Last time I have discussed the construction of dye lasers. Let us discuss the working of dye laser:

The molecules have singlet as well as triplet states. Each electronic  state comprises of several vibrational states and each vibrational state  comprises of several rotational levels.

Due to absorption of light from pumping source, dye molecules get excited from the ground state E1 to upper vibrational rotational levels of excited state E2 which is upper laser level. Most of the dye molecules decay to the lowest vibrational  level L of E2 in a time of about 10^-11 seconds. This process is due to thermal redistribution in level E2, thus, it  is a non-radiative process. Population inversion is achieved at level L.

From level L of E2, the dye molecules decay to any higher lying vibrational sublevel of E1. When this process occurs, then the radiation is emitted. This is termed as fluorescence. The life time of level L for dye molecules is about 10^-9 seconds. As most  of the molecules decay from level E2 by fluorescence, thus laser action occurs at the fluorescence wavelength. Thus laser output is  achieved in between states E2 and E1.

Molecules from the state E2 can also make a non-radiative transition to the triplet level T1.  This transition is known as intersystem crossing. This process of intersystem crossing can limit the laser action because it will lead to reduction of the population of E2 which is upper laser level and thus there will be accumulation of molecules in state T1. As the transition  T1 to T2 is  allowed, and the wavelength corresponding to absorption spectrum of T1 to T2 usually overlaps the  emission spectrum of E2 to E1. Thus, intersystem crossing will  lead to reduction of number of molecules in upper laser level E2 and it will reduce the laser gain or laser output. Sometimes, it may even prevent laser oscillation.

Thus for good laser action, the number of molecules in state E2 should reach the threshold level before a significant number of molecules have dropped to level T1. Therefore, it requires very intense and rapid pumping to maintain population inversion.

Addition of oxygen to solution can also reduce the life time T1. Thus oxygen acts as a triplet quenching additive.

One can read about the construction of dye lasers here.