Exploring the Origin of Lissajous Geometric Modes from the Ray Tracing Model

In this paper, we use the geometric optics and discuss the path of laser beam in a simple laser (concave-plano) cavity with the birefringence crystal. In specific lengths of the laser cavity, we can observe various types of Lissajous-like structural laser modes that can be simulated using our ray tracing model. At the end of this paper, we provide an adjusted ABCD matrix. With the adjusted ABCD matrix and iterative calculation, we can obtain the 3D trajectories which are similar to the experimental results. These structural laser modes can be realized by a (Formula presented.) solid-state laser with off-axis pumping. From the comparison between the experimental data and the numerical data, we clarify the relationship between the 3D Lissajous-like structural laser modes and ray trajectory in the laser cavity.