On the physics of propagating Bessel modes in cylindrical waveguides
  In this paper we demonstrate that using a mathematical physics approach (focusing the attention to the physics and using mathematics as a tool) it is possible to visualize the formation of the transverse modes inside a cylindrical waveguide. In opposition, the physical mathematics solutions (looking at the mathematical problem and then trying to impose a physical interpretation), when studying cylindrical waveguides yields to the Bessel differential equation and then it is argued that in the core are only the Bessel functions of the first kind those who describe the transverse modes. And the Neumann functions are deemed non physical due to its singularity at the origin and eliminated from the final description of the solution. In this paper we show, using a geometrical-wave optics approach, that the inclusion of this function is physically necessary to describe fully and properly the formation of the propagating transverse modes. Also, the field in the outside of a dielectric waveguide arises in a natural way. 
J. E. Gomez-Correa, S. E. Balderas-Mata, V. Coello, N. P. Puente, J.   Rogel-Salazar and S. Chavez-Cerda
physics.optics