## 2 Review of Cryogenic-Cooling Benefits
### Spectroscopic, Lasing, Linear and Nonlinear Optical Parameters
#### 2.7.1 Spectroscopic and Lasing Parameters  
Cryogenic cooling is not just limited to just mitigating thermal effects in solid-state laser media. Laser kinetics and spectroscopy, as well as many important laser parameters are enhanced as well. Of all the rare-earth and transition-metal lasing ions that have been investigated, Yb\({}^{3+}\) is of particular interest. Because of the inefficiency of flashlamp pumping, the development of Yb lasers was delayed until the introduction of laser diode pump sources. Yb based lasers are free of the efficiency limiting processes of concentration quenching, excited-state absorption, and upconversion that plague many other solid-state lasers. Figure 33 shows a typical Yb energy level configuration, in this case for the newer laser material Yb:Lu\({}_{2}\)O\({}_{3}\). Yb has only two manifolds, identified as the ground state manifold \({}^{2}\)F\({}_{7/2}\), and an \({}^{2}\)F\({}_{5/2}\) excited state manifold. Most Yb laser have laser transitions near 1029 nm ((1,1) to (0,3) transition), with some also lasing near 1080 nm ((1,1) to (0,4)) transition. The spectral absorption of Yb laser materials all display bands in the pump regions located near 940 nm, corresponding to the (0,1) to (1,2) transition, and the zero-phonon line absorption near 976 nm from the (0,1) to (1,1) transition. Diode-pumping Yb:YAG near 940 nm can be accomplished using commercially available diode arrays and leads to a small heat fraction of about 9%, whilst pumping near 976 nm results in a very small heat fraction of \(<\) about 5%. Some laser materials like Yb:YAG have such narrow linewidths at 969 nm that they cannot be efficiently pumped at cryogenic temperatures [3]. The recent development of narrow-band Volume Bragg Grating (VBG) diode pump sources at 976 nm does however allow the pumping of the zero-phonon line for some laser materials at room temperature and broader band Yb materials at cryogenic temperatures. It is a significant advantage of Yb laser materials that they display heat fractions that are only a fraction of the more common Nd based materials, primarily because no concentration-quenching occurs, and upconversion is absent. Another compelling advantage of Yb based laser materials is that at cryogenic temperatures the finite Boltzmann population of the ground state (1,3) population is effectively eliminated, and thus the need to intensely pump the material at room temperature just to reach transparency is no longer necessary. We can write the pump power density \(\rho_{P}\) needed to achieve transparency at 1029 nm as  
\[\rho_{P}=\frac{n_{d}f_{03}h\
u_{P}}{(f_{03}+f_{11})\cdot\tau_{f}} \tag{74}\]  
In Equation (74), \(n_{d}\) is the total Yb ion density, \(\
u_{P}\) the frequency of a pump photon, \(\tau_{f}\) the upper level fluorescence lifetime, and \(f_{03}\) and \(f_{11}\) are the Boltzmann occupation factors for the lower laser terminal level, and for the upper laser level respectively (see Figure 33). This equation follows from the transparency condition  
\[f_{11}n_{\mathrm{U}}=f_{03}n_{L} \tag{75}\]  
where \(n_{U}\) and \(n_{L}\)are the total population densities in the upper and lower manifolds respectively. Calculation shows that for Yb:YAG, for example, to pump 1.0 at. % Yb to transparency at room temperature requires a pump density of 1.72 kW/cm\({}^{3}\) for 940 nm pumping. The 940 nm pump power needed to reach transparency at 77 K, however, is only 0.33 W/cm\({}^{3}\) for 1.0 at. % Yb doping. For Yb:YAG, the peak stimulated-emission coefficient \(\sigma_{\mathrm{e}}\) increases by close to a factor of 5 at 77 K while the long metastable lifetime of \(\sim\)951 us at room temperature is maintained at cryogenic temperatures as well. The laser saturation fluence \(J_{L}^{S}\) for Yb:YAG can be calculated from the equation  
\[J_{L}^{S}=\frac{h\
u_{L}}{f_{11}\sigma_{11-03}} \tag{76}\]  
Here, from Figure 33, \(\sigma_{11-03}\) is the 11-03 laser transition stimulated-emission spectroscopic cross-section, and \(J_{L}^{S}\) is inversely proportional to it.