# Radial Structure of OAM-Carrying Fundamental X-Waves
## 3 Discussion  
In the previous section, we have reported on the appearance of an extra ring structure on fundamental X-waves as a function of the spectral index parameter \(n\). To understand the origin of these rings, and to get more insight on the possibility to control their onset, we can look at the structure of the X-wave in the limit of large \(n\) values, i.e., when \(n\to\infty\). In this limit, in fact, it is well-known that the exponentially decaying spectrum can be well-approximated with a Gaussian spectrum centered at the average frequency of the original spectrum [33; 34]. For large \(n\), in fact, as can be seen from Figure 5, we have that:  
\[k^{n}e^{-\alpha k}\quad\to\quad\sqrt{\frac{k}{k_{0}}}e^{-\gamma(k-k_{0})^{2}}, \tag{9}\]  
Figure 4: Plot of the number of visible rings for different values of the spectral index \(n\) (red dots). To calculate the number of rings, we have adopted the algorithm described in Section 4 (Methods). Notice that since we consider only paraxial pulses, we have limited our analysis to only the paraxial region of the pulse, i.e., the region close to its center. For this plot, \(m=1\) and \(\alpha=1\) have been assumed.  
where \(\gamma\simeq\alpha/n\) and \(k_{0}=n/\alpha\). In this limit, the fundamental X-wave can be de-facto considered to be a Bessel-X pulse. Besides establishing a concrete connection between fundamental X-waves and Bessel-X pulses, corroborating the results of reference [34], the result above also sheds some light on the origin of the radial structure emerging on the transverse intensity distribution of fundamental X-waves. The radial structure of Bessel-X pulses, in fact, is a Gaussian-modulated Bessel function, which, in principle, possesses infinite rings (as does the Bessel function). However, the Gaussian damping of the intensity distribution quenches most of them (and therefore allows them to be square integrable, i.e., carry infinite energy) [33]. For fundamental X-waves, this means that as \(n\) grows, the radial structure that appears is nothing but a part of the complete ring structure of a Bessel function, suitably damped by the exponentially decaying radial behavior of the hypergeometric function. For \(n\to\infty\), therefore, the ring structure given by Equation (8) will reproduce the complete ring structure of the Bessel function \(J_{m}(x)\).  
To understand this even further, let us consider the limiting value of the transverse structure of the fundamental X-waves for large values of \(n\). We have in fact that for \(n\to\infty\), since the spectrum of the fundamental X-wave must tend to the Gaussian spectrum of Bessel-X pulses, the hypergeometric radial structure of fundamental X-waves needs to tend to the Bessel function typical of Bessel-X pulses [33], i.e.,  
\[{}_{2}F_{1}\bigg{(}\frac{m+n+1}{2},\frac{m+n+2}{2},m+1,-\frac{\rho^{2}}{\xi^{ 2}}\bigg{)}\quad\to\quad J_{m}\bigg{(}\frac{n\rho}{w(\zeta)}\bigg{)}, \tag{10}\]  
where \(w(\zeta)\) is a suitable function of the co-moving coordinate \(\zeta\), such that \(w(0)=w_{0}\), which represents the _beam waist_ parameter of the Bessel-X pulse. Within this limit, it is not difficult to see, by comparing the radial structure of the two functions above (see Figure 6), that they share the same radial structure, i.e., the position of the zeros (and therefore that of the corresponding rings) is the same.  
Figure 5: Comparison between exponentially decaying spectrum with large \(n\) (blue, solid line) and Gaussian spectrum (dashed, red line), for the case of \(n=100\). As can be seen, the spectrum of fundamental X-waves with a large \(n\) can be very well approximated with that of Bessel-X pulses. For this plot, \(k\) is measured in units of \(\alpha\), which means that \(k_{0}=n=100\) and \(\gamma=1/n=0.01\).