Snell–Descartes law of refraction

Statement

n₁·sin i = n₂·sin r — a light ray crossing from a medium of index n₁ to n₂ bends at the interface, where i is the angle of incidence and r the angle of refraction (measured from the normal).

Why is it true?

Direct consequence of Fermat's principle: light picks the path that minimizes travel time. Because speed in each medium is c/n, the "shortest-time" path is no longer a straight line.

Going from a less to a more refracting medium (n₂ > n₁), the ray bends toward the normal. Conversely, if i exceeds the critical value arcsin(n₂/n₁), no real r exists → total internal reflection — the working mechanism of optical fiber.

Applications

Eyeglasses, camera lenses, microscopes, telescopes, dispersing prisms, internet optical fibers, contact lenses, LASIK eye surgery.

Topics that use this theorem

Step-by-step proofs