With the Helmholtz equation given as:

\begin{equation} \nabla^2{E}+k^2{E} = 0 \end{equation}

we have already found the solution to this equation in Section 2 of this portion of the website via both free space and box modes plane waves. Now, suppose the refractive index become dependent on position, i.e. $n = n(r)$, as might be the case in the atmosphere, GRIN optics, and “thermal lensing”. In this case, we can propose a solution to the Helmholtz equation that appears similar to the plane wave solution but can account for the spatially varying refractive index:

\begin{equation} E(r,t) = A(r)e^{-ik_0S(r)}e^{i\omega{t}} \label{WKB}\end{equation}

where here we can note a few features of this equation:

- The polarization of the wave in xyz-plane can be ignored because it is a scalar wave.
- If the refractive index varies slowly as a function of position, then $A(r)$ is an envelope function that we can assume to also be a slowly-varying function.
- The function “$S(r)$ is the “eikonal” function, and the solution given in Equation \eqref{WKB} is the WKB approach to solving the Helmholtz. The WKB approach is named after Wentzel, Kramers, and Brillouin, and was originally developed for quantum mechanics.

### Like this:

Like Loading...