# Brief Review of Recent Developments in Fiber Lasers
## 3 Rare Earth-Doped Fiber Lasers
### Rare Earth Ions for Laser Applications  
RE atoms include Sc, Y, and two groups: lanthanides with atomic numbers 57 through 71 (characterized by filling the 4f shell); and actinides with atomic numbers 89 through 103 (characterized by filling the 5f shell). In solids, the trivalent (3+) level of ionization is the most stable for lanthanide ions. RE ions have the sharp spectral lines. The sharp spectral lines in the optical spectra of trivalent RE ions are a consequence of the screening of 4f valence electrons by the 5s and 6d electrons due to the effects of the environment. An ion in a solid can be considered an impurity (a dopant) embedded in a host material. The ion replaces the host ion and forms an optically active center. The host materials play a fundamental role in the spectral nature of doped ions. Laser glasses can be divided into oxide, halide, oxyhalide, and chalcogenides [22]. Glasses are amorphous over a long range but contain a local order. Glasses are insulators with bandgaps greater than 5 eV (the deep UV). All transitions in ions occur within the host bandgap. The energy levels of ions typically are identified by three principal quantum numbers: \(L\), \(S\), and \(J\). \(L\) describes the total orbital angular momentum, \(S\) the total spin angular momentum, and \(J\) the total angular momentum. There are several possible ways to obtain a given set of values for \(L\), \(S\), and \(J\). The possible states are referred to by the symbol \({}^{2S+1}L_{J}\), where \(L=0\), \(1\), \(2\), \(3\), \(4\), \(\ldots\) corresponds to letters \(S\), \(P\), \(D\), \(F\), \(G\) \(\ldots\), respectively. Schematical diagrams of energy levels with transitions for Yb\({}^{3+}\), Er\({}^{3+}\), Tm\({}^{3+}\), and Ho\({}^{3+}\) ions are presented in Figure 1.  
Several states can correspond to the same atomic energy level (degeneracy). In RE-doped ions, this degeneracy is lifted through interaction with the local environment. Fiber lasers have been realized with a wide variety of RE ions including ytterbium (Yb\({}^{3+}\)), erbium (Er\({}^{3+}\)), thulium (Tm\({}^{3+}\)), holmium (Ho\({}^{3+}\)), praseodymium (Pr\({}^{3+}\)), and dysprosium (Dy\({}^{3+}\)). The most used RE ions for laser applications are Yb\({}^{3+}\), Er\({}^{3+}\), and Tm\({}^{3+}\). As one can see in Figure 1, only Yb\({}^{3+}\) ions have only one excited state, that is, only Yb\({}^{3+}\) ions are free from excited state absorption, which can serve as a source of heat generation. Figure 1: Diagrams of energy levels and transitions for some Yb\({}^{3+}\), Er\({}^{3+}\), Tm\({}^{3+}\), and Ho\({}^{3+}\) ions. Pump radiation excites electrons from the ground level to higher levels. Excited electrons can undergo spontaneous (radiative or nonradiative) decay or stimulated transition. The spontaneous decay process can be characterized by the lifetime of the excited level \(\tau_{ex}\) as follows:  
\[\frac{1}{\tau_{ex}}=\frac{1}{\tau_{r}}+\frac{1}{\tau_{nr}} \tag{12}\]  
\[\frac{1}{\tau_{nr}}=C\left[\widetilde{\tau}_{r}(T)+1\right]^{p}exp(-\alpha \Delta E) \tag{13}\]  
where \(C\) is an empirical parameter that depends on the host. \[\widetilde{n}(T)=\left[exp\left(\frac{\hbar\Omega}{\kappa_{B}T}\right)-1 \right]^{-1},\quad p=\frac{\Delta E}{\hbar\Omega},\quad\alpha=-\frac{ln( \varepsilon)}{\hbar\Omega} \tag{14}\]  
\(\Delta E\) is the energy gap, \(\varepsilon\) is an electron-phonon coupling coefficient, \(\hbar\Omega\) is the maximum phonon energy of the host, \(\hbar\) is the reduced Planck constant, \(T\) is the temperature of the host, \(\kappa_{B}\) is the Boltzmann constant, and \(p\) is the number of phonons required to bridge the \(\Delta E\) gap. The empirical parameters of relation (13) for some host materials can be found in [23; 24; 25]. The spontaneous decay process can be characterized by the mean fluorescence wavelength, \(\lambda_{F}\), or the mean fluorescence frequency, \(\omega_{F}\).