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We provide a description of the parity space reflection-time reversal (PT) operator acting on the two-dimensional polarization space of light represented by the linear algebra of Jones vectors and matrices. We establish the form of a PT-symmetric Jones matrix. We present two examples of laser resonators whose polarization eigenstates are described by PT-symmetric Jones matrices: one is based on the Faraday effect and a dichroic attenuation, while the other is made of twisted anisotropic mirrors. Both possess a control parameter that experimentally covers the exact, the broken PT-symmetry regions and their boundary, called an exceptional point, where the eigenstates of the resonator coalesce into a single state. The exact PT-symmetric region produces laser polarization modes emitting at the same frequency with different intracavity losses, while the broken PT-symmetric region features polarization modes emitting at distinct frequencies with the same intracavity losses. By applying unitary transformations, the concept of PT-symmetric Jones matrix is extended to matrices that commute with any antiunitary operator, thereby opening the prospect of a larger family of resonators geometries that also feature real or complex-conjugate spectra.

Phase conjugation with stimulated Brillouin scattering can be used to correct wavefront aberrations in high-power laser systems. Here, we consider such process with cylindrical vector lasers beams. By using a vectorial approach and a modal decomposition, we obtain a differential matrix equation that enables the calculation of the structure of Stokes vector eigenmodes and their Brillouin gain. The emphasis is put on the mode with the largest gain, which contributes the most to the Stokes beam. We show that the phase conjugation of orbital angular momentum in cylindrical vector beams is prevented from happening almost everywhere on the higher-order Poincar\'e sphere. This phenomenon is traced to the tendency of the Stokes beam to acquire a polarization structure similar to the incident pump beam, which constraints the possible combinations of topological charges of the two orbital angular momentum modes making up the Stokes cylindrical vector beams. However, near the poles of the higher-order Poincar\'e sphere, the Stokes mode with the largest gain acquires an additional helical phase factor that produces the conjugated topological charge for the stronger circularly polarized component without affecting the polarization structure. Experimental results are found to agree with our theoretical predictions.

Anisotropic mirrors are used to form a laser resonator exhibiting non-Hermitian, parity-time (PT) symmetric, polarization states. The relative angle of the two mirrors principal axes is used to control the degree of non-hermiticity. A sharp symmetry-breaking transition is observed at a specific angle, called the exceptional point, where the two states coalesce into a single polarization state and the interference pattern produced by counter-propagating (CP) waves vanishes. At a smaller angle, in the unbroken PT symmetry regime, the polarization state experiencing higher losses is suppressed. In the broken symmetry regime, the two polarization states coexist, but the orthogonality of the CP waves favors single longitudinal mode emission by suppressing the interference pattern of the standing wave. The two regimes meet at the exceptional point, where a unique polarization state exists in a resonator free from interference intensity pattern. Microchip PT-symmetric lasers operating at the exceptional point are thus an attractive solution to achieve single mode operation from a miniature monolithic device without any intra-cavity element., Comment: 29 pages, 5 figures