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

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. System Architecture

_{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.

_{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

_{1}is the probability of word 012, n

_{1}is the number of the word 012, and the rest of the words are calculated similarly.

^{−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.

_{D}(t) and P

_{R}(t) represent the power of the driver laser and response laser, respectively, and $\langle \xb7\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

_{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

_{R}and P

_{D}are the probability of a specific word of the response laser and drive laser, respectively.

_{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.

_{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.

^{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),

_{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.

_{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

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Zhong, D.Z.; Deng, T.; Zheng, G.L. Manipulation of the complete chaos synchronization in dual-channel encryption system based on polarization-division-multiplexing. Acta Phys. Sin.
**2014**, 63, 070504. [Google Scholar] [CrossRef] - Kanter, I.; Aviad, Y.; Reidler, I.; Cohen, E.; Rosenbluh, M. An optical ultrafast random bit generator. Nat. Photonics
**2010**, 4, 58–61. [Google Scholar] [CrossRef] - Chen, C.Y.; Cheng, C.H.; Pan, D.K.; Lin, F.Y. Experimental generations and analyses of chaos-modulated pulses for pulsed chaos lidar applications based on gain-switched semiconductor lasers subject to optical feedback. Opt. Express
**2018**, 26, 20851–20860. [Google Scholar] [CrossRef] [PubMed] - Rontani, D.; Choi, D.; Chang, C.Y.; Locquet, A.; Citrin, D.S. 6 Compressive sensing with optical chaos. Sci. Rep.
**2016**, 6, 35206. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Huang, Y.; Zhou, P.; Yang, Y.; Chen, T.; Li, N. Time-Delayed Reservoir Computing Based on a Two-Element Phased Laser Array for Image Identification. IEEE Photonics J.
**2021**, 13, 1–9. [Google Scholar] [CrossRef] - Bao, X.; Zhao, Q.; Yin, H. A multiple-input multiple-output reservoir computing system subject to optoelectronic feedbacks and mutual coupling. Entropy
**2020**, 22, 231. [Google Scholar] [CrossRef] [Green Version] - Tao, J.-Y.; Wu, Z.-M.; Yue, D.-Z.; Tan, X.-S.; Zeng, Q.-Q.; Xia, G.-Q. Performance enhancement of a delay-based Reservoir computing system by using gradient boosting technology. IEEE Access
**2020**, 8, 151990–151996. [Google Scholar] [CrossRef] - Fischer, I.; van Tartwijk, G.H.M.; Levine, A.M.; Elsasser, W.; Gobel, E.; Lenstra, D. Fast Pulsing and Chaotic Itinerancy with a Drift in the Coherence Collapse of Semiconductor Lasers. Phys. Rev. Lett.
**1996**, 76, 220–223. [Google Scholar] [CrossRef] [Green Version] - Shastri, B.J.; Nahmias, M.A.; Tait, A.N.; Robriguez, A.W.; Wu, B.; Prucnal, P.R. Spike Processing With a Graphene Excitable Laser. Sci. Rep.
**2016**, 6, 1–12. [Google Scholar] [CrossRef] [Green Version] - Hurtado, A.; Schires, K.; Henning, I.; Adams, M. Investigation of vertical cavity surface emitting laser dynamics for neuromorphic photonic systems. Appl. Phys. Lett.
**2012**, 100, 103703. [Google Scholar] [CrossRef] [Green Version] - Deng, T.; Robertson, J.; Hurtado, A. Controlled propagation of spiking dynamics in vertical-cavity surface-emitting lasers: Towards neuromorphic photonic networks. IEEE J. Sel. Top. Quantum Electron.
**2017**, 23, 1–8. [Google Scholar] [CrossRef] [Green Version] - Xiang, S.Y.; Zhang, H.; Guo, X.X.; Li, J.F.; Wen, A.J.; Pan, W.; Hao, Y. Cascadable neuron-like spiking dynamics in coupled VCSELs subject to orthogonally polarized optical pulse injection. IEEE J. Sel. Top. Quantum Electron.
**2017**, 23, 1–7. [Google Scholar] [CrossRef] - Tait, A.N.; Nahmias, M.A.; Shastri, B.J.; Prucnal, P.R. Broadcast and weight: An integrated network for scalable photonic spike processing. J. Lightwave Technol.
