Suppose I have a three-level system with $E_0$ the ground level, $E_1$ the intermediate and $E_2$ the upper level. In thermal equilibrium they will have a certain probability distribution according to the Boltzmann Statistic, in a laser one needs a population inversion, but that doesn't matter for my question.
My question is this: 3-level laser uses the $E_1 \rightarrow E_0$ transition because the $E_2$ level decays quickly (by design), i.e., emission of an $E_1$ photon is stimulated with an $E_1$ photon. But is there also a stimulated emission of energy $E_1$ for an incident photon of energy $E_2-E_1$? My thinking is that a photon energy of $E_2-E_1$ kicks the electron from $E_1$ in the upper state $E_2$, which will then decay, and every once in a while it should decay into the ground level, not back into the $E_1$ state. Is that correct? And if that is so, is the momentum of the emitted photon with energy $E_2$ aligned with the momentum of the incident photon of energy $E_2-E_1$?