## Design of phase shifting algorithms: fringe contrast maximum |

Optics Express, Vol. 22, Issue 15, pp. 18203-18213 (2014)

http://dx.doi.org/10.1364/OE.22.018203

Acrobat PDF (2730 KB)

### Abstract

In phase shifting interferometry, the fringe contrast is preferred to be at a maximum when there is no phase shift error. In the measurement of highly-reflective surfaces, the signal contrast is relatively low and the measurement would be aborted when the contrast falls below a threshold value. The fringe contrast depends on the design of the phase shifting algorithm. The condition for achieving the fringe contrast maximum is derived as a set of linear equations of the sampling amplitudes. The minimum number of samples necessary for constructing an error-compensating algorithm that is insensitive to the *j*th harmonic component and to the phase shift error is discussed. As examples, two new algorithms (*15*-sample and (*3N* – *2*)-sample) were derived that are useful for the measurement for highly-reflective surfaces.

© 2014 Optical Society of America

## 1. Introduction

1. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. **13**(11), 2693–2703 (1974). [CrossRef] [PubMed]

*R*= 60% at wavelength 632.8 nm) and a fused silica plate (

*R*= 4% at wavelength 632.8 nm), respectively. In case of highly reflective surfaces, such as the surface of semiconductors, the contrast of the fundamental signal is relatively low and, thus, should be kept at a maximum. The fringe contrast is generally changed by the magnitude of the phase shift error. The contrast seems to increase or decrease depending on the phase shift error and it is not necessarily at a maximum when there is no phase shift error.

3. P. Carre, “Installation et utilisation du comparateur photoelectrique et interferential du Bureau International des Poids et Mesures,” Metrologia **2**(1), 13–23 (1966). [CrossRef]

19. R. Hanayama, K. Hibino, S. Warisawa, and M. Mitsuishi, “Phase measurement algorithm in wavelength scanned Fizeau interferometer,” Opt. Rev. **11**(5), 337–343 (2004). [CrossRef]

1. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. **13**(11), 2693–2703 (1974). [CrossRef] [PubMed]

20. K. A. Stetson and W. R. Brohinsky, “Electro-optic holography and its application to hologram interferometry,” Appl. Opt. **24**(21), 3631–3637 (1985). [CrossRef] [PubMed]

*4*-bucket [1

1. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. **13**(11), 2693–2703 (1974). [CrossRef] [PubMed]

*N*+

*1*) [8

8. 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]

*5*[11

11. 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]

*11*[15

15. K. Hibino, “Error-compensating phase measuring algorithms in a Fizeau interferometer,” Opt. Rev. **6**(6), 529–538 (1999). [CrossRef]

*5*[5

5. 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]

7. 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]

*7*[10

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

*7*and

*13*[12

12. 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]

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

*2N*–

*1*) [13

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

*j*th harmonic components and phase shift miscalibration. We will also derive the maximum contrast condition in the Fourier description [21

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

*15*-sample and (

*3N*–

*2*)-sample algorithms were derived that are insensitive to the 6th and (

*N*–

*2*)th order harmonic signals, respectively.

## 2. Fringe contrast maximum condition

### 2.1 Fringe contrast of phase shifting algorithm

*I*(

*x*,

*y*,

*α*) at a designated point (

_{r}*x*,

*y*) with a sinusoidal periodic waveform as a function of a phase shift parameter

*α*:where

_{r}*I*

_{0}(

*x*,

*y*) is the direct current (dc) irradiance and

*γ*is the fringe contrast (also called the modulation or fringe visibility). To simplify the mathematical notation, we will confine our analysis to a single point, although it is equally applicable to all other points of interest. Therefore, the (

*x*,

*y*) dependence of

*I*

_{0}and

*φ*will not be highlighted.

*3*-sample [22

22. A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast levelling,” Meas. Sci. Technol. **9**(5), 744–750 (1998). [CrossRef]

*4*-sample [23

23. K. Patorski, Z. Sienicki, and A. Styk, “Phase-shifting method contrast calculations in time-averaged interferometry: error analysis,” Opt. Eng. **44**(6), 065601 (2005). [CrossRef]

*5*-sample algorithms [23

23. K. Patorski, Z. Sienicki, and A. Styk, “Phase-shifting method contrast calculations in time-averaged interferometry: error analysis,” Opt. Eng. **44**(6), 065601 (2005). [CrossRef]

24. C. S. Vikram, “Phase error effect on contrast measurement in Schwider-Hariharan phase-shifting algorithm,” Optik (Stuttg.) **112**(3), 140–141 (2001). [CrossRef]

