Electrically Stimulated and Frequency-Tunable Photonic Tonic Spiking Neuron Based on a DFB-LD Under Optical Feedback

Abstract

Based on a distributed feedback laser diode (DFB-LD) under optical feedback, a novel scheme for generating neuron-like tonic spiking is proposed, and the characteristics of the generated neuron-like tonic spiking are numerically investigated. Firstly, through adopting the Lang–Kobayashi model to analyze the nonlinear dynamics of the DFB-LD under optical feedback, the switching between different dynamic states is observed by continuously increasing the biased current of the DFB-LD, and the current regions required for driving the DFB-LD into the stable states and period one (P1) states are determined. Next, a rectangular electrical pulse is introduced as a stimulus signal to modulate the DFB-LD, and the lower and upper current values of the rectangular electrical pulse are set at the regions in which the DFB-LD operates at the stable state and P1 state, respectively. Under suitable operation parameters, sub-nanosecond tonic spiking can be generated. Finally, through adjusting the delayed time of optical feedback and selecting the matched rectangular electrical pulse, the frequency of tonic spiking can be detuned within a range exceeding 5 GHz.

1. Introduction

Artificial intelligence (AI), inspired by brain-like biological mechanisms, aims to optimize energy efficiency and enhance parallel processing capabilities in computational systems for handling complex tasks. In order to realize such purposes, artificial neural networks (ANNs), by emulating biological neural dynamics, have evolved into a cornerstone of modern AI. However, existing ANNs based on programmable computer systems face some intrinsic limitations in energy efficiency and parallel scalability. Hence, there is an urgent need to develop neuromorphic hardware systems that can physically emulate neuronal dynamics [1,2].

A photonic neuromorphic system is a promising alternative scheme that possesses some unique advantages, such as ultra-fast speed, low crosstalk, high energy efficiency, and massive parallelism [3]. As the fundamental components in photonic neuromorphic systems, photonic spiking neurons have attracted significant research attention in recent years [4,5]. Tonic spiking, a key phenomenon in biological neural dynamics, has been successfully generated in photonic spiking neurons based on various laser systems. For instance, vertical-cavity surface-emitting lasers (VCSELs) under optical injection and VCSELs with saturable absorbers (VCSEL-SA) under optical injection have been utilized for generating tonic spiking [6,7,8,9,10,11,12,13,14,15]. Similarly, Fabry–Perot lasers with saturable absorbers (FP-SA) have also demonstrated the feasibility of generating tonic spiking [16,17]. In addition, micro-ring resonators [18,19], integrated dual-stage distributed feedback (DFB) lasers with balanced photodetectors [20,21,22], and photopumped micro-pillar lasers with saturable absorbers [23,24] have also been adopted to generate tonic spiking. However, the research mentioned above relies almost entirely on external optical injection to stimulate tonic spiking, which not only needs multiple light sources but also limits the frequency tunable range of tonic spiking.

In this work, based on a distributed feedback laser diode (DFB-LD) under optical feedback, we propose a scheme for generating neuron-like tonic spiking. For such a scheme, a rectangular electrical pulse is adopted to stimulate the tonic spiking. The simulation results demonstrate that under suitable operation parameters, the tonic spiking can be achieved, and its frequency can be tuned within several GHz.

2. Methods and Theory

Figure 1 gives the schematic diagram of the photonic tonic spiking neuron based on a DFB-LD under optical feedback, where the dashed and solid lines represent the electrical and optical paths, respectively. The DFB-LD is driven by a bias current, I_bias, and a rectangular electrical pulse through a Bias Tee. The output of the DFB-LD is directly injected into port 2 of an optical circulator (OC), and the output from port 3 of the OC is divided into two parts by a fiber coupler (FC). One part is used to detect, and the other part is fed back to the DFB-LD after passing through an optical fiber delay line (OFDL), a variable attenuator (VOA), and the OC. The VOA and OFDL are utilized to adjust the feedback strength K_f and feedback delay time τ.

