# Task-Independent Computational Abilities of Semiconductor Lasers with Delayed Optical Feedback for Reservoir Computing

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## Abstract

**:**

## 1. Introduction

## 2. Reservoir Computing with a Semiconductor Laser and Delayed Feedback

#### 2.1. Rate Equation Model

#### 2.2. Pre- and Post-Processing

_{0}is set to $\pi /4$. This is to ensure that the MZM is modulating in its quasi-linear regime. The node separation will be varied throughout the paper.

## 3. Timeseries Prediction

^{−1}and $\eta =7.8$ ns

^{−1}at $I=2.02{I}_{th}$. In this case, the lowest NMSE equals 1%. Repeating the same analysis with other randomly generated masks delivers the same optimal parameter values and a variation of the performance smaller than 0.2%.

## 4. Task-Independent Computational Performance

#### Results

^{−1}. So the optimum that was reached in the benchmark task is also the point in parameter space where the total $CC$ reaches its peak value. Note that the total $CC$ does not reach its ideal value of $N=101$. This reduction of $CC$ will be discussed later. At the optimal point all memory capacities of degrees higher than one, i.e., all nonlinear capacities, have increased. The linear memory capacity (degree 1) remains mainly unaffected by the feedback strength. It is only for higher feedback strengths that the linear memory capacity degrades. These results illustrate that the Santa Fe timeseries prediction task is a very diverse task requiring both nonlinear and linear capacities.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**A schematic of the delay based RC with a semiconductor laser which is modelled by Equations (1)–(4). The data $A\left(t\right)$ is multiplied with mask $M\left(t\right)$, resulting in masked stream $B\left(t\right)$. The masked stream is modulated unto the output $\mathcal{E}$ of an external laser. The Mach–Zehnder modulator is biased by ${\mathsf{\Phi}}_{0}$, such that it remains in its quasi-linear regime. The modulated stream is multiplied by the injection rate $\mu $ and injected into the laser. The feedback strength in the delay line is controlled by the feedback rate $\eta $.

**Figure 2.**Test results obtained from the Bayesian optimisation for the semiconductor laser with delayed feedback trained on a time-series prediction task projected in the plane of (

**a**) feedback rate and pump current or (

**b**) injection rate and pump current. The performance indicator $NMSE$ is coded into the colour and size of the markers in the scatter plot. A better performance corresponds to a bigger marker size and a redder colour. Parameters as in Table 1 and $\theta =$20 ps, $N=200$, $\tau ={\tau}_{M}=$4 ns.

**Figure 3.**Total computational capacity showing the colour-coded contributions of different degrees of nonlinearity vs. the feedback strength η. Parameters as in Table 1 and $\mu =100$ ns

^{−1}, $I=2{I}_{th}$, $N=101$, $\theta =20$ ps and $\tau ={\tau}_{M}=N\theta $.

**Figure 4.**Total computational capacity (CC) showing the colour-coded contributions of different degrees of nonlinearity vs. the node distance θ. Parameters as in Table 1 and $\mu =100$ ns

^{−1}, $\eta =10$ ns

^{−1}, $I=2{I}_{th}$, $N=101$ and $\tau ={\tau}_{M}=N\theta $.

**Figure 5.**Total computational capacity (CC) showing the colour-coded contributions of different degrees of nonlinearity vs. the delay time τ. Parameters as in Table 1 and $\mu =100$ ns

^{−1}, $\eta =10$ ns

^{−1}, $I=2{I}_{th}$, $N=101$ and ${\tau}_{M}=N\theta $.

**Figure 6.**Total computational capacity (CC) showing the colour-coded contributions of different degrees of nonlinearity vs. the delay time τ. Parameters as in Table 1 and $\mu =100$ ns

^{−1}, $\eta =10$ ns

^{−1}, $I=2{I}_{th}$, $N=101$ and ${\tau}_{M}=N\theta $.

Parameters | Designation | Value Used in Bayesian Optimization |
---|---|---|

Linewidth enhancement factor | α | 3.0 |

Loss | Γ | 1 ps^{−1} |

Threshold gain | g | 1 ps^{−1} |

Differential gain | ξ | 5000 s^{−1} |

Spontaneous emission factor | β | 10^{−6} |

Carrier-lifetime | T | 1 ns |

Threshold pump-current | I_{th} | 16 mA |

Pump-current | I | scanned over [I_{th};3I_{th}] |

Feedback rate | η | scanned over [0; 50 ns^{−1}] |

Injection rate | μ | scanned over [0; 100 ns^{−1}] |

Amplitude of injected field | $\mathcal{E}$ | 200 |

Bias voltage of the MZM | Φ_{0} | $\pi /4$ |

Constant feedback phase | Ω | 0 |

Node distance | θ | 20 ps |

Number of nodes | N | 200 |

Delay time | τ | 4 ns |

Mask length | τ_{M} | 4 ns |

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**MDPI and ACS Style**

Harkhoe, K.; Van der Sande, G.
Task-Independent Computational Abilities of Semiconductor Lasers with Delayed Optical Feedback for Reservoir Computing. *Photonics* **2019**, *6*, 124.
https://doi.org/10.3390/photonics6040124

**AMA Style**

Harkhoe K, Van der Sande G.
Task-Independent Computational Abilities of Semiconductor Lasers with Delayed Optical Feedback for Reservoir Computing. *Photonics*. 2019; 6(4):124.
https://doi.org/10.3390/photonics6040124

**Chicago/Turabian Style**

Harkhoe, Krishan, and Guy Van der Sande.
2019. "Task-Independent Computational Abilities of Semiconductor Lasers with Delayed Optical Feedback for Reservoir Computing" *Photonics* 6, no. 4: 124.
https://doi.org/10.3390/photonics6040124