1. Introduction

Time-lens is based on the space-time analogy to that of a spatial lens and is an alternative technique to generate picosecond and even femtosecond ultrashort pulses [1,2]. Ideally, a time-lens imposes a temporal quadratic phase modulation onto the light. In experiments, the quadratic phase modulation can be well approximated by applying a sinusoidal drive signal to electro-optic phase modulators. After dispersion compensation with a chirped fiber Bragg Grating or free-space grating pairs, typically picosecond pulses can be generated from a continuous-wave (CW) laser [2–5]. The pulse width can be further reduced by techniques such as the time-lens loop [5], soliton self-frequency shift in anomalous-dispersion fibers [6], and adiabatic soliton compression in dispersion decreasing fibers [7].

The repetition rate of the time-lens source is determined by the radio-frequency driving signals. For example, if a 10-GHz RF drive is used for phase modulation, the repetition rate of the time-lens output will be 10 GHz, unless other electro-optical (EO) devices such as intensity modulators are used to further reduce repetition rate [8]. This RF-dependent repetition rate lends the time-lens source the remarkable capability of synchronizing to other pulsed laser sources, such as mode-locked lasers. Specifically, if the RF-drives are from optoelectronic detection of the mode-locked laser output, the time-lens source can be synchronized to the mode-locked laser. Based on this principle, recently, synchronized time-lens source has been experimentally demonstrated [9,10]. Besides synchronization, RF delay-tuning, compatibility with intensity modulation, picosecond pulse width, hundreds of milliwatts output power, and even fiber delivery all together make the synchronized time-lens source a very promising source solution for coherent Raman scattering (CRS) imaging, which typically necessitates two-color picosecond synchronized laser source to probe the vibrational transition in biological or chemical samples [11,12]. For comparison, current excitation laser sources for CRS microscopy also include two synchronized mode-locked lasers, optical parametric oscillators (OPOs) synchronously pumped by mode-locked lasers, and two-color sources based on fiber lasers and continuum generation. Synchronized mode-locked lasers and OPOs are bulky and costly, while previously deomonstrated fiber-based sources such as continuum suffers from low output power. Experimentally, synchronized time-lens source has been successfully applied to various modalities of CRS imaging, including single-frequency coherent anti-Stokes Raman scattering (CARS) and stimulated Raman scattering (SRS) imaging [9,10], and hyperspectral SRS imaging [13].

Conventional (single) time-lens source is driven by RF signals from stable RF sources [8]. As a result, the peak power of the time-lens output can be expected to be stable. In the context of a synchronized time-lens source, however, the RF signals generated due to optoelectronic detection may suffer from fluctuation. The reason is that the intrisic timing jitter of the mode-locked laser leads to a fluctuation of the repetition rate, which translates into fluctuation of the RF drive for the phase modulators, including frequency, amplitude and shape. This fluctuation will lead to a fluctuation of the spectrum of individual pulses in the time-lens output pulse train. For example, a larger modulation frequency or amplitude will result in a broader spectrum. While dispersion compensation is fixed and should be optimized for maximum peak power to maximize CRS imaging signals [14], consequently, this spectral bandwidth fluctuation will be translated into the peak power fluctuation. For CRS microscopy, such peak power fluctuation inevitably leads to fluctuation of CRS signals.

In this paper, we performed theoretical investigation of the peak power fluctuation in a synchronized time-lens source, due to the fluctuation of the RF signals caused by the intrinsic timing jitter of the mode locked laser. Our results indicate that a larger intrinsic timing jitter leads to a larger fluctuation of the peak power of the synchronized time-lens source. We further demonstrate that an effective means to reduce this peak power fluctuation is to reduce the bandwidth of the narrowband RF filter, which is used to filter out the RF driving signal for the PM.

2. Synchronized time-lens setup and simulation details

Our investigation is based on the simplified block diagram of a typical synchronized time-lens source [9] (Fig. 1). Similar to the real experimental setup, in our simulation, a fast photodetector converts the 80-MHz mode-locked laser output into an RF pulse train of the same repetition rate. After a RF divider, one branch (80 MHz) drives the Mach-Zehnder intensity modulator (MZ) to carve a synchronized 80-MHz optical pulse train from the CW laser. The other branch is filtered by a narrowband RF filter centered at 10 GHz (125th harmonic) to generate a sinusoid to drive the PM. After dispersion compensation with a dispersion compensator (DC), chirping due to phase modulation can be compensated for and the pulse width can be compressed down to picosecond with highest peak power, suitable for CRS imaging with maximum signal generation [14].

