content_image/437733.jpg
λ = 0 . 5 nm and d = 500 nm. The Talbot length in this case is a very large, z$_{T}$ = 10 6 nm = 1 mm. 

We see in Fig. 10 the Talbot carpets, which lose their forms as z increases. In the vicinity of the grating the Talbot carpets demonstrate a hard fractality - alternation of the caustics and lacu- nae is so dense, that they merge into a gray mass. In the limit λ/d → 0, in particular, the wave function is continuous everywhere, but differen- tiable nowhere [20]. If the wave function con- tains fractal features, its nondifferentiability for- bids a direct calculation of the trajectories. Tun- neling processes taking place in those cases mani- fest themselves as hopping events. Bohmian tra- jectories depicted in this figure show nearby the slits such a Brownian-like movement. As a parti- cle moves away from the grating, its behavior be- comes more predictable - it prefer to pass across the caustics and make the round of the lacunae, see trajectories pointed out by arrows a and b . 

Fig. 11 shows the same interference pattern, as shown in Fig. 10, but here it is extended up to z ≈ 20 z$_{T}$ . We can see, that the Talbot carpets exist only within a narrow strip by width ranging from z = 0 to about z = 4 z$_{T}$ . Next, they disap- pear, and instead of them a hexagonal packing of the interference pattern emerges. It occupies a zone from about z = 9 z$_{T}$ to z = 15 z$_{T}$ . Fi- nal stage is transition to the diffraction in the far-field zone, z > 16 z$_{T}$ . Bohmian trajectories drawn in the upper part of the figure behave by the Brownian-like manner in the vicinity of the slit grating and begin to diverge from each other as z increases. Two divergent trajecto- ries pointed out by arrows a and b demonstrate transition both across the narrow strip occupied by the Talbot-carpets and through the hexagon zone. In the both cases we see different behavior of particles, passing through these zones. In the first case, sharp zigzag behavior takes place. In the second case, we observe wavy behavior. As the particle passes into the far-field region the wavy trajectory become more smooth, until it turns into a straight line asymptotically. 

All three Berry’s conditions for emergence of the Talbot carpets shown above are satisfied. These conditions read [12, 21–23]: (a) paraxial beam; (b) arbitrary large number of slits; (c) ar- 

bitrary small ratio λ/d . In the limits N → ∞ and λ/d → 0 the Talbot carpet tends to fractal interference carpet. 

## V. SIMULATION OF THE FULLERENE MOLECULAR INTERFERENCE 

Atom and molecular interferometry [24] has an important significance in modern lithography technology. In this key, fullerene molecular in- terference experiment was recently presented in arXiv.org [11]. Heavy fullerene molecules C$_{60}$ by passing through a slit grating disclose inter- ference fringes in the near-field zone, Fig. 12. White dots against blue background visible in 

<img src='content_image/437733.jpg'>

Fig. 12(a) are images of fullerene molecules de- posited on a silicon plate. All dots are seen to form vertical strips. Summing over each narrow vertical band number of the dots we disclose in- terference fringe shown in Fig. 12(b). 

Experimental setup and parameters of the ex- periment are described in the article in detail. At simulating the fullerene molecular interference fringes we will use parameters given in this arti- cle. Mass of the fullerene molecule C$_{60}$ is equal to multiplication of the nucleon mass (6 protons and 6 neutrons) to 60 nuclei. That is, the mass 

