## 2 Blind FROG and CRAB  
The first attempt to solve the two-pulse measurement problem in femtosecond regime is called Blind FROG. It was proposed and demonstrated by [PERSON] _et al_. in 1995 [18]. It is based on cross-correlation FROG (XFROG), which uses a known reference pulse to measure an unknown one [31]. A XFROG experimental apparatus implemented with polarization-gate (PG) geometry is shown in Figure 1. A XFROG apparatus becomes a Blind FROG when the reference pulse is an unknown pulse that also needs to be measured. Like a XFROG device, Blind FROG records one FROG trace at the camera. The mathematical expression for the recorded trace is:  
\[I_{Blimd\;FROG}(\omega,\tau)=\left|\int\limits_{-\infty}^{+\infty}dtE_{sig}(t,\tau)\exp\left(i\omega t\right)\right|^{2} \tag{1}\]  
where \(E_{sig}(t)\) represents the signal field from the nonlinear interaction happening inside the nonlinear medium. The expression of \(E_{sig}(t)\) varies depending on the gating geometry in the experimental apparatus. For example, in PG geometry:\[E_{sig}(t,\tau)=E_{1}(t)|E_{2}(t-\tau)|^{2} \tag{2}\]  
where \(E_{1}(t)\) and \(E_{2}(t)\) are the electric field of the two unknown pulses to be measured. The retrieval algorithm begins with guesses of \(E_{1}(t)\) and \(E_{2}(t)\) and generate \(E_{sig}(\omega,\tau)\) by Fourier transforming (2) with respect to \(t\). The magnitude of \(E_{sig}(\omega,\tau)\) is replaced by the square root of the measured Blind FROG trace with the phase unchanged to generate a modified signal field, \(E^{\prime}_{sig}(\omega,\tau)\). \(E^{\prime}_{sig}(t,\tau)\) is generated by an inverse Fourier transform. The method of generalized projections is used to generate new guesses for the fields. An error function, \(Z\), is defined as [18]:  
\[Z=\sum_{t,\tau=1}^{N}\left|E^{\prime}_{sig}(t,\tau)-E_{1}(t)|E_{2}(t-\tau)|^{2} \right|^{2} \tag{3}\]  
Figure 1: The schematic of single-shot polarization-gate (PG) cross-correlation frequency-resolved optical gating (XFROG) setup. The orange beam and red beam represent the known reference pulse and unknown pulse, respectively. The experimental apparatus is essentially a PG FROG setup with the known reference pulse replacing one of the replicas. The apparatus becomes a Blind PG FROG setup when the known reference is replaced by an unknown one. Figure 2: Electric fields of two pulses with different amount of chirp retrieved by second harmonic generation (SHG) Blind FROG (circles) compared with results retrieved by SHG FROG (lines) [18]. On odd iterations, the measured spectrum of pulse 2 is used to replace that of \(E_{2}\), and a new guess for \(E_{1}(t)\) is generated by minimizing \(Z\) with respect to \(E_{1}(t)\). On even iterations, the operations on pulse 1 and pulse 2 reverse. The algorithm continues until the root-mean-square (rms) error between the measured and retrieved Blind FROG traces is minimized. In the original work, the Blind FROG apparatus was implemented with second harmonic generation (SHG) geometry. Figure 2 shows the two pulses with different amounts of chirp retrieved by SHG Blind FROG. The Blind FROG retrieval results were compared with measurement independently made by SHG FROG showing good agreement. However, independently measured spectra of the two pulses were also required in order to achieve convergence. It was shown in the original Blind FROG paper that it suffers from nontrivial ambiguities, which means that multiple pulse pairs could yield Blind FROG traces that are indistinguishable, and [PERSON] _et al_. [32] later considered them in detail. As a result, Blind FROG is in general not very useful in fs-pulse characterization. However, it does find application in attosecond-pulse measurements as a result of the different physical process used to generate the signal field [21,22].