Fiber Optics and Photonics

# Mode propagation in optical fiber

This post answers the question: “What is mode propagation in optical fiber?”.  In order to consider concept of light propagation we will use Maxwell equations for electromagnetic waves.

The Maxwell equations for electric and magnetic fields $E$ and $H$  and electric and magnetic flux densities $D$ and $B$ are:

$\nabla ×E=–\frac{\partial B}{\partial t}\phantom{\rule{0ex}{0ex}}\nabla ×H=\frac{\partial D}{\partial t}\phantom{\rule{0ex}{0ex}}\nabla ·D=0\phantom{\rule{0ex}{0ex}}\nabla ·B=0$

Another relationship between electromagnetic fields and flux densities with permittivity ${\epsilon }_{0}$ and permeability ${\mu }_{0}$, induced electric and magnetic polarizations $P$ and $M$ are:

$D={\epsilon }_{0}E+P\phantom{\rule{0ex}{0ex}}B={\mu }_{0}H+M$

Relation between $P$ and $E$ is , here $\chi \left(r,t\right)$ is a linear susceptibility.

From the first Maxwell equation we can get $\nabla ×\nabla ×E=–\frac{1}{{c}^{2}}\frac{{\partial }^{2}E}{\partial {t}^{2}}–{\mu }_{0}\frac{{\partial }^{2}P}{\partial {t}^{2}}$, here $c=\frac{1}{\sqrt{{\mu }_{0}{\epsilon }_{0}}}$ is a speed of light.

The permittivity $\epsilon \left(r,\omega \right)$ is a complex quantity and can be represented with the refractive index $n$ and the absorption coefficient $\alpha$.  Both these quantities are frequency and linear susceptibility dependent. $n\left(\omega \right)$ is called a chromatic dispersion.

making general simplification we can obtain that ${\nabla }^{2}{E}^{‘}+{n}^{2}\left(\omega \right){{k}_{0}}^{2}{E}^{‘}=0$ this equation is called wave equation, here ${k}_{0}=\frac{2\pi }{\lambda }=\frac{\omega }{c}$ is a free-space wave number and $\lambda$ is a vacuum wavelength of optical field, $\omega$ it’s frequency.

In optical fibers induced magnetic polarization $M=0$, because optical fibers are non-magnetic.

### Optical modes and mode propagation

Optical mode describes specific solution of the wave equation in accordance to the appropriate boundary conditions and whose spatial distribution does not change with propagation.

Types of fiber modes are:

• guided modes;
• leaky modes;
It is very comfortable to use cylindrical coordinates () for our equations, where wave equation will be $\frac{{\partial }^{2}{E}_{z}}{\partial {\rho }^{2}}+\frac{1}{\rho }\frac{\partial {E}_{z}}{\partial z}+\frac{1}{{\rho }^{2}}\frac{{\partial }^{2}{E}_{z}}{\partial {\phi }^{2}}+\frac{{\partial }^{2}{E}_{z}}{\partial {z}^{2}}+{n}^{2}{{k}_{0}}^{2}{E}_{z}=0$, here is a refractive index.
Electric field in cylindrical coordinates . Using these two equations we can divert to three equations by correspondent coordinates . These equations can be resolved using Bessel functions with general solution , here  are Bessel functions, and  are constants. Meantime  are depending on refractive index and free-space wave number , $\beta$ is a propagation constant.
For certain values of parameters  we can find propagation constant ${\beta }_{nm}$ for a given and $m$. Every ${\beta }_{nm}$ corresponds to one possible propagation mode. Here we can introduce mode index  $\frac{\beta }{{k}_{0}}$, it is usually ${n}_{1}>\frac{\beta }{{k}_{0}}>{n}_{2}$ and means that each fiber propagates with an effective refractive index.
The propagation mode is reaching cutoff when $\frac{\beta }{{k}_{0}}={n}_{2}$. Cutoff condition is characterised with  normalised frequency  $V={k}_{0}a\sqrt{\left({{n}_{1}}^{2}–{{n}_{2}}^{2}\right)}$ and normalised propagation constant $b=\frac{\frac{\beta }{{k}_{0}}–{n}_{2}}{{n}_{1}–{n}_{2}}$.