When light travels from a medium with larger refractive index to one with smaller refractive index, the refracted ray bends away from the normal. Beyond a certain incident angle, Snell's law has no transmitted-ray solution. The result is total internal reflection.

Critical angle

Let $n_1>n_2$. At the critical angle $\theta_c$, the refracted ray runs along the interface:

$$\theta_2=90^\circ.$$

Snell's law gives

$$n_1\sin\theta_c=n_2,$$

so

$$\sin\theta_c=\frac{n_2}{n_1}.$$

Above the critical angle

For

$$\theta_1>\theta_c,$$

no propagating transmitted ray exists in the lower-index medium and the wave is reflected back into the first medium.

Conditions

Total internal reflection requires both:

• propagation from higher to lower refractive index;
• incidence greater than the critical angle.

It cannot occur for light entering a higher-index medium from a lower-index one.

Applications

Optical fibres guide light through repeated total internal reflection. Prisms can also use the effect to redirect light with very high efficiency.