Fresnel equations and the refractive index of active media

A class of realizable active media exists for which the refractive index cannot be defined as an analytic function in the upper half-plane of complex frequency. The conventional definition of the refractive index based on analyticity is modified such that it is valid for active media in general, and associated Fresnel equations are proved. In certain active media, the presence of a “backward” wave, for which both phase velocity and Poynting’s vector point towards the excitation source, is demonstrated.

Figure 1

A plane wave incident to a slab of thickness $d$.

Figure 2

The electric field $\mathcal{E}(z,t)$ of the example in Sec. 5B. The frequency ${\omega }_0=1$ (normalized), and $t=30$. The parameters used are $\Gamma ∕{\omega }_0=0.1$ and $F=10$. The field was calculated by a straightforward numerical evaluation of Eq. (14) using $\gamma =0.01{\omega }_0$. The field plot for small $z$ shows the “backward wave,” with phase velocity and Poynting’s vector in the $-z$ direction. The forerunner, located close to $z=30c$, moves at a speed $c$ in the $+z$ direction.

Figure 3

Branch cut and integration paths. The Bromwich path $\mathrm{Im}\omega =\gamma$ can be deformed into a path consisting of two contributions: Integration along the real frequency axis and integration around the branch cut.