## Design of phase-shifting algorithms by fine-tuning spectral shaping |

Optics Express, Vol. 19, Issue 11, pp. 10692-10697 (2011)

http://dx.doi.org/10.1364/OE.19.010692

Acrobat PDF (902 KB)

### Abstract

To estimate the modulating wavefront of an interferogram in Phase Shifting Interferometry (PSI) one frequently uses a Phase Shifting Algorithm (PSA). All PSAs take as input *N* phase-shifted interferometric measures, and give an estimation of their modulating phase. The first and best known PSA designed explicitly to reduce a systematic error source (detuning) was the 5-steps, Schwider-Hariharan (*SH*-PSA) PSA. Since then, dozens of PSAs have been published, designed to reduce specific data error sources on the demodulated phase. In Electrical Engineering the Frequency Transfer Function (FTF) of their linear filters is their standard design tool. Recently the FTF is also being used to design PSAs. In this paper we propose a technique for designing PSAs by fine-tuning the few spectral zeroes of a PSA to approximate a template FTF spectrum. The PSA’s spectral zeroes are moved (tuned) while gauging the plot changes on the resulting FTF’s magnitude.

© 2011 OSA

## 1. Introduction

1. 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]

2. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. **26**(13), 2504–2506 (1987). [CrossRef] [PubMed]

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

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

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

8. J. Schmit and K. Creath, “Window function influence on phase error in phase-shifting algorithms,” Appl. Opt. **35**(28), 5642–5649 (1996). [CrossRef] [PubMed]

12. 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]

*x*-polynomial

*P*(

*x*) associated with a PSA named the Characteristic Polynomial (CP) [9

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

*x*-polynomial closely follows the

*z*-transform, or in Surrel’s words [9

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

*x*instead of

*z*for his

*P*(

*x*) polynomial. Then, Surrel proposed the CP-diagram, to visualize some properties of a PSA based on the (discrete) angular location over the unit circle of these zeroes and their multiplicities [9

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

*x*-polynomials [9

**35**(1), 51–60 (1996). [CrossRef] [PubMed]

**7**(4), 542–551 (1990). [CrossRef]

*z*-transformed digital filters [10

10. J. G. Proakis, and D. G. Manolakis, *Digital Signal Processing*, 4th ed. (Prentice Hall, 2007). [PubMed]

**35**(1), 51–60 (1996). [CrossRef] [PubMed]

**7**(4), 542–551 (1990). [CrossRef]

*x*-polynomial of

*symmetric*PSAs. That is, Burke generated a new visualizing-gauging technique, by combining the discrete CP-diagram [9

**35**(1), 51–60 (1996). [CrossRef] [PubMed]

**7**(4), 542–551 (1990). [CrossRef]

12. 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]

**7**(4), 542–551 (1990). [CrossRef]

*only symmetrical PSAs may be F&K spectrally analyzed*. That is because, F&K analysis needs to: “

*analyze the spectrum without constant or common phase factors*[1

1. 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]

*et al.*[5

5. K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A **9**(10), 1740–1748 (1992). [CrossRef]

12. 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]

*non-symmetric*PSAs. These two drawbacks are serious limitation of the F&K spectral analysis technique.

**17**(24), 21867–21881 (2009). [CrossRef] [PubMed]

*H*(

*ω*) is given. The FTF is just the Fourier transform of the impulse response of a digital filter

*h*(

*t*), that is

*H*(

*ω*)=

*F*[

*h*(

*t*)] [10

10. J. G. Proakis, and D. G. Manolakis, *Digital Signal Processing*, 4th ed. (Prentice Hall, 2007). [PubMed]

*F*[·] is the Fourier transform. The use of the FTF however new in PSI, has been the standard way of spectral analysis in signal processing engineering for decades [10

10. J. G. Proakis, and D. G. Manolakis, *Digital Signal Processing*, 4th ed. (Prentice Hall, 2007). [PubMed]

*H*(

*ω*)| plot, we would not need the CP-diagram, all the PSA’s information is within our |

*H*(

*ω*)| continuous plot.

