Defective Jones matrices: geometric phase and passivity condition

A defective Jones matrix is a 2 x 2 matrix that has only one polarization eigenstate, contrary to the more common case of diagonal izable matrices that have two eigenstates. We analyze the properties of a defective optical system and characterize the optical phase shift psi (dynamic and geometric) introduced by the system. Two scenarios are identified: (a) psi has a finite range, leading to the existence of two particular input states that experience minimum and maximum phase shifts. (b) psi spans the complete range from pi to pi, leading to the existence of two input states whose output states are orthogonal. We solve the inverse problem of designing an optical system for both scenarios. Additionally, we determine the conditions to get physically realizable defective Jones matrices satisfying the passivity condition. Finally, we introduce a new defectivity parameter to characterize the degree of defectiveness of a Jones matrix. (C) 2020 Optical Society of America