# Refined Ptychographic Reconstruction of Attosecond Pulses
## 2 Simulated Spectrograms  
We tested the novel approach by applying it to simulated streaking traces and by comparing it with standard ePIE and VTGPA reconstructions. In this section we will introduce the theoretical model used to obtain the simulated streaking traces. In an attosecond streaking experiment [4], a target is ionized by the energetic XUV photons in the presence of an opportunely delayed phase-locked infrared (IR) pulse. The IR pulse acts as a phase modulator which changes the final photoelectron momentum by an amount that depends on the relative time delay, \(\tau\), between IR and XUV pulses. The collection of photoelectron spectra as a function of pump-probe delay \(\tau\) is called spectrogram. In case the target is an atomic gas and under the strong field approximation (SFA), the spectrogram can be written as [21] (atomic units are used hereafter):  
\[S(p,\tau)=\left|\int dtE_{el}(t,\tau)e^{-i\phi(p,t)}e^{i(\frac{p^{2}}{2}+I_{p} )t}\right|^{2}, \tag{1}\]  
where  
\[\phi(p,t)=\int_{t}^{\infty}dt^{\prime}\left(pA_{IR}(t^{\prime})+\frac{A_{IR}^ {2}(t^{\prime})}{2}\right). \tag{2}\]  
\(I_{p}\) is the target ionization potential, \(p\) is the electron momentum, \(E_{el}(t)\) is the electron burst obtained after ionization by the extreme-ultraviolet (XUV) attosecond electric field \(E_{XUV}(t)\). If no atomic transitions lie within the spectrum of the attosecond pulses, \(E_{el}(t)\) can be expressed in terms of the atomic dipole \(d\) and \(E_{XUV}(t)\) as:  
\[E_{el}(t,\tau)=E_{XUV}(t-\tau)d\left(p+A_{IR}(t)\right)\simeq E_{at}(t-\tau), \tag{3}\]  
where \(E_{at}(t)\) is defined as [15]:  
\[E_{at}(t)=\mathcal{F}^{-1}\left\{\hat{E}_{XUV}(\omega)\sigma_{at}(\omega)e^{i \Phi_{at}(\omega)}\right\}. \tag{4}\]  
Here \(\mathcal{F}^{-1}\) defines the inverse Fourier transform, \(\hat{E}_{XUV}(\omega)=\mathcal{F}\left\{E_{XUV}(t)\right\}\) represents the Fourier transform of the attosecond electric field, \(\sigma_{at}\) is the atomic cross-section for single photon ionization, and \(\Phi_{at}\) is the atomic Wigner-like spectral phase acquired by the electron during the photoemission process [22; 23]. We note that Equations (3) and (4) are valid only in the case of single XUV photon ionization. A different (multiphoton) regime will require a proper and different description of \(E_{el}(t,\tau)\). As multiphoton regimes are less likely to be experimentally implemented and do not improve the pulse reconstruction technique, in this work we limit ourselves to the single XUV photon interaction. In our simulations we chose Ar as a target gas. The Ar cross-section, \(\sigma_{Ar}(\omega)\), is taken from Ref. [24] while the atomic phase \(\Phi_{Ar}(\omega)\) is extracted from Ref. [25]. We assumed a Gaussian envelope for both the attosecond, \(E_{XUV}(t)\), and the femtosecond, \(E_{IR}(t)\), electric fields:  
\[E(t)=A_{0}e^{-\frac{t^{2}}{2\omega^{2}}}\sin(\omega_{0}t), \tag{5}\]  
where \(\omega_{0}\) is the central frequency (\(\omega_{0}=\omega_{IR}\) for the femtosecond IR field, \(\omega_{0}=\omega_{XUV}\) for the attosecond XUV field), and \(A_{0}\) is the field amplitude (\(A_{0}=A_{IR}\) for the IR field, \(A_{0}=A_{XUV}\) for the XUV field). The temporal width \(\sigma\) is linked to the intensity full-width half maximum, \(\Delta t\), by \(\sigma=\Delta t/2\ln(2)\). A quadratic spectral phase term is then added to both pulses by multiplying the field Fourier transform by \(\exp\{iD_{2}(\omega-\omega_{0})^{2}\}\), where \(D_{2}\) is the group-delay dispersion (GDD). In order to mimic realistic experimental conditions, for the IR pulse we set a transform-limited duration \(\Delta t_{IR}=6\) fs, a central frequency \(\omega_{IR}=1.5694\) eV (corresponding to a wavelength of 790 nm), a GDD \(D_{2}^{IR}=10\) fs\({}^{2}\), and a peak intensity \(I_{IR}=5\times 10^{12}\) W/cm\({}^{2}\). For the XUV pulse, instead, we assume a transform-limited duration \(\Delta t_{XUV}=250\) as and a GDD \(D_{2}^{XUV}=0.02\) fs\({}^{2}\).