# Refractive index and electric susceptibility

Suppose we have a complex refractive index $n_{ref}=n+ik$ whose value is given at a precise frequency $\omega_l$ from experimental data. We know that the imaginary part is responsible for the loss/gain mechanism.

Now suppose that we are to use time domain solver(FDTD or DGTD) for Maxwell's equations. One way to incorporate gain/loss in the equation can be done by resorting to a polarization equation.

Indeed Ampere's law would become : $\nabla\times\mathbf{H}=\epsilon_{0}\frac{\partial\mathbf{E}}{\partial t}+\mathbf{J}_{p}$

with the polarization current: $\mathbf{J}_{p}=\frac{\partial\mathbf{P}}{\partial t}$

Where the polarization P is linked to the electric field(in linear medium,isotropic, homogeneous) through the suceptibility :

$\mathbf{P}=\epsilon_{0}\chi_{e}(\omega)\mathbf{E}$

This last formulation will then be solved as an auxiliary differential equation simultaneously with the maxwell's equation.

It terms of frequency domain approach this formulation is somehow equivalent to solving the Helmoltz equation(1D formulation) :

$k^{2}=\epsilon(\omega).\frac{\omega^{2}}{c^{2}}$

Where the imaginary part(related to gain/loss) of the permittivity would be(in the case of a Lorentz model) :

$\epsilon_{2}(\omega)=\frac{\triangle\epsilon\omega_{l}^{2}\gamma_{l}\omega}{(\omega^{2}-\omega_{l}^{2})^{2}+\gamma_{l}^{2}\omega^{2}}$

The question is then how to choose this $\gamma_l$ coefficient so that the modal gain at the resonance frequency $\omega_l$ is the same as the gain that would be obtained for $\omega_l$ with a frequency domain approach(where the experimental complex refractive index would be used)?

• Experimental data will never match the model exactly, as the model is for a single oscillator, but a real system has many. But if your peak is relatively well-separated from other features, you can find $\gamma$ reasonably from the line width of the peak. Are you concerned as well about the magnitude of $\epsilon$? The question is not entirely clear. – garyp Mar 12 '15 at 19:43
• What I currently have is an analytic solution of a 2D mode inside a cylindrical cavity. This TE mode is amplified in the cavity(thanks to stimulated emission).Its complete expression has been determined at a given frequency thanks to an experimental value of the refractive index at this frequency. What I want now is to get the exact same result but from a time domain simulation. So indeed I will need the amplitude of $\epsilon$ as well – Ronan Tarik Drevon Mar 12 '15 at 20:11