Deriving Snell's Law Using Fermat's Principle and the Method of Substitution
Deriving Snell’s Law One of the most elegant results in optics is Snell’s Law, which relates the angles of incidence and refraction at the boundary between two media: $$ n_1 \sin\theta_1 = n_2 \sin\theta_2. $$ Many textbooks derive this result using constrained optimization and Lagrange multipliers. However, there is a simpler and more intuitive approach: eliminate the constraint through substitution and reduce the problem to an ordinary single-variable minimization. This derivation follows directly from Fermat’s Principle, which states that light follows a path for which the optical path length is stationary (typically a minimum). ...