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Abstract
We report the first demonstration of excitability in an all-fiber laser. We present clear experimental evidence in this passively $Q$-switched device with gain and absorber sections for defining properties of excitability, including a threshold-based excitable response and a decreasing reaction delay between input pulse and excitatory response with increasing perturbation amplitude. Our experimental results are consistent with the homoclinic/saddle-loop mechanism that forms the basis of excitability as demonstrated by a study of the Yamada model; they pave the way toward novel and reliable all-fiber architectures for neuro-inspired photonic memory and computing applications.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
The quest for novel alternatives to traditional computing is currently the subject of intense research. One of the most exciting and promising directions is the use of nonlinear systems as the building blocks of neural networks designed to outperform traditional approaches in areas such as language recognition and image processing [1–3]. The current focus of research into neural networks is mainly numerical, but a move toward hardware implementation is crucial for fully exploiting the possibilities. Electronic implementations provide promising platforms to create artificial neural networks [4–7], although they are subject to a fundamental bandwidth-fanout product limit [8]. Optics and photonics, on the other hand, are not limited in the same way and have the added advantage of providing much higher computing speeds. As such, photonics-based neural networks, configured to perform at time scales that are orders of magnitudes faster than their biological and electronic counterparts, have the potential to revolutionize the way we perform computations [9–11].
This work explores a fundamental ingredient for neuro-inspired information processing: the property of excitability. More specifically, we demonstrate the potential of an all-fiber laser with a gain and an absorber section for the excitability-based on-demand generation of optical spikes. Excitability refers to the ability of a system in a rest state to admit two different responses to a single incoming perturbation: (i) for perturbations smaller than a given threshold, the system generates a linear response, generally of small amplitude, proportional to the perturbation, and (ii) for perturbations above this threshold, the system generates an excitable response, generally of large amplitude and effectively independent of any other attributes of the perturbation itself. In both cases, after the response, the system relaxes back to its initial rest state. In optical systems, an excitable response is generally observed as the emission of a single or multiple optical pulses or spikes, which corresponds to a well-defined trajectory in the corresponding phase space [12–15].
Several studies, both experimental and numerical, have demonstrated excitability in different laser systems, including lasers with optical feedback [12,16] and with injected signal [13,17,18]. In particular, it has been shown that the inclusion of a saturable absorber in the cavity leads to excitable behavior when pumping the laser just below its pulsing threshold [2,14,19,20]. Approaches based on potentially integrable components such as semiconductor lasers [21–25], sometimes with polarization effects [26–28], silicon microrings [29], micropillars with integrated saturable absorbers [30,31], and resonant tunneling diodes [32,33] have all been considered.
Two-dimensional (2D) materials, such as graphene, have garnered much interest because of their unique optical [34] as well as electronic and atomic characteristics and performance [35–37]. They have been employed in several applications [38] and, very recently, also as passive saturable absorbers [34,39]. A particular advantage compared to semiconductor saturable absorber mirrors (SESAMs) [38] is that 2D materials come as an extensively broad material family with widely varying morphology and thus bring great flexibility, especially regarding fiber integration [38,40].
More recently, excitability was demonstrated in a $Q$-switched fiber ring laser with graphene as a saturable absorber [8,37]. Using this laser, and leveraging the special configuration described in [14], Shastri et al. were able to demonstrate low-level spike-processing abilities that are critical for higher-level processing such as temporal pattern detection and stable recurrent memory. The use of the 2D material graphene evidently means that this is not an all-fiber configuration. Taking inspiration from Shastri et al., our initial studies on $Q$-switching used graphene as a saturable absorber. However, we soon realized the damage threshold of graphene was low; it burned out easily after long sessions of experiments. This made the experimental setup not very robust, and in light of the difficulty of handling and realigning new graphene saturable absorbers in our cavity, a thulium-doped fiber was adopted. Solid-state saturable-absorber $Q$-switched (SAQS) fibers are able to hold enormous gain generated in the gain section, and their high damage threshold facilitates the generation of high-power $Q$-switched pulses [41].
We provide here the first demonstration to our knowledge of excitability in an all-fiber laser, consisting of a gain section fused to an absorber section in a cavity formed by two fiber Bragg gratings. We present measurements of key properties of excitability, clearly showing that the laser is operating in the excitable regime. This is in qualitative agreement with a numerical analysis of the Yamada model; moreover, our findings are consistent with other experimental studies of excitability in laser systems [20,42,43].
