## Abstract

In this paper the influence of different feedback (FB) and synchronization schemes on the timing phase noise (TPN) power spectral density (PSD) of a quantum-dot based passively mode-locked laser (MLL) is studied numerically and by experiments. The range of investigated schemes cover hybrid mode-locking, an opto-electrical feedback configuration, an all-optical feedback configuration and optical pulse train injection configuration by means of a master MLL. The mechanism responsible for TPN PSD reduction in the case of FB is identified for the first time for monolithic passively MLL and relies on the effective interaction of the timing of the intra-cavity pulse and the time-delayed FB pulse or FB modulation together with an statistical averaging of the independent timing deviations of both. This mechanism is quantified by means of simulation results obtained by introducing an universal and versatile simple time-domain model.

© 2013 OSA

## 1. Introduction

Ultrafast mode-locked laser (MLL) oscillators are very capable low-noise optical and microwave signal sources. In particular solid-state and fiber MLL with pulse widths in the femtosecond range offer ultra-low timing jitter (TJ) [1] enabling demanding applications including ultra-low sampling jitter photonic analog to digital converters [2]. Semiconductor based monolithic passively MLL operating at high repetition rates (RR) and with pulse widths in the picosecond and sub-picosecond range are promising sources for optical signal processing applications [3] offering advantages including compact size, simple fabrication and the ability for hybrid integration to silicon substrates. Hereby, quantum-dot (QD) based MLL offer reduced spontaneous emission rates and low threshold current density which lead to reduced noise [4]. However, the high TJ of the optical pulse-train is the main drawback of such monolithic semiconductor passively MLL. It results from a non-stationary origin but is in particular highly dependent on pulse width, pulse energy and repetition rate having been the subject of theoretical investigations in [5–7]. The frequency-domain distribution of this TJ is represented by the corresponding timing phase noise (TPN) power spectral density (PSD) commonly used in literature. To circumvent the negative issue of the high TJ monolithic semiconductor passively MLL, different schemes for TJ reduction have been proposed. These cover synchronization to an electric reference oscillator by direct electric modulation [8] or by exploitation of a phase-locked loop [9, 10]. An alternative approach was demonstrated by means of an all-optical feedback (AO FB) configuration in [11–13] also theoretically investigated in [14, 15].

In this paper we jointly study the influence of different FB and synchronization schemes on the TPN PSD of a monolithic semiconductor passively MLL in experiment and by simulations. The investigated schemes include hybrid mode-locking (HML), an opto-electrical feedback (OE FB) configuration [16], an all-optical feedback (AO FB) configuration and optical pulse train injection (OPTI) by a master MLL. The hereby utilized simple time-domain model which is based on a random-walk process is capable to reproduce all the measured TPN PSD for all investigated configurations in good agreement and allows to explain the observed effects of the realized feedback and synchronization scenarios for the first time. By comparing the obtained results from the different configurations a comprehensive insight and intuitive understanding of the TJ for monolithic semiconductor passively MLL and the various effects of the controlling techniques is given. The presented experimental scenarios are finally joined in one simple TJ description.

## 2. Devices and experimental setups

The passively mode-locked laser (MLL) under investigation is a monolithic multi-section quantum-dot (QD) based semiconductor laser consisting of 10 InGaAs QD layers separated by GaAs barriers with a total cavity length of 8 mm resulting in a repetition rate (RR) of 5.06 GHz. The waveguide consists of an absorber section and a gain section. The gain section is pumped by a DC current and a reverse bias voltage is applied to the absorber section. The output facet is anti-reflection coated with a reflectivity of 5% whereas the back facet is high-reflection coated with a reflectivity of 95%. For all experiments identical driving conditions of the MLL were maintained resulting in an average output power of 2.5 mW, a pulse width of 4 ps and an output pulse energy of 0.49 pJ. According to the dependence of quantum limited timing jitter (TJ) on pulse width, pulse energy and repetition rate [17, 18] a high value of TJ is expected for the investigated MLL and amounts to an pulse-to-pulse TJ of 152 fs.

To study and compare the timing jitter (TJ) characteristics we implement four experimental schemes including hybrid mode-locking (HML), ML subject to opto-electrical feedback (OE FB), ML subject to all-optic feedback (AO FB), and ML subject to optical pulse train injection (OPTI). Here we used the term OPTI to emphasize the role of the timing interaction of the pulses of master and slave MLL and to distinguish from continuous-wave injection.

