High-resolution Interferometric Diagnostics for Ultrashort Pulses

A. NOISE AND UNCERTAINTYdomain, so that its Fourier-domain amplitude is uniform, and so one can write〈 ˜ζ(t ) ˜ζ ∗ (t )〉 = | ˜ζ| 2 δ(t − t ). (A.4)Substitution into (A.3) gives〈ξ(ω 1 )ξ(ω 2 )〉 = | ˜ζ| 22π ∞−∞dtF(t )F (−t )e i (ω 1−ω 2 )t .(A.5)Since the filter isolates the positive or negative sideband in the quasi-time domain, it satisfies theproperty F (t )F (−t )=0. Therefore 〈ξ(ω 1 )ξ(ω 2 )〉 = 0.The other two-frequency correlation 〈ξ(ω 1 )ξ ∗ (ω 2 )〉 can be written using (A.1) and (A.4) as〈ξ(ω 1 )ξ ∗ (ω 2 )〉 = 〈| ˜ζ| 2 ∞〉dt |F (t )| 2 e i (ω 1−ω 2 )t .2π−∞(A.6)The correlation between frequencies ω 1 and ω 2 is therefore proportional to the Fourier transformof the intensity of the filter response at frequency ω 2 − ω 1 .In SSI, the trace is resampled with spacing Ω, and so the correlation between frequencies separatedby integer multiples of Ω is of interest. The correlation turns out to be zero for a rectangularfilter with passband 2T = 4π/Ω, and will remain zero if this rectangular filter’s response issmoothed by a convolution. Other well-designed filter profiles will produce a small but not identicallyzero correlation. Equation (A.6) is actually a standard result for the cross-spectral density ofa random process subject to a linear system [447, 448].Ignoring correlations in (A.6), the filtered noise is completely characterized by its variance〈|ξ| 2 〉 = 〈| ˜ζ| 2 〉2π A(A.7)where A = ∞−∞ dt |F (t )|2 is the “area” of the filter.The extension of these arguments to a two-dimensional data trace is straightforward, relyingonly on the noise being real-valued and on the filter transfer function having a property analogous222