25. R. Juarez-Salazar, C. Robledo-Sanchez, F. Guerrero-Sanchez, and A. Rangel-Huerta, “Generalized phase-shifting algorithm for inhomogeneous phase shift and spatio-temporal fringe visibility variation,” Opt. Express **22**(4), 4738–4750 (2014). [CrossRef] [PubMed]

*M*-sample phase shifting algorithm, where the reference phases are separated by

*M*– 1 equal intervals of 2π/

*N*rad, where

*N*is an integer. Generally, best phase shift step of

*N*is

*N*=

*j*+ 2, when

*j*is the harmonic order [13

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

14. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A **14**(4), 918–930 (1997). [CrossRef]

*φ**iswhere

*a*and

_{r}*b*are the

_{r}*r*th normalized sampling amplitudes and

*I*(

*α*) is the

_{r}*r*th sampled signal irradiance.

*φ**, respectively, the fringe contrast

*γ*is given bywhere

*A*is the normalization coefficient defined by an averaged intensity:

*A*reduces to the irradiance

*I*

_{0}. The coefficient

*A*changes very slowly and slightly depending on the phase shift error.

### 2.2 Condition for fringe contrast maximum

*α*is a function of the phase shift parameter. The phase shift value for the

_{r}*r*th sample can be denoted by a polynomial of the unperturbed phase shift value

*α*

_{0}

*aswhere*

_{r}*p*is the maximum order of the nonlinearity,

*ε*(1 ≤

_{q}*q*≤

*p*) are the error coefficients, and

*α*

_{0}

*= 2π[*

_{r}*r*– (

*M*+ 1)/2]/

*N*is the unperturbed phase shift. An offset value (

*M*+ 1)/2 for the reference phase is introduced for convenience of notation and contributes only a spatially uniform constant bias to the calculated phase.

*γ*should have a maximum when there is no phase shift error. Here we consider the case in which there is only a miscalibration error

*ε*

_{1}.

*γ*is a function of error

*ε*

_{1}. If we take logarithmic values of Eq. (3) and find small variations for small error

*ε*

_{1}, Eq. (3) can be rewritten to givewhere δ() denotes a small variation. Since the averaged intensity

*A*changes very slowly and slightly compared to each intensity

*I*, we can neglect the first term in the right-hand side of Eq. (6) to first order. If we note that the contrast becomes an extremum at

*ε*

_{1}= 0, the derivative of the contrast with respect to error

*ε*

_{1}can be written as

*A*, Eq. (7) can be rewritten to give

*j*th order and satisfies the orthogonal relations [8

8. 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]

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

14. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A **14**(4), 918–930 (1997). [CrossRef]

20. K. A. Stetson and W. R. Brohinsky, “Electro-optic holography and its application to hologram interferometry,” Appl. Opt. **24**(21), 3631–3637 (1985). [CrossRef] [PubMed]

26. A. Téllez-Quiñones, D. Malacara-Doblado, and J. García-Márquez, “Phase-shifting algorithms for a finite number of harmonics: first-order analysis by solving linear systems,” J. Opt. Soc. Am. A **29**(4), 431–441 (2012). [CrossRef] [PubMed]

*m*, 1) is the Kronecker delta function. Note that Eqs. (10) and (11) for

*m*= 1 also define the normalization of the sampling amplitudes

*a*

_{r}and

*b*

_{r}.

*φ*. This condition is identical to the following equations.

### 2.3 Number of samples necessary for constructing the algorithm

*p*th nonlinearity of the phase shift error and the coupling of phase shift was already investigated by Hibino et al [14

14. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A **14**(4), 918–930 (1997). [CrossRef]

*ε*

_{1}is spatially uniform and the system is not sensitive to a fixed dc component in the calculated phase, the conditions for the sampling amplitudes

*a*and

_{r}*b*to eliminate the phase shift miscalibration are the following 3 equations [14

_{r}**14**(4), 918–930 (1997). [CrossRef]

26. A. Téllez-Quiñones, D. Malacara-Doblado, and J. García-Márquez, “Phase-shifting algorithms for a finite number of harmonics: first-order analysis by solving linear systems,” J. Opt. Soc. Am. A **29**(4), 431–441 (2012). [CrossRef] [PubMed]

*j*– 1 equations [14

**14**(4), 918–930 (1997). [CrossRef]

26. A. Téllez-Quiñones, D. Malacara-Doblado, and J. García-Márquez, “Phase-shifting algorithms for a finite number of harmonics: first-order analysis by solving linear systems,” J. Opt. Soc. Am. A **29**(4), 431–441 (2012). [CrossRef] [PubMed]

*m*= 2, 3, …,

*j*.

