Analytical solution to second-harmonic generation of ultrashort laser pulse

We present an analytical method to evaluate the second-harmonic generation (SHG) process of ultrashort laser pulses in a nonlinear medium within the small-signal regime. Building upon the broadband nonlinear coupled wave theory, we derive an analytical solution that allows one to investigate the fundamental physics and detailed dynamics of the nonlinear interaction between ultrashort laser pulses. We find that SHG for ultrashort laser pulses in a nonlinear crystal arises from a series of self-SHG and self sum-frequency generation processes involving different frequency components. We have applied the analytical method to investigate specific pulse profiles with Gaussian and sech functions and found good agreement between analytical theory and numerical simulations, validating the accuracy of the analytical approach. The analytical method enables one to explore the evolution of the second-harmonic wave under various pulse profiles, pulse widths, and crystal thicknesses, and offers a more efficient and insightful approach to study the nonlinear optical dynamics of ultrashort laser pulses.