**2014**, 32, 3427–3439. [Google Scholar] [CrossRef] - Ma, P.Y.; Shastri, B.J.; de Lima, T.F.; Huang, C.; Tait, A.N.; Nahmias, M.A.; Peng, H.T.; Prucnal, P.R. Simultaneous excitatory and inhibitory dynamics in an excitable laser. Opt. Lett.
**2018**, 43, 3802–3805. [Google Scholar] [CrossRef] - Robertson, J.; Deng, T.; Javaloyes, J.; Hurtado, A. Controlled inhibition of spiking dynamics in VCSELs for neuromorphic photonics: Theory and experiments. Opt. Lett.
**2017**, 42, 1560–1563. [Google Scholar] [CrossRef] [Green Version] - Selmi, F.; Braiv, R.; Beaudoin, G.; Sagnes, I.; Kuszelewicz, R.; Erneux, T.; Barbay, S. Spike latency and response properties of an excitable micropillar laser. Phys. Rev. E
**2016**, 94, 042219. [Google Scholar] [CrossRef] [Green Version] - Hurtado, A.; Javaloyes, J. Controllable spiking patterns in long-wavelength vertical cavity surface emitting lasers for neuromorphic photonics systems. Appl. Phys. Lett.
**2015**, 107, 241103. [Google Scholar] [CrossRef] [Green Version] - Nahmias, M.A.; Shastri, B.J.; Tait, A.N.; Prucnal, P.R. A leaky integrate-and-fire laser neuron for ultrafast cognitive computing. IEEE J. Sel. Top. Quantum Electron.
**2013**, 19, 1–12. [Google Scholar] [CrossRef] - Peng, H.T.; Nahmias, M.A.; de Lima, T.F.; Tait, A.N.; Shastri, B.J.; Prucnal, P.R. Neuromorphic photonic integrated circuits. IEEE J. Sel. Top. Quantum Electron.
**2018**, 24, 6101715. [Google Scholar] [CrossRef] - Tiana-Alsina, J.; Quintero-Quiroz, C.; Masoller, C. Comparing the dynamics of periodically forced lasers and neurons. N. J. Phys.
**2019**, 21, 103039. [Google Scholar] [CrossRef] [Green Version] - Aragoneses, A.; Rubido, N.; Tiana-Alsina, J.; Torrent, M.; Masoller, C. Distinguishing signatures of determinism and stochasticity in spiking complex systems. Sci. Rep.
**2013**, 3, 1778. [Google Scholar] [CrossRef] [Green Version] - Aragoneses, A.; Perrone, S.; Sorrentino, T.; Torrent, M.; Masoller, C. Unveiling the complex organization of recurrent patterns in spiking dynamical systems. Sci. Rep.
**2014**, 4, 4696. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sorrentino, T.; Quintero-Quiroz, C.; Torrent, M.; Masoller, C. Analysis of the spike rate and spike correlations in modulated semiconductor lasers with optical feedback. IEEE J. Sel. Top. Quantum Electron.
**2015**, 21, 561–567. [Google Scholar] [CrossRef] - Sorrentino, T.; Quintero-Quiroz, C.; Aragoneses, A.; Torrent, M.; Masoller, C. Effects of periodic forcing on the temporally correlated spikes of a semiconductor laser with feedback. Opt. Express
**2015**, 23, 5571–5581. [Google Scholar] [CrossRef] [Green Version] - Tiana-Alsina, J.; Quintero-Quiroz, C.; Panozzo, M.; Torrent, M.; Masoller, C. Experimental study of modulation waveforms for entraining the spikes emitted by a semiconductor laser with optical feedback. Opt. Express
**2018**, 26, 9298–9309. [Google Scholar] [CrossRef] - Prucnal, P.R.; Shastri, B.J.; de Lima, T.F.; Nahmias, M.A.; Tait, A.N. Recent progress in semiconductor excitable lasers for photonic spike processing. Adv. Opt. Photonics
**2016**, 8, 228–299. [Google Scholar] [CrossRef] - Masoliver, M.; Masoller, C. Sub-threshold signal encoding in coupled FitzHugh-Nagumo neurons. Sci. Rep.
**2018**, 8, 8276. [Google Scholar] [CrossRef] - Takiguchi, Y.; Fujino, H.; Ohtsubo, J. Experimental synchronization of chaotic oscillations in externally injected semiconductor lasers in a low-frequency fluctuation regime. Opt. Lett.
**1999**, 24, 1570–1572. [Google Scholar] [CrossRef] [PubMed] - Locquet, A.; Masoller, C.; Mirasso, C.R. Synchronization regimes of optical-feedback-induced chaos in unidirectionally coupled semiconductor lasers. Phys. Rev. E
**2002**, 65, 056205. [Google Scholar] [CrossRef] [Green Version]

**Figure 2.**Output of the drive laser with optical feedback under different system parameters. (

**a**) κ

_{D}= 10 ns

^{−1}and I

_{D}= 15 mA, (

**b**) κ

_{D}= 50 ns

^{−1}and I

_{D}= 15 mA, (

**c**) κ

_{D}= 10 ns

^{−1}and I

_{D}= 22 mA, and (

**d**) κ

_{D}= 50 ns

^{−1}and I

_{D}= 22 mA.

