The carrier-envelope phase (CEP) of femtosecond pulses from a mode-locked Ti:sapphire laser is stabilized using a novel method operating in the time domain. This is a direct and relatively simple method for stabilizing the CEP of femtosecond laser pulses, compared to the conventional method based on phase-locked loop. Using this method, we have directly locked the pulse-to-pulse CEP slip to zero with an in-loop phase jitter of 0.05 rad. Out-of-loop measurement has revealed a comparable phase jitter of 0.1 rad, verifying a good performance of this locking technique. The capability of electrical CEP modulation is also demonstrated.
©2005 Optical Society of America
Laser development toward the generation of shorter and shorter pulses has been rapidly progressing in recent years. Laser pulses containing only two optical cycles (~5 fs) can be generated from mirror-dispersion-controlled Ti:sapphire lasers [1, 2]. Amplified high intensity laser pulses can also be shortened to a two-cycle range using a gas-filled hollow-core fiber [3, 4]. In the few-cycle regime, the phase difference between the carrier wave and its pulse envelope, called the carrier-envelope phase (CEP) or the absolute phase, is an important parameter of the laser pulse. For such a pulse, the electric field itself becomes more important than its intensity profile and the shape of their electric field is sensitive to the CEP value. The CEP effects become apparent when few optical cycle pulses are used in high intensity light-matter interactions such as above threshold ionization and high-order harmonic generation (HHG) [5, 6, 7]. In particular, the effect of CEP variation on HHG has been predicted theoretically  and demonstrated experimentally . It is now recognized that such experiments should be carried out with laser pulses whose CEP is stabilized and controlled. Ideally, these experiments require pulses with identical CEP.
The pulse-to-pulse CEP slip of femtosecond pulse train arises [9, 10] from the difference between group and phase velocities inside the oscillator cavity. This CEP slip undergoes random fluctuations due to air currents, temperature, mechanical vibration, and laser intensity variation. Therefore, the output from a typical femtosecond laser is a train of non-identical pulses.
In general CEP stabilization has been done using the method based on phase-locked loop (PLL) [7, 11, 12]. In this method, the carrier-envelope offset frequency (fceo) is measured using a self-referencing technique [9, 11, 13] and then locked to a stable reference frequency. A fixed fceo means a constant pulse-to-pulse CEP slip (Δϕ) given by
where frep is the repetition rate of the laser. In the time domain, a constant Δϕ does not mean identical pulses. In order to produce a pulse train with identical CEP, fceo should be locked to zero, but the PLL method intrinsically requires a non-zero reference frequency. To solve this problem, an additional acousto-optic modulator (AOM) has to be inserted inside the f -to-2f interferometer for a frequency shift, thereby complicating the system .
In this paper, we have demonstrated the CEP stabilization of femtosecond laser pulses using a novel method operating in the time domain that locks Δϕ to zero, which we refer to as direct locking method. To our knowledge, this is the first experimental demonstration of a time-domain feedback method for stabilizing the CEP of the laser pulses. In the direct locking method described here, the beat signal itself, measured in the time domain, is directly used for the CEP stabiliaztioin. Since the beat signal represents the CEP evolution of the laser pulses, the variation of the beat signal from peak to valley corresponds to the CEP change of π rad. The beat signal can be regarded as the ‘error signal’, i.e., signal difference between fceo and reference frequency, of the PLL-based CEP stabilization method. In our method, the reference frequency is replaced by a reference DC level which is usually set as zero-voltage level for simplicity. Thus, any radio-frequency (RF) source providing the reference frequency is not required. A feedback signal can be synthesized from the beat signal (or error signal) to suppress the oscillation of the beat signal itself and ultimately makes it a DC signal. This simple idea is equivalent to ‘zero-frequency locking’ (fceo=0) in the frequency domain and makes it possible to produce the laser pulses with the same CEP value using a simple configuration.
2. Experimental setup
CEP stabilization with the direct locking method is realized using the experimental setup shown in Fig. 1. Sub-10 fs pulses are generated at 82 MHz repetition rate using a mirror-dispersion-controlled Kerr-lens-mode-locked Ti:sapphire laser pumped by a frequency-doubled singlefrequency Nd:YVO4 laser (Verdi, Coherent Inc.). The laser oscillator and the f -to-2f interferometer are placed inside a box and sealed to minimize undesirable effects of air currents, and the entire setup is installed on an isolated optical table to reduce the mechanical vibration which directly contributes to the CEP fluctuation. An AOM is used to stabilize the CEP of laser pulses by controlling the pump power .
