Fraunhofer diffraction

Definition

Let $u_0(\xi, \eta)$ be the complex amplitude of a monochromatic field in the plane $z = 0$, with wavelength $\lambda$ and wavenumber $k = 2\pi/\lambda$. Assume $u_0$ is supported in the disk $\xi^2 + \eta^2 \le a^2$. Assume the observation distance satisfies $z \gg k a^2 / 2$, so that the Fresnel number $a^2/(\lambda z)$ is much smaller than $1$. Then the factor $e^{i k (\xi^2 + \eta^2)/(2z)}$ in the Fresnel diffraction integral is close to $1$, and the field is $$u(x, y, z) = \frac{e^{i k z} \, e^{i k (x^2 + y^2)/(2z)}}{i \lambda z} \, \widehat{u_0}\left( \frac{x}{\lambda z}, \frac{y}{\lambda z} \right),$$ where $\widehat{u_0}(f_x, f_y) = \iint u_0(\xi, \eta) e^{-2\pi i (f_x \xi + f_y \eta)} d\xi \, d\eta$ is the Fourier transform. Fraunhofer diffraction is the diffraction pattern in this regime; its intensity is $\abs{\widehat{u_0}(x/(\lambda z), y/(\lambda z))}^2 / (\lambda z)^2$.