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Abstract
The significance of timing jitter stems from its pivotal role in enhancing the precision of applications like spectroscopy and frequency metrology. In this study, we introduce a comprehensive procedure for achieving low timing jitter values in mode-locked fiber laser systems, highlighting dispersion, intracavity pulse energy, pulse length, and spectral bandwidth as key parameters. Notably, we unveil the influence of fiber amplifier pump power on jitter, a factor neglected in established theories and recent experiments. Applying this procedure to a 200-MHz all-polarization-maintaining (PM) erbium-doped (Er:) nonlinear amplifying loop mirror (NALM) fiber laser system, we demonstrate an exceptionally low timing jitter of 14.25 attoseconds, measured using the balanced optical cross-correlation (BOC) technique and integrated from 10 kHz to 4 MHz. The implementation of our novel method offers the opportunity to improve jitter results in various fiber laser systems and increase the accuracy of fiber laser applications.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The timing jitter stands out as one of the most critical characteristics of ultrafast mode-locked laser sources. In various applications, the temporal jitter of pulses directly impacts achievable performance, such as in timing synchronization for free-electron X-ray lasers [1,2], pulse time-of-flight distance measurements [3], photonic analog-to-digital converters [4], photonic-based radar systems [5], clock distribution networks [6], dual-comb [7,8], pump-probe [9–11] and THz [12] spectroscopy to name just a few examples. Timing jitter is also linked to the linewidth of optical frequency comb lines [13,14].
The availability of fiber lasers with exceedingly low timing jitter introduces novel opportunities for cutting-edge scientific and industrial operations that require not only exceptional levels of precision, resolution, and accuracy but also compactness and resilience. Up until now, the attainment of timing jitter values below the threshold of 100 attoseconds (as) using fiber lasers has been a rarity, with only a few notable exceptions. Among erbium-doped fiber lasers the most remarkable examples are 40 as (integrated from 10$\,\textrm {kHz}$ to 10$\,\textrm {MHz}$) for 72$\,\textrm {MHz}$ Er:NALM [15] and 70 as [10$\,\textrm {kHz}\,-\,{38.8}\,\textrm {MHz}$] achieved with 77.6$\,\textrm {MHz}$ fiber laser using nonlinear polarization evolution (Er:NPE) [16]. Considering fiber lasers with different doping materials, the lowest timing jitter recorded was 14.3 as [10$\,\textrm {kHz}$ - 94$\,\textrm {MHz}$] with a 188 MHz ytterbium-doped fiber laser utilizing nonlinear polarisation rotation (Yb:NPR) [17]. In this work, we demonstrate an unprecedentedly low integrated timing jitter of merely 14.25 as [10$\,\textrm {kHz}$ - 4$\,\textrm {MHz}$] determined with 200$\,\textrm {MHz}$ Er:NALM. To our knowledge, this achievement sets a new record for any free-running mode-locked fiber lasers that are erbium-doped.
To attain this goal, it was imperative to optimize various parameters associated with timing jitter. This optimization encompassed not only the fine-tuning of the utilized 200$\,\textrm {MHz}$ Er:NALM oscillators but also the calibration of the amplifiers, which were essential for conducting measurements employing the balanced optical cross-correlation (BOC).
2. Timing jitter contributions
Timing jitter $\Delta \tau$, or simply jitter, refers to fluctuations in the the temporal alignment of pulses with respect to their intended, regularly spaced temporal instances represented as $t_{i_{0}}$. The variance $\sigma _{\Delta \tau }$ of these deviations for a set of $n$ pulses, each situated at position $t_{i}$, can be expressed as:
The expression defined in the time domain, as given by Eq. 1, corresponds, according to Parseval’s theorem, to the integral of the measured power spectral density (PSD) of the timing jitter $S_{\Delta \tau }$ [18]. The corresponding value of the timing jitter $\Delta \tau$ can be calculated as the square root of this variance [19]. In the subsequent discussion, we will delve into the primary factors that make substantial contributions to jitter.
