# Instantaneous Frequency Representation Used for CPA Laser Simulation
## 3 Instantaneous Frequency Representation for a Realistic CPA Laser Simulation Step by Step  
A typical CPA laser chain is depicted in Figure 5. Here, we follow the scheme presented by [PERSON] et al. [11]. The pulses are derived from a high repetition rate mode-locked oscillator with an initial pulse length order of 25 fs. The laser pulse train typically passes Pockels cells, Faraday isolators and polarizers to decrease the repetition rate and for back-reflection isolation within the laser chain. The repetition rate is reduced for amplification to the order of a few Hz or less, especially in the case of petawatt class systems. After being stretched, typically from a time-bandwidth product of 0.5 to approximately \(10^{5}\), or from 25 fs to \(1\)ns, the laser pulse is amplified and recompressed close to the Fourier limit, which is determined by the spectral shape. The actively controlled spectral shape of the amplified pulse is typically a top hat, which is taken into account in all of the simulations presented here. Since the huge dispersion generated by the stretcher far outweighs the dispersion  
Figure 4: Instantaneous frequency representation of stretched pulses after FWM due to the Kerr effect. The line labeled A is the main pulse, B is the post-pulse, C is the generated pre-pulse, and D is the generated post-pulse. The spectral axis shows the interference in between the four different pulses. of the material in the amplifier chain, we only consider the stretcher and compressor as dispersive elements at this time. Post-pulses are typically generated when the laser pulse passes through elements with plane-parallel surfaces. While for some cases this issue can be avoided by use of wedged components, causing spatio-temporal distortions and degrading the laser pulse quality in the focus, for others, this might not be possible. Anti-reflection coatings reduce the relative pulse level of the generated post-pulse to a certain degree. Plan-convex or plan-concave lenses can also cause post-pulses within a certain angular acceptance. For the simulation discussed here, we considered a rather large post-pulse level of 1% with respect to the main pulse and a delay of 100 ps. Taking this into consideration, the spectral group delay can be expressed as:  
\[\tau(\omega) =\sum_{n=2}^{\infty}\frac{\varphi_{n}}{\left(n-1\right)!}\left[ \left(\omega-\omega_{0}\right)^{n-1}\right]\;\approx\;\varphi_{2}(\omega- \omega_{0})+\frac{\varphi_{3}}{2}(\omega-\omega_{0})^{2}+\frac{\varphi_{4}}{3! }(\omega-\omega_{0})^{3}\] (13) where \[\varphi_{2}\] is the linear chirp, mainly stretching the pulse and called the group velocity dispersion (GVD); \[\varphi_{3}\] is the third-order spectral phase (third-order dispersion TOD), i.e., the first distortion order on the linear chirp; and \[\varphi_{4}\] is the fourth-order spectral phase (fourth-order dispersion FOD), i.e., the second distortion order on the linear chirp. As sketched in Figure 5b, these distortions modify the previously used simplified IFR where only the linear chirp was applied, in contrast to [PERSON] et al. [11]. The pulse and its post-pulse will undergo the Kerr-effect-induced FWM, which is predominantly induced by the amplifier materials [10]. After amplification, the pulses are recompressed by the compressor, such that the main pulse is near-perfect dispersion compensated and compressed as in the previous example. The first simulation considers only pure linear chirp stretching (Figure 6). The overall B-integral is chosen such that the pre-pulse generates yields close to the same power as the post-pulse [12], meaning about \(\sqrt{3}\). As expected, in the output pulse, the Kerr effect through FWM generates a \"time-diffracted\" pre-pulse blue shifted in frequency. A post-pulse should also appear, but it is too weak to be visible on the dynamic scale of 80 dB. In the ideal case of a pure chirp, as already mentioned by [PERSON] et al.