**2.1 Optical schematics of SPIDER**

Our home made SPIDER setup is shown in Fig. 1. Figure 1(a) is the schematic optical path. An input femtosecond laser pulse passes through an aperture A1, and then is reflected by a reflective mirror M1. The reflected pulse is split into two pulses by a beam splitter BS1. One of the split pulse is stretched into a strongly chirped pulse, and anther is fed into a Michelsontype interferometer and then replicate into two replicas. The Michelson-type interferometer consists of two beam splitters BS2, BS3 and two corner mirrors TS2, TS3. A fixed time delay between the replicas is generated by tuning the translation stage TS2. The strongly chirped pulse and the replicas are focused by a paraboloid mirror PM, and then sum-frequency generation occurs in a sum-frequency crystal CR. A translation stage TS1 is used for tuning the relative time delay between the strongly chirped pulse and the two replicas. The two replica pulses interact with different parts of the chirped pulse and are upconverted to two different frequencies. The spectral interferogram of the frequency upconverted pulses is recorded by a fiber spectrometer (HR 4000, Ocean Optics Inc.). The spectral phase of the input pulse can be retrieved from the measured spectral interferogram. Our home-made experimental optical setup is shown in Fig. 1(b), with the optical path superposed.

Fig. 1. Our home-made SPIDER setup and the optical path. (a) Optical schematic, (b) experimental setup.

The spectrum of a pulse can easily be measured with a spectrometer. The pulse would be completely known, if we could, in addition, determine the phase across the spectrum [9]. SPIDER is a technique for measurement spectral phase of femtosecond optical pulses. In SPIDER setup, two replicas of the input pulse to be characterized are generated with a fixed time delay between them. These two replica pulses are then upconverted by sum-frequency mixing with a strongly chirped pulse derived from the same original input pulse. Because the two replica pulses are separated in time domain, they interact with different parts of the chirped pulse and are therefore upconverted to different frequencies. From the interferogram of these spectral shearing pulses, it is possible to extract the amplitude and

Our home made SPIDER setup is shown in Fig. 1. Figure 1(a) is the schematic optical path. An input femtosecond laser pulse passes through an aperture A1, and then is reflected by a reflective mirror M1. The reflected pulse is split into two pulses by a beam splitter BS1. One of the split pulse is stretched into a strongly chirped pulse, and anther is fed into a Michelsontype interferometer and then replicate into two replicas. The Michelson-type interferometer consists of two beam splitters BS2, BS3 and two corner mirrors TS2, TS3. A fixed time delay between the replicas is generated by tuning the translation stage TS2. The strongly chirped pulse and the replicas are focused by a paraboloid mirror PM, and then sum-frequency generation occurs in a sum-frequency crystal CR. A translation stage TS1 is used for tuning the relative time delay between the strongly chirped pulse and the two replicas. The two replica pulses interact with different parts of the chirped pulse and are upconverted to two different frequencies. The spectral interferogram of the frequency upconverted pulses is recorded by a fiber spectrometer (HR 4000, Ocean Optics Inc.). The spectral phase of the input pulse can be retrieved from the measured spectral interferogram. Our home-made experimental optical

**2. Characterization of femtosecond pulses with wavelet-transform for** 

phase of the initial pulse by using an algebraic inversion algorithm [2].

setup is shown in Fig. 1(b), with the optical path superposed.

Fig. 1. Our home-made SPIDER setup and the optical path. (a) Optical schematic,

**spectral phase retrieval** 

**2.1 Optical schematics of SPIDER** 

(b) experimental setup.

### **2.2 Spectral phase retrieval with a wavelet-transfrom**

In the traditional phase retrieval algorithm, spectral phase is extracted from the filtered alternating current component of the Fourier transform [8]. The filter is set by manual selection and adjustment. A problem is that different widths or shapes of the filter produce different spectral phases [10]. Thus the uncertainty of the spectral phase contributes to the uncertainty of the reconstructed pulses.

To reduce the uncertainty component coming from the filter with Fourier transform technique, we introduce a wavelet-transform for spectral phase retrieval of femtosecond optical pulses [11]. The phase is directly extracted from the ridge of wavelet-transform. There are no filters in this procedure so that the uncertainty from the filter is eliminated. In what fallows a demonstration of this procedure will be shown.

We have measured an ultrashort optical pulse train emitted from a Ti:sapphire laser (Micra-5, Coherent Inc.). The average output power of the laser is 360 mW after a pulse compressor. The repeat frequency is 82 MHz, and the central wavelength is 800 nm with a spectral bandwidth (FWHM) of 100 nm. We perform a SPIDER measurement with our home-made SPIDER setup. The measured spectral interferogram is shown in Fig. 2.

Wavelet-transform was applied on the measured spectral interferogram, and the time and frequency distributions are exhibited on a two-dimensional plane. The intensity map and phase map are shown in Figs. 3(a) and 3(b), respectively.

Fig. 2. Measured spectral interferogram.

Fig. 3. Wavelet-transform of spectral interferogram. (a) Intensity topography, (b) phase topography. The ridge of wavelet-transform is indicated with a pink coloured line.

