We demonstrate a novel time domain timing jitter characterization method for ultra-low noise mode-locked lasers. An asynchronous optical sampling (ASOPS) technique is employed, allowing timing jitter statistics on a magnified timescale. As a result, sub femtosecond period jitter of an optical pulse train can be readily accessible to slow detectors and electronics (~100 MHz). The concept is applied to determine the quantum-limited timing jitter for a passively mode-locked Er-fiber laser. Period jitter histogram is acquired following an eye diagram analysis routinely used in electronics. The identified diffusion constant for pulse timing agrees well with analytical solution of perturbed master equation.
© 2017 Optical Society of America
Oscillator instabilities are a fundamental concern for any system that receives or transmits a signal . As the building block for modern information and communication technology, electronic oscillators typically generate clock signals with picosecond timing jitter. This jitter level can be further reduced to a few femtoseconds by using cryogenic sapphire oscillator  or optoelectronic oscillators [1,2]. Recently, passively mode-locked lasers have drawn increasing attention as a candidate for ultralow jitter optical oscillator. It has been known that passively mode-locked lasers emit uniformly spaced femtosecond laser pulse trains with extremely good short term stability [3–6]. The quantum limited timing jitter of a low cost femtosecond Er-fiber laser working at room temperature easily reaches hundreds of attoseconds at high Fourier frequency [3,7]. By phase locking to an optical/rf reference, the mode-locked laser thus serves as an optical master oscillator . Recently, ultralow jitter mode-locked lasers have shown great potential for a number of high precision applications, such as high speed analog digital conversion (ADC) , large-scale timing synchronization , time-of-flight laser ranging [10,11], coherent optical pulse synthesis , and high spectral purity rf signal generation , to name a few. Precise timing jitter characterization of mode-locked lasers is fundamental for the advances in all the fields.
Timing jitter of mode-locked lasers can be simply characterized by direct high-speed photodetection of laser pulse train and a subsequent repetition rate phase noise analysis by using standard rf techniques [14–17]. However, excess phase noise in the photodetection process limits timing resolution to a few femtoseconds. Although it is sufficient to characterize technical jitters at low Fourier frequencies, this method is not capable of measuring sub-femtosecond quantum limited timing jitter of passively mode-locked lasers. The resolution limit of rf techniques has been overcome by optical cross-correlation methods [18–20], which make use of inherent steep rising edge of femtosecond laser pulses as highly sensitive timing error discriminator. Particularly, a balanced optical cross-correlation (BOC) [19,20] setup is most commonly used, which secures amplitude noise insensitive timing information extraction. A record-low timing jitter of 13 as (from DC to Nyquist frequency) from Kerr-lens mode-locked Ti:sapphire lasers generating 10 fs pulses has been demonstrated by using BOC method . Besides well-established BOC method, novel optical heterodyne methods based on comb-line interference [21,22] provides highly sensitive timing discrimination signal as well, enabling attosecond precision timing jitter characterization. Details on the rapidly evolving timing jitter measurement techniques towards attosecond precision can be found in a recent review article .
The existing methods mostly measure timing jitter power spectral density (PSD) for mode-locked lasers. As a complementary approach to spectral analysis, eye diagram analysis has been extensively used for electronic systems as well, where signal timing behavior is visualized in an oscilloscope or digitizer, providing a straightforward online diagnosis in time domain. In this letter, we demonstrate a novel time domain timing jitter characterization method, which is an optical equivalent to eye diagram analysis used at rf frequencies. It essentially takes advantage of an asynchronous optical sampling (ASOPS) approach [24,25], which is capable of linearly stretching ultrafast processes in time such that sub-femtosecond period jitter of optical pulse stream is visible to fast data acquisition electronics. Here, ASOPS is implemented by optical cross-correlation between a pair of passively mode-locked Er-fiber lasers with a fixed or flexible offset repetition rate. The period jitter of one mode-locked Er-fiber laser serving as laser under test (LUT) is resolved with 350 as resolution. This number can be reduced to tens of attoseconds by decreasing the offset repetition rate, which effectively magnifies the time-stretch ratio.
