# Modeling and Analysis of High-Power Ti:sapphire Laser Amplifiers-A Review
## 2 Amplification Model
### Depletion and Pulse Shape Change
An example of a calculation using Equation (5) is shown in Figure 1. The graph based on Equation (5) shows not only the pulse shape change that is induced by the depletion effect but also the total energy increment. However, as depicted by the blue curve in Figure 2a,b, the more simple and practical energy gain equation given by Equation (6) merely results in an average increment in the pulse energy. Figure 1: (**a**) A square seed pulse; (**b**) The solid lines show the amplified output pulse shape for a square seed pulse. They depict the depletion effect during the amplification. Figure 2: (**a**) Seed pulse (blue line) and corresponding square pulse train. (**b**) Amplified pulse shapes calculated by using Equation (3) (blue line) and Equations (7) and (8) (red pulse train). Depletion of stored energy in a gain medium as a function of (**c**) time and (**d**) time and position (optical path direction). A numerical method that achieves amplification of the pulse shape and energy simultaneously by using the energy gain equation given in Equation (6) is introduced [19]. In this method, as shown in Figure 2a, the input pulse, \(\eta_{i}n(t)\), is treated as a consecutive train of square-shaped pulses with the same pulse duration but different amplitudes. The amplitude of the \(i\)-th square pulse is \(\eta_{in}^{(i)}=\eta_{in}((i-1/2)\times\delta t)\), where the superscript \((i)\) indicates the sequence of the square pulse, \(i\) is an integer larger than one, and \(\delta t\) is the duration of the small square pulses. In addition, the uniformly distributed inversion density, \(\Delta_{0}\), is renewed for each square pulse by counting the stimulated emitted photons. The equations for the \(i\)-th pulse amplification are  
\[\eta_{out}^{(i)}=\eta_{in}^{(i)}G_{E}^{(i)}=\frac{1}{\gamma\sigma}\ln\left\{1 +\Big{[}\exp(\gamma\sigma\eta_{in}^{(i)})-1\Big{]}\exp\Big{(}\Delta_{0}^{(i)} \sigma L\Big{)}\right\} \tag{7}\]  
\[\frac{\partial n}{\partial t}=\sigma cn\Delta-\frac{\partial n}{\partial x}c. \tag{2}\]0  
Equation (8) is the energy conservation law in the gain medium. Example calculation results using Equations (7) and (8) are shown in \[\frac{\partial n}{\partial t}=\sigma cn\Delta-\frac{\partial n}{\partial x}c. \tag{2}\]1 As shown in Figure 2b, the envelope of the output square pulse train is identical to the calculation result obtained using Equation (3). Although Equation (7) is based on Equation (6), which is derived under the assumption of a square-shaped seed pulse, this numerical method is applicable to an arbitrary input pulse shape as given in Equation (3) because there is no restriction on the envelope shape of the input square pulse train. Furthermore, because the numerical method does not include the integral calculations, which Equation (3) does, the proposed method is more convenient for various complex pulse shapes. Contrastingly, as shown in Figure 2c, the results obtained using Equation (8) exhibit a decrease in the depletion of the inversion density under the assumption of a uniformity in the entire gain medium. To consider an arbitrary inversion density distribution, one can additionally treat the gain medium as a group of consecutive thin mediums. In this case, an additional spatial dimension index, \(j\), can be used for Equations (7) and (8). The initial inversion density of the \(j\)-th medium is \(\Delta_{0}^{(1,j)}=\Delta_{0}((j-1/2)\times\delta L)\), where \(\delta L\) is the thickness of the sliced medium, and \(\eta_{in}^{(i,j+1)}=\eta_{out}^{(i,j)}\). An example of the spatio-temporal depletion calculation result is shown in Figure 2d. The above-described numerical method is based on the Frantz-Nodvik energy gain equation of Equation (6). There also exist modified Frantz-Nodvik equations for more complex amplifier geometries such as a zig-zag slab [26] and multiple passes in a thin disk [27]. In the case of a double-pass in multiple rod-shaped medium, a modified Frantz-Nodvik equation for two inputs [19] can be used with the above-described numerical method (Figure 3).