aries as we tune the signal across the amplifier's gain profile. Equation 6.3 can aid our understanding of optical system design. We start with the apparently trivial case of a passive element, such as an optical fiber of length L (m) and loss coefficient as (m-1), and calculate its noise figure. Being passive, the element generates no ASE and its "gain" is in fact a loss, having the value G = Pout/Pin = exp(-asL). Substitution into equation 6.3 then shows that the noise figure is F = exp(+asL) = 1/G. Therefore, for the purpose of comparing one network with another, we can assign noise figures to the fibers or passive components. A brief explanation is that, as the signal power decreases in magni- tude, the noise due to the random arrival times of the signal photons (i.e., the shot noise) becomes ever more important. Even though no spontaneous emission photons are gener- ated, the ratio of signal to shot noise is reduced merely through signal attenuation. For this reason, it is sometimes said that the signal "sinks into the noise." (Further details of why we assign a noise figure to a loss element are presented in Bristiel [2006].) The Friis Cascade Formula Equation 6.3 can be used to calculate the noise figure of two elements in series. From a quantum mechanical viewpoint, the following derivation is nonrigorous, but fuller treat- ments are presented in Haus (1998, 2000) and Desurvire (1994, 1999). Let there be two amplifiers (or loss elements) of gains G1 and G2 and noise figures F1 and F2. The total gain for two amplifiers in series is Gtot=GG2. If we can assume that the ASE power is not sufficient to cause saturation, and therefore reduce the gain, the total forward-propagating ASE output power is PASE-tot = G2PASE-1 + PASE-2. This equation applies because the second amplifier amplifies the noise of the first one and adds some noise of its own. The overall noise figure of the two amplifiers in series is Ftot = (1/Gtot) (1 + PASE-tot/hvs BO). By making the appropriate substitutions, it follows that Flot=F1+(F2