1. Introduction

In the past few decades, mode-locked lasers have proven themselves as one of the most reliable tools for applications requiring highest temporal resolution in frequency metrology [1] and laser spectroscopy [2]. With the invention of self-referenced mode-locked lasers [3,4], the era of optical clocks [5,6] and frequency combs started, which has pushed optical frequency measurement instabilities down to the 19th decimal place [7]. Nowadays, researchers are able to compare independent optical clocks located in different countries with a statistical uncertainty of 1 × 10⁻¹⁵/(τ/s)^1/2 (τ: integration time) [8], which opens doors to new horizons in fundamental science such as chronometric geodesy [9] and quantum networks [10]. Analogously in the time domain, a well-defined optical frequency comb presents itself as a pristine temporal periodicity of the pulse envelope, which can be used as an optical flywheel providing a very precise timing reference [11,12]. One of the most exciting timing applications of mode-locked lasers is the synchronization of X-ray free-electron lasers [13–15] and laser-based attoscience facilities [16] to generate ultrashort X-ray pulses [17] for the benefit of creating super microscopes with subatomic spatiotemporal resolution [18].

Main architectures of time and optical frequency transfer applications are quite similar: the signal from the master oscillator is transmitted to distant locations with a stabilized optical fiber or free-space link network, whose output is used for the synchronization of remote mode-locked lasers. So far, there have been numerous publications on time and optical frequency transfer over fibers [19–23] demonstrating exceptionally low fractional instabilities down to 10⁻¹⁹ for frequency transfer [24] and 10⁻²¹ for timing transfer (i.e., daily sub-fs jitter) [25,26]. Furthermore, recent implementations of optical free-space links also give promising results synchronizing remote optical clocks with ~40 fs daily drift below 0.1 Hz [27]. Since current applications are highly satisfied with the provided precision, investigation of jitter and phase noise limitations in time and frequency dissemination systems has been largely ignored. However, as optical clocks continue to evolve and shorter time scales are probed in attoscience facilities, it becomes necessary to determine their limiting noise factors.

In this paper, we present a timing jitter investigation method based on feedback loop analysis of pulsed, optical timing-distribution and remote-laser synchronization systems. In our experimental testbed, we synchronize two mode-locked lasers operating at different central wavelengths and repetition rates in a master-slave configuration both locally and remotely over a timing stabilized fiber link network. Local synchronization experiment reveals the inherent timing jitter of the slave laser, which amounts to 2.1 fs RMS integrated from 45 MHz down to 20 kHz (limited by the locking bandwidth). Feedback model of the local synchronization provides a powerful method to estimate the free-running jitter of the slave laser for offset frequencies lower than the locking bandwidth. Remote synchronization of the lasers over the fiber link network shows an out-of-loop jitter of 8.55 fs RMS integrated from 1 Hz to 1 MHz. Our feedback loop analysis of the complete system is in excellent agreement with the measured precision and identifies seven uncorrelated noise sources, out of which the slave laser’s jitter dominates with 8.19 fs RMS. This analysis method will prove invaluable in large-scale scientific facilities where synchronization of many lasers over large distances is required. Moreover, it can easily be extended to cover different synchronization experiments and is extremely useful in not only identifying sources of timing jitter but also how they depend on the parameters of the network (e.g. feedback gain of the various stabilization loops). This paper starts with the timing jitter characterization of the slave laser and then discusses the remote laser synchronization experiment together with their individual feedback loop analysis.