2.1 Environmental noise
Beyond the consideration of physical phenomena, it is essential to account for the influence of mechanical disturbances introduced by the components. As explained in Supplement 1, the used NALM oscillators are equipped with an output coupler mirror that allows for the adjustment of the resonator length and, consequently, the repetition rate. Vibrations of such a mirror contribute to timing jitter due to fluctuations in the intracavity round-trip time with an dependence on frequency $f$ of approximately $f^{-4}$ [18]. Another source of environmental noise is the refractive index change caused by thermal fluctuations [18,20,21], which has been mitigated in the used oscillator by temperature stabilization using a Peltier element.
2.2 Spectral fluctuations
The variations of the optical spectrum additionally affect the timing jitter, with the strength of this effect primarily dependent on chromatic dispersion [22,23]. This indirect contribution to the temporal fluctuations was first theoretically derived in 1986 and is referred to as the Gordon-Haus effect [24]. The PSD of this indirect component can be expressed as demonstrated [25,26]:
The expression takes into consideration the FWHM gain bandwidth $\nu _\mathrm {g}$, intracavity gain $\mathrm {G}$ and the FWHM pulse length $\tau _\mathrm {p}$. These formulas illustrate that the Gordon-Haus effect contributes to the timing jitter with $S_{\Delta \tau, \mathrm {indirect}}(f)$ $\approx$ $f^{-4}$ for $f>1/(2\pi \tau _\mathrm {cf})$. This implies that at high frequencies, the influence of central frequency fluctuations decreases significantly compared to other effects.
In the context of this study, it is particularly relevant to determine which of the variables listed above can be varied during the experiments to achieve the lowest values of timing jitter. Most factors are determined by the choice of mode-locking mechanism (NALM), active fiber (erbium), and repetition rate (200$\,\textrm {MHz}$) and therefore cannot be adjusted. However, the remaining quantities related to pulse bandwidth, intracavity pulse energy, and group delay dispersion can be altered. Fine-tuning the GDD is particularly challenging as it requires intervention in the cavity and precise adjustment of the length of the active fiber in the micrometer range. For this reason, a theoretical analysis regarding dispersion was conducted, examining the correlation of GDD values with experimentally determined results for timing jitter from various scientific sources.
From Eq. 2, it is evident that larger values of group delay dispersion lead to increased timing jitter. If the GDD were to be zero, the contribution from the Gordon-Haus effect would be eliminated, potentially resulting in a significant reduction in timing jitter.
However, generating stable conventional solitons requires an unavoidable condition of anomalous GDD [40–42]. Consequently, it is necessary to find a negative group delay dispersion value that is sufficient to counteract the effect of self-phase modulation (SPM) but low enough not to cause significant timing jitter. The values of timing jitter obtained from the theoretical analysis for different GDD values are summarized in Table 1 for various fiber lasers and depicted on Fig. 1 in logarithmic representation. It is important to note that all the results considered were achieved using the balanced optical cross-correlation (BOC) technique, which was also employed in the experimental part of this study.
Table 1. Timing jitter $\Delta \tau$ values with their respective integration intervals reported across different group delay dispersion (GDD) settings in fiber lasers. Results obtained in this work (*) are shown for comparison.
From Table 1 and Fig. 1, it can be concluded that achieving the lowest timing jitter requires an anomalous intracavity group delay dispersion with a value of less than -0.01$\textrm {ps}^2$. For the Er:NALM oscillators with a 200$\,\textrm {MHz}$ repetition rate used in this work, this condition was met. This, among other factors, contributed to achieving one of the lowest known integrated jitter values of only 14.25 as [10$\,\textrm {kHz}$ - 4$\,\textrm {MHz}$] (depicted in Fig. 1 with the symbol ${\color{green}{\triangledown}}$).
Another parameter that can be reduced to achieve lower timing jitter is the spectral half-width of the pulse. This goal can be accomplished by inserting a bandwidth-limiting filter into the cavity [43]. However, such a filter introduces additional dispersion and alters the pulse energy, making it challenging to estimate the final impact on jitter [44]. At the same time, the spectral pulse bandwidth undergoes changes during the adjustment of the pump power due to SPM. Additionally, the bandwidth is limited by the round-trip time in the cavity, which should remain constant to preserve the repetition rate [45]. For these reasons, it was not possible to isolate the effect of this parameter in our study.
