High-resolution Interferometric Diagnostics for Ultrashort Pulses

3. PHASE RECONSTRUCTION ALGORITHM FOR MULTIPLE SPECTRAL SHEARINGINTERFEROMETRYence equation between the phasesArgD k (ω)=Γ k (ω)+η k − 2πu k (ω), (3.6)where Γ k (ω)=φ(ω + Ω k ) − φ(ω) are the usual phase differences. Here, u k (ω) ∈ are the unwrappingintegers required because the principal value ArgD k (ω) 1 is only recoverable over the domain[−π,π). In a single-shear reconstruction of a pulse whose duration is lower than the Nyquistdictatedupper bound, the phase differences are also within this interval (−π,π], so that the principalvalue of the argument of D(ω) can be used for Γ(ω). In certain cases, correct retrievals caneven be achieved for pulses which exceed the temporal window if D(ω) is unwrapped by removing2π discontinuities. However, in a multiple shear reconstruction, neither approach will workin general: larger shears can sample phase differences of absolute value much greater than π, yetperforming an unwrapping procedure on each shear individually may produce mutually inconsistentdata. Like the absolute phase issue, this is another subtlety of multiple shear reconstructionswhich must be taken into account by the algorithm.3.4 Sampling and uniqueness of solutionsI shall temporarily set aside the absolute phase and unwrapping issues, and develop some othergeneral features of the algorithm.I assume that the products {D k (ω)} are well sampled by the spectrometer resolution, whichmeans that they can be resampled via interpolation to any desired set of frequencies. Here I derivea linear least-squares method which is noniterative, efficient and a logical generalization ofthe concatenation algorithm. Like the concatenation algorithm the unknown phase is sampledat frequencies {ω n }, whose regular spacing Ω determines the temporal window of the unknownpulse. The strategy is to resample the {D k (ω)} to these points, take the argument to yield the phasedifferences {Γ k (ω)} and then write the difference equations (3.6) at each frequency and shear toproduce a set of equations which are solved simultaneously. To express the difference equations1I use Arg to denote the principal value, within [−π,π), and arg to denote the local analytic continuation so that thesmall fluctuations caused by noise do not cause 2π phase jumps in any evaluation of the statistics.70