2. Timing jitter characterization of the slave laser

2.1 Experimental setup

In our experiments, we use two different mode-locked lasers synchronized in a master-slave configuration. The master laser is a commercially available mode-locked laser (Origami-15 from Onefive GmbH) operating at 1554-nm center wavelength and 216.667-MHz repetition rate. In our earlier work [28], we have measured its free-running timing jitter, which is less than 0.4 fs RMS above 1 kHz and less than 0.17 fs RMS above 10-kHz offset frequency. The slave laser, on the other hand, is a home-built titanium-sapphire (Ti:sa) Kerr-lens mode- locked laser operating at 800-nm center wavelength and 1.0833-GHz repetition rate. As depicted in Fig. 1(a), the Ti:sa laser is built as a ring cavity with two pairs of broadband dispersion-matched double-chirped mirrors (DCMs) which follow the design of our earlier work [29]. The Ti:sa crystal is placed at Brewster angle between the two dispersion matched concave DCMs with a radius of curvature of 25 mm. A 0.7-mm BaF₂ plate is used to compensate the intra-cavity dispersion. The output coupler is a BaF₂ wedge with a thickness of 1.43 – 1.77 mm, which is located in between the dispersion-matched flat DCMs to fine-tune the intra-cavity dispersion. With 6-W pump power at 532 nm and 4% output coupling ratio, the Ti:sa laser yields 650-mW average output power.

 

Fig. 1 (a) Schematic of the Kerr-lens mode-locked Ti:sa laser. (b) Experimental setup for characterizing the timing jitter of the Ti:sa laser. The right panel shows the measured TC-BOC response as a function of the time delay between the two input pulse trains. Abbreviations: cDCM: concave DCM; fDCM: flat DCM; Ti:sa: titanium-sapphire crystal; BaF₂: barium fluoride plate; OC: output coupler; PZT: piezo-electric transducer; λ/2: half-wave plate; SM: silver mirror; DBC: dichroic beam combiner; DBS: dichroic beam splitter; BBO: beta barium borate crystal; IF: interference filter; BPD: balanced photodetector.

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Ti:sa based laser systems are widely used in time and optical frequency transfer applications such as free-electron laser synchronization, pump-probe experiments and clock comparison. Therefore, their jitter characterization holds a great importance to anticipate the temporal resolution of the complete system. By utilizing the low noise properties of the master laser, we characterize the inherent timing jitter of our Ti:sa laser using the balanced optical cross-correlator (BOC) method [30,31]. This method is intrinsically immune to excess amplitude-to-phase noise conversion and delivers extremely low noise floors down to 10⁻¹² fs²/Hz [11]. As displayed in Fig. 1(b), the outputs of these two lasers are combined by a dichroic beam combiner, and then sent into the two-color BOC (TC-BOC). Optical cross-correlation between the input pulses is realized with type-I sum-frequency generation between 800-nm and 1550-nm central wavelengths in a beta barium borate crystal. The pulses propagate through the crystal in a double-pass configuration at different group velocities and generate different amounts of light at their sum-frequency on the forward and backward pass. The difference in the sum-frequency light detected by the balanced photodetector (BPD) yields an output voltage signal depending on the initial timing difference between the input pulses. The most important characteristic of the BOC output is the slope around its zero voltage crossing, which gives the jitter-to-voltage conversion factor (i.e., timing sensitivity) of the detector (see the right panel in Fig. 1(b)). The mean value of the TC-BOC’s timing sensitivity is 1.96 mV/fs from five consecutive measurements resulting in ± 0.05 mV/fs standard deviation due to slope uncertainty (for measurement methods refer to Appendix 1). Once the Ti:sa laser is locked to the master laser at the zero crossing, the TC-BOC cancels the amplitude noise and reads solely the timing jitter between the pulse trains outside the locking bandwidth of the feedback loop [30,31].

2.2 Experimental results and feedback loop analysis

Figure 2 shows the jitter characterization results of the Ti:sa laser from 1 kHz to 45 MHz (high-frequency limit is due to the BPD bandwidth). The gray curve shows the detector noise floor of the TC-BOC, which is below 10⁻¹¹ fs²/Hz for offset frequencies higher than 10 kHz. To ensure jitter characterization at the zero crossing of the TC-BOC, we measure the sum-frequency power with the individual photodetectors of the BPD each time when the lasers arelocked. Both photodetectors read 0.15 V resulting in a measurement uncertainty of 1.25 × 10⁻¹⁰ fs²/Hz due to shot-noise (black dashed line in Fig. 2) which confirms both balanced detection and broad-range shot-noise limited measurement (for details refer to Appendix 1).

 

Fig. 2 Timing jitter characterization results of the Ti:sa laser. Left axes: timing jitter spectral density S_jitter and its integrated jitter δ_jitter for different feedback gains. The color map on the top indicates the feedback gain value. Right axes: equivalent single-sideband (SSB) phase noise L(f) and its integrated phase error δ_phase scaled to 10-GHz carrier frequency.

