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A novel variant of spectral phase interferometry for direct electric-fieldreconstruction (SPIDER) is introduced and experimentally demonstrated. Otherthan most previously demonstrated variants of SPIDER, our method is based on athird-order nonlinear optical effect, namely self-diffraction, rather than thesecond-order effect of sum-frequency generation. On one hand, self-diffraction(SD) substantially simplifies phase-matching capabilities for multi-octavespectra that cannot be hosted by second-order processes, given manufacturinglimitations of crystal lengths in the few-micrometer range. On the other hand,however, SD SPIDER imposes an additional constraint as it effectively measuresthe spectral phase of a self-convolved spectrum rather than immediatelymeasuring the fundamental phase. Reconstruction of the latter from the measuredphase and the spectral amplitude of the fundamental turns out to be anill-posed problem, which we address by a regularization approach. We discussthe numerical implementation in detail and apply it to measured data from aTi:sapphire amplifier system. Our experimental demonstration used 40-fs pulsesand a 500 $\mu$m thick BaF${}_2$ crystal to show that the SD SPIDER signal issufficiently strong to be separable from stray light. Extrapolating thesemeasurements to the thinnest conceivable nonlinear media, we predict thatbandwidths well above two optical octaves can be measured by a suitably adaptedSD SPIDER apparatus, enabling the direct characterization of pulses down tosingle-femtosecond pulse durations. Such characteristics appear out of rangefor any currently established pulse measurement technique.

Phase retrieval via regularization in self-diffraction based spectral interferometry