2. Measurement principle
The concept is based on direct sub-femtosecond precision pulse timing determination by ASOPS, as depicted in Fig. 1. Two mode-locked lasers with repetition periods (repetition rates) of () and () serve as LUT and local oscillator (LO), respectively. The LUT sends an ultrashort pulse stream to a free space pulse interleaver which divides a single laser pulse into a pulse pair. Here, the timing for the front pulse peak is set as origin, while the timing for the rear pulse peak is . The LUT pulse pairs are sampled against LO for high precision rear pulse timing acquisition. Every repetition period (), the LO pulse shifts by relative to the LUT pulses, and samples a slightly different portion of the LUT pulse pair. An entire scan of the LUT pulse pair is updated every , corresponding to an update rate of . The so called ASOPS process effectively stretches real femtosecond timescale by a factor of . As a result, the timing for the rear pulse peak can be determined by , where is the acquired rear pulse timing in the magnified timescale. Ideally, the consecutively acquired would be invariant, determined by the differential optical path length between the two arms of the pulse interleaver. Nevertheless, ASOPS process will inevitably introduce uncertainty in in the presence of timing jitter. Suppose the LO is noise free, while the LUT is quantum limited, meaning that the LUT emits an optical pulse train undergoing random walk. The optical period jitter , defined by the deviation from the ideal time interval between the and the pulse, exhibits normal distribution with variance of . Given that a LO pulse exactly overlaps the peak of LUT front pulse, the sample of rear pulse peak is expected after M repetition periods with . During this time interval, the random walk of LUT pulse train will bring a summed timing error with variance . As a result, the consecutively acquired in the magnified timescale will fluctuate following a normal distribution with variance
3. Numerical simulation
We test the concept by a numerical model used in Ref . A noise free Gaussian shaped LO pulse train with pulse duration of 100 fs and repetition rate of 100 MHz is generated at first. With the same pulse shape and a pulse duration of 1 ps, a LUT pulse train having a lower repetition rate of 99.999 MHz is constructed in the following. A Gaussian white period jitter with variance of fs2 is added to the pulse of LUT, thereby inducing a lumped temporal shift of with respect to its ideal timing position. By this means, the LUT pulse train takes on a random walk character. A copy of this pulse train is delayed by and is combined with the original pulse train, representing the interleaved LUT. Finally, a sum frequency generation (SFG) based intensity cross-correlation between LO and interleaved LUT and a subsequent square law photo-detection [11,26] produce repetitive pulse pairs in a stretched timescale.
The waveforms of a series of 4000 stretched pulse pairs are superimposed such that the front pulses are aligned, as displayed in Fig. 2. This is analogous to creating an eye diagram with an oscilloscope or digitizer in a long-persistence display mode. The density of waveforms reflects a timing error histogram, as shown in the inset of Fig. 2. A visual timing jitter variance of ns2 can be directly derived and the retrieved period jitter variance is 0.982 fs2 considering and N = 105. This number matches well with variance of white period jitter added on LUT in advance.