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We measure the timing jitter spectral density of the Ti:sa laser for different feedback gains. Ideally, one has to apply as low feedback gain as possible to decrease the locking bandwidth and reveal the laser’s inherent jitter for low offset frequencies. Here, we are able to decrease our feedback gain by ~5 dB (see red and blue curves in Fig. 2). Beyond this point, the TC-BOC’s linear range (~200 fs, as can be inferred from Fig. 1(b)) is not able to keep the two lasers locked due to the jitter caused by the acoustic and environmental disturbances on the Ti:sa laser cavity. Note that the Ti:sa laser has only a plastic cover; a more sophisticated acoustic insulation could decrease its environmental sensitivity. Nonetheless, we are able to resolve the laser’s inherent jitter from 20 kHz to 45 MHz as all curves in Fig. 2 overlap with each other in this frequency range. The integration of each spectrum results in ~2.1-fs RMS timing jitter for 20 kHz – 45 MHz, which is only 130-µrad phase error for a 10-GHz carrier. Below 20 kHz, each measured spectrum has a distinct peak value whose magnitude and frequency depend on the applied feedback gain. Therefore, the total integrated jitter for the complete offset frequency range varies from 5.5 fs (red curve) to 10.4 fs (blue curve).

To investigate the peaks observed in the jitter spectra and identify individual noise contributions of the setup elements, we develop a feedback model for BOC-based laser synchronization as illustrated in Fig. 3. In this model, timing jitter between the master and the slave laser is converted to a voltage signal by the BOC transfer function H_BOC, and amplifiedby the BPD transimpedance gain H_BPD together with the detector noise E_BOC. Then, the voltage signal is fed to the proportional-integral (PI) controller with a transfer function H_PI in a negative feedback configuration. Finally, the PI output, along with its electronic noise E_PI, are amplified and converted back to a time delay by the piezoelectric transducer (PZT) with the transfer function H_PZT mounted on an intracavity mirror of the slave laser. Based on this model, the relative timing jitter between the master and the slave laser is given by:

 

Fig. 3 Feedback model for BOC-based laser synchronization. J_M and J_S: free-running jitter of the master and the slave laser; J_O: relative jitter between the two lasers at the BOC input; H_BOC, H_BPD, H_PI and H_PZT: transfer functions of the BOC, BPD, PI controller and slave laser’s PZT, respectively; E_BOC and E_PI: electronic noise of the BOC and the PI controller, respectively.

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Fig. 4 Simulation results of BOC-based laser synchronization. (a) Transfer coefficient |1/(1 + H)|² of the master and slave laser’s inherent jitter. (b) Transfer coefficient |H/(1 + H)|² of the electronic noise jitter. The color map on the top indicates the feedback gain values for individual curves in (a) and (b). (c) (i) Calculated free-running jitter of the Ti:sa laser using Eq. (6) and (ii) measured jitter between the master and slave laser at 0-dB gain. Integrated jitter is shown on a logarithmic scale.

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The transfer coefficients indicate that the inherent timing jitter of each laser is highly suppressed by the feedback loop for low offset frequencies (Fig. 4(a)), whereas the electronic noise jitter is entirely transferred to the output (Fig. 4(b)). More remarkably, both of the transfer coefficients exhibit peak values above unity gain around the locking bandwidth, which shift with the varying feedback gain. Therefore, J_M, J_S and J_N are amplified at these offset frequencies, which explain the jitter peaks observed in the experimental results and their behavior in response to the varied feedback gain (see Fig. 2). Furthermore, using Eq. (5) we can estimate the free-running jitter of the Ti:sa laser as:

In Fig. 4(c), we compare the simulated free-running jitter of the Ti:sa laser (gray curve) and the measured jitter between the two lasers at 0-dB feedback gain (i.e., identical to the red curve in Fig. 2). As both curves nicely overlap with each other for offset frequencies larger than 20 kHz, this result confirms the validity of our argument earlier that the dominant jitter contribution at the output of the TC-BOC stems from the slave Ti:sa laser. Since the feedbackloop suppresses the noise between the master and the slave laser for frequencies lower than the locking bandwidth, the free-running jitter of the Ti:sa laser cannot be resolved experimentally below 20 kHz. However, thanks to our feedback model we estimate a total free-running jitter of 1.3 ps RMS integrated from 100 Hz to 1 MHz.