HML is commonly implemented by modulating the absorber with a frequency corresponding to the RR of the MLL by means of a bias-T and a frequency synthesizer as shown in the upper part of Fig. 1(c). Here, no impedance matching measures of the MLL and the electrical system were taken. To implement the OE FB shown in Fig. 1(a), a part of the output beam of the MLL is coupled into a fast photo-detector via a long optical fiber, is amplified, delayed and subsequently applied to the absorber section by means of a bias-T. The total round-trip delay length of the OE FB amounts to 28.1 m (in vacuum) which corresponds to a frequency of 10.7 MHz. The bandwidth of the electrical system covers only the RR of the MLL. The AO FB configuration is implemented as shown in Fig. 1(b). There, a part of the output beam is coupled into a long fiber, reflected back by means of a mirror, polarization aligned, and redirected into the MLL. The round-trip length of the AO FB external-cavity amounts to 31.8 m (in vacuum) corresponding to a frequency of 9.4 MHz. The optical FB ratio is defined as the inverse ratio of output power to the returning optical power excluding coupling losses. The OPTI setup is depicted in Fig. 1(c). The output beam of the master MLL is coupled into a fiber, polarization aligned, variably attenuated and coupled into the slave MLL. The master MLL is identical to the slave MLL in terms of specifications, however it does not exhibit sufficiently similar RR and is therefore hybridly mode-locked to match the RR of the slave MLL.

Estimation of timing phase noise (TPN) power spectral density (PSD) and TJ bases on a well-known method developed for actively MLL [19]. The power spectrum *P*(*f*) of the optical output power of the MLL is measured by means of an optical high-frequency detector and a high-frequency electrical spectrum analyzer (ESA) and evaluated properly. This method is well applicable to MLL exhibiting a significant amount of TJ. However, ultra-low noise MLL require a different method, which is not limited by the noise of the electrical components [20]. According to [19] and assuming only timing fluctuations the (single-sided) TPN PSD *L*(*f*) is then given by

*ν*

_{0}being the frequency of the RR line,

*P*the total integrated power of the RR line and

_{tot}*RBW*the resolution bandwidth used. It is evident, that

*L*(

*f*) is a normalized power spectrum

*P*(

*f*). For estimation of

*P*the peak power obtained using a RBW much wider than the RR line-width could also be used. This integrated

_{tot}*P*corresponds to the so called carrier power of an actively driven oscillating system. In the measurements, the total TPN PSD is composed of multiple aligned PSD each covering one order of frequency range. The final PSD is obtained by linearly averaging 10 measured PSD with the detection mode of the ESA set to sample-mode. The common integrated TJ (

_{tot}*σ*) can be obtained from the TPN PSD

_{int}*L*(

*f*) by

*f*

_{1}and

*f*

_{2}being the desired integration bandwidth limits.

However, this TPN PSD estimation method is only partially valid for passively MLL as pointed out by [5, 6]. The so measured TPN PSD has a finite value at zero Hz, however an infinite value is expected because for low frequencies which correspond to long time-scales a free-running oscillator tends to accumulate unbounded timing deviation [6]. Intuitively, the RR line-width represents a measure of the TJ for passively MLL. However, calculating *σ _{int}* for a frequency offset from zero to half of the RR yields a constant value which is independent of the RR line-width thus indicating the limitation of Eq. (1) for passively MLL. Therefore a tailored TPN PSD and pulse-to-pulse TJ (

*σ*) estimation method was proposed specifically addressing free-running passively MLL [21]. There the TPN PSD

_{ptp}*L*(

*f*) and the pulse to pulse TJ

*σ*is obtained by

_{ptp}*ν*being the RR line-width. Based on this relation it is emphasized, that for passively MLL the

*σ*and RR line-width are equivalent and a fully sufficient measure for TJ if no additional instabilities are existent. In other words,

_{ptp}*σ*is, besides the RR, the only necessary universal parameter to describe the TJ of a stably running and undisturbed passively MLL. Both specifications

_{ptp}*σ*as well as

_{ptp}*σ*are root-mean-square (RMS) based. It should be noted that a stably running passively MLL exhibits a Lorentzian shaped RR line. Figure 2 shows calculated TPN PSD for various RR line-widths. Plots (a) are obtained using Eq. (1) and a Lorentzian shaped RR line and plots (b) using (3). As plots (b) represent the correct TPN PSD for a passively MLL it can be seen that the evaluation based on Eq. (1) is only valid for frequency offsets well above the RR line-width. It is essential to be aware of this limitation when performing TPN PSD measurements of passively MLL, as calculating

_{int}*σ*for a frequency integration range starting below the RR-line width would lead to a significant underestimation of the TJ.

_{int}## 3. Model description

In the following a simple model will be presented which addresses the dynamics of TJ subject to time-delayed FB and synchronization. It is meant to provide a simplified insight into the mechanisms governing the external control of TJ. Up to now, several full-featured or specialized models were developed covering the dynamics of monolithic MLL [22–24] and the effects of AO FB on MLL were numerically studied in [14, 15, 25]. Here, we provide a simple unified model for all the presented different configurations which allows to reproduce the experimentally obtained TPN PSD.

