# Features of a Self-Mixing Laser Diode Operating Near Relaxation Oscillation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Simulations and Experiments

_{0}) is 35 cm long. From Figure 1 we can see three regions, i.e., stable, semi-stable and unstable respectively.

- The RO-SM signals exhibit the form of high frequency oscillation with its amplitude modulated by a slow-varying signal. Interestingly, the slow-varying envelopes are similar to the conventional SM signal characterized by the same fringe structure. It can be seen from the left column in Figure 2, that there are nearly six fringes corresponding to the peak-peak displacement ($3{\lambda}_{0}$) of the target. That is, each fringe in the RO-SM signals also corresponds to a target displacement ${\lambda}_{0}/2$, and hence the RO-SM can also be used to measure the displacement with the same resolution as the conventional SMI operating in the stable region.
- Although having the same fringe structure, the RO-SM signals are very different from the conventional SM signals in their frequencies. The right column in Figure 2, shows that, the spectrum of a conventional SM signal (Figure 2k) locates in the relatively low frequency range, stopping at 0.2 GHz for the case with C = 1.5. However, the dominated frequency components associated with a RO-SM signal locate in a much higher frequency range, with a central frequency of 2.3 GHz (denoted by f
_{RO}). f_{RO}is generated due to the relaxation oscillation when the SMI system operates above its stability boundary. - The Peak-Peak (P-P) value of a conventional SM signal (in Figure 2f) located in the stable region is around 0.012 while the P-P values of the RO-SM signals in semi-stable region are about 6.64, 5.04 and 4.12 respectively as shown in Figure 2c–e . Hence the RO-SM signals are much stronger (more than 300 times stronger) than the conventional SM signal. This implies that an SMI system working at a semi-stable region has potential for achieving sensing with improved sensitivity.
- When the SMI system enters the unstable region shown in Figure 2b,g, the laser output will be unstable, characterized by a much wider frequency spectrum. In this case there is not an obvious relationship between the target movement and the laser output, and hence the system is not suitable for such waveform based sensing.

- Initially, set the PZT stationary, and adjust the attenuator so that an RO signal can be observed by the oscilloscope.
- Apply a control signal to the PZT actuator using the PZT controller. The signal is a 30.0 V DC offset superposed with a sinusoidal voltage signal (200 Hz and 3.9 V P-P). The corresponding displacement (P-P) generated by the PZT is $2.08\text{}\mathsf{\mu}\mathrm{m}$.
- Adjust the attenuator to vary the feedback strength from low to high, and record three RO/SM waveforms at different feedback levels, as shown in the left column in Figure 4. Meanwhile, for each case, the frequency spectrum is measured using the function provided by the oscilloscope. The corresponding frequency spectra are shown in the right column.

_{0}/2 of the LD. Note that, due to the use of a very high sampling frequency (i.e., 12.5 GS/s), we are only able to take a very short time duration (indicated in Figure 4b) of the signals for FFT. Hence, the spectrum in Figure 4 can only show where the central frequency (f

_{RO}) is. At the semi-stable region, a dominant frequency component can be found at the location of 2.19 GHz, as shown in Figure 4e,f. The envelope in the signal disappears in Figure 4d and its corresponding spectrum is significantly broadened. In this case, the SMI system is working in the unstable region. The observed phenomena from the experiments are consistent with the simulations results. Note that there exists a frequency component at 1.19 GHz across all the spectra shown Figure 4 and Figure 5; the reason for its existence will be explained below.

_{0}= 830 nm, P = 30 mW) and we did the experiments under the conditions of J = 46 mA and L

_{0}= 12.5 cm for both stable and semi-stable cases. The experimental results are presented in Figure 6. For the same target movement, we noticed that RO-SM signals are always much stronger (about 93 times higher) than a conventional SM signal.

_{0}= 785 nm, P = 20 mW). The results are shown in Figure 7. The conventional SM signal is obtained with L

_{0}= 11 cm, J = 60 mA and the RO-SM signal with L

_{0}= 14 cm, J = 46 mA. It shows that the P-P value of RO-SM signal is 27 times that of the conventional SM signal. The spectrum obtained for a small piece from the RO-SM is shown in Figure 7e with f

_{RO}= 4.3 GHz. In the semi-stable region, we also recorded the RO signal and its spectrum is shown in Figure 7d,f, respectively, when the target is stationary. It can be seen that f

_{RO}is same as the one in Figure 7e. This implies that an RO-SM signal has a central frequency that is the same as the relaxation oscillation of the LD system with optical feedback.

