Numerical methods for the bidimensional Maxwell–Bloch equations in nonlinear crystals

Olivier Saut (1) , Antoine Bourgeade (2)

1 IMB - Institut de Mathématiques de Bordeaux
2 CESTA - Centre d'études scientifiques et techniques d'Aquitaine

Two numerical schemes are developed for solutions of the bidimensional Maxwell–Bloch equations in nonlinear optical crystals. The Maxwell–Bloch model was recently extended [C. Besse, B. Bidégaray, A. Bourgeade, P. Degond, O. Saut, A Maxwell–Bloch model with discrete symmetries for wave propagation in nonlinear crystals: an application to KDP, M2AN Math. Model. Numer. Anal. 38 (2) (2004) 321–344] to treat anisotropic materials like nonlinear crystals. This semi-classical model seems to be adequate to describe the wave–matter interaction of ultrashort pulses in nonlinear crystals [A. Bourgeade, O. Saut, Comparison between the Maxwell–Bloch and two nonlinear maxwell models for ultrashort pulses propagation in nonlinear crystals, submitted (2004)] as it is closer to the physics than most macroscopic models. A bidimensional finite-difference-time-domain scheme, adapted from Yee [IEEE Trans. Antennas Propag. AP-14 (1966) 302–307], was already developed in [O. Saut, Bidimensional study of the Maxwell–Bloch model in a nonlinear crystal, submitted (2004)]. This scheme yields very expensive computations. In this paper, we present two numerical schemes much more efficient with their relative advantages and drawbacks.

Mots clés: Nonlinear optics; Harmonic generation; Quantum description of light and matter; Nonlinear optical crystal; Numerical schemes

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]; Optique [physics.optics]

Olivier Saut, Antoine Bourgeade. Numerical methods for the bidimensional Maxwell–Bloch equations in nonlinear crystals. Journal of Computational Physics, 2006, 213, pp.Issue 2, Pages 823-843. ⟨10.1016/j.jcp.2005.09.003⟩. ⟨hal-00131958⟩