The basic idea is to consider a MLL as a free-running oscillator subject to uncorrelated timing noise, namely the optical pulse oscillating in the laser cavity and exhibiting timing deviations within each round trip. The main source of timing noise is spontaneous emission directly coupling to the optical pulse [5, 6, 26, 27]. This spontaneous emission is uncorrelated and thus white in frequency. After each round-trip the timing deviation is accumulated leading to a Wiener process or a random walk. Numerically [7, 24] or analytically [6] applying such a series of accumulated timing deviations to an ideal equidistant pulse train directly leads to a RR mode comb, Lorentzian shaped RR lines and the well-known dependence of TPN PSD on the squared harmonic-number of the investigated RR line in the case of absence of amplitude jitter as experimentally shown in [12]. This dependence is valid for all kinds of timing noise or frequency modulation.

A convenient form of controlling TJ is possible by HML. It affects the timing of the intra-cavity pulse directly by modulation of the absorber section [26] with a frequency corresponding to the RR. This modulation may be viewed as a continuous gating of the intra-cavity pulse. Also for the OE FB configuration the optically detected pulse train is transformed to an electrical signal which in turn modulates the absorber section. This time-delayed self-modulation influences the timing of the intra-cavity pulse. It therefore represents an interaction of the timing of intra-cavity pulse and delayed FB pulse. Based on this an interaction of the timing of intra-cavity pulse and the returning FB pulse for AO FB configuration is assumed and could be explained by means of the MLL gain dynamics. However, another possibility would be in an interaction based on interference [28]. Such timing interaction is expected to lead to an averaged timing per round-trip which was proposed in [11].

The timing dynamics of the MLL pulse are implemented by a numerical finite-difference approach [29] with the time interval corresponding to the inverse RR of the MLL. The timing of a free running MLL is described by a discretized random walk process with white noise being the dominant noise source. HML is incorporated by an additional linearized timing-restoring term [30], the AO or OE FB is incorporated by a weighted time-delayed term and the OPTI is implemented by an independent and weighted Wiener process as additional source of timing noise. The following phenomenological equations describe the discretized time evolution of the timing deviations *T*(*n*) of the optical pulses with respect to the inverse RR. HML is described by Eq. (5), EO and AO FB by Eq. (6) and OPTI by Eq. (7).

*σ _{ptp}* denotes directly the pulse-to-pulse TJ value and is also the direct representation of the noise source strength, Γ is a normally distributed normalized noise term,

*γ*denotes the timing interaction strength,

*n*denotes the discretized time delay of the FB in terms of number of pulses,

_{d}*T*

_{2}(

*n*) denote the absolute timing deviation of an independent master MLL and

*W*(Δ

*T*) is a weighting function. The weighting function is a Gaussian function with a FWHM of Δ

*τ*for the AO FB to account for the assumption that pulses cannot interact if their timing deviation Δ

*T*is larger then their pulse width. For OE FB

*W*(Δ

*T*) is set to 1 due to the large period of the electrical modulation frequency corresponding to the inverse RR. The denominator is obligatory for the timing averaging effect. As a consequence of the finite-difference approach, integer matching of RR and inverse external-delay time is always fulfilled leading to a constant and unchanged RR. Therefore the RR-snapping effect [11] can not be modeled.

These equations are solved numerically and iteratively. As FB is switched on at *t* = 0*s* evaluation is performed after a sufficient waiting time to guarantee a relaxation of the system. The resulting TJ time-series is Fourier transformed by a fast Fourier transform algorithm (FFT) using a Hanning window and squared to obtain the timing-noise power spectrum *L*_{Δt} (*f*). In general, the term power spectrum denotes the squared Fourier-transform of a quantity which is also the case for the experimentally measured ESA (power) spectrum *P*(*f*). The TPN PSD *L*(*f*) can then be obtained from the timing-noise power spectrum by *L*(*f*) = (2*πν*_{0})2*L*_{Δt} (*f*) [24, eq. (2)]. It is essential to use a windowing function for the FFT. A properly normalized Hanning window has proven to provide reasonable results. In addition, as a FFT of a non-stationary or non-periodic signal is performed, namely a random walk process, the obtained power spectrum must be scaled by a factor proportional to the length of the time-series to obtain correct spectral amplitudes. In our case the scaling factor for the power spectrum is *N*/(*ν*_{0}*π*) with *N* being the number of the pulses in the time-series. The presented calculated spectra are obtained by averaging the spectra of 10 simulation runs each containing a series of 6.4 · 10^{5} pulses. A noise floor was added to the calculated spectra to account for the noise floor of the whole experimental setup.

## 4. Results and discussion

First we show the influence of external electrical modulation on the timing phase noise (TPN) power spectral density (PSD) of the free-running mode-locked laser (MLL) which represents hybrid mode-locking (HML). Then this external oscillator is substituted by the time delayed repetition rate (RR) signal of the MLL itself leading to an opto-electric feedback (OE FB) configuration. Then we switch from this OE FB to all-optical feedback (AO FB) and show the correspondence of both FB schemes in terms of TPN PSD reduction. Finally this optical self-injection is changed to external optical pulse train injection by a master MLL leading to the effect of TJ locking.

