# Algorithm to calculate dispersion and effective index in mode solution

#1

Hello,
I like to know, how Mode solution calculate the group velocity dispersion? Is it a fit? What is the algorithm to do it?
I also like to know, what is the algorithm in Mode solution to calculate “effective index”.

Thanks,

#2

FDE uses Maxwell’s equations to solve the eigenvalue problem for a given geometry and boundary condition at a specific frequency/wavelength. Details of the work can be found in the link below:

https://kb.lumerical.com/solvers_feem.html

After solving the equations, the effective index can be calculated from $n_{eff}=c_0\beta/\omega$.

To calculate the group index or dispersion, you need to enable “detailed dispersion calculation” in the Frequency analysis tab. Solver then will shift $\omega = \omega+\delta\omega$ and will calculate new $\beta$ values to calculate $d\omega/d\beta$.

The link below might be useful to view as well:
Group Velocity Dispersion Calculation in MODE Solutions

For the completeness of this thread, you can also calculate the phase and group velocity from field vectors (from “Optical Waveguide Theory” by Allan Synder and John Love, page 230, table 11-1):

$v_p = \omega/\beta=\frac{1}{\mu_0}\frac{\int n^2|E|^2dA}{\int n^2E\times H^*\cdot\hat{z}dA}$

$v_g = d\omega/d\beta=\frac{1}{\epsilon_0}\frac{\int E\times H^*\cdot\hat{z}dA}{\int n^2|E|^2dA}$

Where integrals are calculated over the entire simulation region.

#3

Thanks @bkhanaliloo.
It does make sense.