# Instantaneous Frequency Representation Used for CPA Laser Simulation
## 2 Instantaneous Frequency Representation for a Simple CPA Simulation  
Large time-bandwidth product pulses are difficult to represent. In general, there are no other options other than using a very large temporal excursion that covers the full duration along with a large bandwidth that covers the spectral amplitude of the pulse. As both domains are linked by Fourier transformation, the temporal resolution is inversely proportional to the spectral excursion and vice versa. To visualize such relationships and large time-bandwidth product pulses, time-frequency representations, such as the Wigner-Ville distribution or spectrogram, are commonly used. Here, we introduce the instantaneous frequency representation (IFR). The instantaneous frequency \(\omega(\tau)=\partial\varphi/\partial\tau\) weighted by the associated spectral amplitude is fully representative of a laser pulse. Furthermore, in terms of physics, the line that represents the spectral distribution in the temporal domain can be convoluted by the Fourier-transformed pulse profile temporal amplitude. The IFR of a Fourier transform-limited pulse is shown in Figure 1. For a Wigner-Ville representation, both domains would have the same number of points to easily calculate the Fourier transformation, as represented by the grey grid in Figure 1. Thus, the number of points \(\mathrm{N}\), resolutions (\(\mathrm{sf}\), \(\mathrm{\delta\tau}\)) and excursions (\(\mathrm{\Delta F}\), \(\mathrm{\Delta T}\)) of both domains are fully determined:  
\[\mathrm{\Delta F}\mathrm{\delta\tau}=\mathrm{\Delta T}\mathrm{\delta f}=1\ \mathrm{and}\ \frac{\mathrm{\Delta F}}{\mathrm{\delta f}}=\frac{\mathrm{\Delta T}}{\mathrm{ \delta\tau}}=\mathrm{N}=\frac{1}{\mathrm{\delta f}\mathrm{\delta\tau}} \tag{4}\]  
For Fourier transform-limited pulses, the time-bandwidth product is minimal and close to 0.5:  
\[\mathrm{\Delta\omega}_{\mathrm{pulse}}\mathrm{\Delta t}_{\mathrm{pulse}}\geq 0.5 \tag{5}\]  
where \(\mathrm{\Delta\omega}_{\mathrm{pulse}}\) and \(\mathrm{\Delta t}_{\mathrm{pulse}}\) are statistical widths (root mean squares). On the IFR, similar to the Wigner-Ville representation, the spectral or temporal amplitude can be recovered by the projection of the curve along one dimension (Figure 1(1,2)). A Fourier-limited pulse is represented as a vertical line. The instantaneous frequency curve \(\tau(\omega)=(\partial\varphi/\partial\omega)_{\omega_{1}-\omega_{0}}=0\)) is represented by the black-dotted line in Figure 1 (3). Applying linear filters, such as spectral transmission and dispersion to the pulse, can be performed directly on the IFR with line-by-line modification. A pure spectral amplitude corresponds to a simple multiplication of the line \(\omega_{\rm i}\) by the spectral amplitude \(\mathrm{T}(\omega_{\rm i})\). After line-by-line multiplication, the new spectral amplitude is used to calculate the new temporal Fourier limit. This Fourier limit temporal profile is then applied onto the temporal pulse by convolution. In this case, the constraint of having the same number of points with the temporal excursion limited by the spectral resolution is not limiting, as the pulse time-bandwidth product is still minimal. However, in a CPA laser, the pulse is stretched. Its time-bandwidth product rises to about \(10^{5}\). The temporal excursion of the pulse is then greater than that used for the simulation, as shown by the grey grid in Figure 2a (3). By using Fourier transformation, such as in Wigner-Ville or spectrogram simulation, the temporal domain is restricted to this area. The simulation then presents temporal aliasing that is harmful to the result, as shown in Figure 2a (3), as well as by the blue curve in Figure 2a (1). For conventional large-scale CPA lasers, the initial temporal domain is so small that it becomes nearly invisible on the IFR (Figure 2b (3)). With a conventional Fourier transform, in order to avoid this temporal aliasing, the only solution is to proportionally increase the number of samples on both axes.