A Rayleigh vs Raman scattering for low saturation

[BillKet]

Hello! I have the following situation: I have a 3 level system, with 2 ground states, call them $g_1$ and $g_2$ and an excited state, $e$, with energies $E_{g1}<E_{g2}$ and $E_e$. I have a driving field with frequency $\omega$ such that $\Gamma \ll \Delta \ll E_{g2}-E_{g1} \ll E_e - E_{g1}$, where $\Gamma$ is the linewidth of the excited state and $\Delta$ is the detuning of the excited state from the $g_1\to e$ transition. So basically the laser is very detuned from any transition in the system and we can assume that the laser power is small enough such that the saturation parameter, $s$ is much smaller than 1, so the probability of the atom getting excited to $e$ is virtually zero. I found in AMO books that in this case Rayleigh scattering dominates i.e. coherent scattering, compared to incoherent scattering i.e. decays from $e$ and the ratio of the 2 rates is $\sim s$. However, as far as I can tell, these derivations don't take into account Raman transitions to $g_2$ (assuming we start in $g_1$) in which $e$ doesn't get excited. These kinds of transitions are not Rayleigh (as the light frequency changes), but they are also not incoherent, as there is still a clear phase between the driving field and the emitted photon (but they have different frequencies). So, given my situation, how can I calculate the Rayleigh scattering rate vs Raman scattering rate (i.e. with both of them coherent processes and assuming that the dipole moment coupling between $e$ and $g_1$ is the same as the one between $e$ and $g_2$)? Thank you!