# Brief Review of Recent Developments in Fiber Lasers
## 2 Heat Generation in Fiber Lasers  
For most lasers, the laser wavelength, \(\lambda_{L}\), is longer than the pump wavelength, \(\lambda_{P}\), (Stokes shift). This is unavoidable for all optically pumped schemes where one pump photon can generate one laser photon. The energy difference between the pump and laser photons can be estimated using the so-called quantum defect:  
\[\eta_{q}=1-\frac{\lambda_{P}}{\lambda_{L}} \tag{1}\]  
The quantum defect is an unavoidable source of heat generation in lasers. The process of generating coherent radiation through laser action is never 100% effective. The quantum defect causes a substantial part of the pump energy to be converted into phonons instead of photons. Heating due to the quantum defect manifests itself in many ways including transverse-mode instability, thermal lensing, and catastrophic fiber damage. Heat management is one of the most critical issues for scaling higher output laser powers. Although the very high surface-to-volume ratio and optical guidance have provided tremendous progress in the power scalability of high-power fiber lasers, limitations on the power handling capacity of fiber lasers are now encountered. Under the steady-state operation (\(\partial/\partial t=0\)), the heat conduction equation in an isotropic medium can be written as follows:  
\[\
abla^{2}T(r,z)=-\frac{Q(r,z)}{\kappa} \tag{2}\]  
where \(T\) is the temperature, \(\kappa\) is the thermal conductivity, and \(Q\) is the dissipated heat density per unit volume. In the center of the fiber core (\(r=0\)), the following equation  
\[\left.\frac{\partial T_{co}}{\partial r}\right|_{r=0}=0 \tag{3}\]must be satisfied. Here, \(r\) is the fiber radius, and \(T_{co}\) is the temperature of the fiber core. The core and cladding temperatures and their derivatives must be continuous across the borders. \[T_{co}(r_{co})=T_{cl}(r_{co}) \tag{4}\]  
\[\kappa_{co}\left.\frac{\partial T_{co}}{\partial r}\right|_{r=r_{co}}=\kappa_{cl }\left.\frac{\partial T_{cl}}{\partial r}\right|_{r=r_{co}} \tag{5}\]  
\[\kappa_{cl}\left.\frac{\partial T_{cl}}{\partial r}\right|_{r=r_{cl}}=h[T_{c} -T_{cl}(r_{cl})] \tag{6}\]  
where \(T_{cl}\) is the temperature of the fiber cladding. \(r_{co}\) and \(r_{cl}\) are the core and cladding radii, respectively. \(\kappa_{co}\) and \(\kappa_{cl}\) are the thermal conductivities of the core and cladding, respectively. \(h\) is the heat transfer coefficient. \(T_{c}\) is the environmental temperature. Considering \(Q(r,z)\cong Q(z)\) and taking into account the boundary conditions (3)-(6), one can estimate the temperature of the fiber [17]. \[T_{co}(r,z)=T_{0}(z)-\frac{Q(z)r^{2}}{4\kappa_{co}},\ \ \text{if}\ \ 0<r<r_{co} \tag{7}\]  
\[T_{cl}(r,z)=T_{0}(z)-\frac{Q(z)r_{co}^{2}}{4\kappa_{co}}-\frac{Q(z)r^{2}}{2 \kappa_{cl}}ln\left(\frac{r}{r_{co}}\right),\ \ \text{if}\ \ r_{co}<r<r_{cl} \tag{8}\]  
\[T_{0}(z)=T_{c}+\frac{Q(z)r_{co}^{2}}{2 hc}+\frac{Q(z)r_{co}^{2}}{4\kappa_{co}}+ \frac{Q(z)r_{co}^{2}}{2\kappa_{cl}}ln\left(\frac{r_{cl}}{r_{co}}\right) \tag{9}\]  
Here, \(T_{0}\) is the temperature at the core center. As seen from (7)-(9), the temperature profiles are significantly affected by pump evolution along the fiber length [17]. The temperature distribution is uneven along the fiber. In the forward pump mode, the maximum temperature is attained at the input side of the laser cavity, which corresponds to the intense amplification of the laser signal, accompanied by a tremendous increase in the number of the phonons (\"superfluous\" phonons), causing heating of the laser medium [17; 18]. In the two-end pump mode, the temperature evolution is more even. Under these conditions, the maximum temperature is attained on both sides of the laser cavity [17; 19]. The radial distribution of the temperature profile is more even than the axial distribution. It can be ignored while significant heat is traveling through the axial direction of the fiber [17; 19].