2. Experimental setup, results and analysis

The experimental setup is shown in Figure 1. The system is driven by a Yb:yttrium aluminum garnet (YAG) laser with 0.8 ps pulse duration, 740 μJ pulse energy and 1030 nm center wavelength. The BOC technology is adopted to characterize the time fluctuation. The cross-correlation intensity of BOC measurement has a linear characteristic within a specific delay range, which is attributed to the overlap of two laser pulses in time. It is worth noting that the slope of the linear region represents the measurement accuracy. A shorter pulse duration makes it easier to acquire steep slopes and improves measurement accuracy^[ Reference Shi, Song, Wang, Zhao and Hu^25]. Thus, further compressing the laser pulse duration is an effective way to increase measurement accuracy.

A Herriott-type multi-pass cavity (MPC) is employed for pulse duration compression, as depicted in Figure 1. It is made up of two concave mirrors with a 600-mm curvature radius and 2-inch diameter. The MPC is built in atmosphere, and air is used as the nonlinear medium. Taking into account the ionization threshold of air and the safe operation zone of the MPC, the pulse energy is set to 650 μJ and the distance between cavity mirrors is 750 mm. The spectrum is continuously broadened due to self-phase modulation (SPM) when the laser pulse performs 22 round trips in the MPC, equating to a distance of 33 m in the air. The dispersion is compensated by a pair of dispersive mirrors, which offer a dispersion of −12,000 fs².

The pulse temporal profile is characterized by a second harmonic generation frequency-resolved optical gating (SHG-FROG), as shown in Figure 2. The measured and retrieved FROG traces are shown in Figures 2(a) and 2(b) with a FROG error of 0.9%. As shown in Figure 2(c), the measured (black line) and retrieved (red line) spectra are in good agreement. The pulse duration at full width at half-maximum (FWHM) is 95 fs, as shown in Figure 2(d). The output pulse energy is 590 μJ, yielding about 90% transmission efficiency.

Figure 2 Temporal characterization after compression. (a) Measured FROG traces. (b) Retrieved FROG traces. (c) Retrieved spectrum (red line) and phase (blue line) together with the measured spectrum (black line). (d) Temporal intensity (red line) and phase (blue line).

The compressed laser pulses are split by a beam splitter (BS1) and imported into different paths. After approximately 10 m of propagation, the timing fluctuation can arise from several factors, with the major contributions typically being atmospheric turbulence, mechanical vibrations of the mirror holders and thermal drift caused by environmental temperature. Environmental temperature, in particular, is a significant source of timing errors, as the refractive index of air varies with temperature, leading to changes in the optical path length. Mechanical vibrations are also important, as even minor displacements of optical elements can alter the optical path length and introduce timing jitter. While atmospheric turbulence also plays a role, ensuring mechanical stability and a constant temperature environment is often the main concern in minimizing timing errors. Therefore, the timing fluctuation needs to be actively compensated through feedback control systems. Subsequently, the beams are reflected by BS2 and BS3 into a BOC for time fluctuation measurement. In addition, the BOC signal is analyzed to control a feedback loop for the synchronization of pulse envelopes.

Figure 3(a) depicts the detailed setup of the noncollinear BOC. A 50:50 beam splitter splits the incident laser pulses into two arms of the BOC. The laser beams are focused noncollinearly on 0.5-mm-thick type-I ( $\theta ={23.4}^{\circ },\varphi ={0}^{\circ }$ ) beta barium borate (BBO) crystals for sum frequency generation (SFG). The noncollinear setup eliminates the influence of second harmonics. A glass plate is inserted into one of the transmitted beams for the reference of time offset. The SFG signals are subtracted by a balanced photodetector (PDB450, Thorlabs Inc.) to characterize the timing jitter. Measurement error caused by energy drift is significantly reduced by balanced detection.

Figure 3 (a) Schematic of the noncollinear BOC. M, mirror; BS, beam splitter; GP, glass plate; BBO1 and BBO2, beta barium borate crystals; PhD, photodetector. (b) Measured cross-correlation curves at 0.8 ps and 95 fs, respectively. (c) Timing drift of 0.8 ps when the feedback loop is off (gray line) and on (black line) together with the timing drift of 95 fs when the feedback loop is on (red line).