4. Experiment with stabilized ultrafast lasers
4.1 Experimental setup
This method has been applied to characterize period jitter of a passively mode-locked fiber laser. The experimental setup is shown in Fig. 3. Two NPR mode-locked Er-fiber lasers working at 1550 nm serve as LUT and LO, respectively. Both LUT and LO operate with a close-to-zero cavity dispersion, delivering an output pulse duration of 905 fs and 440 fs, respectively. Longer pulse duration from LUT is attributed to an 8 nm bandpass filter just before the output. The limited output laser bandwidth is necessary to prevent aliasing in the following ASOPS stage . Note that, the intracavity filtering also changes the pulse dynamics of LUT from stretched pulse regime to dissipative soliton formation, bearing an increased timing jitter . Under this situation, LO timing jitter is negligible in comparison with LUT. The repetition rates of the two lasers are ~100 MHz. The LUT repetition rate and offset repetition rate are referenced to external rf standards by feedback control on piezo mirror actuators in LUT and LO, respectively. The locking bandwidth is ~300 Hz. Therefore, magnification factor of M and N would keep constant and thus be precisely determined. A Michelson-based pulse interleaver is used to produce reference pulse pairs, which are combined with LO on a PBS for type-II periodically poled KTiOPO4 (PPKTP) crystal based SFG. The generated intensity cross-correlation signals are received by an avalanche photodetector (Thorlabs, APD120A), low-pass filtered, and then continuously sampled by a 14 bit 100 MHz digitizer (National Instrument, PXIe-5122), thus accomplishing ASOPS. Inset in Fig. 3 shows a full sample of reference pulse pair by using ASOPS. The individual pulses are fitted with Gaussian functions and the retrieved peak-to-peak time interval accounted for the measured .
4.2 Experimental results
Figure 4 summarizes the experimental results at a fixed offset repetition rate of 2 kHz, where is scanned from 7 ps to 93 ps by adjusting the displacement of the movable mirror in the Michelson pulse interleaver from 1 mm to 14 mm. Accordingly, reads from 0.35 μs to 4.65 μs considering a time-stretch factor of N = 50000 by ASOPS. Note that, in the microsecond acquisition time, the impact of technical noise can be ignored, and the mode-locked lasers are regarded as quantum noise limited. The obtained standard deviation (STDEV) of visual timing jitter versus is plotted in Fig. 4(a). Each data point is acquired from histogram analysis based on 5500 successive measurements. The inset gives one histogram at an arbitrary ~4.68 mm movable mirror displacement, agreeing well with a normal distribution. The constant visual timing jitter STDEV of = 0.35 ns for <25 ps (corresponds to a 3.75 mm movable mirror displacement) reflects the sensitivity of electronic data acquisition system, mainly determined by the limited signal noise ratio (SNR) of the photo-detected cross-correlation signals as well as the uncertainty of Gaussian-fit based pulse timing extraction. Visual timing jitter STDEV increases versus for >25 ps, see the red-dashed guide curve in Fig. 4(a), in accordance with Eq. (1). Obviously, increased temporal delay is favorable for an enhanced timing jitter measurement sensitivity.
Real period jitter STDEV has been obtained when the measured visual timing jitter STDEV is divided by . A period jitter STDEV of 0.626 fs with uncertainty of 16 as has been retrieved based on the measurements for > 25 ps, as shown in Fig. 4(b). When period jitter STDEV is measured, the diffusion constant B for pulse timing can be directly obtained by and we can predict a timing jitter PSD, shown as red line in the insert of Fig. 4(b). The grey area in the insert of Fig. 4(b) shows the estimated jitter PSD according to the analytical model in Ref . The laser parameters are from LUT used in experiment, where the reference pulse width τ0 is 480 fs, the saturated gain g is 0.46 calculated from a roundtrip pulse energy attenuation of ~60%, the gain bandwidth Ωg is 1.57 × 1013 rad/s, the intracavity pulse energy w is 600 pJ, the saturable absorption coefficient α is 0.2, the net cavity dispersion coefficient D is + 5500 fs2, and the excess spontaneous emission factor Θ is supposed to be between 2 and 10. The notations are from Ref . The experiment matches well with analytical model, confirming the validity of this timing jitter measurement method.
The same experiment is conducted by changing the offset repetition rate to 1.5 kHz. The measurement results are compared with that conducted under 2 kHz offset repetition rate, as shown in Fig. 5(a). When is fixed, the obtained visual timing jitter STDEV at 1.5 kHz offset repetition rate is apparently larger than that at 2 kHz offset repetition rate, in accordance with Eq. (1). This means that, lower offset repetition rate is also favorable for enhanced jitter measurement sensitivity. The retrieved optical period jitter is identical at different offset repetition rates. This is easy to understand because slight repetition rate change will not affect the laser noise character.
