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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 23 — Nov. 8, 2010
  • pp: 24368–24378
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Phase-shift extraction for phase-shifting interferometry by histogram of phase difference

Jiancheng Xu, Yong Li, Hui Wang, Liqun Chai, and Qiao Xu  »View Author Affiliations


Optics Express, Vol. 18, Issue 23, pp. 24368-24378 (2010)
http://dx.doi.org/10.1364/OE.18.024368


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Abstract

We propose a non-iterative approach to extract the unknown phase shift in phase shifting interferometry without the assumption of equal distribution of measured phase in [0,2π]. According to the histogram of the phase difference between two adjacent frames, the phase shift can be accurately extracted by finding the bin of histogram with the highest frequency. The main factors that influence the accuracy of the proposed method are analyzed and discussed, such as the random noise, the quantization bit of CCD, the number of fringe patterns used and the bin width of histogram. Numerical simulations and optical experiments are also implemented to verify the effectiveness of this method.

© 2010 OSA

1. Introduction

Phase-shifting interferometry (PSI) has been widely used in surface testing of optical elements, especially the large-aperture mirrors in astronomical telescopes or satellite cameras. Owing to the large size of the measured surface, the length of optical path is set to be several meters or even longer to complete the testing. However, the mechanical vibration and the air turbulence will change the preset phase shifts and cause inevitable errors to the measurement [1

1. L. L. Deck, “Suppressing phase errors from vibration in phase-shifting interferometry,” Appl. Opt. 48(20), 3948–3960 (2009). [CrossRef] [PubMed]

]. To reduce the phase-shifting errors caused by the mechanical vibration and the air turbulence, on the one hand, dynamic (vibration-insensitive) phase-shifting interferometry has been developed over the years [2

2. A. A. Freschi and J. Frejlich, “Adjustable phase control in stabilized interferometry,” Opt. Lett. 20(6), 635–637 (1995). [CrossRef] [PubMed]

5

5. G. Rodriguez-Zurita, C. Meneses-Fabian, N.-I. Toto-Arellano, J. F. Vázquez-Castillo, and C. Robledo-Sánchez, “One-shot phase-shifting phase-grating interferometry with modulation of polarization: case of four interferograms,” Opt. Express 16(11), 7806–7817 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-11-7806. [CrossRef] [PubMed]

]. Phase feedback methods [2

2. A. A. Freschi and J. Frejlich, “Adjustable phase control in stabilized interferometry,” Opt. Lett. 20(6), 635–637 (1995). [CrossRef] [PubMed]

,3

3. J. Hayes, “Dynamic interferometry handles vibration,” Laser Focus World 38, 109–116 (2002).

] with mechanical phase-shifting or acousto-optic modulation can compensate vibrations of small amplitude but suffer from low-contrast interferograms. One-shot or simultaneous phase-shifting interferometers [4

4. M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005). [CrossRef] [PubMed]

,5

5. G. Rodriguez-Zurita, C. Meneses-Fabian, N.-I. Toto-Arellano, J. F. Vázquez-Castillo, and C. Robledo-Sánchez, “One-shot phase-shifting phase-grating interferometry with modulation of polarization: case of four interferograms,” Opt. Express 16(11), 7806–7817 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-11-7806. [CrossRef] [PubMed]

] succeeded in reducing time varying noises by capturing the interferograms at the same time with different parts of the same detector. However, complicated phase modulators are needed, and the spatial resolution will be limited with only one detector.