Time and optical frequency transfer applications mostly depend on the stable transmission of the master signal to the remote lasers using long optical fibers. Therefore, in the following section we present our experimental and simulation results on the remote synchronization of our Ti:sa laser on a km-scale timing link network.

3. Synchronization of the slave laser on a km-scale timing link network

3.1 Experimental setup

Figure 5 describes the experimental setup built for the Ti:sa laser synchronization on a 4.7-km timing link network. We use the same master laser as in Section 2 and lock the repetition rate to an RF reference to reduce its drift below ~200 Hz. Then we split the output of the master laser into two separate timing links. Timing link 1 consists of a 3.5-km polarization-maintaining (PM) dispersion-compensated fiber spool, a PM fiber stretcher, and a fiber-coupled motorized delay line with 560-ps range. Similarly, the components of timing link 2 include a 1.2-km PM fiber spool, a PM fiber stretcher, and a free-space motorized stage with 100-ps range. At the end of each link, there is a fiber-coupled mirror reflecting 10% of the optical power back to the link input. Here, we also use a bidirectional erbium-doped fiber amplifier (EDFA) in order to provide sufficient power for the back-propagating signal and the link output required for link stabilization and remote synchronization, respectively. At the link inputs, we combine the round-trip pulses with newly emitted ones in one-color (OC)-BOCs. OC-BOCs operate at 1554-nm wavelength and realize the cross-correlation with the birefringence between two orthogonally polarized input pulses. OC-BOCs measure the propagation delay fluctuations in the links and generate error voltages, which control the fiber stretchers and the motorized delays to compensate for fast jitter and long-term drift, respectively. Here, we employ a third feedback control on the pump current of the EDFAs to stabilize the link power fluctuations. As we have reported in [26], this feedback mechanism ensures drift-free link stabilization with attosecond precision by suppressing timing errorstriggered by the link nonlinearities due to power fluctuations. The Ti:sa laser is placed at the output location of the timing links. As the master laser and Ti:sa laser operate at different central wavelengths, we build two TC-BOCs between each link output and the Ti:sa laser output. TC-BOC1 synchronizes the Ti:sa laser with link 1 output by tuning the repetition rate via its intracavity PZT mirror. Finally, the free-running TC-BOC2 evaluates the timing precision between the synchronized Ti:sa laser and timing link 2 output.

 

Fig. 5 (a) Experimental setup for the synchronization of the Ti:sa laser on a timing link network with a total length of 4.7 km. (b) Individual elements of the timing stabilized fiber links. Abbreviations: RF: RF reference; FC: fiber collimator; MDL: motorized delay line; PM-FS: polarization-maintaining fiber stretcher; PM-DCF: PM dispersion-compensated fiber; EDFA: bidirectional erbium-doped fiber amplifier; PRM: partially reflecting fiber mirror.

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3.2 Experimental results and feedback loop analysis

Since environmental effects such as temperature and humidity act in long time scales (>1 s), timing drift measurements below 1 Hz are ideal to study their influence on our system. Figure 6(a) shows the out-of-loop timing drift between the remotely synchronized Ti:sa laser and timing link 2 output. We are able to keep the complete system synchronized for 8 hours continuously, which is limited by the PZT range of the Ti:sa laser. This duration can easily be extended by adding a slow feedback mechanism, e.g., control of the cavity temperature. Nonetheless, the observed drift is only 25-fs peak-to-peak and 3.65 fs RMS for the complete duration without any excess locking volatility. We also calculate the relative timing instability (i.e., time error in terms of overlapping Allan deviation) from the drift data to investigate the system behavior for different averaging times. As Fig. 6(b) illustrates, the relative timing instability is only 1.2 × 10⁻¹⁵ in 1-s averaging time (τ) and falls to 3.36 × 10⁻¹⁹ at 10000 s following a deterministic slope very close to τ⁻¹.

