The one-dimensional ray equation, the differential description of Fermat’s principle, is deduced directly from Snell’s law using two methods. In the first method, we obtain the ray equation from a differential equation relating the spatial coordinates derivative with the index of refraction field. In the second method, the ray equation is deduced from the proper generalization of Snell’s law for a refractive field, that is, a differential equation relating the index of refraction field and the refraction angle. Additionally, we used an intermediate expression of the first method to find a straightforward analytical solution of the ray path to an inferior mirage.

We present a novel phase-based filtering algorithm designed to retrieve the Degree of Linear Polarization (DoLP) and Angle of Linear Polarization (AoLP) by leveraging the intrinsic properties of a pixelated polarization camera. Unlike conventional intensity-based filters, our approach utilizes complex phase values to estimate polarization parameters, offering a flexible and computationally efficient alternative. Each point is treated as an independent measurement, dependent solely on the kernel size, which enables the potential for real-time processing. Experimental and simulation evaluations under Gaussian noise conditions validate the robustness of our approach, demonstrating a high degree of consistency with standard methods. The ANOVA analysis results reflect this consistency across datasets, as indicated by the sum of squares (SS), mean squares (MS), and F-statistic values. This reinforces the reliability of the proposed algorithm and highlights its practical applicability in noise-affected environments. Our findings suggest that the proposed method provides a stable and adaptable solution for polarization parameter extraction, making it well-suited for applications in fields such as biomedical imaging, remote sensing, and industrial inspection, where real-time performance and noise resilience are critical.

We propose a demodulation algorithm based on the calculus of the complex Fourier coefficients; we used a dual rotating polarizer-analyzer polarimeter to show the feasibility of our proposal. Our demodulation algorithm considers the frequency response obtained by the system, and its possible to calculate the total retardation, fast axis orientation and ellipticity of a sample. Our proposal does not require recovering the full Mueller matrix from getting those parameters. In addition, as the proposal does not use retarders for the measurement, the system presents potential applications for multi-wavelength measurements on phase retardation samples. We show experimental results showing the capabilities of our proposal in characterizing a polarization retardance sample.

The predictive synthesis of graded-index (GRIN) optical materials via physical vapor deposition (PVD) is challenged by the non-linear coupling between multi-source flux dynamics and curved substrate geometries. This work introduces a computational framework for the inverse design of source-specific shadow masks in an electron-beam PVD system, moving beyond traditional empirical approaches. Our methodology solves an ill-posed inverse problem to yield prescribed two-dimensional composition profiles on complex aspheric substrates. The forward engine integrates the Knudsen emission law, high-resolution rotation kinematics, and a stochastic penumbra model. A hybrid Particle Swarm Optimization (PSO) and Nelder-Mead algorithm minimizes a multi-objective cost function targeting specific thickness and stoichiometric gradients. Applied to a MgF2/TiO2 co-deposition system on a 24 cm diameter aspheric substrate, the decoupled masks achieve an 88.0% improvement in simulated thickness uniformity over natural deposition, yielding a global RMSE of 4.175 nm. Furthermore, the framework produces a GRIN profile with an index RMSE of 0.1417 relative to the physically attainable target. To establish the reliability of these predictions, a rigorous uncertainty quantification is conducted; a 10,000-iteration bootstrap resampling establishes a 95% confidence interval for the expected RMSE at [5.283, 6.163] nm, suggesting significant stability under manufacturing perturbations. This statistically grounded framework provides a generalizable theoretical pathway for the rational design of high-performance GRIN nanostructures, serving as a predictive roadmap for future experimental validation in vacuum systems.