## 4 Results
### THz 3D Beam Image  
In order to realize a 3D visualization of THz beam emitted from the PCA, we spatially selected the beam and acquired _fixed DL_ images at different \(z\) positions. We used a 1.20 \(\pm\) 0.05 mm diameter metallic pinhole, moved along \(x\), \(y\) and \(z\) directions. Each image (60 px \(\times\) 60 px, 15 mm \(\times\) 15 mm) was acquired in-plane (\(x\),\(y\)) at fixed \(z\) positions, collecting a \(z\) stack-image. The \(z\) step was set to 1 mm. The spot size for each image was extracted by a Gaussian fit. According to the following equation, the spot radius \(r_{s}\) is defined as  
\[r_{s}=2\sigma \tag{8}\]  
In Figure 5, we report the procedure for the image located at \(z=-3\) mm (3 mm before the THz beam focal plane), as an example. Figure 5a represents a crop of the acquired image at \(z=-3\) mm in grey look-up table (LUT). Figure 5b shows its intensity profile as a function of distance (black dots) and the Gaussian fit (green curve). The 3D Volume Viewer tool by ImageJ was used to convert the \(z\)-stack image into a 3D volume, in Figure 6.  
As expected, the maximum intensity locates where the THz beam has its smallest diameter, i.e., the focal region. Figure 6 is formed of a stack of _fixed DL_ images, therefore no information about THz frequencies spatial distribution is known. The 3D image in Figure 6 is a result of all frequency contributions, each one with its own weight (See Figure S2 in Supplementary Materials for THz power spectrum vs. frequency). The knife edge method was adopted for reconstructing the 1D THz beam profile [46; 52; 53] and for comparing the results with the pinhole method. A sharp metallic edge was placed vertically within the THz beam and was moved along \(x\) and \(z\) directions (the edge position along \(y\) was kept fixed for all the knife edge measurements). Each _fixed  
Figure 5: (**a**) Crop of 2D acquired image (60 px \(\times\) 60 px, 15 mm \(\times\) 15 mm) of THz beam at \(z=-3\) mm (3 mm before the focal point). (**b**) Intensity plot as a function of distance. The scale bar is 1 mm. \(DL\) image was acquired by varying \(x\) position, at different \(z\)-fixed positions. The \(z\) step was set to 1 mm. In general, a sharp edge moving in a light beam having electric field \(E\), can be modeled by a function containing the error function \(erf(x)\)[52]:  
\[E(x,z)=\frac{E_{MAX}}{2}\left[1+erf\left(\frac{\sqrt{2}(x-x_{50\%})}{W(z)} \right)\right] \tag{9}\]  
where \(E(x,z)\) is the THz electric field as a function of distances along \(x\) and \(z\) directions, \(E_{MAX}\) is the maximum THz electric field, \(x_{50\%}\) the position along \(x\) corresponding to \(E_{MAX}/2\) and \(W(z)\) is the THz beam waist as a function of distance \(z\), along the propagation axis. This equation weights all the frequency contributions because no frequency is selected in _fixed_\(DL\) images. By fitting the data points with Equation (9), the green curve is obtained and displayed in Figure 7.  
This was performed for each image at each \(z\) position. The beam waist size \(W(z)\), at different \(z\) positions, have been extracted from the fitting parameters, using Equation (9). Figure 8 shows the results for the two methods adopted for the beam profile reconstruction. Figure 6: 3D image of our THz beam emitted from PCA. The percentual scale is referred to the intensity of the THz beam. The image shows a THz beam double-cone shape, where the higher intensity values are in its focal region. Scale bars: 2.5 mm along \(x\) and \(y\), 5 mm along \(z\). Figure 7: Electric field profile with knife edge method at \(z=-3\) mm. Black: electric field data points as a function of distance along the \(x\) direction (perpendicular to the beam propagation \(z\) axis). Green: fitting curve using Equation (9). In particular, the quantity \(2W(z)\) is plotted in Figure 8, and therefore the THz beam 1D projection is shown 1.2 cm before and after the focal region. The blue circles represent the pinhole method, the oranges show the knife edge method: they are well in agreement.