As we saw from Beer’s Law, in order to have optical amplification occur, we must have the density of atoms in the upper energy state , or if the degeneracies are equal to one, just . As previously mentioned, this is called inversion. Inversion requires pumping – an injection of energy into the atoms. This pumping could take many forms. In gas lasers, an electrical discharge is a common pumping mechanism. It is also possible to pump with light (optical absorption); however, if we are pumping with light, there must be at least one more atomic energy state involved besides and to make it work. Why?
Imagine we have a collection of atoms which are all initially in the lower energy state . For simplicity, take . Let’s try to pump atoms into the upper energy state using a light source whose frequency . This will work fine initially – all the incoming light will be absorbed by the atoms in state 1, and those atoms will be promoted to state 2. However, as the density of atoms in state 2 climbs, eventually we will reach a point where . At this point, an incoming photon is equally likely to be absorbed or cause stimulated emission. Therefore, the light won’t be able to push any higher, at least in steady state. If we briefly manage to get , stimulated emission will dominate over absorption, and will be driven back down.
So, we must introduce a third energy state into our consideration. We could pump atoms from to , as shown in the diagram below. We would be wise to choose a set of states for which there is a fast relaxation process that quickly drops atoms from state to . In this way, will always be close to zero, and pump photons whose frequency will always be absorbed. At the same time, we would like atoms to linger in state a long time (a so-called ‘metastable’ state). If we satisfy these conditions, we can increase until it is greater than using the pump from to . Once , we’ve achieved inversion and therefore amplification for photons whose frequency .
The three-level scheme above has a weakness. It is relatively easy to increase , but we would also like to decrease , in order to maximize the inversion . If the state is the ground state, the pump rate must be extremely large to lower significantly (a very intense pump would be required). One way to address this challenge is to introduce a fourth state into the pumping scheme, as diagrammed below. In this case, we want a pump tuned to the transition from to , a fast relaxation process from to , a fast relaxation process from to , and a metastable state . This scheme is designed to amplify light with frequency . Atoms are pumped up to and quickly drop to , where they linger. After a stimulated emission event drops the atom to state , it quickly relaxes to state , ensuring that will remain small. In this way, we can maximize the inversion .
Of course, in engineering as in life, no improvement comes without a price. The 3- and 4-level pumping schemes make it easier to achieve a large inversion. The price we pay is an inherent loss of efficiency. If we are pumping with a single photon of energy , at best we will get out a single photon of energy . The energy difference represents unavoidable energy loss in this system.