Total internal reflection critical angle calculator

This total internal reflection critical angle calculator finds the boundary angle at which a ray traveling from a higher-index medium toward a lower-index medium begins to reflect completely. Enter the refractive index of the incident medium as n1 and the refractive index of the second medium as n2. The calculator evaluates the inverse sine of n2 divided by n1, then reports the result in both degrees and radians. It also validates the physical requirement that n1 must be greater than n2, helping prevent a plausible-looking but impossible total internal reflection result.

Understand what the critical angle represents

Total internal reflection can occur when light travels from a medium with a higher refractive index into one with a lower refractive index. As the angle of incidence increases, the refracted ray bends farther away from the normal. At one particular incidence angle, the refracted ray reaches ninety degrees and runs along the boundary.

That incidence angle is the critical angle. At angles greater than it, the ideal ray no longer passes into the second medium and is reflected internally. This calculator uses n1 for the incident medium and n2 for the second medium, so n1 must be strictly greater than n2.

Both values are dimensionless refractive indices and should describe the materials at the same wavelength and under compatible conditions. The returned angle is measured from the normal to the interface, not from the surface itself. That distinction matters because an angle quoted from the surface is the complement of the conventional optics angle.

The output includes degrees for ordinary interpretation, radians for mathematical workflows, and the index ratio so you can audit the intermediate value. This is an ideal planar-interface calculation; it does not account for roughness, absorption, graded-index materials, or wavelength-dependent changes unless those effects are already represented in the indices you enter.

Apply the inverse-sine formula correctly

The calculation follows directly from Snell's law. At the critical condition, the refraction angle is ninety degrees, whose sine equals one. Rearranging the law gives sin(theta critical) = n2 / n1, and taking the inverse sine gives theta critical = asin(n2 / n1).

For example, an ideal glass-to-air boundary can be approximated with n1 equal to 1.5 and n2 equal to 1.0. The ratio is two thirds, and its inverse sine is about 0.729728 radians or 41.810315 degrees. Do not reverse the materials merely to make the input pass.

Instead, enter the actual direction of travel. If light travels from air into glass, total internal reflection does not occur at that interface direction, so there is no applicable critical angle even though the same two materials can support it in the reverse direction.

Use the result in experiments and applications

A critical-angle result is useful as a threshold, not as a complete ray trace. In a laboratory exercise, compare a measured onset of internal reflection with the calculated angle while keeping in mind the precision of the material indices and angle scale. In a prism check, confirm that the internal incidence angle is greater than the returned critical angle if the design depends on total internal reflection.

For an optical fibre, the same boundary concept helps explain confinement, although practical fibre calculations may also require core and cladding geometry, numerical aperture, launch conditions, and modal behavior. Because refractive index depends on wavelength and sometimes temperature, choose indices for the conditions that matter to your problem rather than treating a handbook value as universally exact. The calculator does not estimate those inputs.

It answers the narrower question reliably: given two valid indices in the correct order, what angle results from asin(n2 / n1)? Preserve enough precision in n1 and n2 for your intended comparison, and avoid implying more significant figures than the source measurements justify.