Reflexion and refraction
 |
When a wave arrives at the boundary between two media, part of it will be reflected back to the old medium (reflexion or reflection) and part of it will enter the new medium but propagate in an altered direction (refraction). These processes are much simpler to analyse if we represent the waves simply with their direction of propagation. We call these directional lines rays. For a quantitative description of these processes, we need to define the normal, which is a line that is perpendicular to the surface between the media at the point where the ray is incident. The angle of incidence (\(\theta_{1\mathrm{i}}\) in the figure) is the angle the incident ray makes with the normal; the angle of reflexion (\(\theta_{1\mathrm{r}}\)) and the angle of refraction (\(\theta_{2}\)) are the angles between the normal and the reflected and refracted rays, respectively. |
Law of reflexion
The incident ray, the normal and the reflected ray lie in the same plane. Furthermore, the angle of reflexion equals the angle of incidence:
\[\theta_{1\mathrm{r}} = \theta_{1\mathrm{i}}.\]
Law of refraction: Snell’s law
The incident ray, the normal and the refracted ray lie in the same plane. Furthermore, the ratio of the sine of the angle of refraction to the sine of the angle of incidence equals the ratio of the speed of propagation in the new medium to the speed of propagation in the old medium:
\[\frac{\sin\theta_{2}}{\sin\theta_{1\mathrm{i}}} = \frac{v_2}{v_1}\]
Refractive index
The refractive index \(n\) of a medium is the ratio of the speed of propagation of the wave in a reference medium (eg, vacuum in the case of light), \(c\), to the speed of propagation of the wave in the given medium, \(v\):
\[n := \frac{c}{v}.\]
The relative refractive index of medium 1 with respect to medium 2, \(n_{21},\) is defined as the ratio of the refractive indices of the two media:
\[n_{21} := \frac{n_2}{n_1}= \frac{c / v_2}{c / v_1} = \frac{v_1}{v_2}.\]
Media in which the wave propagates slower have greater refractive index.
Snell’s law using refractive indices
Snell's law can be stated using the refractive indices of the media:
\[\frac{\sin\theta_2}{\sin\theta_{1\mathrm{i}}} = \frac{v_2}{v_1} = \frac{c / n_2}{c / n_1} = \frac{n_1}{n_2} =:n_{21},\]
that is,
\[n_1 \sin\theta_{1\mathrm{i}} = n_2 \sin\theta_2.\]