Design principles of nonlinear optical materials for Terahertz lasers

We have investigated both inter-band and intra-band second order nonlinear optical conductivity based on the velocity correlation formalism and the spectral expansion technique. We propose a scenario in which the second order intra-band process is nonzero while the inter-band process is zero. This occurs for a band structure with momentum asymmetry in the Brillouin zone. Very low-energy photons are blocked by the Pauli exclusion principle from participating in the inter-band process; however, they are permitted to participate in the intra-band process, with the band smeared by some impurity scattering. We establish a connection between the inter-band nonlinear optical conductivity in the velocity gauge and the shift vector in the length gauge for a two-band model. Using a quasiclassical kinetic approach, we demonstrate the importance of intra-band transitions in high harmonic generations for the single tilted Dirac cone model and hexagonal warping model. We confirm that the Kramers-Kronig relations break down for the limit case of ($\omega$, $-\omega$) in the nonlinear optical conductivity. Finally, we calculate the superconducting transition temperature of NbN and the dielectric function of AlN, and the resistance of the NbN/AlN junction. The natural non-linearity of the Josephson junction brings a Josephson plasma with frequency in the Terahertz region.