  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.