2.3 Temporal fluctuations
Prior to addressing the impacts of the remaining two variables, namely, the pulse energy and duration, the effects of fluctuations directly induced in the temporal domain are presented. These variations are influenced by the same parameters and exhibit a comparable level of dependence. The PSD of timing jitter arising from the direct influence of amplified spontaneous emission (ASE) can be expressed as follows for a sech$^2$-shaped pulse with FWHM length $\tau _\mathrm {p}$ [25,26]:
The inverse square dependence ($\approx 1/f^2$) observed in Eq. 5 clearly highlights that temporal fluctuations induced by ASE surpass the impact of the Gordon-Haus effect at higher frequencies. Furthermore, the pulse duration exerts a substantial influence on the jitter attributed to direct ASE noise. The heightened jitter observed with prolonged pulses can be elucidated by the fact that a broader temporal distribution of the pulse results in a lengthier exposure of the amplified spontaneous emission noise to the pulse [46]. The PSD of direct timing jitter demonstrates an inverse relationship with intracavity pulse energy, similar to the indirect impact of ASE. This observation aligns with recent experimental research findings [47] and is consistent with expectations, as relative quantum fluctuations tend to decrease at higher power levels, where the photon count per pulse is greater [18]. This phenomenon can be elucidated by recognizing that the perturbations encountered by each individual photon decrease as the pulse photon count increases [48]. However, it’s worth noting that increasing pulse energy as a means of reducing relative quantum noise has limitations. Higher pulse energy implies greater intensity, which, in turn, results in heightened nonlinear effects in optical fibers, contributing to jitter [25,49,50]. These nonlinear effects exhibit varying dependencies on pump power, rendering the precise identification of timing jitter causes significantly more intricate.
The relationships between various physical parameters and timing jitter, as depicted in Eq. 2 and Eq. 5, were originally predicted through numerical simulations [22–24,44,51] and several years later were corroborated by various experiments [48,52–54]. These experimental findings have underscored the critical role of spectral fluctuations and temporal variations in the context of timing jitter in optical fibers. Consequently, in the analysis of the measured spectral power density of timing jitter in the next section, the theoretical curves of the PSD for directly and indirectly induced jitter were presented as a means of comparison with the empirical data.
3. Experimental details
Measurements were performed using erbium nonlinear amplifying loop mirror (NALM) fiber lasers equipped with polarization maintaining (PM) fibers. The design of the implemented NALM oscillators is explained in Supplement 1. NALM-based lasers are advantageous due to their durability, stability, and self-starting mode-locking mechanism. Additionally, they provide a wide optical spectrum and short pulse duration as a result of their fast effective relaxation time, establishing the capability for very low timing noise. In recent work, we reported novel frequency comb fiber oscillators with extraordinarily low free-running comb linewidths of sub 1$\,\textrm {kHz}$ [55]. To determine the timing jitter in the attosecond range of these low-phase noise femtosecond sources we implemented the balanced optical cross-correlator (BOC) method [56]. This technique overcomes the restrictions of radio frequency approaches measuring the timing noise [57], and it enables a high-resolution, drift-free, and temperature-independent observation of jitter between two optical pulses with attosecond resolution while staying unresponsive to laser amplitude noise [8,58,59], which makes BOC auspicious compared to all-electronic techniques.
