Total Internal Reflection and the Critical Angle: Definition and Formula

Light moving from one material into another usually refracts: it changes direction, but still makes it into the new medium. Past a certain point, though, that stops happening.

The exact point where that limit is reached is called the critical angle, or limiting angle: it’s the angle of incidence that makes the refracted ray exit at 90° from the normal, meaning it skims along the boundary between the two media, according to HyperPhysics. Once that value is exceeded, outward refraction stops completely: all the light stays inside the original medium, reflected as if by a mirror. This is total internal reflection. The value is calculated with Snell’s law, applied to the limiting case of a ray exiting at 90°: what matters is the ratio between the refractive index of the less dense medium and that of the denser one, as College Physics (OpenStax) explains.

The two known values show how much the phenomenon depends on the material. Light crossing from water into air hits its critical angle at 48.6°. For diamond, with a much higher refractive index (2.417, according to HyperPhysics), it drops to just 24.4°: the window of angles at which light can still get out is far narrower.

The refractive index values are approximate and don’t account for dispersion, meaning the small variation that depends on the wavelength of light, HyperPhysics notes. For glass, the sources used here don’t report a critical angle: only the water and diamond values remain.

Practical example: a cut diamond relies exactly on this difference. With a critical angle of only 24.4°, most of the light that enters the stone hits its inner faces at an angle above the critical one and gets sent back inside instead of passing through: according to College Physics (OpenStax), it’s the combination of this very low critical angle and the very high refractive index that explains why diamonds sparkle more than other transparent materials.

The same physics, put to a different use, is behind optical fibers. Their operation relies on the total internal reflection of the rays traveling inside them, HyperPhysics explains: a ray that enters at the right angle keeps reflecting off the fiber’s inner walls with almost no loss, covering long distances this way. Bundles of optical fibers make it possible to get images from otherwise inaccessible places, as in endoscopes: each individual fiber works as a light conduit, carrying the light that corresponds to a single point of the image along its own path. The same principle shows up in the polarizing prisms inside optical instruments too, again according to HyperPhysics.

There’s a mistaken use of this principle that’s just as common as the correct one: explaining mirages, such as the puddle that seems to appear on hot asphalt, as a case of total internal reflection. According to the Department of Astronomy at San Diego State University, that idea is wrong: light bends gradually as it passes through layers of air that get less dense going upward, not through a single bounce off a surface.

The Recap on light refraction covers the general case of this same phenomenon. Here, past that same critical angle, the ray simply has nowhere left to go, so it keeps bouncing inside the medium it started in.