Snell's Law Refraction Calculator
Calculate the refraction angle when light passes between two media using Snell's Law. Use it in optics coursework, lens design, or to find the critical angle for total internal reflection.
About this calculator
Snell's Law describes how light bends when it crosses the boundary between two transparent media with different refractive indices. The law states: n₁ · sin(θ₁) = n₂ · sin(θ₂), where n₁ and n₂ are the refractive indices of the first and second medium, and θ₁ and θ₂ are the angles of incidence and refraction measured from the normal to the surface. Rearranging gives the refracted angle: θ₂ = arcsin((n₁ · sin(θ₁)) / n₂). When light travels from a denser medium to a less dense one (n₁ > n₂) and the incident angle exceeds the critical angle θ_c = arcsin(n₂/n₁), the quantity (n₁ · sin θ₁)/n₂ exceeds 1, making arcsin undefined — this is total internal reflection, which underlies fiber optics and diamond brilliance. Refractive indices also vary slightly with wavelength, a phenomenon called dispersion.
How to use
Suppose light travels from glass (n₁ = 1.5) into water (n₂ = 1.33) at an incident angle of 30°. Step 1: Compute n₁ · sin(θ₁) = 1.5 × sin(30°) = 1.5 × 0.5 = 0.75. Step 2: Divide by n₂: 0.75 / 1.33 ≈ 0.5639. Step 3: Since 0.5639 < 1, no total internal reflection occurs. Step 4: θ₂ = arcsin(0.5639) ≈ 34.3°. The light bends away from the normal as it moves into the less dense medium, which is the expected result when n₁ > n₂.
Frequently asked questions
What happens when the critical angle is exceeded in Snell's Law?
When light travels from a denser medium (higher n) to a less dense medium (lower n) and the incident angle surpasses the critical angle θ_c = arcsin(n₂/n₁), no refracted ray can exist. All the light is reflected back into the original medium — a phenomenon called total internal reflection (TIR). This is not just a curiosity: TIR is the operating principle behind optical fibers that carry internet data across continents, and it is why diamonds are cut at specific angles to trap and redirect light internally for maximum sparkle. The calculator flags this condition when (n₁ · sin θ₁)/n₂ ≥ 1.
How does refractive index vary with wavelength and why does it matter?
Refractive index is not a single fixed value for a material — it changes slightly with the wavelength (color) of light, a property called dispersion. Shorter wavelengths (violet, blue) are slowed more by a medium than longer wavelengths (red, orange), so they refract at a slightly larger angle. This is why a glass prism splits white light into a rainbow spectrum and why chromatic aberration occurs in simple lenses. For precise optical design, engineers use the Abbe number to quantify a material's dispersion and choose lens glass accordingly. The wavelength field in this calculator lets you account for these differences.
What are the refractive indices of common materials used in Snell's Law calculations?
Vacuum is defined as n = 1.0000 exactly, and air is so close (n ≈ 1.0003) that it is treated as 1.0 for most calculations. Water has n ≈ 1.333 at visible wavelengths, standard borosilicate glass is around n ≈ 1.52, dense flint glass reaches n ≈ 1.70, and diamond is an impressive n ≈ 2.42. Optical fiber cores typically use silica glass at n ≈ 1.46 surrounded by cladding at a slightly lower index to enable total internal reflection. Using accurate refractive indices is essential because even small errors propagate significantly when computing refraction angles near the critical angle.