etric power flux expression [44], and assuming that the narrowband signal can be extracted from the wavelength integral with respect to the wideband G(@), the expression of the optical power flux can be written as (4.61) where WO is the signal central angular frequency Q(t,z) is the quantum noise instantaneous power, whose average value is given by (4.62) where B/2 is the signal frequency bandwidth (remember that B was measured in angular frequency). Telecommunication Networks The presence of the quantum noise in a real amplifier is superimposed to the presence of the thermal noise, due to the black body emission of the fiber material and of the back- ground. However, in almost all the optical applications, the dominant noise is the quan- tum noise and often the contribution of the thermal noise can be neglected. 4.3.1.2 Stationary Behavior of a Two-Level Amplifier Once the role of quantum noise is clarified, we are ready to present a simple, but general theory of optical amplification. This will be based on the analysis of a two-level amplifier; thus, its results describe only qualitatively the behavior of real amplifiers. However, it will be useful to understand real amplifier behavior. A two-level amplifier is a system where a material interacts with the light field only through one discrete transition between two stationary energy levels: the ground level at energy E and an excited state at energy E. In order to simplify the model as far as possible, we will assume perfectly resonant interaction with a field of angular frequency h w = E - E. In order to generate optical gain, a pump creates population inversion in the medium SO that the incoming radiation is amplified through stimulated emission. We will assume that, as in all practical cases in telecommunication, the amplification occurs during single-mode guided propagation (e.g., in an optical fiber). In this case, neglecting the transversal mode shape and applying to the wave equation the usual rotating wave and slowly varying approximations [16,44], the 