**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:

$\nabla \times E=\u2013\frac{\partial B}{\partial t}\phantom{\rule{0ex}{0ex}}\nabla \times H=\frac{\partial D}{\partial t}\phantom{\rule{0ex}{0ex}}\nabla \xb7D=0\phantom{\rule{0ex}{0ex}}\nabla \xb7B=0$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 $P(r,t)={\epsilon}_{0}{\int}_{\u2013\infty}^{\infty}\chi (r,t\u2013{t}_{1})E(r,{t}_{1})d{t}_{1}$, here $\chi (r,t)$ is a linear susceptibility.

From the first Maxwell equation we can get $\nabla \times \nabla \times E=\u2013\frac{1}{{c}^{2}}\frac{{\partial}^{2}E}{\partial {t}^{2}}\u2013{\mu}_{0}\frac{{\partial}^{2}P}{\partial {t}^{2}}$, here $c=\frac{1}{\sqrt{{\mu}_{0}{\epsilon}_{0}}}$ is a speed of light.

The permittivity $\epsilon (r,\omega )$ 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*. $\epsilon ={(n+\frac{i\alpha c}{2\omega})}^{2},\alpha =\frac{\omega}{nc}Im\chi ,n=\sqrt{(1+Re\chi )}$.

making general simplification we can obtain that ${\nabla}^{2}{E}^{\u2018}+{n}^{2}\left(\omega \right){{k}_{0}}^{2}{E}^{\u2018}=0$ 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;
- radiation modes.

It is very comfortable to use cylindrical coordinates ($\rho ,\phi ,z$) 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 $n=\left\{\begin{array}{l}{n}_{1},\rho \le a\\ {n}_{2},\rho a\end{array}\right.$is a refractive index.

Electric field in cylindrical coordinates ${E}_{z}(\rho ,\phi ,z)=F\left(\rho \right)\Phi \left(\phi \right)Z\left(z\right)$. Using these two equations we can divert to three equations by correspondent coordinates $\rho ,\phi ,z$. These equations can be resolved using Bessel functions with general solution $F\left(\rho \right)=\left\{\begin{array}{l}A{J}_{m}\left(p\rho \right)+{A}^{\u2018}{Y}_{m}\left(p\rho \right),\rho \le a\\ C{K}_{m}\left(q\rho \right)+{C}^{\u2018}{I}_{m}\left(q\rho \right),\rho a\end{array}\right.$, here ${J}_{m},{Y}_{m},{K}_{m},{I}_{m}$ are Bessel functions, and $A,{A}^{\u2018},C,{C}^{\u2018}$ are constants. Meantime $p,q$ are depending on refractive index and free-space wave number ${p}^{2}={{n}_{1}}^{2}{{k}_{0}}^{2}\u2013{\beta}^{2},{q}^{2}={\beta}^{2}\u2013{{n}_{2}}^{2}{{k}_{0}}^{2}$, $\beta $ is a propagation constant.

For certain values of parameters ${k}_{0},a,n$ we can find propagation constant ${\beta}_{nm}$ for a given $n$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* $V={k}_{0}a\sqrt{({{n}_{1}}^{2}\u2013{{n}_{2}}^{2})}$ and *normalised propagation* constant $b=\frac{{\displaystyle \frac{\beta}{{k}_{0}}}\u2013{n}_{2}}{{n}_{1}\u2013{n}_{2}}$.

More educational material is available in out Reddit community **r/ElectronicEasy**.