## 4 Double Blind FROG  
Recently, another two-pulse measurement technique, a variation of the original Blind FROG, was demonstrated and is called Double Blind FROG (DB FROG) [25,26]. Note the lack of a hyphen between \"Double\" and \"Blind\" in its name to avoid confusion with the medical-experiment technique in which neither the experimenter nor the patient has knowledge of the sample in question. DB FROG's name actually implies that it is effectively two Blind FROG setups. Indeed, the experimental apparatus of DB FROG is modified from Blind FROG so that an extra FROG trace is measured. DB  
Figure 8: The reconstructed signal 1 (**A**) and signal 2 (**B**) of two unknown pulses corresponding to the VAMPIRE spectrogram shown in Figure 7[24]. Figure 7: The measured (**left**) and reconstructed (**right**) VAMPIRE spectrograms [24]. FROG is best implemented in PG geometry, in particular, a PG XFROG setup (see Figure 1). The only required modifications to a standard Blind FROG or XFROG apparatus are an additional analyzer oriented at \(-45^{\circ}\). Also, if desired, an additional spectrometer and camera can be used to record the extra FROG trace, although it is quite reasonable to record both traces using a single spectrometer and camera. The two unknown pulses interact with each other in a \(\chi^{(3)}\) nonlinear medium, such as a fused silica window. The polarizations of pulse 1 and pulse 2 are set to be \(45^{\circ}\) relative to each other, for example, pulse 1 at \(0^{\circ}\) and pulse 2 at \(45^{\circ}\), as shown in Figure 9. When the two pulses with \(45^{\circ}\) relative polarization interact in a \(\chi^{(3)}\) nonlinear medium, polarization rotation occurs due to birefringence induced via the third-order nonlinear polarization. This interaction is automatically phase-matched for any crossing-angles, wavelengths and bandwidths. Both pulses experience polarization rotation and will pass through the crossed-polarizer pair in each arm. A spectrometer and camera are placed in each beam path to record a spectrogram (FROG trace), one from each pulse, so two FROG traces are generated in one measurement. The two measured DB PG FROG traces are given by:  
\[I_{1}(\omega,\tau) =\left|\int E_{1}(t)|E_{2}(t-\tau)|^{2}\exp(-i\omega t)\,dt\right|^ {2} \tag{8}\] \[I_{2}(\omega,\tau) =\left|\int E_{2}(t)|E_{1}(t-\tau)|^{2}\exp(-i\omega t)\,dt\right|^ {2} \tag{9}\]  
where \(I_{1}(\omega,\tau)\) represents trace 1, with pulse 1 and pulse 2 acting as the unknown and the known reference pulses, respectively. In trace 2, the roles of pulse 1 and 2 are reversed. Each of the DB PG FROG traces is essentially a PG XFROG trace, but with the known reference pulse replaced by an unknown pulse. In the standard generalized-projections XFROG phase-retrieval algorithm, the known reference pulse is the gate pulse, and the unknown is retrieved by using this gate pulse. DB FROG has two unknown pulses, instead of one known and one unknown pulse. The DB FROG retrieval problem  
Figure 9: The schematic of single-shot Double Blind (DB) PG FROG for measuring an unknown pulse pair [25]. consists of two linked XFROG retrieval problems: in order to retrieve one pulse correctly, the other pulse must be known, and _vice versa_. The DB FROG retrieval algorithm begins with random initial guesses for both \(E_{1}(t)\) and \(E_{2}(t)\). In the first half of the cycle, the DB FROG retrieval algorithm assumes \(E_{1}(t)\) is the unknown pulse and \(E_{2}(t)\) is the gate pulse (even though a random guess is usually not close to the correct gate, it is a sufficient starting point). The algorithm retrieves pulse 1 from trace 1 as a PG XFROG problem. After the first half cycle, the retrieved pulse 1 is not the correct pulse, because pulse 2 is not the correct gate pulse to begin with, but it is a better estimate of it than the initial guess, because this estimate more closely satisfies the trace \(I_{1}(\omega,\tau)\). In the second half of the cycle, the roles of 1 and 2 are switched.