Analytical solutions of the susceptibility for Doppler-broadened three-level atomic systems

We present an analytical calculation of the susceptibility in Doppler-broadened three-level atomic systems using a diagrammatic method. The optical coherences responsible for the susceptibility of the probe field are obtained from the optical Bloch equations for Ladder (lower or upper transition probed), V and Lambda-type three-level atoms up to the first (arbitrary) order in the probe (coupling) field Rabi frequency, and are then averaged over a Maxwell-Boltzmann velocity distribution. Finally, we obtain analytical solutions of the susceptibility for four different three-level atomic systems. In particular, the general expressions of the susceptibility are obtained where the wavelengths of the probe and coupling fields are different from each other.