# Modeling and Analysis of High-Power Ti:sapphire Laser Amplifiers-A Review
## 2 Amplification Model
### Pumping and Spontaneous Emission  
As described in Section 2.1, the Frantz-Nodvik equation is derived under the assumption of an initial population-inverted gain medium [18]. In addition, the stimulated emission is considered as the only process that induces the depopulation of the inversion. Specifically, in the original Frantz-Nodvik calculation, the effects of instantaneous pumping and spontaneous emission during the pulse amplification process are ignored. However, instantaneous pumping or a spontaneous emission is ignorable only when the seed pulse duration is much shorter than the pump pulse duration or the fluorescence lifetime of the gain medium, respectively. To consider the instantaneous pumping and a spontaneous emission into account in the amplification calculation, one should add the two terms representing pumping and emission in the coupled energy conservation equation, Equation (8). The modified result is Equation (18) [43]. \[\Delta_{0}^{(i+1)}=\Delta_{0}^{(i)}-(\eta_{out}^{(i)}-\eta_{in}^{(i)})\frac{1} {L}+W_{p}^{(i)}\delta t-\frac{1}{\tau_{f}}\Delta_{0}^{(i)}\delta t \tag{18}\]  
where \(W_{p}^{(i)}\) is the effective pump rate [cm\({}^{-3}\) s\({}^{-1}\)] and \(\tau_{f}\) is the fluorescence lifetime [s] of the gain medium. Equation (18) is relevant even when the seed and the amplified pulses do not exist. When there is no seed pulse, in Equation (18), \(\eta_{in}\) is zero and \(\eta_{out}\) is also vanishes based on Equation (7) or Equation (14). As shown in Figure 6a, as the pump pulse is absorbed by the gain medium, the number of inversion ions \(\Delta_{0}\), i.e., the effective stored energy, increases. As shown in Figure 6b, the stored energy decreases with the spontaneous emission even when there is no stimulated emission from the seed pulse. The spontaneous decay time is characterized as the fluorescence lifetime of the medium. By performing a simulation similar to the example shown in Figure 6, one can estimate the maximum stored energy and time required for the stored energy to reach the maximum. The stored energy is typically lower than the effective total pump energy because of the spontaneous emission loss, and the instantaneous moment at which the stored energy is highest is the best seed pulse injection timing in the single-pass amplifier. A Ti:sapphire amplifier is widely used in terawatt(TW) or petawatt(PW) CPA laser systems [8]. Figure 7a shows an example of a PW-class, tens of Joule pulse energy, fs pulse width, and multi-pass amplifier. In case of Figure 7, for 2.2 PW of peak power, 120 (\(16\times 7.5\)) J of pump energy, 2-stage of 3-pass amplifiers, 17.7 fs of pulse width, and 67.9 (48.6 after compression) J of pulse energy are assumed. In such a CPA system, the stretched seed pulse duration is a few nanoseconds and it takes tens of nanoseconds to pass through the multi-pass power amplifier [3; 17; 44]. Spontaneous emission would be negligible because the duration of amplification is much shorter than the fluorescence lifetime of the Ti:sapphire medium (3.2 us [28]). However, it is important to consider instantaneous pumping in a multi-pass amplification simulation because, as shown in Figure 7b,c, it allows tracing the stored energy changes in the entire amplification process. Figure 7b,c shows two simulation results for the amplifier with different seed injection times and delays between the passes and pump pulses. Other pump and seed beam conditions, such as the total energy and beam diameter are the same. The results of Figure 7b,c exhibit two relevant differences. First, the residual stored energy is different. This suggests that the extraction efficiency of the amplifier can be optimized by a simulation including the instantaneous pumping. Second, the peak values of the stored energy reached during the amplification processes are different. Large Ti:sapphire crystals, with diameters from 10 to 15 cm, are commonly used in high-power amplifiers to achieve a high pulse energy from tens to hundreds of joules [3, 17, 44, 45, 46, 47].