# A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus

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## Abstract

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## 1. Introduction

## 2. Measurement Principle

#### 2.1. Measurement Formula for Young’s Modulus

#### 2.2. Vibrating Signals Generated by the Test Specimen

#### 2.3. Capture $y(t)$ Using Fiber-Coupled SMLD

## 3. System Design

#### 3.1. Mechanical Supporting for the Specimen

#### 3.2. Steel Ball for Stimulation

_{s}roughly as:

_{0}can thus be approximately determined by

_{0}and the ball’s size ${R}_{steel}$. In our design, the ball moves down along a guided tube and hits onto the center of the specimen. The set-up for the mechanical excitation part is shown is Figure 4. The tube is installed with a tilt angle with respect to the specimen’s plane. When the ball hits onto the specimen, an impulsive force (denoted by $F$) will be generated and thus cause a corresponding ${A}_{0}$. For the given specimen with the dimension shown in Figure 1 and ${A}_{0}$ determined above, F is expressed as below by solving the bending moment equations [29],

#### 3.3. Requirements for SMLD

- Step 1: Measure the stability boundary of the SMLD system and from which to determine a suitable external cavity length to place the tested specimen.
- Step 2: Estimate the maximum magnitude ${A}_{0}$ by Equation (12). Note that a low ${f}_{RO}$, e.g., can be used for the estimation.
- Step 3: Calculate the size of the steel ball ${R}_{steel}$ using Equations (13) and (15) and ${A}_{0}$.

## 4. Simulations

## 5. Experiments

#### 5.1. Experimental Set-up and Results

- Step 1: Install the LD onto a laser mount; set the bias current on the laser controller (LTC100-B from THORLABS) as 52.5 mA and the temperature on the temperature controller (TED200C from THORLABS) is stabilized to 25 ± 0.1 °C.
- Step 2: Install a specimen to be tested and use a coupler (PAF-X-2-B from THORLABS) connected with a step-index multimode fiber optic patch cable (M67L02 from THORLABS) with an adjustable aspheric FC collimators (CFC-2X-B from THORLABS) at the other end to adjust the distance between the specimen and the LD to form an external cavity with 0.5 m long.
- Step 3: Adjust the LD mount so that the fiber-coupled SMLD can be operated in a moderate feedback level by observing the waveform of the SMI signal.
- Step 4: Place the steel ball on the up end of the guided tube and release it. As a result, the specimen is stimulated into vibration. Correspondingly, an SMI signal is produced by the SMLD and recorded by the oscilloscope and the computer through the DAQ card. A LabVIEW script programmed for sampling the SMI signal is set to wait for collecting the signal.

#### 5.2. Comparison with Tensile Testing

^{−3}/s. The load values were recorded by the load cell of the Instron machine. To ensure the measurement accuracy of Young’s modulus, DANTEC digital image correlation (DIC) system was adopted to record the displacement of tensile specimens during the tests. Before testing, random speckle patterns were generated on the specimen surfaces by spray painting. The overall displacement of the entire gauge regions of the specimens was recorded by two high speed cameras facing the speckled surfaces at a frame rate of 5 Hz. The images were 2448 by 2448 pixels with an 8-bit dynamic range. ISTRA 4D software was used to analyze the images and obtain extension values of the gauge regions. The load obtained from the Instron machine and the extension obtained from the DIC system were used to calculate stress and strain values. The stress–strain curves were plotted afterwards. Young’s modulus was obtained from the elastic deformation region of the stress–strain curves.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

SMLD | Self-Mixing laser diode |

LD | Laser Diode |

PD | Photodiode |

SMI | Self-mixing interferometry |

FFT | Fast Fourier Transform |

DAQ | Data Acquisition |

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**Figure 3.**(

**a**) Normalized 1st-order mode vibration of a free-free rectangular specimen; and (

**b**) mechanical support for achieving 1st-order vibration.

**Figure 4.**(

**a**) Set-up for the generation of the excitation; and (

**b**) left-view of the excitation system.

**Figure 7.**FFT spectrum of $G(t)$ under different feedback level. (

**a**) Signal zoomed in from 0.8 s (

**b**) Amplitude spectrum of FFT.

Parameters | Physical Meaning | Unit |
---|---|---|

$t$ | Time index. | s |

${\varphi}_{F}(t)$ | Laser phase with feedback | rad |

${\varphi}_{0}(t)$ | Feedback level factor | rad |

$C$ | Line-width enhancement factor | - |

$\alpha $ | Interference function which indicates the influence of the optical feedback | - |

$G(t)$ | Interference function which indicates the influence of the optical feedback | - |

$m$ | Modulation index for the laser intensity (typically $m\approx 0.001$) | - |

${P}_{0}$ | Laser intensity emitted by the free running LD | - |

$P(t)$ | Laser intensity when LD with optical feedback | - |

Specimen | Aluminum 6061 | Brass | |||
---|---|---|---|---|---|

Times (N) | ${f}_{RO}$ (Hz) | $E$ (GPa) | ${f}_{RO}$ (Hz) | $E$ (GPa) | |

1 | 599 | 70.2 | 451 | 116.6 | |

2 | 598 | 70.0 | 450 | 116.1 | |

3 | 599 | 70.2 | 451 | 116.6 | |

4 | 598 | 70.0 | 451 | 116.6 | |

5 | 597 | 69.7 | 452 | 117.1 | |

6 | 598 | 70.0 | 451 | 116.6 | |

7 | 599 | 70.2 | 451 | 116.6 | |

8 | 598 | 70.0 | 452 | 117.1 | |

9 | 599 | 70.2 | 451 | 116.6 | |

10 | 598 | 70.0 | 451 | 116.6 | |

Mean (μ) | 598 | 70.0 | 451 | 116.7 | |

Standard deviation (σ) | 0.68 | 0.16 | 0.57 | 0.29 |

Times (N) | 1 | 2 | 3 | 4 | 5 | 6 | Mean (μ) | Standard Deviation (σ) | Accuracy (σ/ μ%) | |
---|---|---|---|---|---|---|---|---|---|---|

Specimen | ||||||||||

Aluminum 6061 | 60.6 | 64.4 | 76.2 | 67.0 | 73.9 | 63.0 | 67.6 | 6.2 | 9.2 | |

Brass | 120.3 | 125.6 | 133.4 | 118.6 | 109.6 | 119.4 | 121.1 | 7.9 | 6.5 |

© 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/).

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**MDPI and ACS Style**

Lin, K.; Yu, Y.; Xi, J.; Li, H.; Guo, Q.; Tong, J.; Su, L.
A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus. *Sensors* **2016**, *16*, 928.
https://doi.org/10.3390/s16060928

**AMA Style**

Lin K, Yu Y, Xi J, Li H, Guo Q, Tong J, Su L.
A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus. *Sensors*. 2016; 16(6):928.
https://doi.org/10.3390/s16060928

**Chicago/Turabian Style**

Lin, Ke, Yanguang Yu, Jiangtao Xi, Huijun Li, Qinghua Guo, Jun Tong, and Lihong Su.
2016. "A Fiber-Coupled Self-Mixing Laser Diode for the Measurement of Young’s Modulus" *Sensors* 16, no. 6: 928.
https://doi.org/10.3390/s16060928