## 6 Multi-Slab Laser Amplifier
### Numerical Modelling of Laser Amplifiers  
One of the issues investigated by the HiLASE team is the optimization of the parameters of laser amplifiers in order to enhance their performance and to suppress the deleterious effects, like parasitic amplified spontaneous emission (ASE) oscillations, thermal wavefront aberrations and thermal-stress-induced birefringence. Thermally-induced beam distortions can be suppressed by a proper design of the amplifier geometry, including the width of an unpumped gap between the active area and the ASE absorbing layer, as well as the thickness and absorption coefficient of the absorbing layer itself [21; 57]. An advanced numerical code for the evaluation of energetics and thermo-optical performance has been developed to provide the capability to quantify these effects and to design and optimize the amplifier for high performance. The energetics modeling of the amplifiers is provided by our in-house-developed numerical code [20]. The code works in discrete time steps. During each step, the energy delivered to the amplifier by the pump beam using its ray tracing through the amplifier head is evaluated. This evaluation takes into account the pump pulse temporal profile, as well as the spectrum of the pump radiation. The absorbed energy is evaluated according to the absorption probability given by the number of unexcited active ions and the wavelength-dependent absorption cross-section. Spontaneously-emitted randomly-polarized photons are then generated by the Monte Carlo method according to the excited active ions density and emission cross-section. Bunches of the spontaneously-emitted photons are then traced through the amplifier volume giving rise to ASE. All of the surfaces of the amplifier can be considered to be Fresnel reflecting, anti-reflection (AR) coated, absorbing or diffusive. Various geometries, including rods, disks and slabs, can be investigated, and also, the ASE absorbing clad used to suppress parasitic oscillations can be involved. Using this code, the excited ion density, gain and heat deposition can be calculated. In the case of the lasers being developed at HiLASE, particular attention has been paid to the Yb:YAG disk amplifiers and Yb:YAG ceramics slab amplifiers with the Cr:YAG ceramics cladding. The heat deposition within the amplifier volume obtained from the energetics modeling is consequently used as the input parameter for the thermo-optical modeling, which quantifies the thermally-induced wavefront aberrations and thermal-stress-induced birefringence leading to the laser beam depolarization. The base calculation of the temperature and stress distribution is done by the three-dimensional finite-element method using COMSOL Multiphysics software. The lateral surfaces of the amplifiers can be considered to be cooled by direct thermal contacts with the heat sink or by forced or natural convection cooling by liquid or gas [21; 58; 59; 60]. The non-isothermal flow of the cooling medium is calculated separately in a 3D or simplified 2D, depending on the geometry, model resulting in the spatially-resolved heat transfer coefficient, which is then used as a boundary condition for fully 3D thermal stress-strain calculation. The temperature, mechanical stress and amplifier deformation are subsequently used for the calculation of the refractive indices' changes, which result in wavefrontaberrations. Jones calculus is used for the calculation of the beam depolarization induced by the thermal-stress birefringence. The typical results of the energetics and thermo-optical calculations are shown in Figure 17 and Figure 18. These results were obtained for a square-shaped slab with dimensions of a 45 \(\times\) 45 mm\({}^{2}\) Yb:YAG active area surrounded by a 20 mm-thick Cr:YAG absorber layer. The slab was pumped by 1-ms pulses with an intensity of 5 kW\(\cdot\)cm\({}^{-2}\).