The Phase Effect in perturbative nonlinear optics

Waveform control is essential for ultrafast nonlinear optical processes such as high-order harmonic generation (HHG). For example, a sawtooth-like waveform can enhance the kinetic energies of the electrons such that the cutoff of HHG is extended. In this work, we show that relative phase of the two-color driving laser can affect the outcome of perturbative nonlinear optical processes such as lower-order harmonic generation. Consider the third-harmonic signal generated in argon by the fundamental and second-harmonics of a pump laser with frequencies of $\omega _{1}$, and $\omega _{2}$. A cross-term emerges due to interference of four-wave mixing signals of ($\omega _{1}+\omega _{1}+\omega _{1})$ and ($\omega _{2 + }\omega _{2}-\omega _{1})$. When the intensities of two-color pump at $\omega _{1}$ and $\omega _{2}$ are equal, the modulation in the third-harmonic signal by the cross-term is about 30{\%} of the DC term. As the relative phase between $\omega _{1}$ and $\omega _{2}$ varies, a sinusoidal modulation in output intensity at 3$\omega _{1 }$ is expected. We have also calculated the phase effects for fifth, seventh and ninth harmonic generation, which show more complicated behavior.