Geometrically the only way for the wavelength to decrease but the wave to remain continuous is if the angle of propagation changes. Snell's law defines the relationship between the incident angle $\theta_1$ and the refracted angle $\theta_2$ (sometimes called the transmitted angle $\theta_t$).

*NOTE: All angles are measured with respect to the normal to the surface.*

$n_1 \sin{\theta_1} = n_2 \sin{\theta_2}$

With this relationship if light travels into a medium with a higher index of refraction (transmission), like from air into water, then $\theta_1>\theta_2$ and the light bends towards the normal. If the light instead traveled in the opposite direction, so from a higher to a lower index of refraction, the light would bend away from the normal.

Total Internal Refraction (TIR)

In the right geometry, with the right relationship of index of refraction, light can actually be forced to stay within a medium and not be refracted at all. Imagine traveling from a higher to a lower index material.

The refracted (transmitted) angle will always be larger than the incident. As $\theta_1$ increases and approaches 90º, $\theta_2$ will reach 90º before $\theta_1$ does. At the critical angle ($\theta_c$), the refracted angle is 90° and $\sin(90)=1$, so all the light is reflected and none of it is refracted.

if $n_1>n_2$, $\sin(\theta_{c}) = \frac{n_1}{n_2}$

Fiber optics, HAM radio waves that travel around the Earth, and Lightboards work on the principle of Total Internal Refraction.

Now, take a look at the pre-lecture reading and videos below.