Unlocking ultra-high holographic information capacity through nonorthogonal polarization multiplexing

The study addresses the fundamental limitation imposed by polarization orthogonality in photonics, where the inner product of two output fields based on orthogonal bases constrains polarization multiplexing to two channels (2×2 Jones matrix). While orthogonality ensures perfect isolation in applications like polarization imaging and encryption, it limits the maximum number of interference-free channels. Existing space- or time-division multiplexing expands channels at the cost of spatial/temporal resolution and typically reaches only up to four orthogonal polarization channels. Introducing other degrees of freedom (e.g., wavelength, orbital angular momentum) often increases crosstalk and noise. Even with advanced eigen-polarization modulation, reliance on orthogonal bases persists, restricting high-capacity applications such as dynamic holography and information transmission. The research question is how to overcome orthogonality constraints to achieve many low-crosstalk polarization channels with full use of polarization degrees of freedom, without sacrificing other dimensions like spatial or temporal resolution.

Metasurface technologies have enabled expanded holographic information capacity. Prior work cascaded metasurfaces to realize spatially varying Jones matrices for complex amplitude and phase control (Bao et al.), and high-efficiency metasurfaces achieved complex vectorial holography via polarization multiplexing (Wang et al.). Engineered noise has been used to break traditional polarization multiplexing limits, reaching up to 11 independent holographic images on a single metasurface (Xiong et al.). Broader efforts incorporated wavelength and orbital angular momentum to increase channel count, but typically at the expense of increased crosstalk and noise. Despite progress, full exploitation of metasurface capabilities for high-capacity, low-crosstalk polarization multiplexing remains an open challenge, especially for applications requiring extensive data encoding and secure transmission.

Core concept and theory: The authors propose nonorthogonal polarization-basis multiplexing by engineering spatially varying eigen-polarization states p1(x,y), p2(x,y) in subwavelength metaatoms to collectively produce globally nonorthogonal output polarization channels with minimal crosstalk. The metasurface is modeled with a Jones matrix J = [e1, e2]† diag(A1, A2) [e1, e2]. For arbitrary input/output polarization pairs pi, pj, the metaatom response O(x,y) = pj† A pi depends on local eigen-polarizations, enabling selective channel extraction. In circular basis, the response decomposes into eigen (co-polarized) and conjugate (cross-polarized) components, allowing arbitrary input/output combinations. A position-dependent response matrix O(x,y) over chosen polarization states forms a symmetric matrix due to reciprocity, and the global channel response is obtained by spatial integration, which is optimized to maximize isolation across channels. Optimization via Vectorial Diffraction Neural Network (VDNN): A VDNN framework is built where different polarized lights form the input layer, the metasurface acts as the hidden layer, and the image plane is the output layer. The network jointly optimizes metaatom parameters (e.g., phases φx, φy and orientation θ) to match multiple target holograms for distinct nonorthogonal polarization channels while minimizing crosstalk. Training uses the Adam optimizer with loss terms for each target pattern; correlation coefficients measure similarity between outputs and targets. Wavelength is 3 µm; diffraction distance (metasurface to image plane) is 500 µm. Example calibration sets incident linear polarization angles 0°, 60°, 120° mapped to output 60°, 120°, 0°, generating three gray-scale holograms (puppy, kitten, mouse). For extended systems, ten linear input angles and ten output angles (0° to 162° in 18° steps) define a 10×10 channel grid, with 55 lower-triangular channels used. Device design and fabrication: Metasurfaces consist of anisotropic nano-elliptic silicon metaatoms on a periodic lattice. Example device: 400×400 pixels, 1.5 µm period; metaatom height 4 µm; length/width 0.3–1.2 µm. A larger device uses 800×800 metaatoms (640,000 total) for 55-channel demonstrations. Fabrication steps: electron-beam evaporation of 50 nm Cr on double-polished Si; spin-coat photoresist and bake; define patterns by electron beam lithography (JBX-6300FS); develop in 300-MIF; rinse and dry; inductively coupled plasma (ICP) dry etching to pattern Cr and Si. Experimental setup and measurement: Mid-IR source Electro MIR 4.8 laser, filtered to 3 µm (200 nm bandwidth). Light passes a polarizer and a full-wave liquid crystal retarder (Thorlabs LCC1113-MIR, LCC25) before the sample; the retarder is omitted for linear polarization experiments. Imaging optics: 4 mm aspheric + 25 mm lens; output captured by a homemade 640×512-pixel MCT focal plane array cooled to ~80 K, after an analyzer. Some 55-channel images are recorded at slightly different z positions to optimize resolution. Simulation and training details: Metaatom transmittance and phase are computed by 3D FDTD (Lumerical). Hologram design uses Python 3.8.13, PyTorch 1.12.1, Adam optimizer (learning rate 0.015); the angular spectrum method is used for diffraction in training. Iterations: 300 for linear/elliptic/circular models; 800 for a three-wavelength nine-channel model; 1500 for the 55-channel model. Hardware: 2×NVIDIA RTX 3090, AMD EPYC 7513 (32-core), 256 GB RAM, Windows 10. Large-array (800×800) holograms are simulated using vector integration based on Rayleigh–Sommerfeld diffraction to overcome FDTD limits. Demonstrations: (1) Tri-fold cyclic nonorthogonal linear polarization holography (0°→60°, 60°→120°, 120°→0°). (2) Tri-fold cyclic channels involving linear–circular and arbitrary elliptical polarization states by setting Jones vector parameters α, β (e.g., β=π/2 with α=0°, 90°, 135° for letters A, B, C; β=π/3 with α=30°, 70°, 140° for deer, squirrel, wolf). (3) A 10×10 Jones-matrix-dimensional system using ten input and ten output linear polarizations, yielding 55 distinct channels optimized by VDNN.

