Adiabatic geometric phase in fully nonlinear three-wave mixing

Abstract

In a nonlinear three wave mixing process, the interacting waves can accumulate an adiabatic geometric phase (AGP) if the nonlinear coefficient of the crystal is modulated in a proper manner along the nonlinear crystal. This concept was studied so far only for the case in which the pump wave is much stronger than the two other waves, hence can be assumed to be constant. Here we extend this analysis for the fully nonlinear process, in which all three waves can be depleted and we show that the sign and magnitude of the AGP can be controlled by the period, phase and duty cycle of the nonlinear modulation pattern. In this fully nonlinear interaction, all the states of the system can be mapped onto a closed surface. Specifically, we study a process in which the eigenstate of the system follows a circular rotation on the surface. Our analysis reveals that the AGP equals to the difference between the total phase accumulated along the circular trajectory and that along its vertical projection, which is universal for the undepleted (linear) and depleted (nonlinear) cases. Moreover, the analysis reveals that the AGPs in the processes of sum-frequency generation and difference-frequency generation have opposite chirality. Finally, we utilize the AGP in the fully nonlinear case for splitting the beam into different diffraction orders in the far field.

Emulating spin transport with nonlinear optics, from high-order skyrmions to the topological Hall effect

“…Even simple extensions, such as time-variance ( ), can increase the range of effects produced by our system to enable the study of transient phenomena such as melting 39 or spin waves 40 . Allowing pump depletion 54 , i.e., permitting the pump intensity to be affected by the signal/idler waves, will describe the full nonlinear dynamics under which Berry curvature is known to persist 55 , but will complicate the calculations. In the latter scenario, the effective magnetization will be perturbed by the pseudospin current, much like in spin-transfer torque 56 .…”

The geometric phase in nonlinear frequency conversion

“…Soon after, Karnieli et al analyzed theoretically the undepleted adiabatic geometric phase in three wave mixing [59] and later observed it experimentally [60]. The analysis of the adiabatic geometric phase in the fully nonlinear regime of both three wave mixing [61] and FWM [62], soon followed. Topological effects in nonlinear gauge fields based on the geometric phase were proposed [63][64][65], contributing to the fast-growing field of nonlinear topological photonics [66].…”

“…Further analysis of the geometric description of the nonlinear interaction was presented by Luther et al [52], while a comprehensive physical model of adiabatic three-wave mixing was developed later by several researchers, for example Porat and Arie [134], Phillips et al [135][136][137], Heese et al [138][139][140], and Yaakobi et al [141,142]. In more recent works, Li et al [61,62], Lü et al [143], and Zhao et al [144] have shown how one can generate and control the adiabatic geometric phase in the fully nonlinear regime of both threeand four-wave mixing. Below, we detail the geometric representations of fully-nonlinear three wave mixing and its corresponding geometric phase.…”

“…Ref. [61] discussed such circular adiabatic motion of B in its parameter space, around a unit vector with an angle Θ with the ∆Γ/2 axis (similar to the circular rotation introduced in the undepleted case). Figure 12 manifests the results of the calculation of the AGP through Eq.…”

“…The dashed curves are the theoretical predictions for the undepleted case, i.e., γ = π(1 − cos Θ). Reproduced from Ref [61]…”