d fibers where the dopant is uniformly distributed (without the clustering effect that practically inhibits the interaction with light of the great part of dop- ant ions) and that are stable in time. Several rare earth ions exhibit transitions in the useful wavelength range, and erbium ions, whose electronics energy levels are shown in Figure 4.23, can be inserted in the glass matrix in a uniform and stable way. Thus, erbium-doped fibers can be used as active elements in architectures like that in Figure 4.24. In particular, the transition between the levels I15/2 and I13/2 is around 1531 nm, thus about at the center of the third fibers transmission window. w/2n (1013 Hz) Ground state Excited state 4F512 absorption absorption 4F712 2H11/2 4S3/2 4F9/2 Phonons 900 nm mediated 419/2 decay 4111/2 Stimulated I13/2 1480 nm emission Ground 4115/2 state Ground state FIGURE 4.23 Energy levels of the erbium ion in a glass matrix (a). The density of states in the various states bands is indicated qualitatively with the line thickness. Quantum transitions used in EDFA amplifiers (b). Telecommunication Networks Fiber in Fiber out FIGURE 4.24 Block scheme of a single-stage EDFA. DP, doped fiber; PL, pump laser; DC, power supply; F, filter; I, isolator. The photons-electrons cross section is sufficiently big to foresee a good stimulated emis- sion efficiency and the excited level lifetime is sufficiently long to guarantee stationary amplification at all the interesting bit rates. Once the erbium ion is inserted in the glass matrix, due to the interaction with the glass and with nearby ions, there is a split of the electronic levels, each of which becomes a thin level bandwidth. This is another important characteristic of erbium ions, allowing wide bandwidth amplifiers, suitable for DWDM systems. The choice of erbium ions as dopants of a single-mode glass fiber to realize fiber ampli- fiers is then natural. The first step to do to design a practical amplifier is to define an efficient and practical pumping scheme. From Figure 4.