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Snell's Law

n1*sin(θ1) = n2*sin(θ2), where n1 and n2 are the refractive indices of the two media and θ1 and θ2 are the angles of incidence and refraction.
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The statement of the theorem

Let r(s)\mathbf{r}(s) be a parameterized path in R3\mathbb{R}^3 connecting points AA and BB. Define the refractive index n(r)n(\mathbf{r}) as a scalar field representing the local speed of light variation. The optical path length LL is given by the integral: L=∫ABn(r) dsL = \int_{A}^{B} n(\mathbf{r}) \, ds where dsds is the infinitesimal arc length element. By Fermat's Principle, the path r(s)\mathbf{r}(s) must satisfy the variational condition δL=0\delta L = 0. In a planar interface separating two media, M1M_1 and M2M_2, with refractive indices n1n_1 and n2n_2, and assuming the interface lies in the xyxy-plane, the path minimization leads to the generalized form of Snell's Law. If θ1\theta_1 and θ2\theta_2 are the angles of incidence and refraction, respectively, measured relative to the normal vector k^\mathbf{\hat{k}}, the condition is: n1sin⁡(θ1)=n2sin⁡(θ2)n_1 \sin(\theta_1) = n_2 \sin(\theta_2) This relationship is derived from the conservation of the component of the wave vector k\mathbf{k} tangential to the interface, specifically k∣∣=nk0csin⁡(θ)k_{||} = n \frac{k_0}{c} \sin(\theta), where k0k_0 is the vacuum wave number.
Source: Wikipedia