On the other hand, numerous self-correcting algorithms have also been developed over the years [6

6. Y.-Y. Cheng and J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24(18), 3049–3052 (1985). [CrossRef] [PubMed]

13

13. 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]

] to compensate for the phase shift errors, which are originated primarily from the imperfect piezo transducers (PZT) used to shift the reference mirror. By using considerably more than four frames, it is possible for self-correcting algorithms to compensate for deterministic shift errors (such as the first, the second and the third order nonlinearities of PZT) and non-sinusoidal fringe profiles. The self-correcting algorithms use a special constant phase shift in each step (such as 2π/K, where K is a positive integer), which imposes a strict requirement on the environmental stability. To deal with the problem of the random phase shifts caused by the environmental vibration, iterative algorithms based on the least-square method have been proposed to determine the phase-shift amounts and the phase distribution simultaneously [14

14. K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3-4), 118–124 (1991). [CrossRef]

17

17. Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004). [CrossRef] [PubMed]

]. In addition, Dobroiu et al proposed the statistical self-calibrating algorithms to extract the phase distribution on the assumption of a constant fringe contrast [18

18. A. Dobroiu, P. C. Logofatu, D. Apostol, and V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. 8(7), 738–745 (1997). [CrossRef]

], a quasi-uniformly distributed phase in the range of [0, 2π] [19

19. 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]

] or an uniform illumination [20

20. A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9(9), 1451–1455 (1998). [CrossRef]

]. Both the least-square-based iterative algorithms and the statistical self-calibrating algorithms need three or more frames of interferogram and require an iteration process that will lead to substantial computation loads.

To further simplify the measurement and improve the computing efficiency, Cai et al [21

21. L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004). [CrossRef] [PubMed]

], Xu et al [22

22. X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008). [CrossRef] [PubMed]

,23

23. X. F. Xu, L. Z. Cai, Y. R. Wang, and D. L. Li, “Accurate Phase Shift Extraction for Generalized Phase-Shifting Interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010). [CrossRef]

], Meng et al [24

24. X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, J. P. Guo, and Y. R. Wang, “Wavefront reconstruction and three-dimensional shape measurement by two-step dc-term-suppressed phase-shifted intensities,” Opt. Lett. 34(8), 1210–1212 (2009). [CrossRef] [PubMed]

] and Gao et al [25

25. P. Gao, B. Yao, N. Lindlein, J. Schwider, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett. 34(22), 3553–3555 (2009). [CrossRef] [PubMed]

] introduced some algorithms to directly extract the unknown phase shifts on the assumption that the measured surface has an equal distribution in [0, 2π] over the whole interferogram. If the assumption is not fully met, iterative calculation is also needed to improve the extraction of phase shifts.

In this paper, we report a non-iterative approach to extract the unknown phase shift directly by the histogram of phase difference without the assumption of equal distribution of measured phase in [0,2π]. Thus, the phase shifts and measured phase can be extracted accurately without consideration of optics quality. Firstly, we obtain the cosine of the phase distribution according to the maximum and the minimum of temporal intensity in each pixel, which is similar to that proposed by Chen et al [26

26. X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. 39(4), 585–591 (2000). [CrossRef]

,27

27. X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shift calibration algorithm for phase-shifting interferometry,” J. Opt. Soc. Am. A 17(11), 2061–2066 (2000). [CrossRef]

]. Secondly, according to the histogram of the phase difference between two adjacent frames, the phase shift is accurately extracted by finding the bin of histogram with the highest frequency. Our method is based on the statistical properties of the phase shift and the random noise, so it is less sensitive to random noise than Chen’s max-min algorithm [26

26. X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. 39(4), 585–591 (2000). [CrossRef]

,27

27. X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shift calibration algorithm for phase-shifting interferometry,” J. Opt. Soc. Am. A 17(11), 2061–2066 (2000). [CrossRef]

],which is based on two-point correlation. In this paper, we will first discuss the principle of our method, and then give its verification by computer simulation and experiment.

2. Principle

Suppose N frames of random phase-shifting interferograms are collected and the intensity of an arbitrary pixel (x, y) in the nth interferogram is expressed as
I(x,y,n)=A(x,y)+B(x,y)cos[φ(x,y)+θ(n)]
(1)
whereA(x,y)andB(x,y)represent the background intensity and the modulation amplitude in pixel (x, y), respectively. φ(x,y)is the measured phase distribution andθ(n)is the random phase shift for the nth frame. θ(n) contains two parts: θ(n)=θ1(n)+θ2(n),whereθ1(n)is the preset phase shift, andθ2(n) is a random phase shift but be the same for all pixels, which is brought by the mechanical vibration or the error of PZT.