**Figure 3.**Word probability (

**a**) and pulse rate (

**b**) of the drive laser at different feedback strengths.

**Figure 4.**Output of response laser under different pump currents. κ

_{D}= κ

_{R}= 50 ns

^{−1}. Left column: I

_{D}= 15 mA. Middle column: I

_{D}= 16 mA. Right column: I

_{D}= 17 mA. (

**a**–

**c**): I

_{R}< I

_{D}. (

**d**–

**f**): I

_{R}= I

_{D}. (

**g**–

**i**): I

_{R}> I

_{D}.

**Figure 5.**Output of response laser under different injection strengths. I

_{D}= I

_{R}= 15 mA, κ

_{D}= 50 ns

^{−1}. (

**a**) κ

_{R}= 10 ns

^{−1}, (

**b**) κ

_{R}= 40 ns

^{−1}, (

**c**) κ

_{R}= 50 ns

^{−1}, and (

**d**) κ

_{R}= 100 ns

^{−1}.

**Figure 6.**Parameter relationships that can make the response laser generate the LFF dynamic. (

**a**) The relationship of I

_{R}and I

_{D}and (

**b**) the relationship between κ

_{R}and κ

_{D}.

**Figure 7.**Effect of pump current mismatch on system transmission. (

**a**) Event probability error, (

**b**) pulse rate, and (

**c**) correlation coefficient.

**Figure 8.**Effect of linewidth enhancement factor mismatch on system transmission. (

**a**) Event probability error, (

**b**) pulse rate, and (

**c**) correlation coefficient.

**Figure 9.**Effect of external parameter mismatch on (

**a**) the correlation coefficient and (

**b**) the difference of the average spike rate.

**Figure 10.**Effect of injection strength on the LFF transmission performance. The pump current and linewidth enhancement factor are 15 mA and 5, respectively. (

**a**) Correlation coefficient, (

**b**) average spike rate difference.

Symbol | Parameter | Value |
---|---|---|

G | Gain coefficient | 1.5 × 10^{4} m^{3}/s |

τ_{n} | Carrier lifetime | 2 × 10^{−9} s |

τ_{p} | Photon lifetime | 2 × 10^{−12} s |

N_{0} | Carrier density at transparency | 1.5 × 10^{8} m^{−3} |

ε | Gain saturation coefficient | 0.05 |

τ | Feedback delay time | 1 × 10^{−9} s |

V | Active region volume | 1.5 × 10^{−16} m^{3} |

Serial Number | Word | Relation | Quantity |
---|---|---|---|

1 | 012 | $\Delta t\left(i\right)<\Delta t(i+1)<\Delta t(i+2)$ | n_{1} |

2 | 021 | $\Delta t\left(i\right)<\Delta t(i+2)<\Delta t(i+1)$ | n_{2} |

3 | 102 | $\Delta t\left(i+1\right)<\Delta t(i)<\Delta t(i+2)$ | n_{3} |

4 | 120 | $\Delta t\left(i+1\right)<\Delta t(i+2)<\Delta t(i)$ | n_{4} |

5 | 201 | $\Delta t\left(i+2\right)<\Delta t(i)<\Delta t(i+1)$ | n_{5} |

6 | 210 | $\Delta t\left(i+2\right)<\Delta t(i+1)<\Delta t(i)$ | n_{6} |

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**MDPI and ACS Style**

Dou, X.; Qiu, S.; Wu, W.
Numerical Demonstration of the Transmission of Low Frequency Fluctuation Dynamics Generated by a Semiconductor Laser with Optical Feedback. *Photonics* **2022**, *9*, 483.
https://doi.org/10.3390/photonics9070483

**AMA Style**

Dou X, Qiu S, Wu W.
Numerical Demonstration of the Transmission of Low Frequency Fluctuation Dynamics Generated by a Semiconductor Laser with Optical Feedback. *Photonics*. 2022; 9(7):483.
https://doi.org/10.3390/photonics9070483

**Chicago/Turabian Style**

Dou, Xinyu, Shimeng Qiu, and Wanqing Wu.
2022. "Numerical Demonstration of the Transmission of Low Frequency Fluctuation Dynamics Generated by a Semiconductor Laser with Optical Feedback" *Photonics* 9, no. 7: 483.
https://doi.org/10.3390/photonics9070483