Numerical Demonstration of the Transmission of Low Frequency Fluctuation Dynamics Generated by a Semiconductor Laser with Optical Feedback

Abstract

In this paper, the transmission mechanism of the spike information embedded in the low frequency fluctuation (LFF) dynamic in a cascaded laser system is numerically demonstrated. In the cascaded laser system, the LFF waveform is first generated by a drive laser with optical feedback and is then injected into a response laser. The range of crucial system parameters that can make the response laser generate the LFF dynamic is studied, and the effect of parameter mismatch on the transmission of LFF dynamics is explored through a method of symbolic time-series analysis and the index, such as the spike rate and the cross-correlation coefficient. The results show that the mismatch of the pump current has a more significant influence on the transmission of LFF waveforms than that of the internal physical parameter of the laser, such as the linewidth enhancement factor. Moreover, increasing the injection strength can enhance the robustness of LFF transmission. As spikes of the LFF dynamic generated by lasers with optical feedback is similar to the spike of neurons, the results of this paper can help understanding the information transporting and processing inside the photonic neurons.

1. Introduction

Semiconductor lasers with optical feedback can generate various nonlinear physical phenomena and can be applied in secure communication, random number generation, lidar, comprehensive sensing, and reservoir computing, etc. [1,2,3,4,5,6,7]. Moreover, the low frequency fluctuation (LFF) dynamic induced by the optical feedback architecture has been found to be a type of excitable behavior [8], which can be exploited for photonic neurons [9,10,11,12,13] and neuromorphic computations such as pattern recognition, logic operations, and calculations [14,15,16,17,18,19]. In the LFF region, the laser output displays abrupt and irregular power dropouts, and this type of spike has been proven to be similar to the spikes generated by biological neuron models such as the FitzHugh–Nagumo (FHN) model [20]. As the optical feedback scheme can avoid the optical isolator and is suitable for photonic integrated circuits (PICs), lasers with optical feedback are a promising candidate in the field of photonic neural computing.

There have already been studies on LFF dynamics in the literature. A. Aragoneses et al. implemented numerical simulations and experimental studies for the output signal of a directly modulated optical feedback semiconductor laser with optical feedback and investigated the external locking mechanism of the external modulation signal on the laser output signal through timing analysis (event analysis) of the output waveform [21]. They also proposed a minimal model to simulate the output of an optical feedback laser [22]. T. Sorrentino et al. investigated the pulse rate and pulse correlation of the output pulses of a laser with optical feedback [23]. Moreover, the periodic locking mechanism of the external periodic signal on the output of the optical feedback laser has been studied [24,25]. According to [26], the study of photonic neurons will evolve to the transmission of spikes in a photonic neural system composed of multiple photonic neurons. However, there are few studies on the transmission of various spike information embedded in LFF dynamics. Therefore, in this paper, the transmission of the LFF waveform in a cascaded laser system consisting of a laser with optical feedback and another laser is studied, and the effects of various system parameters on the output of the system are analyzed and discussed.

The reminder of this paper is organized as follows. In Section 2, the system structure, theory model, and tools for describing the system output characteristics are introduced. In Section 3, the effect of various system parameters on the transmission of LFF dynamics is studied and analyzed. The discussion of the results and conclusions are provided in Section 4.

2. Materials and Methods

2.1. System Architecture

Figure 1 shows the schematic diagram of a cascaded laser system composed of two semiconductor lasers. One laser is subjected to optical feedback and it is named the drive laser (LD-1 in Figure 1). The output of the drive laser is injected into the other laser, which is called the response laser (LD-2 in Figure 1).

