# Instantaneous Frequency Representation Used for CPA Laser Simulation
## 2 Instantaneous Frequency Representation for a Simple CPA Simulation
This makes it difficult to use Fourier transformations for very strongly stretched pulses with huge time-bandwidth products, as the required number of points grows quadratically alongside the stretching rate. However, by using IFR, it is possible to stretch the temporal scale without restriction (Figure 2a,b), which makes the method very useful for large-stretched CPA laser pulses. This stretching corresponds to a translation of the points on the instantaneous curve,  
Figure 1: Instantaneous frequency representation of a Fourier-limited pulse in (1) temporal intensity, (2) spectral intensity (\(\hat{\mathrm{I}}(\omega)=\left|\exp\left(-\left(\frac{\hat{t}}{0.426}\right)^ {80}\right)(1-0.3\cdot\exp\left(-\frac{(\hat{t}-0.05)^{2}}{0.002}\right)\right| ^{2}\)) and (3) instantaneous frequency with typically resolutions required in Fourier transformations (grid) for comparison. To avoid symmetric spectral shape while keeping a good visibility on the IFR, the pulse chosen exhibits a dip on its spectral intensity. depending on the frequency in time. For each \(\omega_{i}\), the position of the new point is modified by the relative delay from the central pulsation:  
\[\mathrm{d}\tau(\omega_{i})=(\partial\varphi/\partial\omega)\omega_{i-}\omega_{0}. \tag{6}\]  
The operation has no temporal limitation and does not need a regular temporal pattern. It is a pure line-by-line spectral operation. The output temporal scale is no longer linked to the spectral one by a regular point-to-point pattern. The temporal amplitude estimation will then require either resampling of the curve on a regular temporal pattern or consideration of the irregular pattern in the amplitude calculation. The IFR will then be compatible with any time-bandwidth product. Pulse replicas are also due to linear operations but combine both spectral amplitude and spectral phase. As an example, a post-pulse delayed by \(\mathrm{t}_{\mathrm{d}}\) is simply introduced by replicating the IF curve and translating the full curve by \(\mathrm{t}_{\mathrm{d}}\). Here, the absolute phase difference at \(\omega_{0}\) between the pulses is ignored. It can be considered by using an additional  
Figure 2: Instantaneous frequency representations of (**a**) a chirped pulse with moderate chirp (\(\varphi^{(2)}=\frac{2}{(\pi\Delta f)^{\prime}}8.5\cdot 10^{3}\)), where \(\pi\Delta f\)is the frequency range, and (**b**) a typical chirped pulse with stretching factor of 105, typical for CPA lasers (\(\varphi^{(2)}=\frac{2}{(\pi\Delta f)^{\prime}}1.6\cdot 10^{6}\)). In (a3), the grey area is a reduced time window, typically used for Fourier transformations, and it is depicted together with the resulting aliasing. This leads to a distorted temporal representation (blue inset in (a1)). In (b1), this temporal span is even smaller for large stretching (blue line). array within the phase values or by using complex weights rather than purely real ones. The calculation of the spectral amplitude is then more complex as it requires the integration of all the different \(\omega_{\mathrm{i}}\) components. The sum of these components includes the phase terms that are frequency dependent \(\varphi(\omega_{\mathrm{i}})=\omega_{\mathrm{i}}\mathrm{f}_{\mathrm{d}}\). The spectral amplitude is then obtained through the following integration:  
\[\widetilde{\mathrm{A}}(\omega_{\mathrm{i}})=\int_{-\mathrm{T}}^{\mathrm{T}} \mathrm{A}(\tau)\mathrm{e}^{\mathrm{i}\omega_{\mathrm{i}}\tau}\mathrm{d}\tau. \tag{7}\]  
By symmetry, the temporal intensity is also recovered by integration as:  
\[\mathrm{A}(\tau_{\mathrm{i}})=\int_{-\infty}^{\infty}\widetilde{\mathrm{A}}( \omega)\mathrm{e}^{\mathrm{i}\omega\tau_{\mathrm{i}}}\mathrm{d}\omega. \tag{8}\]  
As expected by its linear nature, this operation is still performed line by line (spectral amplitude) or column by column (temporal amplitude). Combining the main pulse with a post-pulse, generated, e.g., by partial internal reflection in planar transmission optics, and stretching results in what usually occurs in CPA systems (Figure 3).