High-resolution Interferometric Diagnostics for Ultrashort Pulses

3.6 Implications for the relative phase ambiguityThis requirement for a precise registration of each shear carries implications for the use ofmulti-shear in determining the relative phase between disjoint spectral regions. The situation ismost simply explained by considering the spectral intensity to be binary valued, so that the amplitudeof the interferometric product |D k (ω)| = |E (ω + Ω k )E ∗ (ω)| is either 1 (measurable signal) or 0(signal swamped by noise). Consider a two-shear reconstruction of a spectrum with two separatespectral regions. To have any chance of measuring the relative phase of the spectral regions, thesecond shear must be larger than the separation. However, for the registration of the second shearto succeed, one of the spectral regions must also be wider than the separation. The two cases aredepicted in Fig. 3.3. In Fig. 3.3(a), one of the lobes is broad enough to allow the larger shear to beregistered, whereas in Fig. 3.3(b), this is not the case. Note that these conclusions are consistentwith Keusters et al. in their previous discussion of this ambiguity. In their analysis of SPIDER theyconsidered only the situation of Fig. 3.3(b) and concluded that resolving the ambiguity was impossible.For SPIDER with a well-characterized external reference (denoted X-SPIDER in [81]), theyconcluded that the ambiguity could be resolved. My analysis concurs; a well-characterized externalreference would mean that the absolute phases of the ancillae are known and the registrationprocedure is unnecessary.For more complicated situations, involving more than two separate spectral regions, the reasoningabove can be applied inductively. For example, Fig. 3.4 shows how a spectrum with threespectral lobes could be characterized. Note that the gap between the second and third lobe, 3units, is larger than the three lobes (2, 1 and 1 units respectively), so that if a two shear measurementwere attempted then the shear of size 3 required to bridge the gap between the second andthird lobe could not have its absolute phase registered. However, if three shears are used, of size 1,2, and 3 units respectively, then the reconstruction succeeds. The second shear can be registeredon the first spectral lobe (arrow R 2 ), and used to bridge the gap between the first and second lobes(arrow GC 2 ), as depicted in Fig. 3.4(a). The third shear can then be registered using the “compoundlobe” formed by the first and second lobes, whose relative phase is now known (arrow R 3 ). It thendetermines the relative phase of the second and third lobes (arrow GC 3 ).79