1. Introduction

Passively mode-locked quantum dot (QD) semiconductor lasers are promising short-pulse sources for time critical applications, such as ultra-fast optical sampling and frequency comb generation [1–3]. Their main limitation towards these targeted applications is their high pulse timing jitter [4, 5]. Besides optimization of the device design, the most efficient concepts that exist so far to reduce the pulse timing jitter are hybrid mode-locking [6, 7], dual-mode optical injection [8] and single-cavity optical feedback [1,9]. The latter has already been the subject of intensive research both experimentally [9–17] and by simulations [17,19–24]. From an analytic treatment the reduction of the timing jitter for resonant single cavity feedback can be directly related to the increase in the memory of the system, which results in the pulse positions being correlated over longer times [19]. Besides an improvement in timing jitter, single-cavity optical feedback induces unwanted noise peaks in the side-bands around the mode-locked repetition rate frequency in the power spectrum [14]. These noise induced resonances correspond to the round-trip frequency of the external cavity and represent increased fluctuations of the pulse timing at this frequency. Recently, in [18], the first numerical predictions of a dual-cavity optical feedback on the emission properties of a passively mode-locked semiconductor laser have been suggested. Initial experimental investigations to reduce these side-band signatures exist for harmonically mode-locked fiber lasers [25]. In [26] a passively mode-locked quantum well laser subject to optical feedback from two fixed-length external cavities has been investigated where the backreflected light has been re-injected into the absorber section. In these works it is already indicated that dual-feedback has a significant potential to reduce the timing jitter and to partially reduce the distinct noise-peaks in the power spectrum. Hereby, both cavities were adjusted manually to achieve lower noise resonances as compared to the single feedback condition. This approach could have neglected the most optimal adjustment for both cavities. Furthermore, important parameters including TJ and RR were not addressed.

Thus, in this work we demonstrate the change of dynamical properties including TJ, RR, amount of satellite pulses and side band suppression of a passively mode-locked semiconductor laser subject to optical feedback from two external cavities by tuning the full delay-phase of the pulse-to-pulse period of both external cavities separately for the first time for such a dual feedback configuration. We confirm with these wide range of experimentally accessible dynamical laser properties the numerical model presented in [18] by an excellent agreement of experimental results and simulations carried out specifically for the laser investigated here.

2. Experimental set-up for dual optical feedback and full delay-range tuning

The QD laser in the experiment is a monolithic two-section mode-locked laser with a total cavity length of 3 mm, corresponding to a repetition rate of 13.54 GHz. The absorber section is 0.3 mm long. The active region consists of 5 layers of InAs/GaAs QDs [28] embedded in a 440 nm GaAs waveguide surrounded with Al_0.35Ga_0.65As claddings. The ridge width is 4 μm. The gain section is pumped by a low noise DC current and a reverse bias voltage is applied to the absorber section. Both facets are cleaved and the emission wavelength is around λ = 1.25 μm. The developed experimental set-up is depicted in Fig. 1. The light emitted by the QD laser is collimated and directed by beam splitter BS1 towards a retro-reflecting mirror (HR1) mounted on a high-precision linear translation stage (TS1). The back-reflected light can be detected by a photo-detector (PD1) and power meter (PM) at BS1 and fine-adjusted by a variable neutral density filter (Att1). A shutter (Sh1) allows to turn off and on the optical feedback before each individual measurement to avoid any hysteresis of the laser dynamics, which is possible according to results reported in [20]. The second external cavity is realized equivalently. The microscopic lengths of both optical feedback sections can be adjusted independently of each other. The emission of the laser is directed through an optical isolator (isolation > 60 dB) to the diagnosis equipment light path to avoid possible back-reflections. A fiber-coupled semiconductor optical amplifier (SOA) is used to amplify the light. To identify the generation of optical pulses and multi-pulse or satellite pulse emission, the light is directed to an intensity auto-correlator. To investigate the mode-locking repetition rate and timing jitter a fast photodiode with 50 GHz bandwidth connected to an electrical spectrum analyzer (ESA) (9 kHz to 50 GHz bandwidth) is utilized. The experimental set-up is located inside of a closed metal box for thermal isolation and acoustic noise reduction. The lengths of the two investigated optical feedback cavities are approximately 3.18 m and 1.84 m corresponding to repetition rates of f₁ = 94.3 MHz and f₂ = 163 MHz, respectively. The corresponding round-trip times of the long and the short optical feedback cavities amount to approximately $\delta {\tau }_1^{\mathit{mac}.}=143.5\cdot T_0$ and $\delta {\tau }_2^{\mathit{mac}.}=83\cdot T_0$, respectively, with T₀ = 75 ps being the round-trip time of the monolithic mode-locked laser. The additional variable microscopic fine-delay of the optical feedback, $\delta {\tau }_1^{\mathit{mic}.}$ and $\delta {\tau }_2^{\mathit{mic}.}$, can be adjusted by the translation stages and is given in ps in the experiment. The total delay is a sum of the macroscopic and microscopic delay. The back-reflected light returning from two fine-delay cavities enters the laser emission facet. The feedback ratio, i.e., the ratio of optical output power to optical feedback power, amounts to about 5% for each cavity and does not include coupling losses. The laser is operated at a gain injection current density of J = 629.6 A/cm² and at an absorber voltage of U = −3.19 V, and is temperature stabilized at T = 23°C. As a measure of the timing jitter we use the long-term jitter, which we denote σ_lt,0 for the solitary laser and σ_lt for the laser subject to feedback, and which is experimentally calculated by

