Last updated: 2026-07-01

The critical angle

The critical angle is the angle of incidence at which light, travelling from a denser medium toward a rarer one, refracts exactly along the boundary between them. Push the angle of incidence any further and the light no longer escapes: it reflects entirely back into the denser medium. That phenomenon is called total internal reflection, and the critical angle is the threshold where it begins.

Where the formula comes from

Refraction is governed by Snell's law, which relates the angles a ray makes on either side of a boundary to the refractive indices of the two media:

Here $n_1$ is the index of the medium the light starts in and $n_2$ that of the medium it enters. As the angle of incidence $\theta_1$ grows, the refracted ray bends further from the normal. The largest possible refraction angle is $\theta_2 = 90°$, where the refracted ray runs along the surface. Setting $\theta_2 = 90°$ and $\sin 90° = 1$ defines the critical angle $\theta_c$:

which rearranges to

For this to give a real angle the ratio $n_2/n_1$ must be at most one, so the formula only makes sense when $n_1 > n_2$ — that is, when light moves from a denser medium into a rarer one.

Worked example

A ray of light inside a block of crown glass ($n_1 = 1.50$) approaches the boundary with air ($n_2 = 1.00$).

Any ray inside the glass that strikes the surface at more than 41.8° from the normal cannot leave; it is totally reflected. A ray at exactly 41.8° grazes along the surface, and a ray at less than 41.8° refracts out into the air.

Some common interfaces

The higher the index of the denser medium, the smaller the critical angle, and the more readily light is trapped inside.

Applications

Optical fibres. A fibre's core is made of glass with a slightly higher refractive index than the surrounding cladding. Light launched within the acceptance cone strikes the core–cladding boundary above the critical angle and reflects totally, zig-zagging down the fibre over kilometres with very little loss. This is the backbone of modern telecommunications and endoscopy.

Diamonds and gemstones. Diamond's exceptionally high index of about 2.42 gives a critical angle near 24°. Light entering a well-cut stone bounces internally many times before exiting through the top, producing the brilliance and fire that gemstones are prized for. Cutters choose facet angles specifically so that light is reflected internally rather than leaking out the back.

Prisms. Right-angle prisms in binoculars, periscopes, and single-lens reflex cameras use total internal reflection instead of silvered mirrors. Because the reflection is total, there is no loss of brightness and no coating to tarnish.

Refractometers. Instruments that measure the refractive index of a liquid — used to check sugar content in food or the concentration of a solution — observe the critical angle directly, since it depends on the unknown index.

Limits of the model

The relationship $\theta_c = \arcsin(n_2/n_1)$ assumes a sharp, flat interface between two transparent, non-absorbing media, and a single wavelength of light. Because refractive index varies slightly with wavelength, the critical angle differs a little from one colour to another. Absorbing materials such as metals have complex refractive indices, for which the simple formula does not apply. Real surfaces are never perfectly smooth, so a small amount of light can still escape through scattering even above the nominal critical angle.