For the following investigations a monolithic MLL with a total cavity length of 8 mm is used leading to a relatively low RR of 5.06 GHz to allow for an effective electrical modulation. For all experiments the applied reverse bias voltage of the absorber amounts to 6.00 V and the gain current is set to 165.4 mA. The measured optical pulse-width is 4.0 ps, the spectral width is 3.4 nm and the free-running RR line-width is 19 kHz which corresponds to a *σ _{ptp}* of 152 fs. On the one hand these biasing conditions are intentionally chosen to ensure a local minimum of the TJ but on the other hand still to provide a relatively high TPN to be not limited by the setup noise floor. In particular it is essential to ensure that the MLL does not exhibit non-regular mode-locking operation or any sort of instabilities.

A common approach to influence TJ of a semiconductor MLL is HML [8]. Here the absorber section is modulated electrically with a fixed low-noise frequency signal matching the RR of the free-running MLL thus effectively providing a continuous gating of the pulse timing in the absorber section. Experimental and corresponding simulated TPN PSD for the hybridly MLL are shown in Fig. 3 for different modulation powers *P _{mod}* and corresponding timing interaction strengths

*γ*, respectively. In experiment, already for the weakest applied modulation power of 12.2 dBm a RR locking is achieved together with a broad-band TPN PSD reduction up to a frequency of around 100 kHz. With increasing modulation power the TPN PSD plateau decreases thus also extending to higher frequencies of up to around 1 MHz for a modulation power of 31.2 dBm. Above this frequency the free-running TPN PSD is unchanged. Therefore HML significantly reduces timing fluctuations on slow time-scales but due to the wide electric gating window which generally amounts to multiples of the optical pulse-width the high-frequency TPN PSD is not affected. In accordance with [31] we also find a negatively linear dependence of TPN plateau level on modulation power. It is noteworthy, that identical TPN characteristics can also be achieved by utilizing a phase-locked-loop configuration [10].

By properly selecting the timing interaction strength *γ* in the simulations based on Eq. (5) the numerically obtained TPN PSD results accord with the experimental results in the whole frequency range. A comparison of modulation power *P _{mod}* in experiment and corresponding timing interaction strength

*γ*in the simulations is shown in Fig. 4 and yields a square-root dependence of timing interaction strength

*γ*on modulation power. The exponent of 0.5 is expected to be valid for HML in general while the scaling factor of 9.6 · 10

^{−4}results from the realized electrical setup. This estimated dependence is necessary in the following to obtain the correct timing interaction strength

*γ*for an applied modulation power

*P*in the OE FB configuration simulation.

_{mod}Besides exploiting the HML configuration for TJ reduction and RR locking also a corresponding all-optical dual-mode injection approach is established. In [28,32] two optical modes are injected into a passively MLL. These modes are locked by means of a fixed modulation frequency which matches the RR of the passively MLL thus allowing to transfer this fixed modulation frequency onto the optical mode comb of the passively MLL. This approach may be regarded as optical HML.

To realize the idea of statistical interaction of the timing of the optical pulses now the OE FB configuration is implemented. The electrical modulation of the absorber represents a continuous time domain gating of the timing of the intra-cavity pulse. In relation to HML we now replace the reference oscillator with the time-delayed RR signal of the laser output itself. The electrical delay is tuned to match the free running RR of the MLL. The experimental and corresponding simulated TPN PSD for the MLL in OE FB configuration are shown in Fig. 5 for different modulation powers *P _{mod}* and corresponding timing interaction strengths

*γ*, respectively. In the experiment a broadband TPN reduction of 5 dB is achieved up to a frequency of around 3 MHz for the maximum applied modulation power of 31.2 dBm. At around 6 MHz and its multiples weak and broad peaks are evident which originate from the OE FB delay-time. These peaks are noise resonances which are related to super-mode noise peaks in harmonically mode-locked external cavity lasers [29]. Concluding the observations, the OE FB configuration presented here is able to successfully reduce the TPN without the use of a stable reference oscillator. In comparison to HML the TPN PSD reduction of the OE FB configuration is inferior at low frequencies below 800 kHz as expected but surprisingly the reduction at higher frequencies between 800 kHz and 4.5 MHz is superior. This OE FB scheme shows a dependence of TPN PSD on frequency following a

*f*

^{−2}law at frequencies below the noise resonance peak frequency which indicates that this configuration is still subject to a random walk.

