Mapping Full-Stokes Parameters to Metasurface Design via Globally Engineered Disorder

Mapping Full-Stokes Parameters to Metasurface Design via Globally Engineered Disorder | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Mapping Full-Stokes Parameters to Metasurface Design via Globally Engineered Disorder Changyuan YU, Zhi Cheng, Zhou Zhou, Zhuo Wang, Yue Wang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6293050/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The ability to achieve comprehensive control over all Stokes parameters, including both the state of polarization (SoP) and the degree of polarization (DoP), is fundamental to advancements in quantum optics, polarization imaging, and optical communications. While metasurfaces have demonstrated remarkable capabilities in polarization control, existing approaches often struggle to simultaneously manipulate SoP and DoP with high flexibility. Here, we introduce a paradigm shift in metasurface-based polarization engineering by proposing a globally engineered disordered metasurface that enables a one-to-one correspondence between structural parameters and the full-Stokes polarization space. Unlike conventional metasurfaces that rely solely on unit-cell-level deterministic phase profiles, our approach incorporates a statistical design principle, introducing a spatial statistical parameter: the meta-atom quantity ratio. By uniformly distributing two distinct types of meta-atoms with controlled ratios, we effectively decouple the design parameters, enabling independent control over all Stokes parameters. Specifically, the azimuthal and elevation angles of the SoP on the Poincaré sphere are governed by the rotation and size of individual meta-atoms, while the DoP is precisely tuned through global disorder engineering via the quantity ratio of the meta-atoms. This approach establishes a direct mapping between metasurface design and polarization space, revealing new physical insights into disorder-assisted polarization control. A computationally efficient algorithm optimizes the metasurface arrangement, achieving a polarization similarity (evaluated by Stokes Euclidean Distance) of 0.93 in theory and 0.90 in experiment. Our findings advance the development of metasurfaces that harness disorder as a functional design strategy, enabling enhanced flexibility in full-Stokes polarization engineering. Physical sciences/Optics and photonics/Optical materials and structures/Metamaterials Physical sciences/Physics/Electronics, photonics and device physics/Photonic devices Metasurface polarization control degree of polarization Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 1. Introduction The capability to precisely manipulate the full-stokes parameters of light, encompassing both the state of polarization (SoP) and degree of polarization (DoP), is a cornerstone of photonics research and development. In deterministic systems, polarization refers to the orientation of the electric field vector, which defines the SoP. In stochastic systems, polarization evolves over time, giving rise to light that may be unpolarized or partially polarized. This variability is quantitatively described by the DoP, which measures the fraction of light that remains polarized. Partially polarized light is thus a combination of fully polarized (dominant polarization) and unpolarized components, with the DoP indicating the relative contribution of the polarized component. Controlling SoP and DoP is vital for applications such as optical communication 1 – 3 , quantum optics 4 – 7 , imaging 8 , 9 , remote sensing 10 , and spectroscopy 11 – 13 . SoP manipulation is commonly achieved through wave plates, which introduce controlled phase shifts between orthogonal polarization components. On the other hand, DoP modulation typically involves converting fully polarized or unpolarized light into a desired partially polarized state. Achieving this requires differential transmission between orthogonal polarization states. This is often implemented experimentally using time-varying optical retarders or wavelength-dependent wave plates. Simultaneous control over both SoP and DoP, however, presents significant engineering challenges. Traditional approaches for such control often rely on complex and bulky optical setups, which can impede the integration and miniaturization of photonic devices. These limitations highlight the need for more compact and efficient solutions capable of fully manipulating the polarization properties of light. Recently, metasurfaces—comprising arrays of nanoscale structures—have garnered significant attention as flat optical devices capable of manipulating light's amplitude, phase, and polarization at subwavelength scales. These capabilities have catalyzed advancements in fields such as imaging 14 – 19 , polarization control 20 – 36 , quantum optics 37 – 41 , and communications 42 – 45 . Importantly, metasurfaces facilitate the miniaturization and integration of photonic devices, offering enhanced functionality within compact platforms. As shown in Fig. 1 (a), in the context of polarization control, a mapping relationship between the metasurface design parameter space and the polarization control parameter space can be established through careful design. The homogeneous metasurface composed of birefringent meta-atoms, shown in Fig. 1 (b), functions as a nano-waveplate widely used in polarization modulation 46 – 48 . When the transmittance is set close to unity, the system becomes effectively lossless, yielding a unitary Jones matrix. This allows the metasurface to convert one SoP into another by redistributing phase delay between orthogonal polarization components. However, in dynamic systems with incoherent light, time-averaging equalizes intensity differences between these components, rendering this approach unsuitable for controlling the DoP. We adjust the gap distance between two types of meta-atoms within the 'weak coupling' region 25 , 26 . This adjustment induces far-field radiation interference in the diatomic structure shown in Fig. 1 (c). This design could provide at least six degrees of freedom, enabling the realization of any arbitrary Jones matrix for polarization conversion 20 – 25 . The flexibility of this method allows for the creation of non-unitary Jones matrices, which generate intensity differences between orthogonal states in a time-varying system. However, this results in an intrinsic coupling between the metasurface design parameters, specifically the dominant SoP and the DoP within the polarization space, as shown in Fig. 1 (c). This relationship can be further understood using the phasor diagram presented in Fig. S1 . Wang et al. recently tackled this challenge by introducing an inverse-designed meta-atom, optimized through topological methods, capable of converting unpolarized light into partially polarized light 32 . However, the complex geometries of these optimized structures require high-precision fabrication, which presents significant challenges for practical implementation. In this study, we introduce a novel approach to decouple the SoP and DoP using a disordered metasurface. This non-periodic metasurface, illustrated in Fig. 1 (d), is composed of two types of birefringent meta-atoms uniformly distributed across its surface. By leveraging far-field interference effects, we introduce polarization-dependent loss (PDL) to precisely modulate the DoP. The DoP is controlled by the differential conversion of orthogonal polarization components, determined by the ratio of different meta-atoms. SoP control is achieved by adjusting the azimuth and elevation angles on the Poincaré sphere through the rotation and phase difference of individual meta-atoms. We developed an algorithm based on a greedy search and a 2D bin-packing problem to generate the disordered metasurface. This design allows access to all Stokes parameters, enabling systematic control of polarization states on the Poincaré sphere. As a proof of concept, we theoretically and experimentally demonstrate that varying a single parameter can produce polarization distributions along latitude, longitude, and radial directions. It is also shown that the design possesses a degree of tolerance, allowing different meta-atom combinations to function effectively within the same framework. Furthermore, we demonstrate the flexibility of this design approach in enabling arbitrary polarization state conversion applications. 2. Results The physical mechanisms that simultaneously and independently control DoP and SoP using metasurfaces are illustrated in Fig. 2 . Since unpolarized light can be viewed as an incoherent superposition of two orthogonal polarization states, analyzing this mechanism requires focusing on the conversion process between these orthogonal states. Each metasurface is capable of converting ( \(\:|{\alpha\:}^{*}\rangle\:\) , \(\:|{\beta\:}^{*}\rangle\:\) ) into ( \(\:|\alpha\:\rangle\:\) , \(\:|\beta\:\rangle\:\) ), with distinct phase shifts for the respective states. This results in constructive interference for the \(\:|\alpha\:\rangle\:\) state and destructive interference for the \(\:|\beta\:\rangle\:\) state. When the two metasurfaces differ in their proportional contribution, the degrees of constructive interference for \(\:|\alpha\:\rangle\:\) and destructive interference for \(\:|\beta\:\rangle\:\) are no longer balanced, leading to an unequal ratio of \(\:|\alpha\:\rangle\:\) and \(\:|\beta\:\rangle\:\) in the final incoherent light output. In essence, differential conversion between \(\:|\alpha\:\rangle\:\) and \(\:|\beta\:\rangle\:\) is achieved, resulting in partially polarizing the unpolarized light. In what follows, the derivation of the necessary Jones matrices, the mechanism for controlling DoP, and the design methodology for the metasurfaces are discussed. We begin with the birefringent meta-atoms which are commonly used for polarization control. The behavior of these meta-atoms is described by the following Jones matrix: $$\:{J}_{atom}=R\left(\theta\:\right)\left(\begin{array}{cc}{A}_{f}{e}^{i{\phi\:}_{f}}&\:0\\\:0&\:{A}_{s}{e}^{i{\phi\:}_{s}}\end{array}\right)R(-\theta\:)$$ 1 This Jones matrix accounts for the phase retardation along the fast and slow optical axes, denoted by \(\:{\phi\:}_{f}\) and \(\:{\phi\:}_{s}\) , respectively, with A f and A s representing the transmittances along these axes. The rotation matrix R(θ) , defined \(\:R\left(\theta\:\right)=\left[\begin{array}{cc}\text{c}\text{o}\text{s}\left(\theta\:\right)&\:-\text{s}\text{i}\text{n}\left(\theta\:\right)\\\:\text{s}\text{i}\text{n}\left(\theta\:\right)&\:\text{c}\text{o}\text{s}\left(\theta\:\right)\end{array}\right]\) , describes the orientation of the birefringent axes with respect to the incident polarization. With transmittances close to unity, the system becomes effectively lossless, yielding a unitary Jones matrix. To introduce differential transmission between orthogonal states, a non-unitary Jones matrix is required, which can be constructed as a sum of unitary matrices. On the other hand, the Jones vectors of an arbitrary orthogonal polarization pair can be expressed in terms of azimuth angle ψ and elevation angle χ on the Poincare sphere as: $$\:|\alpha\:\rangle\:=R(\psi\:-{45}^{\circ\:}\left)\right(\begin{array}{c}{e}^{-i\chi\:}\\\:{e}^{i\chi\:}\end{array}),|\beta\:\rangle\:=R(\psi\:-{45}^{\circ\:})\left(\begin{array}{c}{e}^{-i\chi\:}\\\:-{e}^{i\chi\:}\end{array}\right)$$ 2 Here, \(\:|\alpha\:\rangle\:\) is located at coordinates (2 ψ , 2 χ ) on the Poincaré sphere, while \(\:|\beta\:\rangle\:\) is positioned at (2 ψ − 180°, -2 χ ), symmetrically opposite about the origin. It is important to note that, in the polarization states ( \(\:|\alpha\:\rangle\:\) , \(\:|\beta\:\rangle\:\) ), the x- and y-components are complex conjugates, except for the negative sign in \(\:|\beta\:\rangle\:\) . This indicates that by applying a phase delay to the x - and y -components, as described by the Jones matrix in the form of Eq. ( 1 ), it is possible to convert between the polarization states ( \(\:|{\alpha\:}^{*}\rangle\:\) , \(\:|{\beta\:}^{*}\rangle\:\) ) and their conjugate states ( \(\:|{\alpha\:}^{*}\rangle\:\) , \(\:|{\beta\:}^{*}\rangle\:\) ) (Supplemental Information Note II). Therefore, the Jones matrix for converting the orthogonal conjugate states can be expressed in a form similar to Eq. ( 1 ): $$\:{J}_{1}=R(\psi\:-{45}^{\circ\:})\left(\begin{array}{cc}{e}^{-2i\chi\:}&\:0\\\:0&\:{e}^{2i\chi\:}\end{array}\right)R({45}^{\circ\:}-\psi\:)$$ 3 This Jones matrix converts the conjugate orthogonal states ( \(\:|{\alpha\:}^{*}\rangle\:\) , \(\:|{\beta\:}^{*}\rangle\:\) ) into ( \(\:|\alpha\:\rangle\:\) , \(\:|\beta\:\rangle\:\) ) while introducing the same global phase. This means that there is no relative phase shift between the two states. Another Jones matrix can be constructed to achieve the conversion between these two pairs of conjugate orthogonal states with a relative phase difference of π (see Supplemental Information Note II). The Jones matrix for this transformation can be expressed as: $$\:{J}_{2}=R\left(\psi\:\right)\left(\begin{array}{cc}1&\:0\\\:0&\:-1\end{array}\right)R(-\psi\:)=R(\psi\:-45)\left(\begin{array}{cc}0&\:1\\\:1&\:0\end{array}\right)R(45-\psi\:)$$ 4 This matrix flips the x- and y-components of the Jones vector in Eq. ( 2 ). The negative sign in the y-component of the \(\:|\beta\:\rangle\:\) state introduces a global phase of π during the conjugate state conversion process. By combining the Jones matrices from Eqs. ( 3 ) and ( 4 ) with different coefficients, differential transmission between orthogonal polarization states can be achieved, as shown in Fig. 2 (a). The resulting composite Jones matrix can be expressed as follows: $$\:J={J}_{1}+{J}_{2}=\frac{{n}_{1}}{N}R\left(\psi\:-{45}^{\circ\:}\right)\left(\begin{array}{cc}{e}^{-2i\chi\:}&\:0\\\:0&\:{e}^{2i\chi\:}\end{array}\right)R\left({45}^{\circ\:}-\psi\:\right)+\frac{{n}_{2}}{N}R\left(\psi\:\right)\left(\begin{array}{cc}1&\:0\\\:0&\:-1\end{array}\right)R\left(-\psi\:\right)$$ 5 To ensure energy conservation, the sum of the coefficients must be equal to unity. Therefore, the coefficients are written in the form of n i /N , where i = 1, 2, and N = n 1 + n 2 . The state \(\:|{\alpha\:}^{*}\rangle\:\) is fully converted into \(\:|\alpha\:\rangle\:\) with an amplitude of 1/2. However, the conversion between \(\:|{\beta\:}^{*}\rangle\:\) and \(\:|\beta\:\rangle\:\) undergoes destructive interference due to the π-phase delay, resulting in an amplitude of ( n 1 - n 2 )/2 N . Consequently, the transmission matrix for converting ( \(\:|{\alpha\:}^{*}\rangle\:\) , \(\:|{\beta\:}^{*}\rangle\:\) ) into ( \(\:|\alpha\:\rangle\:\) , \(\:|\beta\:\rangle\:\) ) is given by: $$\:{J}_{t}=\left(\begin{array}{cc}{t}_{{\alpha\:}^{*}\alpha\:}&\:{t}_{{\beta\:}^{*}\alpha\:}\\\:{t}_{{\alpha\:}^{*}\beta\:}&\:{t}_{{\beta\:}^{*}\beta\:}\end{array}\right)=\left(\begin{array}{cc}1&\:0\\\:0&\:\frac{{n}_{1}-{n}_{2}}{N}\end{array}\right)$$ 6 To determine the DoP in the time domain, it is necessary to apply a time-averaged operation to account for the random phase in unpolarized light. The intensity of the outgoing light can then be expressed as: $$\:\begin{array}{c}\langle\:{I}_{out}\rangle\:=(1-{t}_{{\beta\:}^{*}\beta\:}^{2})\langle\:\alpha\:\rangle\:+{t}_{{\beta\:}^{*}\beta\:}^{2}(\langle\:\alpha\:\rangle\:+\langle\:\beta\:\rangle\:)\\\:=(1-{t}_{{\beta\:}^{*}\beta\:}^{2})FP+2{t}_{{\beta\:}^{*}\beta\:}^{2}UP\end{array}$$ 7 where the brackets denote to time average. According to the definition of DoP, let η = n 1 /n 2 , it is calculated as: $$\:p=\frac{1-{t}_{{\beta\:}^{*}\beta\:}^{2}}{1+{t}_{{\beta\:}^{*}\beta\:}^{2}}=\frac{2\eta\:}{1+{\eta\:}^{2}}$$ 8 As a result of the preceding derivation, a one-to-one correspondence between the structural parameters of the metasurface and the coordinates on the Poincaré sphere has been established, as illustrated in Fig. 2 (b). To achieve the Jones matrix in Eq. ( 5 ), we propose a disordered metasurface composed of various types of meta-atoms. These adjacent meta-atoms are arranged with appropriate gap distances in the weak coupling regime. In this case, the Jones matrix is the summation of the matrices of each meta-atom, represented in the following form: $$\:{J}_{DM}=\sum\:_{i=1}^{\text{M}}\frac{{n}_{i}}{N}{J}_{atom}^{i}$$ 9 where n i ( i = 1,2… M ) represents the number of each meta-atom, and N denotes the total number of atoms. When M = 2 and n 1 = n 2 , the disordered metasurface degenerates into the simpler diatomic design shown in Fig. 1 (c), where a periodic unit cell within wavelength scale is constructed across the metasurface. However, when n₁ ≠ n₂ , if we continue to follow the conventional periodic design approach, multiple types of meta-atoms must be arranged within a supercell, resulting in a lattice constant larger than the wavelength. This oversized lattice constant can introduce energy losses by diverting energy away from the intended polarization conversion, leading to the generation of higher-order diffraction modes. Additionally, while a periodic design does not inherently produce a phase gradient, it is more prone to introducing one without careful optimization. In the presence of a phase gradient, the periodicity can amplify this effect, leading to stronger higher-order diffraction and further deflection of the outgoing beam as shown in Fig. S2. More critically, the periodic design complicates the ability to maintain a uniform distribution of meta-atoms across the metasurface. The translational symmetry inherent to periodic structures hinders the preservation of a consistent ratio of meta-atoms near the boundaries, thereby diminishing polarization conversion efficiency (see Supplemental Information Note III). Building on this understanding, the disordered metasurface, as illustrated in Fig. 2 (c), consists of n 1 meta-atoms of type I and n₂ meta-atoms of type II. Each type of meta-atom is uniformly arranged, with their spacing optimized for the weak-coupling region to ensure effective interaction. This design ensures far-field radiation coherence among the meta-atoms, which is essential for the validity of Eq. ( 9 ), where the total response of the metasurface is expressed as the summation of the individual meta-atoms Jones matrices. Close meta-atom placement causes strong coupling, altering the propagation phase. Conversely, if they are too far apart, the meta-atoms interact weakly, and no coherent interference is observed. The uniform arrangement ensures that the local ratio of meta-atoms within each effective coupling region remains consistent with the target ratio n 1 / n 2 across the entire surface. Unlike conventional interleaved designs, where different meta-atom types function independently, the meta-atoms in the disordered metasurface work cooperatively. The intentional disorder breaks the translational symmetry, effectively suppressing strong higher-order diffraction modes. Through this design, the Stokes parameters (S1, S2, S3) and the DoP are independently controlled. The polarization orientation, represented by S1, S2, and S3, can be mapped to the azimuth and elevation angles on the Poincaré sphere, which is governed by the rotation angle and phase delay of the meta-atoms, as described by Eq. ( 5 ). Meanwhile, the DoP, corresponding to the radius in the Poincaré sphere, is controlled by the ratio of the two types of meta-atoms ( n 1 / n 2 ) within the disordered metasurface. This decoupled control allows for independent tuning of the polarization state’s orientation and its purity, enabling full manipulation of all relevant Stokes parameters. To obtain the desired arrangement of meta-atoms, we developed a two-dimensional bin-packing algorithm based on a greedy search heuristic (see Supplemental Information Note IV). Our empirical calculations show that the weak coupling region typically spans about 1/10 of the operating wavelength. We define the effective working region of each meta-atom using the enclosing rectangles, as shown by the dashed lines in Fig. 2 (c). These rectangles are tessellated uniformly across the metasurface by the algorithm, ensuring that the arrangement maintains both uniformity and the adaptive spacing required for coherent interactions. While our approach yields strong results, other advanced algorithms could also be employed to achieve similarly optimized layouts for disordered metasurfaces. We validate the proposed concept by demonstrating full control of the SoP and DoP across the entire Poincaré sphere through simulations conducted using the Finite-Difference Time-Domain (FDTD) method, as shown in Fig. 3 . For fully polarized light, the polarization state is represented by points on the surface of the Poincaré sphere, while partially polarized light is depicted by points within the sphere. Complete polarization control requires the manipulation of three parameters on the Poincaré sphere: azimuthal angle, elevation angle, and radius, which correspond to the Stokes parameters S1, S2, S3, and the DoP. By independently varying these parameters, we demonstrate precise control over the latitude, longitude, and radius of the Poincaré sphere, allowing for arbitrary manipulation of partially polarized light. In the simulation, 500 points are uniformly sampled across the surface of the Poincaré sphere as shown in Fig. 3 (a), with near-equal Euclidean distances between adjacent points, to represent the input of unpolarized light. The random phase component inherent in unpolarized light is neglected, as it does not influence the results in time-averaged incoherent superposition calculations. The polarization states corresponding to the sampled points are converted by the designed metasurface, resulting in a non-uniform distribution of Stokes vectors across both the surface and interior of the Poincaré sphere, as shown in Fig. 3 (b). This unique distribution on the Poincaré sphere serves as a distinctive fingerprint of partially polarized light, encapsulating its statistical nature and providing profound insights into its transformation through the metasurface. The incoherent sum of these transformed polarization states yields the DoP and the dominant SoP. Figure 3 (c)-(f) demonstrates the evolution of polarization states on the Poincaré sphere, showcasing how the azimuth, elevation angles, and DoP can be independently controlled. The latitude and longitude lines on the Poincaré sphere passing through the polarization state at 2 χ = 45° and 2 ψ = 45°, respectively. These polarization states are achieved by varying the size or orientation of two types of meta-atoms. Only half of the longitude is depicted, as the other half can be obtained by rotation of the meta-atoms. To enhance visualization clarity, the dominant SoP is chosen as (2 χ = -60°, 2 ψ = 225°), located in the southern hemisphere, when demonstrating DoP control through adjustments to the meta-atom quantity ratio. For the latitude data set, the DoP is set to 0.65, while for the longitude data set, it is set to 0.75, corresponding to quantity ratios of 2.714 and 2.214, respectively. In the metasurface design, the target DoP values are approximated by quantity ratios of 19:7 and 31:14 for the two types of meta-atoms. The arrangement coordinates and structural parameters of the employed meta-atoms are detailed in Supplemental Information Note V. In Fig. 3 , the green points represent the target polarization states on the Poincaré sphere, while the blue points