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 jth harmonic component and to the phase shift error is discussed. As examples, two new algorithms (15-sample and (3N2)-sample) were derived that are useful for the measurement for highly-reflective surfaces.

© 2014 Optical Society of America

1. Introduction

Phase shifting interferometry has been used for testing the surface shape and the optical thickness variation of optical components. In phase shifting interferometry [1], the phase difference between a sample wave front and a reference wave front is changed linearly or stepwise, and the resulting signal irradiance distribution is stored at each step, or bucket [2], in the frame memory of a computer. The phase can be calculated from the arctangent of the ratio between the two combinations of the observed signal irradiances, according to the phase shifting algorithm used. The fringe contrast that determines the maximum measurable range is commonly used for noise reduction and phase unwrapping. When the contrast at a measuring position is smaller than a threshold value, commercial interferometers would omit this position as an “outside or undetectable position” of the observing aperture. Figures 1(a) and 1(b) show the interferograms in the tests of a highly-reflective mirror (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.

 

Fig. 1 Interferogram of (a) mirror (R = 60% at wavelength 632.8 nm) and (b) fused silica plate (R = 4% at wavelength 632.8 nm).

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Phase shift errors are the most common sources of systematic errors in phase shifting measurements [2]. Phase shifts also become spatially non-uniform in testing a high numerical-aperture spherical concave, which results in an inevitable error. Many phase shifting algorithms have been developed that eliminate the phase shift errors [319]. However, the condition of maximum fringe contrast has so far not been taken into account when designing these algorithms. It is interesting to note that synchronous detection [1, 20] gains or loses the contrast, depending on the sign of the phase shift miscalibration. In several algorithms, such as 4-bucket [1], Larkin and Oreb (N + 1) [8], Schmit and Creath 5 [11], and Hibino 11 [15], the contrast is not optimized, while other algorithms such as Scwider-Hariharan 5 [5, 7], Hibino 7 [10], de Groot 7 and 13 [12, 16], and Surrel (2N1) [13] the contrast is maximized.

In this paper, we derive the condition that the fringe contrast is at a maximum when there is no phase shift error. We discuss the minimum number of samples necessary for constructing the new algorithm that satisfies the contrast maximum condition and has immunity to the jth harmonic components and phase shift miscalibration. We will also derive the maximum contrast condition in the Fourier description [21] and evaluate the conventional algorithms visually. As examples, two new algorithms, 15-sample and (3N2)-sample algorithms were derived that are insensitive to the 6th and (N2)th order harmonic signals, respectively.

2. Fringe contrast maximum condition

2.1 Fringe contrast of phase shifting algorithm

Consider the signal irradiance I(x, y, αr) at a designated point (x, y) with a sinusoidal periodic waveform as a function of a phase shift parameter αr:

I(x,y,αr)=I0(x,y){1+γcos[αrφ(x,y)]},
where I0(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 I0 and φ will not be highlighted.

The rigorous expressions for fringe contrast of 3-sample [22], 4-sample [23], and 5-sample algorithms [23, 24] have already been reported. Spatio-temporal fringe contrast has also been reported [25]. However, these studies did not discuss the condition of maximum fringe contrast.

Consider an 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, 14]. A general expression for the calculated phase φ* is

φ*=arctanr=1MbrI(αr)r=1MarI(αr),
where ar and br are the rth normalized sampling amplitudes and I(αr) is the rth sampled signal irradiance.

If we note that the numerator and denominator of Eq. (2) are proportional to the sine and cosine of phaseφ*, respectively, the fringe contrast γ is given by

γ=1A[r=1MarI(αr)]2+[r=1MbrI(αr)]2,
where A is the normalization coefficient defined by an averaged intensity:

A=1Mr=1MI(αr).

When there is no phase shift error, the coefficient A reduces to the irradiance I0. The coefficient A changes very slowly and slightly depending on the phase shift error.

2.2 Condition for fringe contrast maximum

When the phase shift is nonlinear, each phase shift value αr is a function of the phase shift parameter. The phase shift value for the rth sample can be denoted by a polynomial of the unperturbed phase shift value α0r as

αr=α0r[1+ε1+ε2α0rπ+ε3(α0rπ)2++εp(α0rπ)p1],
where p is the maximum order of the nonlinearity, εq (1 ≤ qp) are the error coefficients, and α0r = 2π[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.

