The geometric algebra of optical polarization continues to be a source of new insights and connections with mathematical physics. For instance, the four Stokes parameters (S₀, S₁, S₂, S₃) of a partially-polarized light beam satisfy S₀² ≥ S₁² + S₂² + S₃², analogous to the 4-vector relation (ct)² ≥ x² + y² + z² for the relativistic displacement between two causally-connected events in Minkowski space. Equality in the first case represents a state of pure polarization (a point on the Poincaré sphere, rather than inside it), and in the second, a relativistic null vector; this leads to many connections between the formalisms of special relativity and polarization optics. In the past few decades the algebra of relativistic velocity addition (as found by Einstein) has become appreciated as the algebraic structure of so-called gyrovectors. Here, the author shows how the polarization of a light beam experiencing a partial polarizer is transformed using gyrovectors. The input Poincaré sphere is found to transform into an ellipsoid that has somewhat surprising mathematical properties, especially when the polarizer’s parameters take on extreme values. This representation could be useful in the design of polarization technologies and could introduce optical scientists to a new mathematical tool that offers a route to a yet deeper understanding of polarization optics.

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