In this work, we experimentally demonstrate a novel mechanism for nonlinear spatial self-cleaning of a highly multimode (i.e., comprising a large number of spatial frequency components) optical beam into the plane wave mode or component, based on the process of second-harmonic generation (SHG) in a quadratic nonlinear optical crystal.

The generation of the SH of a ruby laser in crystalline quartz was the first nonlinear optical experiment, reported by Franken et al. as early as 1961 [1]. Since then, SHG has grown to be the most established nonlinear optical effect, with widespread use across laser technologies. Nevertheless, the potential of SHG for spatial beam self-cleaning has so far not been fully appreciated.

The existence of nonlinear self-sustained beams and associated beam reshaping at negative mismatch values was earlier demonstrated for quadratic spatial solitons [2,3], and, more recently, for polychromatic filaments [4]. Surprisingly, here we show that nonlinear beam cleanup may occur for initially fully speckled beams, albeit in a limited region around phase-matching, for both signs of the linear phase mismatch.

In order to show that, we employed a coherent, quasi-continuous wave (CW) laser beam, spatially scrambled by propagation in a short segment of highly multimode optical fiber rod. Next, this beam was coupled to a quadratic nonlinear potassium titanyl phosphate (KTP) bulk crystal (Fig.1a). Whenever the phase-matching for SHG is exactly (or nearly) satisfied, and the laser beam energy is sufficiently high, we observed that the output beam undergoes a spontaneous recovery of its spatial quality (Fig.1c,d).

Numerical simulations, based on the general model of three-wave mixing in bulk media, have been performed, and show a good qualitative agreement with our observations. Close to phase-matching of the SHG process, propagation of the FF beam can be described in terms of an equivalent third-order non-local nonlinear response [5]. This permits us to conjecture a possible connection between the mechanism of self-cleaning in instantaneous Kerr media (e.g., multimode optical fibers), and beam self-cleaning in SHG. However, future work, with the aim of developing a general understanding of the complex nonlinear dynamics of multimode wave systems, is required.

We further analyse the mechanism of SHG-induced beam cleaning by characterizing, in the strong conversion regime around the phase-matching condition, the nonlinear spatial response of the crystal with a quasi-plane wave beam. We show that the SH conversion efficiency broadens with fundamental frequency (FF) power, in agreement with a simple plane-wave model (Fig. 2a,c). Moreover, our experiments reveal that the FF beam experiences spatial compression or self-focusing for both positive and negative wave vector mismatch values (Fig. 2b). In addition, FF beam cleaning occurs in a regime where the SHG efficiency is strongly reduced by nonlinear saturation effects (Fig. 2d), owing to the interplay of beam self-focusing and walk-off [6].

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