## Design of algorithms for phase shifting interferometry using self-convolution of the rectangle window |

Optics Express, Vol. 19, Issue 15, pp. 14671-14681 (2011)

http://dx.doi.org/10.1364/OE.19.014671

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### Abstract

The objective of this paper is to design phase shifting algorithms error-resistant to the nonlinearity of phase-shift error and photoelectric detector simultaneously. An effective construction approach is proposed based on self-convolution of the rectangle window to design algorithms with perfect zero point distribution, according to the fact that the error-resistant capability is entirely determined by the number and order of zero points of Fourier transform of the related window function. Theoretical analysis and numerical simulations compared to the commercial 13-frame algorithm demonstrate the validity of the approach to design algorithms with enhanced error-resistant capability not only to CCD-caused harmonics but also to PZT ramping nonlinearity.

© 2011 OSA

## 1. Introduction

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

5. K. Hibino, R. Hanayama, J. Burke, and B. F. Oreb, “Tunable phase-extraction formulae for simultaneous shape measurement of multiple surfaces with wavelength-shifting interferometry,” Opt. Express **12**(23), 5579–5594 (2004). [CrossRef] [PubMed]

6. P. Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. **34**(22), 4723–4730 (1995). [CrossRef] [PubMed]

7. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE **66**(1), 51–83 (1978). [CrossRef]

8. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting for non-sinusoidal waveforms with phase shift errors,” J. Opt. Soc. Am. A **12**(4), 761–768 (1995). [CrossRef]

9. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. **35**(1), 51–60 (1996). [CrossRef] [PubMed]

10. D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. **36**(31), 8098–8115 (1997). [CrossRef] [PubMed]

11. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express **17**(24), 21867–21881 (2009). [CrossRef] [PubMed]

12. J. C. Estrada, M. Servin, and J. A. Quiroga, “Easy and straightforward construction of wideband phase-shifting algorithms for interferometry,” Opt. Lett. **34**(4), 413–415 (2009). [CrossRef] [PubMed]

## 2. Principle

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

*s*and

_{n}*c*are the coefficients of algorithms,

_{n}*w*is the complex coefficients,

_{n}*I*is the

_{n}*n*th fringe,

3. H. Zhang, M. J. Lalor, and D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. **38**(9), 1524–1533 (1999). [CrossRef]

5. K. Hibino, R. Hanayama, J. Burke, and B. F. Oreb, “Tunable phase-extraction formulae for simultaneous shape measurement of multiple surfaces with wavelength-shifting interferometry,” Opt. Express **12**(23), 5579–5594 (2004). [CrossRef] [PubMed]

9. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. **35**(1), 51–60 (1996). [CrossRef] [PubMed]

10. D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. **36**(31), 8098–8115 (1997). [CrossRef] [PubMed]

*k*th-order harmonics;

*v*

_{0}is the PZT ramping velocity and

*q*equidistant multiple roots of order

*p*+1 except at

*k*= −1. Namely it can filter out the unwanted components with the first negative fundamental frequency passed. An example is shown in Fig. 2 . The greater the

*p*and

*q*values satisfy Eq. (8), the stronger the ability of the algorithm error-resistant to the PZT and CCD nonlinearity will be, but always at the cost of more needed frames. In what follows, a self-convolution construction method is presented.

## 3. Design method based on self-convolution

7. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE **66**(1), 51–83 (1978). [CrossRef]

7. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE **66**(1), 51–83 (1978). [CrossRef]

*p*+ 1 power of a sinc function,where

*p*+1 at

*p*+1 at

*w*(

*t*) should be sampled. Suppose the sampling function is,

*r*(=

*q*+2) dimensions; secondly self-convolve the vector

*p*times; at last multiply the self-convolved vector with a phase factor to shift the frequency components of the Fourier domain. Then the coefficients of the algorithm are the real and imaginary parts respectively, and the phase shift step is

*p*+1)

*r*-

*p*. Usually

*r*=

*q*+2 is chosen to design the algorithm with the least number of fringes.

