Snell's law (law of refraction)

Overview

Snell's law is the fundamental relation that governs refraction, the bending of waves when they pass from one material into another with a different propagation speed. In optics it is usually written in the form n₁ sin θ₁ = n₂ sin θ₂, where θ₁ and θ₂ are the angles measured from the normal to the interface and n₁, n₂ are the refractive indices of the two media. Equivalently, sin θ₁/sin θ₂ = v₁/v₂ = n₂/n₁, linking geometry to wave speed and material properties. The refractive index n is defined as n = c/v, where c is the speed of light in vacuum and v is the speed in the medium. For a vacuum n = 1.

Mathematical statement and reasoning

In practical terms the law means that if light enters a denser medium (higher n) at an oblique angle it bends toward the normal; if it enters a less dense medium it bends away. From the wave viewpoint this follows because the component of wavefront advance parallel to the interface must match on both sides, while the perpendicular component changes according to speed, producing the trigonometric relation above. A common derivation uses Fermat's principle: among all possible paths joining two points, light follows the one that minimizes travel time. Applying that principle to a piecewise uniform path gives Snell's formula directly. Another derivation uses Huygens' construction of secondary wavelets.

History and attribution

The relation is named for Willebrord Snellius (Snell), who discovered the law in the early 17th century. The law was later incorporated into the broader optical framework by René Descartes and given a variational foundation by Pierre de Fermat. Historical accounts note that the empirical relation had been observed before a full theoretical explanation became standard in classical optics.

Important consequences and related phenomena

• Total internal reflection: When light attempts to pass from a medium with higher n to one with lower n, there is a critical angle beyond which no refracted ray exists and all energy is reflected back into the original medium. This principle is the basis for optical fibers and many light-guiding devices.
• Dispersion: Because refractive index typically depends on wavelength, different colors refract by different amounts. Prism dispersion, rainbows, and chromatic aberration in lenses all arise from this wavelength dependence.
• Angle conventions: Angles are measured from the normal (perpendicular) to the surface; small-angle approximations lead to useful paraxial formulas for lens design.

Applications and examples

Snell's law underlies the design and analysis of lenses, microscopes, cameras, eyeglasses and prisms. In telecommunications it explains how light remains confined in a fiber core by internal reflection. In geophysics and atmospheric science it helps describe the apparent displacement of celestial objects near the horizon and the bending of seismic waves at material interfaces. Simple laboratory demonstrations—such as a straw appearing bent in a glass of water—illustrate the principle clearly for students.

Further notes and references

For more on the experimental basis and mathematical derivations, see introductory texts on refraction and optics, or accounts of Fermat's principle. Modern treatments extend Snell's law to electromagnetic waves using boundary conditions on the fields and to anisotropic or graded-index media where the simple scalar refractive index is replaced by a tensor or a spatially varying function. Additional practical guidance and optical constants can be found in specialized references: optical materials, prism and lens design, and wave propagation in media.