# Instantaneous Frequency Representation Used for CPA Laser Simulation
## 2 Instantaneous Frequency Representation for a Simple CPA Simulation
Without nonlinearities, the pulses are then amplified with minor dispersions and minor spectral narrowing due to amplification. With proper dispersion management, the main pulse is perfectly recompressed by the compressor, and the output and input pulses look very similar to one another. PW-class CPA systems are designed to maximize the extracted laser pulse power, resulting in an operation close to or within an intensity range of the stretched pulse that causes significant nonlinearities. This results in laser pulse degradation that is deleterious for ultra-high laser applications. The main effect is the Kerr effect, globally characterized by the B-integral. The most deleterious impact of this effect on the temporal contrast is the pre-pulse generation from a post pulse by temporal diffraction [6,10,11]. Let us consider the two pulses of Figure 3, \(\mathrm{E}_{1}\) representing the main pulse and \(\mathrm{E}_{2}\) a post-pulse time, delayed by \(\mathrm{t}_{\mathrm{d}}\). The third-order nonlinear polarization due to the Kerr effect [11] is  
\[\mathrm{P}_{\mathrm{NL}}(\tau)=\chi_{3}[\mathrm{E}_{1}(\omega_{1}(\tau))+ \mathrm{E}_{2}(\omega_{2}(\tau))][\mathrm{E}_{1}^{*}(-\omega_{1}(\tau))+ \mathrm{E}_{2}^{*}(-\omega_{2}(\tau))][\mathrm{E}_{1}(\omega_{1}(\tau))+ \mathrm{E}_{2}(\omega_{2}(\tau))], \tag{9}\]  
where it gives rise to four-wave mixing (FWM), self-phase modulation (SPM), cross-phase modulation (XPM) and cross-polarized wave generation (XPW). In particular, the terms \(\mathrm{E}_{1}(\omega_{1}(\tau))\mathrm{E}_{2}^{*}(-\omega_{2}(\tau))\mathrm{E }_{1}(\omega_{1}(\tau))\) and \(\mathrm{E}_{2}(\omega_{2}(\tau))\mathrm{E}_{1}^{*}(-\omega_{1}(\tau))\mathrm{ E}_{2}(\omega_{2}(\tau))\) are the FWM processes of interest, giving the new frequencies \(2\omega_{1}-\omega_{2}\), and \(2\omega_{2}-\omega_{1}\). We see that a pre-pulse is generated \(\mathrm{t}_{\mathrm{d}}\) before the main pulse and \(8\omega_{1}=\left(2\omega_{1}-\omega_{2}\right)_{\tau}-\omega_{1}=\mathrm{t} _{\mathrm{d}}\), so it  
Figure 1.0 Instantaneous frequency representation of the stretched main and post-pulse. The main pulse (A) is followed by a post-pulse (B) with a delay of 100 ps. The combination of two pulses results in a spectral interferometric pattern on the spectral axis. is constantly blue-shifted from \(\omega_{1}(\tau)\) as b > 0 in PW-class CPA. Similarly, a post-pulse is also generated \(t_{d}\) after the post-pulse and \(\delta\omega_{2}=(2\omega_{2}-\omega_{1})_{\tau}-\omega_{2}=-bt_{d}\) is red shifted from \(\omega_{2}(\tau)\). Here, \(\delta\omega_{1}\) indicates its origin from the main pulse \(\omega_{1}\), and \(\delta\omega_{2}\) indicates its origin from the post-pulse \(\omega_{2}\). The FWM process is efficient as long as all spectral components are phase matched. The phase matching is kept since the components \(\omega_{1}(\tau)\) and \(\omega_{2}(\tau)\) are nearly equal. On the IFR, this temporal effect is simulated column by column. For any \(\tau_{i}\), the temporal amplitude is calculated by using the inverse Fourier transform:  
Figure 1.1  
Then, the nonlinear effect is applied in this time domain:  
Figure 1.2  
Finally, the spectral amplitude at any \(\tau_{i}\) is obtained by Fourier transform:  
Figure 1.3  
The pre-pulse and post-pulse generation appears naturally from this process as shown in Figure 1.4  
If \(\forall\tau_{i},\;i\in\{1..N_{\tau}\},\;b(\tau_{i})\) is large enough, then one can approximate: \(A(t,\tau_{i})\approx\Big{[}\frac{\widetilde{A}(\omega,\tau_{i})}{b(\tau_{i})} \Big{]}\). Thus, \(\widetilde{A_{pre-pulse}}(\omega,\tau_{i})\approx\alpha\Big{|}\widetilde{A}( \omega,\tau_{i})\Big{|}^{2}\widetilde{A_{post-pulse}}(\omega,\tau_{i})\), where \(\alpha\) is the coupling factor from post-pulse to pre-pulse [12].