Fresnel diffraction

In optics, the Fresnel diffraction equation for near-field diffraction is an approximation of the Kirchhoff–Fresnel diffraction that can be applied to the propagation of waves in the near field.^[1] It is used to calculate the diffraction pattern created by waves passing through an aperture or around an object, when viewed from relatively close to the object. In contrast the diffraction pattern in the far field region is given by the Fraunhofer diffraction equation.

The near field can be specified by the Fresnel number, F, of the optical arrangement. When ${\displaystyle F\ll 1}$ the diffracted wave is considered to be in the Fraunhofer field. However, the validity of the Fresnel diffraction integral is deduced by the approximations derived below. Specifically, the phase terms of third order and higher must be negligible, a condition that may be written as

$${\displaystyle {\frac {F\theta ^{2}}{4}}\ll 1,}$$

where ${\displaystyle \theta }$ is the maximal angle described by ${\displaystyle \theta \approx a/L,}$ a and L the same as in the definition of the Fresnel number. Hence this condition can be approximated as ${\textstyle {\frac {a^{4}}{4L^{3}\lambda }}\ll 1}$.

The multiple Fresnel diffraction at closely spaced periodical ridges (ridged mirror) causes the specular reflection; this effect can be used for atomic mirrors.^[2]