High-resolution Interferometric Diagnostics for Ultrashort Pulses

3. PHASE RECONSTRUCTION ALGORITHM FOR MULTIPLE SPECTRAL SHEARINGINTERFEROMETRYThe error matrix may be readily computed from the covariance matrixQ (k ) (ω i ,ω j )=V (k )i,i+ V (k )j,j− 2V (k )i,j. (3.26)3.5.3 Pseudocode algorithmFor a precise summary, this section presents a pseudocode algorithm.for k = 1 to M doif k = 1 thenCompute η 1 using (3.11)Unwrap {D 1 (ω n )} to compute {u 1,n } for n = 0,1,...,N − 1elseCompute {Γ k ,n } for n = 0,1,...,N − 1 using (3.12)Compute {σ 2 } n = 0,1,...,N − 1 using (3.14)Γ k ,nCompute ρ 2 k ,nusing (3.16)end ifForm B (k ) and ¯F (k ) using (3.23)Compute V (k ) using (3.25)Compute ¯φ (k ) = V (k ) ¯F (k )end for3.6 Implications for the relative phase ambiguityThe registration of the absolute phase using (3.18) for a given shear k requires the previous reconstructionφ (k −1) to precisely determine the phase difference between at least one pair of frequenciesω n and ω n+Ck — that is, to imply a value of Γ k ,n with which the measured Γ k ,n can be directlycompared so as to determine η k . Mathematically, such a region will then have a low value of ρ k ,nvia (3.16), so that it will contribute significantly to the sum (3.18). If no such region exists, thenthe argument in (3.18) will give an essentially random value, the η k will not be determined, andfurther reconstruction will be inaccurate.78