Based on the Lang–Kobayashi model [25], the carrier density N(t) and slowly varying field amplitude E(t) in the DFB-LD under optical feedback can be described as follows:

$${\displaystyle \frac{dE}{dt}}={\displaystyle \frac{1}{2}}\left(1+i\alpha \right)\left(\Gamma v_{g}g\left(N\right)-{\displaystyle \frac{1}{{\tau }_p}}\right)E\left(t\right)+K_{\mathrm{f}}E(t-\tau )e^{-i\omega \tau }$$

(1)

$${\displaystyle \frac{dN}{dt}}={\displaystyle \frac{I\left(t\right)}{e{V}_{act}}}-R(N)-{\displaystyle \frac{v_{g}g\left(N\right){\left|E\left(t\right)\right|}^2}{1+ϵ{\left|E\left(t\right)\right|}^2}}$$

(2)

where α is the linewidth enhancement factor; Γ is the optical confinement factor; v_g is the group velocity; τ_p is the photon lifetime; ω is the angular oscillation frequency of the laser; I(t) is the time-varying current; e is the electron charge; V_act is the volume of the active region; R(N) is the carrier recombination rate, and ε is the nonlinear gain coefficient. g(N) is the material gain, which is described as

$$g\left(N\right)={\displaystyle \frac{dg}{dN}}(N-N_{\mathrm{t}\mathrm{r}})$$

(3)

where dg/dN is the differential gain, and N_tr is the transparency carrier density. The carrier recombination rate R(N) is expressed as

$$R\left(N\right)=AN+B{N}^2+C{N}^3$$

(4)

where A, B, and C are the non-radiative recombination coefficient, spontaneous recombination coefficient, and Auger recombination coefficient, respectively. To mimic the stimulus signal of neurons, a rectangular electrical pulse I(t) is loaded into the laser, which is expressed as

$$I\left(t\right)=I_{\mathrm{b}\mathrm{i}\mathrm{a}\mathrm{s}}+I_{\mathrm{m}}\ast rect\left({\tau }_{\mathrm{r}}\right)$$

(5)

where I_bias is the biased current set as the lower current value (I_lo = I_bias); I_m is the amplitude of the rectangular pulse rect (τ_r) of duration τ_r, and the upper current level is I_up = I_lo + I_m = I_bias + I_m. Considering that the light beam outputs from the FC, the output power can be expressed as

$$P\left(t\right)={\eta }_{\mathrm{f}}{\displaystyle \frac{h\omega V_{\mathrm{a}\mathrm{c}\mathrm{t}}}{2\pi \Gamma {\tau }_{\mathrm{p}}}}{\left|E\right(t\left)\right|}^2$$

(6)

where η_f is the product of the differential quantum efficiency and coupling efficiency, and h is Planck’s constant.

Equations (1)–(6) can be numerically solved using the fourth-order Runge–Kutta method. The parameters used in the numerical simulation are listed in

Table 1. Summary of main simulation parameters.

Parameters	Descriptions	Values	Units
Γ	Optical confinement factor	0.06	-
$N_{\mathrm{t}\mathrm{r}}$	Transparency carrier density	1.3 × 1024	m−3
$V_{\mathrm{a}\mathrm{c}\mathrm{t}}$	Volume of active region	1.53 × 10−17	m3
A	Non-radiative recombination coefficient	2.8 × 108	s−1
B	Spontaneous recombination coefficient	1.5 × 10−16	m3s−1
C	Auger recombination coefficient	9.0 × 10−41	m6s−1
${\displaystyle \frac{dg}{dN}}$	Differential gain	4.38 × 10−20	m2
${\tau }_{\mathrm{p}}$	Photon lifetime	2.17	ps
$\alpha$	Linewidth enhancement factor	3	-
$ϵ$	Nonlinear gain coefficient	1.97 × 10−23	m3
${\eta }_{\mathrm{f}}$	Product of the differential quantum efficiency and coupling efficiency	0.17	-

[26].

