# Radiation-Balanced Lasers: History, Status, Potential
## 3 Theory of Radiation-Balanced Lasing
### Athermal Lasing
#### 3.2.2 Athermal Lasing in Real Systems  
In the above section, perfect laser materials were considered, which are free from background absorption and are optically thin. Let us consider more realistic laser systems taking into account quenching, radiative trapping, and background absorption. These effects are present in all laser systems. Quenching of the laser excitation could occur through electron-phonon deactivation and through ion-ion energy transfer [27]. As a result, the radiative lifetime of an RE ion, \(\tau_{R}\), must be replaced by a fluorescence lifetime, \(\tau_{\rm F}\). Energy transfer losses can be arranged with a characteristic quenching lifetime, \(\tau_{\rm Q}\)[27]. Radiation trapping occurs as a result of fluorescence reabsorption and its reflection from the sample boundaries. It can complicate the modeling of the laser systems. Radiation trapping can be taken into account with the effective values for mean fluorescence wavelength, \(\widetilde{\lambda}_{F}\), and lifetimes \(\widetilde{\tau}_{\rm F}\), \(\widetilde{\tau}_{\rm Q}\). These values can be obtained experimentally. For example, for an optically thin Yb\({}^{3+}\):KGW sample with the Yb\({}^{3+}\) ion concentration \(N_{T}=2.2\times 10^{20}\) cm\({}^{-3}\) the mean fluorescence wavelength and the fluorescence lifetime are \(\lambda_{F}=997\) nm and \(\tau_{F}=0.275\) ms, respectively. For the 2 mm \(\times\) 8 mm, Yb\({}^{3+}\):KGW sample fluorescent trapping increases these values to \(\widetilde{\lambda}_{F}=1011\) nm and \(\widetilde{\tau}_{F}=0.75\) ms. The rate equation for the radiation trapped system has the form  
\[\frac{dN_{2}}{dt}=\frac{I_{P}}{\hbar\omega_{P}}\left(N_{T}\sigma_{a}^{P}-N_{2} \left(\sigma_{a}^{P}+\sigma_{e}^{P}\right)\right)+\frac{I_{L}}{\hbar\omega_{L }}\left(N_{T}\sigma_{a}^{L}-N_{2}\left(\sigma_{a}^{L}+\sigma_{e}^{L}\right) \right)-\frac{N_{2}}{\widetilde{\tau}_{F}}-\frac{N_{2}^{2}}{N_{T}\widetilde{ \tau}_{\rm Q}} \tag{16}\]  
It must be used to simulate heat generated with the quantum defect. Unavoidable impurities in the laser host material result in background absorption, which can be described by the background absorption coefficient \(\alpha_{B}(\lambda)\). Even small background absorption can significantly deteriorate the laser cooling process. The heat power density generated in the laser by quenching can be described as  
\[P_{heat}^{\rm Q}=\hbar\omega_{P}N_{T}\widetilde{N}_{2}\left(\frac{1}{\tau_{ F}}-\frac{1}{\tau_{R}}+\frac{\widetilde{N}_{2}}{\widetilde{\tau}_{\rm Q}} \right), \tag{17}\]  
where  
\[\widetilde{N}_{2}\approx\frac{N_{2}}{N_{T}}\left(1+\frac{\left(1-\tau_{R}/ \widetilde{\tau}_{\rm F}\right)-N_{2}\tau_{R}/\left(N_{T}\widetilde{\tau}_{Q }\right)}{\widetilde{I}_{P}+\widetilde{I}_{L}+\tau_{R}/\widetilde{\tau}_{\rm F }}\right), \tag{18}\]  
and  
\[\widetilde{I}_{P,L}=I_{P,L}\frac{\tau_{R}\left(\sigma_{a}^{P,L}+\sigma_{e}^{P, L}\right)}{\hbar\omega_{P,L}}. \tag{19}\]  
The heat power density caused by background absorption has the form  
\[P_{heat}^{a}=\frac{\hbar}{\tau_{R}}\left(\frac{\widetilde{i}_{P}\omega_{P} \alpha_{P}}{\sigma_{a}^{p}+\sigma_{e}^{p}}+\frac{\widetilde{i}_{L}\omega_{L} \alpha_{B}(\lambda_{L})}{\sigma_{a}^{L}+\sigma_{e}^{L}}\right), \tag{20}\]where \(\widetilde{i}_{P,L}=I_{P,L}/\widetilde{I}_{P,L}\). These values must be added to the heat power density generated with the quantum defect if one wants to simulate the heat power density generated by different effects in the realistic laser system.  
Summarizing the theory of radiation-balanced lasing, it is important to emphasize that laser materials suitable to radiation-balanced lasing must have two widely separated sets of discreet energy levels (manifolds) such that nonradiative transitions between them are unlikely. They also must have negligible background absorption. A properly arranged reliable pump system, the shape of a laser gain medium, and material selection criteria can result in little or no heat generation within solid-state lasers.