r the noise equation and solve it to find the noise spectral density. Since the evolution of pumps and signal are known, it is not a difficult task and the following equation is obtained: (4.88) The values of the ASE spectral density obtained with this equation are plotted in Figure 4.39 versus wavelength considering a single-pump-concentrated amplifier made by a DCF km long whose parameters are reported in [68]. The results are compared with the sim- ulated ASE spectral density reported in [68]. The simulation has been carried out using model similar to that reported in Appendix D, taking into account a wide variety of phenomena. The spectral density in the signal bandwidth is very well modeled by Equation 4.88 in a very large part of the spectrum. Near the pumps, the ASE spectral density is in reality much higher than that foreseen by Equation 4.88 due to the fact that parametric amplifica- tion caused by the pumps, which is not included in the simple model we are considering, amplifies ASE up to quite high powers. As a consequence, even if the gain should be sufficient, no signal has to be within a guard bandwidth of about 50 nm from the pumps in order to avoid a high ASE zone. If this rule is respected, Equation 4.88 model the ASE spectral density very well, and it can be used to evaluate the Raman amplifier noise factor obtaining Telecommunication Networks Signal wavelength 1550 nm Simulation FIGURE 4.39 Theory ASE noise spectrum for a single-pump Raman amplifier. (After Bristiel, B. et al., IEEE Photonics Technol. Lett., 18(8), 980, 2006.) Wavelength (nm) (4.89) A more accurate analytical solution of the noise equations can also be found taking into account time fluctuations (thus removing the stationary assumptions). In this way, a more accurate representation of the ASE noise can be achieved that is also valid for very high-gain amplifiers [68], where Equation 4.88 is less accurate. An example of noise figure calculation versus gain is reported in Figure 4.40 for an amplifier whose param- eters are report