# Measuring Linewidth Enhancement Factor by Relaxation Oscillation Frequency in a Laser with Optical Feedback

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## Abstract

**:**

## 1. Introduction

## 2. Measurement Theory

## 3. Simulation Test

- Starting from $L=15.0\text{}\mathrm{cm}$, increase the cavity length and set it with 6 different locations.
- At each location, apply a micro displacement $\mathsf{\Delta}L$ onto the external cavity with 0.8 wavelength shown on Figure 2a. Correspondingly, we can use the SMI model described in by Equations (4) and (7) to plot an SMI signal shown in Figure 2b, on which we can locate the accurate locations for case 1 and case 2.
- With the results obtained at step 2 for case 1 and case 2, generate the corresponding laser intensity $E{(t)}^{2}$ by numerically solving the L-K equations. E.g., the laser transient waveform for case 1 shown in Figure 3, from which, the RO frequency can be obtained.
- Repeat steps 2 and 3 for the 6 different locations, we get 6 RO frequencies for each case, denoted by ${f}_{RO1i}$ and ${f}_{RO2i}$, i = 1,2, … 6. The relationship between the relative RO frequency ($({f}_{ROi}-{f}_{RO-zero})/{f}_{RO-zero}$) and the cavity length L for the two cases are depicted on Figure 4. Note that the gradient can be determined by only 2 points. In order to reduce the measured error, we prefer to use more than 2 points, e.g., 6 points to do line fitting to get the gradients at each case. From the gradients of these two fitting lines, $\alpha $ can be calculated by using Equation (15).

_{1}= 0.0050 and S

_{2}= 0.0017, then $\alpha \text{}=\text{}2.94$ which is close to the preset value of 3. Under the same operation condition, we change the preset value of $\alpha $ with different values and measure it by using the above method. The results are shown in Table 2. Relative error is used to measure the measurement performance, calculated by $\left|\alpha -\widehat{\alpha}\right|/\alpha $, where $\alpha $ is the preset true value and $\widehat{\alpha}$ is calculated using the proposed method. It can be seen that the performance is satisfactory. Then we change the injection current with $J=1.3{J}_{\mathrm{th}}$, Table 3 shows the measured results with the similar relative error as in Table 2. It can be concluded that the proposed method can work for different $\alpha $ values including small $\alpha $ with $\alpha <1$. This method does not need to know any internal or external parameters related to the SL, also does not need the external target having a symmetric reciprocate movement. We achieved $\alpha $ measurement by using RO frequencies without relying on the SMI waveform.

## 4. Experiments

_{PZT}). Note that in our experiment, each 0.1 V of the V

_{PZT}corresponds to 27 nm travel length of the PZT. Figure 6a shows the control signal applied on the PZT and Figure 6b is the corresponding SMI signal. After signal processing on the raw experimental signals, we are able to determine the locations for case 1 and case 2. We then set the SL working under quasi-continuous wave (QCW) mode. In this case, by using the method in [25], the transient laser intensity can be captured by using the external fast photodetector and oscilloscope. Figure 7 shows one of the experimental signals for the transient laser intensity. Still, we apply digital signal processing on the raw experimental signal and make the signal clearer. Then, the period of the transient laser intensity can be measured to get the required RO frequency.

_{0}= 15 cm under moderate feedback regime, it gives α = 2.89 which is close to the result obtained by the proposed method.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Simulation results with $L=15.0\text{}\mathrm{cm}$, $\kappa \text{}=\text{}0.00003$, $J=1.5\text{}{J}_{\mathrm{th}}$ and $\alpha =3$. (

**a**): ΔL vs. Time; (

**b**): An SMI signal.

**Figure 4.**Relationship between the relative relaxation oscillation (RO) frequency difference and the external cavity length L.

**Figure 5.**Experimental set-up. BS: beam splitter; PD: photodiode; SL: semiconductor lasers; PZT: piezoelectric transducer.

