**2.5 Uncertainty discussions**

In the theory of inverse Fourier transform, reconstruction of pulse profile needs two parameters: spectrum and spectral phase. Broad spectrum of femtosecond optical pulse is measured with a spectrometer, which can be accurately calibrated by a black body or a standard lamp. Therefore, the uncertainty of spectral phase contribute majority of the uncertainty of pulse reconstruction. Uncertainty budget of spectral phase measurement is complex. We assume the laser source runs in a stable status, and the fluctuation of a modelocked pulse is ignorable. The effects of the temperature and humidity of the surroundings on the chromatic dispersion of the measurement instrument is also ignorable. Therein we simply discuss the three main components for the uncertainty generation: measurement instrument (SPIDER setup), measured spectral interferogram, and algorithm of phase retrieval.

The chromatic dispersion of the beam splitter and the nonlinear crystal of the SPIDER setup generates an additional phase of the measured pulse. This additional phase shows an unnegligible effects in measurement of short pulses, especially in the case of pulse width less than 5 fs. Therefore, the thichness of beam splitters and the nonlinear crystals must be very small. To reduce the spectral phase retrieval error, the optical paths need carefully alignment for a high signal to noise ratio (SNR) interferogram recording. The spectral phase retrieval from interferograms with different replicas separations have been analyzed in section 2.4. The analysis and comparison of spectral phase retrieval error with wavelettransform and Fourier transform with different filter widths is simulated and calculated in reference [10].