The simulation results based on Eq. (6) with a constant weighting function *W*(Δ*T*) = 1 and using the estimated parameters from the HML simulation [Fig. 4] coincide with the experimental results very well except for the significantly lower noise resonance peaks. A time-domain representation of these noise peaks will be given later. Due to the good accordance between experiment and simulation the assumption of a timing interaction of intra-cavity pulse and time-delayed absorption gating is validated. Reduction of TPN PSD or TJ can be attributed to an statistical averaging of the independent timing deviations of the intra-cavity pulse and the time-delayed absorption gating leading to a reduced TJ. [11]

Now this timing interaction in the OE FB configuration which is based on absorption gating by delayed electrical self-modulation is transferred to the all-optical domain. For the AO FB configuration also an timing interaction of intra-cavity pulse and the returning time-delayed pulse is assumed and could be explained by means of the MLL gain dynamics. For example a weak injected Gaussian shaped FB pulse preceding the main intra-cavity Gaussian shaped pulse experiences higher gain than the main pulse due to the non-linearity of the dynamical gain and thus increases in intensity. This changes the center of mass of the total optical pulse forwards in time. Finally after a few roundtrips the gain and absorber sections reshape this pulse back to a stable Gaussian shape while keeping the changed timing. Despite the all-optical interaction a limited timing interaction range which amounts to the optical pulse width is assumed and is represented by the weighting term *W*(Δ*T*). However, another possible explanation for the timing interaction could base on interference effects.

Such an AO FB configuration is widely exploited to reduce TPN PSD and RR line-with of passively MLL, by means of reflecting a small fraction of the optical output power back into the MLL [11, 12, 33]. Here a combined free-space and fiber based setup is chosen to allow for a convenient and precise control of FB ratio and delay length. It was found that to obtain the highest TPN reduction the external delay length had to be slightly detuned resulting in a change of +0.1 MHz of the RR with respect to the free-running RR. Experimental and simulated TPN PSD for the MLL in the realized AO FB configuration are shown in Fig. 6 for different FB ratios and corresponding timing interaction strengths *γ*, respectively. In the experiment a broadband TPN reduction of 15 dB is achieved up to a frequency of around 3 MHz for the maximum applied FB ratio of 1.6 · 10^{−3}. At 8.7 MHz and its harmonics distinct noise resonance peaks are evident which originate from the AO FB delay with a frequency of 9.4 MHz. For a high FB ratio of 1.6 · 10^{−3} the frequency of these observed narrow resonance peaks corresponds quite well to the delay length whereas for a weak FB ratio of 4 · 10^{−4} these peaks are located at a lower frequency of 8.4 MHz and are strongly broadened. In addition it was found that the dependence of the spectral amplitude of the resonance peaks on harmonic number of the RR lines shows a square-law thus proving that these peaks are pure timing fluctuations [19]. At very high FB ratios instabilities or a coherence collapse is expected [34].

To estimate the correct values of the timing interaction strengths *γ* for this AO FB configuration this parameter is varied to obtain a matching of the TPN PSD in the low frequency regime up to 1 MHz while keeping the full-width at half maximum Δ*τ* of the Gaussian shaped timing weighting function *W*(Δ*τ*) in Eq. (6) at 4 ps corresponding to the measured Gaussian shaped optical pulse-width of 4 ps. This variation is performed for FB ratios higher than 1 · 10^{−4}. As can be seen in Fig. 6 the simulation results coincide with the experimental results very well except for the significantly lower noise resonance peaks which is more clearly shown in Fig. 7. In addition, it is found that the dependence of *γ* on FB ratio yields a simple power-law with an exponent of around *a* = +0.45. This found relation represents the connection of model and experiment.

A detailed depiction of the first noise resonance peak for a FB ratio of 1.6 · 10^{−3} in experiment and two corresponding simulated TPN PSD is shown in Fig. 7. In the presented simulation results two combinations of timing interaction strengths *γ* and interaction widths Δ*τ* are chosen which both match the TPN PSD reduction up to 3 MHz. The parameter combination *γ* = 3.0 · 10^{−2} and Δ*τ* = 4 ps match the measured noise resonance peak frequency of 8.7 MHz exactly and the amplitude much better than the parameter combination *γ* = 4.8 · 10^{−3} and Δ*τ* = **∞**. This better accordance implies that a timing interaction width is indeed necessary to reproduce well the experimentally obtained results and substantiates the chosen timing-interaction approach. The deviation of measured and simulated noise resonance frequency of 8.7 MHz from the realized delay-frequency of 9.4 MHz is a direct consequence of the timing-mechanism because with increasing timing interaction strength *γ* in simulation or increasing FB ratio in experiment the resonance-peak frequency converges towards the realized delay-frequency. Also like the OE FB scheme this AO FB scheme shows a dependence of TPN PSD on frequency following a *f*^{−2} law at frequencies below the resonance peak frequency which indicates that this configuration is still subject to a random walk. From the overall accordance of TPN PSD and resonance-peak frequency in experiment and simulation it can be deduced that the assumption of an optical timing interaction between intra-cavity pulse and returning time-delayed FB pulse is valid similar to the OE FB case. In is emphasized, that the FB pulses still carry the timing noise generated inside the cavity and are still able to reduce the TJ of the intra-cavity pulses. To depict the efficiency of the AO FB it is pointed out that the returning delayed pulse with a power of much less (due to unknown coupling losses) then 1.6 · 10^{−3} times the intra-cavity pulse power influences the timing of the intra-cavity pulse by significant 3% per round-trip which is more than one order of magnitude higher than the FB ratio of 1.6 · 10^{−3} itself.

