High-resolution Interferometric Diagnostics for Ultrashort Pulses

5. COMPACT SPACE-TIME SPIDERy(a) y(b) y(c) y(d)ωωωωFigure 5.1: Spatio-spectral phase reconstruction with a spatially resolved wavefront measurement(red) combined with a one-dimensional spectral phase measurement (subfigures (a) and (b)) anda spatially resolved spectral phase measurement (subfigures (c) and (d)). The greyed out pointrepresents a frequency which is missing from the pulse. In (a), a single frequency-point is missing.In (b) and (c) the beam is spatially chirped. In (d), the beam consists of two spectrally and spatiallydisjoint regions.somewhat artificial and one could simply choose a different point to perform the spectral phasemeasurement, there are realistic situations where this is not possible. Figure 5.1(b) depicts a pulsewith some spatial chirp. Regardless of where the spectral phase is measured, not all frequencieswill be present.One way of avoiding this limitation is to perform a spatially resolved spectral phase measurement,which returns φ(ω,x T )+g (x). Combining this with the spatially resolved wavefront measurement,the phase at missing frequencies can be obtained, as depicted in Fig. 5.1(c). Note thatthis still does not guarantee an ambiguity-free reconstruction — if several spectrally and spatiallydisjoint regions are present, as depicted in Fig. 5.1(d), then their relative phase will not be returned.However, a much larger class of pulses can be measured. In particular, all pulses with asimply connected spatio-spectral intensity profile can be measured with this approach. Anotherpotential advantage is the introduction of redundant information to the retrieval. In general, thisadds noise robustness and flags measurement errors, turning the reconstruction algorithm into anoptimization problem of an overdetermined system. This raises the possibility of using algorithmswhich weight the signal according to its local signal-to-noise ratio (SNR). Such approaches havefound success in the related problem of two-dimensional phase unwrapping [295], particularlyaround signal dropouts which occur, for example, near optical vortices.Spectral shearing interferometry is well suited to the acquisition of a spatially resolved spec-106