1. Introduction

Growing interest in ultrafast electronic processes of atoms, molecules, or condensed mattter has been the driving force behind efforts to shorten the duration of ultrashort-pulse lasers. Recent demonstrations of attosecond pulse generation by high-order harmonics have opened a new frontier to the so-called attosecond science [1, 2]. Fourier synthesis of optically phase-locked multicolor pulses [3–6] is another attractive method for attosecond pulse generation because of its scalability of repetition rate or pulse energy. It would be possible to synthesize arbitrary electric-field waveform including attosecond pulse shape, by superimposing several phase-locked ultrashort pulses with separate spectral components that span far beyond an octave. In recent years, optical phase locking among multicolor pulses has been realized by a femtosecond optical parametric oscillator [7, 8] and two-color synchronized mode-locked lasers [9–12]. Obviously, stable phase locking and reproducible waveform synthesis require tight synchronization of pulses with a very low timing jitter. Until now, several groups have reported synchronization of two mode-locked lasers with active [13–17] and passive [18–21] schemes. In an active scheme, the laser cavity is actively controlled with electronic feedback circuits to minimize the relative timing jitter. Schibli et al. demonstrated active synchronization of Ti:sapphire and Cr:forsterite mode-locked lasers with a timing jitter as low as 300 attoseconds (as) [17].

In contrast, passive synchronization is all-optical, in that cross-phase modulation is utilized for synchronization. The two laser cavities are designed to cross on one of the laser crystals. The cross-phase modulation causes the wavelength shifts, and the change in round-trip group delays in combination with intracavity group-delay dispersion. The two pulses are self-synchronized with the negative dispersion in typical mode-locking conditions [22, 23]. Since the feedback bandwidth is not limited by that of electronic circuits, a low timing jitter can be expected. Nevertheless there are still two remaining causes of jitter on the observation point. First, the path lengths to the observation point fluctuate with environmental disturbances. Second, the spectral shifts induced by the mechanism of passive synchronization in combination with extracavity dispersion causes jitter. In a previous work, we achieved a timing jitter of 126 as by applying active control to passively synchronized Ti:sapphire and Cr:forsterite mode-locked lasers [24]. Recently, we also demonstrated long-term phase locking between the two-color lasers with a phase noise of 0.43 rad [25]. Since the relative optical-phase slip between two lasers is very sensitive to changes in the cavity-length difference, optical-phase stabilization plays a role in stabilizing cavity-length fluctuation. Therefore one can expect that the optical phase stabilization helps suppress the abovementioned spectral shifts and subsequently reduces the timing jitter. However, the influence of optical phase locking on the timing jitter has never been addressed to our knowledge.

 

Fig. 1. Experimental setup. EOM, electro-optic modulator; Pol., polarizer; PZT, piezoelectric transducer; PMT’s, photomultiplier tube; CM’s, chirped mirrors; WP’s, half-wave plates; SHG’s, second harmonic generators; SFG’s, sum-frequency generators; DP, delay plate; APD, avalanche photodiode.

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In this paper, we demonstrate a timing-jitter reduction of the passively synchronized two-color mode-locked lasers by locking the relative optical phase slip to an rf reference. Because of reduction by a factor of 1.7, an rms timing jitter of 123 as was achieved in a frequency range from 10 mHz to 1 MHz.

