Snell's Law Calculator
When a wave, whether it is light, sound, or any other type, crosses the boundary between two media with different wave speeds, it bends. This bending is called refraction, and its rules are governed by Snell's Law, which states that n1 times the sine of the angle of incidence equals n2 times the sine of the angle of refraction, where n1 and n2 are the refractive indices of the two media and the angles are measured from the normal to the surface (a line perpendicular to the boundary). A refractive index is a dimensionless number that tells you how much slower light travels in a material compared with a vacuum. Air has a refractive index of approximately 1, water is about 1.33, and standard glass is about 1.52. Light bends toward the normal when it enters a denser medium (higher n) and away from the normal when it exits into a less dense medium. When moving from denser to less dense, there is a special angle called the critical angle at which the refracted ray lies exactly along the surface. Beyond this angle, no light can cross the boundary and all of it reflects back: total internal reflection. This principle is what makes optical fibres work, allowing light to travel around bends without leaking out. This calculator solves Snell's Law for the refracted angle, calculates the critical angle (if the geometry allows for total internal reflection), and tells you whether total internal reflection is occurring. Common material presets are provided for the refractive indices.
Angles are measured from the normal to the surface. TIR only occurs when light travels from higher-n to lower-n medium above the critical angle.
How it works
Snell's Law: n1 × sin(θ1) = n2 × sin(θ2). Rearranging: sin(θ2) = n1 sin(θ1) / n2; then θ2 = arcsin(n1 sin(θ1) / n2). If the quantity n1 sin(θ1) / n2 exceeds 1, total internal reflection occurs and no refracted ray exists. Critical angle θc = arcsin(n2/n1) when n1 > n2; this angle only exists when going from a denser to a less dense medium. Speed ratio v1/v2 = n2/n1.
Worked example
Light travels through air (n1 = 1.0) and strikes a glass surface (n2 = 1.5) at an angle of incidence θ1 = 30 °. Using Snell's Law: sin(θ2) = 1.0 × sin(30°) / 1.5 = 0.5 / 1.5 = 0.3333. So θ2 = arcsin(0.3333) = 19.47 °. The light bends toward the normal as expected. The critical angle for this glass-to-air interface (n1=1.5, n2=1) is arcsin(1/1.5) = 41.81 °. These match the default values pre-filled above.
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