 

Fig. 1 Schematic of the experimental set-up for investigating the impact of dual optical feedback on the mode-locked QD laser. The optical feedback round-trip frequency amounts to 163 MHz and 94 MHz for the short and long delay case, respectively. (LD - laser diode, LDC - laser diode driver, TEC - temperature controller, U - absorber voltage source, BS - beam splitter, PM - power meter, Att - optical attenuator, Sh - shutter, HR - high reflective mirror, TS - translation stage, PD1 - photo-diode, OI - optical isolator, SMF - single-mode fiber, PC - polarization controller, SOA - semiconductor optical amplifier, BB - beam blocker, AC - auto-correlator, FPD - fast photo-diode, Amp - electrical amplifier, ESA -electrical spectrum analyzer).

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3. Experimental results and simulation validation

In the following, the laser repetition rate, phase noise spectra and auto-correlation time traces have been measured and will be discussed in the following. The long-term timing jitter of the free running laser amounts to σ_lt,0 = 4.4 fs and the pulse width amounts to 4.3 ps. The specific driving parameters of the laser (gain current, absorber voltage and temperature) are chosen to ensure that the strongest jitter suppression by optical feedback is accompanied by nearly zero repetition rate change. This is important to ensure highest reduction of timing jitter because otherwise a single longer cavity would change the repetition rate by a certain amount which would not be the same amount as caused by a single shorter cavity. No common repetition rate solution would be possible. An unchanged repetition rate is denoted as resonant feedback condition. The experimentally obtained fundamental repetition rate as a function of both optical feedback delays is depicted in Fig. 2(a). The resonant feedback condition is located around $\delta {\tau }_1^{\mathit{mic}.}=22.5\hspace{0.17em}\text{ps}$ and $\delta {\tau }_2^{\mathit{mic}.}=15\hspace{0.17em}\text{ps}$ indicated by the unchanged repetition rate of ν₀ = 13.54 GHz. We find that the repetition rate is strongly influenced by both delay values. A two-fold periodic structure can be observed. The periodicity in both delay directions, which are the horizontal or vertical profiles of the map, amounts to the repetition period of the free running mode-locked laser and amounts to T₀ = 75 ps. Several diagonal regimes of continuously changing repetition rate are evident. Between these regimes the repetition frequency changes abruptly from positive to negative deviations from the free running repetition rate, and vice versa. The tilt angle can be estimated by ϕ = arctan(163 MHz/94 MHz) = 60°. For two identical macroscopic optical feedback lengths the tilt angle would amount to 45°. We attribute the increase or decrease of the repetition rate to a leading or trailing re-injection of the optical feedback pulses into the laser cavity, which results in a speed-up or slow-down of the intra cavity-pulse. The observed tilt results from the larger impact of changes in the length of the short feedback cavity on the repetition rate as compared to the long feedback cavity [18]. To validate these findings by modeling, we use the model as given in [18]. Parameters used are γ = 0.93 ps⁻¹, κ = 0.1, T = 75 ps, γ_g = 1ns⁻¹, α_g = 0, D = 0, γ_q = 75ns⁻¹, α_q = 0, k = 0.5, J_g = 0.013 ps⁻¹, r_s = 25, K = 0.1, J_q = 0.1 ps⁻¹, ΔΩ = 0, C₁ and C₂ = 0. It is noted that our model does not take into account the sophisticated details of the carrier scattering dynamics in QD lasers [32] but uses only one equation for the carrier inversion, i.e., as usual for quantum-well lasers. This is justified by simulation results that depict that the dynamic effects induced by optical feedback do not crucially depend on the material model [24]. As in the experiment, we simulate a 3 mm long passively mode-locked two-section laser. The cold cavity round-trip time amounts to 75 ps, the resulting repetition rate period is T₀ = 77 ps and the pulse widths are of the order of 10 ps. For the feedback delay lengths we used values of τ₁ = 146T₀ + δτ₁ and τ₂ = 84T₀ + δτ₂. The feedback strength is approximately 