To show the quantitative dependence of TPN PSD reduction on optical FB ratio Fig. 8 shows the TPN PSD at 120 kHz as a function FB ratio in experiment and simulation. Based on Eq. (3) the TPN PSD is proportional to the RR line-width. Therefore, the presented dependence applies for the RR line-with in the same way. In the experiment, with increasing FB ratio the TPN PSD is steadily reducing and above a FB ratio of 1 · 10^{−4} the TPN PSD should be sufficiently spaced from the free running TPN PSD to allow for meaningful evaluation. Hereby a power-law-fit describing the dependence of TPN PSD on optical FB ratio yields an exponent of *a* = −0.90 which also corresponds to reported experimental results covering AO FB in [35, Fig. 4(b)]. The simulated TPN PSD corresponds to the measured data quite well supporting the following found dependence of *γ* on FB ratio:

Combing these both dependencies leads to the dependence of TPN PSD on *γ* which yields a power-law with an exponent of *a* = −2.1 which matches quite well the expected exponent of *a* = 2 which directly arises from theory. With the found relations and assuming a *γ* of 1 which corresponds to an interaction strength of 100%, a corresponding FB ratio of 4.3 is found by means of Eq. (8). An interaction strength *γ* of 1 should correspond to a FB ratio of 100%. As coupling losses are not included in the given FB ratio the inverse of 4.3 amounting to 23% should reflect these coupling losses. This value of 23% lies in an experimentally reasonable range when laser to fiber coupling or the other way round is regarded which is the case for the realized setup.

To further emphasize the time-domain based picture of TJ, in the following the AO FB or in other words optical self-injection scheme is extended towards an external optical pulse train injection by means of a master and a slave MLL. Such an injection approach was qualitatively investigated in [36] and was applied in [37] to reduce the optical pulse-width. Here, a hybridly MLL is used as the master MLL, to force the RR of the master MLL onto the RR of the slave MLL and to be able to clearly see a potential TPN PSD reduction due to the completely different characteristics of the TPN PSD of master and slave MLL. Biasing conditions of the master MLL are adjusted to allow for a spectral overlap with the slave MLL to ensure the possibility of direct optical interaction. The master MLL is operated at a temperature of 14.6 °C, a reverse bias voltage of 4.40 V, a gain current of 84.5 mA and a modulation power of 16 dBm. The *σ _{ptp}* of the free running master MLL amounts to 46 fs corresponding to a RR line-width of 1.7 kHz. In Fig. 9 the measured and corresponding calculated TPN PSD of the slave MLL in OPTI configuration are shown for different injection ratios and corresponding timing interaction strengths

*γ*, respectively. Injection ratio is defined as the ratio of master MLL input power to slave MLL output power. In the experiment, at a low injection ratio of 1.5 · 10

^{−3}a broad band reduction of TPN PSD in the low frequency region is observed with a reduction of 5 dB up to a frequency of 1 MHz. At an increased injection ratio of 3.0 · 10

^{−3}the TPN PSD reduction is significantly higher with a reduction in the range of 20 dB up to a frequency of around 200 kHz and at higher frequencies with a reduction in the range of 5 dB. Finally at an injection ratios above 3.0 · 10

^{−3}the TPN PSD of the slave MLL completely matches the TPN PSD of the master MLL thus successfully demonstrating optical locking of the timing jitter. In addition, OPTI represents an experimentally convenient way to optically transfer and distribute the TJ characteristics from one MLL with low TJ to multiple slave MLL.

In the simulations the master TPN PSD is characterized by a *σ _{ptp}* of 46 fs and a

*γ*of 3 · 10

^{−4}for HML operation. The timing interaction width Δ

*τ*is kept at 4 ps. The parameter

*γ*was varied to obtain best possible matching of TPN PSD in simulation and experiment. In general the accordance of experimental and simulated TPN PSD is good and the TJ locking effect can be well reproduced. However, clear differences are evident in the frequency range above 300 kHz which may originate in the difference of performance of the two MLL. First the master and slave MLL have a different spectral width of 0.5 nm and 3.4 nm respectively and second the pulse width of the master MLL could not be estimated due to non-sufficient output power for nonlinear auto-correlation. Nevertheless simulations reproduce the discontinuous transition from TPN PSD reduction to complete TJ locking above an interaction strength of

*γ*= 3.55 · 10

^{−2}thus extending the validity of the timing-interaction approach.

To substantiate the correspondence of the all-optical AO FB and OPTI Fig. 10 shows the interaction strength *γ* as a function of the corresponding FB ratio or input ratio of the AO FB or OPTI configuration, respectively. Both optical configurations yield almost identical dependency of *γ* and input ratio thus validating the proposed TJ interaction also in the OPTI case.

