Snell's Law Calculator

When a wavefront moves from one transparent medium into another, its speed changes in proportion to the refractive index of each material. The refractive index n is defined as the ratio of the speed of light in a vacuum (c ≈ 3 × 10⁸ m/s) to the speed of light in the medium (v), so n = c/v. A higher refractive index means light travels more slowly in that medium. At a flat interface between medium 1 (index n₁) and medium 2 (index n₂), the boundary condition for the wave's electric field enforces that the tangential phase velocity must be continuous — which is exactly what Snell's Law encodes.

The governing equation is n₁ sin(θ₁) = n₂ sin(θ₂), where θ₁ is the angle of incidence and θ₂ is the angle of refraction, both measured from the normal (a line perpendicular to the surface). When light passes from a lower-index medium into a higher-index medium (e.g., air into glass), it bends toward the normal (θ₂ < θ₁). The reverse is also true: moving from a denser optical medium to a less dense one bends the ray away from the normal. This calculator also computes the critical angle θ_c = arcsin(n₂/n₁), which applies when n₁ > n₂. At incidence angles equal to or greater than θ_c, no transmitted ray exists and total internal reflection occurs — the physical principle behind optical fiber communication. Additionally, the Fresnel reflectance for both TE (s-polarized) and TM (p-polarized) light is computed using the Fresnel equations, giving the fraction of intensity reflected at the interface.

Practical applications span every branch of optics: designing camera lenses and microscope objectives requires precise ray tracing through multiple refracting surfaces, each governed by Snell's Law. In ophthalmology, corrective lens prescriptions depend on the refraction of light in the eye. In geophysics, the same law governs seismic wave refraction at subsurface boundaries. In fiber optics, total internal reflection keeps light trapped inside the core, enabling gigabit data transmission over thousands of kilometers.

Snell's Law in its standard form applies strictly to planar, isotropic, homogeneous interfaces between non-absorbing media. For absorbing materials (metals, semiconductors), the refractive index becomes complex, and the angle of refraction is no longer a simple real number — a generalized form using complex angles or transfer-matrix methods is needed. In anisotropic crystals (such as calcite or quartz), the refractive index depends on the polarization direction, leading to birefringence and double refraction that Snell's Law alone cannot describe. Very thin films, diffraction gratings, and meta-surfaces can redirect light in ways governed by modified phase-matching conditions rather than the classical law. Finally, at extremely high intensities (ultrafast lasers), nonlinear optical effects alter the effective refractive index dynamically. This calculator assumes lossless, isotropic media and angles strictly between 0° and 90°; entering an angle of 0° (normal incidence) is valid and produces θ₂ = 0°. If the computed sin(θ₂) exceeds 1, the result is undefined — indicating total internal reflection.