Taking into account the impact of both and , the ultimate resolvable optical period jitter STDEV in the presence of electronic limited measurement sensitivity () can be expressed asFigure 5(b) maps the resolvable optical period jitter STDEV by altering and at the present = 0.35 ns. Under a 2 kHz offset repetition rate, the lowest measurable period jitter at = 80 ps reads 0.35 fs. This number can be reduced to 43.75 as by lowering the offset repetition rate to 500 Hz while keeping invariant.
5. Experiment with free running ultrafast lasers
The above measurements require actively repetition-rate-locked ultrafast lasers for the sake of a fixed time-stretch ratio. In fact, the same measurement can be conducted with free running ultrafast lasers as long as and can be updated in real time. This will significantly simplify the measurement setup and results in a practical sub-femtosecond jitter analysis device since the complex phase-locked loop is not necessary.
The experimental setup is based on Fig. 3, while the phase-locked loops are removed. By tuning the offset repetition rate to roughly 2 kHz, stream of ASOPS waveforms can be observed by an oscilloscope and recorded by a digitizer. The offset repetition rate is readily determined by recording update period (), which is no more than the time interval of neighboring repetitive ASOPS waveforms. A frequency counter (Agilent, 53220A) is used to measure the LUT repetition rate . The counter’s gate time is set to one second (1 Hz update rate) for the sake of a 14-digit resolution. Note that, this update rate is much slower than that of . In order to evaluate the impact of slow update of frequency counter on jitter measurement precision, we characterize the repetition rate stability of the LUT. The measured repetition rate Allan deviation of the free running LUT at 1 second observation time is merely 285 mHz. According to Eq. (1), this uncertainty will impose negligible impact on the precision of the acquired optical period jitter. A jitter measurement experiment is conducted when the digitizer and frequency counter are referenced to the same frequency standard. The measured results based on ~3700 frames of ASOPS waveforms are depicted in Fig. 6. The acquired STDEV of optical period jitter is 0.579 fs with 20 as uncertainty. The slight difference from the result with repetition rate phase-locking is due to a slight variation in mode-locking condition between the two experiments.
Note that, digitizers require large data storage depth for jitter measurement based on free running lasers because the measurement of relies on uninterrupted collection of samples. While, by phase-locking both and , we only need to record the sample data within Tp during each ASOPS period (). Considering the limited data storage depth of digitizer, more sample frames can be acquired for jitter statistics by using stabilized lasers. The slight fluctuations of the retrieved optical period jitter versus tp in Fig. 6 is merely due to shortage of sample frames. It can be more severe when offset repetition rate is further decreased, thus limits the obtainable jitter measurement resolution.
In conclusion, we directly characterize the quantum limited timing jitter of femtosecond lasers in time domain with ASOPS method, enabling real-time visualization of attosecond period jitter of mode-locked lasers based on histogram analysis. In comparison with the routinely used BOC technique, the measurement is less dependent on LUT pulse duration since the measurement resolution is mainly determined by time-stretch ratio. Attosecond sensitivity can be preserved even with picosecond laser pulses, which effectively enlarge the measurement dynamics range as well. Particularly, jitter measurement can be conducted without complex phase-locking loops, which is indispensable for most of existing attosecond resolution jitter measurement techniques. As a result, this simple setup is expected to become a routine approach to attosecond precision jitter measurement for ultrafast optical pulse trains, analogous to eye diagram analysis for digital signals.
National Natural Science Foundation of China (NSFC) (11274239, 11527808, 61535009); Program for Changjiang Scholars and Innovative Research Team in University (IRT13033); National High Technology Research and Development Program of China (2013AA122602).
We acknowledge fruitful discussions with Prof. Jungwon Kim from Korea Advanced Institute of Science and Technology (KAIST), South Korea.
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