Eikonal equation - Snell's law

Question

I am interested in equations of the form $|\nabla d|= F(x)$, where $F(x)$ is piecewise constant and $d(x) = 0$ on $\Gamma_D$, a subset of the boundary. In particular, like in the figure, one can consider $F(x)$ taking two values, delimited by an interface $\Sigma$ (which can be considered smooth).

Snell's law gives a relation between the incidence angles of a ray passing through the interface: $\sin \theta_+/\sin \theta_- = V_+/V_-$. I cannot find any clear references proving that Snell's law holds for the eikonal equation described above. Therefore I arrive at my question:

1. Does Snell's law hold across the interface $\Sigma$?

2. If the answer to the first question is affirmative, can you indicate references where a proof can be found?

Answer 1

Yes, Snell's law holds for any smooth interface, it follows directly from the eikonal equation, see for example section 2.4 of González-Acuña and Chaparro-Romo - Stigmatic Optics. I summarize the derivation:

We substitute into the eikonal equation the definition of a ray, being a unit vector ${\bf v}$ in the direction of $\nabla d$. Then the eikonal equation $|\nabla d|=F$ can be rewritten in vectorial form as $\nabla d=F{\bf v}$, and hence for any closed contour $C$ one has $$\oint_C F\,{\bf v}\cdot {\rm d}{\bf l}=\oint_C\nabla d\cdot {\rm d}{\bf l}=0.$$ If you take a contour as in the figure (with tangential unit vector $\hat{\bf t}$ and infinitesimal perpendicular thickness), you arrive at $$\hat{\bf t}\cdot(V^+{\bf v}_+-V^-{\bf v}_-)=0,$$ which implies Snell's law.