# Instantaneous Frequency Representation Used for CPA Laser Simulation
## 3 Instantaneous Frequency Representation for a Realistic CPA Laser Simulation Step by Step
[11], the generated and compressed pre-pulse appears exactly with the mirrored delay of the original post-pulse but with significant frequency shifts (Figure 6a-c). This blue shift can be significant compared to the pulse bandwidth and can lead to an underestimated temporal deterioration by third-order cross-correlators with limited bandwidths [13]. The generated post-pulse can also be resolved on the temporal intensity display. Figure 5: Schematic diagram of a typical laser chain, including post-pulse generation. The simulation was performed on a standard personal computer with a 2000-point array in a few seconds without any code optimization. The chirp stretched the pulse by a factor of 40,000 (1 ns stretched pulse), so the delay of 100 ps is still observable (Figure 3). Smaller or larger delays could be used without significant impact on simulation time and precision of the result. In the more interesting case of distortions of the linear chirp, commonly present in the CPA stretcher and compressor realizations, the generated pre-pulse is modified and not recompressed completely at the output by the compressor. In the case of third-order spectral phase distortions, the pre-pulse is chirped (Figure 7) and its peak power can be decreased by two orders of magnitude. In contrast, the generated post-pulse appears unaffected, changing the relative pulse height of the pre- and post-pulse. The apparent delay of the pre-pulse is also shifted to a smaller delay than the equivalent delay of the post-pulse by symmetry. For a fourth-order spectral phase distortion, the pre-pulse exhibits a third-order spectral phase distortion (Figure 8). Its maximum peak power is also decreased but less than for the third-order distortion. The generated post-pulse disappears below \(-80\) dB. In both cases, interestingly, the peak power of the pre-pulse is decreased by the stretching and detuning of the pure linear chirp. This pre-pulse temporal intensity distortion is similar to the one observed by [PERSON] et al. [6]. A proper stretching distortion could thus be intentionally applied to decrease the pre-pulse peak power and enhance the temporal contrast of the laser. This observation can be understood by the effect of the blue shift on the compression. Without the blue shift, the compressor fully compensates for the dispersion of the system. With the blue shift, the compensation is incomplete:  
\[ \begin{array}{rl}\delta\tau_{8\omega_{1}}(\omega)&=\sum_{n=2}^{\infty}\frac {\varphi_{n}}{(n-1)!}\left[\begin{array}{c}(\omega-\omega_{0})^{n-1}\\ -(\omega+\delta\omega_{1}-\omega_{0})^{n-1}\end{array}\right]\\ &=\sum_{n=2}^{\infty}\frac{\varphi_{n}}{(n-1)!}(-\delta\omega_{1})\left[\sum_{ k=0}^{n-2}\sum_{j=0}^{k}\left(\begin{array}{c}\mathrm{k}\\ \mathrm{j}\end{array}\right)(\delta\omega_{1})^{\mathrm{j}}(\omega-\omega_{0}) ^{\mathrm{k}}\right]\end{array}  \tag{14}\]  
Figure 6: (**a**) Instantaneous frequency representation after compression of a linear chirp, (**b**) temporal intensity on log scale (dB) and (**c**) normalized pre-pulse spectrum.