High-resolution Interferometric Diagnostics for Ultrashort Pulses

3. PHASE RECONSTRUCTION ALGORITHM FOR MULTIPLE SPECTRAL SHEARINGINTERFEROMETRYphase of the first interferogramη 1 = Arg |D 1 (ω n )| 2 exp[i ArgD 1 (ω n )]. (3.11)nSubtracting this absolute phase sets the group delay of the reconstructed pulse, weighted by thespectral intensity, to zero, and makes our procedure invariant against the absolute phase. Theunwrapping integers of the first shear {u 1,n } are chosen to remove 2π phase discontinuities usinga standard unwrapping procedure. One can thus compute {Γ 1,n } via (3.6).3.5.1.2 Second and higher shears(k −1)From the previously reconstructed phase {φ n }, one obtains the expected phase differences {Γ k ,n }for the present shear using(k −1) (k −1)Γ k ,n = φn+C k− φn. (3.12)One also computes the variance of the expected phase differences from the covarianceσ 2 ∆(φ (k −1)=Γ k ,n n+C k− φ(k −1)(k −1)n) 2(3.13)(k −1)(k −1)= Vn,n+ Vn+C k ,n+C k− 2Vn,n+C k. (3.14)One now finds the absolute phase η k that makes the measured {Γ k ,n } most consistent with theexpected {Γ k ,n }. Specifically, one solvesΓ k ,n − ArgD k ,n = η k mod 2π (3.15)for η k . Since (3.15) represents a set of equations — one for each n — with only one unknown, η k ,it is massively overdetermined. Therefore one must perform some form of optimization; here Ichoose a least-squares method. One must also take into account the precision with which {Γ k ,n }and {ArgD k ,n } are known. For example, Γ k ,n will be essentially random if a region of low SNR —most commonly due to a spectral null — occurs between ω n and ω n + Ω k , and has not yet been“bridged” by one of the smaller shears. This is simply the relative phase ambiguity at work. In this74