2. EXPERIMENTAL RESULTS
The experimental setup of the laser system is depicted in Fig. 1; it is similar to that presented in Ref. [44]. The laser is a linear Fabry–Pérot cavity made up of two sections of differently doped fibers, namely, a gain section of 0.9 m of erbium-doped fiber and a saturable absorber section of 1.48 m of thulium-doped fiber. The cavity is bounded by two fiber Bragg gratings: a high-reflectivity grating that reflects 89% of the 1.5 µm lasing signal and a 0.1 nm wide low-reflectivity grating that reflects 23% of the lasing signal. The Fabry–Pérot cavity is optically pumped by a 980 nm semiconductor laser diode with power denoted by ${P_{{\rm in}}}$. The Bragg gratings are transparent to the 980 nm pump signal.
Fig. 1. Experimental schematic for demonstrating excitability of the fiber laser, consisting of 0.9 m of erbium-doped fiber, pumped by a 980 nm pump diode, spliced to ${L_B} = 1.48\,\,\rm m$ of thulium-doped fiber; the high- and low-reflecting fiber Bragg gratings have reflectivities of 89% and 23%, respectively. The perturbation ${\rm P}[t]$ is generated by modulating a CW semiconductor laser diode.
To perturb the system, a signal is generated from a second 980 nm CW semiconductor laser diode. The intensity of the perturbation is modulated with an iXblue NIR-MX-LN-10 intensity Mach–Zehnder modulator (MZM) to provide the pulsed perturbation signal ${\rm P}[t]$ with amplitude AP. An arbitrary waveform generator (AWG) is used to drive the modulator, allowing control over excitation/perturbaton pulse parameters. After the MZM, 10% of the perturbation signal is tapped off for monitoring while the remainder is coupled into the laser cavity. One meter long patch cords connect the lasing signal and the perturbation signal to the scope for monitoring. The optical signals are monitored with Thorlabs detectors, model DET01CFC, with responsivities of 0.94 A/W (0.047 V/mW) connected to a 500 MHz LeCroy 650A waverunner oscilloscope acquiring 8 million points at $200\,\,{\rm MS.{\rm s}^{- 1}}$. Using an average power meter, we calibrated both the pump diode and the response of the photodetector. To this end, the peak powers can be obtained from the average powers and the duty cycle.
A. Identifying the Excitability Threshold
There are two independent parameters for exploring excitability: the primary input pump ${P_{{\rm in}}}$ provided to the gain section and the perturbation signal ${\rm P}[t]$. We know from our earlier work in [44,45] that the excitable regime is located close to but below the lasing/pulsing threshold, which was experimentally found to be at 65.75 mW. We therefore set ${P_{{\rm in}}} = 50\;{\rm mW} $ for all experiments reported here so that the laser is in the off state. We set the perturbation ${\rm P}[t]$ generated by the AWG to consist of square pulses with a rise time of 90 ns, a variable amplitude of AP of duration 0.3 ms, and a repetition rate of 43 Hz. Although the background signal level is calibrated at the beginning of every experiment, there is a small CW present in ${\rm P}[t]$ that causes a negligible off-centering around zero.
Figure 2 shows evidence of an excitable response. For amplitudes of perturbation (AP) up to 80 mW, the output of the laser remains negligible, meaning that the laser remains effectively in the off state in spite of receiving input perturbations; the typical laser response to a single input pulse in this entire range is illustrated in Fig. 2(a) for ${\rm AP} = 75\,{\rm mW}$. For AP above 80 mW, one finds output pulses in reaction to input perturbation pulses. An example is illustrated in Fig. 2(b) for ${\rm AP} = 90\;{\rm mW} $, where there is a high-intensity narrow pulse (of about 0.63 µs) approximately 0.5 ms after the start of the perturbation pulse. In other words, for a sufficiently high perturbation amplitude the perturbation pulse is strong enough to perturb the laser from its off state across the excitability threshold, resulting in an excursion in phase space that generates the output pulse before the laser returns to its off state. Overall, Fig. 2 demonstrates the existence of an excitability threshold of the system, above which an output pulse can be triggered.
Fig. 2. Laser output (blue) in response to a perturbation pulse ${\rm P}[t]$ (orange) (a) in the sub-threshold regime for an amplitude of perturbation AP of 75 mW, where no excitable response is recorded, and (b) above threshold for AP of 90 mW with an excitable response of amplitude 6.8 W, which is several orders of magnitude larger than the corresponding input perturbation amplitude; see also the insets and note the two different scales.
