Apply Snell's law to find refraction angle, critical angle, Brewster's angle, and Fresnel reflectance at any optical interface.
Snell's law — n₁ sin θ₁ = n₂ sin θ₂ — is the cornerstone of geometrical optics, governing how light bends when crossing the boundary between two materials with different refractive indices. Discovered independently by Ibn Sahl (984 AD), Willebrord Snellius (1621), and René Descartes, it applies universally from fiber optics to diamond gemstone design to underwater photography.
When light travels from a denser medium to a less dense one (n₁ > n₂), there exists a critical angle beyond which all light is reflected — total internal reflection (TIR). This phenomenon is the operating principle of optical fibers, prisms, and retroreflectors. At the Brewster angle, reflected light becomes perfectly polarized, a fact exploited in polarizing filters and laser windows.
This calculator computes the refracted angle, detects total internal reflection, calculates Brewster's and critical angles, evaluates Fresnel reflectance for both s- and p-polarizations, and determines the wavelength and speed of light in the second medium. Comprehensive angle and material comparison tables let you explore refraction across all common optical materials and incident angles.
Use this calculator when you need refraction, critical-angle, and reflectance values from the same interface without juggling separate formulas.
It is useful for fiber optics, prism design, underwater imaging, and quick checks of whether a beam will transmit, partially reflect, or hit total internal reflection. It also keeps the wavelength and speed adjustments visible when you want to compare the same interface at a different incident angle or material pair.
Snell's law: n₁ sin θ₁ = n₂ sin θ₂. Critical angle: θ_c = arcsin(n₂/n₁). Brewster's angle: θ_B = arctan(n₂/n₁). Fresnel: R_s = ((n₁cosθ₁ − n₂cosθ₂)/(n₁cosθ₁ + n₂cosθ₂))².
Result: θ₂ = 19.27°
Air to glass at 30°: sin θ₂ = (1.0003 × sin 30°)/1.5168 = 0.3297 → θ₂ = 19.27°. Brewster angle = 56.6°. No TIR possible (going into denser medium).
Snell's law is most useful when you connect angle change to the broader interface behavior. The same pair of refractive indices determines the refracted angle, whether total internal reflection can occur, and how strongly the interface reflects.
The most common mistake is applying the formula without checking the direction of travel or the refractive-index ordering. Another is assuming a single index value is exact across all wavelengths, because real materials disperse and red and blue light do not bend by exactly the same amount. Coatings and polarization effects can also change the practical reflection result at a real surface.
When light hits a less dense medium (n₁ > n₂) at an angle greater than the critical angle, 100% is reflected. This is how optical fibers guide light over kilometers.
The angle at which reflected light is completely p-polarized (zero p-reflectance). θ_B = arctan(n₂/n₁). Laser windows are tilted at Brewster's angle to minimize reflection losses.
Yes, but the refractive index changes with wavelength (dispersion). This calculator uses a single n value; for precise work, use the Sellmeier equation for your wavelength.
When going from less dense to more dense media, sin θ₂ < sin θ₁, so θ₂ < θ₁ — refraction always produces a valid angle. TIR only occurs going from dense to less dense media.
They give the exact fraction of light reflected and transmitted at an interface, separately for s-polarization (perpendicular) and p-polarization (parallel). At normal incidence, both give R = ((n₁−n₂)/(n₁+n₂))².
Light slows down: v = c/n. In glass (n≈1.5), light travels at about 200,000 km/s instead of 300,000 km/s. The wavelength also shrinks: λ_medium = λ_vacuum/n.