5% for each cavity. The achieved dependence of the repetition rate on the lengths of two delay cavities by simulations is shown in Fig. 2(b). We can verify the two-fold periodical structure with a tilt angle of 60° as observed in the experiment and the difference of maximum and minimum repetition rate amounts to 290 MHz for both, underlining the good qualitative agreement between our simulation and the experiment. The shift of the repetition frequency along the diagonal regions is caused by the system adapting to non-resonant delay lengths [18, 20]. In both the experimental and numerical results the base repetition frequency remains approximately equal to 1/T₀. Despite this the dynamics can vary greatly within the depicted delay ranges. Within small ranges near the resonant feedback delay lengths (integer multiples of the pulse repetition period) self-locking occurs. Outside these self-locking regions satellite pulses can form, or the dynamics can become quasi-periodic or chaotic. To be able to distinguish between the self-locking regions with only one pulse in the laser cavity and the dynamical regimes with feedback induced satellite pulses, time traces or auto-correlation signals need to be measured.

 

Fig. 2 (a) Fundamental of the repetition rate as a function of the two microscopic delay times $\delta {\tau }_1^{\mathit{mic}.}$ and $\delta {\tau }_2^{\mathit{mic}.}$ for the dual cavity feedback configuration and fixed macroscopic delay times of $\delta {\tau }_1^{\mathit{mac}.}=(143.5\pm 3.5)\cdot T_0$ and $\delta {\tau }_1^{\mathit{mac}.}=(83\pm 3)\cdot T_0$. (b) Corresponding simulation results as a function of δτ₁ and δτ₂, with total delay times τ₁ = 145.8T₀ + δτ₁ and τ₂ = 83.9T₀ + δτ₂.

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Experimentally, partial information on the dynamical regimes was obtained by measuring the auto-correlation signal at zero delay and T₀/2 = 37.5 ps, which corresponds to the first harmonic repetition rate. The experimentally determined ratio of the auto-correlation amplitude at these two delay times is depicted in Fig. 3(a). A ratio of 1 indicates two equal pulses in the cavity (harmonic mode-locking) and values of more than ≈ 2 (light blue color) and above indicate a single pulse in the cavity (fundamental mode-locking). Values in between indicate satellite pulses. It can be seen that in a large region, fundamental mode-locking occurs. In the white regions, no auto-correlation signals could be detected. We attribute this to the weak auto-correlation signal and emission wavelength dependence of the laser on the feedback delay lengths. Close to the discontinuous repetition rate transition a region with satellite pulses and even harmonic mode-locking can be observed. This can be understood by the theoretically predicted bistability between harmonic and fundamental mode-locking at delay times that are offset from the main resonances by T₀/2. In the fundamental mode-locking regions satellite pulses with offsets other than T₀/2 could still be present. In the simulations the various dynamical regimes are identifiable from the time traces. Figure 3(b) shows a map of the number of pulses in the laser cavity for the feedback delay ranges corresponding to Fig. 2(b). The blue regions indicate the locking ranges where there is one pulse in the laser cavity. In the regions in white the pulse streams are irregular, exhibiting quasi-periodic or chaotic dynamics. Violet and green indicate two or three pulses within the cavity (i.e. one or two satellite pulses). The structure of Fig. 3(b) is slightly more complex as compared to the experimental results as the dynamical regimes are more fully characterized and not influenced by a limited sensitivity of the measurement device. However there is good qualitative agreement between the locking regions and two pulse regions. The simulation results suggest that three pulse dynamical regions (i.e. two satellite pulses) are also likely in the experiment. A comparison with Fig. 2(a) shows that fundamental mode-locking occurs along the smooth repetition rate transitions whereas the discontinuous repetition rate transitions are accompanied by satellite pulses.

