Huygens–Fresnel principle

Remarks

Huygens proposed that each point of a wavefront emits a wavelet and that the new wavefront is their envelope. Fresnel added that the wavelets interfere, with their phases $k r$, and this explains diffraction fringes. The principle in this form is heuristic; Kirchhoff's diffraction formula derives it from the Helmholtz equation and gives $K(\chi) = (1 + \cos\chi)/2$ when $S$ is a spherical wavefront from a point source. The factor $1/(i\lambda)$ makes the formula consistent: applied to a plane wave and a plane $S$, it reproduces the plane wave at $P$, by the method of stationary phase. For the sharp form of Huygens' principle for the wave equation in odd dimensions, see Huygens principle.

Going deeper

For a student

In diffraction problems one takes $S$ to be the plane of an aperture and assumes that $u$ on the aperture equals the incident wave. The integral then becomes the starting point of the Fresnel and Fraunhofer approximations, which simplify the phase $k r$ for distant observation points. Kirchhoff's integral theorem and the Rayleigh–Sommerfeld formulas are the rigorous versions of the principle.