# Radial Structure of OAM-Carrying Fundamental X-Waves
## 5 Conclusions  
In this work, we have studied the effect of the spectral index \(n\) on the transverse structure of fundamental X-waves, and we have discussed its origin and its link with Bessel-X pulses. We have shown that due to their innate spatio-temporal correlation, structuring the spectrum of such pulses results in the appearance of several rings in their intensity distribution, with a structure similar to those of Bessel beams, or, in the paraxial regime, Laguerre-Gaussian beams, with fixed OAM and a varying \(p\) index. Furthermore, we have shown how the number of rings increases in suitable steps of the spectral index,  
Figure 7: Example of application of the algorithm for retrieving the number of rings of a fundamental X-wave, given a specific value of \(n\). For this specific example, we used \(n=25\). Panel (**a**) depicts the phase of the radial derivative \(\partial_{\rho}\phi_{m}^{(n)}\). The red areas correspond to regions where \(\text{Arg}\{\partial_{\rho}\phi_{m}^{(n)}\}=0\), while the yellow areas correspond to regions where \(\text{Arg}\{\partial_{\rho}\phi_{m}^{(n)}\}=\pi\). Once a value of \(n\) is chosen (blue, dashed line, representing \(n=25\) in this case), by counting the number of times red regions cross the line of the constant \(n\) (blue, dashed line in panel (**a**)), we can retrieve the number of rings possessed by \(|\phi_{m}^{(n)}|^{2}\) in a given radial interval, which in this case is \(\rho\in[0,0.5]\). As can be seen, while in panel (**a**) we count three full crossing of the blue line with red areas, we also see, in panel (**b**), that the transverse correspondent transverse distribution of intensity \(|\phi_{1}^{(25)}|^{2}\) possesses three rings, which amounts to one ring because of the \(m=1\) units of OAM carried by the pulse, plus two extra rings deriving from the spectral structure of the pulse itself. as shown in Equation (8). We have then finally discussed the origin of this radial structure in detail, and concluded that extra rings appear in the transverse structure of fundamental X-waves as \(n\) increases, as a progressive step into adapting their transverse shape to that of Bessel-X pulses, which constitute the natural limit of fundamental X-waves for large \(n\) values. Nevertheless, we have shown how it is possible, even with relatively low values of \(n\), to introduce an extra degree of freedom into fundamental X-waves that could be used to tailor their transverse (and longitudinal) structure. The recent technological advances in spectral shaping [36], realization of long-range Bessel beams [37], and the control of the spatiotemporal structure of electromagnetic waves [15; 16] envisions the possibility to achieve great control over the spectral structure of Bessel-X pulses, and, possibly, to realize fundamental X-waves with different values of the spectral index \(n\), thus being able to exploit this extra degree of freedom encoded in their spectral structure. With this at hand, OAM-carrying X-waves could be used for a number of different applications, ranging from communication (where they could be coupled with OAM-carrying waves in optical fibers to realize long-range diffraction-resilient communication systems [38]), sensing (where the spectral index could be used as a means to tune the central frequency of the X-wave spectrum to a specific compound/specimen that needs to be detected), and material processing (where the different field configurations obtained by controlling the spectrum of the X-wave could result in different geometries and possibilities for, e.g., laser drilling, and quantum information). For the latter, controlling the onset of an additional radial structure on fundamental X-waves by controlling their spectrum could provide an interesting alternative, or a complementary resource, to LG beams for the realization of high-dimensional quantum information protocols [39]. To this aim, an agile combination of X-waves or a combination with LG modes can improve the performance as well [40].