 

Fig. 3 (a) Measured pulse amplitude ratio of main pulse and secondary pulse as a function of the two feedback delays $\delta {\tau }_1^{\mathit{mic}.}$ and $\delta {\tau }_2^{\mathit{mic}.}$. (b) Simulated number of pulses as a function of the two feedback delay tuning times δτ₁ and δτ₂, with total delay times τ₁ = 145.8T₀ + δτ₁ and τ₂ = 83.9T₀ + δτ₂. The color code indicates the number of pulses in the laser cavity and regions in white indicate irregular dynamics.

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To uncover the impact of dual optical feedback and full delay-range tuning on the timing jitter on a QD laser, the long-term timing jitter has been characterized experimentally and plotted in units of the long-term jitter of the free running laser σ_lt/σ_lt,0 in Fig. 4(a) as a function of both optical feedback delays. We find that dual-feedback increases the timing jitter for most delay settings. However, narrow areas exhibiting a timing jitter reduction (colored blue) exist and are separated by delays corresponding to the repetition period of the laser of 75 ps. These regions also exhibit a tilt angle of 60°, as observed in Fig. 1, and are located within the smooth repetition rate regions. These results indicate that a delicate tuning of both delays is required to maximize the jitter reduction. Furthermore, at delays of $\delta {\tau }_1^{\mathit{mic}.}=22\hspace{0.17em}\text{ps}$ and $\delta {\tau }_2^{\mathit{mic}.}=60\hspace{0.17em}\text{ps}$ a very small stable region is observed which corresponds to harmonic mode locking as evident in Fig. 3(a). The highest timing jitter reduction is generally observed around the resonant feedback condition ( $\delta {\tau }_1^{\mathit{mic}.}=22.5\hspace{0.17em}\text{ps}$ and $\delta {\tau }_2^{\mathit{mic}.}=15\hspace{0.17em}\text{ps}$) and along the tilted path. The highest reduction of timing jitter is obtained by optimizing the optical delay lengths to $\delta {\tau }_1^{\mathit{mic}.}=92\hspace{0.17em}\text{ps}$, $\delta {\tau }_2^{\mathit{mic}.}=80\hspace{0.17em}\text{ps}$. For these values the feedback induced noise resonances, i.e. the side bands, fall below the noise level of the ESA. In this case a jitter reduction from σ_lt = 4.4 fs to σ_lt = 1.1 fs by a factor of 4 is observed. The corresponding numerically calculated long-term timing jitter σ_lt is shown in Fig. 4(b). Here, the regions of low jitter correspond to the locking regions with one pulse in the laser cavity. Very high timing jitter regions are those with underlying quasi-periodic or chaotic dynamics. The free-running laser exhibits a non-stationary timing jitter which results in a Lorentzian RF line and a f⁻² timing phase noise power spectral density (free-running timing phase noise power spectral density in Fig. 5(a)). The main source for semiconductor lasers is the direct projection of amplified spontaneous emission on the pulse timing and mainly depends on three parameters: Pulse width, pulse energy and round-trip gain [33]. Timing jitter could be reduced by reducing pulse width, increasing pulse energy or reducing output losses. It is difficult to reduce the source of timing jitter but the non-stationarity can be influenced by feedback. Applying ideal feedback conditions of both delays results in strongly reduced timing phase noise power spectral density and a reduced diffusion of the timing jitter statistics (ideal timing phase noise power spectral density in Fig. 5(a)). Hereby it is noted that the noise source and thus the pulse-to-pulse jitter is hardly altered by the feedback. Applying non-ideal feedback conditions even leads to increased timing phase noise power spectral density.

