Dual behavior of caustic optical beams facing obstacles

A full propagation analysis on both fold-type and cusp-type caustic optical beams under various setups of obstructions is theoretically and experimentally performed. It is demonstrated that the self-healing property of caustic optical beams that include the famous Airy beam is a quite relative property. In fact, fold-type and cusp-type beams cannot only behave as self-healing beams by blocking the main intensity peak, but also behave as self-breaking ones in a nonintuitive manner: by blocking a lateral side of the beam without touching the central intensity peak. The regeneration and rupture processes of caustic beams follow a nonlocal propagation dynamic unlike the other conventional beams. Moreover, deep differences between fold and cusp caustic beams are pointed out once facing certain obstructions. The cusp-caustic beam can be broken down by the obstacle placed in a dark zone outside the caustic region, while the fold-type one remains unaltered. This beam rupture confirms the key role of a hidden propagating field in the shadow region for cusp beams that coexist with the evanescent one. The obtained results cast down the established idea that the Airy beam is a robust self-healing beam since any caustic beam can behave in a dual manner depending on the obstruction location. These facts open up different perspectives for the applications in which the self-healing properties of the beam are relevant.

Figure 1

(${\mathrm{a}}_1$) Intensity distribution normalized to its maximum value, $I/I_{\max }$, vs spatial coordinates $s$ and $\xi$ for AB. The intensity scale [0,1] corresponds to $[0,I/I_{max}]$. (${\mathrm{a}}_2$) Spatial ray distribution for AB [orange rays (those coming from the left) for $K>0$ and green rays (those coming from the right) for $K<0$]. (${\mathrm{b}}_1$),(${\mathrm{b}}_2$) Same, but for SAB. White and black curves in the undulatory and geometric representations, respectively, are the caustic curves that are tangent to each of the rays in one point.

Figure 2

$I/I_{max}$ and ray distribution vs $(s,\xi )$ in the presence of an obstacle placed at the Fourier plane ($\xi =0$) partially blocking the main lobe in the interval $-1\le s\le 1$, (a) for AB and (b) for SAB. Notice that the blocking limit rays [(${\mathrm{a}}_1$),(${\mathrm{b}}_1$) (orange and green) thin lines and (${\mathrm{a}}_2$),(${\mathrm{b}}_2$) (orange and green) thick lines] delimit the caustic curve cut off by the obstacle [(white and black) curved lines in undulatory and geometric configurations, respectively].

Figure 3

$I/I_{max}$ and ray distribution vs $(s,\xi )$ in the presence of an obstacle placed at the Fourier plane ($\xi =0$) blocking the left lateral sideways of the beam in the interval $-10\le s\le -2$, (a) for AB and (b) for SAB. Unlike Fig. 2, only the blocking limit rays for $K>0$ delimit the caustic curves cut off by the obstacle [those coming from the left side going to the right side represented by (${\mathrm{a}}_1$),(${\mathrm{b}}_1$) (orange) thin lines and (${\mathrm{a}}_2$),(${\mathrm{b}}_2$) (orange) thick lines].

Figure 4

Experimental normalized intensity distribution ($I/I_{max}$), in terms of the transverse and longitudinal spatial coordinates, $x$ and $z$, from the Fourier plane ($z=0$) for (a) AB representing a fold-caustic beam and (b) SAB representing a cusp-caustic beam. (${\mathrm{a}}_1$),(${\mathrm{b}}_1$) Obstacle-free propagation. (${\mathrm{a}}_2$),(${\mathrm{b}}_2$) Propagation in the presence of an obstacle situated at the beam center. (${\mathrm{a}}_3$),(${\mathrm{b}}_3$) Propagation in the presence of an obstacle blocking the left lateral side of the beams. The red (thick) segment on the Fourier plane indicates the location and size of the obstacle.