Spontaneous polarization effects on solid high harmonic generation in ferroelectric lithium niobate crystals

High-order harmonic generation (HHG) in ferroelectric lithium niobate (LiNbO$_{3}$) is investigated theoretically by solving the semi-conductor Bloch equations. Because of the spontaneous polarization, even-order harmonics are produced in the HHG spectra of the LiNbO$_{3}$ crystal driven by a monochromatic multi-cycle 3300-nm laser. Our numerical calculation shows that they are originated from the suppression of one half-optical cycle HHG process in each cycle of the driving field due to the spontaneous polarization. We also illustrate that the spontaneous polarization will increase the harmonic yield and extend the maximally attainable cutoff energy at the same time. We further report that the carrier-envelope phase dependence of HHG spectra changes from a minimum period of $\pi$ rad to 2$\pi$ rad when the laser polarization direction is parallel/anti-parallel to the spontaneous polarization direction in LiNbO$_{3}$ crystal. This is promising to be utilized as an isolated attosecond pulse (IAP) gating mechanism. Moreover, the two-color relative phase dependence of HHG in LiNbO$_{3}$ is also investigated and shows broken inversion-symmetry.