3. Results and Discussion

Firstly, we analyze the nonlinear dynamics of the DFB-LD with optical feedback under different biased currents. Through extracting the extreme values of the output power, the dynamic bifurcations under the delayed-time τ = 1 ns and different feedback strength K_f are shown in Figure 2, where (a–d) are for K_f = 1.45 ns⁻¹, 1.55 ns⁻¹, 1.65 ns⁻¹, and 1.75 ns⁻¹, respectively. From this diagram, it can be seen that the DFB-LD under optical feedback may exhibit different dynamical states, such as a stable state, a period one (P1) state, a multi-periodic state, and a chaotic state. When I_bias is set at a relatively small value, the laser primarily exhibits a multi-periodic state. However, for I_bias with a relatively large value, the laser primarily exhibits a stable state and a P1 state. By adjusting the current, the state switching between the P1 state and the stable state can be realized, which is the basis for the spiking dynamics in photonic neurons [9,14]. Considering that a large K_f is beneficial for enhancing the amplitude envelope and expanding the range of the P1 state, we fix K_f = 1.75 ns⁻¹ in the following discussion unless otherwise specified. Under this case, there are three regions (named I, II, and III, respectively) for which the laser behaves as the P1 state, and the laser exhibits a stable state behavior between two adjacent regions. The laser switches to a stable state from the first P1 state region I at point A, and the stable state lasts until point B1. At points B2–B4, the laser behaves as a P1 state again. At point B5, the laser operates in a stable state. The corresponding currents for such points are given in Figure 2d.

Next, we adopt a rectangular electrical pulse with τ_r = 40 ns to load into the DFB-LD for generating tonic spiking. Here, the lower current of the pulse I_lo is fixed at 21.8 mA, which is equal to the current for point A, and the upper current of the pulse I_up is set at 22.0 mA, 23.3 mA, 23.8 mA, 24.3 mA, and 25.8 mA corresponding to the current values for points B1–B5, respectively. Figure 3 gives the rectangular electrical pulse signals I(t) (top row) and the corresponding stimulated responses (bottom row). As shown in Figure 3a2, for I_up taken as the current value at point B1, the pulse envelope amplitude gradually decreases over time, and the tonic spiking cannot be generated. For I_up taken as the currents at points B2–B4 (Figure 3b2–d2), the tonic spiking analogous to biological neurons can be observed. However, for I_up taken as the current at point B5 (Figure 3e2), the result is similar to that shown in Figure 3a2. The simulated results demonstrate that for I_up set within region I or region III, the tonic spiking can also be observed. Therefore, in order to generate the tonic spiking analogous to biological neurons, I_lo and I_up are required to make the DFB-LD operate at a stable state and a P1 state, respectively. Notably, the generated tonic spiking possesses a pulse width of sub-nanoseconds, which is faster than the biological neurons with a width of milliseconds.

Then, we discuss the characteristics of the tonic spiking generated under different operation parameters. As mentioned above, under optical feedback with τ = 1 ns, the tonic spiking can be generated for I_up located within regions I, II, or III, and Figure 4a–c shows corresponding results. Here, for each region, three representative results are selected for demonstration. As shown in Figure 4a, for I_lo set at 18.9 mA and three I_up (19.4 mA, 20.0 mA, and 20.9 mA) within region I, the frequencies of the tonic spiking are 3.47 GHz, 3.55 GHz, and 3.65 GHz, respectively. As shown in Figure 4b, for I_lo = 21.8 mA and three I_up (23.3 mA, 23.8 mA, and 24.3 mA) within region II, the frequencies of the tonic spiking are 4.44 GHz, 4.51 GHz, and 4.58 GHz, respectively. As shown in Figure 4c, for I_lo set at 25.4 mA and three I_up (27.4 mA, 28.4 mA, and 28.9 mA) within region III, the frequencies of the tonic spiking are 5.47 GHz, 5.54 GHz, and 5.63 GHz, respectively. As a result, for I_up located at the same P1 region, the tonic spiking generated under larger current values possesses a higher frequency. Due to the discontinuity of P1 regions, the frequency of generated tonic spiking can only be adjusted in discrete segments for given τ = 1 ns. However, through varying the feedback delay time, the frequency of generated tonic spiking can be tuned continuously. Figure 4d gives the frequency of the tonic spiking as a function of I_up under different τ. From this diagram, it can be seen that for a given value of τ, the frequency of the generated tonic spiking exhibits a piecewise–continuous increasing trend with the increase in I_up. For τ taken at different values, the current ranges required to achieve continuous frequency increase in tonic spiking are inconsistent. Therefore, through changing the value of τ from 0.5 ns to 1.2 ns and selecting the matched value within (15.0 mA, 33.0 mA) for I_up, the tonic spiking with the frequency located within (1.12 GHz, 6.30 GHz) can be generated. Under this case, the maximal frequency tuning range of the tonic spiking achieves 5.18 GHz.