The photonic crystal fiber (PCF)  generates the octave-spanning spectrum that simultaneously contains fn (1064 nm) and f 2n (532 nm) components. The long-wavelength portion (fn) is frequency doubled (2fn) at a 1-mm-thick potassium-titanyl phosphate (KTP) crystal and temporally interfered by the original short-wavelength portion (f 2n). The beams from the f and 2f arms of the interferometer are recombined in a polarizing beamsplitter PBS1 and detected using the two avalanche photodiodes APD1 and APD2. A polarizer PBS2 placed in front of APD1 allows us to detect the interference signal I 1 given by
where A and B are the coefficients representing the DC terms of the components at f 2n (532 nm), i.e., , and frequency-doubled fn (1064 nm), i.e., , respectively, and C that of the interference term while ϕ 0 is a constant phase.
We concern only the third term in Eq. (2) because it is the oscillating beat signal containing the fceo information. This term can be easily extracted using an electric high-pass filter when the locking system is operating at a high frequency (>1 MHz) as in the conventional PLL-based method. However, the direct locking method can be initiated only when fceo is close to zero or small enough to be suppressed by an electronic feedback loop (whose operational bandwidth is ~100 kHz in our setup). In case that fceo is small, the fluctuation of two DC components in Eq. (2) can be mixed up with the beat signal term because all three terms are in the low-frequency region. Therefore, we have to decouple the two DC components from the interference signal, I 1, to obtain a pure beat signal. For this purpose, we separately acquire the DC components using APD2 as follows:
where D and E are the coefficients representing the DC terms originating from the f 2n and 2fn components, respectively. In the experiments, coefficients A and D (B and E) are adjusted to have the same magnitude with the help of half-wave plates placed in the f (2f) arms of the f -to-2f interferometer. The signal (I 2) from APD2 is used to cancel out the DC terms contained in the output (I 1) of APD1 using a simple subtraction circuit, and then a pure beat signal is obtained. As mentioned above, this beat signal is used as the error signal in the feedback loop, controlling the AOM that modulates the pumping power. The overall setup has become more simplified than that of conventional PLL-based systems because several RF instruments, such as frequency synthesizer (or frequency divider) and an additional AOM , are not necessary for this time-domain feedback technique.
3. CEP locking process and in-loop analysis
The procedure of CEP stabilization can be described as follows. First, we observe the fceo using an RF spectrum analyzer connected to APD3, and then adjust fceo to be within the operational bandwidth of the servo loop by varying the prism insertion in the laser. The beat signal at fceo in the time domain can be observed using an oscilloscope connected to the servo output. Figure 2(a) shows the pure beat signal extracted from the interference signal (APD1). By balancing the DC levels in APD1 and APD2, DC terms in the interference signal are removed.
The servo loop is turned on after the above procedure. Figure 2(b) shows the beat signal obtained after the servo loop is turned on at time zero. Once the servo loop is turned on, the beat signal amplitude is reduced to a DC signal except for small fluctuations. This means that the CEP of the mode-locked laser pulses is now locked to an unknown but constant value which can be obtained from additional CEP-sensitive experiments [7, 16, 17].
For the confirmation of the CEP locking, histograms of the beat signal before and after locking, which represent the occurrence probability of a certain amplitude value in the beat signal, are plotted as shown on the right side of Fig. 2(b). Before locking, the histogram (blue) appears concentrated at the top and bottom, showing the sinusoidal nature of the beat signal. After locking, however, the histogram (red) appears concentrated near the center of oscillation, thus showing that the CEP of the laser is now stabilized. The CEP jitter, obtained by taking the ratio of the root-mean square (RMS) of the beat signal after locking and the amplitude of the beat signal Vp before locking over a 8-second period, is 0.05 rad. We have observed that this CEP-stabilized operation is easily maintained for more than 40 minutes and its noise characteristics can be further improved by minimizing external perturbations.