3.1 Experimental setup
The experimental setup for a measurement employing BOC is depicted in Fig. 2. To implement the chosen measurement method, two identical laser systems were built, with each consisting of an 200$\,\textrm {MHz}$ Er:NALM oscillator and an amplifier. More detailed information about these components is provided in Supplement 1. Oscillators and amplifiers were individually stimulated by their respective pump diodes operating at a wavelength of 980$\,\textrm {nm}$. The use of the amplifier was necessary to sufficiently increase the signal intensity and induce nonlinearity in the periodically poled potassium titanyl phosphate (PPKTP) crystal, thereby generating the second harmonic (SHG). A combination of the oscillator and the amplifier was referred to as the "Reference Laser" (REF), while the other was named the "Laser Under Test" (LUT). For quasi-phase matching (QPM) and the SHG in PPKTP, it is essential to have orthogonal polarization of the two laser beams. This is accomplished in the setup by using $\lambda /2$ waveplates along the propagation path of reference laser. To prevent back reflections, two optical isolators (ISO) are employed, one for each laser system. Light from REF and LUT is combined using a polarization beam splitter (PBS). The PBS divides the laser light, directing a portion of it to a photodiode (PD) for power measurement, while the other part is sent to the dichroic mirror (DM). The light at the fundamental wavelength of 1560$\,\textrm {nm}$ (depicted in red) is transmitted through the DM and then directed through a lens to the nonlinear crystal. In the birefringent PPKTP crystal with type II QPM, when the two laser signals overlap, second harmonic (SH) light (shown in green) is generated. The fundamental wavelength light is reflected at the dichroic layer on the backside of the crystal and has the opportunity to generate SH light again during its second pass through the PPKTP. The forward SH signal is measured by the first photodiode (PD-1) of a balanced receiver, while the backward second harmonic signal is captured by the other diode (PD-2). At the output of the balanced photodetector (BPD), the difference between the signals measured at PD-1 and PD-2 is obtained.
Fig. 2. Experimental setup for the BOC-based measurement (DM: dichroic mirror, PBS: polarization beam splitter, PD: Photodiode, ISO: optical isolator).
3.2 Operation
Depending on the relative timing between the two input pulses, the balanced photodetector generates an output signal [58]. The signals captured by the photodiodes PD-1 and PD-2, as well as the resulting BOC output signal, are illustrated in Fig. 3.
Fig. 3. Generation of the BOC output signal during the forward and backward propagation through the PPKTP crystal.
For small temporal deviations between the two laser pulses, the difference in photodetector currents shows an almost proportional relationship to the temporal separation of the two pulses as they enter the crystal [8,56]. This implies that the power difference of the signals serves as an indicator of the temporal alignment between the two pulses [60]. The detected signal is free from background noise, i.e., when the two pulses do not overlap in time, the detector signal is completely eliminated [58,61]. The zero crossing occurs when the two pulses of fundamental harmonic exactly coincide at the end of the crystal [59], and the generated SH light exhibits equal power during both forward and backward propagation, as illustrated in Fig. 3. In this case, the detector provides a perfectly balanced signal [56]. As evident from Fig. 3, the measured BOC signal is displayed on a time scale $t_\mathrm {osc}$ corresponding to the time resolution of the utilized oscilloscope, with units in microseconds. To convert these values to the temporal pulse offset, a calibration of the time axis is necessary.
To perform this calibration, a defined difference between the repetition rates ($\Delta f_\mathrm {rep}$) of the two lasers must be set. This was achieved by controlling the temperature and thus the optical length of the oscillator cavity. To experimentally derive the relationship between the deviation in repetition frequencies and the temporal signal on the oscilloscope, several consecutive BOC signals were recorded for various $\Delta f_\mathrm {rep}$ values, as depicted in Fig. 4. This representation shows that the temporal intervals between the consecutive signals visible on the oscilloscope precisely correspond to the inverse value of the repetition rate difference, such that $t_{\mathrm {osc}} = 1/\Delta f_\mathrm {rep}$. Since the temporal separation between pulses at a constant repetition rate is the round-trip period $T_\mathrm {rt}=1/f_\mathrm {rep}$, the factor $k$ needed for the time axis calibration can be calculated from the ratio of these two temporal quantities:
Fig. 4. Distances between individual BOC signals for different $\Delta f_\mathrm {rep}$ values. Conducted with the set oscillator pump power of 636$\,\textrm {mW}$ and amplifier pump power of 630$\,\textrm {mW}$.