represent the simulation results. The latitude and longitude evolution data sets are also shown in Fig. 3 (d) and Fig. 3 (e), with top-view and side-view perspectives, respectively. The average error in the elevation angle is 2.65°, while the average error in the azimuth angle is 2.98°, demonstrating the accuracy of this method in controlling the polarization state. In Fig. 3 (f), error bars indicate the range of simulated DoP values for the latitude and longitude data sets, with the central sphere symbol representing the average DoP. The green shaded area corresponds to a DoP error range of ± 0.1, while nearly all data points fall within the tighter range of ± 0.05. As the elevation angle increases, we observe a rise in error. This is attributed to the limited size of the meta-atom database (see Supplemental Information Note V) and the relatively large size difference between the two types of meta-atoms in the high-latitude regions. This discrepancy complicates the task of finding an appropriate spacing to satisfy both weak coupling and coherence conditions. The mismatch between the propagation phase of the selected meta-atoms and the target phase could also contribute to the observed errors. Expanding the meta-atom database could help mitigate both issues and reduce the error. Experimental validation of the proposed approach is presented in Fig. 4 . Figure 4 (a) illustrates the experimental setup, where a polarization scrambler (PS) was employed to generate randomly polarized light, simulating natural light input with a modulation frequency of 2 Hz. The polarization analyzer operated at a sampling rate of 15 Hz, significantly exceeding the Nyquist sampling frequency, ensuring accurate measurement of the output polarization states. Detailed information on the experimental setup is provided in the Materials and methods section. In our experiment, we fabricated and measured 15 samples. Among them, 8 samples were designed to demonstrate polarization evolution along the latitude on the Poincaré sphere, with a step size of 45°. Three samples were used to illustrate changes along the longitude, covering linear polarization (LP), elliptical polarization (EP), and right-handed circular polarization (RCP). The remaining 4 samples were dedicated to demonstrating DoP control. All samples were fabricated using a standard CMOS-compatible process, with details provided in the Materials and methods section. Representative experimental results from each of the three groups, specifically samples No. 6, 11, and 15, are presented in Fig. 4 . Additional data, arranged sequentially, can be found in Supplementary Information Note VI. Figures 4 (c)–4(e) illustrate the distribution of the experimentally measured polarization states on the Poincaré sphere for three selected samples, accompanied by the corresponding SEM images of the fabricated metasurfaces. The selected samples correspond to polarization states with parameters 2ψ = 225°, 2χ = 90°, and DoP = 0.9, respectively. Figures 4 (f)–4(h) depict the measured Stokes parameters for all polarization states in each sample. Based on the theoretical analysis, the distribution of partially polarized light on the Poincaré sphere is expected to show a clustering effect around the dominant polarization state as the DoP increases. This trend is confirmed by the experimental results, where the target Stokes parameters are marked with white dashed lines. The measured polarization states exhibit a higher density of points near the dominant polarization state, supporting the theoretical predictions. The accuracy of the partially polarized light conversion is quantified by the Stokes Euclidean Distance (SED), which is defined as 1 − distance, where "distance" represents the Euclidean distance between the measured and target Stokes vectors. A value of SED = 1 indicates perfect alignment (i.e., zero Euclidean distance) between the two Stokes vectors. Figure 4 (i) presents the SED values for all 15 experimental samples, with an average SED of 0.90, compared to the theoretical simulation average of 0.93. The accuracy of experimental results for the three representative groups of samples is measured as 0.942, 0.901, and 0.969, respectively, demonstrating the reliability of our proposed method. To further demonstrate the feasibility and flexibility of this approach, we showcase the ability to generate different partially polarized light using the same metasurface arrangement. Since the coherent pixels operate in a weak coupling regime, small variations in the gap distance between adjacent meta-atoms have a negligible effect on the modulation behavior. This introduces a tolerance in the gap distance, allowing the same metasurface design to be applied across different modulation scenarios. For instance, the layout designed to achieve a DoP of 0.65, with a quantity ratio of 2.714, as previously illustrated in Fig. 2 (c). In this layout, the effective sizes of the meta-atoms are (850, 650) nm, and (760, 850) nm. The dominant polarization state is initialized at 2 ψ = 300° and 2 χ = 45°. The distribution of SoP at this point is shown in Fig. 5 (a), obtained through numerical simulations. The corresponding SED is calculated to be 0.989, confirming the high accuracy of the designed metasurface in generating the target polarization state. Starting from the initial point, we varied the sizes and rotation angles of the meta-atoms, causing the partially polarized light to move along both the longitude and latitude lines passing through the initial point. Figure 5 (b) illustrates the operational range of the disordered metasurface, defined as the effective polarization conversion range in terms of latitude and longitude on the Poincaré sphere. Stokes parameters S1-S3 are also given of each point in the Fig. 5 (c) and (d). In Fig. 5 (b), the intersection of these two lines represents the initial point. The tolerance in the azimuthal angle (meta-atom rotation) is approximately 60° (with the range of SED exceeding 0.95), while the elevation angle (meta-atom size) tolerance is less than 10°. This discrepancy arises because adjusting the elevation angle requires a complete change in the meta-atom combination, for which the initial effective sizes are not suitable. Although the physical gap between different meta-atom combinations may exhibit varying tolerances, the specific values presented here are not universal. Nonetheless, these results confirm that the metasurface design provides a tolerance margin, allowing the same layout to be used for different meta-atom configurations. This tolerance is crucial for practical applications, as it enables flexible and independent control of both the DoP and SoP. 3. Discussion In the article, we propose a disordered metasurface that converts unpolarized light into partially polarized light, achieving independent control over both the SoP and DoP. This approach harnesses far-fi`1eld interference between meta-atoms and introduces a novel design parameter—the meta-atom quantity ratio—to enable arbitrary and flexible manipulation of polarization states. By generalizing this method, we derive an analytical solution that directly maps the optical parameters of meta-atoms to each element of an arbitrary Jones matrix, as detailed in Supplemental Information Note VII using a triatomic design. Unlike inverse design methods, our approach offers a more intuitive insight into the underlying physics, establishing a clear and concise relationship between meta-atom properties, the Jones matrix, and full Stokes parameter control. The additional degrees of freedom introduced through this disordered metasurface design, particularly via the meta-atom quantity ratio, significantly enhance the flexibility of polarization control beyond conventional techniques. This advancement empowers independent tuning of all Stokes parameters, opening up possibilities for advanced photonic applications where polarization must be tailored independently, thus paving the way for next-generation systems requiring sophisticated multi-dimensional light manipulation. Materials and methods Fabrication of the disordered metasurface The metasurface was fabricated on a commercially available 940-nm-thick α-Si layer (PECVD) deposited on a sapphire substrate. The structure was patterned on a ZEP520A resist using an E-beam writer (Raith E-line, 50 kV, 20 nA). After developing the resist, the pattern was transferred onto a 100-nm-thick chromium film. Subsequently, the silicon layer was etched using inductively coupled deep reactive ion etching (DRIE), with the chromium layer serving as a hard mask. The resist was then removed with acetone, followed by rinsing with IPA and DI water. Finally, the remaining chromium mask was removed using a chromium etchant. Experimental setup The experimental setup is designed to precisely control and analyze the polarization state of light interacting with the sample. A polarization scrambler (Luna PSY 201) is used to control the input polarization state, ensuring a well-defined and tunable polarization condition. The optical path begins with Objective 1 (10X, NA = 0.17), which functions as a condenser lens to focus the incident light near the sample. Objective 2 (50X Mitutoyo Plan Apo NIR Infinity Corrected Objective, NA = 0.42) is then used for light collection and microscopic imaging of the sample. A 4f imaging system, composed of a tube lens and a Fourier lens, is implemented to relay the optical field while maintaining spatial coherence. At the conjugate sample plane within this system, an iris diaphragm is placed to block unwanted light from outside the sample area, improving signal quality. An imaging lens is used to project the sample image onto a CCD camera, allowing direct observation and adjustment of the iris diaphragm size. The polarization state of the transmitted light is analyzed using a polarization analyzer (Thorlabs PAX1000IR2). Declarations Acknowledgements We thank Prof. Cheng-wei Qiu for valuable comments and feedback. The authors thank the support of Hong Kong Research Grants Council (GRF 15209321 B-Q85G). Author Contributions Z.C. conceived the methodology, developed the software, performed validation, conducted visualization, and wrote the original draft. Z.Z. contributed to visualization and assisted with writing and editing. Z.W. and Y.W. supported validation and manuscript review. C.Y. conceived the project, provided conceptual guidance, supervised the research, secured funding, and contributed to project administration. Conflict of Interest statement The authors declare that they have no conflict of interest. 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Conversion between Arbitrary Amplitude, Phase, and Polarization with Minimal Degrees of Freedom of Metasurface. Laser Photonics Rev. 16 , 2100280 (2022). https://doi.org/10.1002/lpor.202100280 Wu, T. et al. Dielectric Metasurfaces for Complete Control of Phase, Amplitude, and Polarization. Adv. Opt. Mater. 10 , 2101223 (2022). https://doi.org/10.1002/adom.202101223 Feng, C. et al. Diatomic Metasurface for Efficient Six‐Channel Modulation of Jones Matrix. Laser Photonics Rev. 17 , 2200955 (2023). https://doi.org/10.1002/lpor.202200955 Bao, Y. et al. Coherent Pixel Design of Metasurfaces for Multidimensional Optical Control of Multiple Printing‐Image Switching and Encoding. Adv. Funct. Mater. 28 , 1805306 (2018). https://doi.org/10.1002/adfm.201805306 Mustafa, M.-E., Amin, M., Siddiqui, O. & Tahir, F. A. Quasi-Crystal Metasurface for Simultaneous Half- and Quarter-Wave Plate Operation. Sci. Rep. 8 (2018). https://doi.org/10.1038/s41598-018-34142-y Shi, Z. et al. Continuous angle-tunable birefringence with freeform metasurfaces for arbitrary polarization conversion. Sci. Adv. 6 , eaba3367 (2020). https://doi.org/10.1126/sciadv.aba3367 Wang, S. et al. Arbitrary polarization conversion dichroism metasurfaces for all-in-one full Poincare sphere polarizers. Light Sci. Appl. 10 , 24 (2021). https://doi.org/10.1038/s41377-021-00468-y Bao, Y. et al. Observation of full-parameter Jones matrix in bilayer metasurface. Nat. Commun. 