Snell's Law Calculator — Refraction Angle

Snell's Law (also called the Snell–Descartes law) was formulated by Dutch mathematician Willebrord Snellius around 1621 and independently by René Descartes in 1637. It describes the relationship n₁·sin(θ₁) = n₂·sin(θ₂) governing wave refraction at an interface. It applies not only to visible light but also to sound waves, seismic waves, and any wave crossing a boundary between media of different propagation speeds.

The refractive index n of a medium is defined as n = c / v, where c ≈ 2.998 × 10⁸ m/s is the speed of light in vacuum and v is the phase speed of light in the material. For water, v ≈ 2.25 × 10⁸ m/s, giving n ≈ 1.333. A higher n means light travels more slowly in that medium and bends more sharply when entering it from air.

Total internal reflection (TIR) occurs when light travels from a denser medium (higher n) to a rarer one (lower n) and the angle of incidence exceeds the critical angle θ_c = arcsin(n₂/n₁). For a glass-air interface with n₁ = 1.52, θ_c ≈ 41.1°. Beyond this angle, 100% of light reflects back — this principle is the foundation of fiber optic telecommunications, carrying data at the speed of light with virtually no loss.

Light slows down when entering a denser medium (higher n). The wavefront arriving at an angle hits the interface progressively — one edge slows before the other — causing the wave to pivot toward the normal, just as a car turning onto a muddy road slows one wheel first and curves inward. Mathematically, since n₂ > n₁, the ratio sin(θ₁)/sin(θ₂) = n₂/n₁ > 1, so θ₂ < θ₁ always.

No — refractive index is wavelength-dependent, a phenomenon called dispersion. For example, borosilicate glass (BK7) has n = 1.5168 at 589 nm (yellow), 1.5220 at 486 nm (blue), and 1.5143 at 656 nm (red). This means blue light refracts at a slightly different angle than red light, which is why glass prisms split white light into a spectrum. This dispersion is quantified by the Abbe number V = (n_D − 1) / (n_F − n_C).

Air at standard temperature and pressure (0 °C, 1 atm) has n ≈ 1.000293 at 589 nm — close to 1 but not exactly. For most practical optics calculations, n_air = 1.000 introduces negligible error (< 0.03%). However, in precision metrology (e.g., interferometry at NIST), the exact air refractive index must be calculated using the Edlén equation, which accounts for temperature, pressure, humidity, and CO₂ concentration.

Yes. Snell's Law applies to all waves obeying Fermat's principle of least time. In acoustics, it governs sound refraction at boundaries (e.g., air-to-water: angle changes because sound travels at 343 m/s in air vs. 1,480 m/s in water). In seismology, it predicts how P-waves and S-waves bend through Earth's mantle layers — a key tool for locating earthquakes and mapping interior Earth structure.

The critical angle is θ_c = arcsin(n₂ / n₁), valid only when n₁ > n₂. Examples: glass (n=1.52) to air → θ_c = arcsin(1/1.52) ≈ 41.1°; diamond (n=2.417) to air → θ_c = arcsin(1/2.417) ≈ 24.4°. Diamond's low critical angle means most light striking interior facets at typical angles undergoes TIR, which is why well-cut diamonds appear exceptionally bright and brilliant.

This apparent depth illusion is a direct consequence of Snell's Law. Light from a submerged object refracts away from the normal as it crosses from water (n=1.333) into air (n=1.000). Your eye traces the refracted ray backward in a straight line, locating the object at an apparent depth = real depth / n_water ≈ real depth / 1.333. A fish 1.33 m deep appears to be only ~1.0 m deep — an important correction for spearfishing and underwater surveying.