High-resolution Interferometric Diagnostics for Ultrashort Pulses

3.8 Signal-to-noise ratio benefit of multi-shearThe retrieval is formulated as the least squares minimization of the deviation between theunknown phase and the observed phase differences, summing over all shears:M N−1R = |φ n+Ck − φ n − Γ k ,n | 2 Ω. (3.27)k =1 n=0Using Parseval’s theorem and the discrete Fourier transform one hasR =MN−1k =1 m =0| ˜φ m (e i Ω k t m− 1) − ˜Γ k ,m | 2 2π/B. (3.28)Here, the quasi-time points t m = m2π/B, m = − N −12 ,..., N where B = N Ω is the measured2bandwidth, { ˜φ m } and {˜Γ k ,m } are the discrete Fourier transforms of {φ n } and {Γ k ,n } respectively,and M is the number of shears. Minimization of R with respect to ˜φ m then gives˜φ m = Mk =1 ˜Γ k ,m e−it m Ω k − 12 Mk =1 1 − cosΩ k t m. (3.29)Equation (3.29) gives the discrete Fourier transform of the unknown phase in terms of the discreteFourier transform of the observed phase differences. For a single shear, it reduces to˜φ m =˜Γ m(e it m Ω − 1)(3.30)which upon inverse Fourier transforming becomes concatenation in the frequency domain, thetraditional single-shear algorithm. For multiple shears, (3.29) can be thought of as a weightedsum of the information from the various shears, with the contribution of each reflecting its SNRat each point in the quasi-time domain. The zero in the denominator at t m = 0 corresponds tothe absolute spectral phase, which cannot be retrieved with spectral shearing or any other selfreferencingenvelope-based technique.The behaviour of the coefficient of ˜Γ k ,m in (3.29),β k ,m = e−it m Ω k − 12 Mk =1 1 − cosΩ k t m, (3.31)83