Continuation Finite Element Simulation of Second Harmonic Generation in Photonic Crystals
Gang Bao 1, Zhengfu Xu 2, Jianhua Yuan 3*1 Department of Mathematics, Zhejiang University, Hangzhou 310027, China; and Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA.
2 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA.
3 Department of Mathematics, Beijing University of Posts and Telecommunications, Beijing 100876, China.
Received 15 July 2010; Accepted (in revised version) 29 September 2010
Available online 7 March 2011
A computational study on the enhancement of the second harmonic generation (SHG) in one-dimensional (1D) photonic crystals is presented. The mathematical model is derived from a nonlinear system of Maxwell's equations, which partly overcomes the shortcoming of some existing models based on the undepleted pump approximation. We designed an iterative scheme coupled with the finite element method which can be applied to simulate the SHG in one dimensional nonlinear photonic band gap structures in our previous work. For the case that the nonlinearity is strong which is desirable to enhance the conversion efficiency, a continuation method is introduced to ensure the convergence of the iterative procedure. The convergence of our method is fast. Numerical experiments also indicate the conversion efficiency of SHG can be significantly enhanced when the frequencies of the fundamental and the second harmonic wave are tuned at the photonic band edges. The maximum total conversion efficiency available reaches more than 50% in all the cases studied.AMS subject classifications: 35Q60, 78M10, 65Z05, 78A60, 65L60
PACS: 42.65.Ky, 42.70.Qs, 02.70.Dh, 02.60.Cb
Key words: Photonic crystals, second harmonic generation, photonic band gap, conversion efficiency, finite element methods, fixed-point iterations, continuation method.
Email: email@example.com (G. Bao), firstname.lastname@example.org (Z. Xu), email@example.com (J. Yuan)