OP: Collaborative Research: Compatible Discretizations for Maxwell Models in Nonlinear Optics

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Nonlinear optics is the study of the behavior of light in nonlinear media. This field has developed into a significant branch of physics since the introduction of intense lasers with high peak powers. Compared with the huge amount of literature on simulations of Maxwell's equations in linear optical media, developing mathematically well-understood computational tools for space-time models in nonlinear optical media is relatively less tackled by the computational math community. Major advancement in this aspect can provide the scientific community reliable and accurate tools to simulate and to understand nonlinear optical phenomena, which hence can be better harnessed for practical applications. The objective of the collaborative research program is to make significant advances in the understanding and simulations of Maxwell models in nonlinear optics with the aim of: (1) providing robust simulation tools for the nonlinear optics community, (2) developing novel mathematical and numerical techniques that are specifically tailored for different types of nonlinear models. The specific technical aspect includes the development of energy-stable time discretizations as well as two classes of spatial discretizations, discontinuous Galerkin methods and mimetic finite difference methods, for the propagation of electromagnetic waves in nonlinear (dispersive) optical media. Both macroscopic phenomenological and microscopic quantum descriptions will be considered for modeling the nonlinear material responses. Applications involving femtosecond soliton propagation, harmonic generation, self focusing, among others will be simulated and compared to existing time domain methods. This collaborative program is strengthened by a cohesive research plan that relies on the complementary expertise of each principal investigator. The educational components are integrated through the training of graduate students.