Applied numerical linear algebra
Applied numerical linear algebra
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A computer program is developed for numerical simulations of diode lasers operating well above threshold. Three-dimensional structures of typical single-mode lasers with the goal of efficient fibre coupling are considered. These devices have buried waveguides with high indices of refraction. The so-called round-trip operator constructed by a beam propagation method is non-linear due to gain saturation and thermal effects. Thus, the problem for numerical modelling of lasing is the eigenvalue problem for the non-linear round-trip operator. Fox-Li iterative procedure is applied for calculation of a lasing mode. A large size 3D numerical mesh is employed to discretize a set of equations describing (a) propagation of two counter-propagating waves using Pade approximation, (b) lateral diffusion of charge carriers within a quantum well, and (c) thermal conductivity. So, many important non-linear effects are properly accounted for: gain saturation, self-focusing, and thermal lens. A serious problem arising for operation far above threshold is the appearance of additional lasing modes that usually cause degradation in optical beam quality. To calculate a critical electric current, at which additional modes appear, the numerical code incorporates a subroutine that calculates a set of competing modes using gain and index variations produced by the oscillating mode. The corresponding linear eigenproblem is solved by the Arnoldi method. The oscillating mode has an eigenvalue equal to 1, while higher-order modes have eigenvalues of amplitude less than 1, which grow with injection current. These eigenvalues are calculated at several values of electric current, and the eigenvalues of the sub-threshold modes are then extrapolated to 1, at which point the device becomes multimode. Results of numerical simulations for typical experimental conditions will be presented.