High-resolution Interferometric Diagnostics for Ultrashort Pulses

3.4 Sampling and uniqueness of solutionssimply in terms of the unknown phase at the sampled points, one requires that for every combinationof n and k , ω n + Ω k must be a sampled frequency ω m for some m — that is, the frequenciesconnected by every equation must be sampled frequencies. Otherwise, expressing φ(ω n + Ω k ) in(3.6) would require interpolation, complicating the expressions. The shears must therefore be aninteger multiple of the sampling rate, so that one may write Ω k = C k Ω. Imposing this restrictionon the shears permits the use of simple linear difference equations, of the formφ n+Ck − φ n = Γ k ,n (3.7)for all n and k where φ n = φ(ω n ) and Γ k ,n = Γ k (ω n ). One also imposes φ 0 = 0 as a boundarycondition since the absolute phase is not defined.I now consider the uniqueness of the solution to (3.7). Although not used in practice, a simplesolution method is to start at the boundary condition ω 0 , and to use superposition to “visit” othersampling points. One may determine which of the other sampling points can be visited by applyingseries of jumps, forwards or backwards, of the different shear sizes. If one applies a 1 jumps ofshear Ω 1 , a 2 jumps of shear Ω 2 , and so on (the {a k } may be positive or negative) then one will havemoved a total displacement ofω − ω 0 = a 1 Ω 1 + a 2 Ω 2 + ... (3.8)Therefore, one can reach frequency ω n if and only if there exist integer solutions a 1 ,a 2 ,... toa 1 C 1 + a 2 C 2 + ...= n. (3.9)By Bézout’s Identity [287], this occurs if and only if n is an integer multiple of the greatest commondivisor of {C k }. If this condition is not satisfied, then there is no way that any linear combination ofthe difference equations can refer ω n to ω 0 , and hence φ n is not determined by the experimentaldata. Therefore, a unique solution for all n requires that the {C k } be relatively prime. Examplesof the two cases are illustrated in Fig. 3.2. I proceed assuming the shears are relatively prime, inwhich case one has N unknowns and approximately NM equations, where M is the number of71