Imax(x,y)=A(x,y)+B(x,y)
(2)
Imin(x,y)=A(x,y)B(x,y)
(3)

SoA(x,y)andB(x,y)can be derived from
A(x,y)=[Imax(x,y)+Imin(x,y)]/2
(4)
B(x,y)=[Imax(x,y)Imin(x,y)]/2
(5)
with A(x,y)andB(x,y)already known, the normalized fringe pattern can be obtained. Then the phase of the nth frame, denoted as Φ(x,y,n)=φ(x,y)+θ(n), can be solved from the normalized fringe pattern with arc cosine function ((x, y) are omitted for brevity)

Φ0,π(n)=arccos[I(n)AB]
(6)

Here, Φ0,π(n)means the unrecovered phase whose range is (0, π), since the range of arc cosine function goes from 0 to π. It is usually time-consuming to recover Φ0,π(n)to its principal phaseΦπ,π(n), whose range goes from –π to π. In addition, it is easy to cause phase error at the position where Φ0,π(n)approaches 0 or π [28

28. J. A. Quiroga and M. Servín, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003). [CrossRef]

]. However, if the phase shift between adjacent frames, denoted asΔθ(n)=θ(n)θ(n1), would be extracted accurately, then the principal phase can be easily derived from two adjacent frames [23

23. X. F. Xu, L. Z. Cai, Y. R. Wang, and D. L. Li, “Accurate Phase Shift Extraction for Generalized Phase-Shifting Interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010). [CrossRef]

],

Φπ,π(n)=tan1[cot(Δθ(n))I(n)Asin(Δθ(n))(I(n1)A)]
(7)

So now, the question is focused on how to extract the phase shift Δθ(n) from the obtainedΦ0,π(n). Firstly, we define the phase difference between the nth and the (n-1)th frames as

ΔΦ0,π(n)=Φ0,π(n)Φ0,π(n1)
(8)

Figure 1(a)
Fig. 1 (a) The relation between two phases and their phase difference in one dimension and (b) the histogram of the absolute value of the phase difference.
shows the phases of the nth and the (n-1)th frames and the phase difference between these two frames. SinceΦ(n)=φ+θ(n),ΔΦ0,π(n) equals to Δθ(n) for the place where both Φπ,π(n)ανδΦπ,π(n1)are in (0, π), and ΔΦ0,π(n) equals to Δθ(n) for the place where both Φπ,π(n)ανδΦπ,π(n1)are in (-π, 0). For the other place, ΔΦ0,π(n)varies nearly uniform fromΔθ(n) toΔθ(n).Here we assume that Δθ(n)>0,which can be easily controlled by selecting proper value of the preset phase shift θ1(n). From Fig. 1(a) we can see that ΔΦ0,π(n)=0.5 for the place 320<x<650,ΔΦ0,π(n)=0.5 for the place 190<x<300 and ΔΦ0,π(n) varies nearly uniform from −0.5 to 0.5 for the place300<x<320.

Secondly, we assume that Δθ(n) is less than π/2 and the peak-to-valley (PV) value of the measured phase φ is larger than π, which is usually met in practical experiment. Thus there is always more than half of the area over the interferogram where |ΔΦ0,π(n)|=Δθ(n). In the other parts of the interferogram, |ΔΦ0,π(n)|varies nearly uniform from zero toΔθ(n). Thus, the frequency of|ΔΦ0,π(n)|=Δθ(n)is much larger than that of |ΔΦ0,π(n)| equaling any other values. Therefore, if the air turbulence and the calculation error are considered, then the relation between |ΔΦ0,π(n)| and Δθ(n)can be approximately expressed as
|ΔΦ0,π(x,y,n)|=Δθ(n)+η(x,y,n)
(9)
where η(x,y,n) is the additive white Gaussian noise with mean zero and probability density function of normal distribution. Equation (9) shows that the frequency of|ΔΦ0,π(n)| gets to its maximum when the value of |ΔΦ0,π(n)| equals toΔθ(n).