Of interest now is the determination of the optimal difference in pulse repetition frequencies, which will be used for time axis calibration and sensitivity calculation. For this purpose, the BOC measurements were conducted six times each for various $\Delta f_\mathrm {rep}$ values in the range of 125 to 1000$\,\textrm {Hz}$ and evaluated in terms of amplitude and sensitivity. The means and standard deviations of the measured BOC signals, after time axis calibration using the chosen repetition rate deviations according to the expression 6, are presented in Fig. 5.
Fig. 5. Influence of set $\Delta f_\mathrm {rep}$ on BOC signal. Conducted with the set oscillator pump power of 636$\,\textrm {mW}$ and amplifier pump power of 630$\,\textrm {mW}$.
The enlarged section in Fig. 5 illustrates the difference in the number of points detected by the photodetector for small and large deviations in repetition frequency. It is evident that an increase in $\Delta f_\mathrm {rep}$ leads to a reduction in the recorded measurement points per time interval, negatively affecting the reliability of the obtained data. At the same time, a larger $\Delta f_\mathrm {rep}$ results in a decrease in the number of pulses overlapping in the crystal, reducing the amplitude of the BOC signal. This is illustrated in Fig. 6. However, the difference in repetition rates cannot be arbitrarily small, as the accuracy of controlling the repetition frequency by the Peltier element is limited by the precision of the repetition rate measurement at $\pm$ 4$\,\textrm {Hz}$ per oscillator. This means that in the worst case, the deviation between the set and actual value of $\Delta f_\mathrm {rep}$ between the two oscillators is $\pm$ 8$\,\textrm {Hz}$. To ensure the accuracy of the evaluation, it is therefore necessary for the set difference to be at least 160$\,\textrm {Hz}$ to keep the relative error below 5%.
Fig. 6. Influence of $\Delta f_\mathrm {rep}$ on the (a) amplitude and (b) sensitivity of the BOC signal. The evaluation is based on Fig. 5, with according colors of the data points.
The goal of this work is to achieve low sensitivity of the BOC, despite the initial apparent contradiction. This is because a low sensitivity value is associated with a higher signal amplitude, which in turn leads to an increased number of overlapping pulses in the PPKTP crystal. This relationship is illustrated in Fig. 6.
After the conducted analysis, a repetition frequency difference of 250$\,\textrm {Hz}$ was selected for all further measurements. This choice ensures, on the one hand, a high signal amplitude and, consequently, low sensitivity. On the other hand, the relative error remains at only 3%, which falls within an acceptable range for measurement accuracy. With the chosen value for $\Delta f_\mathrm {rep}$ of 250$\,\textrm {Hz}$, the time axis calibration was performed, and the sensitivity was determined. An example of such an evaluation of the balanced optical cross-correlation signal is shown in Fig. 7. To ensure the accuracy of the calculated sensitivity, the BOC measurements were performed five times before and after determining the spectral power density of the timing jitter. Figure 7 shows the mean of the ten measurements along with the standard deviation.
Fig. 7. Determination of the sensitivity of the BOC signal from 10 measurements at $\Delta f_\mathrm {rep}$ of 250$\,\textrm {Hz}$. (a) Measured BOC signal with (b) the linear fit for sensitivity determination. Conducted with the set oscillator pump power of 636$\,\textrm {mW}$ and amplifier pump power of 350$\,\textrm {mW}$.
From the observed BOC curve, depicted in Fig. 7, it is possible to calculate the inverse of the slope in the linear segment of the curve, which is referred to as the sensitivity. This value is employed for the conversion between the measured PSD in V$^2$Hz$^{-1}$ and the desired PSD in s$^2$Hz$^{-1}$. The determination of the slope of the BOC curve at the zero crossing was performed by fitting a linear function in the time range between -45 and 45 fs. This yielded 10 slope values, from which the mean and the error of the BOC sensitivity were calculated. The determined value of the BOC sensitivity was used for the calculation of temporal fluctuations, and its standard deviation was included in the error estimate of the timing jitter.
