Fig. 1. Schematic representation of the experimental setup, also adopted in ref.²³, allowing high-resolution PSHG measurements in stationary, raster-scanned samples.Abbreviations, as met by the laser fundamental pulse: HWP: zero-order halfwaveplate, L: lens, GM: galvanometric mirrors, M: mirror, D: dichroic, O: objective, S: sample, C: condenser, F: filters, LP: linear polarizer, PMT: photomultiplier tube. The linear polarization of the excitation electric field E starts horizontal in the sample plane and is rotated clockwise with an angle φ (see also Fig. 3).

Fig. 2. Schematic representation of the structure of 2D TMDs, containing three sublattices, with a plane of metal atoms being hexagonally packed between two planes of chalgogen atoms.

Fig. 3. The experimental configuration, showing the laboratory X-Y-Z, and the crystalline x-y-z coordinate systems.Angles φ, θ, and ζ describe, respectively, the orientation of the rotating fundamental linear polarization, crystal armchair, and linear polarizer, with respect to X laboratory axis.

Fig. 4. Simulated PSHG modulation presented in polar diagrams, as function of the linear polarization orientation φ, with φ ϵ [1°, 360°], for fixed polarizer at angle (a) ζ=0° and (b) ζ=90°.The orientation of the fourfold pattern rotates for different crystal armchair directions θ.

Fig. 5. Snapshots of experimental PSHG images of a WS₂ flake, CVD-grown on a sapphire substrate.The white double arrow shows the constant angle, ζ=0°, of the linear polarizer, while the orange double arrow shows the rotating angle, φ, of the excitation linear polarization. Here, the rotation of φ ϵ[0°–360°] with step 40°, clearly shows the switching on and off of the SHG signal. The supplementary material includes a movie capturing the complete experimental PSHG modulation, for φ ϵ[0°–360°] with step 1°.

Fig. 6. (a) Integration of the experimentally detected PSHG intensity from the WS₂ island, for φ ϵ[0°–90°] with step 1°, presented upon marking three POIs and two LOIs for further analysis. The POIs are actually single pixels of the 1200×1200 original image, magnified here, for illustration purposes. (b) Intensity profile of the experimental PSHG modulation presented in (a), along the LOIs shown there. As may be seen for LOI i, the intensity in the central, brighter area is magnified by a factor of ~4, which suggests the presence of second layer²⁹. (c) Polar diagrams of the experimental PSHG modulation for φ ϵ[0°–360°] with step 1°, for the POIs illustrated in (a). We show with red color the raw data, and with blue the fitting using Eq. (3). We also present the retrieved values of the armchair orientation θ and the quality of fitting R².

Fig. 7. Mapping in (a) 2D diagram and (b) histogram of the armchair orientation distribution of the WS₂ flake (with < θ > being the mean vale), based on the pixel-by- pixel fitting (R²≥0.88) of Eq. (3) on the experimental PSHG modulation for φ ϵ[0°–360°] with step 1°.Significant color changes in the map, or equivalently, large standard deviation (∆θ) of the histogram, denote inhomogeneity in either crystalline orientation, material thickness, or layer stacking.