Parameters of decentered Gaussian beams

A decentered Gaussian beam is a recently introduced generalization of an ordinary Gaussian beam, the propagation of both of which through an arbitrary centered paraxial optical elements can be described by ABCD matrices of paraxial optical rays. The paper shows that the M2 beam quality factor of the decentered Gaussian beam is the same as that of the ordinary Gaussian beam -- equal to 1. We further find out general conditions (for elements with pure real ABCD matrices) which must be fulfilled in order to accomplish the transformation of the Gaussian beam into decentered one or vice versa. The particular cases under specific conditions are presented in the work as well.