## 5 Numerical Simulation
### Computation of the IG Beams at \(p=4\)  
Figure 3 shows intensity distributions of the IG beams with \(p=4\) for several values of the parameter \(\varepsilon\).  
As seen in Figure 3, when the value of the parameter \(\varepsilon\) is small, IG modes align with their corresponding LG modes: the beam \(\mathbf{IG}_{4,0}^{\varepsilon}\) has an intensity with one central light spot and two surrounding light rings, and the beam \(\mathbf{IG}_{4,2}^{\varepsilon}\) has an intensity with four petals on each of these two rings. The intensity of the beam \(\mathbf{IG}_{4,4}^{\varepsilon}\) has 8 petals located on a ring, whereas the intensity of the beam \(\mathbf{IG}_{4,2}^{o}\) also contains 8 petals, but they are divided into 4 petals on each of the two rings. The beam \(\mathbf{IG}_{4,4}^{o}\) has an intensity with 8 petals located on a single ring.  
Additionally, Figure 3 confirms that for large values of the parameter \(\varepsilon\), IG beams align with the HG beams. Therefore, intensity patterns of the IG beams at \(\varepsilon\gg 1\) align with respective HG beams. The intensity of the beam \(\mathbf{IG}_{4,0}^{\varepsilon}\) has 5 petals that are located on the vertical axis and separated by four horizontal zero-intensity lines; the intensity of the beam \(\mathbf{IG}_{4,2}^{\varepsilon}\) has 9 petals separated by two horizontal and two vertical zero-intensity lines; the intensity of the beam \(\mathbf{IG}_{4,4}^{\varepsilon}\) has 5 petals, which are located on the horizontal axis and separated by four vertical zero-intensity lines; the beam \(\mathbf{IG}_{4,2}^{o}\) has an intensity with 8 petals separated by one vertical and three horizontal zero-intensity lines, and the beam \(\mathbf{IG}_{4,4}^{o}\) has the same intensity as the beam \(\mathbf{IG}_{4,2}^{o}\), but rotated by \(\pi/2\).  
A comparison of Figure 2 for \(p=3\) and Figure 3 for \(p=4\) reveals that if \(p\) is even, then changing the sign of the parameter \(\varepsilon\) leads to rotation of the intensity pattern (Figure 3), whereas if \(p\) is odd, then changing the sign of \(\varepsilon\) leads not only to rotation of the intensity pattern, but also to a change in the pattern (Figure 2). This finding from  
Figure 2: Intensity distributions (here and in all figures below, black color means 0 and yellow-white colors mean maximum) of the IG modes \(\mathbf{IG}_{3,1}^{\varepsilon}(\mathbf{r},\varepsilon)\), \(\mathbf{IG}_{3,3}^{\varepsilon}(\mathbf{r},\varepsilon)\), \(\mathbf{IG}_{3,1}^{\varepsilon}(\mathbf{r},\varepsilon)\), and \(\mathbf{IG}_{3,3}^{\varepsilon}(\mathbf{r},\varepsilon)\) (from top to bottom) for some values of the parameter \(\varepsilon\). The other parameters are as follows: wavelength \(\lambda=532\) nm, Gaussian beam waist radius \(w_{0}=1\) mm.  
Equation (21). This equation indicates that for even values of \(p\), the IG functions are not themselves changed; instead, only their arguments are swapped: \(\mathbf{IG}^{e}_{p,2m}(x,y,-\varepsilon)=(-1)^{m}\mathbf{IG}^{e}_{p,2m}(y,x,\varepsilon)\) or \(\mathbf{IG}^{o}_{p,2m}(x,y,-\varepsilon)=(-1)^{m+1}\mathbf{IG}^{o}_{p,2m}(y,x,\varepsilon)\). Due to this exchange of arguments, the intensity rotates by \(\pi/2\) (Figure 3). By contrast, for odd values of \(p\), when \(\varepsilon\) changes sign, not only are the arguments swapped, but the functions are changed (an even to an odd, or vice versa): \(\mathbf{IG}^{o}_{p,2m+1}(x,y,-\varepsilon)=(-1)^{m}\mathbf{IG}^{e}_{p,2m+1}(y,x,\varepsilon)\).