The fringe contrast 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 ε1.

In this case, the contrast γ is a function of error ε1. If we take logarithmic values of Eq. (3) and find small variations for small error ε1, Eq. (3) can be rewritten to give

δγγ=δAA+δ{[r=1MarI(αr)]2+[r=1MbrI(αr)]2}2[r=1MarI(αr)]2+[r=1MbrI(αr)]2,
where 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 ε1 = 0, the derivative of the contrast with respect to error ε1 can be written as

dγdε1|ε1=0=0.

Substituting Eq. (6) into Eq. (7) and use the approximation of neglecting δA, Eq. (7) can be rewritten to give

r=1Marcos(φα0r)r=1Mα0rarsin(φα0r)+r=1Mbrcos(φα0r)r=1Mα0rbrsin(φα0r)=12(r=1Mα0rarsinα0rr=1Mα0rbrcosα0r)+12cos2φ(r=1Mα0rarsinα0r+r=1Mα0rbrcosα0r)=0,
where we assumed that the algorithm is insensitive to harmonic components of the signal up to the jth order and satisfies the orthogonal relations [8, 10, 14, 20, 26] of
r=1Marsin(mα0r)=0form=1,2,,j,
r=1Marcos(mα0r)=δ(m,1)form=0,1,,j,
r=1Mbrsin(mα0r)=δ(m,1)form=1,2,,j,
r=1Mbrcos(mα0r)=0form=0,1,,j.
where δ(m, 1) is the Kronecker delta function. Note that Eqs. (10) and (11) for m = 1 also define the normalization of the sampling amplitudes ar and br.

Note that Eq. (8) needs to be zero for an arbitrary value of phase φ. This condition is identical to the following equations.

r=1Mα0rarsinα0r=0.
r=1Mα0rbrcosα0r=0.

Equations (13) and (14) are the necessary and sufficient conditions for the fringe contrast to be maximized when there is no miscalibration error of phase shift.

2.3 Number of samples necessary for constructing the algorithm

Here we discuss the minimum number of samples necessary for constructing the algorithm that satisfies the contrast maximum condition and has immunity to the jth harmonic component, phase shift miscalibration, and the coupling error between the harmonic signal and phase shift error. When a phase shifting algorithm is insensitive to harmonic components of the signal up to the jth order, the sampling amplitudes ar and br satisfy Eqs. (9)(12), a total of 4j + 2 equations. Equations (9)(12) are the common conditions for synchronous detection [1]. The number of samples necessary to eliminate the harmonic signals up to the jth order has already been investigated by several authors [10, 13, 14, 20]. The minimum number of samples is j + 2, when the interval equals 2π/(j + 2) rad.

The necessary condition for constructing an algorithm that is insensitive to the pth nonlinearity of the phase shift error and the coupling of phase shift was already investigated by Hibino et al [14]. When the error coefficient ε1 is spatially uniform and the system is not sensitive to a fixed dc component in the calculated phase, the conditions for the sampling amplitudes ar and br to eliminate the phase shift miscalibration are the following 3 equations [14, 26] as

r=1Mα0r(arcosα0r+brsinα0r)=0,
r=1Mα0r(arcosα0rbrsinα0r)=0,
r=1Mα0r(arsinα0r+brcosα0r)=0.

Similarly, in order to suppress the coupling error between the phase shift miscalibration and the harmonic components, the amplitudes should satisfy the following 4j – 1 equations [14, 26]:

r=1Mα0rarsin(mα0r)=0,
r=1Mα0rarcos(mα0r)=0,
r=1Mα0rbrsin(mα0r)=0,
r=1Mα0rbrcos(mα0r)=0,
for m = 2, 3, …, j.

If we assume the following symmetric and asymmetric properties of sampling amplitudes and phase shift parameter αr as

ar=aM+1r,br=bM+1r,αr=αM+1r,
Equations (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 j – 1, when the phase shift interval is 2π/(j + 2) rad [14].

Finally, the total number of samples necessary for the immunity to the jth harmonic component, phase shift miscalibration and the coupling error, and satisfying the contrast maximum condition, is then calculated to give

M=(j+2)+2+(j1)=2j+3=2N1,
where the phase shift interval is 2π/(j + 2) and N = j + 2.

