E is the reference electrical field state in the Heisenberg representation P1 and Po represent the populations of the excited and ground states, respectively and the population inversion term is inserted in Equation 4.65 phenomenologically to take into account a realistic condition (for a more rigorous derivation on the ground of the den- sity matrix equations see [44,45]). To describe the average evolution, Equation 4.63 can be averaged and added to the com- plex conjugate, obtaining an equation for the power density flux: (4.66) If the input power Pin is much smaller than the local saturation power Psat, the solution of Equation 4.66 can be obtained in the following form, where the amplifier gain definition G =Pout/Pin has been used to eliminate the output power in favor of the amplifier gain (4.67) where the unsaturated gain Go is given by Go = =exp[g(w) P)L], L being the amplifier length. The main phenomena related to the stationary working of an optical amplifier can deduced from Equation 4.67. The first point is gain saturation. As far as the input power increases, the amplifier passes from linear working, where the output power is simply proportional in average to the input one through the unsaturated gain Go, to saturated working where the output power is constant and it is the gain that decreases while increasing the input power. The gain behavior versus the input power is shown in Figure 4.13 for Gn=18dB and Psat=10mW. = From Figure 4.13 it is also clear that the so-called macroscopic input saturation power Ps does not coincide with the microscopic saturation power Psat* Macroscopic saturation power = -10.2 dBm Input power (dBm) FIGURE 4.13 Gain saturation in an ideal two-level amplifier. Telecommunication Networks As a matter of fact, defining Ps as the input power at which the gain G is reduced to half the linear gain Go, we get (4.68) and the output saturation power is given by This is essentially due to the fact that the macroscopic amplifier can be seen as the cas- cade of a huge number of microsco