At the zero crossing of the above mentioned BOC curve, the cancellation of intensity noise occurs, allowing for the extraction of only the timing information from the input pulses. At this point the low-bandwidth repetition-rate lock of lasers can be performed, with consequent measuring of the relative timing jitter PSD outside the locking bandwidth with the spectrum analyzer. Given that the oscillators possess uniform characteristics, the timing jitter of the single laser source can be determined by dividing the measured PSD by two [17,33,37], conducting integration within the specified interval, and then extracting the square root of the resulting value.
3.3 Synchronization of repetition rates
After determining the sensitivity of the BOC signal, $\Delta f_\mathrm {rep}$ is reset so that both lasers have the same repetition frequency. Using the BOC output, synchronization of the repetition rates of both lasers can now be achieved through a low-bandwidth feedback loop. For this purpose, the resonator length of the laser was controlled using a piezoelectric transducer (PZT), which was driven by the BOC error signal using a proportional-integral-differential (PID) controller. To accomplish this, digital laser lock module (DigiLock, TOPTICA) was utilized. This system is schematically depicted in Fig. 8. After synchronizing the repetition rates with a low bandwidth of about 1$\,\textrm {kHz}$, the power spectral density (PSD) of the remaining timing jitter at higher frequencies was measured using a Rohde & Schwarz FSV-13 spectrum analyzer.
Fig. 8. Representation of the control loop employed for synchronization of the repetition rates. By inputting the BOC error signal into a proportional–integral–derivative (PID) controller, low-bandwidth synchronization of the repetition rates between the two lasers is achieved. This is accomplished by adjusting the resonator length of the REF laser using a piezoelectric transducer (PZT).
4. Results and discussion
4.1 Measuring timing jitter
The result of measuring the PSD of temporal fluctuations with an oscillator pump power (OPP) of 636$\,\textrm {mW}$ and an amplifier pump power (APP) of 350$\,\textrm {mW}$ is illustrated in Fig. 9(b). For comparison with theoretical expectations, we utilized Paschotta’s well-established analytic model, which addresses both the direct and indirect contributions of ASE to timing jitter. This state-of-the-art model, as presented in the previous sections and equations 2 and 5, has been successfully implemented in recent publications on timing jitter [29,30,47,62,63] and aligns qualitatively with our experimental findings. The background curve depicted in blue represents the signal level in the case of complete absence of light and serves as an absolute lower limit of the measurement, dictated by the utilized measurement configuration. Prior to conducting the integration within the selected interval, this background signal was subtracted from the measurement data. Taking this into account, the integration was consistently carried out until the frequency at which the data reaches the background floor, preventing further measurement. Conversely, the shot noise corresponds to the lower measurement boundary set by the signal acquired by the photodetector and was computed using Eq. (S1) from Supplement 1.
Fig. 9. (a) BOC signal and the determined sensitivity. (b) Measured PSD of timing jitter and background, along with calculated contributions of jitter due to shot noise, direct ASE, and Gordon-Haus effect.
Figure 9(b) reveals the alignment between experimentally obtained data and the theoretical predictions. The variations in the magnitude of the curves can be ascribed to the assumed pulse shape and indirectly measured intracavity pulse duration and energy. Simply halving the FWHM pulse duration would lead to an almost one-order-of-magnitude reduction in the theoretical curves. This highlights that while the theoretical curves serve as valuable benchmarks for predicting the anticipated jitter PSD, they do not indicate an exact measurement minimum. At high frequencies, the graph encounters the shot noise limit, impeding further diminishment. Additionally, steep drops can be observed in the experimental data at specific frequencies. These artifacts precisely correlate with the resonance frequencies of the employed piezoelectric transducer (PZT) used for synchronizing repetition rates. In contrast, the two small peaks in Fig. 9(b) are entirely arbitrary. In other measurements, these peaks were observed at different positions, appeared at varying heights, or not at all. This confirms that these measurement artifacts are caused by environmental influences and have no connection to the used oscillators or the measurement setup.
