Phase Memory Preserving Harmonics from Abruptly Autofocusing Beams

We demonstrate both theoretically and experimentally that the harmonics from abruptly autofocusing ring-Airy beams present a surprising property: They preserve the phase distribution of the fundamental beam. Consequently, this “phase memory” imparts to the harmonics the abrupt autofocusing behavior, while, under certain conditions, their foci coincide in space with the one of the fundamental. Experiments agree well with our theoretical estimates and detailed numerical calculations. Our findings open the way for the use of such beams and their harmonics in strong field science.

Figure 1

Approximation of the ${\mathrm{Ai}}^2$ function.

Figure 2

Autofocusing positions of ring-Airy harmonics. Note that even orders exhibit multiple foci (vertical lines are guides to the eye to identify each harmonic order). Curves that follow the points are $f=2^j{f}_{\mathrm{Ai}}^{\omega }$ $(j=0,-1,\dots )$; the factors $2^j$ are indicated on each curve.

Figure 3

Focusing of a ring-Airy beam by a thin lens. As in the case of all Janus waves, two foci are observed [23].

Figure 4

Schematic representation of the experimental setup. The ring-Airy beam is generated at the FP of a lens after being phase modulated by a SLM. A type I BBO crystal generates the second harmonic. The fundamental and its second harmonic either propagate freely and autofocus (a) or are focused by a lens (b). The transverse profile of the beam is captured by a CCD camera at different distances along the propagation axis $z$.

Figure 5

Free propagating fundamental and second harmonic ring-Airy beams (a),(b) together with numerical simulations (c),(d).

Figure 6

Fundamental and second harmonic ring-Airy beams after being focused by a lens. (a),(b) Experimental results and (c),(d) numerical simulations.