## Phase shifting speckle interferometry for dynamic phenomena

Optics Express, Vol. 16, Issue 7, pp. 4665-4670 (2008)

http://dx.doi.org/10.1364/OE.16.004665

Acrobat PDF (5469 KB)

### Abstract

The paper presents an algorithm able to retrieve the phase in speckle interferometry by a single intensity pattern acquired in a deformed state, provided that the integrated speckle field is resolved in the reference condition in terms of mean intensity, modulation amplitude and phase. The proposed approach, called throughout the paper “one-step”, can be applied for studying phenomena whose rapid evolution does not allow the application of a standard phase-shifting procedure, which, on the other hand, must be applied at the beginning of the experiment. The approach was proved by an experimental test reported at the end of the paper.

© 2008 Optical Society of America

## 1. Introduction

3. J. M. Huntley, “Automated fringe pattern analysis in experimental mechanics: a review,” J. Strain Analysis Eng. Design **33**, 105–125 (1998). [CrossRef]

8. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transorm method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. **72**, 156–160 (1982). [CrossRef]

11. M. Kujawinska and D. W. Robinson, “Multichannel phase-stepped holographic interferometry,” Appl. Opt. **27**, 312–320 (1988). [CrossRef] [PubMed]

## 2. Theory

*can be mathematically described by the complex intensity*

**P***(*

**I***) [16]:*

**P***m*(

*) is the mean intensity, ‖*

**P***(*

**A***)‖ the modulation amplitude and*

**P***φ*(

*) the phase. The bold type is used for the vector quantity*

**P***and for the complex numbers*

**P***(*

**I***) and*

**P***(*

**A***). The physical quantity detected by the light sensor is the light intensity, that is the real part of the complex intensity. If a phase variation occurs, as it happens when a phase-shifting algorithm is applied or when the observed surface undergoes a deformation, the light intensity can be expressed as:*

**P***Δφ*is the phase variation and

*i*is the light intensity. In Eq. (2) the dependence on the spatial coordinates

*is omitted for the sake of brevity. Figure 1 shows graphically the Eq. (1) and (2) in the complex plane, where the vectors represent the complex intensity and the projections on the real axis represent the detected light intensity.*

**P***m*, ‖

*‖ and*

**A***φ*, and this procedure must be applied before and after the deformation occurs. If we suppose that the mean intensity and modulation amplitude do not change during the deformation of the investigated object, after the speckle field is resolved in the reference condition the light intensity depends only on the phase variation, which is the sole unknown. However, as widely reported in literature, this unknown cannot be directly retrieved from the light intensity, due to the ambiguity on the sign, and at least two acquisitions must be used.

*j*-th pixel of the region around the

*k*-th pixel as follows:

*m*(

_{j}*t*

_{0}) and

*(*

**A**_{j}*t*

_{0}) are the mean intensity and the modulation amplitude in the initial condition of the

*j*-th pixel, whereas Δ

*φ*(

_{k}*t*) is the phase variation of the whole region after the deformation. If

*N*pixels fall in this region around the

*k*-th pixel Eq. (3) can be arranged in the following matrix form:

*φ*occurs. The real and the imaginary parts of the complex modulation in the reference condition are the two columns of the matrix

_{k}**C**

_{k}, the difference between the light intensity in the deformed configuration and the mean intensity in the reference condition is the known vector Δ

**I**

_{k}and it is the real part of the complex modulation in the deformed configuration, as shown in Fig. 2.

*k*-th pixel can be retrieved by applying the two-argument

**ArcTan**function implemented in most of the programming languages [18,19] and able to take into account which quadrant the angle is in, and so to overcome the ambiguity on the sign.

**C**

_{k}depends only on the complex modulation (i.e. the modulation amplitude and the phase) in the reference condition, then the phase of the deformed surface can be retrieved by multiplying the pseudoinverse of the matrix

**C**

_{k}by the vector Δ

**I**

_{k}, which is calculated by subtracting the light intensities of the single step acquired in the deformed configuration by the corresponding mean intensity in the reference condition. Obviously this operation must be carried out for each pixel.

## 3. Application of the procedure and experimental results

*µm*(the speckle size is about half of the pixel size). In the application, after the speckle was resolved in the initial condition by a conventional phase-shifting procedure, a 2.63

*s*video was acquired, corresponding to 79 frames; these were then analyzed one by one by the onestep algorithm. Figure 4 reports three representative frames of the multimedia file (linked to the figure) generated by the processing of the aforementioned video. The region of interest of each frame is an area of 640×640 pixels.

