# Radial Structure of OAM-Carrying Fundamental X-Waves
## 2 Results
### Fundamental X Waves with OAM  
Let us consider the exponentially decaying spectrum \(g_{exp}(k)\) given by Equation (5), and remember that \(n\in\mathbb{N}\) is a constant representing the spectral order. The X-waves constructed with this spectrum are often called _fundamental_ X-waves, and represent the most common type of localized waves normally studied. First, let us notice that as Figure 1 suggests, as \(n\) increases, the spectrum acquires a bell-like shape, which shifts to higher frequencies with increasing \(n\). In particular, we observe that as \(n\) grows, the maximum shifts linearly with \(n\) towards higher frequencies, and the full width half maximum (FWHM) of the spectrum changes according to:  
\[\Delta k_{FWHM}=-\frac{2n}{\alpha}\mathrm{W}_{0}\Big{(}-\frac{\alpha}{n\,2^{ 1/n}}\Big{)}, \tag{6}\]  
where \(W_{0}(x)\) is the Lambert W-function [32]. Notice that \(\Delta k_{FWHM}\) grows quite slowly with \(n\), and, in a first approximation, it remains roughly constant. Moreover, it asymptotically tends to a constant value for \(n\to\infty\). In addition, for very big values of \(n\), the exponentially decaying spectrum can be approximated by a Gaussian spectrum (centered around the maximum of \(k^{n}\) exp \((-\alpha k)\)), and thus fundamental X-waves with very high \(n\) values can be easily described in terms of Bessel-X pulses [33; 34]. To find the explicit expression of the electric field of a fundamental X-wave, we substitute Equation (5) into Equation (4) and use the relation 6.621.1 of reference [35], to obtain:  
\[\phi_{m}^{(n)}(\mathbf{r},t)=C_{m,n}\,e^{im\theta}\,\rho^{m}\,_{2}F_{1}\Big{(} \frac{m+n+1}{2},\frac{m+n+2}{2};m+1,-\rho^{2}/\xi^{2}\Big{)}, \tag{7}\]  
where \(\rho=R\sin\theta_{0}\), \(\xi=(\alpha-i\xi)\), \(C_{m,n}=(m+n)!/(2^{m}g^{cm+n+1}m! )\), and \({}_{2}F_{1}(a,b;c,x)\) is the Gauss hypergeometric function [32]. An example of the transverse structure of fundamental X-waves for several values of the spectral index \(n\) is given in Figure 2. The longitudinal structure of fundamental X-waves as a function of the spectral index \(n\) is instead given in  
Figure 1: Exponentially decaying spectrum of fundamental X-waves, as given by Equation (5), for different values of the parameter \(n\), namely \(n=0\) (blue), \(n=2\) (red), \(n=4\) (green), and \(n=6\) (black). As can be seen, as \(n\) grows, the spectrum shifts its maximum towards higher frequencies. For this plot, \(k\) is measured in units of \(\alpha\). In particular, from Figure 2 it seems that the pulses acquire a radial structure very similar to that of LG beams or, equivalently, Bessel beams. This result can be directly compared to previous ones concerning assigning a second index to monochromatic Bessel beams [30]. However, while in the case of Bessel beams with two indices, the second index, and therefore an LG-like radial structure, was realized by constructing suitable superpositions of Bessel beams with different transverse wave vectors [30], in this case, the additional radial structure is due to the form of the spectrum. In a sense, one could say that by suitably superimposing Bessel beams of different frequencies, it is possible to induce a radial structure to them. To some extent, this is another manifestation of the spatio-temporal correlation typical of localized waves. [17]. To better understand the connection between spectral order and radial structure of fundamental X-waves, we used the algorithm described in Section 4 (Methods) to calculate the number of rings acquired by the fundamental X-waves as a function of the spectral index \(n\), which is depicted in Figure 4.  
Figure 3: Normalized longitudinal intensity distribution of fundamental X-waves carrying \(m=1\) units of OAM, for different values of the spectral index \(n\), namely \(n=0\) (**a**), \(n=20\) (**b**), and \(n=50\) (**c**). As can be seen, as \(n\) grows, the initially X-shaped function breaks down into two wings (symmetric with respect to the plane \(R=0\).