4. Out-of-loop measurement
Due to extracavity phase noises, the actual CEP stability of the laser oscillator can be, in general, different from the results shown in Fig. 2 above. Thus, in order to verify our results, we have performed an out-of-loop measurement using an independent f -to-2f interferometer with the same optical configuration as the in-loop setup (see Fig. 1) .
Figure 3 shows the results of the out-of-loop measurement. The stabilized beat signals of the two loops are shown in Fig. 3(a). The difference in the locking positions is due to the difference in optical path lengths in the two loops. When the locking system is working, the out-of-loop beat signal becomes constant, as can be seen in the in-loop measurement. However, extracavity perturbations cause low-frequency phase fluctuations.
We have analyzed the CEP jitter in both the loops by comparing their power spectral density (PSD) and accumulated phase . The PSD curve is calculated from the Fourier transform of the stabilized beat signal. The accumulated phase is the RMS of fluctuations (ΔϕRMS) in the CEP and is obtained by integrating the PSD curve (Sϕ(ν)):
where τobs is the observation time. The accumulated phase, calculated from PSD, is logically same as the one mentioned in section 3. Figure 3(b) shows the PSD curves and the accumulated phases in both the loops. The phases accumulated over a 3.5-second period are 0.05 rad for in-loop analysis and 0.1 rad for out-of-loop measurement, respectively. The accumulated phase of the out-of-loop (blue) is ~2 times larger than that of the in-loop (black) since the phase variation is not actively compensated for by the feedback loop. The coherence time , defined as the duration over which the CEP is confined within 1 rad, is over 150 seconds. The above results show that the direct locking method works well for the CEP stabilization of the laser.
5. Electrical CEP modulation
We have also tested the capability of electrical control of the CEP. Figure 4(a) shows a constant change of the CEP value by changing the reference DC level. When a DC signal (red) of 2 V is applied to the stabilized beat signal in the servo loop at time t 1, the locking position shifts from zero to 2 V. From this shifted value and the peak value (5.6 V) of the beat signal before locking, the relative change in the CEP is estimated to be 0.37 rad. In this way, we can control the relative CEP value by controlling the input level. Therefore we can avoid additional beam modulation due to optical components, such as thin wedges, used for the external CEP adjustment. This is one of the unique characteristics of the direct locking method compared to conventional PLL-based method.
For more dramatic example, a saw tooth signal (red) is applied to the stabilized beat signal at time t 2 as shown in Fig. 4(b). The stabilized beat signal exactly tracks the applied waveform. From above two example, we have shown that the electrical CEP modulation allws an intuitive and control of the CEP. This technique will also be useful in the amplification of CEP stabilized pulses in a chirped-pulse amplification (CPA)  laser, since any additional phase error coming from the amplification process can be eliminated by simply adding an appropriate signal to the servo loop.
We proposed and demonstrated a direct locking method for stabilizing the CEP of femtosecond laser pulses. The pulse-to-pulse CEP slip was locked to zero with a phase jitter of 0.05 rad. The out-of-loop measurement using an independent f -to-2f interferometer verified the good operational performance of the proposed locking method. We also demonstrated the capability of an electrical CEP modulation.
The direct locking method implemented in the time domain is complementary to the conventional PLL-based method in the frequency domain, but it has several advantageous features as follows: First, the direct locking method maintains the same CEP for all pulses and thus it can be directly applied to the amplification of CEP-stabilized pulses in a femtosecond CPA laser without the need for selecting pulses that have the same CEP. Second, the instrumental requirement for the locking system is not as tight as that of PLL-based system because frequency counter, phase detector, frequency synthesizer, or frequency divider are not necessary. Third, our method allows an electrical control of the CEP value and thus makes it possible to avoid additional modulation of laser pulses in the optical components, such as thin wedges, used for external CEP adjustment. Fourth, all the data for the CEP stabilization process can be acquired using just an oscilloscope and can be easily analyzed using Fourier transform method. Moreover, the oscilloscope data themselves intuitively visualizes the CEP evolution of the laser pulses because the CEP change is tracked in the time domain.
In conclusion, we believe that the direct locking method proposed and demonstrated in this paper is an excellent and convenient tool for CEP stabilization and control.
The authors gratefully acknowledge Dr. L. Vetrivel for helpful discussions. This work was supported by the Korea Science and Engineering Foundation through the Creative Research Initiative Program.
References and links
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