13 (2022). https://doi.org/10.1038/s41467-022-35313-2 Chen, J. et al. Polychromatic full-polarization control in mid-infrared light. Light Sci. Appl. 12 (2023). https://doi.org/10.1038/s41377-023-01140-3 Wang, S., Wen, S., Deng, Z. L., Li, X. & Yang, Y. Metasurface-Based Solid Poincare Sphere Polarizer. Phys Rev Lett 130 , 123801 (2023). https://doi.org/10.1103/PhysRevLett.130.123801 Xiong, B. et al. Breaking the limitation of polarization multiplexing in optical metasurfaces with engineered noise. Science 379 , 294-299 (2023). https://doi.org/10.1126/science.ade5140 Zhang, G. et al. Retrieving Jones matrix from an imperfect metasurface polarizer. Adv. Photonics Nexus 3 (2024). https://doi.org/10.1117/1.Apn.3.2.026005 Wang, J. et al. Unlocking ultra-high holographic information capacity through nonorthogonal polarization multiplexing. Nat. Commun. 15 (2024). https://doi.org/10.1038/s41467-024-50586-5 Li, S. et al. Metasurface Polarization Optics: Phase Manipulation for Arbitrary Polarization Conversion Condition. Phys. Rev. Lett. 134 , 023803 (2025). https://doi.org/10.1103/PhysRevLett.134.023803 Huang, T. Y. et al. A monolithic immersion metalens for imaging solid-state quantum emitters. Nat Commun 10 , 2392 (2019). https://doi.org/10.1038/s41467-019-10238-5 Kan, Y. et al. Metasurface-Enabled Generation of Circularly Polarized Single Photons. Adv Mater 32 , e1907832 (2020). https://doi.org/10.1002/adma.201907832 Li, L. et al. Metalens-array-based high-dimensional and multiphoton quantum source. Science 368 , 1487-+ (2020). Solntsev, A. S., Agarwal, G. S. & Kivshar, Y. S. Metasurfaces for quantum photonics. Nat. Photonics 15 , 327-336 (2021). https://doi.org/10.1038/s41566-021-00793-z Fan, Y. et al. Metalens array for quantum random number. Appl. Phys. Rev. 11 (2024). Li, L., Zhao, H., Liu, C., Li, L. & Cui, T. J. Intelligent metasurfaces: control, communication and computing. Elight 2 , 7 (2022). Shlezinger, N., Alexandropoulos, G. C., Imani, M. F., Eldar, Y. C. & Smith, D. R. Dynamic metasurface antennas for 6G extreme massive MIMO communications. IEEE Wireless Commun. 28 , 106-113 (2021). Zhang, L. et al. A wireless communication scheme based on space-and frequency-division multiplexing using digital metasurfaces. Nat. Electron. 4 , 218-227 (2021). Wang, J. et al. Terabit free-space data transmission employing orbital angular momentum multiplexing. Nat. Photonics 6 , 488-496 (2012). Ding, F., Deng, Y., Meng, C., Thrane, P. C. & Bozhevolnyi, S. I. Electrically tunable topological phase transition in non-Hermitian optical MEMS metasurfaces. Sci. Adv. 10 , eadl4661 (2024). Ding, F., Wang, Z., He, S., Shalaev, V. M. & Kildishev, A. V. Broadband high-efficiency half-wave plate: a supercell-based plasmonic metasurface approach. ACS Nano 9 , 4111-4119 (2015). Hu, J. et al. All-dielectric metasurface circular dichroism waveplate. Sci. Rep. 7 , 41893 (2017). Additional Declarations There is no conflict of interest Supplementary Files SupplementalInformationPlaintext.docx Supplemential material Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-6293050","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":433096099,"identity":"ab4d3fe3-0d02-464a-b3c4-78f943db665d","order_by":0,"name":"Changyuan YU","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABDklEQVRIiWNgGAWjYDACCWTOBwYGGTCDh4EhQQKrejQtjDPAiknRwsxDjBb+2c3PHn5tOyxvzsBjJm27w4bH4EYC44O3bQx5kg04LLlzzNxYtu2w4c4GoJbcM2kgLcyGc9sYiqVx2GIgkWAmLdl2mHHDAZCWtsMgLWzSvG0MifNwakn/BtJiD9ZiCdHC/hu/lhwzyY9thxPBWhihtjCDtMzGoUXiRk6ZNMO59OQNh9mKLXuBfpE887BZcs45icSZOLzPPyN9m+SPMmvbDcebN974ucNGju948sEPb8psEmccwGENEDDzsoFIDgMGRqDBCgdAJAPOiAQDxh9/QBT7A7AWeRzuGQWjYBSMgpELAOKcWfl2TUv8AAAAAElFTkSuQmCC","orcid":"https://orcid.org/0000-0002-3185-0441","institution":"The Hong Kong Polytechnic University","correspondingAuthor":true,"prefix":"","firstName":"Changyuan","middleName":"","lastName":"YU","suffix":""},{"id":433096100,"identity":"2016f1ed-3966-40ae-a2f9-dd3ae8195044","order_by":1,"name":"Zhi Cheng","email":"","orcid":"https://orcid.org/0000-0002-3466-3609","institution":"Department of Electrical and Electronic Engineering, the Hong Kong Polytechnic University","correspondingAuthor":false,"prefix":"","firstName":"Zhi","middleName":"","lastName":"Cheng","suffix":""},{"id":433096101,"identity":"4f6e9e8c-20cd-4884-a3b2-386dfa21dde1","order_by":2,"name":"Zhou Zhou","email":"","orcid":"https://orcid.org/0009-0000-8405-541X","institution":"National University of Singapore","correspondingAuthor":false,"prefix":"","firstName":"Zhou","middleName":"","lastName":"Zhou","suffix":""},{"id":433096102,"identity":"f824dc66-dac5-4064-b486-5122141a256e","order_by":3,"name":"Zhuo Wang","email":"","orcid":"","institution":"School of Optoelectronic Science and Engineering, South China Normal University","correspondingAuthor":false,"prefix":"","firstName":"Zhuo","middleName":"","lastName":"Wang","suffix":""},{"id":433096103,"identity":"a1b97cf4-23c1-4506-bcf2-4fe90cab57f8","order_by":4,"name":"Yue Wang","email":"","orcid":"","institution":"Department of Electrical and Electronic Engineering, The Hong Kong Polytechnic University","correspondingAuthor":false,"prefix":"","firstName":"Yue","middleName":"","lastName":"Wang","suffix":""}],"badges":[],"createdAt":"2025-03-24 08:20:29","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6293050/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6293050/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":80045131,"identity":"43974449-8b78-493a-94a6-1ec564946bfe","added_by":"auto","created_at":"2025-04-07 09:41:54","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":210780,"visible":true,"origin":"","legend":"\u003cp\u003e(a) The mapping between metasurface parameter space and polarization control space. The metasurface parameters include the complex transmission coefficients of the fast and slow axes, as well as the rotation angle. A point on or inside the Poincaré sphere is defined by the azimuth \u003cem\u003eψ\u003c/em\u003e, elevation angles \u003cem\u003eχ\u003c/em\u003e, and radius \u003cem\u003er\u003c/em\u003e. The symbol ⊕ represents the incoherent superposition of light. PP: partially polarized, UP: unpolarized, FP: fully polarized. (b) The homogeneous metasurface with one meta-atom in a unit cell, cannot convert unpolarized light into partially polarized light. (c) The diatomic metasurface with two meta-atoms in a supercell, converts unpolarized light into partially polarized light. (d) The proposed disordered metasurface could partially polarize natural light with the independent control of the dominant full-polarized light and DoP.\u003c/p\u003e","description":"","filename":"FigureCMYK1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-6293050/v1/7a4a41d8b80f0f8abce1ebf9.jpg"},{"id":80045133,"identity":"28a9e2b0-f4ee-4ac8-a034-c282f035fd02","added_by":"auto","created_at":"2025-04-07 09:41:54","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":246883,"visible":true,"origin":"","legend":"\u003cp\u003e(a) The physical mechanism of partially polarizing natural light with the independent control over the SoP and DoP using metasurfaces. (b) Corresponding relationship between metasurface parameters on Poincar\u003cstrong\u003eé\u003c/strong\u003e sphere. (c) The construction of the disordered metasurface. The green and blue rectangles represent the top views of different meta-atoms. The white dashed box represents the effective range of each meta-atom.\u003c/p\u003e","description":"","filename":"FigureCMYK2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-6293050/v1/42c066430485fd8ecbc0be37.jpg"},{"id":80046226,"identity":"a62c9311-0686-471c-8ac9-9c871e915951","added_by":"auto","created_at":"2025-04-07 09:49:54","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":662325,"visible":true,"origin":"","legend":"\u003cp\u003eDemonstration of independent control of the azimuth and elevation angles of the dominantpolarization state and the DoP. (a) 500 uniformly sampled points on the Poincaré sphere, simulating natural light with arbitrary polarization states. (b) 500 points distributed inside and on the surface of the Poincaré sphere, representing partially polarized light after metasurface conversion. Color indicates light intensity. Theoretical parameters in this conversion are 2\u003cem\u003eψ\u003c/em\u003e = 300°, 2\u003cem\u003eχ\u003c/em\u003e = 45°, and DoP = 0.65, respectively. (c) Evolution of the polarization on the Poincare sphere along the latitude at 2\u003cem\u003eχ\u003c/em\u003e = 45° with DoP=0.65, longitude at 2\u003cem\u003eψ\u003c/em\u003e = 45° with DoP = 0.75, and radius at 2\u003cem\u003eχ\u003c/em\u003e = -60°, 2\u003cem\u003eψ\u003c/em\u003e=225°, respectively. The green dots represent the target polarization while the blue dots represent the simulation results. (d) The top view of (c) illustrates the polarization evolution along the latitude. (e) The side view of (c) illustrates the polarization evolution along the longitude. (f) The DoP values of all calculation points. The scatter points are along the radius, and the other two error bars are along the longitude and latitude, respectively. The green shaded area represents the range of DoP error ±0.1.\u003c/p\u003e","description":"","filename":"FigureCMYK3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-6293050/v1/3e63a02bf61c7793d7051296.jpg"},{"id":80045132,"identity":"4e46522f-f406-4c58-8527-3c3633d2f9e1","added_by":"auto","created_at":"2025-04-07 09:41:54","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":4577202,"visible":true,"origin":"","legend":"\u003cp\u003e(a) Experimental setup employed for measuring partially polarized light. PS: polarization scrambler; PM fiber: polarization-maintaining fiber; FC: fiber collimator; OL: objective lens; MS: metasurface; TL: tube lens; FL: Fourier lens; IL: imaging lens; BS: beam splitter; PA: polarization analyzer. (b) Experimental results illustrating measured polarization states represented within the Poincaré sphere. (c)–(e) Measured SoP distributions within the Poincaré sphere for samples 6, 11, and 15, respectively. Insets show corresponding scanning electron microscope (SEM) images of each metasurface. (f)–(h) Experimentally measured Stokes parameters for samples 6, 11, and 15, respectively. (i) Calculated Stokes Euclidean Distances (SED) for all fabricated metasurfaces.\u003c/p\u003e","description":"","filename":"FigureCMYK4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-6293050/v1/e9aa66bf28288bc5071ff2bd.jpg"},{"id":80046224,"identity":"c51ce561-650e-4815-902a-251b97a42ae0","added_by":"auto","created_at":"2025-04-07 09:49:54","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":329185,"visible":true,"origin":"","legend":"\u003cp\u003e(a) Distribution of all polarization states of partially polarized light (2\u003cem\u003eψ\u003c/em\u003e=300°, 2\u003cem\u003eχ\u003c/em\u003e=45°, DoP=0.65) in the time domain on a solid Poincare sphere. The color represents the light intensity and is normalized. (b) Robustness of disordered metasurface design. Taking the position of partially polarized light represented by (a) on the Poincare sphere as the origin, the disordered metasurface is constructed using the same layout along the longitude and latitude directions. The intersection of the two lines is the state represented by (a). (c) and (d) Stokes parameters of the selected points in (b).