Gaussian–Schell model sources in one-dimensional first-order systems with loss or gain

A detailed examination of the propagation of Gaussian-Schell model sources in one-dimensional, possibly nonlossless, first-order systems is constructed. The laws of focusing are derived. The conditions for periodicity of the Gaussian-Schell model source are derived. This result generalizes the well-known result -2 </= A + D </= 2 for confinement of a perfectly coherent Gaussian beam to the partially coherent nonlossless case. When loss or gain is present several conditions must be satisfied simultaneously for periodicity. The self-consistent solutions are derived and the perturbation stability of the solutions is studied. A physical realization of an arbitrary nonlossless one-dimensional ABCD system is derived, which yields a convenient formula for deciding whether the ABCD system has loss or gain. Special attention is devoted to real and ripple systems.