Spatial beam self-cleaning in second-harmonic generation

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.

Fig. 1: Conditions for quadratic spatial beam self-cleaning. (a) FF beam at the output of the multimode fiber rod for an energy of 0.25 mJ; the speckled beam shows that no nonlinear beam reshaping occurs in the MMFR; (b) FF beam at the KTP crystal output for a FF beam energy of 0.06 μJ and θ = 1◦; FF beam at the KTP crystal output for a FF beam energy of 0.25 mJ, for θ = 1◦ (c) and θ = 3.5◦ (d).

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.

Fig. 2: Beam self-compression. Relative efficiency of SHG upon phase mismatch at low input intensity (black curve 0.13 GW/cm2) and at high intensity (blue curve 0.9 GW/cm2) in KTP crystal. Panel (a) shows the result with respect to the input angle θ. Panel (b) shows the corresponding diameter at FF. Panel (c) shows the same results of panel (a) upon an abscissa proportional to κ. Input beam diameter 250 μm. Panel (d): experimental results of SHG efficiency vs. input beam energy, at the phase-matching point θ = 0◦; here arrows indicate the two values of input FF intensity corresponding to the black squares and blue dots of curves in panels (a–c).

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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