The Three-Level System

The population inversion in a three-level system as a function of the pump rate is given by (i).

......(i)

If we multiply both sides of this equation by the stimulated emission cross section σ₂₁, we obtain

.......(1)

where α₀ = σ₂1ntot is the absorption coefficient of the material when all atoms exist in the ground state. In the absence of pumping, (1) simply becomes g₀ = −α₀. In order to simplify our analysis, we assumed g₂ = g₁.
We assume now that the pump rate W_P is a linear function of lamp input P_in,

........(2)

Introducing (2) into (1) yields

.......(3)

In a four-level system, the small signal gain is directly proportional to the pump rate and therefore to the input power of the pump source. we obtain

.........(4)

Equation (3) and (4) are plotted in Fig. 1. Since both lasers are assumed to be pulsed, the input power P_in has been replaced by the flashlamp input energy E_in = P_int_p, where t_p = 1ms is the pump pulse width. The other parameters are (η/I_SA) = 10⁻⁶W⁻¹, α₀ = 0.2 cm⁻¹, and = 15 cm.

FIGURE 1. Gain versus lamp input for a four-level and a three-level system.