\u003c/p\u003e","description":"","filename":"FigureCMYK5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-6293050/v1/2515e048fd9970639571d24f.jpg"},{"id":81893100,"identity":"9c37221c-233b-4457-9e3c-4ad2c719893f","added_by":"auto","created_at":"2025-05-04 02:43:39","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":6645612,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6293050/v1/44cf486a-b087-434e-ab04-2ae20f9b2a25.pdf"},{"id":80046227,"identity":"98889410-3e74-4b23-ace5-68e053f3d7e2","added_by":"auto","created_at":"2025-04-07 09:49:54","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":20119562,"visible":true,"origin":"","legend":"Supplemential material","description":"","filename":"SupplementalInformationPlaintext.docx","url":"https://assets-eu.researchsquare.com/files/rs-6293050/v1/09104b81b107ee81e9f19f33.docx"}],"financialInterests":"There is no conflict of interest","formattedTitle":"Mapping Full-Stokes Parameters to Metasurface Design via Globally Engineered Disorder","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eThe capability to precisely manipulate the full-stokes parameters of light, encompassing both the state of polarization (SoP) and degree of polarization (DoP), is a cornerstone of photonics research and development. In deterministic systems, polarization refers to the orientation of the electric field vector, which defines the SoP. In stochastic systems, polarization evolves over time, giving rise to light that may be unpolarized or partially polarized. This variability is quantitatively described by the DoP, which measures the fraction of light that remains polarized. Partially polarized light is thus a combination of fully polarized (dominant polarization) and unpolarized components, with the DoP indicating the relative contribution of the polarized component.\u003c/p\u003e \u003cp\u003eControlling SoP and DoP is vital for applications such as optical communication \u003csup\u003e\u003cspan additionalcitationids=\"CR2\" citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e\u003c/sup\u003e, quantum optics \u003csup\u003e\u003cspan additionalcitationids=\"CR5 CR6\" citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e\u003c/sup\u003e, imaging \u003csup\u003e\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e,\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e\u003c/sup\u003e, remote sensing \u003csup\u003e\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u003c/sup\u003e, and spectroscopy \u003csup\u003e\u003cspan additionalcitationids=\"CR12\" citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e\u003c/sup\u003e. SoP manipulation is commonly achieved through wave plates, which introduce controlled phase shifts between orthogonal polarization components. On the other hand, DoP modulation typically involves converting fully polarized or unpolarized light into a desired partially polarized state. Achieving this requires differential transmission between orthogonal polarization states. This is often implemented experimentally using time-varying optical retarders or wavelength-dependent wave plates. Simultaneous control over both SoP and DoP, however, presents significant engineering challenges. Traditional approaches for such control often rely on complex and bulky optical setups, which can impede the integration and miniaturization of photonic devices. These limitations highlight the need for more compact and efficient solutions capable of fully manipulating the polarization properties of light.\u003c/p\u003e \u003cp\u003eRecently, metasurfaces\u0026mdash;comprising arrays of nanoscale structures\u0026mdash;have garnered significant attention as flat optical devices capable of manipulating light's amplitude, phase, and polarization at subwavelength scales. These capabilities have catalyzed advancements in fields such as imaging \u003csup\u003e\u003cspan additionalcitationids=\"CR15 CR16 CR17 CR18\" citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e\u003c/sup\u003e, polarization control \u003csup\u003e\u003cspan additionalcitationids=\"CR21 CR22 CR23 CR24 CR25 CR26 CR27 CR28 CR29 CR30 CR31 CR32 CR33 CR34 CR35\" citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e\u003c/sup\u003e, quantum optics \u003csup\u003e\u003cspan additionalcitationids=\"CR38 CR39 CR40\" citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e41\u003c/span\u003e\u003c/sup\u003e, and communications \u003csup\u003e\u003cspan additionalcitationids=\"CR43 CR44\" citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e45\u003c/span\u003e\u003c/sup\u003e. Importantly, metasurfaces facilitate the miniaturization and integration of photonic devices, offering enhanced functionality within compact platforms. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(a), in the context of polarization control, a mapping relationship between the metasurface design parameter space and the polarization control parameter space can be established through careful design. The homogeneous metasurface composed of birefringent meta-atoms, shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(b), functions as a nano-waveplate widely used in polarization modulation \u003csup\u003e\u003cspan additionalcitationids=\"CR47\" citationid=\"CR46\" class=\"CitationRef\"\u003e46\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e48\u003c/span\u003e\u003c/sup\u003e. When the transmittance is set close to unity, the system becomes effectively lossless, yielding a unitary Jones matrix. This allows the metasurface to convert one SoP into another by redistributing phase delay between orthogonal polarization components. However, in dynamic systems with incoherent light, time-averaging equalizes intensity differences between these components, rendering this approach unsuitable for controlling the DoP. We adjust the gap distance between two types of meta-atoms within the 'weak coupling' region \u003csup\u003e\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e,\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e\u003c/sup\u003e. This adjustment induces far-field radiation interference in the diatomic structure shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(c). This design could provide at least six degrees of freedom, enabling the realization of any arbitrary Jones matrix for polarization conversion \u003csup\u003e\u003cspan additionalcitationids=\"CR21 CR22 CR23 CR24\" citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e\u003c/sup\u003e. The flexibility of this method allows for the creation of non-unitary Jones matrices, which generate intensity differences between orthogonal states in a time-varying system. However, this results in an intrinsic coupling between the metasurface design parameters, specifically the dominant SoP and the DoP within the polarization space, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(c). This relationship can be further understood using the phasor diagram presented in Fig. \u003cspan refid=\"MOESM1\" class=\"InternalRef\"\u003eS1\u003c/span\u003e. Wang et al. recently tackled this challenge by introducing an inverse-designed meta-atom, optimized through topological methods, capable of converting unpolarized light into partially polarized light \u003csup\u003e\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e\u003c/sup\u003e. However, the complex geometries of these optimized structures require high-precision fabrication, which presents significant challenges for practical implementation.\u003c/p\u003e \u003cp\u003eIn this study, we introduce a novel approach to decouple the SoP and DoP using a disordered metasurface. This non-periodic metasurface, illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(d), is composed of two types of birefringent meta-atoms uniformly distributed across its surface. By leveraging far-field interference effects, we introduce polarization-dependent loss (PDL) to precisely modulate the DoP. The DoP is controlled by the differential conversion of orthogonal polarization components, determined by the ratio of different meta-atoms. SoP control is achieved by adjusting the azimuth and elevation angles on the Poincar\u0026eacute; sphere through the rotation and phase difference of individual meta-atoms. We developed an algorithm based on a greedy search and a 2D bin-packing problem to generate the disordered metasurface. This design allows access to all Stokes parameters, enabling systematic control of polarization states on the Poincar\u0026eacute; sphere. As a proof of concept, we theoretically and experimentally demonstrate that varying a single parameter can produce polarization distributions along latitude, longitude, and radial directions. It is also shown that the design possesses a degree of tolerance, allowing different meta-atom combinations to function effectively within the same framework. Furthermore, we demonstrate the flexibility of this design approach in enabling arbitrary polarization state conversion applications.\u003c/p\u003e"},{"header":"2. Results","content":" \u003cp\u003eThe physical mechanisms that simultaneously and independently control DoP and SoP using metasurfaces are illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. Since unpolarized light can be viewed as an incoherent superposition of two orthogonal polarization states, analyzing this mechanism requires focusing on the conversion process between these orthogonal states. Each metasurface is capable of converting (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|{\\alpha\\:}^{*}\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|{\\beta\\:}^{*}\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e) into (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|\\alpha\\:\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|\\beta\\:\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e), with distinct phase shifts for the respective states. This results in constructive interference for the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|\\alpha\\:\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e state and destructive interference for the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|\\beta\\:\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e state. When the two metasurfaces differ in their proportional contribution, the degrees of constructive interference for \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|\\alpha\\:\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e and destructive interference for \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|\\beta\\:\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e are no longer balanced, leading to an unequal ratio of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|\\alpha\\:\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|\\beta\\:\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e in the final incoherent light output. In essence, differential conversion between \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|\\alpha\\:\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|\\beta\\:\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e is achieved, resulting in partially polarizing the unpolarized light. In what follows, the derivation of the necessary Jones matrices, the mechanism for controlling DoP, and the design methodology for the metasurfaces are discussed.