Surrel reported that (2N2) samples are necessary for constructing an algorithm that is insensitive to the jth harmonic component, phase shift miscalibration, and their coupling error [13]. For computational efficiency, Surrel added one more sample to his 2N2 algorithm and derived another 2N1 algorithm. It will be shown in Sect. 4, this 2N1 algorithm is one of the solutions that satisfy the contrast maximum condition.

3. Fourier representation of contrast maximum

The contrast maximum condition can be visualized if we take a Fourier representation of the sampling functions of the algorithm [8, 1114, 21]. The sampling functions of the numerator and the denominator of a phase shifting algorithm given by Eq. (2) are defined by

f1(α)=r=1Mbrδ(ααr),
f2(α)=r=1Marδ(ααr),
where δ(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 αr, as for Eqs. (26) and (27).
F1(ν)=r=1Mbrexp(iαrν)=ir=1Mbrsin(αrν),
F2(ν)=r=1Marexp(iαrν)=r=1Marcos(αrν),
where i is the imaginary unit and ν is the frequency variable. F1 is the pure imaginary and F2 is the real function, by the symmetric and asymmetric properties of the sampling amplitudes ar and br, and the phase shift parameter αr, mentioned Eq. (22).

Freischlad and Koliopoulos showed that for any phase shifting algorithm the Fourier transforms of two sampling functions have matched values at the fundamental frequency [21]. Larkin and Hibino have shown that when a phase shifting algorithm is insensitive to a miscalibration error of phase shift, the first derivatives of the Fourier transforms of the sampling functions also have matched values at the fundamental frequency [8, 10, 14]. In Sect. 2.3, we have derived new conditions, Eqs. (13) and (14), for a phase shifting algorithm to show the maximum value of fringe contrast when there is no phase shift miscalibration. Here, we derive a Fourier representation of these new conditions.

Differentiating the sampling functions of Eqs. (26) and (27) and assigning the value ν = 1, we obtain the equations

diF1dν|ν=1=r=1Mα0rbrcosα0r,
dF2dν|ν=1=r=1Mα0rarsinα0r.

Comparing Eqs. (28) and (29) with Eqs. (13) and (14), we find that the condition is identical to the requirement that the first derivative of the sampling functions iF1 and F2 are zero at the fundamental frequency.

Figure 2 shows the sampling functions of conventional algorithms: (a) synchronous detection (4-sample) [1], (b) Schwider-Hariharan 5-sample [5, 7], (c) Larkin-Oreb N + 1 algorithm (N = 6) [8], (d) Surrel 2N1 algorithm (N = 6) [13], (e) Hibino 11-sample algorithm [15], and (f) de Groot 13-sample algorithm [16]. The sampling functions have zero slopes at the fundamental frequency (ν = 1), in Figs. 2(b), 2(d) and 2(f). These three algorithms have maximum contrast when there is no phase shift miscalibration.

 

Fig. 2 Sampling functions of (a) synchronous detection (4-sample), (b) Scwider-Hariharan 5-sample, (c) Larkin-Oreb N + 1 (N = 6), (d) Surrel 2N1 (N = 6), (e) Hibino 11-sample, (f) de Groot 13-sample algorithm.

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Figure 3 shows the fringe contrast for the Schwider-Hariharan 5-sample algorithm [5, 7] as a function of phase shift miscalibration ε1. The contrast decreases by 21% for ε1 = ± 0.3. A miscalibration of ε1 = –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.

 

Fig. 3 Fringe contrast of Schwider-Hariharan 5-sample algorithm.

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4. Characteristic polynomial and fringe contrast maximum condition

4.1 An example of 15-sample algorithm

In this section, we discuss the fringe contrast maximum condition in the characteristic polynomial representation [13]. First we derive an algorithm, as an example, that satisfies the fringe contrast maximum condition for the case of j = 6 and p = 1. From Eq. (23), this algorithm consists of the 15-sample algorithm.

