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2. Population inversion versus pumping in an "upper-level" three-level laser sys- tem. It is possible (though not likely in practice) to have a three-level laser system which is pumped on the 1 - 3 transition and in which ew laser action takes place on the 3 - 2 rather then the 2 - 1 transition (no such real system is known). Suppose that level 3 in such a system is long-lived with lifetime 73; level 2 has a short relaxation time to the ground state; and the system is pumped with transition probability Wp on the 1 -+ 3 transition. Carry through the rate-equation analysis (in the optical approximation) necessary to find the population inversion on the 3 - 2 transition as a function of pumping power Wp. Compare the AN versus Wp curve for this system to to those for the four-level and three-level models shown in Figure 6.4. Four-level Three-level System system Laser threshold FIGURE 6.4 Laser population inversion Inverted population AN inversion versus normalized pumping Normalized pump rate, Wer rate for idealized four-level and three-level laser systems