**2.6 Comparison of wavelet-transform with Fourier transform**

The comparison of spectral phase retrieval procedure with wavelet-transform and that with Fourier transform is plotted in Fig. 9. Figure 9(a) is the flow chart of phase retrieval by Fourier transform, and Fig. 9(b) is that by wavelet-transform.

From Fig. 9, we can see that phase retrieval procedure with wavelet-transform is much simpler. The phase reconstruction by Fourier transform includes three steps: Fourier transform, filtering, and inversed Fourier transform. In contrast, phase information can be read directly after the wavelet-transform, which saves two steps in the phase reconstruction, thus simplified the procedure. This simplified process eliminated the uncertainty coming from the filter with traditional Fourier transform; therefore more accurate spectral phase is retrieved. Another advantage of this simplified procedure is that the manual process of selection and adjustment filter is eliminated; therefore, the spectral phase compensation and control can realize automatic operation.

Figure 8(a) shows that with wavelet transform, spectral phases are retrieved from a large range of replicas separations. With replicas separations from 0.22 ps to 1.82 ps, the relative difference of spectral phase is within 0.5 rad. In Fig. 8(b), the full width at half maximum (FWHM) of the reconstructed pulse profiles from replicas separations of 0.22 ps, 0.55 ps, 0.83 ps, 1.58 ps, and 1.82 ps are 28.50 fs, 27.30 fs, 27.77 fs, 28.75 fs, 28.47 fs, 26.81 fs, 26.65 fs, 27.60 fs, and 28.03 fs, respectively. The maximum relative difference is 2.76%, which demonstrates

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

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

The comparison of spectral phase retrieval procedure with wavelet-transform and that with Fourier transform is plotted in Fig. 9. Figure 9(a) is the flow chart of phase retrieval by

From Fig. 9, we can see that phase retrieval procedure with wavelet-transform is much simpler. The phase reconstruction by Fourier transform includes three steps: Fourier transform, filtering, and inversed Fourier transform. In contrast, phase information can be read directly after the wavelet-transform, which saves two steps in the phase reconstruction, thus simplified the procedure. This simplified process eliminated the uncertainty coming from the filter with traditional Fourier transform; therefore more accurate spectral phase is retrieved. Another advantage of this simplified procedure is that the manual process of selection and adjustment filter is eliminated; therefore, the spectral phase compensation and

the accuracy of the spectral phase retrieval with wavelet-transform.

**2.6 Comparison of wavelet-transform with Fourier transform** 

Fourier transform, and Fig. 9(b) is that by wavelet-transform.

control can realize automatic operation.

**2.5 Uncertainty discussions** 

retrieval.

reference [10].

In principle, Fourier transform mixes all the time information of the spectral interferogram at different frequency position no mater whether there are signal or not, therefore, it is more prone to bring noise into signal information and result in pseudo phase.

Fig. 9. Comparison of spectral phase retrieval flows between by Fourier transform and by wavelet-transform. (a) Flow chart of phase retrieval by Fourier transform, (b) by wavelettransform.

However, the major advantage of phase reconstruction using wavelet-transform technique is that the wavelet-transform transfers the spectral interferogram into a two-dimensional graph that separates the signal from noise naturally. It does not need a filter to obtain phase information, but just probe the phase at the maximum of the transform. Therefore the error coming from the filter and the envelope noise is avoided; because there is no filter that needs to be judged by human, all the calculation can be processed automatically and is more suitable for video speed display.