We search for the maximum values from the intensity topography along each frequency column. Connecting the positions of the maximum values at each frequency point constructs the ridge line of the wavelet-transform, which is superposed on Fig. 3(a) with a pink coloured line. Then we project the position of ridge from intensity topography (Fig. 3(a)) on the phase topography, as is shown in Fig. 3(b).

The phase of the spectral interferogram was directly extracted from the phase topography at the position of the ridge. With the extracted interferometry phase, the spectral phase was obtained with a concatenation algorithm [8], as is shown in Fig. 4.

Fig. 4. Measured spectrum and retrieved spectral phase.

Fig. 3. Wavelet-transform of spectral interferogram. (a) Intensity topography, (b) phase topography. The ridge of wavelet-transform is indicated with a pink coloured line.

the phase topography, as is shown in Fig. 3(b).

obtained with a concatenation algorithm [8], as is shown in Fig. 4.

Fig. 4. Measured spectrum and retrieved spectral phase.

We search for the maximum values from the intensity topography along each frequency column. Connecting the positions of the maximum values at each frequency point constructs the ridge line of the wavelet-transform, which is superposed on Fig. 3(a) with a pink coloured line. Then we project the position of ridge from intensity topography (Fig. 3(a)) on

The phase of the spectral interferogram was directly extracted from the phase topography at the position of the ridge. With the extracted interferometry phase, the spectral phase was

#### **2.3 Pulse waveform reconstruction and autocorrelation simulation**

Figure 4 shows the measured spectrum with a spectrometer (HR4000 CG-UV-NIR, Ocean Optics Inc.) and the retrieved spectral phase with wavelet-transform technique. The electric field (Fig. 5(a)) and waveform (Fig. 5(b)) of the femtosecond optical pulse are reconstructed from the spectrum and spectral phase with an inverse Fourier transform technique. The pulse width (FWHM) shown in Fig. 5(b) is 18.2 fs.

Fig. 5. Reconstructed electric field and waveform. (a) Electric field, (b) waveform.

From the reconstructed electric field in Fig. 5(a), we simulated the autocorrelation traces of the pulses. The simulated interferometric autocorrelation and intensity autocorrelation are shown in Fig. 6(a) and 6(b), respectively. The intensity autocorrelation in Fig. 6(b) shows the width of autocorrelation (FWHM) is 26.6 fs. With the reconstructed pulse waveform in Fig. 5(b) and the simulated intensity autocorrelation in Fig. 6(b), we can obtain the ratio of autocorrelation width *<sup>c</sup>* to the pulse width *p* is *c<sup>p</sup>* = 1.462.

Fig. 6. Simulated autocorrelation traces with reconstructed pulse electric field. (a) Interferomatric autocorrelation, (b) intensity autocorrelation.

#### **2.4 Pulse waveform reconstruction from different replicas separations**

We have investigated the effects of pulse replicas separation on the spectral phase retrieval of femtosecond optical pulses. By tuning the translation stage TS2, a serious of spectral interferograms with different replicas separations was recorded. Figures 7(a), 7(c), 7(e), 7(g), and 7(i) are five measured spectral interferograms with replicas separations of 0.22 ps, 0.55 ps, 0.83 ps, 1.58 ps, and 1.82 ps respectively. We retrieved the spectral phases from the five measured interferograms with wavelet-transfrom. The extracted spectral phases are shown in Figs. 7(b), 7(d), 7(f), 7(h), and 7(j).

We have investigated the effects of pulse replicas separation on the spectral phase retrieval of femtosecond optical pulses. By tuning the translation stage TS2, a serious of spectral interferograms with different replicas separations was recorded. Figures 7(a), 7(c), 7(e), 7(g), and 7(i) are five measured spectral interferograms with replicas separations of 0.22 ps, 0.55 ps, 0.83 ps, 1.58 ps, and 1.82 ps respectively. We retrieved the spectral phases from the five measured interferograms with wavelet-transfrom. The extracted spectral phases are shown

**2.4 Pulse waveform reconstruction from different replicas separations** 

in Figs. 7(b), 7(d), 7(f), 7(h), and 7(j).

Fig. 7. Measured spectral interferograms with different replicas separations and retrived spectral phases from the interferograms. (a), (c), (e), (g), and (i) are measured interferograms with replicas separations of 0.22 ps, 0.55 ps, 0.83 ps, 1.58 ps, and 1.82 ps respectively; (b), (d), (f), (h), and (j) are the retrieved spectral phases from interfrograms in (a), (c), (e), (g), and (i) respectively.

With the retrieved spectral phases and the measured spectrum, the waveform of the pulse can be reconstructed. Figure 8(a) is the five retrieved spectral phases with different replicas separations and the spectrum. Figure 8(b) is the reconstructed pulse profiles from the spectrum and spectral phases with inverse Fourier transform.

Fig. 8. Retrieved spectral phase and reconstructed pulse profile with different replicas separations. (a) Spectrum and retrieved spectral phases, (b) reconstructed pulse profiles.

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