# Laser Beam Positioning by Using a Broken-Down Optical Vortex Marker
## 5 Pseudo-Phase Analysis Method  
The spiral feature of the vortex beam can be extracted from the intensity images by applying the Laguerre-Gauss filter to obtain a so-called composite signal [36]. The recorded intensity is convolutional with the filter in the form of  
\[fLG(r,\theta)=i\cdot\pi^{0.2}d^{4}r\exp\left(-\pi^{0.2}r^{2}d^{2}\right)\exp( i\cdot m\theta\ ) \tag{3}\]  
where \(r\) and \(\theta\) are the polar coordinates, and \(d\) serves as a bandwidth of the LG filter.  
The results obtained by the convolution can be generally written as  
\[I_{con}(r,\theta)=I(r,\theta)\ \bigotimes\ fLG(r,\theta)=|I_{con}(r,\theta)| \exp[i\varphi(r,\theta)] \tag{4}\]  
where \(\varphi(r,\theta)\) is called the pseudo-phase.  
Figure 3: The separated optical vortices with the addition of the plane wave, (**a**,**c**); the phase map for the separated optical vortices, (**b**,**d**). Cases (**a**,**b**) were generated for \(m=+3\); (**c**,**d**) were generated for \(m=-5\). The plane wave slope direction was the same in all cases.  
Figure 4 shows the arguments and result (only phase \(\varphi\)) of the convolution operation of an LG filter with an intensity image. After performing the convolution operation described by Equation (4) and extracting the phase \(\varphi\), we obtain the pseudo-phase maps shown in Figure 4c.  
Colored and superimposed pseudo-phase and original phase maps are presented in Figure 5. Vortex positions are indicated by the red circles, which are at the point where the spiral connected with the \(2\pi\) phase jump ends.  
The presented pseudo-phase maps differ depending on the number of optical vortices. The next figure (Figure 6) shows an example for five optical vortices.  
Figure 4: (**a**) Intensity of the vortex beam with a topological charge of \(m=3\) after separation into three vortices with topological charges of \(m=1\), (**b**) phase plotted as \(\arg(fLG(r,\theta))\), (**c**) pseudo-phase, plotted as \(\arg(I_{con}(r,\theta))\) for the vortex beam with a topological charge of \(m=3\). The figure shows the convolution operation.  
Figure 5: Pseudo-phase, plotted as \(\arg(I_{con}(r,\theta))\) for the vortex beam with a topological charge of \(m=3\) after separation into three vortices with topological charges of \(m=1\), with the original phase map colored and superimposed. The red circles indicate the locations of vortex points on both types of maps.  
The initial location of vortex points can be determined from the intensity minima. Having defined these areas, we cut out small areas from the pseudo-phase maps around the initially found position. These maps contain the beginning of the phase jump. This phase jump line can be treated as an edge and can be extracted by the [PERSON] method [37] of edge finding, which is used in image processing. This method gives a location accuracy of 2 pixels on the CCD matrix. Location accuracy can be significantly increased by using a learned artificial neural network [38]. It is shown that the vortex can be localized with subpixel accuracy also on the experimental phase maps. A measurement accuracy on the order of \(10^{-5}\) pixels was possible after the application of a feedforward artificial neural network technique with three hidden layers. The hidden layers had a hyperbolic tangent activation function, and the output layer was characterized by a linear activation function [39]. The close neighborhoods of vortex points on pseudo-phase maps are similar to the phase maps of the Laguerre-Gauss mode. Learned artificial neural networks work in the same way for both kinds of phase maps. Moreover, the algorithm based on neural networks is less sensitive to experimental disturbances related to wavefront noise.