In order to determine the phase shiftΔθ(n)from the phase difference|ΔΦ0,π(n)|, we divide |ΔΦ0,π(n)| into many groups, named as bins in the context of histograms. By setting the bin width ε, we can calculate the histogram of |ΔΦ0,π(n)|. Figure 1(b) is the histogram of Fig. 1 (a) and it shows the frequency distribution of the absolute value of the phase difference. From the histogram, we can easily find the bin with the highest frequency, e.g., (m0.5)ε<|ΔΦ0,π(n)|(m+0.5)ε, where m is positive integer and means the mth bin of histogram. Because the phase shown in Fig. 1 (a) has a very small noise, ε is chosen to be a small value (ε = 0.001 rad) when calculating the histogram. From the Fig. 1(b), we can obtain that the 500th bin has the highest frequency and its range is (0.4995, 0.5005).However, if the extracted phase Φ0,π(n) is noisy, then ε should be chosen larger, such as ε = 0.01 rad. According to the property of histogram of|ΔΦ0,π(n)|, we know that the Δθ(n) would be located in the bin with the highest frequency. So the phase shiftΔθ(n)can be estimated as the average of |ΔΦ0,π(n)| in the mth bin.

3. Numerical simulation and discussion

Numerical simulations are carried out to test the performance of the proposed method. N frames are generated according to Eq. (1) by setting the parameters as follows. A=130exp[0.02(x2+y2)] , B=120exp[0.02(x2+y2)], and φ=2πcos[(x2+y2)+3x] where1x1 and 1y1 . θ1(n)=nπ/4 and θ2(n) is random rational number ranging from −0.3 to 0.3. Here the phase-shift extraction error is defined as the difference between the extracted phase shift and the given phase shift. The average phase-shift extraction error is defined as the average of the absolute value of phase-shift extraction error over N frames, where N is the number of fringe patterns used in the simulation. Then the main factors that influence the accuracy of the proposed method are analyzed and discussed as follows.

3.1 Influence of random noise

The measurement is invariably sensitive to noise, thus, it is important to study the robustness of our method in the presence of noise. The noise usually consists of two parts. One part is back noise that is static and can be viewed as a part of the background intensity, and the other part is random noise that is dynamic. Here we define the signal-to-noise ratio (SNR) as the ratio of the average of the modulation amplitude to the root mean square of the random noise. We assume that ε equals to 0.01 rad, the number of fringe pattern used (N) is 50 and the bit of the CCD is 10. Then we perform numerical simulations at different signal-to-noise ratios to obtain the phase-shift extraction error. The relation between the average phase-shift extraction error and the SNR of interferogram is shown in Fig. 2(a)
Fig. 2 The relation between the phase-shift extraction error and SNR. (a) The average phase-shift extraction errors for different SNRs and (b) the phase-shift extraction errors of 50 frames when SNR = 60dB.
.It shows that the average phase-shift extraction error decreases with the SNR increasing. When SNR = 60dB, the phase-shift extraction error of the N frames is shown in Fig. 2(b).It shows that the phase-shift extraction error is less than 0.02rad.

3.2 Influence of quantization error

The fringe-intensity error caused by quantization isΔIn=InINT(In), where INT indicates the nearest integer representation. We assume that the maximum of INT(In)to be 2t-1, where t means the quantization bit of CCD. Figure 3(a)
Fig. 3 The relation between the phase-shift extraction error and the bit of CCD. (a) The average phase-shift extraction errors for different bits of CCD and (b) the phase-shift extraction errors of 30 frames when the bit of CCD is 8.
shows the average phase-shift extraction errors for different bits of CCD when N = 30, SNR = 100dB and ε = 0.01rad. It shows that the average phase-shift extraction error decreases as the bit of CCD increases. To reduce the influence of quantization error, the bit of CCD should be chosen no less than 8 and the intensity of laser source should be chosen to make the maximum of INT(In)close to 2t-1. When the bit of CCD is 8, the phase-shift extraction errors of these 30 frames are shown in Fig. 3(b).It shows that the phase-shift extraction error is less than 0.01rad.

