# A Radial-In-Plane Sensitivity Interferometer with Divergent Illumination for Displacement Measurement

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

## 3. Experimental Results and Discussion

## 4. Discussion, Advantages, and Disadvantages

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The proposed optical arrangement for ESPI interferometry with radial-in-plane sensitivity and divergent illumination. (

**b**) The scheme of optical arrangement, where SiL is the source of the illumination laser; fc, one-quarter wave retarder film; fl, linear polarizer; Of, opaque filter; Mf, flat mirror; Mc, conical mirror; Spc, surface specimen; $\overrightarrow{e}$ is the sensitivity vector; and ${\widehat{n}}_{A}$ and ${\widehat{n}}_{B}$ are unitary vectors related to the illumination beams.

**Figure 2.**Reflection geometry for the calculation of the sensitivity vector with divergent illumination.

**Figure 6.**Interferograms obtained in distinct relative angles of the linear polarizer, (

**a**) 0 rad, (

**b**) $\frac{2\pi}{5}$ rad, (

**c**) $\frac{4\pi}{5}$ rad, (

**d**) $\frac{6\pi}{5}$ rad, (

**e**) $\frac{8\pi}{5}$ rad, (

**f**) wrapped phase in module $2\pi $ terms, (

**g**) $\Delta \varphi $ optical phase in rad, (

**h**) position of the heating source.

**Figure 8.**(

**a**) The geometrical model, (

**b**) used mesh in the simulation, and (

**c**) results of the simulation by finite element analysis.

**Figure 9.**The comparison between ESPI radial-in-plane sensitivity measurement and the calculation of the deformation by finite element analysis.

Laser Illumination Data | Wavelength, $\lambda =552$ nm Laser Power: 1 W |
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Digital camera CCD data | Resolution: 804 × 556 pixels |

Divergent illumination source coordinates, $SiL\left({r}_{s},0,{z}_{s}\right)$ | ${r}_{s}=0$ |

${z}_{s}=265$ mm | |

Conic mirror dimensions | Outer radius: 37.5 mm |

Inner radius: 25.0 mm | |

High: 20 mm | |

Conic mirror angle: $\delta =32\xb0$ | |

Illuminated surface radius | Radius: 25.0 mm |

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**MDPI and ACS Style**

Parra-Michel, J.R.; Gutiérrez-Hernández, D.; Martínez-Peláez, R.; Escobar, M.A.
A Radial-In-Plane Sensitivity Interferometer with Divergent Illumination for Displacement Measurement. *Appl. Sci.* **2020**, *10*, 908.
https://doi.org/10.3390/app10030908

**AMA Style**

Parra-Michel JR, Gutiérrez-Hernández D, Martínez-Peláez R, Escobar MA.
A Radial-In-Plane Sensitivity Interferometer with Divergent Illumination for Displacement Measurement. *Applied Sciences*. 2020; 10(3):908.
https://doi.org/10.3390/app10030908

**Chicago/Turabian Style**

Parra-Michel, Jorge R., David Gutiérrez-Hernández, Rafael Martínez-Peláez, and Marco A. Escobar.
2020. "A Radial-In-Plane Sensitivity Interferometer with Divergent Illumination for Displacement Measurement" *Applied Sciences* 10, no. 3: 908.
https://doi.org/10.3390/app10030908