High-resolution Interferometric Diagnostics for Ultrashort Pulses

A. NOISE AND UNCERTAINTYUsing the identity Im(z 1 )Im(z 2 )= 1 2 Re z 1 z ∗ 2 − z 1z 2, one writes〈∆Γ(ω 1 )∆Γ(ω 2 )〉 = 1 2 Re 〈ξ(ω1 )ξ ∗ (ω 2 )〉| ¯D(ω 1 ) ¯D(ω 2 )| − 〈ξ(ω 1)ξ(ω 2 )〉¯D(ω 1 ) ¯D(ω 2 )(A.11)factoring ¯D(ω) out of the expectation brackets because it is constant. From section A.1 below, thesecond term is zero and for a choice of filter appropriate for SSI, correlations are small. The noiseintensity is also assumed to be frequency-independent. One therefore hasσ 2 Γ (ω)= 〈|ξ|2 〉2| ¯D(ω)| 2 . (A.12)Equation (A.12) also assumes that 〈|ξ| 2 〉| ¯D(ω)|. Although this is false around spectral nulls, itstill leads to an appropriate de-weighting which is the main purpose of this formalism.A.4 Minimizing root-mean-square variation in a set of fields in the presenceof trivial ambiguitiesIt is often necessary, when evaluating the variation of an ensemble of pulse reconstructions, toremove zeroth and first order phase (some of the “trivial ambiguities”) so that only fluctuations ofhigher order would affect the statistics. One method of accomplishing this is by minimizing theroot-mean-square (RMS) field variation over the ensemble [283, 284] ∞ 1/2ε = |∆E (t )| 2 dt . (A.13)−∞Here, ∆ denotes fluctuation about the mean value. The norm may also be equivalently defined viaParseval’s theorem in the frequency domain. A procedure for assigning the zeroth- and first-orderphase to minimize ε for a pair of pulses is given in Appendix A in ref. [283]. To generalize thisprocedure to a set of pulses, I used an iterative algorithm. Each iteration consists of two steps. Inthe first step, the average pulse 〈E (t )〉 is computed. In the second step, the procedure of ref. [283]is individually applied between each pulse and the average pulse. Generally, only a few iterationswere required for convergence.224