Nodes in Optical Fibers
The number of nodes in an optical fiber with paraxial guided rays is equal to the number of roots of the characteristic equation obtained from the Helmholtz equation in cylindrical coordinates. These are plotted at the bottom of the picture. At the top, the corresponding Bessel functions are plotted, as well as views of the cross section of the fiber.
The characteristic equation is
and are Bessel functions of the first and second kind, respectively, where and are normalized geometrical parameters.
 B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, New York: John Wiley & Sons, 1991.