2. Optical-phase stabilization

The passively synchronized Ti:sapphire and Cr:forsterite laser system is described in Ref. 24. Both lasers have a common repetition frequency of 100 MHz. The frequencies of Ti:sapphire (f _TiS) and Cr:forsterite (f _CrF) mode-locked pulse trains are expressed as f _TiS=δ _TiS+mf _rep and f _CrF=δ _CrF+m ^′ f _rep, where f _rep is the repetition frequency, δ_TiS and δ_CrF are the carrier-envelope offset frequencies of Ti:sapphire and Cr:forsterite lasers, and m and m ^′ are integers. Since the second harmonic of Ti:sapphire and the third harmonic of Cr:forsterite are in the same spectral range (~410nm), the two harmonic-frequency combs generate heterodyne beats. The beat frequencies are given by f _beat=|2 f _TiS-3f _CrF|=|δ+n f _rep|, where δ=2δ _TiS-3δ_CrF and n is an integer. The cavity-length variation leads to a change in pulse-to-pulse slip of relative carrier-envelope phase and results in a change in beat frequency. Since the changes in repetition frequencies are negligible for a cavity-length change of several microns, the changes in carrierenvelope phase slips of Ti:sapphire (Δϕ_TiS) and Cr:forsterite (Δϕ_CrF) laser pulses are given by Δϕ_TiS=2π·2Δl _TiS/λ_TiS and Δϕ_CrF=2π·2Δl _CrF/λ _CrF, respectively, where Δl _TiS and Δl _CrF are the cavity-length changes of Ti:sapphire and Cr:forsterite lasers, and λ _TiS and λ_CrF are their wavelengths. Thus the changes in phase slips cause changes in carrier-envelope offset frequencies of Ti:sapphire (Δδ _TiS) and Cr:forsterite (Δδ _CrF) lasers given by Δδ _TiS=2Δl _TiS f _rep/λ _TiS and Δδ _CrF=2Δl _CrF f _rep/λ _CrF. Hence, the beat-frequency shift (Δf _beat) caused by a change in the cavity-length difference (Δl) is given by [26],

where we used some relations written as Δl=Δl _TiS-Δl _CrF and λ _TiS : λ _CrF=2 : 3 for derivation of Eq. (1). This equation suggests that the cavity-length fluctuation can be suppressed by the phase locking in a sensitive manner.

Figure 1 depicts the experimental setup. Ti:sapphire and Cr:forsterite laser pulses were split with half mirrors: some beams were then used for phase locking and the others for jitter measurement. Three BBO crystals were used to obtain the second harmonic of the Ti:sapphire laser and the third harmonic of the Cr:forsterite laser by the sum-frequency mixing of the fundamental and its second harmonic. Half-wave plates were positioned to get two harmonics with the same polarizations. The beat frequency was locked to a 40-MHz rf reference signal by active control of the cavity length of the Ti:sapphire laser with a piezoelectric transducer, and by control of the pump power of Ti:sapphire laser with an electro-optic modulator [25].

3. High-resolution jitter measurement

A cross-correlation trace is utilized in typical jitter measurements, in which the timing jitter can be estimated from the fluctuation of the cross-correlation signal at the slope of half maximum. When the cross-correlation trace is written as I _cc=I ₀ g _cc(τ), where I ₀ is the peak intensity, g _cc(τ) is the normalized cross-correlation function, and τ is the relative delay between two pulses, the fluctuation of the correlation signal is written as

The first term represents the fluctuation caused by the jitter (Δτ), while the second term represents the amplitude noise (ΔI ₀). If the delay is set to g _cc(τ)=1/2 and typical parameters in our experiment are assumed (full width of correlation ~70 fs, ampliutude noise ~0.1%), the second term is comparable to the first term in a sub-100-as regime, which limits the resolution of this method.

To overcome this problem, the balanced cross-correlator proposed and demonstrated by Schibli et al [17] was used. The balanced cross-correlator consists of two cross-correlators with different relative delays between two pulses, as illustrated in Fig. 1, and opposite slopes of correlation traces are used to monitor the fluctuation. A 3-mm-thick fused-silica delay plate was inserted on one branch to shift its delay to the opposite slope. The balanced cross-correlation is defined as the difference of two correlations, namely,

where τ₀ is the relative delay given by the delay plate. The fluctuation of the signal is

 

Fig. 2. Measured timing jitter at bandwidths of (a) 1 kHz, (b) 400 kHz, and (c) 1 MHz. The top and bottom figures are the results with and without phase locking, respectively.

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Similarly, the first term represents contribution of the jitter (Δτ), and the second term represents the amplitude noise (ΔI ₀). In this case, the amplitude noise can be excluded by choosing a delay to satisfy g _cc(τ+τ₀/2)=g _cc(τ-τ₀/2), which is τ=0 for symmetric correlation functions. In other words, since the amplitude noise affects the two correlations in the same way when the condition is satisfied, the effects are cancelled by taking the difference of two correlations. The two correlation signals were detected simultaneously with two photomultiplier tubes and measured with a multichannel digital oscilloscope. The balanced signal was obtained by calculated difference of two traces.

