High-resolution Interferometric Diagnostics for Ultrashort Pulses

3.5 Main algorithmfrom all the shears and the reconstruction is complete.Throughout the algorithm, one must keep track not only of the reconstructed phase at eachstep, but also the covariance of the reconstructed phase. This summarises one’s knowledge of theuncertainty in reconstructed phase samples and is essential for the preprocessing step. The uncertaintyis assumed to arise from detector noise, which is represented as an additive contributionof variance σ 2 ξ to the measured interferometric products D k (ω). The value of σ 2 ξmay be estimatedfrom the raw data in a number of ways, depending on the encoding method used in theexperimental implementation. Appendix A.2 presents one method, suitable for Fourier-transforminterferometry. Using σ 2 ξ, the covariance is calculated through elementary, if somewhat involved,error propagation analysis.3.5.1 PreprocessingThe preprocessing step for shear k computes the absolute phase η k and the unwrapping integers{u k ,n } for that shear. For k = 1, these are chosen in a similar manner to the single-shear case. Fork ≥ 2, the preprocessing step of shear k relies on a reconstruction performed using only the subsetof shears 1,2,...,k − 1, which is denoted by φreconstruction: that is(k −1)n,mV(k −1)n. It requires as its input the covariance of this−1) (k −1)= 〈∆φ(kn∆φm〉 (3.10)where ∆ denotes fluctuations about the mean value.Both of these inputs — the reconstructed phase and its covariance — are computed in theprevious generalized concatenation step. I therefore take these as given in the description of thisstep.3.5.1.1 First shearThe first shear is the sampling rate Ω 1 = Ω. Its absolute phase is arbitrary — the other absolutephases will be set accordingly. It is now convenient to set it to the intensity-weighted average73