Snell's Law Calculator
Snell's Law describes how a ray of light bends when it crosses the boundary between two transparent media with different refractive indices. This calculator solves for the angle of refraction given the index of the first medium, the index of the second medium, and the angle of incidence measured from the normal. It also flags total internal reflection and reports the critical angle whenever light moves toward a less dense medium, so you can model fiber optics, lenses, and water surfaces with confidence.
Formula
n1·sin(theta1) = n2·sin(theta2); theta2 = arcsin(n1·sin(theta1) / n2); theta_c = arcsin(n2 / n1)
- n1
- Refractive index of the incident medium
- n2
- Refractive index of the second medium
- theta1
- Angle of incidence from the normal (degrees)
- theta2
- Angle of refraction from the normal (degrees)
- theta_c
- Critical angle, defined only when n1 > n2
How it works
- Enter the refractive index of the incident medium (n1), the refractive index of the second medium (n2), and the angle of incidence in degrees measured from the surface normal.
- The calculator applies Snell's Law, n1·sin(theta1) = n2·sin(theta2), and solves for the refraction angle as theta2 = arcsin(n1·sin(theta1) / n2).
- When n1 is greater than n2, it also computes the critical angle, arcsin(n2 / n1). If the incident angle reaches or exceeds that critical angle the ratio exceeds one, arcsin is undefined, and the result is reported as total internal reflection instead of a refraction angle.
Worked example
Light passes from air (n1 = 1.00) into water (n2 = 1.33) at an incident angle of 30 degrees.
- sin(30 degrees) = 0.5, so n1·sin(theta1) = 1.00 × 0.5 = 0.5.
- Divide by n2: 0.5 / 1.33 = 0.37594.
- theta2 = arcsin(0.37594) = 22.0826 degrees.
The ray refracts to 22.0826 degrees, bending toward the normal because water is denser than air. There is no critical angle here since n1 < n2.
Frequently asked questions
- What is total internal reflection and when does it happen?
- Total internal reflection occurs when light travels from a denser medium to a less dense one (n1 > n2) and strikes the boundary at or beyond the critical angle. At that point all the light reflects back into the first medium instead of refracting, because arcsin(n1·sin(theta1) / n2) has no solution. The critical angle is arcsin(n2 / n1); for water to air it is about 48.75 degrees, which is why fiber optics can trap light along a glass core.
- How should I measure the angle of incidence?
- The angle of incidence is measured between the incoming ray and the normal, an imaginary line perpendicular to the boundary surface, not from the surface itself. A ray hitting the surface straight on has an angle of incidence of 0 degrees and passes through without bending, while a grazing ray approaches 90 degrees.
- Why does light bend toward or away from the normal?
- Light bends toward the normal when it enters a denser medium (n2 > n1) because it slows down, and away from the normal when it enters a less dense medium. The refractive index encodes how much a medium slows light relative to a vacuum, so a larger index difference produces a larger bend.