## Two-frame algorithm to design quadrature filters in phase shifting interferometry |

Optics Express, Vol. 18, Issue 24, pp. 24405-24411 (2010)

http://dx.doi.org/10.1364/OE.18.024405

Acrobat PDF (1189 KB)

### Abstract

The main purpose of this paper is to present a method to design tunable quadrature filters in phase shifting interferometry. From a general tunable two-frame algorithm introduced, a set of individual filters corresponding to each quadrature conditions of the filter is obtained. Then, through a convolution algorithm of this set of filters the desired symmetric quadrature filter is recovered. Finally, the method is applied to obtain several tunable filters, like four and five-frame algorithms.

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## 1. Introduction

1. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. **34**(19), 3610–3619 (1995). [CrossRef] [PubMed]

6. M. Afifi, K. Nassim, and S. Rachafi, “Five-frame phase-shifting algorithm insensitive to diode laser power variation,” Opt. Commun. **197**(1-3), 37–42 (2001). [CrossRef]

1. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. **34**(19), 3610–3619 (1995). [CrossRef] [PubMed]

8. J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “Calculus of exact detuning phase shift error in temporal phase shifting algorithms,” Opt. Express **17**(18), 15766–15771 (2009). [CrossRef] [PubMed]

9. J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “A method to design tunable quadrature filters in phase shifting interferometry,” Opt. Express **17**(18), 15772–15777 (2009). [CrossRef] [PubMed]

9. J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “A method to design tunable quadrature filters in phase shifting interferometry,” Opt. Express **17**(18), 15772–15777 (2009). [CrossRef] [PubMed]

*M*is given by [1

1. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. **34**(19), 3610–3619 (1995). [CrossRef] [PubMed]

9. J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “A method to design tunable quadrature filters in phase shifting interferometry,” Opt. Express **17**(18), 15772–15777 (2009). [CrossRef] [PubMed]

*N*and

*D*are the desired numerator and denominator row vectors; then, for a symmetrical filter the corresponding temporal impulse response

6. M. Afifi, K. Nassim, and S. Rachafi, “Five-frame phase-shifting algorithm insensitive to diode laser power variation,” Opt. Commun. **197**(1-3), 37–42 (2001). [CrossRef]

**17**(18), 15772–15777 (2009). [CrossRef] [PubMed]

*N*and

*D*the Eqs. (1) and (2) are recovered easily. In the previous work it was proved that the Fourier transform of

*h*(

*t*) is the real function

7. 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]

8. J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “Calculus of exact detuning phase shift error in temporal phase shifting algorithms,” Opt. Express **17**(18), 15766–15771 (2009). [CrossRef] [PubMed]

*α*step, any quadrature filter satisfies the two conditions

*ω*= 0 and

*ω*=

*α*frequencies. Therefore, the condition for a filter tuned onto the right side and insensitive to the

*m*order phase shift detuning error is [1

^{th}**34**(19), 3610–3619 (1995). [CrossRef] [PubMed]

7. 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]

*m*gives the order of insensitivity to the phase shift error, and

*m*derivate of

^{th}*ω*. In the same way, the condition to be satisfied by a filter which is insensitive to the

*m*order bias variation error

^{th}**17**(18), 15772–15777 (2009). [CrossRef] [PubMed]

## 2. Convolution algorithm

*n*and

*m*order filters respectively, the individual estimated phases

*ϕ*and given bythat is, the convolution algorithm can simply be represented as

*n + m*-1) frame filter from two individual filters is obtained. Likewise, the design of a tunable quadrature filter is seen as an algebraic problem without the use of Fourier formalisms. The convolution properties allow this case to be extended for three or more filters.

## 3. Design of tunable filters

*M*implies that only

*M*-1 parameters are free, two of which are the quadrature conditions and the other

*M*-3 are used to compensate some errors.

