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The eigenvalue equation for LP modes in Optical Fibers is:

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

where:

This equation can be graphically solved. But how to plot it? It includes functions of two different variables, $u$ and $w$.

In some examples, $J_{\ell - 1}(u) / J_{\ell}(u)$ is plotted using the $u$-axis as abscissa, and then the RHS is superimposed in the same plane. How is it possible?


This is so different from the dielectric slab modes, where the equations used are of the form

$$\begin{cases} w = u \tan (u)\\ v^2 = u^2 + w^2\end{cases}$$

$u$ is considered as the abscissa and $w$ the ordinate. Intersections between the two curves are easily determined.

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  • $\begingroup$ This is a function in implicit form. A CAS can help to plot such functions. $\endgroup$
    – Jon
    Nov 29 '19 at 11:02
  • $\begingroup$ @Jon yes, of course, but what is the abscissa and what is the ordinate? It is weird that $J_{\ell - 1}(u) / J_{\ell}(u)$ is the ordinate. $\endgroup$
    – BowPark
    Nov 29 '19 at 11:15
  • $\begingroup$ There is no difference. It is up to you and your needs. $\endgroup$
    – Jon
    Nov 29 '19 at 11:22
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One example of visualization of solutions of this equation in the plane (u,w) using Mathematica 12 Figure 1

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  • $\begingroup$ So, this is the locus defined by the implicit equation $\frac{J_{\ell - 1}(u)}{J_{\ell}(u)} + \frac{w}{u} \frac{K_{\ell - 1}(w)}{K_{\ell}(w)} = 0$ in the first quadrant of the $(u, w)$ plane, ultimately obtaining $w$ as a function of $u$? $\endgroup$
    – BowPark
    Nov 29 '19 at 21:14
  • $\begingroup$ Yes, in numerical calculations we can define w as function u for some branches in the given region in a first quadrant of the $(u, w)$ plane. $\endgroup$ Nov 30 '19 at 12:23
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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

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

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.

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