High-resolution Interferometric Diagnostics for Ultrashort Pulses

2. BACKGROUNDφ 2 − φ 1 between E 1 and E 2 to the intensities ascos(φ 2 − φ 1 )= |E S| 2 − |E 1 | 2 − |E 2 | 2. (2.24)2|E 1 ||E 2 |However, the inverse cosine is not single-valued, producing a sign ambiguity in the relative phase.Another concern with this direct approach is that it requires precise separate measurements ofthe individual intensities and is sensitive to imperfections (such as nonlinearity) in the detectorresponse.In FTSI, this problem is solved by the use of a carrier phase caused by a time delay τ betweenthe pulses, which in the frequency domain amounts to an additive phase i ωτ. Figure 2.3 illustratesthis. The superposition is of the formB(ω)=|E 1 (ω)+E 2 (ω)e i ωτ | 2 (2.25)= |E 1 (ω)| 2 + |E 2 (ω)| 2 + 2|E 1 (ω)||E 2 (ω)|cos[Γ(ω)+ωτ] (2.26)where Γ(ω) =φ 2 (ω) − φ 1 (ω) is the relative phase. The cosine term produces fringes of nominalperiod 2π/τ in the frequency domain, whose spacing is modulated by the relative phase. For thefringes to be well-defined, they must be on a finer scale than any phase or amplitude variation ofthe two fields. The spirit of this condition is expressed by the following equation1∂ E 1 (ω)|E 1 (ω)| ∂ω τ (2.27)and likewise for E 2 (ω). Absent in (2.27) is a quantitative specification of the required separation ofscales — this depends on the desired accuracy. A useful practical criterion, which naturally arisesfrom the time-domain processing procedure, shall be described below.Given a well-defined fringe pattern, various algorithms exist for extracting the relative phaseΓ(ω). Intuitively the most direct is to pick the peaks and compare their positions with those expectedwhen Γ(ω)=0. However a more robust and widely used method which is adopted here isFourier-domain filtering [92], which utilises the separation of the fringe pattern from the individ-28