## 2. Spectral analysis of PSI algorithms based on the FTF

**17**(24), 21867–21881 (2009). [CrossRef] [PubMed]

*N*phase-shifted interferometric data as,where the background illumination is

*a*(

*x,y*), the fringe contrast is

*b*(

*x,y*), and the carrier frequency is

*ω*

_{0}(radians/interferogram). The Fourier transform

*F*[•] of this signal over

*t*is,To know the interesting phase

*φ*(

*x,y*) one needs a filter

*h*(

*t*), to wipe-out the

*aδ*(

*ω*) term and one

*b*/2 term. To this end, the measured signal

*I*(

*t*) is introduced into a general

*N*-steps PSA as,This PSA may be seen as the following

*N*-steps, quadrature-filter

*h*(

*t*) tuned at

*ω*

_{0}rad/sample,The data

*I*(

*t*) in Eq. (1) and the quadrature-filter

*h*(

*t*) may be convolved as,The complex number

*S*is the value of

*I*(

*t*)*

*h*(

*t*) evaluated at the middle-point

*t*=

*N*-1, in which the filter

*h*(

*t*) and the data

*I*(

*t*) fully overlap. Finally, the searched phase is

*φ*(

*x,y*)=

*angle*[

*S*].

*H*(

*ω*)| as [12

**17**(24), 21867–21881 (2009). [CrossRef] [PubMed]

*H*(

*ω*)=

*Re*(

*ω*)+

*i Im*(

*ω*), being

*Re*(

*ω*) and

*Im*(

*ω*) real-valued functions. The magnitude of

*H*(

*ω*) has the good properties of being invariant to local oscillator’s (exp[

*iω*

_{0}

*t*]) phase-shifts, and have a well-defined spectral plot for either a symmetric or non-symmetric PSAs.

## 3. Characteristic polynomial (CP) associated to a PSI algorithms

**35**(1), 51–60 (1996). [CrossRef] [PubMed]

*P*(

*x*) is the CP

*x*-polynomial, the data

*I*(

*k*) in Eq. (5) is formally substituted by

*x*[9

^{k}**35**(1), 51–60 (1996). [CrossRef] [PubMed]

*P*(

*x*) as a product of

*N*-1 monomials (

*x*-

*d*).

_{k}*M*roots at

*d*means that the PSA is robust to detuning at

_{k}*d*up to order

_{k}*M*. Finally, the zeroes

*d*, and their multiplicities are plotted in a CP-diagram [9

_{k}**35**(1), 51–60 (1996). [CrossRef] [PubMed]

**35**(1), 51–60 (1996). [CrossRef] [PubMed]

## 4. Fine-tuning spectral-zeroes for approximating a desired PSA spectrum

*ω*

_{0}is [10

*Digital Signal Processing*, 4th ed. (Prentice Hall, 2007). [PubMed]

*h*(

*t*) as,This is the basic building-block of our zero-based, spectral-shaping design of PSAs. Note that, the equivalent

*x*-monomial is

*P*(

*x*)=[

*x*-exp(-

*iω*)]. We will combine several first-order blocks (Eq. (10)), freely tuning their zeroes, to approximate a desired PSA spectral template.

_{0}*π*/2 than a 5-step Schwider-Hariharan (

*SH*) PSA [1

1. 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]

2. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. **26**(13), 2504–2506 (1987). [CrossRef] [PubMed]

*SH*-PSA has the following FTF,The

*x*-polynomial of the

*SH*-PSA is (

*x*-1)(

*x*+

*i*)

^{2}(

*x*+1). The CP-diagram and the plot of |

*H*(

_{SH}*ω*)| are shown in Fig. 1 . This PSA has a second-order zero at –

*π*/2, and two first-order zeroes at 0 and

*π*. The second-order zero gives (second-order) detuning robustness at -

*π*/2.

*π*/2, and the one at

*π*, to flatten the response around -

*π*/2. This is done while gauging the |

*H*

_{5}(

*ω*)| plot. After some iterations we settle on the following FTF, which corresponds to a non-symmetric PSA,The equivalent (CP)

*x*-polynomial is

*H*

_{5}(

*ω*)| are shown in Fig. 2 . By inverse-Fourier transforming

*H*

_{5}(

*ω*), or expanding

*P*(

_{5}*x*), (both options being equally easy) we find the desired

*non-symmetric*PSA as,

*ω*=-π/2, and also robustness to bias illumination’s variation; 2-zeroes at

_{0}*ω*=0. In Fig. 4 the CP-diagram of

*P*

_{9}(

*x*)=(

*x*-1)

^{2}(

*x*+

*i*)

^{6}, and the plot of |

*P*

_{9}(

*e*)| are shown,

^{iω}*P*

_{9}(

*e*) around -

^{iω}*ω*

_{0}. A flatter than

*P*

_{9}(

*e*) spectral response is obtained by spreading-out its 8 available zeroes. This is done while gauging the |

