Focusing Properties of Partially Coherent Controllable Dark-Hollow Beams through a Thin Lens

Analytical propagation formulas are derived for partially coherent controllable dark-hollow beams (CDHBs) through a thin lens based on the generalized Huygens–Fresnel integral. The expressions of the position for maximum irradiance on-axis and the relative focal shift are evaluated by the analytical propagation formulas. Our numerical results show that both the relative focal shift and the effective beam width of focused partially coherent CDHBs are mainly determined by the initial transverse coherence width δ_g and the Fresnel number N_w, which are also affected by the changes of both the dark-size adjusting parameter p and the order N of CDHBs.