Engineering
Snell's Law refraction angle & Total Internal Reflection critical angle
Calculate refracted ray angles DEGREES(ASIN((n1/n2)*SIN(RADIANS(theta1)))) and fiber-optic Total Internal Reflection critical angles safely in degrees.
Convert degrees to radians AND guard ASIN against Total Internal Reflection!
For an ideal interface between transparent lossless media, Snell’s law is n1*sin(theta1)=n2*sin(theta2). Angles are measured from the surface normal, not from the surface. Enter positive refractive indices and an incidence angle from 0 to 90 degrees.
Total internal reflection occurs above the critical angle when n1>n2. At the critical angle the refracted ray is at 90 degrees. The displayed 41.81 degrees is rounded; do not use it as an exact boundary. The formula clamps numerical roundoff within 1e-12 of a transmitted sine of 1.
| Optical Ray Scenario | Formula | Output |
|---|---|---|
| Air (1.00) -> Crown Glass (1.50) at 30° | DEGREES(ASIN((1.0/1.5)*SIN(RADIANS(30)))) | 19.47° refracted |
| Glass (1.50) -> Air (1.00) Critical Angle | DEGREES(ASIN(1.00 / 1.50)) | 41.81° critical angle |
| Glass (1.50) -> Air (1.00) at 45° (> 41.81°) | (1.5/1.0)*SIN(45°) = 1.0607 > 1 | "Total Internal Reflection" |
Refracted angle in degrees with Total Internal Reflection guard
A2=n1 (1.00), B2=n2 (1.50), C2=IncidenceAngleDeg (30).
=IF(OR(COUNT(A2:C2)<>3,A2<=0,B2<=0,C2<0,C2>90),"Invalid inputs",LET(transmittedSine, (A2 / B2) * SIN(RADIANS(C2)), IF(transmittedSine > 1+1E-12, "Total Internal Reflection", ROUND(DEGREES(ASIN(MIN(1,transmittedSine))), 2))))Critical angle for Total Internal Reflection (when n1 > n2)
Returns the critical angle rounded to two decimals for n1>n2. At equality with the unrounded critical angle the ray is grazing; total internal reflection occurs above it.
=IF(OR(COUNT(A2:B2)<>2,A2<=0,B2<=0),"Invalid inputs",IF(A2 <= B2, "No TIR (n1 <= n2)", ROUND(DEGREES(ASIN(B2 / A2)), 2)))Try the example
Use the original values or change them to test your understanding.