The sampling amplitudes should satisfy the simultaneous Eqs. (9)(14) and Eqs. (18)(21). We can also assume the following symmetric and asymmetric sampling amplitudes as for Eq. (22):

ar=(a1,a2,a3,a4,a5,a6,a7,a8,a7,a6,a5,a4,a3,a2,a1),br=(b1,b2,b3,b4,b5,b6,b7,b8,b7,b6,b5,b4,b3,b2,b1),
where the unperturbed phase shift value is defined by α0r = π(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

a1=264,a2=0,a3=3264,a4=18,a5=5264,a6=0,a7=7264,a8=14,b1=264,b2=116,b3=3264,b4=0,b5=5264,b6=316,b7=7264,b8=0.

The resultant algorithm is

tanφ*=(I1I15)+22(I2I14)+3(I3I13)5(I5I11)62(I6I10)7(I7I9)(I1+I15)3(I3+I13)42(I4+I12)5(I5+I11)+7(I7+I9)+82I8,
which is nothing but the 2N1 phase shifting algorithm proposed by Surrel when N = 8 [14].

Surrel showed that when a phase shifting algorithm has matched slopes at the fundamental frequency (ν = 1), the characteristic polynomials have double roots at ζ−1 = exp(−2πi/N) on the unit circle [13]. Algorithms that satisfy the fringe contrast maximum condition have matched zero slopes and, thus, have double roots at ζ−1. In contrast, algorithms which have double roots at ζ−1 do not necessarily satisfy the contrast maximum condition.

However, it is worth to note that when the algorithm has double roots or triple roots on the all positions on the unit circle 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.

In the next subsection, we derive another example of the 3N2 algorithm that has triple roots and confirm this conclusion.

4.2 Characteristic polynomials and 3N – 2 algorithm

The second example is a (3N2)-sample algorithm (j = N – 2, p = 2) which has triple roots at all the positions on the unit circle in the characteristic diagram [13]. By expanding the characteristic polynomials, a new windowed DFT phase shifting algorithm is obtained by the Eq. (33).

ar=2Nwrcos2πN(r3N12),br=2Nwrsin2πN(r3N12),
where we denote the sampling weights by

wr=1N2[12r(r+1)](1rN),wr=1N2[12N(N+1)+(rN)(2Nr1)](N+1r2N2),wr=1N2[12(3Nr1)(3Nr)](2N1r3N2).

Figure 4 shows the Fourier transforms iF1 and F2 for (a) 15-sample and (b) (3N2)-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 3N2 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.

 

Fig. 4 Fourier transforms of the sampling amplitudes for (a) 15-sample algorithm, (b) 3N2 algorithm (N = 15).

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5. Conclusion

The condition for a phase shifting algorithm to satisfy the fringe contrast maximum requirement was derived as a set of linear equations of the sampling amplitudes. We discussed the minimum number of samples necessary for constructing the error-compensating algorithm that satisfies the contrast maximum condition and has immunity to the jth 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 (3N2)-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. 13(11), 2693–2703 (1974). [CrossRef]   [PubMed]  

2. K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1988).

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]  

4. 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]  

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]  

6. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23(4), 350–352 (1984). [CrossRef]  

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]  

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]  

9. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32(19), 3598–3600 (1993). [CrossRef]   [PubMed]  

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]  

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]  

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]  

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]  

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

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

17. K. Hibino, B. F. Oreb, and P. S. Fairman, “Wavelength-scanning interferometry of a transparent parallel plate with refractive-index dispersion,” Appl. Opt. 42(19), 3888–3895 (2003). [CrossRef]   [PubMed]  

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. 43(6), 1241–1249 (2004). [CrossRef]   [PubMed]  

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]  

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]  

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

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]  

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]  

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]  

References

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  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]
  2. K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1988).
  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]
  4. 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]
  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]
  6. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23(4), 350–352 (1984).
    [CrossRef]
  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]
  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]
  9. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32(19), 3598–3600 (1993).
    [CrossRef] [PubMed]
  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]
  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]
  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]
  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]
  15. K. Hibino, “Error-compensating phase measuring algorithms in a Fizeau interferometer,” Opt. Rev. 6(6), 529–538 (1999).
    [CrossRef]
  16. P. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39(16), 2658–2663 (2000).
    [CrossRef] [PubMed]
  17. K. Hibino, B. F. Oreb, and P. S. Fairman, “Wavelength-scanning interferometry of a transparent parallel plate with refractive-index dispersion,” Appl. Opt. 42(19), 3888–3895 (2003).
    [CrossRef] [PubMed]
  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. 43(6), 1241–1249 (2004).
    [CrossRef] [PubMed]
  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]
  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]
  21. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-shifting algorithms,” J. Opt. Soc. Am. A 7, 542–551 (1990).
    [CrossRef]
  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]
  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]
  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]

