Modal Dispersion - Introduction

Demonstration of modal dispersion of a multi mode fiber (MMF).

Constants:
$c_0=299792458~\mathrm{m/s}$ (speed of light)

Parameters (with default values):
step index fiber
$l = 10~\mathrm{km}$ (fiber length)
$R_s = 1~\mathrm{MBd}$ (symbol rate)
$n_\mathrm{K}=1.46$ (core)
$n_\mathrm{M}=1.445$ (cladding)
$d=50~\mathrm{\mu m}$ (core diameter)
$\lambda=1550~\mathrm{nm}$ (wave length)

plastic optical fiber (POF)
$l = 100~\mathrm{m}$
$R_s = 200~\mathrm{MBd}$
$n_\mathrm{K}=1.49$
$n_\mathrm{M}=1.4036$
$d=900~\mathrm{\mu m}$
$\lambda=800~\mathrm{nm}$

Formula [1]:

$A_\mathrm{N}=\sqrt{n_\mathrm{K}^2-n_\mathrm{M}^2}$    (numerical aperture)
$\gamma_\mathrm{L,krit}=\arcsin A_\mathrm{N}$    (critical angle, air)
$\gamma_\mathrm{K,krit}=\arcsin \frac{\sin \gamma_\mathrm{L,krit}}{n_\mathrm{K}}$    (critical angle, core)
$\Delta t_{\mathrm{Mo}}= \frac{n_\mathrm{K}\cdot l}{c}(\frac{n_\mathrm{K}}{n_\mathrm{M}}-1)$    (modal dispersion)
$V=\frac{d \pi}{\lambda}A_\mathrm{N}$    (structual parameter)
$M \approx 4(\frac{V}{\pi})^2$   (number of modes)

Assumption: Input power is equally distributed among the $M$ modes between $-\gamma_\mathrm{K,krit}$ and $\gamma_\mathrm{K,krit}$. Fiber propagation delay of received signal neglected.

References:
[1] Schiffner, Gerhard: Optische Nachrichtentechnik. Teubner, 1. Auflage, 2005