# Instantaneous Frequency Representation Used for CPA Laser Simulation
## 1 Introduction  
Ultra-intense laser systems are used for a variety of applications in the fields of high-energy-density physics and relativistic laser plasma physics [1]. Achievable contrast ratios are in the range of \(10^{-12}\) on temporal scales between ns and a few 10 ps prior to the main pulse. Amplification of stretched pulses (i.e., chirped pulse amplification, CPA [2]) is conventionally performed in regenerative and multipass amplifiers. Despite its capacity to lower nonlinearities in the amplifiers, the CPA technique, when used for an extremely high-power system (PW range), still accumulates nonlinear distortions due to the Kerr effect, which is characterized by the maximum value of the B-integral. The B-integral represents the nonlinear temporal phase shift acquired after propagation through the system. Typical levels of the B-integral are in the range of 0.1 to 1 [3; 4; 5; 6]. To simulate the consequence of this nonlinearity on the temporal profile of the output beam, the simulation of nonlinear effects on stretched pulses with large time-bandwidth products is required. As an example, recent observations [6] have shown that the temporal characteristics of a pre-pulse may differ significantly from the main and post pulses from which it was generated via temporal diffraction. These observations, significantly affecting the interaction of high-power laser pulses with matter, have renewed the interest in intuitive simulation capabilities of complex pulse shapes that are developing through the CPA process. In the classical finite-difference time-domain and split-step Fourier methods, large time-bandwidth product laser pulses, such as the ones used in CPA, are difficult to handle. Although these methods solve the pulse propagation problem, they are time- and memory-consuming in computation, as the temporal resolution has to be in the order of the Fourier transform-limited pulse width, while the excursion should remain larger than the pulse width. For typical CPA conditions, this requires a minimum of \(10^{4}\) or \(10^{5}\) points. For the sake of simplicity, we restricted our attention to a one-dimensional model for a scalar field and used [PERSON]'s wave equation in the form [7]:  
\[\frac{\partial^{2}\mathrm{E}}{\partial z^{2}}-\frac{1}{\mathrm{c}^{2}}\frac{ \partial^{2}\mathrm{D}}{\partial\tau^{2}}=0, \tag{1}\]where \(\mathrm{E}=\mathrm{E}(\tau,\mathrm{z})\) is the electric field, \(\mathrm{D}\) is the electric displacement, and \(\mathrm{c}\) the velocity of light in a vacuum. The constitutive relation between \(\mathrm{D}\) and \(\mathrm{E}\) takes into account both the dispersion and nonlinearity of the medium:  
\[\mathrm{D}=\epsilon\mathrm{E}+\mathrm{P}_{\mathrm{NL}}, \tag{2}\]  
where \(\epsilon\) is the dielectric permittivity (\(\epsilon=\mathrm{n}^{2}\) with \(\mathrm{n}\) being the refractive index), and \(\mathrm{PNL}\) the nonlinear polarization. In our case, we will only consider the third-order nonlinear polarization enabling effects, such as four-wave mixing, self-phase modulation and cross-phase modulation: \(\mathrm{P}_{\mathrm{NL}}=\chi^{(3)}|\mathrm{E}|^{2}\mathrm{E}\), where \(\chi^{(3)}\) is the third-order nonlinear susceptibility. The first term of the electric displacement includes all linear terms and has a simple form in the frequency domain:  
\[\widetilde{\mathrm{E}}_{\mathrm{out}}(\omega)=\widetilde{\mathrm{H}}(\omega) \widetilde{\mathrm{E}}_{\mathrm{in}}(\omega), \tag{3}\]  
where \(\widetilde{\mathrm{E}}_{\mathrm{out}}(\omega)\) and \(\widetilde{\mathrm{E}}_{\mathrm{in}}(\omega)\) are the Fourier transforms of the output and input signals, and \(\omega=2\pi\mathrm{f}\) is the optical pulsation where \(\mathrm{f}\) is the optical frequency. The frequency response function \(\widetilde{\mathrm{H}}(\omega)\) includes the spectral transmission \(\mathrm{T}(\omega)\) as its amplitude and the spectral phase dispersion \(\varphi(\omega)\) as its argument, \(\widetilde{\mathrm{H}}(\omega)=\mathrm{T}(\omega)\mathrm{e}^{\mathrm{i} \varphi(\omega)}\).