*α*asEquations (15) and (16) are reduced to trivial equations. Also, Eq. (17) will be the trivial equation from Eqs. (13) and (14). The number of independent equations among Eqs. (18)–(21) is equal to

_{r}*j*– 1, when the phase shift interval is 2π/(

*j*+ 2) rad [14

**14**(4), 918–930 (1997). [CrossRef]

*j*th harmonic component, phase shift miscalibration and the coupling error, and satisfying the contrast maximum condition, is then calculated to givewhere the phase shift interval is 2π/(

*j*+ 2) and

*N*=

*j*+ 2.

*2N*–

*2*) samples are necessary for constructing an algorithm that is insensitive to the

*j*th harmonic component, phase shift miscalibration, and their coupling error [13

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

*2N*–

*2*algorithm and derived another

*2N*–

*1*algorithm. It will be shown in Sect. 4, this

*2N*–

*1*algorithm is one of the solutions that satisfy the contrast maximum condition.

## 3. Fourier representation of contrast maximum

8. 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]

11. 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]

**14**(4), 918–930 (1997). [CrossRef]

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

*α*) is the Dirac delta function and the other parameters are as defined in Sect. 2.1. The Fourier transforms of these two functions are simplified by the symmetric and asymmetric properties of the sampling amplitudes and phase shift parameter

*α*

_{r}_{,}as for Eqs. (26) and (27). where

*i*is the imaginary unit and

*ν*is the frequency variable.

*F*

_{1}is the pure imaginary and

*F*

_{2}is the real function, by the symmetric and asymmetric properties of the sampling amplitudes

*a*and

_{r}*b*and the phase shift parameter

_{r,}*α*mentioned Eq. (22).

_{r,}21. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-shifting algorithms,” J. Opt. Soc. Am. A **7**, 542–551 (1990). [CrossRef]

**9**(10), 1740–1748 (1992). [CrossRef]

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

**14**(4), 918–930 (1997). [CrossRef]

*ν*= 1, we obtain the equations

*iF*

_{1}and

*F*

_{2}are zero at the fundamental frequency.

*4*-sample) [1

**13**(11), 2693–2703 (1974). [CrossRef] [PubMed]

*5*-sample [5

5. 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]

7. 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]

*N*+

*1*algorithm (

*N*= 6) [8

**9**(10), 1740–1748 (1992). [CrossRef]

*2N*–

*1*algorithm (

*N*= 6) [13

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

*11*-sample algorithm [15

15. K. Hibino, “Error-compensating phase measuring algorithms in a Fizeau interferometer,” Opt. Rev. **6**(6), 529–538 (1999). [CrossRef]

*13*-sample algorithm [16

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

*ν*= 1), in Figs. 2(b), 2(d) and 2(f). These three algorithms have maximum contrast when there is no phase shift miscalibration.

*5*-sample algorithm [5

5. 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]

7. 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]

*ε*

_{1}. The contrast decreases by 21% for

*ε*

_{1}= ± 0.3. A miscalibration of

*ε*

_{1}= –0.3 is common in the spherical concave test, in which the phase shift of the oblique ray component becomes smaller than that of the axial ray. We can see that the fringe contrast maximum condition is satisfied in this algorithm.

## 4. Characteristic polynomial and fringe contrast maximum condition

### 4.1 An example of 15-sample algorithm

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

*j*= 6 and

*p*= 1. From Eq. (23), this algorithm consists of the 15-sample algorithm.

*α*

_{0}

*= π(*

_{r}*r*– 8)/4. Substituting Eq. (30) into Eqs. (9)–(14) and Eqs. (18)–(21), we obtain 16 equations for sampling amplitudes. The solutions are unique and given by

*2N*–

*1*phase shifting algorithm proposed by Surrel when

*N*= 8 [14

**14**(4), 918–930 (1997). [CrossRef]

*ν*= 1), the characteristic polynomials have double roots at

*ζ*

^{−1}= exp(−2π

*i*/

*N*) on the unit circle [13

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

*ζ*

^{−1}. In contrast, algorithms which have double roots at

*ζ*

^{−1}do not necessarily satisfy the contrast maximum condition.

*ζ*

^{-}

*(*

^{m}*m*= …, −2, −1, 0, 2…) in the characteristic diagram, the sampling functions in Fourier space become symmetric around the fundamental frequency (

*ν*= 1). Then the sampling functions have zero slopes at the fundamental frequency and automatically satisfy the contrast maximum condition. This conclusion is confirmed by the above

*15*-sample algorithm.

*3N*–

*2*algorithm that has triple roots and confirm this conclusion.

