## 2 Blind FROG and CRAB
Specifically, in single-attosecond-pulse characterization, instead of using a nonlinear optical interaction to generate the signal field, a weak fs probe pulse is used to probe the phase information of the attosecond-pulse by photoionization of atoms induced by the attosecond-pulse. This technique, called complete reconstruction of attosecond bursts (CRAB), uses the idea of Blind FROG to retrieve both the fs probe pulse and attosecond pulse. In the original work, the photoelectron spectra, \(S(\omega,\tau)\), are measured with different delay times, \(\tau\), between the probe pulse and the attosecond-pulse. An iterative generalized-projections reconstruction method is employed to determine the electric field of the attosecond-pulse. This approach has been modified to also obtain the fs probe pulse, in effect evolving into a Blind FROG technique. The mathematical expression for the reconstructed photoelectron spectra, \(S_{R}(\omega,\tau)\), is:  
\[S_{R}(\omega,\tau)=\left|\int_{-\infty}^{+\infty}dt\;X(t-\tau)G(t)\exp(i\omega t )\right|^{2} \tag{4}\]  
where \(X(t)\) is the reconstructed signal and \(G(t)\) is the gate. They are related (without the normalization factor) to the electric field of the attosecond-pulse, \(E_{X}(t)\), and the vector potential of the probe pulse, \(A(t)\), by:  
\[ \begin{split} X(t)\approx\int_{-\infty}^{+\infty}d(\omega) \omega_{e}^{1/4}\left[\int_{-\infty}^{+\infty}E_{X}(t^{\prime})\exp\;(i \omega t^{\prime})dt^{\prime}\right]\exp(-i\omega t)\;d\omega\\ G(t)\approx\exp\left[-i\int_{t}^{+\infty}dt\{\
u_{o}A(t)+A^{2}( t)\}\right]\end{split}  \tag{5}\]  
where \(\omega_{e}\) is the electron energy and \(\
u_{o}\) is a constant obtained by the reconstruction algorithm that minimizes the error between \(S(\omega,\tau)\) and \(S_{R}(\omega,\tau)\). The algorithm starts with setting \(X^{(i=0)}(t)\) and \(G^{(i=0)}(t)\) to unity, where \(i\) represents the iteration number. \(S_{R}^{(i)}(\omega,\tau)\) is generated from the \(X^{(i)}(t)\) and \(G^{(i)}(t)\). The intensity of \(S_{R}^{(i)}(\omega,\tau)\) is replaced by the experimentally measured \(S(\omega,\tau)\). New guesses for \(E_{X}^{(i+1)}(t)\) and \(G^{(i+1)}(t)\) are generated from \(S_{R}^{(i)}(\omega,\tau)\) by a generalized projections approach. The algorithm continues until the rms error between measured and reconstructed photoelectron spectra has stabilized. Two sets of experimental photoelectron spectra are obtained at two laser intensities, \(1.6\times 10^{14}W/cm^{2}\) and \(3.2\times 10^{14}W/cm^{2}\). Figure 3 shows the measured and reconstructed photoelectron spectra for a 33 fs probe pulse and attosecond-pulse trains with rms error of 0.014 and 0.012 with low- and high-intensity cases, respectively. The reconstructed temporal profiles are shown in Figure 4 which are consistent with the prediction made by numerical simulations. Figure 4: Reconstructed temporal profiles of attosecond-pulse train corresponding to the cases of low (Figure 3a) and high intensities (Figure 3b), as indicated by the blue and red solid lines, respectively. The temporal profile of the reconstructed probe pulse is shown by the red dashed line. The duration of each pulse in the train is shown by squares and circles for low and high intensities, respectively [22]. Figure 3: Photoelectron spectra of He ionized by attosecond harmonic pulses with a probe laser pulse. High harmonics were obtained with a driving laser at an intensity of **(a)**\(1.6\times 10^{14}\)W/cm\({}^{2}\) and **(b)**\(3.2\times 10^{14}\)W/cm\({}^{2}\). The reconstructed spectra are shown in **(c)** and **(d)** for cases **(a)** and **(b)**, respectively [22].