High-resolution Interferometric Diagnostics for Ultrashort Pulses

ANoise and uncertaintyThis appendix derives several results concerning noise and uncertainty in interferometry andpulse characterisation.A.1 Correlations of filtered additive white noiseThis section derives some properties of additive detector noise after the Fourier-domain filteringalgorithm used in interferometry. I shall treat one-dimensional spectral interferometry forconcreteness. The results may be easily adapted to other domains and/or higher dimensions.The additive real detector noise ζ(ω) is filtered to produce complex noise ξ(ω), which contaminatesthe true value of the interferometric product ¯D(ω) =E 2 (ω)E1 ∗ (ω). One therefore writesD(ω) = ¯D(ω)+ξ(ω), where D(ω) is the value of the interferometric product obtained by Fourierfiltering. I shall determine the two-frequency correlations 〈ξ(ω 1 )ξ ∗ (ω 2 )〉 and 〈ξ(ω 1 )ξ(ω 2 )〉. Thesewill be used in the next section to derive the statistics of the fluctuations in the extracted phaseΓ(ω)=argD(ω).One writes ξ(ω) as being the result of Fourier filtering the detector noiseξ(ω)= 1 2π ∞−∞dt ˜ζ(t )F (t )e i ωt(A.1)where ˜ζ(t ) is the Fourier transform of the detector noise and F (t ) is the response of the filter in thequasi-time domain. Thenξ(ω 1 )ξ(ω 2 )= 1 ∞ ∞dt dt ˜ζ(t ) ˜ζ(t )F (t )F (t )e i ω 1t +i ω 2 t .2π−∞ −∞(A.2)One now applies the expectation brackets and uses ˜ζ(t )= ˜ζ ∗ (−t ) since ζ(ω) is real, yielding〈ξ(ω 1 )ξ(ω 2 )〉 = 1 ∞ ∞dt dt 〈2π˜ζ(t ) ˜ζ ∗ (−t )〉F (t )F (t )e i ω 1t +i ω 2 t .−∞ −∞(A.3)Assuming that the noise intensity in the frequency domain is uniform, the two-time correlation isa delta function. I also assume the detector noise is white i.e. it is uncorrelated in the frequency221