 

Fig. 1 Block diagram of the synchronized time-lens source for CRS imaging. PM: phase modulator, MZ: Mach-Zehnder intensity modulator, DC: dispersion compensator, ML laser: mode-locked laser. The optical and electrical signals are in red and blue, respectively.

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To account for intrinsic timing jitter due to the mode-locked laser, the modulated electric field of the time-lens source is given by [15]

To simulate intrinsic timing jitter of the mode-locked laser, in accordance with our previous procedure [15], a Gaussian white noise generator was used to introduce intrinsic timing jitter (σ_ML) of different root-mean-square (RMS) values to the optical pulse train from the mode-locked laser. Specifically, without any timing jitter σ_ML, the peak positions of the mode-locked laser output pulse train were precisely separated by 12.5 ns for a repetition rate of 80 MHz. With the introduction of σ_ML, the peak positions of the pulse train were deviated from the original 12.5-ns separation randomly, introduced by the Gaussian white noise generator. The RMS deviation of all the pulses in this pulse train was given by the statistical quantity σ_ML. Pulse trains of 64 optical or electrical pulses were simulated.

3. Simulation results

3.1. Peak power fluctuation due to the intrinsic timing jitter of the mode-locked laser: origin

First we elucidate the problem of peak power fluctuation due to variation of the PM driving signals. Due to the intrinsic timing jitter of the mode-locked laser, the actual repetition rate will fluctuate around 80 MHz (Fig. 2(a)). This repetition rate directly translates into the fluctuation of the modulation frequency of the sinusoidal PM driving signal. For example, quantitatively, if the repetition rate of the mode-locked laser is 80 ± 0.0064 MHz, corresponding to an intrinsic timing jitter of ± 1 ps, the resultant frequency of the PM driving signal will be 10 ± 0.0008 GHz (Fig. 2(b)). Besides, due to the finite bandwidth of the narrowband RF filter, the PM driving signal is not strictly a sinusoid with fixed peak voltage for each synchronized time-lens output optical pulse, but rather with varying amplitude (Fig. 2(c)) and shape. As was demonstrated theoretically, dispersion compensation is fixed for maximum peak power of the compressed pulse. Consequently, it can be expected that the resultant pulse will suffer from peak power fluctuation due to these combined effects.

 

Fig. 2 The intrinsic timing jitter of the mode-locked laser (a) results in the fluctuation of the modulation frequency (b) or the amplitude (c) of the sinusoidal PM driving signal. The figures are for illustrative purpose and not to scale.

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3.2. Quantitative results for 50-MHz narrowband RF filtering

Next we perform quantitative investigation of this peak power fluctuation for various intrinsic timing jitter (σ_ML) of the mode-locked laser. In accordance with experiment, the bandwidth of the narrowband RF filter is fixed at 50 MHz. Figure 3(a) shows that due to the existence of σ_ML, peak power P_p of the synchronized time-lens source is no longer a constant, but varying on a pulse-to-pulse basis. A comparison of P_ps for smaller (0.2 ps) and larger (0.5 ps) σ_ML (upper and lower panel in Fig. 3(a)) clearly indicates that a larger σ_ML leads to a larger fluctuation of P_p from a statistical point of view (we note that the ordinates are different in Fig. 3(a)). Two representative intensity profiles from the compressed time-lens output pulse train are shown in Fig. 3(b), corresponding to the 7th and 46th pulse indicated by arrows in Fig. 3(a), respectively. In this example, P_p of the 7th pulse is 96.8% of that of the 46th pulse, indicative of pulse-to-pulse peak power fluctuation.

 

Fig. 3 Peak power fluctuation of the time-lens source of two typical simulation runs for σ_ML = 0.2 ps (upper panel) and σ_ML = 0.5 ps (lower panel). (b) The compressed temporal intensity profiles corresponding to the 7th and 46th pulse indicated by arrows.

