Angle-to-retardance converter and universal polarization-state synthesizer

A compact analytical framework for designing tunable retarders and universal polarization-state generators from short cascades of linear retarders is presented. Closed-form expressions are derived for the parameters of the polarimetrically equivalent system associated with any pair of linear retarders, including the singular case of two half-wave plates acting as a pure rotator. For a symmetric three-plate cascade formed by an arbitrary linear retarder sandwiched between two identical linear retarders, we obtain explicit expressions for the effective parameters of the equivalent single linear retarder. In particular, the configuration with a central half-wave plate and two quarter-wave plates yields a fixed-axis retarder with retardance Δ = 4θ, where θ is the relative orientation of the half-wave plate. This configuration implements a perfect angle-to-retardance converter and, when combined with a fixed linear polarizer, functions as a universal polarization-state synthesizer providing full coverage of the Poincaré sphere with straightforward angular control. The analytical formulation complements existing geometric and numerical approaches and facilitates the design, calibration, and implementation of polarization devices for advanced polarimetric instrumentation.