Inspired by [10.1126/science.1126493].
Conformal transformation and Helmholtz equation
Consider the Helmholtz equation describing a scalar phasor field $\psi$ in a 2D space filled by a isotropic medium of index $n(\mathbf{r})$. $$ (\nabla^2 + n^2k_0^2)\psi = 0. $$ the phasor field is corresponding to a given light polarization. We can introduce the complex variable $z=x+iy$, and we have the relations $$ \partial_x = \partial_z+\partial_{\bar{z}}, $$ $$ \partial_y = i\left(\partial_z-\partial_{\bar{z}}\right). $$ By substitution, $\nabla^2 = 4\partial_z\partial_{\bar{z}}$. Consider now an analytical function $w(z)$. We have $$ \partial_z\partial_{\bar{z}} = \left|\partial_z w\right|^2 \partial_w\partial_{\bar{w}}. $$ We can map the Helmholtz equation into $$ (4\partial_w\partial_{\bar{w}} + n’^2k_0^2)\psi = 0 $$ where the new index is $n’ = n |\partial_z w|$. Then, we can map again the Helmholtz equation into new variables $u, v$, having $w=u+iv$, obtaining $$ (\partial_u^2+\partial_v^2 + n’^2k_0^2)\psi = 0 $$ therefore mapping the previous electromagnetic problem into a new problem with a transformed index.
Ray equation
We can perform the Geometrical optics approximation to Helmholtz equation. We define $$ \psi(\mathbf{r}) = C(\mathbf{r})e^{ik_0S(\mathbf{r})} $$ With $C$ and $S$ real functions. Substituting into Hemholtz equation, and approximating $C$ as a slowly varying function of space, namely $\nabla C(\mathbf{r})\ll k_0 C(\mathbf{r})$, we obtain the Eikonal equation (from the greek $\varepsilon \iota \kappa \tilde{\omega}\nu$, “image”), $$ |\nabla S|^2 = n^2. $$ Introducing a curvilinear coordinate $u$ (the coordinate along the ray), with versor $\hat{u} = \frac{d u}{d\mathbf{r}}$. The versor $\hat{u}$ is everywhere normal to the constant-phase surface, therefore, $$ \nabla S = n \hat{u}, $$ from which, after some vector algebra (Someda sec. 5.4), we can derive the ray equation, $$ \frac{d}{du} \left(n\frac{d\mathbf{r}}{du}\right) = \nabla n. $$ We can solve for rays in both the original and the mapped problem! Also, the mapping function $w(z)$ can have branch cuts, and therefor it can be interesting to see what happens to rays going through Riemann sheets of the mapped space.