Optical chaos in nonlinear photonic crystals

Abstract

We examine a spatial evolution of lightwaves in a nonlinear photonic crystal with a quadratic nonlinearity when simultaneously a second harmonic and a sum-frequency generation are quasi-phase-matched. We find the conditions of a transition to Hamiltonian chaos for different amplitudes of lightwaves at the boundary of the crystal. PACS: 42.65.Sf,05.45.Ac,42.70.Mp,42.65.Ky Wave mixing in nonlinear optical materials is a basis of modern optical sciences and technologies. Cascading several wave-mixing processes in the same low-loss material one can in principle achieve a high efficiency using a large value of the lowest-order optical nonlinearity. The theoretical investigations of cascading of several scalar optical three-wave-mixing processes in the bulk materials with χ (2) nonlinearity has a long history [1]. In particular, Akhmanov and co-workers have found the efficiency of a third harmonic generation (THG) via cascading of a second harmonic generation (SHG) and a sum-frequency mixing (SFM) in a quadratic medium [2], while Komissarova and Sukhorukov have described an efficient parametric amplification at a high-frequency pump in the same system [3]. Obviously, the observation of these nonlinear effects demands a simultaneous satisfaction of phase-matching conditions for several parametric processes as perfectly as possible. On other hand, it has been shown later that the systems, for which several optical wave-mixing processes can be simultaneously phase-matched, are in general nonintegrable; therefore a competition of two (or more) parametric processes can often result in a chaotic spatial evolution of lightwaves [4,5]. However, until nowadays it was unclear how to achieve a phase-matching for several processes in homogeneous medium employing traditional techniques, such as using a birefringence in ferroelectric crystals.The solution of this problem has been found rather recently [6,7,8]; it consists in an introduction of the different types of artificial periodicity of a nonlinear medium resulting in a formation of nonlinear 1D and 2D superstructures termed optical superlattices [9] or nonlinear * E-mail: Kirill.Alekseev@oulu.fi photonic crystals (NPCs) [10]. In NPCs there is a periodic (or quasiperiodic) spatial variation of a nonlinear susceptibility tensor while a linear susceptibility tensor is constant.In these engineering nonlinear materials a phase mismatch between the interacting lightwaves could be compensated by the Bragg vector of NPC. The idea of such kind of quasi-phase-matching (QPM) was introduced by Bloembergen and co-workers many years ago [11]. However, only recently the rapid progress in a fabrication of high quality ferroelectric crystals with a periodic domain inversion made the QPM method very popular [9,12]. We should stress that the conditions for QPM may be fulfilled for several wave-mixing processes simultaneously; the QPM also has an advantage as using of largest nonlinear coefficient.Nowadays there are several experiments on an observation of third and fourth harmonics i...

Multistep parametric processes in nonlinear optics

“…show that the same parametric interaction can be used for 100% conversion from 2ω wave into 3ω wave. Considering the situations with nonzero SH and TH boundary conditions it has to be taken into account that, as shown by Alekseev and Ponomarev [2002], the spatial evolution of three light waves participating simultaneously in SHG and SFM under the conditions of QPM double phase matching becomes chaotic at large propagation distances for many values of the complex input wave amplitudes. Thus, the possibility of transition to chaos exists in the application of an additional pump at frequency 2ω in order to increase the efficiency of THG [Egorov and Sukhorukov [1998]].…”