Why light bends towards the normal when it travels from rarer to denser medium. Explain with diagram.

The light propagation is described, in the simplest case, by Asin(k⃗ r⃗ −2πνt)$Asin(\overrightarrow{k}\overrightarrow{r}-2\pi \nu t)$. The quantity ϕ=k⃗ r⃗ −2πνt $\varphi =\overrightarrow{k}\overrightarrow{r}-2\pi \nu t$ is called phase, and a surface on which at a given time t$t$ the phase is constant, is called wave-front. The phase-velocity is the ratio of the distance between two neighbor wave-fronts of the same phase ϕ$\varphi$, and the time T$T$ needed for the light to travel a distance equal to the distance d$d$between two neighbor wave-fronts. This time is given by

Note that the light frequency ν$\nu$ doesn't change from medium to medium, therefore T$T$ doesn't change.

For illustration of the situation imagine the following scenario:

Consider a front wave of phase ϕ$\varphi$ (red line) touching at the time t0$t_0$ the point A$A$ on the separation surface between the two media, then at a time t1$t_1$ the point B$B$ of the front wave touches a point B1$B_1$ of the separation surface, and at a time t2$t_2$ the point C$C$ of the front wave touches the point C2$C_2$ of the separation surface. Let the points A,B,C$A,B,C$ be chosen so as t1=t0+T$t_1=t_0+T$, and t2=t1+T$t_2=t_1+T$. As the velocity of the wave is smaller in M2$M_2$,

the distance that the wave can travel during the time T$T$ is smaller,

(hence the distance between two consecutive wave-fronts is smaller.)

In consequence, the angle θ$\theta$ of the wave-front with the separation surface is smaller in M2$M_2$ than in M1$M_1$. Finally, note that the angle between the front wave in M2$M_2$ and the separation surface is exactly equal to the refraction angle (i.e. between the normal to the separation surface and the normal to the wave-front).