Another observation is the hill spanning frequencies from e4 to e5$\,\textrm {Hz}$, which might also have contributed to the deviation from the theoretically predicted timing jitter magnitude in this frequency region. One might assume that this hill corresponds to the so-called "servo bump". Servo bumps are typical for proportional–integral–derivative (PID) control loops and occur when disturbances can no longer be compensated in a phase-opposed manner, causing the phase delay of the feedback signal to exceed $\pi$ at a specific frequency [64]. As a result, noise at this frequency is amplified rather than suppressed. However, servo bumps are narrow, their frequency changes with different settings of the PID controller, and they occur within the control bandwidth [65]. In the present case, the observed hill-like feature does not meet any of these criteria since it extends across a broader frequency range, remains in the same position during measurements with various settings, and is notably outside the control bandwidth of approximately e3$\,\textrm {Hz}$. To eliminate the employed spectrum analyzer as the potential cause, the measurement was repeated using a second identical device, yielding an identical curve.
According to the literature, the relaxation oscillation peak of the erbium-doped fiber laser should appear exactly in this frequency region. Theoretical studies predict the relaxation oscillation frequency of the erbium fiber laser to be 56$\,\textrm {kHz}$ [66] or "around 10$\,\textrm {kHz}$ to 50$\,\textrm {kHz}$" [67], while experimental measurements lead to a value of 68$\,\textrm {kHz}$ [68]. Since the precise value is contingent on parameters such as the length of the erbium-doped fiber and the concentration of erbium ions [66], we can infer that our relaxation frequency should fall within a similar range. Moreover, the trajectory of the curve calculated and observed in the aforementioned papers matches the one measured in our study. Furthermore, with increasing amplifier pump power, this peak exhibits a slight shift towards higher frequencies, as demonstrated in Supplement 1, Fig. S3 and S4. This phenomenon is attributed to the high-pass filtering of the oscillator’s relative intensity noise (RIN) [69] and corresponds to the expected behavior of the relaxation oscillation peak [70–72]. This specific coupling to timing noise arises from Amplitude-Modulation-to-Timing-Modulation (AM-TM) mechanism [18,63]. Certainly, at such a low level of jitter, it is plausible that additional effects, previously regarded as negligible, may have also contributed to influencing the results.
4.2 Optimizing timing jitter
As elucidated in the theoretical section, the aim was to enhance the intracavity pulse energy, in conjunction with shorter pulse duration and a narrower spectral bandwidth. Increasing the pulse energy could have been accomplished by elevating the pump power of the oscillator. Nonetheless, this approach would have also heightened the influence of SPM, which, based on our simulations, would result in spectral broadening and a reduction in pulse duration. As this method still met two of the three desired conditions, we conducted jitter measurements for OPP of 588, 612, and 636$\,\textrm {mW}$, maintaining a constant APP of 700$\,\textrm {mW}$. The integration interval remained consistent across all three measurements. Further augmentation of the OPP was unfeasible due to the constraints imposed by the pump diode. Since the measurements were conducted under similar but not yet optimized conditions, we show the results on Fig. 10(a) in arbitrary units (a.u.). Nevertheless, this presentation suffices to observe the anticipated outcome of reducing timing jitter through increased intracavity energy. The consistency of the BOC sensitivity within the margin of error can be attributed to the negligible impact of the slight increase in OPP on the power level after passing through the amplifiers. As a consequence, this leads to an identical amplitude of the BOC signal and the same sensitivity. The standard deviation of the BOC sensitivity was computed based on ten measurements of the BOC curve, with five taken prior to the measurement of timing jitter PSD and the remaining five taken afterward. This calculation was also taken into consideration when determining the error associated with the jitter value.
Fig. 10. Influence of (a) oscillator pump power and (b) the amplifier pump power on the timing jitter (
As a next step, the influence of the APP on timing jitter was investigated while maintaining a constant OPP at the optimal value of 636$\,\textrm {mW}$. The observed outcomes have been summarized in Table 2 and are visually depicted in Fig. 10(b). Analytical formulas do not consider the influence of amplifier pump power on timing jitter since, in cases where oscillator timing jitter is already high, the minor effect of the amplifier becomes negligible and cannot be independently observed. However, our measurements demonstrate that when oscillator timing jitter is low, the APP does play a discernible role. Increasing the amplifier pump current has a direct effect on the output power and consequently on the sensitivity of the BOC signal. Therefore, reducing sensitivity due to a higher signal amplitude should lead to a improvement in the accuracy of jitter measurement. Nonetheless Fig. 10(b) illustrates that an increase in APP corresponds to higher jitter values.
