Efficient Adaptive Step Size Method for the Simulation of Supercontinuum Generation in Optical Fibers

The use of an adaptive step size selection significantly reduces the computational effort for the numerical solution of the generalized nonlinear Schrödinger equation (GNLSE). The most commonly employed adaptive step size method is based on the estimation of the local error by applying step size doubling and local extrapolation. While this method works well in combination with the globally second-order split-step Fourier (SSF) integration scheme, it can be significantly improved when the highly accurate fourth-order Runge-Kutta in the Interaction Picture (RK4IP) method is used for integration, which was recently introduced into the nonlinear optics field. It is demonstrated that the local error can then be estimated using a conservation quantity error (CQE) without the necessity of step size doubling. The CQE method for solving the GNLSE is explained in detail, and in addition the concept is transferred to the normal nonlinear Schrödinger equation and extended to include linear loss. The RK4IP-CQE combination proves to be the most efficient algorithm for the modeling of ultrashort pulse propagation in optical fiber, reducing the computational effort by up to ~ 50 % relative to the local error method.