The transformation of a Gaussian beam by a thin lens is an important component of the laser principles course, as it enhances students' understanding of laser optical system design. However, because the course usually omits the general propagation of arbitrary laser beams through linear optical systems, students often raise questions when they try to extend their knowledge. This paper derives, under the second-order moment definition, the waist diameter, waist position, and far-field divergence angle of a general laser beam after passing through a linear optical system represented by a ray-transfer matrix. In addition, we retrace the beam-parameter transformation with the q-parameter method, demonstrating the consistency of the two approaches and thereby reinforcing students' understanding of laser-beam behavior. The general beam-parameter transformation theory can be expected to find direct application in the design of multi-lens beam-transformation systems.
CHEN et al. (Mon,) studied this question.