Effect of laser focus in two-color synthesized waveform on generation of soft x-ray high harmonics

Synthesis of multi-color laser pulses has been developed as a promising way to improve low conversion efficiency of high-order harmonic generation (HHG). Here we systematically study the effect of laser focus in a two-color waveform on generation of macroscopic HHG in soft x-rays. We find that the dependence of HHG yields on laser focus at low or high gas pressure is sensitive to the characteristics of single-atom harmonic response, in which "short"- or "long"-trajectory emissions can be selectively controlled by changing the waveform of two-color synthesized laser pulse. We uncover the phase-matching mechanism of HHG in the gas medium by examining the propagation of the two-color waveform and the evolution of time-frequency emissions of high-harmonic field. We further reveal that the nonlinear effects, such as geometric phase, atomic dispersion, and plasma defocusing, are responsible for modification of two-color waveform upon propagation. This work can be used to find better macroscopic conditions for generating soft x-ray HHG by employing two-color optimized waveforms.