1. Introduction

Carrier-envelope phase (CEP) stabilization of femtosecond laser pulses has become one of crucial techniques in attosecond science [1–3] as well as in frequency metrology [4–6]. The CEP stabilization technique has been developed on the basis of the phase-locked loop (PLL) method [7–10], operating in the frequency domain, and widely used for the generation of a stable optical frequency comb. The PLL method stabilizes pulse-to-pulse CEP slip to be constant. On the other hand, the direct locking (DL) method is a different kind of CEP stabilization technique operating in the time domain [11]. DL directly uses the beat signal from an f-to-2f interferometer as an error signal and quenches it through a negative feedback. As the DL method generates CEP-stabilized pulses with zero CEP slip, the CEP of every pulse is same. It is, thus, beneficial to applications requiring femtosecond pulses with identical CEP, compared to the conventional PLL method necessitating a special scheme to achieve zero CEP slip [5]. Successful operation of DL has been confirmed with an out-of-loop measurement [11]. Recently, the DL technique has been improved further using homodyne balanced detection (HBD) and double feedback techniques, achieving low CEP jitter and long-term CEP stabilization [12]. Furthermore, the CEP of amplified pulses in a kHz femtosecond Ti:Sapphire laser was successfully stabilized using DL and applied to high harmonic generation with CEP-stabilized femtosecond laser pulses [13].

In this paper, we present a practical solution to implement the DL method for the CEP stabilization of femtosecond laser pulses, achieving continuous CEP stabilization for 24 hours without any optical realignment. The CEP stabilization is realized using commercially available standard feedback electronics containing the functions of proportional, integral, and derivative (PID) operations. By testing PID functions an optimum solution to realize the DL has been obtained. In addition we show that the DL can be realized without using any frequency-related equipment such as an RF spectrum analyzer, making it also an economical solution for CEP stabilization of femtosecond lasers.

2. Practical implementation of the direct locking method

CEP stabilization means the control of pulse-to-pulse CEP variation of a femtosecond laser. The information on CEP variation of a femtosecond laser can be obtained from an f-to-2f interferometer [14, 15]. The DL method first measures a beat signal between f _2n and 2f_n components of an octave-spanned laser spectrum due to CEP variation, in which f _2n and 2f_n are the fundamental and frequency-doubled 532-nm components of the laser spectrum, respectively. The beat signal is then sent to a feedback servo as an error signal. The CEP stabilization by the DL method is achieved by quenching the beat signal by the feedback servo. We have realized the feedback servo using PID controllers, commonly used for stabilizing system operation, such as constant temperature maintenance.

 

Fig. 1. Schematic diagram of the direct locking setup for CEP stabilization. (AOM: acoustooptic modulator, PID: analog PID controller, PCF: photonic crystal fiber, DB: dichroic beam splitter, SHG: second-harmonic generation crystal, HWP: half-wave plate, BPF: band-pass filter, PBS: polarizing beam splitter at 532 nm, APD: avalanche photodiode, BPD: balanced PIN photodiode)

Download Full Size | PDF

We applied DL for the CEP stabilization of a femtosecond oscillator with a set of intracavity prisms. Figure 1 shows the schematic diagram of the direct locking setup that consists of the f-to-2f interferometer, photodiodes, and locking servo [11]. For the generation of an octave spanning spectrum a tapered photonic crystal fiber (Femtowhite 800, Crystal-fibre) was used, which greatly improved the alignment sensitivity compared to a bare photonic crystal fiber. The HBD setup in Fig. 1 optically eliminates any signal fluctuations, not originating from CEP variation, for the generation of a pure beat signal. The half-wave plate and polarizing beam splitters (PBS3, PBS4) were installed in front of the balanced photodiodes (BPD) to realize HBD. The pure beat signal generated from BPD was then sent to the feedback servo (PID1). The feedback signal, sent to an acousto-optic modulator (AOM), adjusted the pump power to the oscillator, and the CEP stabilization was achieved by directly quenching the beat signal [15, 18].