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eWe begin with the birefringent meta-atoms which are commonly used for polarization control. The behavior of these meta-atoms is described by the following Jones matrix:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:{J}_{atom}=R\\left(\\theta\\:\\right)\\left(\\begin{array}{cc}{A}_{f}{e}^{i{\\phi\\:}_{f}}\u0026amp;\\:0\\\\\\:0\u0026amp;\\:{A}_{s}{e}^{i{\\phi\\:}_{s}}\\end{array}\\right)R(-\\theta\\:)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThis Jones matrix accounts for the phase retardation along the fast and slow optical axes, denoted by \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\phi\\:}_{f}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\phi\\:}_{s}\\)\u003c/span\u003e\u003c/span\u003e, respectively, with \u003cem\u003eA\u003c/em\u003e\u003csub\u003e\u003cem\u003ef\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eA\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e representing the transmittances along these axes. The rotation matrix \u003cem\u003eR(θ)\u003c/em\u003e, defined \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:R\\left(\\theta\\:\\right)=\\left[\\begin{array}{cc}\\text{c}\\text{o}\\text{s}\\left(\\theta\\:\\right)\u0026amp;\\:-\\text{s}\\text{i}\\text{n}\\left(\\theta\\:\\right)\\\\\\:\\text{s}\\text{i}\\text{n}\\left(\\theta\\:\\right)\u0026amp;\\:\\text{c}\\text{o}\\text{s}\\left(\\theta\\:\\right)\\end{array}\\right]\\)\u003c/span\u003e\u003c/span\u003e, describes the orientation of the birefringent axes with respect to the incident polarization. With transmittances close to unity, the system becomes effectively lossless, yielding a unitary Jones matrix. To introduce differential transmission between orthogonal states, a non-unitary Jones matrix is required, which can be constructed as a sum of unitary matrices.\u003c/p\u003e \u003cp\u003eOn the other hand, the Jones vectors of an arbitrary orthogonal polarization pair can be expressed in terms of azimuth angle \u003cem\u003eψ\u003c/em\u003e and elevation angle \u003cem\u003eχ\u003c/em\u003e on the Poincare sphere as:\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:|\\alpha\\:\\rangle\\:=R(\\psi\\:-{45}^{\\circ\\:}\\left)\\right(\\begin{array}{c}{e}^{-i\\chi\\:}\\\\\\:{e}^{i\\chi\\:}\\end{array}),|\\beta\\:\\rangle\\:=R(\\psi\\:-{45}^{\\circ\\:})\\left(\\begin{array}{c}{e}^{-i\\chi\\:}\\\\\\:-{e}^{i\\chi\\:}\\end{array}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eHere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|\\alpha\\:\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e is located at coordinates (2\u003cem\u003eψ\u003c/em\u003e, 2\u003cem\u003eχ\u003c/em\u003e) on the Poincar\u0026eacute; sphere, while \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|\\beta\\:\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e is positioned at (2\u003cem\u003eψ\u003c/em\u003e \u0026minus;\u0026thinsp;180\u0026deg;, -2\u003cem\u003eχ\u003c/em\u003e), symmetrically opposite about the origin. It is important to note that, in the polarization states (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|\\alpha\\:\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|\\beta\\:\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e), the x- and y-components are complex conjugates, except for the negative sign in \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|\\beta\\:\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e. This indicates that by applying a phase delay to the \u003cem\u003ex\u003c/em\u003e- and \u003cem\u003ey\u003c/em\u003e-components, as described by the Jones matrix in the form of Eq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e), it is possible to convert between the polarization states (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|{\\alpha\\:}^{*}\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|{\\beta\\:}^{*}\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e) and their conjugate states (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|{\\alpha\\:}^{*}\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|{\\beta\\:}^{*}\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e) (Supplemental Information Note II). Therefore, the Jones matrix for converting the orthogonal conjugate states can be expressed in a form similar to Eq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e):\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:{J}_{1}=R(\\psi\\:-{45}^{\\circ\\:})\\left(\\begin{array}{cc}{e}^{-2i\\chi\\:}\u0026amp;\\:0\\\\\\:0\u0026amp;\\:{e}^{2i\\chi\\:}\\end{array}\\right)R({45}^{\\circ\\:}-\\psi\\:)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThis Jones matrix converts the conjugate orthogonal states (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|{\\alpha\\:}^{*}\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|{\\beta\\:}^{*}\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e) into (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|\\alpha\\:\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|\\beta\\:\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e) while introducing the same global phase. This means that there is no relative phase shift between the two states. Another Jones matrix can be constructed to achieve the conversion between these two pairs of conjugate orthogonal states with a relative phase difference of π (see Supplemental Information Note II). The Jones matrix for this transformation can be expressed as:\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\:{J}_{2}=R\\left(\\psi\\:\\right)\\left(\\begin{array}{cc}1\u0026amp;\\:0\\\\\\:0\u0026amp;\\:-1\\end{array}\\right)R(-\\psi\\:)=R(\\psi\\:-45)\\left(\\begin{array}{cc}0\u0026amp;\\:1\\\\\\:1\u0026amp;\\:0\\end{array}\\right)R(45-\\psi\\:)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThis matrix flips the x- and y-components of the Jones vector in Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). The negative sign in the y-component of the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|\\beta\\:\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e state introduces a global phase of π during the conjugate state conversion process. By combining the Jones matrices from Eqs.\u0026nbsp;(\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e) and (\u003cspan refid=\"Equ4\" class=\"InternalRef\"\u003e4\u003c/span\u003e) with different coefficients, differential transmission between orthogonal polarization states can be achieved, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e(a). The resulting composite Jones matrix can be expressed as follows:\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$\\:J={J}_{1}+{J}_{2}=\\frac{{n}_{1}}{N}R\\left(\\psi\\:-{45}^{\\circ\\:}\\right)\\left(\\begin{array}{cc}{e}^{-2i\\chi\\:}\u0026amp;\\:0\\\\\\:0\u0026amp;\\:{e}^{2i\\chi\\:}\\end{array}\\right)R\\left({45}^{\\circ\\:}-\\psi\\:\\right)+\\frac{{n}_{2}}{N}R\\left(\\psi\\:\\right)\\left(\\begin{array}{cc}1\u0026amp;\\:0\\\\\\:0\u0026amp;\\:-1\\end{array}\\right)R\\left(-\\psi\\:\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eTo ensure energy conservation, the sum of the coefficients must be equal to unity. Therefore, the coefficients are written in the form of \u003cem\u003en\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e/N\u003c/em\u003e, where \u003cem\u003ei\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1, 2, and \u003cem\u003eN\u0026thinsp;=\u0026thinsp;n\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u003cem\u003e+\u0026thinsp;n\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e. The state \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|{\\alpha\\:}^{*}\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e is fully converted into \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|\\alpha\\:\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e with an amplitude of 1/2. However, the conversion between \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|{\\beta\\:}^{*}\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|\\beta\\:\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e undergoes destructive interference due to the π-phase delay, resulting in an amplitude of (\u003cem\u003en\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e - \u003cem\u003en\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e)/2\u003cem\u003eN\u003c/em\u003e. Consequently, the transmission matrix for converting (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|{\\alpha\\:}^{*}\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|{\\beta\\:}^{*}\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e) into (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|\\alpha\\:\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:|\\beta\\:\\rangle\\:\\)\u003c/span\u003e\u003c/span\u003e) is given by:\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$\\:{J}_{t}=\\left(\\begin{array}{cc}{t}_{{\\alpha\\:}^{*}\\alpha\\:}\u0026amp;\\:{t}_{{\\beta\\:}^{*}\\alpha\\:}\\\\\\:{t}_{{\\alpha\\:}^{*}\\beta\\:}\u0026amp;\\:{t}_{{\\beta\\:}^{*}\\beta\\:}\\end{array}\\right)=\\left(\\begin{array}{cc}1\u0026amp;\\:0\\\\\\:0\u0026amp;\\:\\frac{{n}_{1}-{n}_{2}}{N}\\end{array}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eTo determine the DoP in the time domain, it is necessary to apply a time-averaged operation to account for the random phase in unpolarized light. The intensity of the outgoing light can then be expressed as:\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$\\:\\begin{array}{c}\\langle\\:{I}_{out}\\rangle\\:=(1-{t}_{{\\beta\\:}^{*}\\beta\\:}^{2})\\langle\\:\\alpha\\:\\rangle\\:+{t}_{{\\beta\\:}^{*}\\beta\\:}^{2}(\\langle\\:\\alpha\\:\\rangle\\:+\\langle\\:\\beta\\:\\rangle\\:)\\\\\\:=(1-{t}_{{\\beta\\:}^{*}\\beta\\:}^{2})FP+2{t}_{{\\beta\\:}^{*}\\beta\\:}^{2}UP\\end{array}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere the brackets\u0026thinsp;\u0026lt;\u0026thinsp;\u0026gt;\u0026thinsp;denote to time average. According to the definition of DoP, let \u003cem\u003eη\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003en\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e/n\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e, it is calculated as:\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$$\\:p=\\frac{1-{t}_{{\\beta\\:}^{*}\\beta\\:}^{2}}{1+{t}_{{\\beta\\:}^{*}\\beta\\:}^{2}}=\\frac{2\\eta\\:}{1+{\\eta\\:}^{2}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAs a result of the preceding derivation, a one-to-one correspondence between the structural parameters of the metasurface and the coordinates on the Poincar\u0026eacute; sphere has been established, as illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e(b).\u003c/p\u003e \u003cp\u003eTo achieve the Jones matrix in Eq.\u0026nbsp;(\u003cspan refid=\"Equ5\" class=\"InternalRef\"\u003e5\u003c/span\u003e), we propose a disordered metasurface composed of various types of meta-atoms. These adjacent meta-atoms are arranged with appropriate gap distances in the weak coupling regime. In this case, the Jones matrix is the summation of the matrices of each meta-atom, represented in the following form:\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$$\\:{J}_{DM}=\\sum\\:_{i=1}^{\\text{M}}\\frac{{n}_{i}}{N}{J}_{atom}^{i}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003en\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e (\u003cem\u003ei\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1,2\u0026hellip;\u003cem\u003eM\u003c/em\u003e) represents the number of each meta-atom, and \u003cem\u003eN\u003c/em\u003e denotes the total number of atoms. When \u003cem\u003eM\u003c/em\u003e\u0026thinsp;=\u0026thinsp;2 and \u003cem\u003en\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003en\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e, the disordered metasurface degenerates into the simpler diatomic design shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(c), where a periodic unit cell within wavelength scale is constructed across the metasurface. However, when \u003cem\u003en₁\u003c/em\u003e \u0026ne; \u003cem\u003en₂\u003c/em\u003e, if we continue to follow the conventional periodic design approach, multiple types of meta-atoms must be arranged within a supercell, resulting in a lattice constant larger than the wavelength. This oversized lattice constant can introduce energy losses by diverting energy away from the intended polarization conversion, leading to the generation of higher-order diffraction modes. Additionally, while a periodic design does not inherently produce a phase gradient, it is more prone to introducing one without careful optimization. In the presence of a phase gradient, the periodicity can amplify this effect, leading to stronger higher-order diffraction and further deflection of the outgoing beam as shown in Fig. S2. More critically, the periodic design complicates the ability to maintain a uniform distribution of meta-atoms across the metasurface. The translational symmetry inherent to periodic structures hinders the preservation of a consistent ratio of meta-atoms near the boundaries, thereby diminishing polarization conversion efficiency (see Supplemental Information Note III).