 

Fig. 6 Out-of-loop measurements between the remotely synchronized Ti:sa laser and timing link 2 output. (a) Timing drift below 1 Hz. (b) Calculated relative timing instability from the drift data. (c) Jitter spectral density S_jitter and its integrated jitter δ_jitter; right axes: equivalent SSB phase noise L(f) and its integrated phase δ_phase scaled to a 10-GHz carrier frequency. The gray curve shows the noise floor of the free-running TC-BOC2.

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Timing jitter spectral density for offset frequencies larger than 1 Hz is measured with a baseband analyzer, which Fourier transforms the TC-BOC2 output. The red curve in Fig. 6(c) shows the out-of-loop jitter between the remotely synchronized Ti:sa laser and timing link 2 output. The integrated jitter for 1 Hz – 1 MHz is 8.55 fs RMS corresponding to a phase error of 0.5 mrad for a 10-GHz carrier.

As there are four BOCs and numerous locking elements used in this experiment, we pursue a comprehensive feedback loop analysis to identify the factors influencing the observed jitter. Figure 7 shows the feedback flow diagrams of each BOC-based timing measurement. In the feedback models of the timing links (Figs. 7(a) and 7(c)), the timing jitter of the link J_I is detected by the BOC and converted to a voltage signal by the BOC transfer function H_BOC. Then the BOC signal is amplified by H_BPD with the detector noise E_BOC. The BPD signal is fed to the PI controller H_PI in a negative feedback configuration. The output of the PI controller together with its electronic noise E_PI is amplified and converted to a jitter value J_F by the fiber stretcher H_PZT, which acts as a compensating jitter to maintain the lock. Since we perform error analysis upon round-trip link propagation, delayed copies of the compensation jitter J_F, master laser’s inherent jitter J_M and the environmental jitter imposed on the link J_E are also added to the link jitter beside the original sources. Round-trip link delay 2τ_L is considered for J_F and J_M, whereas only one-way link delay τ_L is applied to J_E as it already includes one-way integrated environmental jitter by its definition.

 

Fig. 7 Feedback model for (a) stabilization of timing link 1, (b) remote synchronization of the Ti:sa laser via timing link 1, (c) stabilization of timing link 2, and (d) out-of-loop jitter measurement between the Ti:sa laser and timing link 2. J_E: total environmental jitter imposed on the link for one-way travel; J_I: timing jitter detected by the in-loop BOC; J_F: equivalent jitter generated by the feedback loop to maintain the lock; J_O: out-of-loop jitter between the Ti:sa laser and timing link 2 output; τ: one-way fiber link travel time, s = jω: complex frequency. Subscripts L1, L2, and S after each abbreviation refer to timing link 1, timing link 2 and the Ti:sa laser, respectively.

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The feedback flow for the remote synchronization of the Ti:sa laser (Fig. 7(b)) is quite similar to the one for BOC-based laser synchronization discussed in Section 2. However, this time we have to consider two additional jitter sources: (i) environmental jitter J_E,L1 and (ii) compensation jitter of timing link 1 J_F,L1 beside the inherent jitter of the master and the slave laser. Then again, one-way link delay must be applied to the inherent jitter of the master laser and to the link compensation jitter because they are actuated at the link input location. Therefore, starting with the feedback model of timing link 1 (Fig. 7(a)) we get:

and from the remote-locking of the Ti:sa laser (Fig. 7(b)):

Stabilization of timing link 2 yields (Fig. 7(c)):

The out-of-loop measurement results in (Fig. 7(d)):

