Snell's law

Snell's law states that at an interface between two transparent media of refractive indices n₁ and n₂, an incident ray making angle θ₁ with the surface normal refracts into a transmitted ray at angle θ₂ satisfying n₁ sin θ₁ = n₂ sin θ₂. The transmitted ray bends toward the normal when entering a denser medium (n₂ > n₁, so θ₂ < θ₁) and away from the normal when entering a less dense medium. Above the critical angle θ_c = arcsin(n₂/n₁) when going from denser to less dense, no transmitted ray exists and total internal reflection takes over.

The law can be derived three ways: from Fermat's principle of least time applied to a path crossing the interface; from matching the tangential component of the wavevector k across the boundary (k_∥ must be continuous, giving n₁ sin θ₁ = n₂ sin θ₂ for light of fixed frequency); or from Huygens's secondary-wavelet construction at the boundary. Snell derived it experimentally in 1621 by meticulous angle-measurement across glass, water, and other media; he never published. Descartes independently derived and published in 1637, and French texts called it "Descartes's law" for two centuries; Huygens rediscovered Snell's original derivation in the 1670s and the Snell name prevailed in English and Dutch usage. Every geometrical-optics analysis — lens design, fibre-optic coupling, underwater optics — starts from Snell's law.