Plot the eigenvalue equation for optical fibers LP modes

The choice about the quantities represented by the $x$ and $y$ axes is arbitrary. A convenient choice is the following one.

Rewrite the eigenvalue equation as

$$u frac{J_{ell - 1}(u)}{J_{ell}(u)} = - w frac{K_{ell - 1}(w)}{K_{ell}(w)}$$

Actually, the solution of this equation is the solution of the system

$$left{begin{matrix} displaystyle u frac{J_{ell - 1}(u)}{J_{ell}(u)} = - w frac{K_{ell - 1}(w)}{K_{ell}(w)} u^2 + w^2 = v^2end{matrix}right.$$

In fact, $u$ and $w$ are bounded by the second equation of the system, where

$$v^2 = a^2 k_0^2 (n_1^2 - n_2^2) = a^2 omega^2 mu_0 epsilon_0 (n_1^2 - n_2^2)$$

$k_0$ is the propagation constant of the current signal if it was a plain wave in the vacuum space. Note that $k_0$ is proportional to the frequency $omega$ of this signal. $n_1$ and $n_2$ are the refractive indices (respectively) of the core and the cladding, so $n_1 > n_2$ and $v^2$ is a positive quantity.

Rewrite the second equation of the system as follows:

$$w = sqrt{v^2 - u^2}$$

$v$ is a parameter, depending on the signal frequency $omega$. Choose a value for it. Now, consider $u$ as the abscissa, an independent quantity.

Plot the Left Hand Side of the eigenvalue equation, which only depends on $u$. The Right Hand Side also depends only on $u$ through the relation $w = sqrt{v^2 - u^2}$. Note, however, that (unlike the Left Hand Side) the Right Hand Side plot depends on the parameter $v$, which in turn depends on the signal frequency $omega$. If $v$ changes, also the Right Hand Side shape will change.

Plot the Right Hand Side and evaluate the value (or values) of $u$ where an intersection between the branches of the Left Hand Side and the Right Hand Side occurs. These are the solutions of the system. In particular, the $u$ values of the intersections can be graphically determined; then, the corresponding $w = sqrt{v^2 - u^2}$ is obtained through the second equation of the system.

If $u$ and $w$ are determined, the mode is defined.