Why do TM and TE modes coexist in glass fiber? I have read that TM (transverse magnetic) and TE (transverse electric) modes of the electromagnetic field of light travelling in a glass optic fiber cable coexist. I want to know the precise mathematical justification for this (not physical intuition). Is this a banal observation or a deeper fact? Does it follow immediately from the refractive indices and Snell's law? I just do not see how at the moment. Can someone explain it? 
EDIT: People asked for clarifications. I talk about light propagating in a multi-mode optical fiber with total internal reflection, assuming perfect rotational symmetry.
 A: Let's consider an optical fiber as a rotationally symmetric structure consisting of a core with a slightly higher refractive index as the surrounding cladding. We want to find the solutions of Maxwell's equations for the electromagnetic field that can propagate along this structure. First one would express Maxwell's equations in cyllindrical coordinates and then solve the resulting set of differential equations with the separation of variables technique. The solutions would then consist of a product of Bessel functions for the radial coordinate, harmonic functions for the azimuthal coordinate and an exponential function for the propagation direction. Since all the field components are related to each other via Maxwell's equations, one usually considers only one of these components for the electric and magnetic fields, respectively, namely the $z$-component, which point in the direction of propagation.
Next, one would impose the boundary conditions between the core and the cladding to ensure that all the tangential components of the fields are continuous across this boundary. This fixes all but one parameter (apart from the modal index): the propagation constant. To find the propagation constant one composes a charateristic equation, which needs to be solved numerically for all the different values of the modal index, because it is rather complicated.
When one sets the modal index to zero one finds that the charateristic equation factors into two parts. So now either one of the two factors can become zero independent of the other. This is a specially situation that only applies for the zero modal index. In this case it turns out that the two solutions, respectively, represents the TM and TE solutions. In other words, these solutions either do not have an electric field component in the $z$-direction (it is purely transverse, hence TE) or they do not have a magnetic field component in the $z$-direction (hence TM). For all other values of the mode index both the electric and the magnetic fields have $z$-components. They are therefore called hybrid modes.
It is important to know that the TE and TM modes are not the modes that propagate in single mode fiber. The mode that propragates in single mode fibre is a hybrid mode HE$_{11}$.
This is an explanation in words. The mathematics is quite involved and for that I'd suggest you consult a proper text book on the topic.
A: Maxwell's equations are linear. As a consequence, if some TE modes and some TM modes can propagate in a given guide (not necessarily a fiber), then their sum can as well.
Note that optic fibers, unlike rectangular waveguides but similarly to two-conductors guides (coaxial or not), can propagate TEM modes. But a TEM mode is not a superposition of TE and TM modes.
