Intermediate · 13 min
Calculating refraction angles
Apply Snell's law using angles measured from the normal.
Relate the two angles
Snell's law connects a ray crossing an interface between two transparent media: n₁ × sin θ₁ = n₂ × sin θ₂. Both angles are measured from the normal, not the surface. The n values are refractive indices. If the second index is larger, the refracted angle is smaller for an oblique ray. Use degree mode when entering angles in degrees on a calculator.
Worked example
If n₁ = 1, n₂ = 1.5, and sin θ₁ = 0.6: sin θ₂ = 1 × 0.6 ÷ 1.5 = 0.4
Check whether a refracted ray exists
A sine value for a real angle must lie between −1 and 1. For light going from a higher index to a lower index, a sufficiently large incidence angle would require sin θ₂ greater than 1. There is then no propagating refracted ray in the simple ray model: total internal reflection occurs. At the critical angle, the refracted ray lies along the interface. Reflection can also occur alongside refraction below that angle.
Worked example
For n₁ = 1.5 and n₂ = 1, an incident sine of 0.8 would require sin θ₂ = 1.2. That is impossible for a real angle, indicating total internal reflection.
Check a refraction result
- For light entering index 1.5 from index 1 with incident sine 0.75, calculate the refracted sine.
- Reverse the media and keep the incident sine at 0.75. Calculate the value Snell's law would require and explain whether a propagating refracted ray is possible.