*p*= 3,

*q*= 2, the discrete approach can be done as follows: A four dimensional vector

## 4. Theoretical analysis

*p*

_{1},

*p*

_{2}and

*p*

_{3}are the PZT miscalibration error, second-order and third-order nonlinearity coefficients respectively. The coefficients

*a*

_{0},

*a*

_{1}and

*a*

_{2}of CCD curve will affect the amplitude of harmonics of an interference signal. Because the coefficients

*p*

_{1},

*p*

_{2}and

*p*

_{3}are close to zero, taking first-order Taylor series approximation like Eq. (5) is reasonable.

**66**(1), 51–83 (1978). [CrossRef]

5. K. Hibino, R. Hanayama, J. Burke, and B. F. Oreb, “Tunable phase-extraction formulae for simultaneous shape measurement of multiple surfaces with wavelength-shifting interferometry,” Opt. Express **12**(23), 5579–5594 (2004). [CrossRef] [PubMed]

**66**(1), 51–83 (1978). [CrossRef]

10. D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. **36**(31), 8098–8115 (1997). [CrossRef] [PubMed]

## 5. Numerical simulations

4. P. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. **39**(16), 2658–2663 (2000). [CrossRef] [PubMed]

## 6. Summary

## References and links

1. | H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in |

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

3. | H. Zhang, M. J. Lalor, and D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. |

4. | P. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. |

5. | K. Hibino, R. Hanayama, J. Burke, and B. F. Oreb, “Tunable phase-extraction formulae for simultaneous shape measurement of multiple surfaces with wavelength-shifting interferometry,” Opt. Express |

6. | P. Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. |

7. | F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE |

8. | K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting for non-sinusoidal waveforms with phase shift errors,” J. Opt. Soc. Am. A |

9. | Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. |

10. | D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. |

11. | M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express |

12. | J. C. Estrada, M. Servin, and J. A. Quiroga, “Easy and straightforward construction of wideband phase-shifting algorithms for interferometry,” Opt. Lett. |

13. | Zygo is a registered trademark of Zygo Corporation. |

**OCIS Codes**

(070.4790) Fourier optics and signal processing : Spectrum analysis

(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: March 22, 2011

Revised Manuscript: May 20, 2011

Manuscript Accepted: May 23, 2011

Published: July 15, 2011

**Citation**

Zhenguang Shi, Jian Zhang, Yongxin Sui, Ji Peng, Feng Yan, and Huaijiang Yang, "Design of algorithms for phase shifting interferometry using self-convolution of the rectangle window," Opt. Express **19**, 14671-14681 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-15-14671

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### References

- H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing Third Edition, D. Malacara, (New York, 2007), pp. 547–666.
- K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990). [CrossRef]
- H. Zhang, M. J. Lalor, and D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38(9), 1524–1533 (1999). [CrossRef]
- P. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39(16), 2658–2663 (2000). [CrossRef] [PubMed]
- K. Hibino, R. Hanayama, J. Burke, and B. F. Oreb, “Tunable phase-extraction formulae for simultaneous shape measurement of multiple surfaces with wavelength-shifting interferometry,” Opt. Express 12(23), 5579–5594 (2004). [CrossRef] [PubMed]
- P. Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34(22), 4723–4730 (1995). [CrossRef] [PubMed]
- F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66(1), 51–83 (1978). [CrossRef]
- K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting for non-sinusoidal waveforms with phase shift errors,” J. Opt. Soc. Am. A 12(4), 761–768 (1995). [CrossRef]
- Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996). [CrossRef] [PubMed]
- D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. 36(31), 8098–8115 (1997). [CrossRef] [PubMed]
- M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009). [CrossRef] [PubMed]
- J. C. Estrada, M. Servin, and J. A. Quiroga, “Easy and straightforward construction of wideband phase-shifting algorithms for interferometry,” Opt. Lett. 34(4), 413–415 (2009). [CrossRef] [PubMed]
- Zygo is a registered trademark of Zygo Corporation.

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