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Our experimental analysis was carried out with a sample of Potassium Titanyl Phosphate, a nonlinear crystal with quadratic susceptibility, commonly known as KTP. The crystal, 30 mm long, was cut to maximize the efficiency of second harmonic generation (SHG) by exploiting its anisotropic nature at room temperature. The nonlinear process is known as type-II SHG, as it involves the nonlinear coupling of the two polarization components of the input FF beam, propagating along the ordinary and extraordinary axis of the crystal, and the SH wave, which is polarized along the extraordinary axis. Light propagation in anisotropic crystals (in the specific case, the KTP is an orthorhombic biaxial crystal) is quite peculiar, since the wavevector (the vector normal to the wave fronts) does not align, in general, with the direction of the energy flow (the Poynting vector). Fig. 1 gives a visual interpretation of the nonlinear frequency conversion process. The input optical beam is linearly polarized: the electric field forms an angle α with the horizontal reference axis. The white arrow represents the wavevector of the FF, while the Poynting vectors are indicated in red for the FF and in green for the SH. The Poynting vector of the FF extraordinary wave (e) has a spatial walk-off angle ρ₁ with the horizontal axis. Similarly, the Poynting vector of the SH wave (e) has a spatial walk-off angle ρ₂. Figure 1 also shows the orientation of the KTP crystallographic axes, X, Y, and Z, as well as the two relevant angles θ and φ with respect to the propagation reference frame, which are used in order to control the phase matching for type-II SHG.

The fundamental frequency (FF) ordinary (o), FF extraordinary (e), and the second harmonic (SH) beams have different walk-off angles, ρ₀, ρ₁, and ρ₂, respectively. The orientation of the reference frame with respect to the crystallographic axes X, Y, and Z is identified by the angles θ and φ. The orientation of the input linear polarization of the light beam is measured by the angle α.

Experiments have been carried out close to the values θ = 90°, φ = 23.5° at the phase matching of type-II SHG. Small angle tilts out of these reference values are a common way to introduce a controlled phase mismatch. In our experiments, we noticed the presence of a spatial asymmetry among the three waves, which we may explain by the presence of an additional walk-off of the (pseudo) ordinary axis, which we indicated by ρ₀. As it is well-known, these walk-off angles and the consequent beam splitting induce limits to the effective nonlinear interaction length along the propagation direction.

In order to demonstrate the generation of a spatial two-dimensional TSB, let us consider first a series of experimental results obtained with a polarization orientation angle α = 47°: such a choice imposes a power unbalance in the FF between the ordinary and the extraordinary axis. Figure 2 summarizes the experimentally obtained spatial beam shapes of the FF and the SH for different input intensities. When the intensity is relatively low, we observed only weak conversion into the SH in absence of any change of shape in the FF (see Fig. 2a for FF and Fig. 2e for SH). When we increased the input FF beam intensity up to 0.07 GW/cm², we observed the formation of a narrow-coupled FF-SH beam trapped at the intersection of the two diverging beams at the FF. Such a self-focused beam, or TSB, has a diameter of ~40 µm, that is about ten times smaller than the diameter of the input FF beam. The TSB, which results from a strong reshaping of the input FF beam, has the main features of a spatial soliton surrounded by radiation:²⁷ it involves both spectral components at the FF and the SH, as shown in Fig. 2b, c, f, g. The maximum peak intensity of the TSB, which is 2.7 times higher than that of the encircling background, is reached for an input FF intensity of 0.64 GW/cm² (see Fig. 2c).

a–d Refer to the fundamental frequency (FF). e–h Show the corresponding second harmonic (SH) intensity profile. Color gradient goes from blue (the lowest light peak intensity) to red (the highest light peak intensity).

Now, surprisingly, we observed that by increasing further the input FF intensity, the TSB suddenly vanished (see Fig. 2d): its existence is thus confined to a limited range of input intensity values. At relatively high intensities, the smooth FF background beam recovers a spatial profile, which is nearly as wide as that observed at low intensities, except for the presence of minor residual distortions. To summarize, upon an increase of the input laser intensity, the TSB is first formed, then it grows up to an intensity elevation, which is nearly three times that of the local background, and then it disappears without leaving a visible trace. The observed TSB dynamics is thus very different from that of spatial solitons, which tend to persist and can even be strengthened upon increasing input FF intensity. It is important to notice that as the FF component of the TSB grows up, its shape is replicated at the SH, with high contrast from the associated background. However, the beam dynamics at the FF and the SH are very different at very high intensities (cf. Fig. 2d, h).

As can be seen, although the high-energy located spot at the FF nearly disappears, beam-breaking occurs at the SH. Specifically, for input intensities of 9.3 GW/cm², the SH breaks up into a speckled beam (Fig. 2h and Fig. SM4 in Supplementary Note 3). The process of beam-breaking at the SH has been studied in detail in a related study³⁷.

Figure 3 collects many different transverse beam sections at the FF vs. the input FF intensity: panel 3(a) clearly shows the appearance and the vanishing of the TSB. Whereas panel 3(b) clarifies the quantitative dependency of the TSB to background intensity contrast vs. the input intensity.

a Twin spotlight beam (TSB) sections of the FF component of the TSB vs. input FF intensity; b peak intensity measuring the intensity elevation of the FF component of the TSB from the local background vs. input FF intensity. R is the peak intensity ratio between the top of the TSB and the encircling background.