4. Results and discussion

Figure 2 presents the timing-jitter results measured with observation bandwidths of 1 kHz, 400 kHz, and 1 MHz. As seen in Fig. 2(a), there is clear evidence of suppression of the drift and fluctuation by the phase locking at 1-kHz bandwidth. As a result, an rms jitter of 194±30 as was reduced to 87±8 as. Likewise, an rms jitter decreased from 108±9 as to 96±4 as at 400-kHz bandwidth [Fig. 2(b)] and from 101±8 as to 88±2 as at 1-MHz bandwidth [Fig. 2(c)], although there was no significant evidence of jitter suppression at the two fastest bandwidths.

Figure 3 depicts the beat-frequency variation measured at the same time as the 1-kHz jitter measurement shown in Fig. 2(a). There was a drift of about 5 MHz in beat frequency, which is similar to the drift in relative delay displayed in the result without phase locking in Fig. 2(a). The beat-frequency drift can be attributed to the drift in cavity length, because the commercial pump sources were power-stabilized and the active control by the electro-optic modulator was not done in the case without phase locking. To confirm this, the cavity length of the Ti:sapphire laser was modulated by swinging the end mirror with the piezoelectric transducer at a frequency of 500 Hz. Figure 4 illustrates the response of (a) beat frequency and (b) relative delay to the modulation. The slow fluctuation seen in the relative delay is a 50-Hz jitter, which originates from the ac power-supply noise. In Fig. 4(a), the modulation amplitude of beat frequency was 5 MHz, which corresponds to a cavity-length change of 10 nm according to Eq. (1). In Fig. 4(b), the modulation amplitude of relative delay was 600 as. The intracavity-path change corresponds to an optical delay of only 30 as, which is not a primary cause of the delay change, because it is considered to be compensated by the passive-synchronization mechanism on a much shorter time scale.

 

Fig. 3. Beat frequency variation at a bandwidth of 1 kHz (a) with and (b) without phase locking.

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Fig. 4. (a) Beat frequency and (b) relative delay response to the cavity-length modulation.

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The observed timing shift originating from the cavity-length change has two possible mechanisms. First, the spectral shift inherent in the passive synchronization mechanism possibly causes a timing shift in combination with the group-delay dispersion on the extracavity optical paths. A shift of 0.3 nm in the Cr:forsterite spectrum was observed, whereas no shift was seen in the Ti:sapphire spectrum. The estimated group-delay change resulting from this spectral shift is 200 as with the estimated extracavity dispersion of 500 fs ². Second, the relative timing difference of two pulses inside the Ti:sapphire crystal varies depending on the cavity-length mismatch, because it is passively tuned to an optimal value to induce the spectral shift which makes the round-trip group delays of two pulses identical. As well as the cavity-length fluctuation, the fluctuation of the pump power and/or the beam pointing would cause the timing jitter, which was not suppressed in the scheme presented in this paper.

Figure 5 shows the power spectral density profiles calculated directly from the time-domain data in Fig. 2 and the integrated rms values. The timing jitter is effectively suppressed in the low-frequency region, especially lower than 1 Hz, whereas no remarkable suppression is seen in the high-frequency region. The results suggest that the cavity-length control with the piezoelectric transducer contributes to the jitter reduction. This fact agrees well with the abovementioned speculation. Finally, the rms jitter integrated over a frequency range from 10 mHz to 1 MHz was 213±31 as without phase locking, which was reduced to 123±8 as by phase locking. The reduction factor was approximately 1.7. The obtained rms jitter is comparable to the previous result with active jitter control in Ref. 24. The scheme presented in this paper requires a simpler setup than the previous one because it allows simultaneous locking of timing and phase with one active control.

 

Fig. 5. Power spectral density and integrated rms jitter (a) with and (b) without phase locking. Each color in power spectral density corrsponds to respective time-domain data with different bandwidths.

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5. Conclusion

Reduction of the timing jitter between passively synchronized Ti:sapphire and Cr:forsterite mode-locked lasers was achieved by the optical-phase stabilization technique. The cavity-length fluctuation was suppressed by locking the beat frequency to an rf reference. As a result of timing-jitter reduction by a factor of 1.7, an rms jitter of 123 as was achieved in a frequency range from 10 mHz to 1 MHz. It would be possible to generate two-color pulses with a fixed carrier-envelope phase relation for every pulse by locking the beat frequency to dc. Fourier synthesis is now feasible with the phase-locked two-color pulses synchronized with 100-as precision.