# Radiation-Balanced Lasers: History, Status, Potential
## 3 Theory of Radiation-Balanced Lasing
### Athermal Lasing
#### 3.2.1 Athermal Lasing in Ideal Systems  
Let us consider the basic concepts of a radiation-balanced (athermal) laser. A solid-state laser of this type can be often referred to as a quasi-three-level laser. The population of each sublevel within a manifold is described by Boltzmann occupation factors. We assume that transitions between these sublevels are purely nonradiative transitions, provided by phonon absorption and emission. This process takes place on a picosecond timescale and is known as thermalization. Let us consider an isotropic laser medium with the total density of the active ions \(N_{T}\). Optical transitions can occur between the ground manifold and the first excited manifolds, giving rise to overlapping, thermally broadened absorption and emission spectra. These spectra are characterized by absorption, \(\sigma_{a}(\lambda,\ T)\), and emission, \(\sigma_{e}(\lambda,\ T)\), cross sections, respectively. They depend on a wavelength, \(\lambda\), and a temperature, \(T\). For radiation-balanced operation of a laser, the mean fluorescence frequency and the pump and laser frequencies must satisfy the following relation: \(\omega_{F}>\omega_{P}>\omega_{L}\) (Figure 4). In this case, the system includes two cycles: the laser cycle and the cooling cycle. The laser cycle includes the pump and laser photons at the frequencies \(\omega_{P}\) and \(\omega_{L}\), respectively. It is accompanied by phonon generation. The cooling cycle can be considered as a cycle including the pump photon and the mean fluorescence photon at the frequencies \(\omega_{P}\) and \(\omega_{F}\), respectively. It is accompanied by phonon absorption. We assume that a bandgap between the excited and ground manifolds is large compared to the energies of the phonons. As a result, the transitions between the excited and ground manifolds are purely radiative. We also assume the absence of excited-state absorption, energy transfer, radiative trapping, and background absorption. In this case, the rate equation takes the form:  
\[\frac{dN_{2}}{dt}=W_{P}-W_{L}-\frac{N_{2}}{\tau_{R}}, \tag{5}\]  
\[N_{1}+N_{2}=N_{T}, \tag{6}\]  
Figure 4: Energy diagram of an Yb\({}^{3+}\)-doped radiation-balanced laser. \(\omega_{P}\) and \(\omega_{L}\) are the pump and laser frequencies, respectively. \(\omega_{F}\) is the mean fluorescence frequency. where \(N_{1}\) is the population of the ground manifold and \(N_{2}\) is the population of the excited manifold. The last term in Equation (5) describes the total spontaneous decay rate of the upper manifold. \(W_{P}\) is a pump rate, and \(W_{L}\) is the stimulated emission rate:  
\[W_{P}=\frac{I_{P}}{\hbar\omega_{P}}\Big{[}N_{T}\sigma_{a}^{P}-N_{2}\Big{(} \sigma_{a}^{P}+\sigma_{e}^{P}\Big{)}\Big{]}, \tag{7}\]  
\[W_{L}=\frac{I_{L}}{\hbar\omega_{L}}\Big{[}N_{2}\Big{(}\sigma_{a}^{L}+\sigma_{e }^{L}\Big{)}-N_{T}\sigma_{a}^{L}\Big{]}. \tag{8}\]  
Here \(I_{P}\) and \(I_{L}\) are the intensities of the pump beam and the laser signal, respectively. \(\sigma_{a,e}^{P}\) and \(\sigma_{a,e}^{L}\) are the absorption \((a)\) and emission \((e)\) cross sections at the pump \((P)\) wavelength, \(\lambda_{P}\), and at the laser \((L)\) wavelength, \(\lambda_{L}\), respectively. Following the law of conservation of energy, one can estimate heat generated in the laser medium. Indeed, the difference between the absorbed power density and the emitted power density must be equal to the local generated heat power density:  
\[P_{heat}=\hbar\omega_{P}W_{P}-\hbar\omega_{L}W_{L}-\hbar\omega_{F}\frac{N_{2} }{\tau_{R}} \tag{9}\]  
Substituting Equations (5)-(8) into Equation (9), one can calculate the heat power density generated at any point of the laser medium. Equation (9) for heat power density includes both the pump, \(I_{P}\), and laser, \(I_{L}\), intensities in the laser medium.