Spatial shaping of the second harmonic (SH) beams generated in a non-linear crystal has been explored in the past few years using several different methods. For example, a nonlinear computer generated hologram can be realized by modulating the quadratic non-linear susceptibility of the crystal itself [1-3]. It is also possible to encode linear holographic patterns at the exit facet of the crystal [4]. However, these methods require special fabrication procedures and in addition each hologram is fixed and is suitable for generating only one specific beam shape. Moreover, owing to the fabrication limitations, the nonlinear computer generated hologram is mainly suitable for one-dimensional shaping. Here we present a new method that circumvents these limitations: The phase profile of the input fundamental beam is dynamically shaped using a spatial light modulator (SLM), thereby generating a SH beam with the desired profile in the far field.

Although the non-linear process is known, there is no solution to the inverse problem. Nevertheless, if the nonlinear crystal is sufficiently short, diffraction is negligible so the phase profile of the SH beam is nearly twice that of the input fundamental beam. This method essentially imprints the phase profile of the SLM (multiplied by 2) on the SH beam, thereby enabling arbitrary two-dimensional shaping of the generated beam at the far field.

For longer crystals, which provide higher conversion efficiency, this approximation no longer holds. However, a genetic algorithm can be utilized in order to solve the inverse problem and determine the desired input fundamental phase profile for arbitrary SH beam shapes.

We experimentally confirmed both methods, by frequency doubling of a 1550 nm pump laser in short (1mm) and long (12 mm) PPKTP crystals. Different SH beams were generated – including two-dimensional Airy beams and high order Hermite Gauss beams – HG01, HG10, HG11, HG02 and HG20. We also generated SH beams with a quadratic phase profile, which acted as lenses, with SLM-controlled focal distances in the range of 20-80mm. In addition, we have determined, through numerical simulations, guidelines for deciding whether the "short crystal" method can be used. The method we presented enables to dynamically shape a beam in a nonlinear process, using standard crystals and optical equipment, and without the need to use any optical element after the nonlinear crystal.

[1] A. Shapira, R. Shiloh, I. Juwiler and A. Arie “Two-dimensional nonlinear beam shaping”, Optics Letters 37, 2136 (2012)

[2] T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz and A. Arie “Nonlinear generation and manipulation of Airy beams”, Nature Photonics 3, 395 (2009)

[3] A. Shapira, I. Juwiler and A. Arie "Nonlinear computer generated holograms", Optics Letters 36, 3015 (2011)

[4] A. Shapira, A. Libster, Y. Lilach and A. Arie “Functional facets for nonlinear crystals”, Optics Communications 300, 244 (2013)