Arago spot

Example

Let a plane wave $e^{ikz}$, with $k = 2\pi/\lambda$, fall normally on an opaque disk of radius $a$ in the plane $z = 0$. Take the field just behind the plane to be $1$ outside the disk and $0$ on it. Compute the field on the axis at distance $z$ with the Fresnel diffraction integral. In polar coordinates $\rho$ in the plane, $$u(0, 0, z) = \frac{e^{ikz}}{i \lambda z} \int_a^\infty e^{i k \rho^2/(2z)} \, 2\pi \rho \, d\rho = \frac{\pi e^{ikz}}{i \lambda z} \int_{a^2}^\infty e^{i k w/(2z)} \, dw,$$ with $w = \rho^2$. The last integral, taken as an Abel limit (with a factor $e^{-\varepsilon w}$, $\varepsilon \to 0^+$), equals $\frac{2 i z}{k} e^{i k a^2/(2z)}$. Since $\frac{\pi}{i\lambda z} \cdot \frac{2 i z}{k} = 1$, one gets $$u(0, 0, z) = e^{ikz} \, e^{i k a^2 / (2z)}, \qquad \abs{u}^2 = 1.$$ So the centre of the shadow is exactly as bright as the unobstructed wave. This bright spot in the middle of the shadow is the Arago spot.