The Lang–Kobayashi (LK) model is always used to describe semiconductor lasers with optical feedback, and the rate equations are as follows:

$$\frac{d{E}_{\mathrm{D}}(t)}{dt}=\frac{1}{2}\left(1+i{\alpha }_{\mathrm{D}}\right)\left\{\frac{G\left[N_{\mathrm{D}}(t)-N_0\right]}{1+\epsilon {\left|E_{\mathrm{D}}\left(t\right)\right|}^2}-\frac{1}{{\tau }_{\mathrm{p}}}\right\}{E}_{\mathrm{D}}\left(t\right)+{\kappa }_{\mathrm{D}}{E}_D\left(t-\tau \right)e^{-i\omega \tau }+\sqrt{2{\beta }_{sp}}\xi ,$$

(1)

$$\frac{d{N}_{\mathrm{D}}\left(\mathrm{t}\right)}{d\mathrm{t}}=\frac{I_{\mathrm{D}}}{\mathrm{q}V}-\frac{1}{{\tau }_{\mathrm{n}}}{N}_{\mathrm{D}}\left(\mathrm{t}\right)-\frac{G\left[N_{\mathrm{D}}(t)-N_0\right]}{1+\epsilon {\left|E_{\mathrm{D}}\left(t\right)\right|}^2}{\left|E_{\mathrm{D}}\left(t\right)\right|}^2,$$

(2)

$$\frac{d{E}_{\mathrm{R}}(t)}{dt}=\frac{1}{2}\left(1+i{\alpha }_{\mathrm{R}}\right)\left\{\frac{G\left[N_{\mathrm{R}}(t)-N_0\right]}{1+\epsilon {\left|E_{\mathrm{R}}\left(t\right)\right|}^2}-\frac{1}{{\tau }_{\mathrm{p}}}\right\}{E}_{\mathrm{R}}\left(t\right)+{\kappa }_{\mathrm{R}}{E}_{\mathrm{D}}\left(t\right)e^{-i\omega t}+\sqrt{2{\beta }_{sp}}\xi ,$$

(3)

$$\frac{d{N}_{\mathrm{R}}\left(\mathrm{t}\right)}{dt}=\frac{I_{\mathrm{R}}}{\mathrm{q}V}-\frac{1}{{\tau }_{\mathrm{n}}}{N}_{\mathrm{R}}\left(\mathrm{t}\right)-\frac{G\left[N_{\mathrm{R}}(t)-N_0\right]}{1+\epsilon {\left|E_{\mathrm{R}}\left(t\right)\right|}^2}{\left|E_{\mathrm{R}}\left(t\right)\right|}^2,$$

(4)

where E_D(t) and E_R(t) in Equations (1) and (3) represent the slowly varying electric field amplitude of the drive laser and response laser, respectively, and N_D(t) and N_R(t) in Equations (2) and (4) denote the carrier density of two lasers, respectively. κ_D is the feedback strength of the drive laser, and κ_R is the injection strength of the response laser. I_D and I_R denote the pump current of two lasers, respectively, and α_D and α_R represent the linewidth enhancement factor of the drive laser and response laser, respectively. β_sp = 10⁻⁴ ns⁻¹ is the noise strength, and ξ is a Gaussian distribution with zero mean and unit variance. The physical meaning and values of the remaining parameters are shown in

Table 1. Parameters and values.

Symbol	Parameter	Value
G	Gain coefficient	1.5 × 104 m3/s
τn	Carrier lifetime	2 × 10−9 s
τp	Photon lifetime	2 × 10−12 s
N0	Carrier density at transparency	1.5 × 108 m−3
ε	Gain saturation coefficient	0.05
τ	Feedback delay time	1 × 10−9 s
V	Active region volume	1.5 × 10−16 m3

.

As is well-known, with different system parameters, such as the pump current I_D and the feedback strength κ_D, the semiconductor laser with optical feedback can generate rich dynamic behaviors. When I_D is 15 mA, which is near the threshold (14.6 mA in our model) and κ_D is small (10 ns⁻¹ for instance), the output of the laser exhibits random oscillations and belongs to the coherent collapse, as shown in Figure 2a. Moreover, when I_D is near the threshold and κ_D is relatively large (50 ns⁻¹ for instance), the output of the laser contains abrupt power dropouts, which have certain probabilistic characteristics, as shown in Figure 2b. At this time, the output of the laser belongs to the LFF waveform. In addition, when I_D is higher than the threshold current (22 mA for instance), the output of the laser belongs to the coherent collapse, no matter whether the feedback strength κ_D is big or small, as shown in Figure 2c,d. Therefore, to generate the LFF dynamic, the pump current should be near the threshold, and the feedback strength should be relatively large. In the rest of this paper, I_D is set to be 15 mA and κ_D is set to be 50 ns⁻¹.