B. Repeated Perturbations and Noise Effects
To obtain a better understanding of the nature of the excitability threshold and the system dynamics near it, we now consider the consistency of the pulse response as a function of the amplitude of perturbation AP. More specifically, we gradually increase AP from 50 mW and record for each value the output of the system to 1000 perturbations.
Figure 3 shows the response of the system to just eight successive perturbation pulses for different values of AP. As is shown in panel (a) for ${\rm AP} = 78\;{\rm mW} $, below the excitability threshold, no excitable responses occur. For ${\rm AP} = 85\;{\rm mW} $ as in panel (b), output pulses may be generated, but this happens only two out of the eight times. As AP is increased to 91 mW as in Fig. 3(c), four output pulses are triggered in the shown time window of eight input pulses. Finally, when AP is sufficiently large, as in panel (d) for ${\rm AP} = 99\;{\rm mW} $, each input perturbation pulse triggers an output excitable pulse. Note from Fig. 3 that the amplitude of the excited response remains effectively the same when AP is kept constant or is increased; this is a key characteristic of excitable systems and essential for using such excitable lasers for computational purposes.
Fig. 3. Time traces showing pulsed perturbation signals (orange) with their corresponding excitable responses (teal) to eight input pulses with amplitude of perturbation AP of (a) 78 mW, (b) 85 mW, (c) 91 mW, and (d) 99 mW.
For each given value of the perturbation amplitude AP, the efficiency of response can be quantified by the response rate $\eta$, defined as the ratio of excitable response events to perturbation events. Determining $\eta$ experimentally from the counts of recorded output signals for different values of AP yields the plot shown in Fig. 4. Clearly, $\eta = 0$ for ${\rm AP} \le 80\;{\rm mW} $, while for ${\rm AP} = 82\;{\rm mW} $ and above output pulses are triggered. Initially the response rate $\eta$ is quite small, and it increases and reaches 1 at ${\rm AP} = 95\;{\rm mW} $. Figure 4 shows that there is a range of AP, between 80 and 95 mW, of increasing efficiency of triggering pulses, rather than a sharp threshold. This is explained by the presence of noise in the system: the closer the perturbation is to the excitability threshold, the more likely noise is to result in triggering an output pulse. Similar observations have been obtained on other excitable systems such as those in [18,43]; the slope of the response rate depends on the shape of the excitability threshold and hence how sensitive the system is to noise. Indeed, for most cases of ${\rm AP} \le 80\;{\rm mW} $, the noise level is insufficient to trigger a pulse, while for ${\rm AP} \ge 95\;{\rm mW} $ an output pulse is always triggered. Note that the response rate at ${\rm AP} \ge 91\;{\rm mW} $ is higher than that at the next data point; this may be a consequence of a net-positive of noise factors that may have nullified perturbations to the system, thus resulting in fewer registered excitable responses. Also note that $\eta$ is actually slightly above 1 for some values of AP, which we determined to be due to occasional and also noise-generated additional pulses in reaction to a single input pulse. Noise factors in the form of imperfect modulator performance, thermal effects, fluctuations in the laser gain medium, or simply from transferred intensity noise from a pump source would contribute to the generation of additional pulses.
Fig. 4. Response rate $\eta$ as a function of AP, determined experimentally from 1000 perturbation events for each recorded value of AP.
C. Excitable Response Delay
While the peak intensity of the excitable response changes only very slightly with the amplitude of the perturbation, the timing of the response is definitely affected by AP. To demonstrate this we show in Fig. 5, overlayed and synchronized to the triggering pulse, the time traces of reactions of the laser to a single input pulse for values of AP varying between 100 and 350 mW and ${P_{{\rm in}}} = 53\;{\rm mW} $. There is clearly a response delay, defined as the time between the input pulse and the output pulse; it decreases as AP is increased. This is illustrated further in Fig. 6, where we show the average or mean response delay for each of the chosen values of AP from the range 100 to 160 mW.
Fig. 5. Time traces in a $600\,\unicode{x00B5}\rm s$ window, synchronized to the triggering pulse, illustrating the response delay between a perturbation event and the corresponding excitable response; the associated values of AP are as given by the color bar.
Fig. 6. Mean response delay between a perturbation, and the associated excitable response as a function of the perturbation amplitude AP.
