Pumping Schemes

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 N_2 > (g_2 / g_1) N_1, or if the degeneracies are equal to one, just N_2 > N_1.  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 E_1 and E_2 to make it work.  Why?

Imagine we have a collection of atoms which are all initially in the lower energy state E_1.  For simplicity, take g_2 = g_1 = 1.  Let’s try to pump atoms into the upper energy state E_2 using a light source whose frequency \nu = (E_2 - E_1) / h.  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 N_2 climbs, eventually we will reach a point where N_2 = N_1.  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 N_2 any higher, at least in steady state.  If we briefly manage to get N_2 > N_1, stimulated emission will dominate over absorption, and N_2 will be driven back down.

So, we must introduce a third energy state E_3 into our consideration.  We could pump atoms from E_1 to E_3, 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 E_3 to E_2.  In this way, N_3 will always be close to zero, and pump photons whose frequency \nu_{pump} = (E_3 - E_1) / h will always be absorbed.  At the same time, we would like atoms to linger in state E_2 a long time (a so-called ‘metastable’ state).  If we satisfy these conditions, we can increase N_2 until it is greater than N_1 using the pump from E_1 to E_3.  Once N_2 > N_1, we’ve achieved inversion and therefore amplification for photons whose frequency \nu_{stim} = (E_2 - E_1) / h.

The three-level scheme above has a weakness.  It is relatively easy to increase N_2, but we would also like to decrease N_1, in order to maximize the inversion (N_2 - N_1).  If the state E_1 is the ground state, the pump rate must be extremely large to lower N_1 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 E_1 to E_4, a fast relaxation process from E_4 to E_3, a fast relaxation process from E_2 to E_1, and a metastable state E_3.  This scheme is designed to amplify light with frequency \nu_{stim} = (E_3 - E_2) / h.  Atoms are pumped up to E_4 and quickly drop to E_3, where they linger.  After a stimulated emission event drops the atom to state E_2, it quickly relaxes to state E_1, ensuring that N_2 will remain small.   In this way, we can maximize the inversion (N_3 - N_2).

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 h \nu_{pump} = (E_4 - E_1), at best we will get out a single photon of energy h \nu_{stim} = (E_3 - E_2).  The energy difference (h \nu_{pump} - h \nu_{stim}) represents unavoidable energy loss in this system.