Linearly Polarized Projections

• Linearly Polarized Projections are methods for mapping, manipulating, and measuring the state of light’s linear polarization using devices like coronagraphs, polarimetric cameras, and metasurfaces.
• They employ mathematical frameworks based on Stokes parameters, Mueller and Jones matrices, and projection operators to accurately characterize and process polarization data.
• Recent advances, including quasi-exceptional-point metasurfaces, enable broadband, high-fidelity control of LP light, driving innovations in astronomy, imaging, and optical communications.

Linearly polarized projections refer to the mapping, manipulation, and measurement of the linear polarization state of light in optical systems, encompassing both its physical transformation through materials and its detection via imaging modalities. This domain encompasses instrumentation such as coronagraphs, polarimetric cameras, and metasurfaces, with applications ranging from solar astronomy to advanced wavefront shaping. Recent progress resolves long-standing theoretical constraints on geometric-phase control for linearly polarized (LP) light, enabling polarization-agnostic projection and high-dimensional control in modern optical devices (Pistellato et al., 2022, Gao et al., 10 Nov 2025).

1. Mathematical Framework for Linearly Polarized Projections

Linearly polarized light is fully characterized by its Stokes parameters $(S_0, S_1, S_2)$, where $S_0$ is the total intensity and $(S_1, S_2)$ encode the degree and orientation of linear polarization. The sensing and transformation of LP states involve linear projections onto analyzer directions defined by physical action (e.g., polarizers, metasurfaces), which can be mathematically described by projection operators (Mueller or Jones matrices).

For projective polarization imaging, the measured intensity $I$ under analyzer orientation $\alpha$ follows: $$I = \tfrac12 \left[ S_0 + S_1 \cos(2\alpha) + S_2 \sin(2\alpha) \right]$$ In metasurface-based projection, the Jones matrix $J$ defines the transformation: $$J = \begin{pmatrix} A & C \ C & B \end{pmatrix}$$ where $A$, $B$, $C$ are complex scattering coefficients tied to the meta-atom geometry and material properties. At or near exceptional points, this system imparts a geometric phase on LP states, generalizing Pancharatnam–Berry effects to the linear domain (Gao et al., 10 Nov 2025).

2. Geometric Models for Polarization Projection in Imaging Systems

Standard shape-from-polarization (SfP) models assume orthographic imaging, failing for projective optics. A physically consistent geometric model incorporates local ray geometry in the imaging plane by:

• Mapping each pixel’s viewing direction to a local polarization reference frame $\mathcal P_j$
• Rotating analyzer axes and effective polarizer directions into $\mathcal P_j$ with a per-pixel rotation matrix $R_j$
• Correcting the measurement via tilted-polarizer Mueller matrices to recover the true Stokes vector for each ray (Pistellato et al., 2022)

Critical pipeline steps include pre-processing raw intensities by correcting for analyzer tilt and solving local linear systems for Stokes vectors, followed by post-processing to rotate projections (e.g., surface normals) back into the global camera frame for physical consistency.

3. Exceptional Point Metasurfaces and Linearly Polarized Phase Control

Conventional geometric-phase metasurfaces rely on circularly polarized states due to the Pancharatnam–Berry mechanism. Quasi-exceptional-point (QEP) metasurfaces introduce engineered singularities—exceptional points—into the scatterer (meta-atom) design, enabling a pure geometric phase $\phi_{geo}=2\theta$ to be imparted on arbitrary linear polarizations, independent of the input eigenstate or analyzer orientation (Gao et al., 10 Nov 2025). The meta-atom’s orientation $\theta$ directly encodes the projected geometric phase: $$|E_{cross}| \propto e^{i\phi_{geo}}, \quad \phi_{geo}=2(y-\theta)$$ where $y$ is the input LP angle.

Meta-atoms are designed using a multi-parameter structural sweep (e.g., widths $\ell_1, \ell_2, \ell_3$, gap $g$), tuned such that at the QEP wavelength, the condition $(A-B)^2 + 4C^2=0$ is approximately met, causing eigenvector coalescence and phase winding.

4. Instrumentation and Measurement Protocols for LP Projections

Polarimetric imaging systems (DoFP sensors, rotating-analyzer cameras) and solar coronagraphs employ symmetric measurement protocols—e.g., three-polarizer systems—to sample the Stokes parameters via projections at distinct analyzer angles (Pistellato et al., 2022). Demosaicing strategies combine neighboring pixel samples to reconstruct full Stokes vectors, enforcing physical constraints such as $I_0+I_{90}=I_{45}+I_{135}$ and $\mathrm{DoLP}\leq1$ for the degree of linear polarization.

In QEP metasurfaces, spatially varying meta-atom orientations encode arbitrary phase and amplitude maps for LP light, enabling broadband, efficient, and fidelity-preserving projections for imaging, beam steering, and holography (Gao et al., 10 Nov 2025).

5. Physical Constraints, Noise, and Data Conditioning

For physical validity, LP projections and their measurement must enforce:

• Nonnegativity ($S_0 \ge 0$)
• Bounded polarization degree ($\sqrt{S_1^2 + S_2^2} \le S_0$)
• Orthogonality relationships between synthesized analyzer outputs

Noise analysis is central in both imaging and metasurface design. In imaging, noise propagation through the tilt-corrected projection operator affects Stokes vector recovery (Pistellato et al., 2022). In metasurfaces, detuning from the exact EP condition impacts phase linearity and efficiency but, within broad parameter windows ("quasi-EP"), the output remains robust for arbitrary LP input.

6. Applications and Implications of Linearly Polarized Projections

Linearly polarized projections underlie advanced solar and heliospheric imaging, photometric colorimetry analogies, and high-dimensional light control for next-generation imaging, communications, and multiplexing. QEP metasurfaces now enable polarization-universal wavefront shaping—previously unattainable for LP states—facilitating

• Crosstalk-free holography across arbitrary LP input angles
• Broadband beam steering and amplitude-phase modulation in the cross-polarized channel
• Projection of vector-valued LP patterns for mode-division multiplexing in optical communications (Gao et al., 10 Nov 2025)
• High-fidelity empirical recovery of 3D surface normals in SfP pipelines for perspective cameras (Pistellato et al., 2022)

A plausible implication is that embedding EP singularities into the projection operator provides an intrinsic mechanism to bypass the helicity constraints of prior devices, unlocking scalable polarization optics suited for modern integrated photonics and computational imaging.

7. Current Limitations and Future Directions

Projection systems for LP light face technical constraints in anisotropy control, noise minimization, and physical channel orthogonality. QEP metasurfaces represent an emerging paradigm that may generalize to other polarization domains (elliptical, hybrid), and scale to multiplexed, amplitude–phase–polarization entanglement. Integration of perspective-corrected geometric models in imaging promises improved interpretability and reconstruction accuracy for polarization vision systems.

These developments mark a notable convergence of non-Hermitian photonics, geometric-phase engineering, and advanced polarization sensing—forming a robust foundational framework for next-generation linear polarization projection technologies (Pistellato et al., 2022, Gao et al., 10 Nov 2025).