# Modeling and Analysis of High-Power Ti:sapphire Laser Amplifiers-A Review
## 2 Amplification Model  
### Depletion and Pulse Shape Change  
In this section, we describe the fundamental aspects and modeling method of a pulse amplification under the following assumptions: a quasi-monochromatic seed pulse, an initially population-inverted gain medium, an infinitely long fluorescence lifetime, a negligible loss, and a uniform transverse distribution of the pump and seed beam. In the following sections, we will expand the model for broadband chirped seed pulse, pumping during amplification, a finite fluorescence lifetime, and non-uniform beam profiles. If we ignore the finite fluorescence lifetime and pumping during the amplification process, then the inversion density, \(\Delta\), is dependent only on the stimulated emission. The time-dependent rate equation for \(\Delta\) will be  
\[\frac{\partial\Delta}{\partial t}=-\gamma\sigma cn\Delta, \tag{1}\]  
where \(\gamma=1+g_{2}/g_{1}\), \(g_{1,2}\) is the degeneracy of the lower and upper energy levels, \(c\) is the speed of light, \(\sigma\) is the emission cross-section, \(n\) is the photon density [1] (ch.4.1.1). The photon density increases as photons are generated by the stimulated emission and decreases as the photons flow out of the gain medium. The time-dependent photon-transfer equation [1] (ch.4.1.1) describes the rate of the photon density change as  
\[\frac{\partial n}{\partial t}=\sigma cn\Delta-\frac{\partial n}{\partial x}c. \tag{2}\]  
[PERSON] and [PERSON] solved the coupled equations given in Equations (1) and (2) and found a solution, i.e., Equations (3) and (4) [18]. This rate equation approach is valid only for the seed pulses for which the pulse duration is longer than the phase relaxation time of the gain medium [24] (Ch.3.2.2). the typical phase relaxation time of a solid laser medium is \(10^{-11}\)-\(10^{-14}\) s [24] (Ch.3.1). Therefore, the amplification modeling of a femtosecond pulse by using the Frantz-Nodvik solution is inappropriate. However, most femtosecond amplifiers achieving a high-energy pulse by using the CPA technique will be introduced in more detail in Section 2.2 [25]. A chirped pulse generally has a nanosecond pulse duration. Consequently, the Frantz-Nodvik solution can also be used for most Ti:sapphire laser amplifier modeling. \[n(x,t)=\frac{n_{0}(t-\frac{x}{c})}{1-\{1-\exp[-\sigma\int_{0}^{x}\Delta_{0}(x ^{\prime})dx^{\prime}]\}\exp[-\gamma\sigma c\int_{-\infty}^{t-x/c}n_{0}(t^{ \prime})dt^{\prime}]} \tag{3}\]  
\[\Delta(x,t)=\frac{\Delta_{0}(x)\exp[-\sigma\int_{0}^{x}\Delta_{0}(x^{\prime}) dx^{\prime}]}{\exp[-\gamma\sigma c\int_{-\infty}^{t-x/c}n_{0}(t^{\prime})dt^{ \prime}]+\exp[-\sigma\int_{0}^{x}\Delta_{0}(x^{\prime})dx^{\prime}]-1} \tag{4}\]  
Equations (3) and (4) are applicable for an arbitrary seed pulse of \(n_{0}(t)\) and an arbitrary initial population inversion distribution of \(\Delta_{0}(x)\). These above arbitrary seed shape and inversion distribution are the functions for of the longitudinal (propagation) direction, and we will consider the transverse profile of for the seed beam and inversion density in Section 2.4. If we assume a square seed pulse of duration \(\tau\) and a homogeneous initial population inversion distribution, then the photon density equation can be further simplified [18]. \[n(x,t)=\frac{n_{0}}{1-\left[1-\exp(-\sigma\Delta_{0}x)\right]\exp[-\gamma\sigma \eta(t-x/c)/\tau]} \tag{5}\]  
where \(n_{0}\) [cm\({}^{-3}\)] is the amplitude of the square pulse, \(\Delta_{0}\) [cm\({}^{-3}\)] is uniform inversion density, and \(\eta\stackrel{{\text{def}}}{{=}}n_{0}c\tau\) [cm\({}^{-2}\)]. By integrating Equation (5) at the input and output of the gain medium, one can determine the energy gain, \(G_{E}\), between the seed and amplified pulse. \[G_{E}=\frac{1}{\gamma\sigma\eta}\ln\{1+\left[\exp(\gamma\sigma\eta)-1\right] \exp(\gamma_{0}\sigma L)\} \tag{6}\]  
where \(L\) is the length of the gain medium.