The TPN PSD can be obtained conveniently in experiment and contains much information about the characteristics of the TJ. The direct time-domain representation of the TJ experimentally investigated in [38] allows for a complementary insight into the different presented mechanisms as well as clarification of the different quantities used for the description of the TJ. A calculated time-domain representation of the absolute timing deviation of the MLL with respect to an ideal oscillator is shown in Fig. 11 for a single simulation run. There the absolute timing deviation of a pulse is shown as a function of the pulse number for the free running MLL with a *σ _{ptp}* of 152 fs, for the AO FB configuration with a FB ratio of 3 · 10

^{−2}(

*γ*= 3.0·10

^{−2}), for the OE FB configuration with a modulation power of 31.2 dBm (

*γ*= 1.10·10

^{−3}) and HML with a modulation power of 31.2 dBm (

*γ*= 1.10 · 10

^{−3}). It can be seen that for of the free-running MLL the timing deviation performs a random walk which may lead to an unlimited accumulation of timing deviation. This is a direct consequence of the accumulation of Gaussian distributed pulse-to-pulse timing deviations driven by direct emission of spontaneous emission onto the optical pulse per round-trip. In the AO FB configuration the walk is significantly reduced on this selected time-scale and in addition a particular periodicity is evident with a period of

*n*= 538 pulses. This periodicity corresponds to the resonance peaks shown in the TPN PSD in Fig. 6 and represents a recurring noise-pattern which is partially preserved and even enhanced by the external delay. This external delay therefore represents an effective timing noise memory of the system. Hints of this periodicity can also be observed in the OE FB case. For the HML operation the timing deviation is also reduced and tends to be located around zero. Calculating the mean of the timing difference of consecutive pulses yields the expected

_{d}*σ*for the passively MLL with a value of around 152 fs. But also for all the other presented schemes the

_{ptp}*σ*amounts to a value around 152 fs. This is clear because from the presented model the noise source (

_{ptp}*σ*), namely the spontaneous emission, is hardly influenced by the timing interaction strength

_{ptp}*γ*which for example for the HML operation is three orders of magnitude smaller (

*γ*= 1.10·10

^{−3}) than the noise source (

*σ*) itself. As a consequence of the unchanged noise source the TPN PSD is unchanged with respect to the free-running case at very high frequencies above 1 MHz which is most clearly seen for HML in Fig. 3. Even for AO FB the TPN PSD is not reduced at very high frequencies above 10 MHz as seen in Fig. 6. To conclude, the

_{ptp}*σ*represents solely the unchanged noise-source-strength of the MLL and the TPN PSD represents in addition the important statistical and spectral distribution of TJ. It should be noted that significant TPN PSD reduction and the occurrence of TPN PSD resonances can only be achieved for ratios of FB length to MLL cavity length higher than 1 · 10

_{ptp}^{2}due to the required long timing memory of the delay line. Therefore too small ratios used in literature to study FB effects may not reveal statistical timing interaction effects.

For a random walk process it is known, that the dependence of the standard deviation *σ* on pulse number *n* follows a square-root law with
$\sigma =\beta \sqrt{n}$ with *β* describing the RMS timing deviation per pulse of a random walk system. The standard deviation *σ* obtained from 100 simulation runs is visualized in Fig. 12 for the free running MLL, for the AO FB configuration, for the OE FB configuration and for HML. For the free running MLL the law is fulfilled yielding a *β* of 144 fs close to the expected value of 152 fs which corresponds to *σ _{ptp}*. For the AO FB and OE FB configuration the standard deviation

*σ*also follows the square-root law therefore both FB schemes still represent a random walk. Hereby a reduced

*β*of 37 fs and 102 fs is obtained for the AO FB and OE FB configuration, respectively which correspond well to the values of 36 fs and 94 fs obtained by utilizing Eq. (4), which is only valid for free running MLL, and the RR line-width while excluding the resonance peaks. For both FB schemes

*β*or the RR line-width reflect a measure of the low-frequency fluctuations but disregard the high-frequency timing noise contributions. Finally, the HML operation exhibits a bounded timing deviation as expected thus not representing a random walk process.

As *σ _{ptp}*, representing the timing noise source strength, is unchanged in all investigated configurations a different strategy has to be pursued to target this timing noise source directly. In [39] it was found, that in a passively MLL a net gain window may exist outside the optical pulse position, either preceding or succeeding the main optical pulse. This net gain window amplifies spontaneous emission thus leading to significantly increased noise. In [40] it was suggested that by means of an AO FB configuration the non-coherent noise preceding the optical pulse can be compensated by a sufficiently strong coherent FB pulse realized by detuning the external FB cavity to a slightly higher frequency. It was also found that such fractional external to internal cavity ratio reduces also amplitude noise [15]. A reduction of the noise source strength, therefore a lower

*σ*, would be clearly identifiable by a reduced high frequency TPN, which is not observed in the AO FB experimental results presented here. In addition in the OPTI configuration which has shown TPN PSD reduction the required detuning is not possible in principle because the slave MLL pulse-train is locked to the timing-phase and RR of the master MLL pulse-train. Therefore, it can be deduced that this mechanism of noise source strength reduction can be excluded for the investigated MLL and the chosen biasing conditions.