During the experiment, we have observed that the measured out-of-loop jitter is highly sensitive to the feedback parameters of TC-BOC1 lock (i.e., synchronization of the Ti:sa laser to the output of timing link 1). With the help of our feedback model, we can provide evidence for this observation by calculating the transfer coefficients for varying TC-BOC1 feedback gain (Figs. 8(a)-8(g)). Except for the environmental noise on timing link 2 and the electronic noise of its stabilization (Figs. 8(d) and 8(g)), all other transfer coefficients are influenced by the synchronization of the Ti:sa laser. By increasing the TC-BOC1 feedback gain, we can suppress the excess noise coming from the inherent jitter of the Ti:sa laser, environmental disturbances imposed on timing link 1 and electronic jitter of its stabilization (see Figs. 8(b), 8(c) and 8(f)). This is highly intuitive because higher feedback gain achieves ideally tighter synchronization between the link output and the Ti:sa laser. However, excess gain causes noise amplification peaks appearing around the PZT resonance of the Ti:sa laser (Figs. 8(b), 8(c) and 8(f)) and introduces more electronic noise from TC-BOC1 lock (Fig. 8(e)).

 

Fig. 8 Calculated transfer coefficients in Eq. (21) for varying TC-BOC1 feedback gain. Coefficients are given as following: (a) master laser’s inherent jitter; (b) slave laser’s inherent jitter; environmental noise imposed on (c) timing link 1 and (d) timing link 2; electronic feedback noise from (e) TC-BOC1 lock, (f) OC-BOC1 lock and (g) OC-BOC2 lock. The color map on the right-hand side is common for all figures.

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Figure 8(a) confirms once more that it is vital to have a master laser with low inherent jitter, since it is transferred to the output entirely even with amplification around the PZT resonance of its intracavity mirror. Figure 8(d) shows that the environmental jitter on timing link 2 is well suppressed below 20 kHz. However, one has to pay attention to the electronic noise coming from its stabilization as it is transferred partially to the output by the feedback (Fig. 8(g)). Now that we have calculated the transfer coefficients, we can determine the jitter spectral densities in Eq. (21) and compare the results with the measured out-of-loop jitter. We obtain the environmental jitter imposed on the links ($\overline{J_{E,L1}^2}$and$\overline{J_{E,L2}^2}$) experimentally by simply measuring their unlocked OC-BOC output voltage with a baseband analyzer. We use the experimental data from our previous publication [28] for the master laser’s inherent timing jitter ($\overline{J_M^2}$), whereas we estimate the inherent timing jitter of the Ti:sa laser ($\overline{J_S^2}$) by using Eq. (6), which is depicted as the gray curve in Fig. 4(c). Finally, we estimate the electronic jitter terms ($\overline{J_{N,S}^2},\overline{J_{N,L1}^2}$ and$\overline{J_{N,L2}^2}$) arising from the TC-BOC1, OC-BOC1 and OC-BOC2 locks respectively as detailed in Appendix 2.

Figure 9(a) shows the simulation results of individual jitter contributions for offset frequencies 100 Hz to 1 MHz where the low frequency limit is set by the free-running jitter of the master laser, which can be resolved down to 100 Hz. The inherent jitter of the Ti:sa laser is by far the largest noise factor with 8.19-fs integrated jitter (dark blue curve). This result points out that it is imperative to pay considerable attention to the low noise properties of the slave lasers before employing them in time and frequency transfer applications requiring higher precision.

 

Fig. 9 (a) Simulation results of individual jitter contributions. The legend on the right shows the color code of the jitter spectral densities and their integrated jitter from 100 Hz to 1 MHz. Integrated jitter in this graph is shown on a logarithmic scale. (b) Comparison between (i) the sum of all simulated jitter sources and (ii) the out-of-loop jitter measurement.

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The inherent jitter of the master laser also plays an important role with 1.67-fs total jitter (light blue curve in Fig. 9(a)). Its contribution becomes more noticeable for offset frequencies below 200 Hz, which justifies the stabilization of its repetition rate with a low-noise RF source. However, one has to maintain a relatively low locking bandwidth here, as the master laser surpasses the RF source in terms of low-noise performance for higher offset frequencies.

Large feedback gain in the link stabilization helps to suppress the environmental noise on the fibers; however, one has to pay attention to the PZT resonances of the fiber stretchers. As can be seen from the dark green curve in Fig. 9(a), the environmental jitter imposed on timing link 1 accumulates 1.5-fs jitter due to the excess noise around 18 kHz caused by its PZT resonance. We can appreciate the advantage of low noise floors provided by the BOCs once more, since the least dominant jitter contributions in our system are the electronic noise coming from the feedback elements with a total integrated jitter below 0.3 fs.