^{iω}*H*

_{9}(

*ω*)| plot. After some trial-and-error iterations we settle on the following

*non-symmetric*PSA,

*P*

_{9}(

*e*). The 8 zeroes have been spread-out around the unit circle. The designer may widen even more the rejection-band at a cost of tolerating bigger ripples; the signal amplitude at

^{iω}*ω*

_{0}is 82 times bigger than the highest ripple. This plot clearly shows the detailed spectral amplitude including the small ripples which cannot be seen in the CP-diagram. Fourier-inverse transforming

*H*

_{9}(

*ω*) we find

*h*

_{9}(

*t*) and from it, the searched 9-steps

*non-symmetric*PSA.

## 5. Signal-to-Noise power ratio (S/N) in PSA designs

*ω*

_{0}. By looking at Fig. 5 (for example) we see that the desired signal at

*ω*=

*π*/2, do not coincide with the spectra’s peak. Let us compute the S/N power-ratio at

*ω*

_{0}, and at its peak 1.16

*ω*

_{0}[12

**17**(24), 21867–21881 (2009). [CrossRef] [PubMed]

*ω*

_{0}from 0.5

*π*(radians/sample) to 0.58

*π*one obtains a 24.5% gain in the S/N power-ratio,

*i.e.*(6.6/5.3)=1.245. We have suggested this carrier increase because we can seen the continuous plot of |

*H*

_{9}(

*ω*)| in Fig. 5. In contrast, observing the CP-diagram alone, it is impossible to see where the location of the PSA’s spectral maximum is. Also, the F&K’ spectral plot [4

**7**(4), 542–551 (1990). [CrossRef]

*non-symmetrical*PSAs.

## 6. Conclusions

*N*-steps PSAs, based on the visualization and gauging of its continuous FTF spectral magnitude |

*H*(

*ω*)|. This technique gives more possibilities to approximate a target spectrum while keeping the size

*N*of the PSA unchanged. This PSA’s design method starts by specifying a desired PSA spectrum’s template. Afterwards, the available (

*N*-1) first-order zeroes are freely moved (fine-tuned), to approximate it. Finally, the inverse transform of the FTF (

*h*(

*t*)) is found, and from it the desired (in general non-symmetric) PSA. We remark, given that we freely move the PSA’s (

*N*-1) zeroes, we end-up (with probability 1) synthesizing

*non-symmetric*PSAs. These non-symmetric PSAs cannot be analyzed using the F&K spectral-plotting technique.

*N*a given PSA needs, and also where its (

*N*-1) zeroes may be located.

## References and links

1. | 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. |

2. | P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. |

3. | D. Malacara, M. Servin, and Z. Malacara, |

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

5. | K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A |

6. | J. Schwider, O. Falkenstörfer, H. Schreiber, H. Zöller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. |

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

8. | J. Schmit and K. Creath, “Window function influence on phase error in phase-shifting algorithms,” Appl. Opt. |

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

10. | J. G. Proakis, and D. G. Manolakis, |

11. | J. Burke, “Extended averaging phase-shifting schemes for Fizeau interferometry on high-numerical aperture spherical surfaces,” Proc. of SPIE |

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

**OCIS Codes**

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

(120.3180) Instrumentation, measurement, and metrology : Interferometry

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: March 7, 2011

Revised Manuscript: April 25, 2011

Manuscript Accepted: April 25, 2011

Published: May 17, 2011

**Citation**

A. Gonzalez, M. Servin, J. C. Estrada, and J. A. Quiroga, "Design of phase-shifting algorithms by fine-tuning spectral shaping," Opt. Express **19**, 10692-10697 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-11-10692

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### References

- 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]
- P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987). [CrossRef] [PubMed]
- D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Taylor & Francis, CRC, 2005).
- K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990). [CrossRef]
- K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9(10), 1740–1748 (1992). [CrossRef]
- J. Schwider, O. Falkenstörfer, H. Schreiber, H. Zöller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993). [CrossRef]
- 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]
- J. Schmit and K. Creath, “Window function influence on phase error in phase-shifting algorithms,” Appl. Opt. 35(28), 5642–5649 (1996). [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]
- J. G. Proakis, and D. G. Manolakis, Digital Signal Processing, 4th ed. (Prentice Hall, 2007). [PubMed]
- J. Burke, “Extended averaging phase-shifting schemes for Fizeau interferometry on high-numerical aperture spherical surfaces,” Proc. of SPIE 7790, (2010).
- 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]

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