## 3. Conclusions

## Author Contributions

## Conflicts of Interest

## References

- Rudd, M.J. A laser Doppler velocimeter employing the laser as a mixer-oscillator. J. Phys. E Sci. Instrum.
**1968**, 1, 723. [Google Scholar] [CrossRef] - Donati, S. Laser interferometry by induced modulation of cavity field. J. App. Phys.
**1978**, 49, 495–497. [Google Scholar] [CrossRef] - Giuliani, G.; Norgia, M.; Donati, S.; Bosch, T. Laser diode selfmixing technique for sensing applications. J. Opt. A Pure Appl. Opt.
**2002**, 4, S283–S294. [Google Scholar] [CrossRef] - Yu, Y.; Giuliani, G.; and Donati, S. Measurement of the linewidth enhancement factor of semiconductor lasers based on the optical feedback self-mixing effect. IEEE Photonics Technol. Lett.
**2004**, 16, 990–992. [Google Scholar] [CrossRef] - Lin, K.; Yu, Y.; Xi, J.; Fan, Y.; Li, H. Measuring Young’s modulus using a self-mixing laser diode. In Proceedings of the SPIE—The International Society for Optical Engineering, San Francisco, CA, USA, 1 February 2014.
- Nikolić, M.; Hicks, E.; Lim, Y.L.; Bertling, K.; Rakić, A.D. Self-mixing laser Doppler flow sensor: An optofluidic implementation. Appl. Opt.
**2013**, 52, 8128–8133. [Google Scholar] - Taimre, T.; Bertling, K.; Lim, Y.L.; Dean, P.; Indjin, D.; Rakić, A.D. Methodology for materials analysis using swept-frequency feedback interferometry with terahertz frequency quantum cascade lasers. Opt. Express
**2014**, 22, 18633–18647. [Google Scholar] [CrossRef] [PubMed] - Yu, Y.; Xi, J.; Chicharo, J.F. Measuring the feedback parameter of a semiconductor laser with external optical feedback. Opt. Express
**2011**, 19, 9582–9593. [Google Scholar] [CrossRef] [PubMed] - Guo, D.; Wang, M.; Tan, S. Self-mixing interferometer based on sinusoidal phase modulating technique. Opt. Express.
**2005**, 13, 1537–1543. [Google Scholar] [CrossRef] [PubMed] - Yu, Y.; Ye, H.; Yao, J. Analysis for the self-mixing interference effects in a laser diode at high optical feedback levels. J. Opt. A Pure Appl. Opt.
**2003**, 5, 117–122. [Google Scholar] [CrossRef] - Norgia, M.; Pesatori, A.; Rovati, L. Self-Mixing Laser Doppler Spectra of Extracorporeal Blood Flow: A Theoretical and Experimental Study. IEEE Sens.
**2012**, 12, 552–557. [Google Scholar] [CrossRef] - Plantier, G.; Bes, C.; Bosch, T. Behavioral model of a self-mixing laser diode sensor. IEEE J. Quantum Electron.
**2005**, 41, 1157–1167. [Google Scholar] [CrossRef] - Lang, R.; Kobayashi, K. External optical feedback effects on semiconductor injection laser properties. IEEE J. Quantum Electron.
**1980**, 16, 347–355. [Google Scholar] [CrossRef] - Tromborg, B.; Osmundsen, J.; Olesen, H. Stability analysis for a semiconductor laser in an external cavity. IEEE J. Quantum Electron.
**1984**, 20, 1023–1032. [Google Scholar] [CrossRef] - Helms, J.; Petermann, K. A simple analytic expression for the stable operation range of laser diodes with optical feedback. IEEE J. Quantum Electron.
**1990**, 26, 833–836. [Google Scholar] [CrossRef] - Mork, J.; Tromborg, B.; Mark, J. Chaos in semiconductor lasers with optical feedback: Theory and experiment. IEEE J. Quantum Electron.
**1992**, 28, 93–108. [Google Scholar] [CrossRef] - Uchida, A. Analysis of Chaotic Laser Dynamics: Example of Semiconductor Laser with Optical Feedback. In Optical Communication with Chaotic Lasers: Applications of Nonlinear Dynamics and Synchronization, 1st ed.; John Wiley & Sons: Hoboken, NJ, USA, 2012; pp. 145–199. [Google Scholar]
- Fan, Y.; Yu, Y.; Xi, J.; Guo, Q. Dynamic stability analysis for a self-mixing interferometry system. Opt. Express
**2014**, 22, 29260–29269. [Google Scholar] [CrossRef] [PubMed] - Fan, Y.; Yu, Y.; Xi, J.; Guo, Q. Stability limit of a semiconductor laser with optical feedback. IEEE J. Quantum Electron.
**2015**, 51, 1–9. [Google Scholar] [CrossRef]

