a frequency of 10.67 GHz, the cross-polarized transmitted light undergoes geometric phase modulation, producing patterns such as J, N, and U. By direct comparison, the transmitted light exhibiting the same polarization as the incident wave shows no vortex beam in other regions, confirming the effectiveness of the geometric phase-based design.

To integrate metasurfaces into a vortex beam encryption system based on geometric phase modulation, this study utilizes various digital metasurface array boards to construct the system. To further evaluate the practical performance of the proposed metasurface, we fabricated a transmissive metasurface by the printed circuit board (PCB) process technology. The size of the metasurface is 400mm×400mm and consists of 144 meta-quaternion arrays. Meanwhile, the RCP horn antenna, which is used for transmitting waves, was located in front of the metasurface, and the LCP waveguide antenna, which was located 12 mm behind the metasurface, received the signal. Both antennas were connected to two ports of a PNA network analyzer (Keysight N5227A) with a minimum scanning step of 2 mm. The probes were moved by a motion controller to scan the x-y plane, and the size of the test area was set to 400mm×400mm to test the amplitude and phase distributions on the probe plane. Figure 7(a) shows the fabricated digital array board and the experimental setup. At 10.67 GHz, the final reception results are observed, as shown in Figs. 7(b) and 7(c). Comparing these results with the simulated amplitude and phase distributions, a high degree of correlation between experimental and simulated results is achieved. These findings demonstrate the feasibility of the proposed design under experimental conditions and realistic physical environments.

We further explore the potential application in secure encryption by designing a vortex light field encryption system based on pixel-level vortex arrays. Inspired by the classical Luoshu nine-grid encryption method and incorporating weight-based superposition processing, we integrate its principles into the proposed metasurface framework to achieve dynamic and highly secure encoding. In this method, randomly generated numerical sequences and target plaintext letter sequences are combined to construct a joint matrix. By embedding a weight-based nine-grid system, disturbances are applied separately to the numerical and alphabetical components of the matrix. The disturbed numerical results are further used to modulate the alphabetical elements, creating a combined layer of encryption. The final encrypted matrix is flattened into a one-dimensional encoded sequence using a zigzag flattening process.

The proposed encryption scheme aims to generate a secure and complex encoded sequence by integrating numerical and alphabetical disturbances, while leveraging the mathematical characteristics of a weighted nine-grid system. This ensures randomness, reversibility, and adaptability throughout the encryption process. The encryption framework is divided into four main steps: construction of a joint matrix, numerical disturbance, alphabetical disturbance, and matrix flattening. Initially, a joint matrix is constructed by combining plaintext letters (“JNU”) with randomly generated numerical sequences [4, 3, 4]. The numerical inputs can originate from optical vortex beam arrays, while the letters represent the target plaintext. This matrix serves as the basis for subsequent disturbance calculations, where each element in the matrix is influenced by position-specific weights.

For example, as shown in Fig. 8, given the plaintext JNU, three numerical sub-array blocks such as 4, 3, and 4 are randomly selected as the initial encryption sequence. These numbers, derived from the metasurface arrays, serve as critical inputs to the encryption matrix, driving subsequent disturbance calculations. The numerical inputs are combined with the plaintext letters to form a joint matrix. By leveraging the fixed weight-based nine-grid system and following the principle of row column consistency, numerical and alphabetical elements are disturbed separately. The resulting disturbed sequence is 3, 8, and 6. The disturbed numerical results are then integrated into the alphabetical disturbance, where the plaintext letters J, N, and U are transformed into U, A, and C, respectively. This completes the disturbance operation. The disturbed numerical and alphabetical results are then merged into the encrypted matrix, which is subsequently flattened using a zigzag pattern to generate the final one-dimensional encoded sequence: [U, 3, 8, A, C, 6]. The core reason for choosing the nine-grid as the weight matrix is its natural orthogonal balance (each row, column, and diagonal sum is 15), which ensures the uniform distribution of the ciphertext during the numerical perturbation process and avoids the information leakage due to the weight bias. The flow of the algorithm is briefly described as follows. In the numerical perturbation stage, each plaintext number is superimposed with the corresponding weight value and dynamic key according to its position in the nine-grid, and the ciphertext is generated by taking the mode, while the key is updated in real time along with the ciphertext result to realize the dynamic correlation of the perturbation parameters. In the alphabetic perturbation stage, the numerical perturbation results are used to locate the coordinates of the nine-grid, extract the positional weights, and perform the weighted mode operation with the alphabetic control code to generate the final ciphertext alphabets. To decrypt, the user must know the specific frequency of the incident light, the corresponding imaging location, and the weight selection principle of the nine-grid system. These parameters enable the reverse operation of the encoded sequence, reconstructing the original plaintext matrix and extracting the decryption target. This approach seamlessly combines the metasurface array’s optical characteristics with classical encryption logic. By leveraging the dynamic nature of optical field generation, the method enhances the randomness and complexity of encryption, demonstrating the innovative potential of metasurfaces in secure communication and information encryption.