The generation of the beat signal requires the overlap of the f _2n and 2f_n components in space and time. The spatial overlap was obtained by careful alignment, but the temporal overlap could be found by cautious monitoring of the beat signal with an oscilloscope. Figure 2 shows how the temporal overlap can be checked with the oscilloscope. The leakage beam containing the 2f_n and f _2n components from PBS3 was detected using an avalanche photodiode (APD) with 100 MHz bandwidth. When the f _2n and 2f_n signals did not temporally overlap, a pulse train with the laser repetition rate appeared, as shown in Fig. 2(a). As the time delay was adjusted, the pulse train exhibited a modulation when the two signals became temporally overlapped, as shown in Fig. 2(b). In this case the modulation frequency was too high for feedback control. In order to make the frequency within the accessible feedback frequency of 100 kHz, the insertion of the intracavity prism was adjusted. Figure 2(c) shows the pulse train with a modulation frequency of about 2.5 MHz. With further adjustment of the prism insertion the modulation frequency could be reduced within the feedback control frequency. This is another benefit of the DL method achieved without using an RF spectrum analyzer.

 

Fig. 2. Monitoring the beat signal, using APD in Fig. 1 and an oscilloscope, to match the time delay between f and 2f signals in the f-to-2f interferometer. The beat signal changed from (a) to (b) to (c) as the time delay became better matched.

Download Full Size | PDF

With sufficiently low frequency modulation below 100 kHz the direct locking servo was activated. For the feedback control an analog PID controller (SIM960, SRS) with 100-kHz operational bandwidth was used. The pure beat signal from the BPD in Fig. 1 was supplied as an error signal, ε. The feedback signal (V_out) generated from the PID controller can be expressed as follows [19]:

where P, I, and D are proportional, integral, and derivative gain, respectively, and V_off is an offset voltage. Using only the ‘P’ function the error signal can be quenched, but it may generate a non-zero error that can be corrected using the ‘I’ function. When the non-zero error contains rapidly varying component, it can be corrected by the ‘D’ function. The result of the feedback control operation is shown in Fig. 3. The pure beat signal in Fig. 3(a) was generated using the HBD technique. As the beat frequency became less than 100 kHz, it could be measured with the BPD consisting of two PIN photodiodes; it is another benefit of the DL method as compared to the conventional PLL method requiring high sensitivity APD’s due to its operation at much higher frequency. When the feedback was performed with only the ‘P’ function, the quenched beat signal contained low-frequency fluctuation, as shown in Fig. 3(b). This fluctuation could be stabilized using the ‘I’ function, as shown in Fig. 3(c), without further applying the ‘D’ function.

 

Fig. 3. Quenching of the beating signal using standard feedback electronics. (a) Pure beating signal monitored by the BPD in the f-to-2f interferometer. Feedback signals obtained using only the ‘P’ function of the PID controller (b) and using the ‘P’ and ‘I’ functions (c).

Download Full Size | PDF

 

Fig. 4. CEP jitter measured with different observation duration. The stabilized beat signals measured for 100 us (a), 1 ms (b), and 10 ms (c). (d) Change of the accumulated CEP jitter with different observation duration.

Download Full Size | PDF

In the DL method, the data analysis is simple and intuitive. CEP jitter can be estimated simply by comparing the BPD signals before and after locking. As the beat signal before locking provides a full sinusoidal swing, the CEP jitter can be easily obtained by converting each point of the beat signal after locking to corresponding angular value [11]. In order to investigate the accumulated phase jitter of CEP-stabilized laser pulses, the root-mean-square CEP jitter was measured with different observation duration, as shown in Fig 4. The CEP jitter increased from 31 mrad for 100-µs observed duration to 35 mrad for 1ms, and to 36 mrad for 10 ms, as shown in Figs. 4 (a), (b), and (c), respectively. The accumulated phase jitter saturated to about 37 mrad for the observed duration of 100 ms or longer, as shown in the Fig. 4 (d). In the DL method this kind of analysis can be done without using a vector spectrum analyzer or frequency counter, one of strong advantages of the DL method.

