High-resolution Interferometric Diagnostics for Ultrashort Pulses

3.9 Numerical comparison of single- and multi-shear retrieval|E |42(a)|D|010.50(b)10∠D/π−1|D|10.50(c)2.2 2.3 2.4 2.5ω (rad/fs)10∠D/π−1Figure 3.7: (Color online) (a) Spectrum of a two-lobed pulse (blue, solid) and spectrally shearedreplicas with small (red, dashed) and large (green, thick line) shears. (b) and (c): Amplitude (blue)and phase (red, dashed) of interference product for the small (b) and the large (c) shear. Thevariation of the interference products with a peak SNR of 50 is shown by the shaded area.phase retardation of π/2 was then applied to the higher frequency lobe. I compared a single shearreconstruction with Ω ≈ 3.14mrad/fs with a two-shear reconstruction with Ω 1 = Ω and Ω 2 = 8Ω ≈25.1mrad/fs. The pulse spectral amplitude and phase are shown in Fig. 3.8(a).Each interferogram D(ω) was contaminated with additive white Gaussian noise with RMS amplitudeequal to 2% of the peak signal amplitude for the single shear case and 4% in the doubleshear case to emulate the worst-case scenario for multi-shear with sequential acquisition. A randomabsolute phase, uniformly distributed over [0,2π), was also added to each interferogram. Thealgorithm of section 3.5 was used for the double shear whilst standard concatenation was used forthe single shear case. Having determined the phase, I reconstructed the complex envelope usingthe spectral intensity of the unknown pulse. The entire process was repeated 1000 times toproduce an ensemble of reconstructions so that the precision could be examined. Once this ensemblewas computed, it was necessary to remove ambiguous zeroth and first order phase fromeach reconstruction so that only fluctuations of higher order would affect the statistics. To do this,87