High-resolution Interferometric Diagnostics for Ultrashort Pulses

5.4 A common-path re-imaging ST-SPIDERin the Fourier filtering step.After the usual Fourier filtering to isolate the sidebands and subtraction of the carrier, thephase of the fundamental interferogram isΛ f (ω,y )=φ(ω,y − Y ) − φ(ω,y ) (5.1)whilst the phase of the upconverted interferogram isΛ up (ω + ω up ,y )=φ(ω − Ω,y − Y ) − φ(ω,y )+η(y ) (5.2)Here, η(y ) is the spatially dependent phase of the upconverted interferogram, which depends onthe ancilla phases. To avoid making assumptions about the uniformity of the ancilla spatial profile,one must treat η(y ) as unknown and incorporate this into the algorithm.In developing a reconstruction algorithm, I note that much research has been done into twodimensionalwavefront reconstruction [295, 297–300] for Shack-Hartmann and shearing interferometers.Expressed in the present notation i.e. in the spatio-spectral domain, these algorithmsact upon discretely sampled orthogonal phase differencesΓ ω (ω m ,y n )=φ(ω m +1 ,y n ) − φ(ω m ,y n ) (5.3)andΓ y (ω m ,y n )=φ(ω m ,y n+1 ) − φ(ω m ,y n ) (5.4)where ω m = ω 0 +m |Ω| and y n = y 0 +n|Y | are the sampling points. To deploy these algorithms, onemust convert the measured interferogram phases (5.1) and (5.2) into the forms of (5.3) and (5.4).The spatial shear component of the upconverted interferogram is removed by subtracting the117