A continuation fixed-point iterative method on harmonic generations with strong nonlinear optical effects in multi-layer structures

作者：Yuan, Jianhua^*; Yang, Jian; Shi, Dan; Ai, Wenbao

来源：COMPUTATIONAL & APPLIED MATHEMATICS, 2017, 36(1): 805-824.

DOI：10.1007/s40314-015-0267-7

• Finite element methods
• Nonlinear Helmholtz equations
• Harmonic generations
• Iterative methods
• Continuation methods

A continuation fixed-point iterative method for harmonic generations with strong nonlinear optical effects in one-dimensional nonlinear multi-layer structures, which are governed by the nonlinear Helmholtz systems, is presented in this paper. Three kinds of harmonic generations via or nonlinear process, which are important phenomena in nonlinear optics, are modeled as the application of our proposed method. In the regime of strong nonlinearity, the theoretical and computational analysis of the model problems encounters enormous challenges due to the discontinuity across layers and the strongly nonlinear coupling of modes. To overcome the difficulties in the nonlinearity and discontinuity, a fixed-point iteration with the finite element method is used to study the nonlinear conversion efficiencies by solving the weak formulations. Furthermore, a continuation technique that depends on the weak formulations to solve three model problems, is introduced to handle the iterative procedure for cases with large nonlinearity. The theoretical framework and numerical algorithm towards a uniform process is offered to study almost all the harmonic generations in or interaction. With the continuation fixed-point iterative technique, the convergence of the iterative procedure can be ensured, even in the presence of strong nonlinearity. Three numerical experiments with high conversion efficiencies are also presented to show the accuracy and efficiency of our proposed method.

• 出版日期2017-3
• 单位北京邮电大学