To compare the BOC measurement accuracy for long and short pulses, the cross-correlation ‘S’ curves are measured with and without pulse duration compression (yellow dashed lines in Figure 1), as shown in Figure 3(b). The corresponding accuracy is 14.57 mV/fs (0.8 ps) and 52.5 mV/fs (95 fs), which indicates a 3.6 times improvement in measurement accuracy by compressing the laser pulse duration from 0.8 ps to 95 fs via an MPC compressor. Analysis control unit 1 includes a data acquisition card, which captures the voltage amplitude output from the balanced photodetector and converts it into timing jitter using the measurement accuracy, thus determining the required movement distance for ODL1. This process forms a closed-loop feedback system. The time fluctuation is corrected by the feedback loop, in which the BOC signal is analyzed and a delay line with 10 nm accuracy is controlled to compensate for the drift. As shown in Figure 3(c), with higher measurement accuracy, the corrected time fluctuation decreases from 8.3 to 1.12 fs RMS in 2 hours, which is shorter than one optical cycle.

On this basis of the BOC-based feedback loop, interferometry-based feedback is applied to further improve the timing accuracy to attosecond level and realize phase synchronization between laser pulses. Interferometry benefits high precision in phase measurement by converting the fluctuation of interference fringes to relative phase drift. Assume that a laser propagates along the z-axis, which is perpendicular to the plane formed by the x-axis (horizontal direction) and the y-axis (vertical direction). The horizontal spatial electric field distribution of the laser when it is incident at a certain angle on the laser beam profiler can be written as follows^[ Reference Liu, Song, Liu, Wang, Huang, Tang, Wang, Liu and Leng^24, Reference Liu, Peng, Wu, Liang and Li^26]:

(1) $$\begin{align}E(x)=A(x)\exp \left[j\left( kx+\phi \right)\right],\end{align}$$

where $A(x)$ , $k$ and $\phi$ represent the amplitude, wave vector and phase of the laser on the x-axis, respectively. When laser beams are incident on a camera with a small included angle, the intensity of interference fringes is as follows:

(2) $$\begin{align}&I(x)=\sum {A}_n{(x)}^2+2{\sum}_n{\sum}_{m,m\ne n}{A}_n(x){A}_m(x)\nonumber\\&\cdot \cos \left[2\pi \alpha x/\lambda +\left({\phi}_n-{\phi}_m\right)\right],\end{align}$$

where $\lambda$ is the laser wavelength, $\Delta \varphi =2\pi \alpha x/\lambda +\left({\phi}_n-{\phi}_m\right)$ is the laser phase difference and $\alpha$ is the angle between the two incident lasers beams. The interference period ${D=\lambda /\sin \alpha}$ . The optical path difference $\delta =d\times \sin \alpha$ , where $d$ is the movement distance of the interference fringes^[ Reference Liu, Song, Liu, Wang, Huang, Tang, Wang, Liu and Leng^24]. Thus, the timing jitter is converted to the movement distance of the interference fringes.

As in the experimental setup shown in Figure 1, the transmissions of BS2 and BS3 are reflected onto a camera (Ophir-Spiricon, BeamGage) with a small included angle to record interference fringes. The fringes are analyzed by analysis and control unit 2, which also includes data acquisition and processing cards, captures the interference fringe data output from the camera and calculates the movement distance $d$ of the fringes between consecutive moments, and optical delay line 2 is controlled to compensate for the laser phase drift.

The interference fringes are shown in Figure 4(a) with a beam pattern as the insert. Phase shift between laser pulses is characterized by the offset of interference fringes, as indicated by the red dotted line in Figure 4(a). The shift of interference fringes is converted to optical path difference by $\delta =d\times \sin \alpha$ . The included angle $\alpha ={0.13}^{\circ }$ is calculated by an interference period of 439 μm. By fine adjustment of the interferometry-based feedback loop, the time fluctuation is compensated to 189 as RMS over 30 minutes, which represents $\lambda /18$ for a laser wavelength of 1030 nm, as shown in Figure 4(b).

Figure 4 (a) Schematic diagram of interference fringe intensity distribution with (red line) and without (black line) phase drift (inset shows the coherently combined). (b) Time difference when the BOC system is ON and when both BOC and interferometry are ON.

In the closed-loop feedback system, feedback bandwidth plays a vital role in ensuring that the system operates accurately, stably and efficiently. The feedback bandwidth mainly depends on the performance of a series of components in the experiment, including the photodetector, data acquisition card, camera acquisition frequency, proportional–integral–derivative (PID) algorithm and the response speed of the motor. Limited by the stepper motor speed and camera acquisition speed, the feedback bandwidth is 1 kHz and 10 Hz in the BOC and interferometry-based feedback loops, respectively. The bandwidth can potentially be further increased by improving the speed of the detection. Furthermore, owing to the features of the optical delay line, the timing jitter can be compensated for up to 66.7 as, which ultimately determines the best compensation effect that the synchronization system can achieve. The timing jitter compensation capabilities can be further enhanced in the future by further compressing the pulse duration as well as improving the feedback bandwidth.