The monotonously decreasing nature of the response delay in Figs. 5 and 6 is characteristic of an excitability threshold that is formed by the stable manifold of a saddle-type steady state near the off state [14,44]. We will now proceed by presenting numerical evidence for this interpretation, which also explains the observed independence of the response amplitude with changing AP.
3. MODEL CHARACTERIZATION OF THE EXCITABLE REGIME
We model the fiber laser system with a modified Yamada model for a passively mode-locked $Q$-switched laser [46–48], which we recently adapted to take into account different relaxation times for the gain and absorber sections [44,45]. In particular, Ref. [45] considers in detail the influence of the ratio of these two relaxation times on the observable $Q$-switched dynamics. We remark that the Yamada model exclusively describes $Q$-switching in a laser; considering the transition from $Q$-switching to mode locking requires a more complicated model, such as the one studied in [49–51].
The Yamada model describes the evolution of gain $G$, absorption $Q$, and optical intensity $I$ of a single-mode laser with a passive saturable absorber, and it is considered here with an external perturbation ${\rm P}[t]$ and a noise term ${\rm noise}[t]$; it can be written in dimensionless form as
Here the parameter $A$ is the pumping strength, ${\gamma _G}$ is the ratio of the photon and carrier lifetimes in the gain section, $B$ is the absorption coefficient, and $a$ is the relative absorption versus gain parameter. The parameter $\sigma$ encodes the different decay times of gain and absorber media and hence their different recovery times. For our system, we determined the values
To model the experimental situation, the perturbation ${\rm P}[t]$ is applied to the gain $G$ and taken to consist of square pulses of height AP and width 2.8750. The noise term ${\rm noise}[t]$ on $I$ models the effects of spontaneous emission noise, and it is generated as the absolute value of a white Gaussian noise signal with a mean noise level of 0.0006; in particular, ${\rm noise}[t]$ breaks the invariance in Eq. (1) of the plane defined by $I = 0$ that contains the off state, so that a perturbation in $G$ is able to trigger a pulse.
Figure 7 shows that the modified Yamada model Eq. (1) features the experimentally observed transition to excitability as the (rescaled) amplitude AP of the perturbation pulses is increased from 0.23 to 0.275, showing no reaction to input pulses for ${\rm AP} = 0.23$, then an increasing response rate, and finally a response pulse for every input perturbation pulse for ${\rm AP} = 0.275$. Indeed, the gradually increasing efficiency of pulse response to input perturbations demonstrates the characteristics of an excitable threshold in the presence of noise, in good qualitative agreement with the experimental equivalent in Fig. 3. This regime is found for a pump level of $A = 6.00$, which occurs in Eq. (1) just below the threshold $A = 6.8$ given by a transcritical bifurcation [45,48].
A. Geometry of the Excitability Threshold
The excitable behavior can be understood by considering the phase portrait of the Yamada model Eq. (1) in the absence of external perturbation and noise term. Figure 8 shows the relevant region in $(G,Q,I)$ space near the stable off state $o$ (with $I = 0$) and the nearby saddle steady state $p$ (with $I \gt 0$). The crucial object here is the two-dimensional stable manifold ${W^s}(p)$, which is formed by the trajectories that end up at the saddle $p$. This surface is invariant under the flow (without perturbation and noise terms) and divides the phase space locally into two regions: points below and to the right of the surface in Fig. 8 follow the lower branch of the one-dimensional unstable manifold ${W^s}(p)$ of $p$ directly to the off state $o$; on the other hand, points above and to the left of the surface follow the upper branch of ${W^s}(p)$ to well outside of the shown region, which corresponds to a spike or pulse in the intensity $I$, and then come back to the off state $o$ from the right in Fig. 8. This geometry identifies the surface ${W^s}(p)$ as the excitability threshold. Any additional perturbation that allows the system to cross ${W^s}(p)$ will lead to a response pulse, while for perturbations that do not cross ${W^s}(p)$ the trajectory relaxes straight back to the off state without a pulse being generated. This is illustrated in Fig. 8 with three trajectories: ${{\rm T}_1}$ and ${{\rm T}_2}$ stay below ${W^s}(p)$ and do not lead to a pulse, and ${{\rm T}_3}$ crosses ${W^s}(p)$ and hence features a large excursion (the peak value of $I$ is several thousand) well beyond the $I$ range of Fig. 8. In the presence of the noise term, this geometry leads to a range of AP with increasing response rate of system Eq. (1) to pulsed inputs as illustrated in Fig. 7.