\u003c/p\u003e \u003cp\u003eBuilding on this understanding, the disordered metasurface, as illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e(c), consists of \u003cem\u003en\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e meta-atoms of type I and n₂ meta-atoms of type II. Each type of meta-atom is uniformly arranged, with their spacing optimized for the weak-coupling region to ensure effective interaction. This design ensures far-field radiation coherence among the meta-atoms, which is essential for the validity of Eq.\u0026nbsp;(\u003cspan refid=\"Equ9\" class=\"InternalRef\"\u003e9\u003c/span\u003e), where the total response of the metasurface is expressed as the summation of the individual meta-atoms Jones matrices. Close meta-atom placement causes strong coupling, altering the propagation phase. Conversely, if they are too far apart, the meta-atoms interact weakly, and no coherent interference is observed. The uniform arrangement ensures that the local ratio of meta-atoms within each effective coupling region remains consistent with the target ratio \u003cem\u003en\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e/\u003cem\u003en\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e across the entire surface. Unlike conventional interleaved designs, where different meta-atom types function independently, the meta-atoms in the disordered metasurface work cooperatively. The intentional disorder breaks the translational symmetry, effectively suppressing strong higher-order diffraction modes. Through this design, the Stokes parameters (S1, S2, S3) and the DoP are independently controlled. The polarization orientation, represented by S1, S2, and S3, can be mapped to the azimuth and elevation angles on the Poincar\u0026eacute; sphere, which is governed by the rotation angle and phase delay of the meta-atoms, as described by Eq.\u0026nbsp;(\u003cspan refid=\"Equ5\" class=\"InternalRef\"\u003e5\u003c/span\u003e). Meanwhile, the DoP, corresponding to the radius in the Poincar\u0026eacute; sphere, is controlled by the ratio of the two types of meta-atoms (\u003cem\u003en\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e/\u003cem\u003en\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e) within the disordered metasurface. This decoupled control allows for independent tuning of the polarization state\u0026rsquo;s orientation and its purity, enabling full manipulation of all relevant Stokes parameters.\u003c/p\u003e \u003cp\u003eTo obtain the desired arrangement of meta-atoms, we developed a two-dimensional bin-packing algorithm based on a greedy search heuristic (see Supplemental Information Note IV). Our empirical calculations show that the weak coupling region typically spans about 1/10 of the operating wavelength. We define the effective working region of each meta-atom using the enclosing rectangles, as shown by the dashed lines in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e(c). These rectangles are tessellated uniformly across the metasurface by the algorithm, ensuring that the arrangement maintains both uniformity and the adaptive spacing required for coherent interactions. While our approach yields strong results, other advanced algorithms could also be employed to achieve similarly optimized layouts for disordered metasurfaces.\u003c/p\u003e \u003cp\u003eWe validate the proposed concept by demonstrating full control of the SoP and DoP across the entire Poincar\u0026eacute; sphere through simulations conducted using the Finite-Difference Time-Domain (FDTD) method, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. For fully polarized light, the polarization state is represented by points on the surface of the Poincar\u0026eacute; sphere, while partially polarized light is depicted by points within the sphere. Complete polarization control requires the manipulation of three parameters on the Poincar\u0026eacute; sphere: azimuthal angle, elevation angle, and radius, which correspond to the Stokes parameters S1, S2, S3, and the DoP. By independently varying these parameters, we demonstrate precise control over the latitude, longitude, and radius of the Poincar\u0026eacute; sphere, allowing for arbitrary manipulation of partially polarized light.\u003c/p\u003e \u003cp\u003eIn the simulation, 500 points are uniformly sampled across the surface of the Poincar\u0026eacute; sphere as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e(a), with near-equal Euclidean distances between adjacent points, to represent the input of unpolarized light. The random phase component inherent in unpolarized light is neglected, as it does not influence the results in time-averaged incoherent superposition calculations. The polarization states corresponding to the sampled points are converted by the designed metasurface, resulting in a non-uniform distribution of Stokes vectors across both the surface and interior of the Poincar\u0026eacute; sphere, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e(b). This unique distribution on the Poincar\u0026eacute; sphere serves as a distinctive fingerprint of partially polarized light, encapsulating its statistical nature and providing profound insights into its transformation through the metasurface. The incoherent sum of these transformed polarization states yields the DoP and the dominant SoP.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e(c)-(f) demonstrates the evolution of polarization states on the Poincar\u0026eacute; sphere, showcasing how the azimuth, elevation angles, and DoP can be independently controlled. The latitude and longitude lines on the Poincar\u0026eacute; sphere passing through the polarization state at 2\u003cem\u003eχ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;45\u0026deg; and 2\u003cem\u003eψ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;45\u0026deg;, respectively. These polarization states are achieved by varying the size or orientation of two types of meta-atoms. Only half of the longitude is depicted, as the other half can be obtained by rotation of the meta-atoms. To enhance visualization clarity, the dominant SoP is chosen as (2\u003cem\u003eχ\u003c/em\u003e = -60\u0026deg;, 2\u003cem\u003eψ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;225\u0026deg;), located in the southern hemisphere, when demonstrating DoP control through adjustments to the meta-atom quantity ratio. For the latitude data set, the DoP is set to 0.65, while for the longitude data set, it is set to 0.75, corresponding to quantity ratios of 2.714 and 2.214, respectively. In the metasurface design, the target DoP values are approximated by quantity ratios of 19:7 and 31:14 for the two types of meta-atoms. The arrangement coordinates and structural parameters of the employed meta-atoms are detailed in Supplemental Information Note V. In Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, the green points represent the target polarization states on the Poincar\u0026eacute; sphere, while the blue points represent the simulation results. The latitude and longitude evolution data sets are also shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e(d) and Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e(e), with top-view and side-view perspectives, respectively. The average error in the elevation angle is 2.65\u0026deg;, while the average error in the azimuth angle is 2.98\u0026deg;, demonstrating the accuracy of this method in controlling the polarization state. In Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e(f), error bars indicate the range of simulated DoP values for the latitude and longitude data sets, with the central sphere symbol representing the average DoP. The green shaded area corresponds to a DoP error range of \u0026plusmn;\u0026thinsp;0.1, while nearly all data points fall within the tighter range of \u0026plusmn;\u0026thinsp;0.05. As the elevation angle increases, we observe a rise in error. This is attributed to the limited size of the meta-atom database (see Supplemental Information Note V) and the relatively large size difference between the two types of meta-atoms in the high-latitude regions. This discrepancy complicates the task of finding an appropriate spacing to satisfy both weak coupling and coherence conditions. The mismatch between the propagation phase of the selected meta-atoms and the target phase could also contribute to the observed errors. Expanding the meta-atom database could help mitigate both issues and reduce the error.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eExperimental validation of the proposed approach is presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. Figure\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e(a) illustrates the experimental setup, where a polarization scrambler (PS) was employed to generate randomly polarized light, simulating natural light input with a modulation frequency of 2 Hz. The polarization analyzer operated at a sampling rate of 15 Hz, significantly exceeding the Nyquist sampling frequency, ensuring accurate measurement of the output polarization states. Detailed information on the experimental setup is provided in the Materials and methods section.\u003c/p\u003e \u003cp\u003eIn our experiment, we fabricated and measured 15 samples. Among them, 8 samples were designed to demonstrate polarization evolution along the latitude on the Poincar\u0026eacute; sphere, with a step size of 45\u0026deg;. Three samples were used to illustrate changes along the longitude, covering linear polarization (LP), elliptical polarization (EP), and right-handed circular polarization (RCP). The remaining 4 samples were dedicated to demonstrating DoP control. All samples were fabricated using a standard CMOS-compatible process, with details provided in the Materials and methods section.\u003c/p\u003e \u003cp\u003eRepresentative experimental results from each of the three groups, specifically samples No. 6, 11, and 15, are presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. Additional data, arranged sequentially, can be found in Supplementary Information Note VI. Figures\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e(c)\u0026ndash;4(e) illustrate the distribution of the experimentally measured polarization states on the Poincar\u0026eacute; sphere for three selected samples, accompanied by the corresponding SEM images of the fabricated metasurfaces. The selected samples correspond to polarization states with parameters 2ψ\u0026thinsp;=\u0026thinsp;225\u0026deg;, 2χ\u0026thinsp;=\u0026thinsp;90\u0026deg;, and DoP\u0026thinsp;=\u0026thinsp;0.9, respectively. Figures\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e(f)\u0026ndash;4(h) depict the measured Stokes parameters for all polarization states in each sample. Based on the theoretical analysis, the distribution of partially polarized light on the Poincar\u0026eacute; sphere is expected to show a clustering effect around the dominant polarization state as the DoP increases. This trend is confirmed by the experimental results, where the target Stokes parameters are marked with white dashed lines. The measured polarization states exhibit a higher density of points near the dominant polarization state, supporting the theoretical predictions.