3.3 Influence of the number of frame used

Our method assumes that the number of fringe patterns used N is large enough and the intensity at (x, y) is ergodic over N frames. If N is not large enough, the intensity at (x, y) is not ergodic, and the obtainedA(x,y)andB(x,y)will have errors which will cause inevitable errors to phase-shift extraction and phase measurement. Thus, the phase-shift extraction error of our method depends on the number of fringe patterns used. Figure 4(a)
Fig. 4 The relation between the phase-shift extraction error and the number of fringe patterns used. (a) The average phase-shift extraction errors for different numbers of frames and (b) the phase-shift extraction errors of 50 frames when N = 50.
the average phase-shift extraction errors for different numbers of fringe patterns used when SNR = 100dB, ε = 0.01rad and the bit of CCD is 8. It shows that the average phase-shift extraction error decreases as the number of fringe patterns used increases. Since the preset phase shift θ1(n)=nπ/4 and the random phase shiftθ2(n)<0.3, the intensity at (x, y) is near ergodic over more than 16 frames. Thus, when N is smaller than 16, the average phase-shift extraction error is reduced effectively as N increases; and when N is larger than 16, the average phase-shift extraction error is reduced as N increases but not significantly. Therefore, in order to reduce the phase-shift extraction error, N should be larger than2π/θ1(1) for small random phase shift. In practical phase measurement, due to mechanical vibration and air turbulence, the random phase shiftθ2(n)may be larger than 0.3 rad, and so N should be about 2~5 times as large as2π/θ1(1).When N = 50, the phase-shift extraction errors of 50 frames are shown in Fig. 4(b). It shows that the phase-shift extraction error is less than 0.015 rad.

3.4 Influence of bin width

In the above three simulations, the bin width of histogram ε is set to 0.01 rad. However, in our method, the phase shiftΔθ(n)is estimated as the average of |ΔΦ0,π(n)| in the mth bin, thus the bin width certainly has effect on the phase-shift extraction error. According to the practical phase measurement condition, we assume that the number of fringe pattern used is 50, the SNR of each fringe pattern is 60dB and the bit of CCD is 8. Then we calculate the average phase-shift extraction errors at different bin widths and the result is shown in Fig. 5
Fig. 5 The relation between the average phase-shift extraction error and the bin width.
. It shows that the average phase-shift extraction error increases with the bin width increasing for ε>0.02 rad but increases with the bin width decreasing for ε<0.01 rad. Thus, a small value of ε may lead to large phase-shift extraction error because our method is sensitive to noise when ε is very small. For the case that N = 50, SNR = 60dB and the bit of CCD is 8, the average phase-shift extraction error is less than 0.01 rad when 0.001rad <ε<0.03 rad.

3.5 Comparison with other method

Our method extracts the unknown phase shift between two adjacent interferograms, which is similar to the algorithms proposed by Cai et al [21

21. L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004). [CrossRef] [PubMed]

], Xu et al [22

22. X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008). [CrossRef] [PubMed]

,23

23. X. F. Xu, L. Z. Cai, Y. R. Wang, and D. L. Li, “Accurate Phase Shift Extraction for Generalized Phase-Shifting Interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010). [CrossRef]

] and Meng et al [24

24. X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, J. P. Guo, and Y. R. Wang, “Wavefront reconstruction and three-dimensional shape measurement by two-step dc-term-suppressed phase-shifted intensities,” Opt. Lett. 34(8), 1210–1212 (2009). [CrossRef] [PubMed]

].We make a comparison between our method and Xu’s method [23

23. X. F. Xu, L. Z. Cai, Y. R. Wang, and D. L. Li, “Accurate Phase Shift Extraction for Generalized Phase-Shifting Interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010). [CrossRef]

