Focusing of ultrashort pulse by small-aperture lenses

From the Helmholtz equation, the optical field analytical expression of an ultrashort pulse propagating through a small aperture lens was acquired with chromatic and spherical aberration. By numerical modulation of the analytical expression, the spatial and temporal distributions of intensity in the paraxial and marginal focal plane and on the axis were obtained, and the impact of spherical aberration on them was also summarized. Because of the interference among the pulses passing through the lens from center to edge, two pulses with apparently different peak values appeared both in the paraxial and in the marginal focal plane. The intensity distribution was basically symmetrical on the axis in the focal region. The peak value didn't appear in the paraxial or marginal focal plane, but at