High-resolution Interferometric Diagnostics for Ultrashort Pulses

3. PHASE RECONSTRUCTION ALGORITHM FOR MULTIPLE SPECTRAL SHEARINGINTERFEROMETRYphases removed using (3.6), so that they may be directly related to the unknown phase via (3.7).This step outputs the reconstructed phase {φ n(k ) } incorporating the information from all of theseshears, as well as the associated covariance.For k = 1, the phase reconstruction reduces to the standard single-shear concatenation procedure.Calculation of the covariance for this step is described in section 3.2. For k > 1, the concatenationprocedure must be generalized for multiple shears. Of course, this generalization worksfor the single-shear case, and so for simplicity of programming it is unnecessary to include thesingle-shear procedure as a special case.The equation set represented by (3.7) is linear and furthermore, the coupling of the unknownsis local — a pair of frequencies ω i and ω j are linked only if they differ by one of the shears. Thissuggests a sparse matrix representation leading to an efficient least-squares solution.The unknown phase, sampled at ω n , n = 0,1,...,N − 1, is written as a column vector ¯φ (k ) =φ (k )0 ,φ(k )1 ,...,φ(k )N −1of length N . The post-processed phase differences, defined by (3.6), are writtenas a set of column vectors {¯Γ j }, j = 1,2,...,k , each of length N −|C j |. These vectors are slightlyshorter than ¯φ since the phase differences must not overrun the range of sampled frequencies.I introduce the shear matrices {G j }, j = 1,2,...,k each of size (N − |C j |) × N , which, when rightmultiplied by ¯φ (k ) , give the corresponding phase differences:⎡⎤. .. . .. 00 −1 0 ··· 0 1 0G j =0 −1 0 ··· 0 1 00 −1 0 ··· 0 1 0⎢⎥⎣.0.. . ⎦ ..(3.20)so that each row consists of a −1 and a 1 separated by C j columns. I also introduce the diagonalweights matrices {W j }, each of size (N − |C j |) × (N − |C j |), with entries (W j ) n,n = σ −1argD j ,n, obtainedby (3.3). Rewriting (3.7) in the present matrix formalism with the weights, one hasW j G j ¯φ(k ) = W j ¯Γ j (3.21)76