3. Long-term CEP stabilization

Long-term CEP stability is crucial for actual applications of CEP-stabilized femtosecond lasers. In the free running mode, the CEP variation contains a slow drift as well as fast variation. The DL quenches both components, but the slow drift may cause the feedback signal to AOM to drift to one direction, eventually disrupting the CEP locking or even the mode-locking of the oscillator itself. The large pump power modulation by AOM was thus the most severe obstacle in the long-term CEP stabilization. This problem has been resolved using the double feedback loop that separately controls fast variation and slow drift using two feedback loops [12].

We have realized the double feedback loop using two PID controllers (PID1 and PID2), as shown in Fig. 1. The fast CEP variation was compensated by the first feedback servo connected to AOM, but continuous CEP drift can cause too large modulation of AOM. Figure 5(a) show the feedback signal to AOM (black line) and the beat signal measured with the detector BPD (gray line) obtained only with the feedback servo PID1. The feedback signal to AOM was continuously decreased and then the CEP stabilization was interrupted after about 20 minutes. In this case the strong modulation of AOM disrupted the CEP locking.

The large modulation of AOM can be prevented by compensating for the slow drift component with the second feedback servo controlling the intracavity prism insertion. Instead of compensating for CEP variation only with the first feedback servo that may cause too large pumping power modulation, the slow CEP drift was compensated for by adjusting the intracavity prism insertion. For the latter the second PID controller (PID2) was employed with the ‘P’ and ‘I’ feedback functions for the prism insertion control. Since the drift was not so fast, a low cutoff frequency in the ‘I’ operation was used. Figure 5(b) shows the change of the feedback signal to AOM with different cutoff frequencies. In the case of 0.3-Hz cutoff frequency, the feedback signal contained a drifting component. With 3-Hz cutoff frequency it was improved, but the feedback signal still shows some variation. So the cutoff frequency was increased to find the optimum frequency of 10 Hz at which the feedback signal to AOM showed fluctuation around a steady value. Figure 5(c) presents the result of 24-hour CEP stabilization achieved without realignment of any optical components. It shows the feedback signals to AOM (black line) for pumping power control and to PZT for the control of prism insertion (gray line). The feedback signal to PZT was continuously adjusted so as to stably maintain the feedback signal to AOM, allowing long-term CEP stabilization. The CEP stabilization for a whole day is a clear demonstration of the robustness of the DL method. As a consequence, the capability of long-term CEP stabilization will make the DL method a powerful tool in investigating light-matter interactions with CEP-stabilized laser pulses, such as attosecond science. In addition the DL method has a good potential also in frequency metrology, since the control of the comb frequency can be achieved by adjusting the cavity length of the oscillator.

 

Fig. 5. 24-hour CEP stabilization achieved by employing the double feedback loop. (a) Performance of the CEP stabilization obtained by controlling only AOM. (b) Optimization of the cutoff frequency in the ‘I’ feedback. (c) Continuous CEP stabilization for 24 hours. Feedback signals to AOM (black) and to PZT (gray) are shown during the 24-hour CEP stabilization.

Download Full Size | PDF

4. Conclusion

Based on the direct locking method CEP stabilization of femtosecond laser pulses for a whole day has been demonstrated. The direct CEP locking that stabilizes CEP of every laser pulse to a same CEP value was realized using two analog PID controllers. The fast CEP variation was stabilized by the first PID controller connected to the AOM, and the slow CEP drift was managed by employing the second PID controller to adjust the intracavity prism insertion. It is noted that the CEP stabilization was achieved without using any frequency-analyzing equipment such as RF spectrum analyzer. As a consequence, the direct locking method is a very practical solution for the CEP stabilization of a femtosecond laser, greatly improving the accessibility of CEP stabilization, and will greatly benefit such areas as attosecond physics.