*Mathematica*

^{®}environment by the two functions reported in Tab. 1. The inputs of first function (

**GlobalPseudoInverse**) are: the kernel (

*ker*) which is a matrix like those reported in Fig. 3, the modulation amplitude (

*ia0*) and the phase (

*phi0*) in the reference condition. This function generates for each pixel of the image a 2×

*N*matrix, which represents the pseudoinverse on the matrix

**C**

*, where*

_{k}*N*is the number of pixels identified by the kernel. These data (

*pseudos*), the kernel (

*ker*), the mean intensity in the reference condition (

*im0*) and the frame in the deformed configuration (

*newIm*) are the inputs of the second function (

**OneStep**), which evalute pixel by pixel the phase by means of the two-argument built-in

**ArcTan**.

## 4. Conclusions

*Mathematica*

^{®}environment, is particularly suitable for studying those phenomena rapidly varying in the time. At the present the procedure is not able to work in real-time, but it can be applied at the end of the measuring session on a stored video. In the paper the approach was used for observing the out-of-plane displacements of a debonding cyclically stressed in order to simulate a transient phenomenon.

## References and links

1. | K. Creath, “Temporal phase measurement methods,” in |

2. | M. Kujawinska, “Spatial phase measurement methods,” in |

3. | J. M. Huntley, “Automated fringe pattern analysis in experimental mechanics: a review,” J. Strain Analysis Eng. Design |

4. | P. A. A. M. Somers and N. Bhattacharya, “Maintaining sub-pixel alignment for a single-camera two-bucket shearing speckle interferometer,” J. Opt. A: Pure and Appl. Opt. |

5. | L. Yang, W. Steinchen, G. Kupfer, P. Mackel, and F. Vossing, “Vibration analysis by means of digital shearography,” Opt. Lasers Eng. |

6. | F. J. Casillas, A. Davilla, S. J. Rothberg, and G. Garnica, “Small amplitude estimation of mechanical vibrations using electronic speckle shearing pattern interferometry,” Opt. Eng. |

7. | S. Liu, D. Thomas, P. R. Samala, and L. X. Yang, “Vibration measurement of MEMS by digital laser microinterferometer,” Proc. SPIE |

8. | M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transorm method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. |

9. | G. H. Kaufmann and G. E. Galizzi, “Phase measurement in temporal speckle pattern interferometry:comparison between the phase-shifting and the Fourier transform methods,” Appl. Opt. |

10. | B. Bhaduri, N. Krishna Mohan, M. P. Kothiyal, and R. S. Sirohi, “Use of spatial phase shifting technique in digital speckle pattern interferometry (DSPI) and digital shearography (DS),” Opt. Express |

11. | M. Kujawinska and D. W. Robinson, “Multichannel phase-stepped holographic interferometry,” Appl. Opt. |

12. | M. Adachi, Y. Ueyama, and K. Inabe, “Automatic deformation analysis in electronic speckle pattern interferometry using one speckle interferogram of deformed object,” Opt. Rev. |

13. | C. C. Kao, G. B. Yeh, S. S. Lee, C. K. Lee, C. S. Yang, and K. C. Wu, “Phase-shifting algorithms for electronic speckle pattern interferometry,” Appl. Opt. |

14. | W. An and T. E. Carlsson, “Speckle interferometry for measurement of continuous deformations,” Opt. Lasers Eng. |

15. | M. Adachi, J. N. Petzing, and D. Kerr, “Deformation-phase measurement of diffuse objects that have started nonrepeatable dynamic deformation,” Appl. Opt. |

16. | L. Bruno and A. Poggialini, “Phase retrieval in speckle interferometry: a one-step approach,” in |

17. | A. E. Arthur, |

18. | A. Knight, |

19. | S. Wolfram, |

20. | L. Bruno, G. Felice, and A. Poggialini, “Design and calibration of a piezoelectric actuator for interferometric applications,” Opt. Lasers Eng. |

**OCIS Codes**

(100.5070) Image processing : Phase retrieval

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

(120.6160) Instrumentation, measurement, and metrology : Speckle interferometry

(350.4600) Other areas of optics : Optical engineering

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: February 13, 2008

Revised Manuscript: March 17, 2008

Manuscript Accepted: March 18, 2008

Published: March 20, 2008

**Virtual Issues**

Vol. 3, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Luigi Bruno and Andrea Poggialini, "Phase shifting speckle interferometry for dynamic phenomena," Opt. Express **16**, 4665-4670 (2008)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-7-4665

