Section 2.4: Rate Equations and Population Inversion
Two-level Atomic System
Let’s first examine the two-level atomic system. Pumping process provides the incident radiation satisfying hn = E2 - E1, E2>E1. Let’s define W12 as the possibility of atoms jumping from E1 to E2 because of stimulated absorption, define W21 as the possibility of atoms jumping from E2 to E1 because of stimulated emission, define w12 and w21 as the corresponding relaxation or decay rate (including both spontaneous radiation rate and non-radiation decay rate). Then the rate equations for two-level atomic system are:
N is the total atom number, ”N=N1-N2 is the population difference. T1 is the population recovery time or energy relaxation time of the system. We also define N10 and N20 as the atom population at thermal equilibrium, ”N0=N10-N20 as the population difference at thermal equilibrium. Note also W12=W21. After some operation, we get:
At steady state, i.e., when ”N doesn’t change with time, we get:
”N0 is bigger than zero, W12 is closely related with the incident signal and the stronger the incident signal, the bigger the W12, W12>0. From the above equation we see, for a two-level atomic system, the incident signal will make the population difference ”N approach zero when the signal is bigger enough, but no population inversion occurs! The best it can go is at steady state or at saturation, the population difference becomes 0.
The conclusion is: to get population inversion, we must use atomic systems with more than two related energy levels.
Figure2.8: Two level atomic system
Three-level Laser System
Now let’s examine the three level laser system.
Figure2.9: Three level atomic system
For the three-level laser system, E1 is the ground state, lasing is between E2 and E1. Ruby laser is a typical three level laser system.
Supposing the pumping process produces a stimulated transition probability between E1 and E3, W13=W31=Wp. Atoms at E3 have fast decay time T32, i.e., atoms at E3 decay to E2 in a very short time T32. Atoms at E2 have a relatively slow transition time T21, i.e., atoms at E2 will take a longer time to change to E1 than atoms from E3 to E2. Also we have N=N1+N2+N3. So we have rate equations for three level laser systems:
At steady state, the population difference between E2 and E1 is:
From above we see, for population inversion to occur, i.e., for N2-N1>0, the pumping rate Wp must satisfy: WpT21>1/(1-b ). If we assume the atoms at E3 decay to E2 immediately, thus b =0, then at steady state, we have:
Conclusion: population inversion is possible for a three level atomic system. The condition is: T32<<T21 and the pumping rate must be bigger than a positive threshold value. Because N1 is the ground level, N1 is always very big in the beginning. Population inversion starts after half of the ground level atoms being pumped to the E2 level. So three-level laser system is not very efficient, present lasers are usually four-level or more level systems.
Now let’s see what advantages the four-level laser systems have over three-level systems.
Four-level Laser System
Nd:YAG laser is a four level laser system. Look at the four level laser model below.
Figure2.10: Four level atomic system
We have E1, N1 at the ground level, pumping process raise the atoms from E1 to E4, pumping rate is Wp=W14=W41. Atoms at E4 have fast decay to E3, decay time is T43. Lasing happens between E3 and E2, the transmission time is Trad. Atoms at E2 then decay very fast to E1, the decay time is T21. We have the relation: N=N1+N2+N3+N4. Then we have rate equations for four-level laser systems:
For simplification, we assume T43 is short enough that the pumped atoms to E4 immediately decay to E3, N4 is nearly 0. Also atoms at E2 decay so fast that we can say N2 is nearly zero. Let’s examine population inversion between N3 and N2. We list the conclusion here:
Since b » 0 for a four level system, N3-N2 is readily bigger than zero. There is almost no threshold for Wp to generate population conversion. The advantage of four-level laser system is very clear now.
Up to now we have discussed population inversion conditions by analyzing the pumping rate equations. We have to make a statement here: we make some simplifications to make the material easily understood. The actual detailed rate equations are far more complex than we see here, we suggest the interested readers refer books on principles of lasers for further questions.
When we get population inversion, our next step is to analyze the amplification of incident waves.
Figure2.11: Population inversion for Three and Four level laser systems