2.2. Tools

We analyzed the timing information of the spike sequence through an ordinal time-series analysis [22]. Firstly, a pulse sequence with a length of N is divided into N-D vectors of length D, as shown in Equation (5). Δt(i) is the time interval of pulse i, and t(i) is the time at which pulse i occurs.

$$\Delta t\left(i\right)=t\left(i\right)-t\left(i-1\right).$$

(5)

Based on the relative length of Δt(i), each time interval is associated with a word composed of D symbols. For example, when D = 2, two types of words exist. Δt(i) < Δt(i + 1) gives the word “01” and Δt(i) > Δt(i + 1) gives the word “10”.

Table 2. Ordinal patterns for D = 3.

Serial Number	Word	Relation	Quantity
1	012	$\Delta t\left(i\right)<\Delta t(i+1)<\Delta t(i+2)$	n1
2	021	$\Delta t\left(i\right)<\Delta t(i+2)<\Delta t(i+1)$	n2
3	102	$\Delta t\left(i+1\right)<\Delta t(i)<\Delta t(i+2)$	n3
4	120	$\Delta t\left(i+1\right)<\Delta t(i+2)<\Delta t(i)$	n4
5	201	$\Delta t\left(i+2\right)<\Delta t(i)<\Delta t(i+1)$	n5
6	210	$\Delta t\left(i+2\right)<\Delta t(i+1)<\Delta t(i)$	n6

shows ordinal patterns for D = 3. Then, by counting the frequency of occurrence of the different words, their probabilities are computed, as shown in Equation (6),

$$P_1=\frac{n_1}{{\displaystyle \sum _{j=1}^6{n}_j}},$$

(6)

where P₁ is the probability of word 012, n₁ is the number of the word 012, and the rest of the words are calculated similarly.

This symbolic transformation has the drawback that it disregards the information about the precise duration of the inter-spike intervals (ISIs), but it has the advantage that it keeps the information about the temporal correlations among them, i.e., correlations in the timing of the optical spikes. Figure 3a shows the words extracted from the output of the drive laser. When the feedback strength increases from 50 ns⁻¹ to 100 ns⁻¹, the probabilities of all of the words occur certain degrees of oscillations. In particular, when the feedback strength was within the region from 70 ns⁻¹ to 80 ns⁻¹, the probabilities of words “012” (blue line) and “201” (light blue line) was lower than the average probability (1/6), while the probabilities of other words were higher than 1/6. According to [27], this phenomenon indicates that the laser may has an encoding effect on the feedback strength.

In addition, the pulse rate can be used to analyze the output waveform of the system, and it can be calculated by Equation (7), where s is the total number of ISIs.

$$rate=\frac{s}{{\displaystyle \sum _{i=1}^s\Delta t(i)}}.$$

(7)

Figure 3b reflects the effect of the feedback strength on the pulse rate. The figure shows that when the feedback strength is enhanced, the average pulse rate of the drive laser tends to decrease, indicating that the greater the feedback strength the longer the pulse interval is. This also means that the LFF dynamic has an encoding effect on the feedback strength [27].

Meanwhile, the performance of the LFF transmission is evaluated by the correlation coefficient between the output waveforms of the drive and response laser, and it can be defined as

$$CC=\frac{\langle \left[P_{\mathrm{D}}\left(t\right)-\langle P_{\mathrm{D}}\left(t\right)\rangle \right]\left[P_{\mathrm{R}}\left(t\right)-\langle P_R\left(t\right)\rangle \right]\rangle }{\sqrt{\langle \left|{\left[P_{\mathrm{D}}\left(t\right)-\langle P_{\mathrm{D}}\left(t\right)\rangle \right]}^{{}^2}\right|\rangle }\sqrt{\langle \left|{\left[P_{\mathrm{R}}\left(t\right)-\langle P_{\mathrm{R}}\left(t\right)\rangle \right]}^{{}^2}\right|\rangle }},$$

(8)

where P_D(t) and P_R(t) represent the power of the driver laser and response laser, respectively, and $\langle ·\rangle$ represents the time average. When CC = 1, the output waveforms of both lasers are identical, and CC is less than 1 if there is a deviation between the output of the lasers.

