# Modeling and Analysis of High-Power Ti:sapphire Laser Amplifiers-A Review
## 2 Amplification Model
### Amplification of Chirped Pulses  
In the previous section, we presented the development of an amplification model under the assumption of a quasi-monochromatic seed pulse. This assumption is valid only when the spectral bandwidth of the seed pulse is much narrower than the gain bandwidth of the medium. The gain bandwidth of Ti:sapphire is 222 nm (FWHM) at room temperature [28; 29]. Based on the time-bandwidth product rule [17; 25; 30], for Gaussian pulses of 100 fs and 10 fs pulse lengths (FWHM), spectral linewidths of at least 9.4 nm and 94 nm are required, respectively. As a femtosecond pulses have a non-negligible spectral bandwidth relative to the gain bandwidth of the Ti:sapphire medium, the amplification model should be improved to avoid using the monochromatic seed pulse assumption. Before describing the amplification model for non-monochromatic pulses, a brief introduction to the ultrashort pulse amplification technique is needed. Because ultrashort pulses have high optical intensity [W/cm\({}^{2}\)], they can easily damage optics residing within the amplifier. Lowering the intensity by expanding the beam size is restrictive because the available sizes of the optical and amplifying media are limited. Therefore, the development of laser intensity was stagnant until a breakthrough was achieved. As can be seen in Figure 4, this breakthrough was the CPA technique [25]. As depicted in Figure 4, in a CPA system, to lower the optical intensity during the amplification process, the pulse duration before the amplification is increased and the amplified pulse is recompressed. In 2018, the first demonstrators of the CPA technique were awarded the Nobel prize [31]. The CPA technique has been used in various spectral phase manipulation devices to control the temporal width of the pulse. In the first demonstration of the CPA technique [25], an optical fiber and Treacy type compressor [32] were used as the stretcher and compressor pair. Subsequently, the fiber stretcher was replaced by [PERSON]'s stretcher [33] to compensate for the third-order dispersion and by a lens-less Offner type stretcher to avoid aberrations [34; 35]. In addition to the stretcher and compressor pair, various dispersion control methods have been developed [36; 37; 38; 39]. Figure 4: (**a**) History of laser intensity [40] and (**b**) a schematic of a chirped pulse amplification (CPA) system [25]. Figure 3: Definitions of inputs and outputs in double-pass geometry. As the CPA technique is commonly used for the amplification of an ultrashort pulse in a Ti:sapphire medium, the wavelength, \(\lambda\), of the simulation model should be considered as a function of time. In addition, the emission cross-section, \(\sigma\), should be a function of the wavelength, \(\lambda\), because an ultrashort pulse has a non-negligible spectral bandwidth. Consequently, the emission cross-section \(\sigma\) becomes a function of time, as represented in Figure 5 and Equation (12). \[\sigma(\lambda)=\sigma\big{(}\lambda(t)\big{)} \tag{12}\]  
The spectral emission cross-section, \(\sigma(\lambda)\), can be obtained from a fluorescence spectrum measurement and the McCumber theory [28; 41], and an example with Ti:sapphire is shown in Figure 5. The time-dependent wavelength, \(\lambda(t)\), depends on the structure and configuration of the stretcher and can be calculated by using a wavelength-dependent group delay function. \[Group\ delay(\lambda)=-\frac{\lambda^{2}}{2\pi c}\frac{\partial\varphi_{st}}{ \partial\lambda} \tag{13}\]  
where \(c\) is the speed of light and \(\phi_{st}\) is the spectral phase of the stretcher. The amplification calculation for a chirped seed pulse, where the cross-section is a variable of time, can be achieved by adding a discretized time index, \((i)\), to \(\sigma\) in Equation (7).