Table 2. Calculated timing jitter $\Delta \tau$ and BOC-sensitivity for different values of amplifier pump power (APP) at constant oscillator pump power (OPP) of 636$\,\textrm {mW}$.
The coherent amplification process in fiber amplifiers leads to the generation of spontaneous-emission noise [24,73], particularly given their composition of long erbium-doped fibers. However, the theoretical expressions describing the power spectral density of ASE presented previously cannot be directly applied to the case of fiber amplifiers due to the single-pass propagation. The amplifier noise tends to rise at higher pump powers [74], which accounts for the increased jitter values observed in our measurements for elevated APPs. The timing jitter PSD plots for all four conducted measurements are available in Supplement 1. Lowering the amplifier pump power led to the measurement data dropping to the background floor at an earlier point, thereby shortening the integration interval. Further reduction of the APP was not feasible due to the insufficient power for second harmonic generation in the nonlinear crystal. The most favorable outcome, recording a timing jitter of 14.25 as [10$\,\textrm {kHz}$ - 4$\,\textrm {MHz}$], was attained in this study with an OPP of 636$\,\textrm {mW}$ and an APP of 350$\,\textrm {mW}$. The corresponding measurement after the subtraction of the background is presented in Fig. 11.
Fig. 11. Measured PSD of the timing jitter and its integrated value displayed after the subtraction of the background shown in Fig. 9(b). Conducted with the OPP of 636$\,\textrm {mW}$, APP of 350$\,\textrm {mW}$, and a BOC sensitivity of 266.69 fs/V.
Figure 11 illustrates that beyond 4$\,\textrm {MHz}$ the power spectral density is primarily influenced by the background noise and/or sensitivity of the instrument. Consequently, it can be inferred that the PSD components above 4$\,\textrm {MHz}$ do not contribute to the integrated jitter and can be neglected.
5. Conclusion
The factors contributing to timing jitter were investigated, and a comparative analysis with the findings of other researchers was conducted to identify the most suitable parameter for intracavity dispersion. In this work we juxtaposed the outcomes of established theoretical models with our observations to determine the influence of oscillator pump power on the timing noise. Furthermore, we evaluated the impact of amplified spontaneous emission (ASE) during the amplification process on timing jitter.
Moreover, we performed calculations and generated plots for direct ASE-Jitter, Gordon-Haus-Jitter, and shot-noise, which demonstrated a good level of agreement between our experimental findings and the theoretical expectations. We have established through the analytical theory and experiments, that the directly induced timing jitter can be reduced through higher intracavity pulse energy. Concurrently, the noise coupled by the center frequency fluctuations can be lowered by choosing the intracavity dispersion to be slightly negative near zero. We achieved our lowest timing jitter value of 14.25 as integrated from 10$\,\textrm {kHz}$ to 4$\,\textrm {MHz}$ by increasing the pump power of the oscillators but reducing the pumping power of amplifiers, which were needed to generate second harmonic in the BOC setup. Despite the fact that decreasing the amplifier output had a negative effect on the sensitivity of the BOC system, it simultaneously diminished the influence of ASE. As a result timing jitter spectral density achieved such low values, that the measurement became limited through the shot noise and the background of the used balanced photodetector. Because of this reason, we subtracted the background from the measured timing jitter and performed the integration till the frequency value of 4$\,\textrm {MHz}$, at which measurement data hits the background floor.
Funding
Deutsche Forschungsgemeinschaft (EXC2089/1-390776260).
Disclosures
Natālija Rīgere, Andreas Brodschelm, and Rafał Wilk are employees of TOPTICA Photonics AG, a company developing high-end laser systems for scientific and industrial applications. The remaining author declares no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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