1. Introduction

Linearly chirped pulse is widely used as probe light today because the ultrafast information in time domain could be mapped to frequency domain linearly with a chirped probe pulse and recorded by a spectrometer without fast response detectors [1,2]. So far, the chirped pulses have played an important role in terahertz detection [3–5], laser driven shock waves [6–9], laser-induced plasmas [10–14], et al. Typically, a linearly chirped pulse is obtained by travelling a femtosecond pulse through a grating pair or highly dispersive material, its chirp characteristics can be measured by streak camera coupled to a spectrometer [13], cross correlation method [15], optical Kerr gate technique (OKG) [16–18], two-photon absorption [19], or cross phase modulation (XPM) [20]. However, streak camera is limited by picosecond time resolution which is unsatisfactory for short-duration pulse measurement. For the other methods, nonlinear optical materials with large nonlinearities and fast response time, and relatively high intensity laser pulses are essential [18]. Consequently, the application of these techniques is limited.

In this paper, we developed a linear optical technique to measure the chirp characteristics of linearly chirped pulses without intensity restrictions and nonlinear optical materials. Using this method, we performed a demonstrative experiment to measure the chirp rate and duration of a picosecond chirped pulse.

2. Measurement principle

Consider a Fourier transform-limited Gaussian pulse whose electric field is

According to the principle of frequency domain interference, after the two pulses above enter into the spectrometer with a time delay$T$, a spectral interferogram would be acquired, and the intensity distribution of the spectral interference fringes is

To show this technique more specifically, the frequency domain interference between a femtosecond pulse and the corresponding picosecond chirped pulse is simulated. In this simulation, the pulse duration is 30fs and 40ps respectively, the chirp rate is$2b=2.2100\times {10}^{-6}{\text{rad/fs}}^{\text{2}}$, the time delay is$T=\text{5ps}$, and the center wavelength is ${\lambda }_0=\text{800nm}$. Figure 1 shows the simulated 1D spectral interference fringes in the wavelength domain. We can see that there is a very wide fringe on the pattern as predicted. However, the extremum of the widest fringe does not locate exactly on the central position ${\lambda }_s$ because $\cos \left(\Delta \varphi \right)$ in Eq. (3) is modulated by the DC background and the modulation amplitude. Fortunately, the modulation becomes weaker and weaker when the fringe width is decreasing. Therefore, compared to determine it by the minimum of the widest fringe, taking the average of symmetric order fringe positions gives ${\lambda }_s$ much more precisely, here we adopted ${\lambda }_s=1/2\left({\lambda }_{-1st}+{\lambda }_{1st}\right)=\text{803}\text{.78nm}$. Put${\lambda }_s$and $T$ in Eq. (6), we recovered a chirp rate $2b=2.2100\times {10}^{-6}{\text{rad/fs}}^{\text{2}}$ which matches very well with the input. Therefore, the central position ${\lambda }_s$ can be determined quite precisely although the extremum position of the widest fringe deviates from it.

 

Fig. 1 The spectral interference fringes between a femtosecond pulse and the corresponding picosecond chirped pulse in the wavelength domain.

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3. Experiment

To confirm our proposal, an experiment was carried out. It was performed on a Ti: sapphire laser oscillator which delivers 30fs, 790nm pulses at a repetition rate of 84MHz, the energy is ~9nJ per pulse. Figure 2 shows the experimental setup for measuring the chirp characteristics of linearly chirped pulses. The incident pulse is split into two by a beam splitter BS1, a portion of the laser pulse travels through a pair of gratings which stretch the pulse temporally, after reflected by BS2 and going on through another beam splitter BS3, the stretched pulse is perpendicularly reflected by a planar mirror M5 which is mounted on a translation stage. The other portion of the laser pulse is used as a reference pulse. These two beams are recombined at BS3, and an imaging spectrometer is used to record the spectral interferogram. The chirp rate and duration of the stretched pulse are then measured by the frequency domain interference technique as described above.

 

Fig. 2 Experimental setup to measure the chirp characteristics of linearly chirped pulses, BS-beam splitter, M-planar mirror, G-grating.

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In this experiment, a sequence of time delays between the chirped pulse and the reference pulse are achieved by moving M5 with a certain step length which has been accurately calibrated by frequency domain interferometry. In the calibration, M6 is inserted into the optical path, and then a Michelson interferometer is formed. The spectral interferogram of two chirped pulses is recorded by the spectrometer. Finally, we get the step length by reading out the fringe width variation between adjacent interferogram, and calibration results show that the step length is $93.02\pm 2.69\mu \text{m}$.

Figure 3(a) shows spectral interferogram of a femtosecond laser pulse and the corresponding chirped pulse at four different time delays. One can see that there is a very wide fringe in each interferogram, and the widest fringe shifts in the same direction as the time delay increases. What is more, fringe widths on both sides are approximately symmetric and decrease gradually when departing from the widest fringes, as theoretically predicted. Integrating in the vertical direction gives an 1D distribution of Fig. 3(a) at 7441.2fs time delay, as shown in Fig. 3(b), revealing a similar fringe distribution to Fig. 1. Figure 4 shows our measurement of the widest fringe central position${\omega }_s$versus time delay. Each central position was measured for ten times, and the inset in Fig. 4 is a zoom-in to show an error bar. There is an approximately linear relationship between the central position and the time delay, which matches well with the effect of grating stretcher, and our fit gives the chirp rate $2b=-4.13\times {10}^{-6}\pm 1.58\times {10}^{-8}{\text{rad/fs}}^{\text{2}}$ (negative chirp for grating-based pulse stretcher). Figure 4 also shows the chirped pulse spectrum whose width is $\Delta \omega =0.0\text{982rad}/\text{fs}$ (FWHM)). Put them in ${\tau }_c=\Delta \omega /\left|2b\right|$ leads to the duration ${\tau }_c\approx \text{23}\text{.8ps}$, the relative uncertainty is 3.44%.

 

Fig. 3 (a) Spectral interferogram at time delays of 7441.2fs, 11781.9fs, 16122.6fs and 20463.3fs. (b) 1D spectral interferogram at the time delay of 7441.2fs.

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Fig. 4 The central position ${\omega }_s$ versus time delay $T$, and the spectrum of the chirped pulse. The inset is a zoom-in to show an error bar.

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Additionally, the reference pulse is not strictly Fourier transform-limited but slightly chirped by inserted optical elements dispersion. One might worry if this would affect the measurement of the chirp characteristics. Actually, the technique is safe as long as ${\tau }_0\ll {\tau }_c$ is satisfied no matter the reference pulse is chirped or not.

4. Conclusion

A diagnostic technique was proposed for measuring the chirp characteristics of linearly chirped pulses that widely used in single shot, ultrafast measurements. Compared to conventional nonlinear techniques, this technique can be carried out without nonlinear optical materials and pulse intensity restrictions. Using this method, we performed the chirp measurement of a picosecond chirped pulse, the experimental results shows that the frequency of the chirped pulse is mapped to time linearly as we expected.