# Modeling and Analysis of High-Power Ti:sapphire Laser Amplifiers-A Review
## 2 Amplification Model
### Effects of Transverse Profiles  
In this section, we will discuss how the transverse profile of a laser affects the amplification process. A one-dimensional(1D) amplification simulation assumes that the transverse profile of the laser is a flat-top, so that the energy is much higher compared to the actual value. The seed and pump beams used in Ti: sapphire amplifiers do not have a uniform spatial distribution and flat-top steep edges. For a residual pump beam [17], the energy of the edge of the pump beam is unused for amplification, which reduces the energy efficiency of the amplifier. To obtain accurate simulation results, the amplification process should be calculated considering both the transverse profile of the seed beam and pump beam. By considering the spatial beam distribution of the seed beam and pump beam in a two-dimensional(2D) amplification simulation, we can achieve a better agreement with experimental results. Spatially expanding the Frantz-Nodvik equation for the CPA used in a 1D calculation allows simple 2D amplification calculations to be performed. Assuming that the thickness of the laser medium is ignored, the gain and output fluence of the amplifier can be calculated from Equations (19) and (20). \(\lambda_{laser}\) and \(\lambda_{pump}\) are the wavelengths of the seed laser and pump laser, respectively, \(G_{0}\) is the gain. \(J_{in}J_{out}\) are the input and output fluences of the seed beam along with the wavelength, respectively, and \(J_{pump}\) and \(J_{sat}\) are the fluence of the pump beam, and saturation fluence, respectively. The length of a chirped pulse is typically about 1 ns, and the optical path difference between the passes in a multi-pass amplifier, which is mainly used as the main amplifier, is longer than this. Therefore the effect of overlapping of the pulses in the medium does not have to be considered. \[J_{out}(x,y,\lambda)=J_{sat}\ln\left[1+G_{0}(x,y,\lambda)\left(\,\exp\left[ \frac{J_{in}(x,y,\lambda)}{J_{sat}(\lambda)}\right]-1\right)\right] \tag{19}\]  
\[G_{0}(x,y,\lambda)=exp[\lambda_{pump}/\lambda_{laser}\times J_{pump}(x,y)/ J_{sat}(\lambda_{laser})] \tag{20}\]  
A 2D amplification simulation can be used for the spatial shaping of the pump beam, which is one of the most important factors in a laser amplifier, as well as for obtaining simulation results similar to the experimental results. The spatial distribution of the output laser beam through the laser amplifier follows that of the pump beam. Therefore, the pump beam should have a high steepness and high uniformity for uniform and efficient amplification. If the steepness is low, it implies a slow rising edge, and the pump beam energy cannot be efficiently transferred to the seed beam owing to the low gain near the edge of the beam, leaving energy in the gain medium. This effect allows the amplifier to have a low energy extraction efficiency. Moreover, the non-uniformity of the pump beam results in an amplified beam with a low spatial beam quality. One approach to minimize this problem and improve the quality of the output beam is to pump a flat-top laser beam into the gain medium through a beam shaping system, such as a beam homogenizer using a lens array or diffractive optics [58; 59]. A beam shaping system increases the edge steepness and spatial uniformity of the pump beam formed in the gain medium and improves the quality of the output beam by making the spatial distribution uniform. By combining beam shaping system design techniques and amplifier simulation, an optimal amplifier condition can be designed to maximize the energy extraction efficiency and spatial uniformity. The edge steepness and beam uniformity criteria described in ISO13694 can be used to optimize the amplifier simply by evaluating the spatial distribution of the beam.