Fig. 7. Computed time traces of Eq. (1) with $A = 6.00$ showing pulsed perturbation signals (orange) with their corresponding excitable responses (teal) to eight input pulses with amplitude of perturbation AP of (a) 0.23, (b) 0.255, (c) 0.257, (d) 0.263, and (e) 0.275; compare with Fig. 3.
Fig. 8. Phase portrait in $(G,Q,I)$ space of the Yamada model Eq. (1), without external perturbation and noise term, in the excitable regime for $A = 6.00$. Shown is the off state $o$ (blue diamond) the saddle steady state $p$ (red diamond) with its unstable manifold ${W^u}(p)$ (red curve) and its stable manifold ${W^s}(p)$ (blue surface). Also shown are trajectories after a perturbation from $o$, namely, ${{\rm T}_1}$ with ${\rm AP} = 0.1$, ${{\rm T}_2}$ with ${\rm AP} = 0.2$, and ${{\rm T}_3}$ with ${\rm AP} = 0.3$. Note that ${{\rm T}_3}$ follows ${W^u}(p)$ in phase space well beyond the shown region to return back to $o$.
Fig. 9. Computed time traces of Eq. (1) for $A = 6.00$, synchronized to a triggering pulse, illustrate the decreasing response delay as AP is increased from 0.27 to 0.35 in steps of 0.01; compare with Fig. 5.
Fig. 10. Response delay of Eq. (1) for $A = 6.00$ between perturbation event and excitable response as a function of the amplitude AP of the perturbation pulse; compare with Fig. 6.
B. Characterization of Excitatory Response Delay
The existence of the saddle $p$ with its stable manifold ${W^s}(p)$ forming the excitability threshold in phase space also provides an explanation for the observed decreasing response delay for increasing amplitude of perturbation. For AP just above the excitability threshold ${W^s}(p)$, the perturbed trajectory is very close to this stable manifold and hence approaches the saddle steady state $p$ closely before leaving it the direction of the unstable manifold ${W^u}(p)$ to generate the response pulse. For increasingly larger values of AP, on the other hand, the perturbed trajectory is increasingly above ${W^s}(p)$ and, as a result, does approach the saddle $p$ less and less closely. Since the closeness of the passage near a saddle directly translates to the time spent near such a point, this explains the decreasing response delay with the perturbation amplitude AP.
Figure 9 illustrates, with plots of computed output pulses synchronized to a triggering pulse, that this geometry of system Eq. (1) indeed induces the observed reduction of the reaction delay with increasing AP. The reaction delay itself is shown in Fig. 10 as a function of AP, where it can be seen to decrease monotonically. These computations for the Yamada model Eq. (1) with noise term show very good qualitative agreement with the experiment; compare with Figs. 5 and 6, respectively.
The study presented here does not include the temporal integration effect in the Yamada model; its further investigation remains an interesting direction for future work.
4. CONCLUSIONS
We have presented experimental and modeling evidence to characterize the excitable properties of a passively $Q$-switched all-fiber laser with regards to the system’s response to external perturbations. The laser system is perturbed with a secondary pump diode that generates pulses of varying amplitude and fixed pulse width. Within this framework, defining properties of excitability have been demonstrated for this fiber laser system. It shows an all-or-none response to a perturbation, which is a strong indicator of excitability. Owing to the unavoidable presence of noise in the experiment, this leads to a monotonous increase of the response rate from 0 to 1 over a well-defined range of perturbation amplitudes. Moreover, we showed a characteristic decrease in the delay of the response of the fiber laser as a function of the amplitude of the triggering perturbation pulse. The good qualitative agreement between modeling and experiment supports the observation that the homoclinic/saddle-loop mechanism [14,45,48] forms the basis of excitability in this system, where the excitability threshold is given by the stable manifold of a nearby saddle steady state. Overall, this work demonstrates the suitability and potential of excitable all-fiber lasers, when feedback mechanisms are included and considered, as elements for future neuro-inspired memory storage and information processing applications. In this context, it would also be of fundamental interest to consider other types of saturable absorbers, both fiber-based and non-fiber-based, including electro-mechanical devices and SESAMs.
Acknowledgment
We thank Stéphane Barland for helpful discussions and encouragement.
Disclosures
The authors declare 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.
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