Deriving Snell's Law Using Fermat's Principle and the Method of Substitution

Sun, 26 Jul 2026 00:00:00 +0000

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).

Geometrical-Optics on Dr. Simon Bance

Deriving Snell's Law Using Fermat's Principle and the Method of Substitution

Deriving Snell’s Law