Optics Wave Optics Huygens Principle
** Huygens Principle**
 Key Idea: Each point on a wavefront acts as a new source of secondary waves, and these waves interfere to produce the new wavefront.
Visualization:
Imagine dropping a pebble in a calm pond. The waves that spread out from the point of impact are secondary waves. Each point on the wavefront acts as a new source of secondary waves, which spread out in all directions. The envelope of these secondary waves gives the new wavefront.
O O O
/ \ / \ / \
/ \ / \ / \
/_____\_/_____\_/_____\_
O O
Mathematical Description:
The Huygens principle can be mathematically described by the FresnelKirchhoff integral, which gives the amplitude (U(P,t)) of a wave at a point (P) in terms of the amplitude (U(Q)) of the wave at all points (Q) on a surface (S): $$U(P,t)=\frac{1}{4\pi}\iint\limits_S \frac{U(\mathbf{Q},t_r)}{r}\cos(\mathbf{n},\hat{\mathbf{r}})d\sigma$$
where

(d\sigma) is an area element of (S) (\mathbf{r}= \overrightarrow{QP}) is the vector from (Q) to (P) (\hat{\mathbf{r}}= \frac{\overrightarrow{QP}}{\overrightarrow{QP}}) is the unit vector in the direction of (r) (t_r= t \frac{r}{v}) is the retarded time (v) is the speed of the wave

(r=\overrightarrow{QP}) is the distance from (Q) to (P).

Applications:

The Huygens principle can explain the laws of reflection and refraction.

The Huygens principle can also explain the phenomena of diffraction and interference.