High-resolution Interferometric Diagnostics for Ultrashort Pulses

7.4 Data processingDivergence (mrad)10−1(a)(b)15 20 25Harmonic (order)15 20 25Harmonic (order)Figure 7.7: Typical phase difference between two interferograms. (a) As obtained from interferograms.(b) After zeroing absolute phase at every frequency to zero.intensity-weighted average, to be zero, effectively placing the centroid of all sources at x = 0. Thiscauses the fringe phase to have predominately spatial variations, as shown in Figure 7.7(b).As Fig. 7.7(b) suggests, the resulting phase differences were fairly homogenous across eachindividual harmonic. (The irregularities of the 21 st and 23 rd harmonics were not reproduced fromshot-to-shot.) To simplify the analysis and improve the signal-to-noise ratio (SNR), I averaged thephase across each harmonic, again using an intensity weighting. Figure 7.8 shows a typical set ofresults for a subset of 7 out the 50 shots taken at each combination of shear and gas jet position.Significant shot-to-shot fluctuation is evident. I performed shot averaging on the complex-valuedinterferometric products, and then took the phase of the result. This produces better results thantaking the phases, then averaging.I also estimated the uncertainty on the shot-averaged phase differences. Without some knowledgeof their uncertainties, it would have impossible to quantify the consistency between the differentshears. I decomposed the fluctuations about the mean into the in-phase and quadraturecomponents. Only the latter cause phase fluctuations. Examination of the quadrature fluctuationsshowed little shot-to-shot correlation, so I assumed that their average converged with thesquare root of the number of samples. The standard error of the quadrature fluctuations was thenconverted into phase noise using the methods of appendix A. The end result of this process was169