2014 (1)

2012 (1)

2005 (1)

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]

2004 (2)

2003 (1)

2001 (1)

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

2000 (1)

1999 (1)

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

1998 (1)

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]

1997 (1)

1996 (1)

1995 (3)

1993 (1)

1992 (1)

1990 (1)

1987 (1)

1985 (1)

1984 (1)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23(4), 350–352 (1984).
[CrossRef]

1983 (1)

1975 (1)

1974 (1)

1966 (1)

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

Apostol, D.

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]

Brangaccio, D. J.

Brohinsky, W. R.

Bruning, J. H.

Burke, J.

Burow, R.

Carre, P.

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

Creath, K.

Damian, V.

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]

de Groot, P.

Dobroiu, A.

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]

Eiju, T.

Elssner, K. E.

Fairman, P. S.

Farrant, D. I.

Freischlad, K.

Gallagher, J. E.

García-Márquez, J.

Greivenkamp, J. E.

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23(4), 350–352 (1984).
[CrossRef]

Groot, P.

Grzanna, J.

Guerrero-Sanchez, F.

Hanayama, R.

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]

Hariharan, P.

Herriott, D. R.

Hibino, K.

Juarez-Salazar, R.

Koliopoulos, C. L.

Larkin, K. G.

Malacara-Doblado, D.

Merkel, K.

Mitsuishi, M.

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]

Nascov, V.

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]

Oreb, B. F.

Patorski, K.

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]

Rangel-Huerta, A.

Robledo-Sanchez, C.

Rosenfeld, D. P.

Schmit, J.

Schwider, J.

Sienicki, Z.

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]

Spolaczyk, R.

Stetson, K. A.

Styk, A.

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]

Surrel, Y.

Téllez-Quiñones, A.

Vikram, C. S.

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

Warisawa, S.

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]

White, A. D.

Wyant, J. C.

Appl. Opt. (12)

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]

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]

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]

Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32(19), 3598–3600 (1993).
[CrossRef] [PubMed]

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]

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]

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

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

K. Hibino, B. F. Oreb, and P. S. Fairman, “Wavelength-scanning interferometry of a transparent parallel plate with refractive-index dispersion,” Appl. Opt. 42(19), 3888–3895 (2003).
[CrossRef] [PubMed]

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. 43(6), 1241–1249 (2004).
[CrossRef] [PubMed]

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]

J. Opt. Soc. Am. A (5)

Meas. Sci. Technol. (1)

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]

Metrologia (1)

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

Opt. Eng. (2)

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]

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23(4), 350–352 (1984).
[CrossRef]

Opt. Express (1)

Opt. Rev. (2)

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

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]

Optik (Stuttg.) (1)

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

Other (1)

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1988).

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Figures (4)

Fig. 1
Fig. 1

Interferogram of (a) mirror (R = 60% at wavelength 632.8 nm) and (b) fused silica plate (R = 4% at wavelength 632.8 nm).

Fig. 2
Fig. 2

Sampling functions of (a) synchronous detection (4-sample), (b) Scwider-Hariharan 5-sample, (c) Larkin-Oreb N + 1 (N = 6), (d) Surrel 2N1 (N = 6), (e) Hibino 11-sample, (f) de Groot 13-sample algorithm.

Fig. 3
Fig. 3

Fringe contrast of Schwider-Hariharan 5-sample algorithm.

Fig. 4
Fig. 4

Fourier transforms of the sampling amplitudes for (a) 15-sample algorithm, (b) 3N2 algorithm (N = 15).