Furthermore, the sensitivity of the frequency tunability to the feedback strength K_f and the linewidth enhancement factor α is discussed. Figure 5a gives the frequency tuning range of the tonic spiking varied with ∆K_f, where ∆K_f = (K_f − K_f0)/K_f0, and K_f0 = 1.750 ns⁻¹. From this diagram, it can be seen that for ∆K_f = −10%, the maximal frequency tuning range of the tonic spiking is about 3.5 GHz. For ∆K_f increasing from −8% to 10%, the maximal frequency tuning range shows a stepwise changing trend. For ∆K_f within (−8% −2%), the maximal frequency tuning range remains at around 4.3 GHz. For ∆K_f within (0–10%), the maximum tuning range remains at around 5.18 GHz. Therefore, through setting K_f at the center of such steps, the maximal tuning range of tonic spiking is robust against the deviation of K_f, which ensures the feasibility of this experiment. Figure 5b gives the maximal frequency tuning range of tonic spiking varied with ∆α, where ∆α = (α − α₀)/α₀, and α₀ = 3. Similarly, a stepwise changing trend can also be observed in this diagram. As a result, for α taken at certain values, the maximal tuning range of tonic spiking can also be insensitive to the deviation of α.

Finally, a comparison of achieved performances in this work with some key alternative photonic neuron implementations is given in

Table 2. Summary of related studies in generating tonic spiking.

Refs.	Years	Photonic Spiking Neurons	Control Methods	Tuning Ranges (GHz)
[15]	2016	VCSELs	pump current and optical injection	~4
[12]	2018	VCSELs	optical injection	~1
[22]	2023	DFB-SA	gain current	~2
[14]	2025	VCSELs	optical feedback	~2
This work	2025	DFB-LD	optical feedback and electrical pulse	~5

. Obviously, our scheme not only improves the operational convenience but also expands the tonic spiking frequency tuning range.

Additionally, it should be pointed out that although this work focuses on the numerical simulation based on physical models, artificial intelligence (AI) [27,28] and machine learning (ML) [29] may be other effective techniques for modeling the complex nonlinear dynamics of the DFB-LD under feedback and optimizing the operation parameters.

4. Conclusions

In summary, a scheme for generating the tonic spiking is proposed and numerically demonstrated. For such a scheme, a DFB-LD under optical feedback is utilized as the photonic tonic spiking neuron, which is electrically stimulated by a rectangular electrical pulse with lower current I_lo and upper current I_up. Via the Lang–Kobayashi theoretical model for analyzing the dynamical performances of the DFB-LD under optical feedback, the current regions required for driving the DFB-LD into stable states and period one (P1) states are determined. A rectangular electrical pulse is further introduced as the stimulated signal for generating the tonic spiking. The simulated results demonstrate that I_lo and I_up of the pulse are set to make the DFB-LD operate at a stable state and P1 state, respectively, and the tonic spiking can be successfully generated. For a fixed feedback delay time τ, the frequency of the generated tonic spiking exhibits a piecewise–continuous increasing trend with the increase in I_up. However, through changing the value of τ from 0.5 ns to 1.2 ns and selecting matched I_up within (15.0 mA, 33.0 mA), the tonic spiking with a frequency located within (1.12 GHz, 6.30 GHz) can be generated. It should be pointed out that though the frequency tuning range is obtained under a given feedback strength, it is hoped to be enlarged by adjusting the value of the feedback strength. Therefore, this proposed scheme for generating tonic spiking possesses an application potential in future ultrafast photonic neuro-morphic computing due to its relatively simple structure and ease of control.