Spectral method for numerical solution of the electric field envelope propagation equation

We report new method for solving propagation equation for the electric field envelope, obtained from the Maxwell equations under the slowly varying envelope approximation and the paraxial approximation. This propagation equation usually represents part of Maxwell-Bloch system of equations. Method is semi-analytical and semi-numerical. We use analytical solution of the Fourier transform of the envelope equation, for a given initial condition, to find numerical solutions on grid nodes in 3+1 dimensions. We have tested our numerical results by comparing them with exact particular solutions of the envelope equation. We show that results of our method are in good agreement with the exact solutions for sufficient density of grid nodes. Solving the propagation equations by using this method is important for modelling many phenomena in areas of quantum and nonlinear optics dealing with the propagation of light through an active optical medium.