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### References

- K. Creath, "Temporal phase measurement methods," in Interferogram Analysis, D. W. Robinson and G. T. Reid, eds., (Institute of Physics Publishing, Bristol, Philadelphia, 1993) Chap. 4, pp. 94-140.
- M. Kujawinska, "Spatial phase measurement methods," in Interferogram Analysis, D. W. Robinson and G. T. Reid, eds., (Institute of Physics Publishing, Bristol, Philadelphia, 1993) Chap. 5, pp. 141-193
- J. M. Huntley, "Automated fringe pattern analysis in experimental mechanics: a review," J. Strain Anal. Eng. Des. 33, 105-125 (1998). [CrossRef]
- P. A. A. M. Somers and N. Bhattacharya, "Maintaining sub-pixel alignment for a single-camera two-bucket shearing speckle interferometer," J. Opt. A: Pure and Appl. Opt. 7, S385-S391 (2005). [CrossRef]
- L. Yang, W. Steinchen, G. Kupfer, P. Mackel, and F. Vossing, "Vibration analysis by means of digital shearography," Opt. Lasers Eng. 30, 199-212 (1998). [CrossRef]
- F. J. Casillas, A. Davilla, S. J. Rothberg, and G. Garnica, "Small amplitude estimation of mechanical vibrations using electronic speckle shearing pattern interferometry," Opt. Eng. 43, 880-887 (2004). [CrossRef]
- S. Liu, D. Thomas, P. R. Samala and L. X. Yang, "Vibration measurement of MEMS by digital laser microinterferometer," Proc. SPIE 5878, 58780C1-9 (2005).
- M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transorm method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am. 72, 156-160 (1982). [CrossRef]
- G. H. Kaufmann and G. E. Galizzi, "Phase measurement in temporal speckle pattern interferometry: comparison between the phase-shifting and the Fourier transform methods," Appl. Opt. 41, 7254-7263 (2002). [CrossRef] [PubMed]
- B. Bhaduri, N. Krishna Mohan, M. P. Kothiyal, and R. S. Sirohi, "Use of spatial phase shifting technique in digital speckle pattern interferometry (DSPI) and digital shearography (DS)," Opt. Express 14, 11598-11607 (2006). [CrossRef] [PubMed]
- M. Kujawinska and D. W. Robinson, "Multichannel phase-stepped holographic interferometry," Appl. Opt. 27, 312-320 (1988). [CrossRef] [PubMed]
- M. Adachi, Y. Ueyama, and K. Inabe, "Automatic deformation analysis in electronic speckle pattern interferometry using one speckle interferogram of deformed object," Opt. Rev. 4, 429-432 (1997). [CrossRef]
- C. C. Kao, G. B. Yeh, S. S. Lee, C. K. Lee, C. S. Yang, and K. C. Wu, "Phase-shifting algorithms for electronic speckle pattern interferometry," Appl. Opt. 41, 46-54 (2002). [CrossRef] [PubMed]
- W. An and T. E. Carlsson, "Speckle interferometry for measurement of continuous deformations," Opt. Lasers Eng. 40, 529-541 (2003). [CrossRef]
- M. Adachi, J. N. Petzing, and D. Kerr, "Deformation-phase measurement of diffuse objects that have started nonrepeatable dynamic deformation," Appl. Opt. 40, 6187-6192 (2001). [CrossRef]
- L. Bruno and A. Poggialini, "Phase retrieval in speckle interferometry: a one-step approach," in Proceedings of Interferometry in Speckle Light - Theory and Applications, P. Jacquot and J. M. Fournier, eds., (Springer, Heidelberg, Germany, 2000), pp. 461-472.
- A. E. Arthur, Regression and the Moore-Penrose Pseudoinverse (Academic Press, New York, 1972).
- A. Knight, Basics of Matlab and Beyond (Chapman & Hall/CRC, Natick, Massachusetts, 2000).
- S. Wolfram, The Mathematica Book, 5th edition (Wolfram Media, Champaign, Illinois, 2003).
- L. Bruno, G. Felice and A. Poggialini, "Design and calibration of a piezoelectric actuator for interferometric applications," Opt. Lasers Eng. 45, 1148-1156 (2007). [CrossRef]

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