Nonparaxial ray tracing is performed to investigate the field scanned out by a single beam through two rotatable thick prisms with different parameters, and a general solution is obtained and then expanded into a power series to establish the third-order theory for Risley prisms that paves the way to investigate topics of interest such as optical distortions in the scan pattern and an analytical solution of the inverse problem of a Risley-prism-based laser beam steering system; i.e., the problem is concerned with how to direct a laser beam to any specified direction within the angular range of the system.