### 3.1 The tunable two-frame algorithm

**17**(18), 15772–15777 (2009). [CrossRef] [PubMed]

**17**(18), 15772–15777 (2009). [CrossRef] [PubMed]

*α*step is,

*N/D*are respectively,

*m*orders phase shift detuning error are respectively where

*m*+ 1 denotes the times that the convolution is applied. From Eq. (13), for

*α*= 0, the other quadrature condition

*m*order bias modulation error is,and from Eq. (15) for

*α*= 0, the corresponding algorithm is

### 3.2 The tunable three-frame algorithm

*H*(0) = 0 and

*H*(

*α*) = 0 is

*ω =*0 and

*ω = α*. Then, in terms of the corresponding two-frame filters, and through the convolution algorithm Eq. (8), the desired

*N*/

*D*becomesand from Eq. (7) the estimated phase is expressed as [1

**34**(19), 3610–3619 (1995). [CrossRef] [PubMed]

### 3.3 The tunable four-frame algorithm

*β*. Then, by applying the corresponding two-frame filters, the result is

*α*step and an arbitrary frequency

*β.*Thus, by using the geometrical condition

*α*, then the derivative at frequency

*α*is zero. That is to say, that the filter becomes insensitive to the linear phase shift detuning error, and gives the reported four-frame algorithm class B [9

**17**(18), 15772–15777 (2009). [CrossRef] [PubMed]

**17**(18), 15772–15777 (2009). [CrossRef] [PubMed]

### 3.4 *The tunable five-frame algorithm*

*α*step and the two arbitrary frequencies

*β*and

*γ*, the Fourier transform of the filter is,

6. M. Afifi, K. Nassim, and S. Rachafi, “Five-frame phase-shifting algorithm insensitive to diode laser power variation,” Opt. Commun. **197**(1-3), 37–42 (2001). [CrossRef]

7. 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]

8. J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “Calculus of exact detuning phase shift error in temporal phase shifting algorithms,” Opt. Express **17**(18), 15766–15771 (2009). [CrossRef] [PubMed]

*Δ*. Therefore, for

*q*= 2, while case 6 has

*q*= 3, and the others cases have the value

*q*= 1.

## 4. Conclusions

## Acknowledgments

## References and links

1. | J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. |

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

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

4. | Y. Surrel, “Design of phase-detection algorithms insensitive to bias modulation,” Appl. Opt. |

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

6. | M. Afifi, K. Nassim, and S. Rachafi, “Five-frame phase-shifting algorithm insensitive to diode laser power variation,” Opt. Commun. |

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

8. | J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “Calculus of exact detuning phase shift error in temporal phase shifting algorithms,” Opt. Express |

9. | J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “A method to design tunable quadrature filters in phase shifting interferometry,” 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: September 7, 2010

Revised Manuscript: October 25, 2010

Manuscript Accepted: October 27, 2010

Published: November 8, 2010

**Citation**

J. F. Mosiño, J. C. Gutiérrez-García, T. A. Gutiérrez-García, and J. M. Macías-Preza, "Two-frame algorithm to design quadrature filters in phase shifting interferometry," Opt. Express **18**, 24405-24411 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-24405

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### References

- J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34(19), 3610–3619 (1995). [CrossRef] [PubMed]
- 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]
- Y. Surrel, “Design of phase-detection algorithms insensitive to bias modulation,” Appl. Opt. 36(4), 805–807 (1997). [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. Afifi, K. Nassim, and S. Rachafi, “Five-frame phase-shifting algorithm insensitive to diode laser power variation,” Opt. Commun. 197(1-3), 37–42 (2001). [CrossRef]
- 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]
- J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “Calculus of exact detuning phase shift error in temporal phase shifting algorithms,” Opt. Express 17(18), 15766–15771 (2009). [CrossRef] [PubMed]
- J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “A method to design tunable quadrature filters in phase shifting interferometry,” Opt. Express 17(18), 15772–15777 (2009). [CrossRef] [PubMed]

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