-1)/G (6.4) Equation 6.4 is the noise figure cascade formula, which is attributed to the Danish radio engineer Harald Friis (Brittain, 1995). It should be remembered that linear units, not decibels, are to be used when substituting into it. Moreover, the formula should not be used in this form when there are very low photon numbers (Haus, 1998, 2000; Desurvire, 1999). Amplifier Noise in Optical Systems The Friis formula enables some simple but important deductions. First, the cascade formula for amplifier gains is commutative: Gtot = G1 G2 = G2 G1, which means that the order in which we place the amplifiers in a system does not affect the power budget. This is not the case with noise figures-equation 6.4 gives a different answer if we swap the order of the two amplifiers. (A simple example illustrates the point: If G1 = 10, G2 = 20, F1 = 5, and F2 = 4, the total noise figure is either 5.3 or 4.2, depending on amplifier order.) The process of amplifying the input signal plus ASE and then adding more ASE is necessarily order dependent. As a second deduction, it can easily be proved by mathematical induction that the total noise figure of a chain of N amplifiers is given by =F1+(F2-1)/G+(F3-1)/(G1G2)+ + (Fx -1)/(GG2...GV-1) (6.5) A noise figure can therefore be assigned to an entire system, in which some of the elements with gains Gi and noise figures Fi are true amplifiers (such as EDFAs) while the others are passive components and transmission fibers. Unfortunately, such a treatment does not provide a complete picture of a network's performance because SO many factors, such as nonlinear effects, chromatic and polarization mode dispersion, and time interval errors also contribute. In nearly every practical case, what matters in digital communications is a sys- tem's BER (the topic of the next section). Equation 6.5 nevertheless allows us to understand which of several network or subsystem designs provides better noise performance, and it is in this spirit that it is applied here. Our third deduction is that, when t