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To statistically quantify this peak power fluctuation, we calculated relative peak power fluctuation ΔP_p/P_p for a pulse train with 64 pulses, where P_p and ΔP_p are the average and standard deviation of the peak powers of these 64 pulses from the synchronized time-lens source. This calculation was repeated for ten pulse trains each with 64 pulses and the same σ_ML. Finally, the mean value ΔP_p/P_p(10 ave) (red circles) and the standard deviation of these ten calculations (error bars) are presented in Fig. 4. Clearly, the larger the intrinsic timing jitter of the mode-locked laser is, the more the peak power of the time-lens output fluctuates. For example, ΔP_p/P_p(10 ave) for σ_ML = 0.5 ps is 5.66 times larger than that for σ_ML = 0.2 ps. At even larger σ_ML of 2 ps (not shown in the figure), peak power fluctuation ΔP_p/P_p(10 ave) can be as high as 2.4% (mean value). Besides, a larger timing jitter also incurs a larger standard deviation of ΔP_p/P_p(10 ave). For example, the standard deviation of ΔP_p/P_p(10 ave) for σ_ML = 0.5 ps is 5.39 times larger than that for σ_ML = 0.2 ps. This physically reflects the fluctuation of peak power on a pulse train to pulse train basis (each comprising 64 pulses).

 

Fig. 4 Peak power fluctuation vs different intrinsic timing jitter of the mode-locked laser for ten successive runs. Red circles: mean values of these ten successive runs; error bars: standard deviations.

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3.3. Dependence of peak power fluctuation on the bandwidth of the narrowband RF filter

As mentioned above, in the synchronized time-lens source, a narrowband RF filter is used to filter out the 10-GHz quasi-sinusoid to drive the PM. It can be speculated that the more this filtered RF drive resembles the sinusoid, the less the peak power fluctuation will be incurred. A “purer” sinusoid can be generated using a RF filter with a narrower pass band, since in the limit of infinitesimal bandwidth, the RF signal will be a genuine sinusoid. We tested this hypothesis for peak power fluctuation reduction in the simulation. Figure 5(a) shows calculated peak powers of the time-lens output with 64 pulses for a filter bandwidth of 10-MHz and 50-MHz, respectively. It can be seen that due to a “narrower” filtering to generate a “purer” sinusoidal RF drive for the PM, the compressed pulse suffers from less peak power fluctuation. With statistics on simulation runs of ten pulse trains, Fig. 5(b) clearly indicates that a narrower RF filter helps to reduce the peak power fluctuation, both mean value and standard deviation. Our quantitative results indicate that, the mean ΔP_p/P_p(10 ave) for 30-MHz and 10-MHz RF filters are only 63.9% and 18.9% of that induced by the 50-MHz RF filter, respectively, for σ_ML = 0.5 ps. While for the same intrinsitc timing jitter, the standard deviations of ΔP_p/P_p(10 ave) for 30-MHz and 10-MHz RF filters are 94.5% and 30.4% of that induced by the 50-MHz RF filter, respectively. This suggests narrowing the bandwidth of the RF filter is an effective means to suppress peak power fluctuation.

 

Fig. 5 (a) Pulse peak power of the time-lens source for various pulse numbers with a 50-MHz bandpass filter (red circles) and a 10-MHz bandpass filter (black squares). σ_ML = 0.5 ps. (b) Peak power fluctuation vs different intrinsic timing jitter of the mode locked laser for different filter bandwidths. Error bars: standard deviation.

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4. Discussion and conclusion

In this paper we performed theoretical investigation of peak power fluctuation in a synchronized time-lens source due to the intrinsic timing jitter of the mode-locked laser, to qualitatively and quantitatively illustrate the physical underlying of this phenomenon and to which extent it could affect the performance of the synchronized time-lens source. The intrinsic timing jitter of the mode-locked laser leads to a fluctuation of the RF driving signal for the PM, including modulation frequency, amplitude and shape of modulation. With fixed dispersion compensation in the form of quadratic spectral phase in frequency, this fluctuation in RF driving signal inevitably leads to a fluctuation of the peak power of the compressed time-lens output.

To suppress this peak power fluctuation, we proposed to generate a “purer” sinusoid to drive the phase modulators. Experimentally, this could be achieved using a RF filter with a narrower pass band. Since in the limit of an infinitely narrow bandpass RF filter, the RF driving signal will be a pure sinusoid without any fluctuation in frequency, amplitude or shape. Our simulation results supported this idea by showing that the narrower the bandwidth of the RF filter is, the smaller the peak fluctuation is. For example, a 10-MHz RF filter reduces the peak power fluctuation to only ~1/5 of that using a 50-MHz filter.

However, we note that the technique of suppressing peak power fluctuation by using a narrow pass band RF filter comes at its own price. As we reported recently [15], a narrower bandpass filter results in a poorer following of the peak position of the synchronized time-lens source to that of the mode-locked laser in the presence of jitter. This leads to a larger relative timing jitter between the time-lens source and the mode-locked laser. So in real applications, there is a comprise between peak power fluctuation and relative timing jitter for the proper choice of the bandwidth of the narrowband RF filter.