Conservative Difference Schemes for Some Problems of Femtosecond Nonlinear Optics

Abstract

Nowadays, mathematical modeling with a wide use of numerical experiments is an efficient methods for solving problems of nonlinear femtosecond optics [1–4]. This leads to the problem on the accuracy of the correspondence between the numerical results and the real physical process. There are two basic approaches to the solution of problems of nonlinear optics. The first is the decomposition method (DM) (e.g., see [2, 3, 5–7]), which is the easiest to implement. The other implies the construction of conservative difference schemes (CDS) [8], which were represented in [9] in the case of various problems of nonlinear optics. In the present paper, we compare the decomposition method and conservative difference systems in connection with modern problems of femtosecond nonlinear optics: the propagation of an optic pulse in a nonlinear photonic crystal (PC) and its self-focusing in a cubically nonlinear medium with regard to the dispersion of the nonlinear response. Note that such a comparison was earlier performed in [10, 11] for problems of self-interaction and generation of the second harmonics by longer (as compared with the femtosecond range) pulses. The advantage of CDS for traditional problems of nonlinear optics was illustrated by numerical modeling results. In what follows, we show that the DM is not effective in numerous cases for these problems. For example, when using computer modeling to obtain the limit distribution of intensity (that is, the distribution remaining constant as the grid increment diminishes) of the propagating femtosecond pulse in a PC for a scheme constructed on the basis of the DM, the time increment should be less than that for the CDS by at least an order of magnitude. It is important to note that the use of the DM does not preserve finite-difference analogs of invariants, and to achieve a sufficient accuracy of their preservation one should substantially diminish the grid increment. However, the decomposition method is effective for problems of linear propagation.