In this paper, we demonstrate that by using a mathematical physics approach—focusing attention on the physics and using mathematics as a tool—it is possible to visualize the formation of the transverse modes inside a cylindrical waveguide. The opposite (physical mathematics) approach looks at the mathematical problem and then tries to impose a physical interpretation. For cylindrical waveguides, the physical mathematics route leads to the Bessel differential equation, and it is argued that in the core of the waveguide there are only Bessel functions of the first kind in the description of the transverse modes. The Neumann functions are deemed non-physical due to their singularity at the origin and are eliminated from the final description of the solution. In this paper, by combining geometric optics and wave optics concepts, we show that the inclusion of the Neumann function is physically necessary to describe fully and properly the formation of the propagating transverse modes. With this approach, we also show that the field outside a dielectric waveguide arises in a natural way.
On the physics of propagating Bessel modes in cylindrical waveguides
J. E. Gómez-Correa, S. E. Balderas-Mata, V. Coello, N. P. Puente, J. Rogel-Salazar, S. Chávez-Cerda; On the physics of propagating Bessel modes in cylindrical waveguides. Am. J. Phys. 1 May 2017; 85 (5): 341–345. https://doi.org/10.1119/1.4976698
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