I have said this multiple times in various places on this website, but this is my bread and butter! It is also a lot of other people’s bread and butter, which is why they are so prolific and why I have also repeated myself numerous times on this website since fiber waveguides pop up everywhere :-). Now if that doesn’t motivate you to keep reading, I don’t know what to do for you!

We can start with the assumption that we have a time-harmonic i.e. monochromatic plane wave solution for our waveguide, which means that we just need to solve the scalar wave equation: the Helmholtz Equation! We can also make this assumption as a mode remains unchanged with propagation, which means that the monochromatic assumption is not entirely invalid. This is given as:

\begin{equation} \nabla^2E + n_j^2k_0^2E = 0 \end{equation}

where $j = 1$ for the cladding index and $j = 2$ for the core index. Before, we found plane wave and box mode solutions to the Helmholtz equation which are valid in free space and in the case of rectangular boundary conditions, respectively. Now, since the propagation in the fiber is governed by the physical dimensions of the fiber, we need to have solutions to this that are in cylindrical coordinates. This means that we can break apart the spatial electric field solution as:

\begin{align} E(r) & = E(r,\theta,z) \\ & = \hat{r}E_r(r,\theta,z) + \hat{\theta}E_{\theta}(r,\theta,z) + \hat{z}E_z(r,\theta,z) \end{align}

We think physically, then, that the “r” and “$\theta$” components cannot be decoupled due to the fact that propagation would “add in” $\theta$-components as the wave is reflected at each boundary. For example, when the electric field begins propagation at the center of the fiber, it is purely defined by its r- and z-coordinates. However, as it continues propagation towards a boundary, the field becomes increasingly defined by its $\theta$ component, with there being zero components of the $r$-coordinate at the boundary. For propagation in the z-direction, though, any purely z-components of the initial propagation will remain in the z-direction.

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