DPG methods for nonlinear fiber optics




Nagaraj, Sriram

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In recent years, the Discontinuous Petrov-Galerkin (DPG) method has been the subject of significant study. It comes with a collection of desirable properties, including uniform/mesh independent stability, localizable test norms via broken test spaces, and a canonical error indicator that is incorporated as part of the solution. In this work, the DPG method is applied to problems arising in fiber optics. Accurate modeling of wave propagation in nonlinear media is an important task in fiber optics applications. Nonlinear Maxwell equations in the context of optical fibers have been studied extensively in the past. Analysis of these intensity-dependent nonlinearities are based on several simplifying approximations which result in a nonlinear Schrodinger (NLS) type equation. The Schrodinger equation from a spacetime DPG perspective is discussed. In particular, a 2nd order L² stable ultraweak formulation of the Schrodinger equation is constructed by introducing the notion of an auxiliary boundary operator. This theoretical device requires an operator-specific conforming element to develop optimal convergence rates. Numerical studies show how, modulo (expected) roundoff issues, the theoretical convergence rates are delivered. Next, the use of the DPG method in modeling and simulating optical fiber laser amplifiers with nonlinear Raman gain is studied. In this application, the interaction of two time harmonic electromagnetic fields (the signal and pump fields) governed by two weakly coupled nonlinear Maxwell equations results in the amplification phenomenon. A novel Raman gain model for describing the phenomenon is proposed and an ultra weak DPG formulation is used for the discretization of the proposed model. The nonlinearity is handled by using simple iterations between the two systems. DPG implementation of a perfectly matched layer (PML) at the exit end of the fiber is essential in this model, as is the use of sum factorization for element computations. The presented results show that the signal field indeed gains power along the fiber, thereby justifying the use of the model. Auxiliary results presented in this dissertation include the construction of DPG Fortin operators for 2nd order problems.


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