In the numerical disturbance step, the numerical elements of the joint matrix are perturbed using a weighted nine-grid system. The nine-grid is a classic magic square where the sum of the numbers in each row, column, and diagonal is equal [8, 1, 6; 3, 5, 7; 4, 9, 2]. The disturbance is calculated using the formula: N′=(N+W)%9, where N is the original numerical input, W is the weight from the corresponding position in the nine-grid, and %9 ensures the result remains within the range of 0 to 8. The disturbed numerical values not only transform the original numbers but also serve as inputs for the subsequent alphabetical disturbance. In the alphabetical disturbance step, the plaintext letters are perturbed by combining the results of the numerical disturbance with the nine-grid weights. The disturbance is performed using the formula: P′=(P+N′+W) %26, where P is the position of the letter in the alphabet (A=0,B=1,…,Z=25), N′ is the result of the numerical disturbance, and W is the nine-grid weight. The %26 operation ensures the disturbed positions remain within the range of the English alphabet. This step introduces dual-layer perturbations to the letters, enhancing the complexity of the encryption process.

Once the numerical and alphabetical disturbances are applied, the elements of the joint matrix are transformed into a new encrypted matrix. To further obfuscate the encryption rules, the encrypted matrix is flattened into a one-dimensional sequence using a zigzag flattening method. This process involves reading rows alternately from left to right and right to left. For instance, the encrypted matrix [U, 3; A, 8; C, 6] is flattened to produce the final encoded sequence [U, 3, 8, A, C, 6]. The weighted nine-grid system plays a pivotal role in this encryption scheme. As a traditional mathematical construct, it provides position-specific weights that drive both numerical and alphabetical disturbances. The weights introduce a high degree of unpredictability to the encryption results. Additionally, the nine-grid system can either be fixed (as a classic structure) or dynamically generated to enhance the encryption’s flexibility and randomness. The use of modular arithmetic ensures the process is fully reversible, enabling decryption through inverse calculations to restore the original matrix.

In summary, the encryption scheme transforms plaintext into a secure encoded sequence through a combination of numerical disturbance, alphabetical disturbance, and matrix flattening. The nine-grid system underpins the disturbance operations by introducing position-dependent perturbations, while the dynamic numerical inputs derived from optical systems add further randomness and complexity. This approach provides a secure and reversible encryption framework suitable for high-security applications in information encoding. In the encryption scheme designed in this work, the mapping relationship between letters and numbers is provided in the mapping table on the left side of Fig. 9. Additionally, the nine-grid used in our design is not fixed for different ciphertexts; modifying any effective number within the grid can lead to variations in the final result. For example, in this case, we employ a different nine-grid from that used in the main text while ensuring that it remains symmetric and follows the classical numerical nine-grid structure. However, when encrypting “JNU” using this modified grid and the given encryption formulas, the resulting ciphertext is “RAF” instead of the previously obtained “UAC.” This demonstrates that the proposed encryption scheme is not entirely rigid but rather flexible, allowing significant variations in ciphertext output by adjusting the underlying grid parameters.

In conclusion, we propose a meta-atom with high transmittance in the X-band based on the geometric phase. At the operating frequency, the transmittance coefficients for the x- and y-polarized components are 0.99 and 0.86, respectively, with a phase difference of approximately 2π. This enables the device to function as an HWP for perfect polarization conversion. We validate this approach through simulations using CST software to analyze the amplitude and phase distributions for various topological charges, thereby confirming the structure’s scalability. Subsequently, the array configuration is further reduced, and its near-field performance is evaluated to yield the desired meta-quaternion array, which serves as the constituent pixel for the target pattern. A GHz device composed of meta-quaternion arrays is fabricated and characterized using RCP light at an operating frequency of 10.67 GHz. The experimental results closely match the simulations, confirming the structure’s effectiveness in reconstructing the contours of the target image. The meta-quaternion arrays can be used as elements for numeric and alphabetic images and integrated with the classical Luoshu nine-grid system and weighted superposition computation technique to encrypt target characters. The proposed method holds promise for applications in electromagnetic beam control, image processing, and information encryption, offering an attractive pathway to advanced photonic and communication technologies. The designed metasurface system also exhibits significant potential for applications in radar systems, information encryption, and next-generation 5G/6G wireless communication networks.