()((2 where the symbols have the following meaning: P is the signal power flux; it is supposed that only one narrowband signal travels the amplifiers in the positive z-direction. Pp is the pump power flux; a single continuous wave pump is assumed; in multi- pump configurations, other pump equations have to be added similar to this one and, when the pump appears in the signal and noise equations, the sum of the pumps have to be inserted. N is the ASE noise power flux in the signal bandwidth; the ASE noise in the pump bandwidth has been neglected. S is the pump directional factor: it is +1 if the pump is copropagating with the sig- nal and -1 if it is counterpropagating. and r's are the superposition integrals between the pump and signal modes, respectively, and the distribution of the erbium ions. Os and oa are the emission and absorption cross sections at the signal wavelengths, respectively. Op is the absorption cross section at the pump wavelength. Up and Vs are the group velocities of the pump and of the signal. In Equations D.1, several approximations are already embedded: The population of the higher energy state is assumed to be zero due to the very fast decay time. This is a condition to have an efficient amplifier. Noise is not present in the pump and signal equations (high signal-to-noise ratio approximation). This is needed to separate the problems: first the problem related to amplification and other deterministic effects, and after that the noise equation becomes approachable once the pump and the signal evolutions are known. The nondepleting pump approximation is also contained in the equations, since the pump evolution does not depend on the signal. Generally the adiabatic approximation is also carried out, which in this case is more than justified by the slow pace of variation of the populations. Appendix D: Fiber Optical Amplifiers: Analytical Modeling Starting from Equations D.1, the main microscopic parameters related to the amplifier func- tionalities can be defined: the local gain g(z) and the