Oh, and note the math used in analyzing the Talbot effect:

Lord Rayleigh showed that the Talbot Effect was a natural consequence of Fresnel Diffraction and that the Talbot Length can be derived using the following formula[2]:
z_T= 2a²/ λ
where a is the period of the diffraction grating and λ is the wavelength of the light incident on the grating.

"The Atomic Talbot Effect

Due to the quantum mechanical wave nature of particles, diffraction effects have also been observed with atoms. Chapman et al.[3] carried out an experiment in which a collimated beam of sodium atoms was passed through two diffraction gratings, the second used as a mask, to observe the Talbot effect and measure the Talbot length. The beam had a mean velocity of 1000 m/s corresponding to a de Broglie wavelength of λdB = 0.017 nm. Their experiment was performed with 200 and 300 nm gratings which yielded Talbot lengths of 4.7 and 10.6 mm respectively. This showed that for an atomic beam of constant velocity, by using λdB, the atomic Talbot length can be found in the same way in as the optical case."

Note that your integral has no bearing on either variant, sir. Ptui, redux.

A