The transfer matrix method (TMM) has been widely used to calculate the linear optical scattering of electromagnetic waves by optical multilayers and photonic crystal slabs. In this paper, we extend the ordinary TMM into a nonlinear TMM and use the method to analyze the problem of nonlinear frequency conversion such as second-harmonic generation (SHG), sum frequency generation, and difference frequency generation (DFG) in a general one-dimensional (1D) nonlinear photonic crystal (NPC) under normal incidence of fundamental wave. The transfer matrix of both the fundamental and nonlinear light waves for an arbitrary individual layer is first derived and the analytical recursion algorithm of the transfer matrix for these waves is presented under the numerically stable scattering matrix formulation. This allows one to construct the overall transfer matrix of nonlinear waves very conveniently for the whole nonlinear optical structure from all the individual layer transfer matrices. The method is efficient, user friendly, and free from numerical instability. We have employed the nonlinear TMM to discuss SHG and DFG in a 1D multilayer NPC with defects. The simulations show that strong confinement of the incident waves at the defect states of the NPC can lead to significant enhancement of SHG and DFG. Moreover, optimization of the polarization direction of ferroelectric crystal within the defective cell can further increase the efficiency of SHG and DFG by more than 1 order of magnitude. By adopting appropriate parameters, a single NPC structure can fulfill high-efficiency SHG and DFG simultaneously.
© 2010 Optical Society of America
Original Manuscript: February 23, 2010
Revised Manuscript: June 4, 2010
Manuscript Accepted: June 8, 2010
Published: July 16, 2010
Ming-Liang Ren and Zhi-Yuan Li, "Enhanced nonlinear frequency conversion in defective nonlinear photonic crystals with designed polarization distribution," J. Opt. Soc. Am. B 27, 1551-1560 (2010)