Ultrashort pulse focusing through a planar interface between free space and a nonlinear medium

We present a method to calculate the spatiotemporal electric field distribution of ultrashort pulses focused by an aberration-free lens through a planar interface between free space and a nonlinear medium. The method combines the Fresnel diffraction integral, which is used to model the propagation of the focused pulse in free space, and the angular spectrumpropagation method, used to propagate the focused pulse within the nonlinear medium by introducing the irradiance-dependent nonlinear refractive index in the angular spectrum propagator.We have modeled the propagation of ultrashort mildly focused pulses through a Ti:sapphire crystal, characterized only by its linear and nonlinear refractive indices, for pulses with different powers and durations, finding that the proposed method is able to reproduce the self-focusing phenomenon observed in nonlinear media. Our results show that the focal spot within the nonlinear medium is closer to the interface, and it is slightly wider for pulses with higher incident power. However, despite the dependence of the effective refractive index of the nonlinear medium on irradiance, which is the power per unit area, and assuming that the group velocity dispersion and the propagation time difference are suitably corrected, the focused pulse duration is essentially unaffected by the incident power and remains virtually constant during propagation in the nonlinear medium. Finally, the proposed method also reproduces the spatiotemporal coupling arising from the intrinsic correlation between spatial and temporal properties of the focused pulse.