## Calculus of exact detuning phase shift error in temporal phase shifting algorithms

Optics Express, Vol. 17, Issue 18, pp. 15766-15771 (2009)

http://dx.doi.org/10.1364/OE.17.015766

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### Abstract

The detuning phase shift error is a common systematic error observed in temporal phase shifting (TPS) algorithms. This error, generally due to miscalibration of the phase shifter, is solved by using a quadrature filter insensitive to this detuning error. To compare algorithms, this error is frequently analyzed numerically. However, in this work we present an exact and analytical expression to calculate such error which is applicable to any kind of filters with real or complex frequency response. Finally, a table with the detuning error for several algorithms is reported.

© 2009 OSA

## 1. Introduction

## 2. Error Detuning in Phase-Shifting Interferometry (PSI)

*r*. That is, from Eq. (15) and the ratio

*r*we can obtain the exact detuning error for any TPS algorithm. Then, from Eq. (15) we can observe that if

7. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase measuring interferometry,” J. Opt. Soc. Am . **A 7**(4), 542–551 (1990). [CrossRef]

*φ*, obtaining the following expression,We emphasize that this Eq. (18) for the maximum detuning error is exact when tuned onto the left side; in this fashion, it coincides exactly with the detuning error that was evaluated numerically [1–7

7. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase measuring interferometry,” J. Opt. Soc. Am . **A 7**(4), 542–551 (1990). [CrossRef]

9. J. F. Mosiño, M. Servin, J. C. Estrada, and J. A. Quiroga, “Phasorial analysis of detuning error in temporal phase shifting algorithms,” Opt. Express **17**(7), 5618–5623 (2009). [CrossRef] [PubMed]

7. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase measuring interferometry,” J. Opt. Soc. Am . **A 7**(4), 542–551 (1990). [CrossRef]

## 3. Some Examples of Error Detuning in Phase-Shifting Interferometry

### 3.1 Three frame algorithm case

### 3.2 Four frame algorithm

### 3.2 Other algorithms

*r,*for some detuning phase shift errors for several TPS algorithms are presented. We notice that many of them have the form tan

*(Δ/2) for*

^{n}*n*integer.

## 4. Conclusions

## Acknowledgments

## References and links

1. | J. Schwider, “Advanced evaluation techniques in interferometry,” in |

2. | J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometry for testing optical surfaces and lenses,” Appl. Opt. |

3. | H. Schreiber, J. H. Brunning, and J. E. Greivenkamp, “Phase shifting interferometry,” in |

4. | M. Servin, and M. Kujawinska, “Modern fringe pattern analysis in Interferometry,” in |

5. | D. Malacara, M. Servin, and Z. Malacara, “Phase Detection Algorithms,” in |

6. | J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. |

7. | K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase measuring interferometry,” J. Opt. Soc. Am . |

8. | J. E. Hernández and D. Malacara, “Exact linear detuning error in phase shifting algorithms,” Opt. comm. |

9. | J. F. Mosiño, M. Servin, J. C. Estrada, and J. A. Quiroga, “Phasorial analysis of detuning error in temporal phase shifting algorithms,” Opt. Express |

**OCIS Codes**

(120.2650) Instrumentation, measurement, and metrology : Fringe analysis

(120.3180) Instrumentation, measurement, and metrology : Interferometry

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

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: June 17, 2009

Revised Manuscript: July 19, 2009

Manuscript Accepted: July 21, 2009

Published: August 20, 2009

**Citation**

J. F. Mosiño, D. Malacara Doblado, and D. Malacara Hernández, "Calculus of exact detuning phase shift error in temporal phase shifting algorithms," Opt. Express **17**, 15766-15771 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-18-15766

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### References

- J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf ed., (North Holland, Amsterdam, Oxford, New York, Tokyo, 1990).
- J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometry for testing optical surfaces and lenses,” Appl. Opt. 13(11), 2693–2703 (1974). [CrossRef] [PubMed]
- H. Schreiber, J. H. Brunning, and J. E. Greivenkamp, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed., (John Wiley & Sons, Inc., Hoboken, New Jersey 2007).
- M. Servin, and M. Kujawinska, “Modern fringe pattern analysis in Interferometry,” in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds., (Marcel Dekker, 2001).
- D. Malacara, M. Servin, and Z. Malacara, “Phase Detection Algorithms,” in Interferogram Analysis for Optical Testing, D. Malacara ed., (Taylor & Francis Group, 2005).
- J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22(21), 3421–3432 (1983). [CrossRef] [PubMed]
- K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase measuring interferometry,” J. Opt. Soc. Am . A 7(4), 542–551 (1990). [CrossRef]
- J. E. Hernández and D. Malacara, “Exact linear detuning error in phase shifting algorithms,” Opt. comm. 180, 9–14 (2000).
- J. F. Mosiño, M. Servin, J. C. Estrada, and J. A. Quiroga, “Phasorial analysis of detuning error in temporal phase shifting algorithms,” Opt. Express 17(7), 5618–5623 (2009). [CrossRef] [PubMed]

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