Equations (34)

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I( x,y, α r )= I 0 ( x,y ){ 1+γcos[ α r φ( x,y ) ] },
φ*=arctan r=1 M b r I( α r ) r=1 M a r I( α r ) ,
γ= 1 A [ r=1 M a r I( α r ) ] 2 + [ r=1 M b r I( α r ) ] 2 ,
A= 1 M r=1 M I( α r ) .
α r = α 0r [ 1+ ε 1 + ε 2 α 0r π + ε 3 ( α 0r π ) 2 ++ ε p ( α 0r π ) p1 ],
δγ γ = δA A + δ{ [ r=1 M a r I( α r ) ] 2 + [ r=1 M b r I( α r ) ] 2 } 2 [ r=1 M a r I( α r ) ] 2 + [ r=1 M b r I( α r ) ] 2 ,
dγ d ε 1 | ε 1 =0 =0.
r=1 M a r cos( φ α 0r ) r=1 M α 0r a r sin( φ α 0r ) + r=1 M b r cos( φ α 0r ) r=1 M α 0r b r sin( φ α 0r ) = 1 2 ( r=1 M α 0r a r sin α 0r r=1 M α 0r b r cos α 0r ) + 1 2 cos2φ( r=1 M α 0r a r sin α 0r + r=1 M α 0r b r cos α 0r )=0,
r=1 M a r sin( m α 0r ) =0form=1,2,,j,
r=1 M a r cos( m α 0r ) =δ( m,1 )form=0,1,,j,
r=1 M b r sin( m α 0r ) =δ( m,1 )form=1,2,,j,
r=1 M b r cos( m α 0r ) =0form=0,1,,j.
r=1 M α 0r a r sin α 0r =0.
r=1 M α 0r b r cos α 0r =0.
r=1 M α 0r ( a r cos α 0r + b r sin α 0r ) =0,
r=1 M α 0r ( a r cos α 0r b r sin α 0r ) =0,
r=1 M α 0r ( a r sin α 0r + b r cos α 0r ) =0.
r=1 M α 0r a r sin( m α 0r ) =0,
r=1 M α 0r a r cos( m α 0r ) =0,
r=1 M α 0r b r sin( m α 0r ) =0,
r=1 M α 0r b r cos( m α 0r ) =0,
a r = a M+1r , b r = b M+1r , α r = α M+1r ,
M=( j+2 )+2+( j1 ) =2j+3 =2N1,
f 1 ( α )= r=1 M b r δ( α α r ) ,
f 2 ( α )= r=1 M a r δ( α α r ) ,
F 1 ( ν )= r=1 M b r exp( i α r ν ) =i r=1 M b r sin( α r ν ) ,
F 2 ( ν )= r=1 M a r exp( i α r ν ) = r=1 M a r cos( α r ν ) ,
di F 1 dν | ν=1 = r=1 M α 0r b r cos α 0r ,
d F 2 dν | ν=1 = r=1 M α 0r a r sin α 0r .
a r =( a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 , a 7 , a 6 , a 5 , a 4 , a 3 , a 2 , a 1 ), b r =( b 1 , b 2 , b 3 , b 4 , b 5 , b 6 , b 7 , b 8 , b 7 , b 6 , b 5 , b 4 , b 3 , b 2 , b 1 ),
a 1 = 2 64 , a 2 =0, a 3 = 3 2 64 , a 4 = 1 8 , a 5 = 5 2 64 , a 6 =0, a 7 = 7 2 64 , a 8 = 1 4 , b 1 = 2 64 , b 2 = 1 16 , b 3 = 3 2 64 , b 4 =0, b 5 = 5 2 64 , b 6 = 3 16 , b 7 = 7 2 64 , b 8 =0.
tanφ*= ( I 1 I 15 )+2 2 ( I 2 I 14 )+3( I 3 I 13 )5( I 5 I 11 )6 2 ( I 6 I 10 )7( I 7 I 9 ) ( I 1 + I 15 )3( I 3 + I 13 )4 2 ( I 4 + I 12 )5( I 5 + I 11 )+7( I 7 + I 9 )+8 2 I 8 ,
a r = 2 N w r cos 2π N ( r 3N1 2 ), b r = 2 N w r sin 2π N ( r 3N1 2 ),
w r = 1 N 2 [ 1 2 r( r+1 ) ]( 1rN ), w r = 1 N 2 [ 1 2 N( N+1 )+( rN )( 2Nr1 ) ]( N+1r2N2 ), w r = 1 N 2 [ 1 2 ( 3Nr1 )( 3Nr ) ]( 2N1r3N2 ).

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