The coherent, lensless self-imaging of light fields was discovered by Henry Fox Talbot in 1936:
When a plane monochromatic wave illuminates a spatially periodic structure, the transmission function of this grating is reproduced as a light-field pattern further downstream, even without any imaging optics.
This phenomenon is a direct consequence of the wave nature of light but not limited to the optical domain.
It has gained technological relevance in all fields where refractive and reflective optics is a challenge, such as X-ray imaging or in particular also matter-wave imaging.
The idea behind the experiments can be easily demonstrated in a low-cost high-school experiment, illustrated in Figure 1: an expanded parallel, monochromatic laser beam is directed from left to rigth and hits a chromium plated glass grating with a period of 200 µm. The transmitted field distribution can be collected by a lens and a web cam. When this detector arrangement is displaced longitudinally, the web cam image retrieves the Talbot Carpet shown in green.
You can follow the self-imaging which occurs at multiples of the Talbot-length, L=d2/λ.
Not all light sources exhibit sufficient spatial coherence to allow observing the Talbot-Effect.
Coherence can, however, be established by virtue of single-slit diffraction in an array illuminator, as first discussed by E. Lau in 1948 
In this arrangement it is sufficient to use a relatively well monochromatic wave without initial spatial coherence. In a high-school experiment this may, for instance, be a simple sodium lamp.
While a single grating prepares coherence, it does not yet produce a intensity pattern on a screen.
This can be established by adding a second grating, at a multiple of the Talbot distance and the detector symmetrically in the same distance further downstream.
Again, self-imaging can be observed. This trick, first proposed for matter-waves by John Clauser  has become the basis for all near-field matter-wave interferometers run by the QNP group at the University of Vienna[3-5].
It has also gained recent technological relevance in other fields where refractive and reflective optics is a challenge, such as X-ray imaging.