High-resolution Interferometric Diagnostics for Ultrashort Pulses

3. PHASE RECONSTRUCTION ALGORITHM FOR MULTIPLE SPECTRAL SHEARINGINTERFEROMETRY[281], using different calibration or reconstruction parameters [282], or by simply taking manymeasurements.Here I use an alternative representation which quantifies errors and their correlations causedby detector noise in SSI. It is based on the realisation that since SSI measures spectral phase differences,the uncertainty is well represented by an error matrixQ(ω i ,ω j )= ∆ φ(ωi ) − φ(ω j ) 2 1/2(3.1)where ω i and ω j are sampled frequencies, ∆ denotes the fluctuation from the mean value and theangle brackets denote expectation over a large number of measurements. A low value of Q(ω i ,ω j )means that the phase difference between ω i and ω j is precisely constrained by the measurement.The error matrix is an intuitive depiction of uncertainty in SSI and some examples will be given inthe experimental demonstrations in chapter 4. Although one may be ultimately concerned withsome other measure of precision, such as the root-mean-square (RMS) field variation [283, 284] orthe variation in the pulse duration, these quantities can be derived from knowledge of Q(ω 1 ,ω 2 )combined with knowledge of any uncertainty in the spectral intensity measurement.For a single-shear reconstruction, the error matrix may be calculated using a Pythagoreansummation of the errors. Concatenation gives the reconstructed phase difference between frequenciesω j and ω i asj −1φ(ω j ) − φ(ω k )= Γ(ω n ). (3.2)n=kThe properties of the errors on Γ(ω) are derived in Appendix A. The main result is that the errorsare independent and their standard deviation is given by (A.12), repeated here:σ 2 Γ (ω)=σ 2 ξ2| ¯D(ω)| 2 (3.3)where σ 2 ξ is the variance of the noise in the filtered sideband, and | ¯D(ω)| = |E (ω + Ω)E ∗ (ω)| is theamplitude of the filtered sideband in the absence of noise. The errors add in Pythagorean fashion68