In recent years, photon orbital angular momentum, as a new degree of freedom, has attracted wide attention. The Laguerre-Gaussian (LG) beam is a commonly used orbital-angular-momentum (OAM) beam and is widely studied. Among a large amount of research, the mode conversion of LG beam attracts much attention from researchers. It is found that an LG beam can be converted into a Hermite-Gaussian (HG) beam with the help of cylindrical lenses. In addition, the nonlinear beam shaping technique can change the condition of phase matching in the nonlinear process to realize wave-front modulation, and harmonic waves with shaped wave-fronts can be achieved by this technique during nonlinear optical processes. By combining these hot topics, we propose a novel mode conversion method for special beams in nonlinear processes, and the local quasi-phase-matching (LQPM) theory is employed to design the required optical superlattice (OSL). The function of mode converters is concentrated on the superlattice structure, and the phase-matching condition is satisfied in the nonlinear mode conversion.

LQPM is used to design the OSL structure required by the nonlinear mode conversion process. Different from the conventional quasi-phase-matching (QPM) method in the reciprocal space, the LQPM is a nonlinear beam manipulation theory in real space. According to the principle of LQPM, the OSL structure function for the nonlinear mode conversion of LG beams can be obtained. The domain structure shows a curved boundary, and the function of cylindrical lenses is integrated into the OSL. Matlab software is used to carry out numerical simulations for the nonlinear mode conversion process. The finite-difference method is employed in the numerical simulation process to calculate the field distribution in the two-dimensional OSL. The field distribution and intensity curves of the second-harmonic wave after nonlinear mode conversion are obtained. It can be found that the designed OSL structure shows positive results on both mode conversion and frequency doubling.

In this paper, we propose an OSL structure that can realize the nonlinear mode conversion of the LG beam, and the curved domain structure is designed based on the LQPM theory. After analyzing the forms of the isophase planes of the structure, two different modes which we called positive mode and negative mode can be obtained (Fig. 1). It is found that both modes can be used to achieve nonlinear mode conversion under the phase-matching condition, but the positive mode has better effect and can achieve perfect mode conversion. The LG beam with p=0 is taken as the fundamental wave to perform functional simulation on the designed OSL, and the converted HG beam image is observed as expected (Fig. 5). The relationship between the LG beam and the HG beam mode index is verified by observing the light and dark fringes. According to the above theoretical calculation and simulation results, it can be seen that the designed OSL integrates the functions of mode conversion and frequency doubling effect at the same time, which can be used to obtain higher-order vortex beams and make the device more compact. This study is expected to promote the research on nonlinear mode converters.

In this paper, we propose a nonlinear mode conversion method based on OSL which can realize both second-harmonic generation and mode conversion in a single device. Compared with the previous mode conversion methods based on cylindrical lenses, in our method, not only the mode conversion function of the cylindrical lens is concentrated on the optical superlattice, but also the nonlinear effect of frequency doubling is realized. In addition, the original linear conversion process is replaced by a nonlinear process, and the phase mismatch is compensated by the LQPM theory. According to the field distribution image of the HG beam obtained by the numerical simulation, it is verified that the optical superlattice can realize the function of nonlinear mode conversion. In a word, a multifunctional OSL is designed, which can realize the nonlinear mode conversion between LG beams and HG beams. Through this study, the beam mode of the fundamental wave can be converted into another form in a nonlinear process with high efficiency, which has an important role in optical communication and other fields.