Finally, we compare the sum of all individual jitter spectral densities with the out-of-loop measurement (Fig. 9(b)). The experiment and the simulation agree very well with each other as the simulation estimates 8.49-fs RMS total integrated jitter, whereas the experiment shows 8.51 fs RMS for 100 Hz – 1 MHz. It even reveals many specific jitter structures such as electronic noise spurs below 2 kHz. However, there are some deviations between the two spectra for offset frequencies 5 kHz – 25 kHz. We suspect that they originate from the imperfect estimation of the PZT transfer function of the Ti:sa cavity mirror. The simulation results can be improved even further by measuring the PZT voltage response around its resonance. Note that above 200 kHz, the simulation result is obscured by the environmental noise and master laser jitter measurements, which fall beyond the detector noise floor.

4. Conclusion

We have presented a comprehensive jitter analysis method for optical timing-distribution and remote-laser synchronization systems based on feedback flow between setup elements. First, we have characterized the timing jitter of a 1.0833 GHz Ti:sa laser experimentally with a shot-noise limited TC-BOC. The inherent timing jitter of the laser is only 2.1 fs RMS for offset frequencies 20 kHz – 45 MHz. We have developed a feedback model for BOC-based laser synchronization, which agrees very well with the measured timing jitter and offers a method to estimate the free-running laser noise for offset frequencies lower than the locking bandwidth.

To examine the jitter properties of our Ti:sa laser when used in time and frequency transfer applications, we have synchronized its pulse repetition rate to a timing network consisting of a low-noise master laser and two stabilized fiber links. The out-of-loop drift below 1 Hz between the Ti:sa laser and the timing network is only 3.65 fs RMS for 8 hours, corresponding to a relative timing instability of 3.36 × 10⁻¹⁹ for 10000 s averaging time. We have also measured the out-of-loop jitter spectral density of the synchronization, which gives 8.55-fs RMS total jitter integrated from 1 Hz to 1 MHz. With the help of a comprehensive feedback model involving all setup elements, we have studied individual noise sources in the system and their transfer to the observed jitter. The feedback loop analysis has identified seven uncorrelated noise sources, out of which the jitter arising from the Ti:sa laser is the largest with 8.19-fs RMS. This suggests that the resolution of the system can be improved down to the sub-fs regime with lower-noise slave lasers.

The presented analysis method can easily be adapted to different scenarios in order to optimize feedback control parameters and identify sources of noise within a complex synchronization network. For instance, besides the atmospheric turbulences and temperature changes acting on long time periods (>1 s), limiting noise sources of free-space optical links [27] in the high-frequency range (>100 Hz) remain still unknown. Therefore, it would be a good scientific practice to compare high-frequency timing jitter of optical fiber and free-space links by employing the presented model in this paper.

Appendix 1 BOC characteristics

When synchronizing the timing of two mode-locked lasers, the two input pulse trains have different repetition rates before the feedback loop is activated. In this case, the response voltage is a train of characteristic BOC traces whose frequency is equal to the repetition rate difference of the input pulses. One can simultaneously record a BOC trace on an oscilloscope and measure the instantaneous repetition rate difference with photodetectors and an electronic mixer. The real time scale of the BOC trace is obtained by multiplying the time scale of the oscilloscope by the ratio of the repetition rate difference to the repetition rate of the master laser. The slope of the measured BOC trace at its zero crossing is the timing sensitivity.

The electronic noise of a BOC is measured (in units of V/√Hz) when there is no input light on the device, and then converted to timing jitter (s/√Hz) by multiplying with the timing sensitivity. The shot noise limit of a BOC is obtained when the BOC-based feedback loop is engaged. In this case, the photocurrent detected by each photodetector is measured from the individual ports of the BPD. The corresponding shot noise is calculated from the measured photocurrent in 1 Hz bandwidth (A/√Hz). Then, the shot noise floor is converted to voltage noise (V/√Hz) by the transimpedance gain of the BPD and finally to timing jitter (s/√Hz) by multiplying with the timing sensitivity.

