High-resolution Interferometric Diagnostics for Ultrashort Pulses

A.2 Estimation of the filtered noise amplitudeto the aforementioned condition for the one-dimensional case.A.2 Estimation of the filtered noise amplitudeThis section explains a convenient means of estimating 〈|ξ| 2 〉 without applying (A.7) directly. Thisis useful because estimating 〈|ζ| 2 〉 straight from the raw data may be difficult if the detector iscompletely filled by the signal. However, the signal rarely occupies the entirety of the Fourierdomain. Therefore, one may use choose a “noise filter” which picks out some region of the Fourierdomain which has no significant contribution from the signal. The intensity of filtered noise isthen integrated, providing an estimate of 〈| ˜ζ| 2 〉A n /(2π) where A n is the area under the noise filter.Multiplying this by A/A n provides an estimate of 〈|ξ| 2 〉 provided that the noise is uniform in theFourier domain.A.3 Amplitude of the phase fluctuationsHere, the aim is to determine the statistics of the extracted phase Γ(ω)=argD(ω). Taylor expansionof the complex logarithm givesΓ(ω) = arg ¯D(ω)+ξ(ω) (A.8)= arg ¯D(ω)+Im[ ξ(ω) ξ(ω) 2¯D(ω) ]+O . (A.9)¯D(ω)Note that the lower-case arg denotes the local analytic continuation, so that branch cuts of theargument function are ignored. I henceforth work to first order in ξ(ω) . I shall obtain the two-¯D(ω)frequency correlation〈∆Γ(ω 1 )∆Γ(ω 2 )〉 =Im[ ξ(ω 1)¯D(ω 1 ) ]Im[ ξ(ω 2)¯D(ω 2 ) ] . (A.10)Throughout this section, ∆ denotes departure from the expected value, and angle brackets denoteaveraging over an ensemble of measurements, each identical except for random detector noise.223