• Established a nonorthogonal polarization-basis multiplexing strategy using spatially varied eigen-polarizations of metaatoms, enabling globally nonorthogonal channels with minimal crosstalk and without sacrificing spatial or temporal dimensions for three-channel cases.
• Demonstrated free-vector holography across three asymmetric nonorthogonal linear polarization channels with high fidelity; simulations and experiments closely match target patterns.
• Expanded the effective Jones matrix dimensionality to 10×10 and experimentally realized 55 distinct holographic channels (selected in a lower triangular arrangement) using ten input and ten output linear polarizations; presented a correlation coefficient matrix indicating strong inter-channel isolation.
• Achieved tri-fold cyclic nonorthogonal holography in circular and elliptical polarization channels, including linear-to-circular conversion (letters A, B, C) and three distinct elliptical channels (deer, squirrel, wolf), showing high correlation between targets, simulations, and experiments.
• Reported measured holographic efficiencies: for the three nonorthogonal linear channels, 26.04% (0°→60°), 25.25% (60°→120°), 27.83% (120°→0°). For Metasurface 1 (linear–circular letters), 19.13% (A), 22.16% (B), 20.12% (C); for Metasurface 2 (elliptical images), 22.08% (deer), 24.61% (squirrel), 21.4% (wolf).
• Identified and analyzed brightness nonuniformity in some 55-channel holograms due to increased metaatom reuse; proposed practical mitigation strategies (increasing metaatom height, using combined or multilayer metaatoms, and increasing metaatom count).

Theoretical analysis (Supplementary Notes 1 and 4) indicates the largest rank of the coefficient matrix for a single-layer metasurface with similar metaatoms is 3, implying a limit of three nonorthogonal polarization channels without compromising other degrees of freedom; beyond three, crosstalk becomes significant. The study overcomes this by leveraging controllable eigen-polarization modulation and optimization via a vectorial diffraction neural network. For scaling to higher channel counts, metaatoms must provide stronger control (broader phase coverage, more design DOF, complex geometries, higher index contrast) or employ multilayer architectures; engineered noise can also assist. The approach demonstrates robust isolation and high fidelity across diverse nonorthogonal channels and enables a practical 10×10 polarization response space from a single metasurface. The methodology is extensible to other dynamic optical functions (e.g., tunable/zoom lenses) by controlling eigen-polarizations and optimizing global responses, potentially expanding the Jones matrix to even higher effective dimensions. This nonorthogonal strategy enhances capacity and security for information transmission in photonics and quantum information science.

This work pioneers nonorthogonal polarization-basis multiplexing by engineering spatially varying eigen-polarizations in metasurface metaatoms and optimizing with a vectorial diffraction neural network. It achieves high-fidelity free-vector holography across three nonorthogonal channels with minimal crosstalk, expands the effective Jones matrix dimensionality to 10×10, and experimentally realizes 55 distinct holographic channels. The framework supports arbitrary polarization forms and asymmetric channels, substantially boosting holographic information capacity. Future directions include enhancing metaatom control (greater phase coverage, complex shapes), multilayer metasurfaces, integration with noise-assisted methods, and extending the approach to dynamic optical components such as adaptive or zoom lenses. These advances will further increase channel counts, reduce crosstalk, and open new applications in secure communications and quantum encryption.

• Fundamental limit: For single-layer metasurfaces with similar metaatoms, the coefficient matrix rank limits nonorthogonal polarization multiplexing to three channels without trading other dimensions; beyond three, crosstalk rises.
• Scaling effects: In the 55-channel system, repeated metaatom use leads to nonuniform brightness and potential interference at the focal plane as channel count increases.
• Device constraints: Achieving more channels requires metaatoms with stronger control (larger depth-to-width ratios, higher refractive index contrast, more complex geometries) or multilayer metasurfaces, which may impact fabrication complexity and efficiency.
• Mitigations suggested: Increase metaatom height, adopt combined/compound metaatoms with more DOF, use multiple metasurface layers, and increase the number of metaatoms to improve control precision and brightness uniformity.