Critical Angle and Total Internal Reflection Calculator

Find the critical angle between two media and see whether a ray refracts or is trapped by total internal reflection.

What it does

Send light into water from below and, past a certain angle, it refuses to leave. The light is trapped. That is total internal reflection, and it is the reason an optical fibre can carry a signal around a corner, the reason a diamond sparkles, and the reason a layer of hot air over a road can throw a mirage at you.

OpenStax describes the mechanism. The critical angle is defined as the incident angle that produces a refraction angle of 90 degrees, and if the incident angle is greater than that, all of the light is reflected back into the first medium. The condition can only be met when the second medium has a lower index of refraction than the first, because the light has to speed up on the way out.

Putting the critical angle in terms of Snell's law gives a short result: the sine of the critical angle is the ratio of the two indices. The OpenStax text works through the consequences with real numbers, noting that the critical angle for a ray going from water to air is 48.6 degrees while the one from diamond to air is only 24.4 degrees, and that a flint glass to crown glass interface gives 66.3 degrees. A large index of refraction means a small critical angle, which is why diamonds are so good at holding light inside.

This page computes the critical angle for any pair of media and then applies Snell's law to the angle of incidence you enter. It tells you whether the ray refracts, where it goes if it does, how much of the light is reflected, and whether you are above or below the critical angle.

The calculation is for a clean flat interface. In a real fibre the core and cladding have slightly different indices, the ray bounces along the core, and the light that is not guided leaks out at bends and where the fibre is not perfectly straight. The angle of acceptance of a real fibre is smaller than the simple critical angle suggests, because the light also has to travel down the fibre at an angle.

How to use

1. Pick the medium the light starts in. It has to be the denser one for total internal reflection to be possible.
2. Pick the medium the light would enter.
3. Enter the angle of incidence, measured from the normal.
4. Read the critical angle for the pair of media.
5. Read the outcome. Above the critical angle the ray is trapped and the reflectance is 100 %.
6. Below the critical angle, read the refraction angle and the reflected fraction, which rises as the angle approaches the critical one.

Formula

theta c

Critical angle of incidence, in degrees.

n1

Refractive index of the medium the light starts in.

n2

Refractive index of the medium the light would enter.

theta i

Angle of incidence, measured from the normal.

theta t

Angle of refraction, measured from the normal.

There is no critical angle when the light is going from a less dense medium into a denser one, because the refraction angle can never reach 90 degrees. The formula then has no solution, and the page says so rather than printing a number. As the incidence angle approaches the critical angle from below, the reflectance rises steeply towards 100 % and the transmitted ray runs along the surface.

Worked example

Reference table

Limits

• The calculation assumes a smooth flat interface with no absorption and no scattering in either medium.
• It describes a single ray, while a real fibre guides a spread of angles and loses light at bends and at joints.
• The wavelength of the light is fixed at the usual indices, and the index of a material shifts a little with wavelength.
• It does not cover the case where the interface is only a thin film, where a frustrated total internal reflection can still leak light across.

Common mistakes

• Measuring the angle from the surface rather than from the normal. The critical angle is quoted from the normal.
• Looking for a critical angle when the light goes from air into water. There is none, because the light slows down and the refraction angle can never reach 90 degrees.
• Swapping the two media. The formula has the denser medium on top, and the answer changes completely if you swap them.
• Forgetting that the critical angle falls as the index of refraction rises. Diamond holds light in far better than glass does.
• Assuming a fibre transmits every ray that is under the critical angle. Only rays close to the axis stay guided for any distance.

Try it yourself

Quick checks. Answers sit tucked under each question.

Questions

Sources

• OpenStax University Physics Volume 3, 1.4: total internal reflection and the critical angle
• OpenStax College Physics 2e, 25.4: total internal reflection
• Physics LibreTexts: critical angle, diamond and flint glass examples

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