_{ptp}The presented explanation and simple model may also be easily extended towards different schemes for example a dual AO FB configuration which is demonstrated in [35]. A schematic of the experimental setup is shown in Fig. 13. There, these two different optical delays, which amount to a round-trip frequency of 3.2 MHz and 9.6 MHz, are reinjected at the back facet of the MLL. The free-running MLL exhibits a RR of 19.7 GHz, a *σ _{ptp}* of 56 fs and a FB pulse width of 2.1 ps. Output power is 1.26 mW, pulse Energy is 64 fJ and FB power is 5 uW (without coupling losses) resulting in a total FB ratio of 4 · 10

^{−3}corresponding to a

*γ*of 4.5 · 10

^{−2}obtained by Eq. (8) whereby an identical coupling efficiency as compared to the measurements presented here has been assumed. For dual AO FB a

*γ*of 4.05 · 10

^{−2}and a

*γ*of 0.45 · 10

^{−2}are assumed for the long and short FB cavity, respectively. In our simulations a timing interaction width of 2.1 ps corresponding to the FB pulse width was used. Hereby, the experimental TPN PSD reduction [35, Fig. 2(b)] can be reproduced using the given parameters and Eq. (8) for the single and dual AO FB configuration despite the significantly higher resonance peaks in simulation shown in Fig. 14. However, the characteristic resonance peak suppression pattern in the dual AO FB configuration is evident in simulations showing the actual benefit of dual AO FB as compared to single AO FB. Based on our model, we attribute this reduction of noise resonances of the dual AO FB configuration to a mean destructive statistical averaging of the timing deviations of the returning noise stemming from the two different FB delay lengths. In addition, we found in simulations that longer FB delays provide a higher TPN reduction or a narrower RR line-width which is not shown here and is also found experimentally in [35, Fig. 4(a)].

## 5. Conclusion

In conclusion the influence of different feedback (FB) and synchronization schemes on the timing phase noise (TPN) power spectral density (PSD) of a monolithic semiconductor passively mode-locked laser (MLL) is studied in experiments and by simulations. The mechanism responsible for timing jitter (TJ) or TPN PSD reduction in the case feedback (FB) is identified and quantified by means of simulation results. This mechanism relies on the effective interaction of the timing of the intra-cavity pulse and the time-delayed FB pulse or FB modulation together with an statistical averaging of the independent timing deviations of both. This interaction is also found to result in the observed locking behavior for the optical pulse train injection (OPTI) configuration. For a stably running MLL the TJ reduction mechanism can be approximately regarded as an isolated process independent of the intrinsic laser dynamics. The presented simple model describes the pulse-to-pulse timing-deviations which are subject to time-delayed FB or timing-synchronization. In general simulated and experimental results are in excellent agreement regarding the TPN PSD reduction and characteristics. However, in the simulations the TPN PSD resonances, related to supermode-noise in literature, are significantly lower as compared to experimental results. It is emphasized that the obtained findings apply to monolithic semiconductor passively MLL exhibiting comparable parameters including pulse width and pulse energy.

In particular, the found description is applicable to the different investigated external TJ control configurations which exhibit peculiar TPN PSD characteristics. Figure 15 shows the calculated TPN PSD for the investigated configurations. The well known HML configuration [Fig. 15(a)] shows the highly efficient limitation of TPN PSD in the low frequency regime. The EO FB [Fig. 15(b)] and the AO FB configuration [Fig. 15(c)] reduces the TPN PSD in the low frequency regime or equivalently the repetition rate (RR) line-width. The hereby observed TPN PSD resonances originate from the timing-noise memory of the FB delay. In particular it was found, that TPN PSD reduction by FB is an all-statistical effect keeping the timing-noise source strength, namely the spontaneous emission noise, unaltered. The timing-interaction is supported by changing this AO FB, which may also be referred to as optical self-injection, to external optical injection by means of a master MLL. Hereby a RR and discontinuous TPN PSD locking was observed [Fig. 15(d)]. Extending the simulations to a dual AO FB scheme [35] a reproduction of the TPN PSD reduction and the noise-resonances suppression characteristics is achieved thus showing the universal potential of the presented explanation and model for semiconductor based monolithic passively MLL. This modeling approach complemented by experiments allows to study numerically the dependence of TPN PSD on various experimentally controllable parameters including FB delay-length and FB strength at various intrinsic pulse-to-pulse TJ values for a monolithic passively MLL with comparable parameters.

## Acknowledgment

This work was supported by the Project FASTDOT, an Integrated Program funded by the European Commission through the 7th ICT-Framework Program, under Grant 224338. The authors gratefully acknowledge the supply of quantum-dot lasers by I. Krestnikov of Innolume GmbH.

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