] published in 2010. 30 frames of random phase-shifting interferograms are generated according to Eq. (1). One of typical interferogram is shown in Fig. 6(a)
Fig. 6 Comparison between our method and Xu’s method. (a) interferogram,(b) the real phase shift between adjacent frames, (c) the histogram of phase difference,(d) the residual phase error of our method,(e) the residual phase error of Xu’s method and (f) the phase-shift extraction errors of our method and Xu’s method.
. The SNR of the interferogram is 60dB and the bit of CCD is 8. The real phase shift θ1(n)+θ2(n)is shown in Fig. 6(b).From Fig. 6(b) we can obtain that the real phase shift between the second and the third frames is 0.6524 rad. The background intensity and the modulation amplitude are calculated from the maximum and the minimum intensities of 30 frames in each pixel. By setting ε = 0.01rad, we calculate the histogram of the absolute value of the phase difference between the second and the third frames, and the result is shown Fig. 6(c). It shows the extracted phase shift is 0.6512 rad and its error is 0.012 rad. According the extracted phase shift and Eq. (7), the final phase is obtained and its residual phase error is shown in Fig. 6(d) with PV of 0.1819 rad and RMS of 0.0203 rad. If Xu’s method is used, the phase-shift extraction error is 0.0826 rad and the residual phase error is shown Fig. 6(e) with PV of 0.3130 rad and RMS of 0.0507 rad. From the comparison between Figs. 6(d) and 6(e), we obviously obtain that the phase shift and the phase extracted by our method are more accurate than that extracted by Xu’s method. In addition, we calculate the phase-shift extraction errors of 30 frames by the two methods, the results are shown in Fig. 6(f). It shows that the phase-shift extraction error of our method is less than 0.015 rad, while the phase-shift extraction error of Xu’s method oscillates with frame index and the oscillation amplitude modulation is more than 0.08 rad. The phase-shift extraction error of Xu’s method depends on how equally the wrapped total phase (φ(x,y)+θ(n)) distributes in (0, 2π) in statistics. It will obtain accurate phase shifts when the PV value of the measured phase is more than 100π rad [21

21. L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004). [CrossRef] [PubMed]

24

24. X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, J. P. Guo, and Y. R. Wang, “Wavefront reconstruction and three-dimensional shape measurement by two-step dc-term-suppressed phase-shifted intensities,” Opt. Lett. 34(8), 1210–1212 (2009). [CrossRef] [PubMed]

]. Thus, the results of comparison clearly indicate that the proposed method provides excellent results, whereas Xu’s method produces significant errors when the assumption of equal distribution of the measured phase in [0, 2π] is not fully met.

4. Experiment

Optical experiments have also been carried out to investigate the performance of our method. An optical flat with aperture of 100mm is measured in a standard phase-shifting Fizeau interferometer with the active vibration isolation workstation turned off. A total of 100 frames of interferograms, whose preset phase shift isnπ/4, are captured at a frame frequency of 20 fps. Two typical adjacent interferograms are shown in Figs. 7(a)
Fig. 7 Optical experiment results:(a)-(b) two typical adjacent interferograms,(c)-(d) the phase recovered by the proposed method and ZYGO’s standard method respectively and (e) the difference between (c) and (d).
and 7(b). The real phase shift between them is calculated by our proposed method and equals to 1.120 rad. According to Eq. (7), the principal phase is obtained. Then by use of phase unwrapping and tilt removal, the recovered phase, with PV and RMS values of 1.1316 and 0.1446 rad, is shown in Fig. 7(c). Meanwhile, the optical flat is also measured by ZYGO interferometer with vibration-isolating platform and calibrated PZT. The recovered phase, with PV and RMS values of 1.1398 and 0.1685 rad, is shown in Fig. 7(d). The difference between Figs. 7(c) and 7(d), with PV and RMS values of 0.2715 and 0.0453 rad, is shown in Fig. 7(e).From the comparison between Figs. 7(c) and 7(d), it shows that the result from our proposed method coincides well with that from standard ZYGO interferometer. However, with the proposed method, we can accurately extract phase shift and thus relax the requirements on the accuracy of PZT and the performance of active vibration isolation workstation, which has potential application for test and measurement of large-aperture optical elements.

