High-resolution Interferometric Diagnostics for Ultrashort Pulses

3. PHASE RECONSTRUCTION ALGORITHM FOR MULTIPLE SPECTRAL SHEARINGINTERFEROMETRY30(a)1.0φ (rad)2010←→0.5|E (ω)|00.01.0RMS err.0.50.0(b)2.2 2.3 2.4 2.5ω (rad/fs)Figure 3.8: (Color online) (a) Spectral phase (red, thick line, left axis) and amplitude (blue, thinline, right axis) of the notched pulse. (b) RMS phase variation for single shear (blue, thin line) anddouble-shear (red, thick line) reconstructions.I minimized the RMS field variation over the ensemble [283, 284] using the procedure describedin section A.4. To verify the accuracy (difference between the reconstructed and original pulse) Ithen adjusted the zeroth and first order phase of the entire ensemble to minimize the RMS fielderror between the average reconstructed pulse and the unknown so that, once again, only errorsof higher order would be observed.To compute the precision of the reconstructed phase, I adopted the following definition of theRMS phase variation which takes into account its modulo-2π nature:σ 2 φ (ω)= exp[i φ(ω)] −〈exp[i φ(ω)]〉 2 . (3.34)Again, angle brackets denote the expectation value over a large number of measurements. Equation(3.34) reduces to the standard RMS when the fluctuations are small, and saturates to σ φ = 1when the fluctuations are so large that the phase is essentially random.The results are that both the single- and double-shear reconstructions are accurate, in thattheir averages do both converge to the unknown pulse. However, the precision of the single-shear88