LAUSR

By analyzing the phase vector evolution of a paraxial optical system (POS) with a variational background refractive index, we obtain a continuous dynamic equation, called state evolution formula (SEF), which simultaneously gives the phase vector transformation and ray trajectory inside and outside the optical elements. Compared with ray transfer matrix method, this phase-vector equation is universal in treating problems about propagation and stability of paraxial rays, since it extends the linear and discrete matrix equation to a differential equation. It takes a consistent form for both continuous and discontinuous cases without considering the special rays, even the input and output states present a nonlinear relation. Based on the SEF, we further propose a rigorous criterion about whether a continuous and non-periodic POS is stable. This formula provides a reference model for the theoretical analysis of ray dynamics in geometric and physical optical systems.