**Figure 1.**Improved stability boundary for describing an SMI system when ${L}_{0}=35\text{}\mathrm{cm}$, $J=1.1J\mathrm{th}$.

**Figure 2.**Modulated laser intensity at different regions and their corresponding spectra, where the laser intensity is scaled by ${E}^{2}(t)/{10}^{20}$ (

**a**) displacement of the external target; (

**b**–

**f**) laser intensity when C = 9, C = 5, C = 3.5, C = 2.5, and C = 1.5 respectively; (

**g**–

**k**) spectra corresponding to (

**b**–

**f**) respectively. Note that the DC component in each case is removed when applying FFT.

**Figure 4.**Experimental signals and their spectra in semi-stable and unstable regions. (

**a**) PZT control signal; (

**b**,

**c**) RO-SM signals in semi-stable region; (

**d**) SM signal in unstable region; (

**e**–

**g**) Spectra corresponding to (

**b**–

**d**) respectively.

**Figure 5.**Experimental signals and their spectra in stable and semi-stable region. (

**a**) PZT control signal; (

**b**) conventional SM signals at stable region; (

**c**,

**d**) RO-SM signals at semi-stable region; (

**e**–

**g**) the spectra corresponding to (

**b**–

**d**) respectively.

**Figure 6.**Experimental signals with J = 46 mA and L

_{0}= 12.5 cm for DL5032-001 (

**a**) PZT control signal; (

**b**) conventional SM signals at stable region; (

**c**,

**d**) RO-SM signals at semi-stable region.

**Figure 7.**Experimental signals for DL4140-001S. (

**a**) PZT control signal; (

**b**) conventional SM signals at stable region; (

**c**) RO-SM signals at semi-stable region; (

**d**) the laser intensity when the target is stationary; (

**e**,

**f**) the spectra corresponding to (

**c**,

**d**) respectively.

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

$t$ | time | |

$E(t)$ | amplitude of the intra-cavity electric field | |

$\varphi (t)$ | phase of the intra-cavity electric field, $\varphi (t)=\left[\omega (t)-{\omega}_{0}\right]t$ | |

$\omega (t)$ | laser angular frequency with feedback | |

${\omega}_{0}$ | Laser angular frequency without feedback | $2.42\times {10}^{15}{\text{}\mathrm{rads}}^{-1}$ |

$N(t)$ | carrier density | |

${G}_{N}$ | modal gain coefficient | $8.1\times {10}^{-13}{\text{}\mathrm{m}}^{3}\xb7{\mathrm{s}}^{-1}$ |

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

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

$\mathrm{\Gamma}$ | confinement factor | $3$ |

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

$\kappa $ | feedback strength | |

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

$\alpha $ | line-width enhancement factor | $6$ |

$J$ | injection current | |

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

$\tau $ | light roundtrip time in the external cavity, $\tau =2L/c$ | |

$L$ | external cavity length |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liu, B.; Yu, Y.; Xi, J.; Fan, Y.; Guo, Q.; Tong, J.; Lewis, R.A.
Features of a Self-Mixing Laser Diode Operating Near Relaxation Oscillation. *Sensors* **2016**, *16*, 1546.
https://doi.org/10.3390/s16091546

**AMA Style**

Liu B, Yu Y, Xi J, Fan Y, Guo Q, Tong J, Lewis RA.
Features of a Self-Mixing Laser Diode Operating Near Relaxation Oscillation. *Sensors*. 2016; 16(9):1546.
https://doi.org/10.3390/s16091546

**Chicago/Turabian Style**

Liu, Bin, Yanguang Yu, Jiangtao Xi, Yuanlong Fan, Qinghua Guo, Jun Tong, and Roger A. Lewis.
2016. "Features of a Self-Mixing Laser Diode Operating Near Relaxation Oscillation" *Sensors* 16, no. 9: 1546.
https://doi.org/10.3390/s16091546