Talbot effect

Remarks

Check: in the Fresnel approximation the order $n$ is multiplied by $e^{ikz} e^{-i \pi \lambda z n^2 / d^2}$, so $u(x, z) = e^{ikz} \sum_n c_n e^{-i \pi \lambda z n^2 / d^2} e^{2\pi i n x / d}$. At $z = z_T$ the extra phase is $e^{-2 \pi i n^2} = 1$ for every $n$, which gives the self-image. At $z = z_T / 2$ it is $e^{-i\pi n^2} = (-1)^n$, so $u(x, z_T/2) = e^{ikz} t(x + d/2)$: the image is shifted by half a period. Non-example: in exact wave optics the order $n$ propagates with the phase $z \sqrt{k^2 - (2\pi n / d)^2}$, which is not quadratic in $n$. The self-images are then only approximate. They are good when $d \gg \lambda$, which is where the paraxial approximation holds.