 

Fig. 4 (a) Measured timing jitter ratio σ_lt/σ_lt,0 as a function of the two optical delay times $\delta {\tau }_1^{\mathit{mic}.}$ and $\delta {\tau }_2^{\mathit{mic}.}$. (b) Corresponding simulation results as a function of the delay times δτ₁ and δτ₂, with total delay times τ₁ = 145.8T₀ + δτ₁ and τ₂ = 83.9T₀ + δτ₂.

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Fig. 5 (a) Selected measured timing phase noise power spectral density spectra for a strong (blue), a non-ideal (grey) timing jitter reduction by the dual-cavity feedback and as obtained at the free-running mode-locked laser (red) (delays: τ₁ = 92 ps, τ₂ = 80 ps, as in Fig. 2(a)). (b) Side-band suppression in the RF domain as a function of the two optical delay times for delay times $\delta {\tau }_1^{\mathit{mic}.}$ and $\delta {\tau }_2^{\mathit{mic}.}$. Color: (Experiment) side-band reduction below the noise floor, white: non-vanishing side-bands.

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As suggested by simulations in [18], in the following the suppression of noise-induced power spectrum side-band peaks is experimentally studied. Already in [26] a suppression of the long-cavity-induced side-band frequency by 38 dB had been reported in Fig. 2 (b,d), whereas less suppression of short cavity induced noise was achieved and taking into account that a completely different experiment had been considered. In our first results, we observe a complete reduction of all long and short cavity induced noise sidebands below the noise floor. These experimentally observed regions of a complete reduction of the side-band signatures are depicted in Fig. 5(b). Colored white are the regimes where the side-band signals are suppressed below the noise floor of the spectrum analyzer. In these regimes, the peak amplitude of the fundamental mode-locking frequency ranges between 25 and 40 dB which indicates the side-peak suppression strength. Colored are the regions that still exhibit side-bands and the power of the side bands is given. We observe that the areas of side-band suppression coincide with the ones of maximum timing jitter reduction as indicated in Fig. 4(a). These results indicate that a delicate adjustment of both microscopic delays is required to ensure highest side-band suppression (as indicated in [26]).

Our findings allow the statement that the overall dynamics of a mode-locked laser subject to optical feedback critically depends on the delay times of the external optical cavities, supporting the modeling predictions in [20, 21] and as experimentally indicated in [27, Fig. 5].

4. Conclusion and outlook

We have experimentally demonstrated the change and improvement of important dynamical properties of a mode-locked laser subject to dual optical feedback within the full fine-delay parameter tuning range for the first time. The dynamical laser observables studied are the repetition rate, the timing jitter, the existence of satellite pulses and the side-band suppression in dependence on the both independently-tuned microscopic fine-delays of a dual cavity feedback configuration for the whole relavant microscopic delay-range. We find complex regimes of repetition rate deviation and timing jitter reduction that exhibit a two-fold periodic dependence on the two feedback delay lengths. The strongest timing jitter reduction in the experiment is found at optical feedback lengths that yield the lowest repetition rate frequency deviation and along a diagonal delay-parameter path representing a delicate balance of both delay values. Within these regions there is only one pulse in the laser cavity, i.e. regions of fundamental mode-locking. This is in agreement with the condition for optimum timing jitter reduction in the case of single-cavity feedback as reported in literature. We find that a careful adjustment of both cavities is required to ensure the highest jitter reduction and sideband suppression. By experiment we identify new dynamical regime in quantum dot passively mode-locked lasers exhibiting satellite pulses that exhibit strongly reduced jitter which is shown in experiment and simulation. Overall, the experiment allows us to confirm the predicted dynamical regimes in predicted in [18]. The periodic dependencies of the measured dynamical observables are reproduced by numerical simulations based on a delay differential equation model with both excellent quantitative and qualitative agreement. The presented results could lead to further improvements of the timing jitter reduction schemes, for example by utilizing a macroscopic distributed-feedback scheme. They could form the basis towards photonically enhanced passively mode-locked semiconductor lasers with low timing jitter, controllable repetition rate and without additional noise induced side-bands.