Observation of spatial self-phase modulation excited by off-axis integer and fractional vortex beams

Spatial self-phase modulation (SSPM) is widely used to characterize the nonlinear refractive index of two-dimensional layered materials and design passive nonlinear photonic devices. In this work, we investigate the SSPM phenomena of black phosphorus dispersed in agarose solid under the excitation of on-axis and off-axis integer and half-integer vortex beams. It is found that with the increase of the off-axis distance of the Gaussian vortex beam, the pattern of the far-field self-diffraction intensity gradually breaks from the initial ring and finally evolves into a tail outside the self-diffraction ring. Moreover, for half-integer vortex beams, the size of the innermost tail of the self-diffraction ring is nearly half smaller than that of the second outer tail. The underlying mechanism of the observed SSPM phenomenon is also analyzed. Interestingly, the trailing number and rotation direction of the self-diffraction ring pattern quantitatively reflect the magnitude and sign of the topological charge (TC) of the off-axis vortex beam, respectively. This work provides a novel method to measure the TC of off-axis vortex beam in the nonlinear optics regime.