# Radial Structure of OAM-Carrying Fundamental X-Waves
## 4 Materials and Methods  
_Number of Rings of Localized Waves_  
To understand how the algorithm we used to extract information about the number of rings possessed by a localized wave and generated by its spectral index \(n\) works, let us consider the case of a fundamental X-wave with \(n=25\) as an example of how we can calculate the number of rings appearing in the transverse cross-section of the X-wave according to its spectral index. First, we calculate the radial derivative of the localized wave, which, in the case of fundamental X-waves, is given by:  
\[\frac{\partial\phi_{m}^{(n)}(\mathbf{r},t)}{\partial\rho} = e^{im\theta}\Bigg{\{}\frac{m\rho^{m-1}}{(\alpha-i\zeta)^{m+n+1} \;2F_{1}\bigg{(}\frac{m+n+1}{2},\frac{m+n+2}{2};m+1,-\frac{\rho^{2}}{(\alpha-i \zeta)^{2}}\bigg{)}}\] \[- \frac{(m+n+1)(m+n+2)\rho^{m+1}}{2(m+1)(\alpha-i\zeta)^{m+n+3}\;2F_ {1}\bigg{(}\frac{m+n+3}{2},\frac{m+n+4}{2};m+2,-\frac{\rho^{2}}{(\alpha-i \zeta)^{2}}\bigg{)}}\Bigg{\}}.\]  
The zeros of the above equation, which we indicate with \(\rho_{\ell,n}\), represent the position, as a function of the spectral index, of the \(\ell\)-th maximum of \(|\phi_{m}^{(n)}(\mathbf{r},t)|^{2}\). We can extract information about the number of rings for a given value of \(n\) directly from the zeros of the above equation if we first notice the following things:  
* In general, Equation (4.1) admits complex solutions for \(\zeta\
eq 0\), while it only possesses real solutions for \(\zeta=0\). In this last case, in particular, the phase of Equation (4.1) will always be either zero or \(\pi\) (the latter occurring when \(\phi_{m}^{(n)}(\mathbf{r},t)\) changes sign);
* Not all the zeros of Equation (4.1) will contribute to the final number of rings. In fact, this depends on how fast the intensity distribution decays radially. If \(\phi_{m}^{(n)}(\mathbf{r},t)\) decays too quickly, only the zeros of Equation (4.1) that find themselves in a small region close to the center of the wave will participate in determining its radial structure. Figure 6: Comparison between the radial structure of a fundamental X-wave with \(n=100\) (blue, solid line) with that of the corresponding limiting form, i.e., a Bessel-X pulse with the same OAM value \(m=1\) (red, dashed line), in the plane \(\zeta=0\). Notice that the scaling of the Bessel function is \(n\rho\), which scales the distributions of zeros of the Bessel function to match that of the hypergeometric function \(\phi_{1}^{(100)}\). The higher-order zeros cannot be easily visualized, due to the rapidly decaying behavior of the hypergeometric function for large \(n\). For this plot, \(\rho\) is measured in units of \(\alpha\) for the X-wave, and \(w_{0}=\alpha\) has been assumed for the Bessel beam. Figure 7 depicts how the search algorithm works. For a given value of \(n\) (in our example \(n=25\)), we count the number of jumps (between 0 and \(\pi\)) of the phase of \(\partial_{\rho}\phi_{m}^{(n)}\) (panel (a)). This number now represents the total number of the rings' intensity distribution \(|\phi_{m}^{(n)}|^{2}\). For the example in Figure 7, the number of zeros is 3. When we plot the intensity distribution for \(|\phi_{1}^{(25)}|^{2}\), however, we notice that the third ring is barely visible (i.e., its intensity is very small compared to that of the other two rings). This is a consequence of the rapidly radially decaying behavior of the hypergeometric function. To correctly count the number of actually appearing rings (i.e., those that have enough intensity to be non-neglectable), we compare the intensity of all the maxima of \(|\phi_{m}^{(n)}|^{2}\) for a given value of \(n\), fit the relative intensity of the various rings with an exponentially decaying function, and neglect all rings whose intensity falls under \(1\%\) of the intensity of the primary OAM ring.