High-resolution Interferometric Diagnostics for Ultrashort Pulses

3. PHASE RECONSTRUCTION ALGORITHM FOR MULTIPLE SPECTRAL SHEARINGINTERFEROMETRY32|βk ,m |10−0.4 −0.2 0 0.2 0.4t m /TFigure 3.5: (Color online) Modulus of the reconstruction coefficient |β k ,m | (eq. (3.31) in the text)for a single shear of Ω = 2π/T (solid blue) and for multi-shear retrieval with Ω 1 = Ω (dashed, red)and Ω 2 = 5Ω (dotted, green).gives further insight into the algorithm. The modulus of this factor is shown in Fig. 3.5 for the twoshearcase Ω 1 = Ω and Ω 2 = 5Ω, and compared with the single shear case. The larger shear (dottedgreen) information is preferred at around t = 0 which corresponds to large scale spectral features.However, at t = ±T /5,±2T /5, the contribution from the larger shear drops to zero because itcontains no information with this periodicity, and all information is obtained from the smallershear, so that its coefficient equals the single-shear case at these points.To evaluate the overall performance, one considers ˜Γ k ,m to be corrupted by uniform noise withunit variance. In the retrieval, the noise, along with the signal, is multiplied by (3.31), so that thetotal noise energy in { ˜φ m }, and equivalently in {φ n }, can be calculated by taking the Pythagoreansum of β k ,m over all shears and points in the quasi-time domainA 2 [{Ω k }]= 1 N−14Tm =1k e−it m Ω k − 1 2k (1 − cost m Ω k ) 2(3.32)One can compare the multiple shear case with simply taking the same number of measurementswith the same shear to obtain a “noise suppression factor”A[Ω] MS =. (3.33)A[{Ω k = Ω}] α SNR84