of optical systems generally it is sufficient good knowledge of the base phenom- ena that occurs in such devices. This is the reason why, more than presenting a very accurate model, in Chapter 4, we have described with the help of simple equations the behavior of amplifiers in many rel- evant circumstances. When designing a very high-performance transmission system how- ever, since all the margins are reduced, it is quite important to have an accurate modeling of the devices that have been selected for the system. This is why we present in this appendix a much more accurate analytical model the behavior of erbium-doped fiber amplifiers (EDFAs) and lumped Raman amplifiers. Generally the complete coupled equations cannot be solved analytically, SO that what we will introduce here is the first step toward an effective simulation of fiber-based amplifiers. In order to set up a simulation engine based on these equations we recommend to exe- cute a study of the literature starting from [1-3] as far as EDFAs are concerned and from [4,5] if a Raman amplifier has to be simulated. D.2 Erbium-Doped Fiber Amplifiers In principle, the theoretical model of an EDFA depends critically on the pumping scheme, both due to the possible presence of exited state absorption and to the fact that pumping at 1480 is near to a two-level system. In practical amplifiers, however, almost only 980 pumps are used; thus, we will assume that the pumps will be at this wavelength SO that the system behave as a three-level sys- tem: an exited state (411/2), a transition state (41/13/2), and a ground state (41/15/2). We will refer to these states with numbers 2, 1, and 0, respectively, and all the variables referring to a particular state will have the state index. Let us call Nj the population of the jth state; the population rate equations are similar to those of excited molecules in gasses, but for the great simplification that erbium ion are included into a glass matrix and cannot move. Thus, the diffusion term is not present. In these conditions