Second-harmonic generation of 2D materials excited by the Laguerre-Gaussian beam

We theoretically study the second-harmonic generation (SHG) of two-dimensional (2D) materials excited by a Laguerre-Gaussian (LG) beam at normal incidence and provide a method to distinguish SHG induced by the electric dipole (ED) interaction and SHG induced by the electric quadrupole and magnetic dipole (EQ-MD) interaction by their different dependence on the LG beam parameters, including the effective spot area v02 and the order of orbital angular momentum (OAM) m. In an approximation of neglecting reflection and taking a beam radius to infinity, the intensity of the ED induced SHG is proportional to F _m/v02 with F_m = 2^-2|m|(2|m|)!/(π(|m|!)²), while the EQ-MD induced one is proportional to (4|m|+2)F _m/v04. An in-plane isotropic substrate can strongly affect the signal amplitude but slightly change the v₀ and m dependence. Our results provide an all-optical way to detect the OAM by SHG, as well as a theoretical basis for studying the EQ-MD induced SHG by the LG beams.