5. Conclusion

In conclusion, we have presented a non-iterative method to extract the unknown phase shift directly without the assumption of equal distribution of measured phase in [0,2π]. The phase shift between two adjacent frames can be accurately extracted by calculating the histogram of the phase difference. Simulated and experimental results demonstrate the effectiveness of the proposed algorithm. In order to improve the accuracy of the proposed method, we should capture more fringe patterns with high signal-to-noise ratio, use the CCD with high quantization bit, and choose the bin width of histogram properly according to the practical experiment condition. The average phase-shift extraction error is less than 0.01 rad when N = 50, SNR = 60dB,0.001<ε<0.03 rad and the bit of CCD is 8.This method is well implemented in existing interferometers simply by incorporating a high-speed camera and choosing a proper preset phase shift. It has potential application for test and measurement of large-aperture optical elements.

Acknowledgments

The authors thank the reviewers for their helpful and valuable suggestions, and thank Qikai Shi for his valuable assistance in practical experiments.

References and links

1.

L. L. Deck, “Suppressing phase errors from vibration in phase-shifting interferometry,” Appl. Opt. 48(20), 3948–3960 (2009). [CrossRef] [PubMed]

2.

A. A. Freschi and J. Frejlich, “Adjustable phase control in stabilized interferometry,” Opt. Lett. 20(6), 635–637 (1995). [CrossRef] [PubMed]

3.

J. Hayes, “Dynamic interferometry handles vibration,” Laser Focus World 38, 109–116 (2002).

4.

M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005). [CrossRef] [PubMed]

5.

G. Rodriguez-Zurita, C. Meneses-Fabian, N.-I. Toto-Arellano, J. F. Vázquez-Castillo, and C. Robledo-Sánchez, “One-shot phase-shifting phase-grating interferometry with modulation of polarization: case of four interferograms,” Opt. Express 16(11), 7806–7817 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-11-7806. [CrossRef] [PubMed]

6.

Y.-Y. Cheng and J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24(18), 3049–3052 (1985). [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]

8.

P. L. Wizinowich, “Phase shifting interferometry in the presence of vibration: a new algorithm and system,” Appl. Opt. 29(22), 3271–3279 (1990). [CrossRef] [PubMed]

9.

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.

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

11.

Y. Ishii and R. Onodera, “Phase-extraction algorithm in laser-diode phase-shifting interferometry,” Opt. Lett. 20(18), 1883–1885 (1995). [CrossRef] [PubMed]

12.

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]

13.

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]

14.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3-4), 118–124 (1991). [CrossRef]

15.

G.-S. Han and S.-W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. 33(31), 7321–7325 (1994). [CrossRef] [PubMed]

16.

B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995). [CrossRef]

17.

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004). [CrossRef] [PubMed]

18.

A. Dobroiu, P. C. Logofatu, D. Apostol, and V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. 8(7), 738–745 (1997). [CrossRef]

19.

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]

20.

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9(9), 1451–1455 (1998). [CrossRef]

21.

L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004). [CrossRef] [PubMed]

22.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008). [CrossRef] [PubMed]

23.

X. F. Xu, L. Z. Cai, Y. R. Wang, and D. L. Li, “Accurate Phase Shift Extraction for Generalized Phase-Shifting Interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010). [CrossRef]

24.

X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, J. P. Guo, and Y. R. Wang, “Wavefront reconstruction and three-dimensional shape measurement by two-step dc-term-suppressed phase-shifted intensities,” Opt. Lett. 34(8), 1210–1212 (2009). [CrossRef] [PubMed]

25.

P. Gao, B. Yao, N. Lindlein, J. Schwider, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett. 34(22), 3553–3555 (2009). [CrossRef] [PubMed]

26.

X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. 39(4), 585–591 (2000). [CrossRef]

27.

X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shift calibration algorithm for phase-shifting interferometry,” J. Opt. Soc. Am. A 17(11), 2061–2066 (2000). [CrossRef]

28.

J. A. Quiroga and M. Servín, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003). [CrossRef]

29.

