Forward degenerate four wave mixing in two-level Doppler broadened absorbers: analytical calculation

Nowadays degenerate four wave mixing (DFWM) with forward geometry (FDFWM) has found increasing usage, for example, in the gas phase detection. In the fields of theoretical development of DFWM, most of the studies have been concentrated on the understanding of phase-conjugated DFWM. However, few theories have dealt with FDFWM. These models calculated the signal directly from the off-diagonal element of the density matrix by assuming optically thin medium. In the paper we obtain a theoretical model for FDFWM in Doppler broadened two-level absorbers including propagation effect and discuss the Doppler effect on the signal intensity and lineshape. Assuming that probe and signal field amplitudes are small compared with those of the pump fields and pump field amplitudes are constant in the interaction volume, we obtain the following coupled equations from the Maxwell equation using the slowly varying envelope approximation and the paraxial approximation.