Appendix 2 Total transfer function H and electronic noise jitter J_N

According to the feedback models in Figs. 3 and 7, each BOC lock has two electronic noise sources E_BOC and E_PI (in units of V/√Hz) and four transfer functions H_BOC, H_BPD, H_PI and H_PZT. H_BOC is simply the timing sensitivity of the BOC converting the timing error to a voltage error signal. Then, the error voltage is amplified by the transimpedance amplifier of the BPD with its 3-dB bandwidth (i.e., H_BPD) and processed by the PI controller (i.e., H_PI). Finally, the PI output is applied to the PZT actuator with a resonance frequency at f_res and a transfer function H_PZT, which converts it to a corrective time delay and closes the feedback-loop. To simplify the mathematical derivation of the jitter sources, we define a total transfer function, H and an electronic jitter term, J_N for each BOC lock:

(22)$$H=H_{BOC}{H}_{BPD}{H}_{PI}{H}_{PZT}$$

(23)$$J_N=-\frac{E_{BOC}}{H_{BOC}}+\frac{E_{PI}}{H_{BOC}{H}_{BPD}{H}_{PI}}$$

As can be inferred from Eq. (23), J_N refers to the timing jitter (in units of s/√Hz) generated by the electronic noise sources (i.e., E_BOC and E_PI) and transferred to the input of the BOC by the feedback loop (hence the division by H_BOC and H_BOCH_BPDH_PI for E_BOC and E_PI respectively). Finally, we use the following transfer functions for our experimental devices to calculate H:

(24)$$\begin{array}{l}{H}_{BOC}=k_{BOC}{H}_{PI}=k_{PI}\left(\frac{s+2\pi f_{PI}}{s}\right)\\ H_{BPD}={\left(1+\frac{s}{2\pi f_{BW}}\right)}^{-1}{H}_{PZT}=\frac{k_{PZT}}{f_{rep}s}\left(\frac{{\left(2\pi f_{res}\right)}^2}{s^2+\zeta s+{\left(2\pi f_{res}\right)}^2}\right)\end{array}$$

k_BOC is the timing sensitivity of the BOC, k_PI and f_PI are the gain and corner frequency of the PI controller, f_BW is the 3-dB bandwidth of the BPD, k_PZT is the slave laser’s PZT sensitivity, f_rep is the repetition rate of the slave laser, f_res is the slave laser’s PZT resonance and ζ is the PZT response parameter. Please note that these individual transfer functions are complex functions acting as commutative linear operators in the frequency domain. For the PZT transfer functions of the fiber stretchers in the timing links, we use voltage-response data provided by the manufacturer.

Furthermore, we have to determine the electronic noise of the BOC and the PI controller to calculate J_N. We measure the BOC electronic noise experimentally and then compare it with the shot-noise limit (see Appendix 1). The larger noise contribution is used as E_BOC in the calculation. Finally, we estimate the electronic noise of the PI controller E_PI using a noise model based on an integrating-operational-amplifier as in Fig. 10:

(25)$$E_{PI}^2=\frac{Z_f^{2}4k_{B}T}{R_{in}}+4k_{B}T{R}_f+{\left(1+\frac{Z_f}{R_{in}}\right)}^2{V}_n^2 \text{where}{Z}_f=\frac{1/R_f -jsC}{1/R_f+s^2{C}^2}$$

Z_f is the impedance of the parallel R_fC circuit, k_B is the Boltzmann constant, and T is the resistor’s absolute temperature. Finally, we can calculate the individual jitter sources using the experimental values listed in Table 1.

 

Fig. 10 Noise model of the PI controller. V_in and V_out: input and output voltage; V_n: input voltage noise of the PI controller; R_in and i_th,Rin: input impedance and its thermal noise current; R_f and i_th,Rf: feedback gain resistance and its thermal noise current; C: integrating capacitance.

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Table 1. Experimental parameters used in the feedback loop analysis

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Funding

European Research Council (ERC Grant Agreement n. 609920), Center for Free-Electron Laser Science at Deutsches Elektronen-Synchrotron, the Hamburg Centre for Ultrafast Imaging of Deutsche Forschungsgemeinschaft.

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