Laser resonator with a defective Jones matrix designed with the polar decomposition theorem

The polar decomposition theorem (PDT) makes it possible to express any Jones matrix as a combination of a pure polarizer and a pure retarder. We establish the relations that these two elements must have to produce a defective Jones matrix, where two polarization eigenstates coalesce into a single state to create an exceptional point (EP). We present a laser resonator whose structure directly derives from the PDT and whose round-trip Jones matrix, J RT , becomes defective for some values of the control parameters. We show conditions wherein J RT commutes with the time-reversal ( T ) operator. J RT transitions from exact T symmetry, where the eigenvectors of the Jones matrix and the time reversal operator are shared and eigenvalues are real, to broken T symmetry, where eigenvectors are distinct, and eigenvalues are a pair of complex conjugates. The EP, located at the boundary between the two regimes, is characterized by a local maximum in the round-trip mode loss and a dip in the degree of polarization of the eigenstates.