The first two equations of the system (4.84) are independent from the noise and can be solved exactly obtaining simple attenuated propagation for the pumps and the following signal expression (4.85) where Ps(0) is the power of the signal at the amplifier input (z = 0) P10 and P20 are the powers of the two pumps injected in the fiber at the amplifier end (z=L) From (4.85), the following simple expression of the gain is obtained (4.86) where it is assumed K=2 and the effective local gain, measured in 1/W, is given by g'i = (gj/4A). This corresponds to the practical condition of unpolarized pumps that is needed in Raman amplifiers for telecommunications due to the random signal fluctuations. Frequently, Raman amplifiers are characterized through the so-called on-off gain GRI instead of the standard net gain G. The on-off gain is defined as the ratio between the output power with pumps on and the output power with pumps off while the net gain is defined as usual as the relationship between the signal at the output and at the input of the amplifier. The relation between the net gain G and the on-off gain can be written as follows, where all the gains and losses are expressed in dB: GR=G-1=0.4343(P10g1+P20g2Le (all in dB) (4.87) where A is the (negative) loss of the Raman fiber with pumps off in dB. Equation 4.87 clearly explains the utility of the concept of on-off gain: as a matter of fact, the Raman amplification depurated from the losses of the fiber where amplification occurs is exactly equal to the on-off gain. Thus, this parameter characterizes the effectiveness of the amplification process without taking into account the losses of the Raman fiber also. Telecommunication Networks Probe and copolarized Probe and pump orthogonal Frequency shift FIGURE 4.33 Polarization dependence of the Raman gain curve. (After Islam, M.N., IEEE J. Sel. Top. Quantum Electron., 8(3), 548, 2002.) Generally, a draft design of a Raman amplifier is carried out starting from (4.86) or a sim- ilar approximation, then it is verified by si