# Radiation-Balanced Lasers: History, Status, Potential
## 2 Laser Heat Management  
Most materials are not optically active. They simply convert absorbed laser power directly into heat. If \(\alpha_{a}\) is an absorption coefficient describing the absorption process and \(I_{L}\) is the laser intensity, the absorbed power density is equal to \(\alpha_{a}I_{L}\). This is not the case for laser media, which are optically active. The laser performance is based on efficient reemitting of absorbed pump light. The amount of heat generated in a laser depends on the pump power and the emission spectra.  
It is well known that population inversion is a necessary condition for lasing. It is impossible to reach the population inversion in a two-level system. Three or four level schemes are required. As a result, the reradiated light is normally red-shifted relative to the pump (Figure 1).  
By this means, every emitted laser photon contributes a fraction of its energy,  
\[\eta_{QD}=1-\omega_{L}/\omega_{P}, \tag{1}\]  
to heating the laser material through phonon generation. Here \(\eta_{QD}\) is the quantum defect, \(\omega_{P}\) and \(\omega_{L}\) are the frequencies of the pump and laser photons, respectively.  
Figure 1: Energy levels and relevant excitation and decay processes for (**a**) three-level and (**b**) four-level lasers.  
A study of thermal effects in a crystal requires the calculation of the temperature field at each point of the crystal. This can be achieved with the heat equation. For analytical solutions to the heat equation to be obtained, some assumptions must be imposed on the laser system. Let us suppose that the pump profile is axisymmetric, and the thermal conductivity of the laser medium does not depend on temperature and can be considered as a constant. In this case, the heat equation has the form [16]:  
\[\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial T}{\partial r} \right)+\frac{\partial^{2}T}{\partial r^{2}}=-\frac{Q}{\text{K}_{\text{c}}}, \tag{2}\]  
where \(T\) is the temperature, \(r\) is the radial coordinate of a point inside the laser medium (see insert in Figure 2), \(\text{K}_{\text{c}}\) is the thermal conductivity, and \(Q\) is the thermal power.  
In a vast majority of laser schemes, the pump beam profile can be described by a super-Gaussian function. The general solution of the heat Equation (2) for a super-Gaussian beam of any order, as well as for Gaussian and top-hat pump beams, can be found in [17; 18; 19; 20]. As one can see in these solutions, heat generated in the laser's interior produces a temperature rise in the center of the laser medium relative to its edge. As an example, normalized temperature distribution for a laser crystal pumped by a top-hat-profile pump beam is presented in Figure 2.  
The temperature rise in the laser medium produces several undesirable effects. It reduces the lifetime of the laser active ions and increases losses. The temperature gradient causes the strain distribution inside the laser medium. The strain-induced modification of the refractive index results in depolarization and degradation in beam quality. It limits the average laser output power [21]. All these effects are known as thermal lensing. They are particularly important in the case of high-power lasers. The very high temperature rise can even melt or burn the laser medium (thermal damage). In most laser structures, beam quality degradation limits laser power below thermal damage.  
As mentioned earlier, the problem of laser power scaling can be solved by reducing or eliminating the heat generated during the optical pumping and lasing process using optical cooling by anti-Stokes fluorescence within the laser medium to balance the heat generated by the Stokes shifted stimulated emission. The theory of this approach is considered in the next section.