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Refraction

The bending of light as it passes from one medium to another, governed by Snell's Law: n1*sin(θ1) = n2*sin(θ2), where n is the refractive index.
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The statement of the theorem

Let r(s)\mathbf{r}(s) be a path parameterized by arc length ss connecting points r1\mathbf{r}_1 and r2\mathbf{r}_2 in two media, M1M_1 and M2M_2, separated by an interface Σ\Sigma. Let n(r)n(\mathbf{r}) be the spatially varying refractive index. The optical path length LL is defined by the action integral SS: \n\nS=∫r1r2n(r) dsS = \int_{\mathbf{r}_1}^{\mathbf{r}_2} n(\mathbf{r}) \, ds\n\nBy Fermat's Principle, the path r(s)\mathbf{r}(s) must satisfy the Euler-Lagrange equations derived from minimizing SS. If the interface Σ\Sigma is defined by r⋅n=0\mathbf{r} \cdot \mathbf{n} = 0, where n\mathbf{n} is the normal vector, and k1\mathbf{k}_1 and k2\mathbf{k}_2 are the wave vectors in M1M_1 and M2M_2 respectively, the continuity of the tangential component of the wave vector across the boundary Σ\Sigma requires that the generalized momentum component perpendicular to Σ\Sigma must satisfy the boundary condition derived from the Hamiltonian formulation of the Eikonal equation. Specifically, if θ1\theta_1 and θ2\theta_2 are the angles between k1\mathbf{k}_1 and k2\mathbf{k}_2 and the normal n\mathbf{n}, then the conservation of the component of the wave vector parallel to Σ\Sigma yields:\n\nn1sin⁡(θ1)=n2sin⁡(θ2)n_1 \sin(\theta_1) = n_2 \sin(\theta_2)
Source: Wikipedia