Q. Hao, Q. Zhu, and Y. Hu, “Random phase-shifting interferometry without accurately controlling or calibrating the phase shifts,” Opt. Lett. 34(8), 1288–1290 (2009). [CrossRef] [PubMed]

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 13, 2010
Revised Manuscript: October 21, 2010
Manuscript Accepted: October 21, 2010
Published: November 5, 2010

Citation
Jiancheng Xu, Yong Li, Hui Wang, Liqun Chai, and Qiao Xu, "Phase-shift extraction for phase-shifting interferometry by histogram of phase difference," Opt. Express 18, 24368-24378 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-23-24368


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References

  1. L. L. Deck, “Suppressing phase errors from vibration in phase-shifting interferometry,” Appl. Opt. 48(20), 3948–3960 (2009). [CrossRef] [PubMed]
  2. A. A. Freschi and J. Frejlich, “Adjustable phase control in stabilized interferometry,” Opt. Lett. 20(6), 635–637 (1995). [CrossRef] [PubMed]
  3. J. Hayes, “Dynamic interferometry handles vibration,” Laser Focus World 38, 109–116 (2002).
  4. M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005). [CrossRef] [PubMed]
  5. G. Rodriguez-Zurita, C. Meneses-Fabian, N.-I. Toto-Arellano, J. F. Vázquez-Castillo, and C. Robledo-Sánchez, “One-shot phase-shifting phase-grating interferometry with modulation of polarization: case of four interferograms,” Opt. Express 16(11), 7806–7817 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-11-7806 . [CrossRef] [PubMed]
  6. Y.-Y. Cheng and J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24(18), 3049–3052 (1985). [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]
  8. P. L. Wizinowich, “Phase shifting interferometry in the presence of vibration: a new algorithm and system,” Appl. Opt. 29(22), 3271–3279 (1990). [CrossRef] [PubMed]
  9. 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. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32(19), 3598–3600 (1993). [CrossRef] [PubMed]
  11. Y. Ishii and R. Onodera, “Phase-extraction algorithm in laser-diode phase-shifting interferometry,” Opt. Lett. 20(18), 1883–1885 (1995). [CrossRef] [PubMed]
  12. 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]
  13. 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]
  14. K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3-4), 118–124 (1991). [CrossRef]
  15. G.-S. Han and S.-W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. 33(31), 7321–7325 (1994). [CrossRef] [PubMed]
  16. B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995). [CrossRef]
  17. Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004). [CrossRef] [PubMed]
  18. A. Dobroiu, P. C. Logofatu, D. Apostol, and V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. 8(7), 738–745 (1997). [CrossRef]
  19. 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]
  20. A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9(9), 1451–1455 (1998). [CrossRef]
  21. L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004). [CrossRef] [PubMed]
  22. X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008). [CrossRef] [PubMed]
  23. X. F. Xu, L. Z. Cai, Y. R. Wang, and D. L. Li, “Accurate Phase Shift Extraction for Generalized Phase-Shifting Interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010). [CrossRef]
  24. X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, J. P. Guo, and Y. R. Wang, “Wavefront reconstruction and three-dimensional shape measurement by two-step dc-term-suppressed phase-shifted intensities,” Opt. Lett. 34(8), 1210–1212 (2009). [CrossRef] [PubMed]
  25. P. Gao, B. Yao, N. Lindlein, J. Schwider, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett. 34(22), 3553–3555 (2009). [CrossRef] [PubMed]
  26. X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. 39(4), 585–591 (2000). [CrossRef]
  27. X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shift calibration algorithm for phase-shifting interferometry,” J. Opt. Soc. Am. A 17(11), 2061–2066 (2000). [CrossRef]
  28. J. A. Quiroga and M. Servín, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003). [CrossRef]
  29. Q. Hao, Q. Zhu, and Y. Hu, “Random phase-shifting interferometry without accurately controlling or calibrating the phase shifts,” Opt. Lett. 34(8), 1288–1290 (2009). [CrossRef] [PubMed]

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