  The work defines the general form of the Jones vector and establishes the
Jones matrix for polarizers, wave plates, Faraday rotators, Q-plates, and
spiral phase plates. We establish the generalized Jones calculus for vortex,
vector, and vortex-vector beam transformations. Systematic formalism presents
the theoretical arrangements in the manipulation of the phase and polarization
of the structured lights. The eigenstates and the time-reversal behaviors of
optical components and systems are discussed through matrix algebra. The
generalized Jones calculus is then used in the characterization of Sagnac
interferometers, and the potential applications are proposed.