3. Results

3.1. Effect of System Parameters on the Output of the Response Laser

Firstly, the range of the parameter that can make the response laser generate the LFF dynamic is investigated. The pump current of the response laser I_R is used as an example. During the simulation, κ_D is set to be 50 ns⁻¹ and I_D is set to be 15 mA, 16 mA, and 17 mA, respectively. For all of these I_D values, the drive laser can generate LFF dynamics. Meanwhile, the injection strength κ_R is set to be 50 ns⁻¹. When I_R is smaller than I_D, the response laser can generate the LFF waveform, as shown in Figure 4a–c. Then, when I_R is equal to I_D, the response laser can also output the LFF dynamic, as shown in Figure 4d–f. However, when I_R is larger than I_D, the response laser no longer displays irregular power dropouts. This means that the output of the response laser does not belong to the LFF region when I_R > I_D. In addition, the range of κ_R is also explored. When κ_R is near, equal, or higher than κ_D, the response laser can generate the LFF waveform, as shown in Figure 5b–d. However, when κ_R is smaller than κ_D, the response laser cannot generate the LFF dynamic, as shown in Figure 5a. Figure 6 clearly demonstrates the influence of parameter relationships on the output of the response laser. When I_R ≤ I_D or κ_R ≥ κ_D, the LFF dynamic from the drive laser can be transmitted to the response laser, and when I_R > I_D or κ_R < κ_D, the transmission is failed.

3.2. Influence of System Parameter Mismatch on the Transmission of LFF Waveforms

In practice, it is difficult to guarantee that both lasers have identical parameters. Thus, in this section, the effect of the mismatch of various physical parameters on the system performance is investigated.

The pump current is studied as the external operating parameter. Firstly, the error of the word probability between the drive and response laser is defined as follows,

$$d=\frac{|P_{\mathrm{R}}-P_{\mathrm{D}}|}{P_{\mathrm{D}}},$$

(9)

where d represents the event probability error, and P_R and P_D are the probability of a specific word of the response laser and drive laser, respectively.

From Figure 7a, certain transmission errors occur for all words when I_R is smaller than I_D. When I_R is between 14.75 mA and 14.8 mA, the transmission of word 021 has the biggest error, and when I_R is larger than 14.85 mA, the transmission of word 210 always has the largest error. When I_R gradually increases to 15 mA, all of the errors converge to the zero point, indicating that the LFF waveform from the drive laser is successfully transmitted to the response laser. In addition, from Figure 7b, the red and black lines are rate_D and rate_R, respectively, and rate_D is fixed at 41.97 MHz. The pulse rate of the response laser increases gradually with the injection current and increases rapidly when the I_R is 14.5 mA and 14.6 mA, indicating that the rate_R is greatly influenced by the injection current at this time. When I_R increases from 14.7 mA to 15 mA, the curve increases smoothly to 41.97 MHz. What is more, Figure 7c shows the overall increase in the correlation coefficient CC with the increasing current.

The mismatch of the internal physical parameters, such as the linewidth enhancement factor α, can also induce a derivation on the LFF transmission. From Figure 8a, the transmission of the word 021 always has a large error when α_R is not equal to 5. For instance, the error is larger than 0.7 when α_R is 4.8 and 5.2, and reaches the largest value when α_R is larger than 5.8. Meanwhile, the transmission of word 210 also always has a large error in the whole range of α_R. In addition, the error of the transmission of word 012 is always small and less than 0.3. In Figure 8b, rate_R increases rapidly and gradually approaches rate_D when α_R increases from 4.2 to 5, while in the range 5 to 6, rate_R decreases steadily and gradually deviates from rate_D. This means that when α_R is smaller than α_D, the LFF dynamic from the drive laser can hardly be mapped to the response laser. While for α_R > α_D, the LFF dynamic can be better transmitted. This can also be proven by the correlation coefficient. In Figure 8c, CC changes steeply when α_R increases from 4.2 to 5, indicating that the LFF transmission is seriously affected in this region. When α_R > α_D, the system is more sluggish to parameter mismatch. These phenomena are consistent with the trend of the spike rate. Thus, α_R < α_D will cause a greater error in the transmission of the LFF waveform.