We observed that the TSB generation is influenced, similarly to quadratic spatial solitons, by the power unbalance between the two polarization components of the beam at the FF, as determined by the polarization orientation angle α. Since the beams carried by each of the two polarization components have slightly different propagation directions (owing to spatial walk-off), by modifying the polarization angle α, the spatial position of the TSB can abruptly hop from one output spatial position to another one. Although the TSB is systematically obtained for all angles from α = 10° to α = 80° for a fixed FF intensity, the elevation of the FF component of the TSB from the surrounding background and its spatial position varies: maximum TSB intensity is reached for two symmetric polarization orientation angles, close to either α = 30° and α = 60° (Fig. 4a): the corresponding spatial shape is given in panel (4b). When the input polarization angle was set to α = 45° (i.e., equal input FF intensities in both polarization axes), we observed the presence of two competing TSBs (Fig. 4c, d). Panel (c) shows an example of spatial switching of the TSB position for three particular values of α; while panel (d) illustrates the sharp nonlinear transition of the spatial TSB position for a set of values of the orientation angle α. Note that analogous dynamics were earlier reported for spatial solitons^38,39: the sharp dependence of the output beam position on the input state of polarization could find possible applications to ultrafast all-optical switching, as polarization comparator²⁴, or even, by extension, as an optical implementation of an activation function for an artificial neuron.

a Peak intensity ratio between the TSB and the input beam; b Example of TSB generation for an input polarization orientation angle α = 60°; c Example of TSB spatial switching for three different input polarization angles; d Spatial position of the TSB vs. the input polarization angle α (input beam intensity I_FF = 0.64 GW/cm²).

Additionally, in the large positive phase mismatch regime, we also observed the presence of multiple TSBs (see Fig. SM1 in supplementary note 1).

In our experimental analysis, we found that the generation of the TSB not only involves the spatial domain, as it can be detected by a slow-time response camera but also has a direct influence on the temporal shape of the FF component of the TSB. For analyzing temporal reshaping effects in detail, we measured the temporal autocorrelation within the TSB area, and we varied the input FF intensity (cf. Fig. 5a) as well as the FF-SH phase mismatch. At low input intensities, the temporal autocorrelation trace has a Gaussian shape with a 41 ps temporal width (compatible with a pulse duration of 29 ps FWHMI), see Fig. 5b. However, once the TSB is formed, the shape of the autocorrelation trace takes a nearly triangular profile of 24 ps (FWHMI), see Fig. 5c. At even higher input powers, where the TSB is close to its vanishing, the temporal autocorrelation trace exhibits several peaks, which can be associated with a temporal break-up of the underlying pulse, see Fig. 5d. Interestingly, different behavior is observed for positive and negative values of the phase mismatch (cf. Figs. SM2,SM3 in the supplementary note 2). Indeed, it is known that temporal modulational instability may occur for both positive and negative values of the phase mismatch in quadratic crystals. However, asymmetric evolutions (i.e., different gain profiles) of the sidebands are obtained when changing the sign of phase mismatch⁴⁰. Note that spectral sideband generation at the FF and the SH have also been experimentally reported in ref. ³⁶. for similar pulse durations.

a Autocorrelation traces at different input intensities for perfect phase matching. Details of the traces: b Autocorrelation of the input laser pulse; c Autocorrelation at 1 GW/cm²; d Autocorrelation taken at high FF intensities, such that the TSB has nearly vanished. The circular inset illustrates the orientation of the linear polarization state of the input beam.

We would like to underline that the nature of the observed extreme TSB is fully deterministic and reproducible. We confirmed its deterministic nature by numerically solving the nonlinear coupled equations involving the three interacting waves, i.e., the two FF beams and the SH wave. In the model, we included all relevant effects, namely diffraction and quadratic susceptibility. We limited our numerical analysis to the case of a simple spatial interaction of the three waves, thus neglecting the temporal dimension. Moreover, our analysis was limited to walk-off angles corresponding to the two extraordinary waves, and we set to zero, for simplicity, the FF ordinary beam angle ρ₀. The input beam had a Gaussian profile with a 400-μm diameter, and we superimposed to it a weak bump, whose center had an offset of 200 μm in one of the spatial coordinates, which was used as a seed.

In Fig. 6, we show a collection of numerically computed output FF components of the TSB for different input intensities. As can be seen by comparing Fig. 6 with Fig. 3a, numerical simulations agree well with experimental results. In particular, the intensity of the FF component of the TSB grows up to an elevation of about 2.5 times the background intensity. Then the TSB progressively disappears when the input FF beam intensity grows larger. A numerical study of the FF beam shape at different positions along the crystal and at fixed input intensity is also given in supplementary note 4 (Fig. SM5).

The images have been collected after propagation of 30 mm. The color bar relates to the light peak intensity (normalized with the encircling background).

In conclusion, we demonstrated that a coupled FF and SH 2D spotlight beam can appear and disappear at the output of a quadratic crystal in the presence of a monotonic increase of the input FF beam intensity. Such a beam bears a strong resemblance to fully localized waves such as the Peregrine temporal soliton of the 1D-NLSE, which also appears and disappears as the input power grows larger, thanks to the equivalence between varying power or propagation length. However, the localized wave that we have unveiled here emerges in the frame of a genuine 2D nonlinear wave coupling process. Besides the resemblance with other types of extreme waves, the full nature of the extreme or rogue wave described in the present work appears to involve a coupling between its spatial and temporal dimensions. Additionally, we showed how the spatial position of the TSB can abruptly switch upon a relatively small change in the linear polarization angle of the input FF beam. Although phase mismatched SHG seems to be the dominant nonlinear process in the early stage of the TSB generation, the observed temporal wave breaking at the FF could provide the mechanism which is responsible for its disappearance at high intensities.