A new method of ray tracing in gradient index optics

As to the nonlinear optical equations,the exact analytical solutions can only be available on the condition that the refractive index of the medium is axial symmetry with the paraxial approximation.Beyond these con ditions,the analysis of the equations should resort to computer simulations.The traditional simulation based on Taylor series is a very intricate method consuming a great number of time.However,by introducing new proper variables,the Runge-Kutta method is an economic method on this problem.In this method,the calculations related to the refractive index on either side of the optical planes are greatly simplified.The Runge-Kutta method is introduced and the comparisons between these two methods are also presented.