Radiation guided along a cylindrical symmetry system according to the refractive index profile.

We aim at finding, from a purely theoretical analysis, the behavior that the refractive index should have within a cylindrical waveguide so that the radiation entering the system in a definite way is guided through it. Based on the criterion we have set in a previous article applying the Fermat's extremal principle in the framework of the geometrical optics, we depict the radiation confinement regions for refractive index profiles often used in the construction of waveguides, one step, multi-step and parabolic, by drawing upon the Legendre transform space as an intermediate resource in the process. We have also studied the possibility of performing the reverse path: for a wanted confinement region, to find the parameters defining the refractive index profile of the waveguide to be built. We conclude that such a process is possible as long as we know the shape of the profile. Under such restriction, our analysis allows us to deduce the characteristics that the guide should have so that the radiation entering with a given angle and at a certain distance from its axis remains confined. The technique can be used in design processes as a resource to limit the parameters that characterize the system.