## 3 Discussion
The main limiting factors to \(\Delta t_{\text{res}}\) will be the duration of the pump pulse, the group velocity mismatch, and the pump-probe timing at the beamline. Optical pulses used for protein excitation at XFELs up to now have been on the order of 100 fs. In addition, the absolute pump-probe time at the interaction point in the experiment, determined through a specific timing tool in SFX experiments, carries an uncertainty of 30-50 fs [64; 65; 66]. Development of hollow core fibers for ultrafast laser pulse generation [67; 68; 69] has enabled production of single-cycle pulses, with a duration of only a few femtoseconds, or attosecond half-cycle pulses [47; 70; 71; 72] that have also been used for ultrafast pump-probe spectroscopy experiments [73; 74]. The implementation of such pulses in pumping sample at SFX experiments would improve the earliest observable time point after excitation. Furthermore, [PERSON]'s original discussion proposed that the pump pulse occur within the X-ray probe, so that each image would contain an internal dark control as well as multiple time points. Laser pulses with attosecond or few-femtosecond durations will be a requirement to have optical excitation occur within the 100-fs XFEL pulse. In the context of using shorter pump pulses and with the idea of pursuing femtosecond time resolution as well as [PERSON]'s original proposal [34] of pumping within the probe pulse, chromatic dispersion of the laser light has to be considered: there is a physical limit dictated by the material dispersion of the optical pump pulse when propagating through the protein crystal and the delivery matrix or solution. Chromatic dispersion is larger in pulses with large bandwidths, and so will become significant if few-cycles ultrashort lasers are used for pumping; the fact that the protein sample is excited by the pump is also important, as group velocity dispersion (GVD) behavior differs at absorption [75]. Though crystal birefringence results in an orientation dependence of the GVD value [76], its contribution is not further considered here. Figure 9 shows final pulse durations given a range of extremely short femtosecond pulses and different group delay dispersion (GDD) values. The output pulse is calculated with the following standard relationship [77]:  
\[t_{\text{out}}=\frac{\sqrt{t_{\text{in}}^{4}+16(\text{ln}2)^{2}\text{GDD}^{2}} }{t_{\text{in}}}, \tag{2}\]  
where GDD is the GVD multiplied by sample thickness and \(t_{\text{in}}\) is the initial pulse duration. As there are (to our knowledge) no published GVD values for visible light absorbing protein crystals, we used the concentration dependence of the GVD for Rhodamine 6G dye solution near its 600-nm absorption resonance [78] to estimate the GDD for a protein microcrystal: for a 5-\(\upmu\)m thick crystal with 60-mM chromophore concentration, the GVD is scaled to 2258 fs\({}^{2}\)/mm and the GDD is calculated to be 11.29 fs\({}^{2}\)This yields an output pulse of 10.85 fs, which is more than three times the 3-fs input pulse. The method of sample delivery will also add to chromatic dispersion: the stretching introduced by a 5-\(\upmu\)m GDVN nozzle jet (GDD = 0.20 fs\({}^{2}\) if the GVD of water is taken to be 39 fs\({}^{2}\)/mm at 600 nm [79]) is on the order of a few attoseconds, so not large for our purposes. Additional stretching, however, becomes significant with delivery methods that introduce more liquid in the beam path, such as the drop-on-demand injector [80]: this produces droplets approximately 170-\(\upmu\)m in diameter that are expected to double the duration of a 3-fs pulse (Figure 9--table). The stretching of the optical pump is clearly relevant for visible light absorbing protein crystals and will therefore have to be evaluated depending on the experimental conditions (such as the distribution of crystal sizes and the injection method available). The same considerations apply to third-order dispersion (TOD), which can contribute to stretching as much as GVD at such short pulse durations.