# Modeling and Analysis of High-Power Ti:sapphire Laser Amplifiers-A Review
## 3 Phase Distortion Analysis  
### Atomic Phase Shift (Population Inversion Induced Phase Shift)  
The atomic phase shift, i.e., a population inversion induced phase shift, produces a temporal nonlinear phase distortion in an amplified pulse. In a CPA system, a temporal phase shift causes a spectral phase shift and is combined with gain depletion, resulting in a nonlinear spectral phase distortion. Gain depletion is a phenomenon in which the population inversion is reduced during the amplification process, and the gain is decreased. A reduction in the gain causes a nonlinear effect that redshifts the center of the spectrum, which in combination with the atomic phase shift induces a spectral phase shift in the chirped pulse.  
\[k_{e}=\frac{\omega}{c}\frac{1}{2}\chi_{e}=\frac{\Delta N(t)\sigma_{0}}{2}\frac {\gamma}{\omega_{0}-\omega-i\gamma/2}=i[g(t,\omega)-i\varphi(t,\omega)]/L \tag{21}\]  
\[g(t,\omega(t))=g(t)=\Delta N(t)\sigma_{0}L\frac{\left(\Delta\omega/2\right)^{ 2}}{\left(\omega_{0}-\omega(t)\right)^{2}+(\Delta\omega/2)^{2}} \tag{22}\]  
Figure 9: Pumping the gain medium using a square beam (**a**) multiple circular beams (**b**) beamlets divided by a lens array (**c**) square-shaped pump beam.  
\[\varphi(t,\omega(t))=\varphi(t)=\frac{\Delta N(t)\sigma_{0}L}{2}\frac{\Delta\omega/2 (\omega_{0}-\omega)}{\left(\omega_{0}-\omega(t)\right)^{2}+(\Delta\omega/2)^{2}} \tag{23}\]  
The atomic phase shift is derived from the classical electron model. Equation (21) expresses the wavenumber, \(k_{e}\), for an excited ion, and the gain and phase shift for the excited ion are obtained from Equations (22) and (23) [23]. In a CPA system, the laser pulse is chirped; specifically, because the frequency is distributed in the time domain, the phase shift and gain, which are functions of time \(t\) and frequency \(\omega\), can be expressed solely as a functions of time \(t\). \(\Delta N\) is the population inversion, \(\sigma_{0}\) is the peak emission cross-section, \(L\) is the length of the laser medium, and \(\omega_{0}\) and \(\Delta\omega\) are the peak angular frequency and bandwidth, respectively. The population inversion over time is considered in Equations (22) and (23); therefore, this model considers the gain depletion effect.  
Because the phase shift in amplification is superimposed, the atomic phase shift effect in a laser amplifier chain is obtained by adding all the phase shifts generated in each amplification pass. To investigate the effect of the pulse distortion caused by the atomic phase shift in an amplifier chain structure, as shown in Figure 10, the amplifier simulation introduced in the previous section and the phase shift have to be combined. Figure 11 shows the output spectrum and accumulated atomic phase shift of each amplifier. The distortion in the phase shift due to the gain depletion effect is reflected in the asymmetric phase shift with respect to the center of the spectrum. The phase shift caused at each amplification stage is superimposed and distort the peak power and pulse width of the compressed pulse in the pulse compression as shown in Figure 12.  
Figure 11: (**a**) The output spectra of the amplifier and (**b**) atomic phase shifts.  
Figure 10: Petawatt laser amplifier chain.