Snell Law Calculator
Refraction angle from two indices, with critical angle and TIR check.
What it does
Light bends at every boundary because speed changes between materials. A straw looks broken in a glass, lenses focus with curved faces, and diamonds sparkle through trapped rays, all on one equation relating the two indices and the two angles.
The tool above runs Snell law both ways. Enter the starting index, the landing index, and the incident angle from the normal to get the refracted angle, the critical angle where relevant, and which way the ray bends. Air to water at 30 degrees lands at 22.1 degrees toward the normal.
The law is textbook standard: MIT electromagnetics lectures derive n1 sin t1 = n2 sin t2 from phase matching at the boundary, and UT Austin lecture notes restate it from wavefront timing with the same sine ratio. Georgia State LibreTexts pages work the identical air water example to 1.33.
Angles need care. Optics measures from the perpendicular normal, not from the glass surface, and degrees here stay below 90. When light leaves dense glass for thin air past about 41 degrees, no refracted ray exists at all, which the tool reports as total internal reflection.
How to use
- Type the starting medium index n1.
- Type the second medium index n2.
- Type the incident angle in degrees from the normal.
- Read the refracted angle first.
- Read the critical angle line for dense to thin travel.
- Watch for the total internal reflection verdict past it.
Formula
- n1, n2
- Refractive indices of the two media.
- t1
- Incident angle from the normal in degrees.
- t2
- Refracted angle from the normal in degrees.
- tc
- Critical angle; beyond it no ray exits.
Entering denser glass bends rays toward the normal; exiting toward air bends them away until the sine would exceed 1, which is impossible and therefore reflects. Equal indices pass straight through. Dispersion (color dependent index) is ignored, so lasers match best.
Worked example
Example: air to water at 30 deg
Defaults n1 = 1.00, n2 = 1.33, angle 30 already filled in.
- Sine of incidence: sin 30 = 0.5.
- Scale: 1.00 x 0.5 / 1.33 = 0.37594.
- Invert: asin 0.37594 = 22.08, shown as 22.1 deg.
- Direction: n2 above n1, so toward the normal.
- Critical: none, since light enters denser medium.
- Tool shows 22.1 deg with no trapped ray.
Reference table
| Boundary | Refracted | Critical |
|---|---|---|
| Air (1.00) to water (1.33) | 22.1 deg | none |
| Air (1.00) to glass (1.52) | 19.2 deg | none |
| Glass (1.52) to air (1.00) | 49.5 deg | 41.1 deg |
| Water (1.33) to air (1.00) | 41.7 deg | 48.8 deg |
Limits
- Single wavelength assumed; prisms split colors this page merges.
- Flat boundary assumed; curved lenses need ray tracing.
- Isotropic media only; crystals split rays in two.
- Intensity split (Fresnel reflection fractions) not computed.
Common mistakes
- Measuring from the surface instead of the normal.
- Expecting a refracted ray past the critical angle.
- Using 1.33 for glass or 1.52 for water.
- Typing radians into a degree field.
- Forgetting the critical angle only exists dense to thin.
Try it yourself
Quick checks. Answers sit tucked under each question.
Glass (1.52) to air at 45 deg. Outcome?
1.52 x sin 45 = 1.0749 above 1: total internal reflection, critical 41.1 deg.
Water (1.33) to air at 20 deg. Refracted?
asin(1.33 x sin 20) = 27.1 deg, away from the normal.
Why does diamond sparkle?
Index near 2.42 traps rays by repeated total internal reflection before exit.
Questions
What is Snell law?
n1 sin t1 = n2 sin t2, linking bend angles to indices.
From where are angles measured?
From the normal line perpendicular to the surface.
What is the critical angle?
The incidence where refraction grazes 90 deg; past it all light reflects.
Why does light bend toward normal entering water?
Slower speed in water shortens the wavelength, swinging the wavefront inward.
Does color matter?
Slightly. Blue bends more than red; this page uses one index per medium.
Where is TIR used?
Fiber optics, binocular prisms, diamond cuts, and car rain sensors.