### 4.2 Characteristic polynomials and 3N – 2 algorithm

*3N*–

*2*)-sample algorithm (

*j*=

*N*– 2,

*p*= 2) which has triple roots at all the positions on the unit circle in the characteristic diagram [13

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

*iF*

_{1}and

*F*

_{2}for (a)

*15*-sample and (b) (

*3N*–

*2*)-sample algorithms (

*N*= 15). We can observe that both algorithms have zero slopes at the fundamental frequency. Note that we did not require the contrast maximum condition in deriving the

*3N*–

*2*algorithm. However, from Fig. 3, we can observe that the algorithm has matched zero slopes at the fundamental frequency. Therefore, we can conclude that when the algorithm has double or multiple roots on the all positions of the unit circle in the characteristic diagram, the algorithm satisfies the contrast maximum condition.

## 5. Conclusion

*j*th harmonic components and phase shift miscalibration. The maximum contrast condition was seen to require zero-derivatives for the sampling functions in the Fourier description. The relation between the contrast maximum condition and characteristic polynomial representation was also discussed. As examples, two algorithms,

*15*-sample and (

*3N*–

*2*)-sample, were derived that are useful for the measurement of the highly-reflective surfaces.

## References and links

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

2. | K. Creath, “Phase measurement interferometry techniques,” in |

3. | P. Carre, “Installation et utilisation du comparateur photoelectrique et interferential du Bureau International des Poids et Mesures,” Metrologia |

4. | J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real-time wavefront correction systems,” Appl. Opt. |

5. | 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. |

6. | J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. |

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

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

9. | Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. |

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

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

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

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

14. | K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A |

15. | K. Hibino, “Error-compensating phase measuring algorithms in a Fizeau interferometer,” Opt. Rev. |

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

17. | K. Hibino, B. F. Oreb, and P. S. Fairman, “Wavelength-scanning interferometry of a transparent parallel plate with refractive-index dispersion,” Appl. Opt. |

18. | K. Hibino, B. F. Oreb, P. S. Fairman, and J. Burke, “Simultaneous measurement of surface shape and variation in optical thickness of a transparent parallel plate in wavelength-scanning Fizeau interferometer,” Appl. Opt. |

19. | R. Hanayama, K. Hibino, S. Warisawa, and M. Mitsuishi, “Phase measurement algorithm in wavelength scanned Fizeau interferometer,” Opt. Rev. |

20. | K. A. Stetson and W. R. Brohinsky, “Electro-optic holography and its application to hologram interferometry,” Appl. Opt. |

21. | K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-shifting algorithms,” J. Opt. Soc. Am. A |

22. | A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast levelling,” Meas. Sci. Technol. |

23. | K. Patorski, Z. Sienicki, and A. Styk, “Phase-shifting method contrast calculations in time-averaged interferometry: error analysis,” Opt. Eng. |

24. | C. S. Vikram, “Phase error effect on contrast measurement in Schwider-Hariharan phase-shifting algorithm,” Optik (Stuttg.) |

25. | R. Juarez-Salazar, C. Robledo-Sanchez, F. Guerrero-Sanchez, and A. Rangel-Huerta, “Generalized phase-shifting algorithm for inhomogeneous phase shift and spatio-temporal fringe visibility variation,” Opt. Express |

26. | A. Téllez-Quiñones, D. Malacara-Doblado, and J. García-Márquez, “Phase-shifting algorithms for a finite number of harmonics: first-order analysis by solving linear systems,” J. Opt. Soc. Am. A |

**OCIS Codes**

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

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.3940) Instrumentation, measurement, and metrology : Metrology

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

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: May 9, 2014

Revised Manuscript: June 21, 2014

Manuscript Accepted: June 27, 2014

Published: July 21, 2014

**Citation**

Yangjin Kim, Kenichi Hibino, Naohiko Sugita, and Mamoru Mitsuishi, "Design of phase shifting algorithms: fringe contrast maximum," Opt. Express **22**, 18203-18213 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-15-18203

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### References

- J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt.13(11), 2693–2703 (1974). [CrossRef] [PubMed]
- K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1988).
- P. Carre, “Installation et utilisation du comparateur photoelectrique et interferential du Bureau International des Poids et Mesures,” Metrologia2(1), 13–23 (1966). [CrossRef]
- J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real-time wavefront correction systems,” Appl. Opt.14(11), 2622–2626 (1975). [CrossRef] [PubMed]
- 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]
- J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng.23(4), 350–352 (1984). [CrossRef]
- 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]
- K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A9(10), 1740–1748 (1992). [CrossRef]
- Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt.32(19), 3598–3600 (1993). [CrossRef] [PubMed]
- K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase-shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A12(4), 761–768 (1995). [CrossRef]
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