Relationship between second-harmonic generation and electric-field-induced second-harmonic generation

We calculate the electric-field-induced second-harmonic-generation (EFISH) susceptibility of a homogeneously broadened two-level system with permanent dipole moments. This susceptibility has contributions from the second-harmonic-generation (SHG) susceptibility of the same system without the presence of a dc field and the four-wave-mixing (FWM) component of the susceptibility ${\gamma }_{\mathit{e}}$(-2ω;ω,ω,0). The magnitude and phase of the various contributions to EFISH arising from SHG and from FWM depend on the frequency of the fundamental optical field, the transition dipole moment ${\mathrm{\mu }}_{\mathit{a}\mathit{b}}$, the permanent dipole moment of the ground-state level ${\mathrm{\mu }}_{\mathit{a}\mathit{a}}$, and the difference of the ground- and excited-state dipole moments Δμ. We present non-rotating-wave-approximation results for both the quasi-steady-state in systems where the dephasing time ${\mathit{T}}_2$ is much shorter than the population decay time ${\mathit{T}}_1$, so the polarization is in a steady state but the population is not, and for the complete steady state. Extraction of the SHG susceptibility from EFISH measurements is considered.