**Figure 6.**Experimental self-mixing interferometry (SMI) signal (

**a**) control signal applied on PZT; (

**b**) corresponding SMI signal.

Symbol | Physical Meaning | Value |
---|---|---|

$\kappa $ | Feedback strength | |

$\tau $ | External cavity round trip time,$\tau =2L/c$, where L is external cavity length, c is speed of light | |

${\omega}_{0}$ | Angular frequency of solitary laser | |

$\alpha $ | Line-width enhancement factor | |

$J$ | Injection current density | |

${J}_{\mathrm{th}}$ | Threshold injection current density | |

${\tau}_{in}$ | Internal cavity round-trip time | $8.0\times {10}^{-12}\text{}\mathrm{s}$ |

${\tau}_{p}$ | Photon life time | $2.0\times {10}^{-12}\text{}\mathrm{s}$ |

${\tau}_{s}$ | Carrier life time | $2.0\times {10}^{-9}\text{}\mathrm{s}$ |

${G}_{N}$ | Modal gain coefficient | $8.1\times {10}^{-13}\text{}{\mathrm{m}}^{3}{\mathrm{s}}^{-1}$ |

${N}_{0}$ | Carrier density at transparency | $1.1\times {10}^{24}\text{}{\mathrm{m}}^{-3}$ |

$\epsilon $ | Nonlinear gain compression coefficient | $2.5\times {10}^{-23}\text{}{\mathrm{m}}^{3}$ |

$\Gamma $ | Confinement factor | $0$ |

$\mathit{\alpha}\left(\mathit{\tau}\mathit{\rho}\mathit{\upsilon}\mathit{\epsilon}\right)\text{}$ | $\widehat{\mathit{\alpha}}\left(\mathit{s}\mathit{i}\mathit{m}\mathit{u}\mathit{l}\mathit{a}\mathit{t}\mathit{e}\mathit{d}\right)\text{}$ | $\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}\text{}\mathbf{\%}\text{}$ |
---|---|---|

0.50 | 0.52 | 3.8% |

1.00 | 1.01 | 0.6% |

2.00 | 2.04 | 1.9% |

3.00 | 2.94 | 2.0% |

4.00 | 3.81 | 4.8% |

5.00 | 4.75 | 5.0% |

$\mathit{\alpha}\left(\mathit{t}\mathit{r}\mathit{u}\mathit{e}\right)\text{}$ | $\widehat{\mathit{\alpha}}\left(\mathit{s}\mathit{i}\mathit{m}\mathit{u}\mathit{l}\mathit{a}\mathit{t}\mathit{e}\mathit{d}\right)\text{}$ | $\mathit{E}\mathit{r}\mathit{r}\mathit{o}\mathit{r}\text{}\mathbf{\%}\text{}$ |
---|---|---|

0.50 | 0.51 | 2.4% |

1.00 | 1.04 | 4.2% |

2.00 | 2.05 | 2.7% |

3.00 | 3.08 | 2.7% |

4.00 | 4.01 | 0.2% |

5.00 | 4.82 | 3.7% |

L (cm) | 15.0 | 15.1 | 15.2 | 15.3 | 15.4 | 15.5 |

${f}_{RO1}\text{}\left(\mathrm{GHz}\right)$ | 4.631 | 4.645 | 4.659 | 4.669 | 4.687 | 4.701 |

${f}_{RO2}\text{}\left(\mathrm{GHz}\right)$ | 4.697 | 4.702 | 4.708 | 4.709 | 4.717 | 4.725 |

${f}_{RO-zero}\text{}\left(\mathrm{GHz}\right)$ | 4.751 | |||||

${S}_{1}=0.139$, ${S}_{2}=0.053$, $\alpha =2.62$. |

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**MDPI and ACS Style**

Ruan, Y.; Liu, B.; Yu, Y.; Xi, J.; Guo, Q.; Tong, J.
Measuring Linewidth Enhancement Factor by Relaxation Oscillation Frequency in a Laser with Optical Feedback. *Sensors* **2018**, *18*, 4004.
https://doi.org/10.3390/s18114004

**AMA Style**

Ruan Y, Liu B, Yu Y, Xi J, Guo Q, Tong J.
Measuring Linewidth Enhancement Factor by Relaxation Oscillation Frequency in a Laser with Optical Feedback. *Sensors*. 2018; 18(11):4004.
https://doi.org/10.3390/s18114004

**Chicago/Turabian Style**

Ruan, Yuxi, Bin Liu, Yanguang Yu, Jiangtao Xi, Qinghua Guo, and Jun Tong.
2018. "Measuring Linewidth Enhancement Factor by Relaxation Oscillation Frequency in a Laser with Optical Feedback" *Sensors* 18, no. 11: 4004.
https://doi.org/10.3390/s18114004