\u003c/p\u003e \u003cp\u003eThe accuracy of the partially polarized light conversion is quantified by the Stokes Euclidean Distance (SED), which is defined as 1\u0026thinsp;\u0026minus;\u0026thinsp;distance, where \"distance\" represents the Euclidean distance between the measured and target Stokes vectors. A value of SED\u0026thinsp;=\u0026thinsp;1 indicates perfect alignment (i.e., zero Euclidean distance) between the two Stokes vectors. Figure\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e(i) presents the SED values for all 15 experimental samples, with an average SED of 0.90, compared to the theoretical simulation average of 0.93. The accuracy of experimental results for the three representative groups of samples is measured as 0.942, 0.901, and 0.969, respectively, demonstrating the reliability of our proposed method.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eTo further demonstrate the feasibility and flexibility of this approach, we showcase the ability to generate different partially polarized light using the same metasurface arrangement. Since the coherent pixels operate in a weak coupling regime, small variations in the gap distance between adjacent meta-atoms have a negligible effect on the modulation behavior. This introduces a tolerance in the gap distance, allowing the same metasurface design to be applied across different modulation scenarios.\u003c/p\u003e \u003cp\u003eFor instance, the layout designed to achieve a DoP of 0.65, with a quantity ratio of 2.714, as previously illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e(c). In this layout, the effective sizes of the meta-atoms are (850, 650) nm, and (760, 850) nm. The dominant polarization state is initialized at 2\u003cem\u003eψ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;300\u0026deg; and 2\u003cem\u003eχ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;45\u0026deg;. The distribution of SoP at this point is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e(a), obtained through numerical simulations. The corresponding SED is calculated to be 0.989, confirming the high accuracy of the designed metasurface in generating the target polarization state. Starting from the initial point, we varied the sizes and rotation angles of the meta-atoms, causing the partially polarized light to move along both the longitude and latitude lines passing through the initial point.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e(b) illustrates the operational range of the disordered metasurface, defined as the effective polarization conversion range in terms of latitude and longitude on the Poincar\u0026eacute; sphere. Stokes parameters S1-S3 are also given of each point in the Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e(c) and (d). In Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e(b), the intersection of these two lines represents the initial point. The tolerance in the azimuthal angle (meta-atom rotation) is approximately 60\u0026deg; (with the range of SED exceeding 0.95), while the elevation angle (meta-atom size) tolerance is less than 10\u0026deg;. This discrepancy arises because adjusting the elevation angle requires a complete change in the meta-atom combination, for which the initial effective sizes are not suitable. Although the physical gap between different meta-atom combinations may exhibit varying tolerances, the specific values presented here are not universal. Nonetheless, these results confirm that the metasurface design provides a tolerance margin, allowing the same layout to be used for different meta-atom configurations. This tolerance is crucial for practical applications, as it enables flexible and independent control of both the DoP and SoP.\u003c/p\u003e"},{"header":"3. Discussion","content":"\u003cp\u003eIn the article, we propose a disordered metasurface that converts unpolarized light into partially polarized light, achieving independent control over both the SoP and DoP. This approach harnesses far-fi`1eld interference between meta-atoms and introduces a novel design parameter—the meta-atom quantity ratio—to enable arbitrary and flexible manipulation of polarization states. By generalizing this method, we derive an analytical solution that directly maps the optical parameters of meta-atoms to each element of an arbitrary Jones matrix, as detailed in Supplemental Information Note VII using a triatomic design. Unlike inverse design methods, our approach offers a more intuitive insight into the underlying physics, establishing a clear and concise relationship between meta-atom properties, the Jones matrix, and full Stokes parameter control. The additional degrees of freedom introduced through this disordered metasurface design, particularly via the meta-atom quantity ratio, significantly enhance the flexibility of polarization control beyond conventional techniques. This advancement empowers independent tuning of all Stokes parameters, opening up possibilities for advanced photonic applications where polarization must be tailored independently, thus paving the way for next-generation systems requiring sophisticated multi-dimensional light manipulation.\u003c/p\u003e "},{"header":"Materials and methods","content":"\u003cp\u003e \u003cb\u003eFabrication of the disordered metasurface\u003c/b\u003e \u003c/p\u003e\u003cp\u003eThe metasurface was fabricated on a commercially available 940-nm-thick α-Si layer (PECVD) deposited on a sapphire substrate. The structure was patterned on a ZEP520A resist using an E-beam writer (Raith E-line, 50 kV, 20 nA). After developing the resist, the pattern was transferred onto a 100-nm-thick chromium film. Subsequently, the silicon layer was etched using inductively coupled deep reactive ion etching (DRIE), with the chromium layer serving as a hard mask. The resist was then removed with acetone, followed by rinsing with IPA and DI water. Finally, the remaining chromium mask was removed using a chromium etchant.\u003c/p\u003e\u003cp\u003e \u003cb\u003eExperimental setup\u003c/b\u003e \u003c/p\u003e\u003cp\u003eThe experimental setup is designed to precisely control and analyze the polarization state of light interacting with the sample. A polarization scrambler (Luna PSY 201) is used to control the input polarization state, ensuring a well-defined and tunable polarization condition. The optical path begins with Objective 1 (10X, NA = 0.17), which functions as a condenser lens to focus the incident light near the sample. Objective 2 (50X Mitutoyo Plan Apo NIR Infinity Corrected Objective, NA = 0.42) is then used for light collection and microscopic imaging of the sample. A 4f imaging system, composed of a tube lens and a Fourier lens, is implemented to relay the optical field while maintaining spatial coherence. At the conjugate sample plane within this system, an iris diaphragm is placed to block unwanted light from outside the sample area, improving signal quality. An imaging lens is used to project the sample image onto a CCD camera, allowing direct observation and adjustment of the iris diaphragm size. The polarization state of the transmitted light is analyzed using a polarization analyzer (Thorlabs PAX1000IR2).\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003eAcknowledgements\u003c/p\u003e\n\u003cp\u003eWe thank Prof. Cheng-wei Qiu for valuable comments and feedback. The authors thank the support of Hong Kong Research Grants Council (GRF 15209321 B-Q85G).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor Contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eZ.C. conceived the methodology, developed the software, performed validation, conducted visualization, and wrote the original draft. Z.Z. contributed to visualization and assisted with writing and editing. Z.W. and Y.W. supported validation and manuscript review. C.Y. conceived the project, provided conceptual guidance, supervised the research, secured funding, and contributed to project administration.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConflict of Interest statement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare that they have no conflict of interest.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData availability statements:\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe data that support the findings of this study are available from the corresponding author upon reasonable request.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eZheng, Y., Zhang, X., Chen, L. \u0026amp; Yang, B. Analysis of degree of polarization ellipsoid as feedback signal for polarization mode dispersion compensation in NRZ, RZ and CS-RZ systems. \u003cem\u003eOpt. 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While metasurfaces have demonstrated remarkable capabilities in polarization control, existing approaches often struggle to simultaneously manipulate SoP and DoP with high flexibility. Here, we introduce a paradigm shift in metasurface-based polarization engineering by proposing a globally engineered disordered metasurface that enables a one-to-one correspondence between structural parameters and the full-Stokes polarization space. Unlike conventional metasurfaces that rely solely on unit-cell-level deterministic phase profiles, our approach incorporates a statistical design principle, introducing a spatial statistical parameter: the meta-atom quantity ratio. By uniformly distributing two distinct types of meta-atoms with controlled ratios, we effectively decouple the design parameters, enabling independent control over all Stokes parameters. Specifically, the azimuthal and elevation angles of the SoP on the Poincar\u0026eacute; sphere are governed by the rotation and size of individual meta-atoms, while the DoP is precisely tuned through global disorder engineering via the quantity ratio of the meta-atoms. This approach establishes a direct mapping between metasurface design and polarization space, revealing new physical insights into disorder-assisted polarization control. A computationally efficient algorithm optimizes the metasurface arrangement, achieving a polarization similarity (evaluated by Stokes Euclidean Distance) of 0.93 in theory and 0.90 in experiment. Our findings advance the development of metasurfaces that harness disorder as a functional design strategy, enabling enhanced flexibility in full-Stokes polarization engineering.\u003c/p\u003e","manuscriptTitle":"Mapping Full-Stokes Parameters to Metasurface Design via Globally Engineered Disorder","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-04-07 09:41:49","doi":"10.21203/rs.3.rs-6293050/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"28989207-767f-4684-8501-e6a8fc402a0b","owner":[],"postedDate":"April 7th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":46116068,"name":"Physical sciences/Optics and photonics/Optical materials and structures/Metamaterials"},{"id":46116069,"name":"Physical sciences/Physics/Electronics, photonics and device physics/Photonic devices"}],"tags":[],"updatedAt":"2025-05-04T02:35:29+00:00","versionOfRecord":[],"versionCreatedAt":"2025-04-07 09:41:49","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-6293050","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6293050","identity":"rs-6293050","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}