As the mismatch of linewidth enhancement factor and the pump current of the response laser can both induce an effect on the output LFF waveforms, to quantify which external operating parameter has a more significant effect on the transmission of the system, the mismatch is analyzed using the mismatch degree, which is defined in Equation (10),

$$Mismatch\left(\gamma \right)=\frac{\left|{\gamma }^{\mathrm{R}}-{\gamma }^{\mathrm{D}}\right|}{{\gamma }^{\mathrm{D}}},$$

(10)

where γ^R and γ^D represent the external parameters of the response laser and the drive laser, respectively. As can be seen in Figure 9a, the red line is always higher than the black line, which means that the mismatch of the linewidth enhancement factor always has a limited impact on the system transmission, while the mismatch of the pump current I_R can cause a more serious decrease in CC. This can also be proven by the difference in the average spike rate Diff, which is defined in Equation (11),

$$Diff=\left|rat{e}_{\mathrm{R}}-rat{e}_{\mathrm{D}}\right|,$$

(11)

where rate_R and rate_D represent the average spike rate of the response and the drive laser, respectively. A larger Diff means a greater difference between the output of the drive laser and the response laser. From Figure 9b, the red line is always lower than the black line, which means that the mismatch in pump current I_R can lead to a more severe increase in Diff. Therefore, the system transmission is more sensitive to the current mismatch, and to ensure the effectiveness of the transmission, efforts should be made to ensure that the pump current of the drive and response laser should be as consistent as possible.

From the above results, it can be found that parameter mismatch significantly influences the transmission quality. To discover methods that can guarantee LFF transmission, we investigated the influence of injection strength κ_R on CC when the parameters of two lasers are mismatched. In Figure 10, the red line denotes α_D = 5 and α_R = 6, while the rest of the parameters are the same. Similarly, the black line represents that I_D = 15 mA and I_R = 14.6 mA. From this figure, CC increases with κ_R for both mismatch conditions. The difference in the average spike rate also decreases when the injection strength is enhanced, as shown in Figure 10b. These results mean that increasing the strength can enhance the robustness of the LFF transmission, even though the parameters of the two lasers are different.

4. Discussion

In this paper, the transmission of the spike information, such as the spike rate and the probability of symbolic patterns embedded in the LFF dynamics in a cascaded laser system, are numerically demonstrated. In the last decades, there have been works studying the LFF synchronization in an open-loop configuration [28,29]. Most works utilize CC to measure the performance of LFF synchronization. However, when using the laser with optical feedback as a photonic neuron, CC is insufficient to describe the spike information because it can hardly reflect the timing of the spikes of LFF dynamics. Therefore, we analyzed the transmission of the LFF dynamic through both CC and the timing of spikes such as the spike rate and the probabilities of symbolic patterns. Firstly, the results show that the performance of the LFF transmission can be improved through increasing the injection strength, which is identical to the existing numerical and experimental results. Moreover, we find that to make the response laser generate the LFF dynamic, the injection strength should be larger than the feedback strength of the drive laser, and the pump current of the response laser should be smaller than that of the drive laser. Meanwhile, the mismatch of the pump current always has a more significant influence on the transmission of LFF waveforms than the internal physical parameters. For the transmission of the timing of spikes, we find that when there is a mismatch of the system parameters, the spike rate of the response laser is smaller than that of the drive laser. In addition, the symbolic pattern 210 always has a large transmission error when the parameters are mismatched. As the LFF waveform of the semiconductor laser is similar to what is produced by the neuron, and the results of this paper are beneficial for the practical application of photonic neurons and offer promising prospects for brain-like optical information transmission. Future work will focus on the LFF waveform of other laser systems, such as mutually coupled laser systems.