The nonlinear propagation of cylindrical vector beams in an optical fiber

•The VFE-BPM has been used to simulate the nonlinear evolution of CVBs.•The CVBs tend to break up into more filaments with higher input power.•The polarization of each filaments is basically linear.•Stability analysis by the approximate analytic stability theory. We simulate the evolution of cylindrical vector beams (CVBs) in a Kerr nonlinear medium by the full-vector finite-element beam propagation method (VFE-BPM). This beams show poor transverse stability which contrasts with the strong transverse stability of fundamental modes. The approximate analytic theory has been used to qualitative analyze the variations of the numerically calculated instability growth rates. We present a reasonable concentric circle model to avoid the influence of triangular grids during simulation.