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]. 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 [2–5]. Phase feedback methods [2,3] 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 [4,5] 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 [6–13] 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 [14–17]. 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], a quasi-uniformly distributed phase in the range of [0, 2π] [19] or an uniform illumination [20]. 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], Xu et al [22,23], Meng et al [24] and Gao et al [25] 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 [26,27]. 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 [26,27],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

If the terms$A(x,y)$and$B(x,y)$are spatially smooth functions, the fringe pattern expressed in Eq. (1) can be normalized by use of n-dimensional quasiquadrature transform [28].But in practical measurement, it is difficult to make the term$A(x,y)$spatially smooth because of the local noise. Thus, to obtain the normalized fringe pattern, the terms $A(x,y)$ and$B(x,y)$should be determined. They can be obtained by many methods, such as direct measurement [21–23], low-pass filtering [24] and so on. Here $A(x,y)$and$B(x,y)$are calculated from N frames of random phase-shifting interferograms by finding the maximum and the minimum intensity in each pixel [26,27]. The intensity at (x, y) can be ergodic over N frames if N is large enough [29]. Then the maximum and the minimum intensity, denoted as $I_{\max }(x,y)$and$I_{\min }(x,y)$, can be easily determined from N frames of interferograms. These extreme values are simply related to $A(x,y)$and$B(x,y)$ in Eq. (1) as

So$A(x,y)$and$B(x,y)$can be derived from

Here, ${\Phi }_{0,\pi }(n)$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 ${\Phi }_{0,\pi }(n)$to its principal phase${\Phi }_{-\pi ,\pi }(n)$, whose range goes from –π to π. In addition, it is easy to cause phase error at the position where ${\Phi }_{0,\pi }(n)$approaches 0 or π [28]. However, if the phase shift between adjacent frames, denoted as$\Delta \theta (n)=\theta (n)-\theta (n-1)$, would be extracted accurately, then the principal phase can be easily derived from two adjacent frames [23],

So now, the question is focused on how to extract the phase shift $\Delta \theta (n)$ from the obtained${\Phi }_{0,\pi }(n)$. Firstly, we define the phase difference between the nth and the (n-1)th frames as

Figure 1(a) shows the phases of the nth and the (n-1)th frames and the phase difference between these two frames. Since$\Phi (n)=\phi +\theta (n)$,$\Delta {\Phi }_{0,\pi }(n)$ equals to $\Delta \theta (n)$ for the place where both ${\Phi }_{-\pi ,\pi }(n)$ανδ${\Phi }_{-\pi ,\pi }(n-1)$are in (0, π), and $\Delta {\Phi }_{0,\pi }(n)$ equals to $-\Delta \theta (n)$ for the place where both ${\Phi }_{-\pi ,\pi }(n)$ανδ${\Phi }_{-\pi ,\pi }(n-1)$are in (-π, 0). For the other place, $\Delta {\Phi }_{0,\pi }(n)$varies nearly uniform from$-\Delta \theta (n)$ to$\Delta \theta (n)$.Here we assume that $\Delta \theta (n)>0$,which can be easily controlled by selecting proper value of the preset phase shift ${\theta }_1(n)$. From Fig. 1(a) we can see that $\Delta {\Phi }_{0,\pi }(n)=-0.5$ for the place $320<x<650$,$\Delta {\Phi }_{0,\pi }(n)=0.5$ for the place $190<x<300$ and $\Delta {\Phi }_{0,\pi }(n)$ varies nearly uniform from −0.5 to 0.5 for the place$300<x<320$.

 

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.

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Secondly, we assume that $\Delta \theta (n)$ is less than $\pi /2$ 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 $\left|\Delta {\Phi }_{0,\pi }(n)\right|=\Delta \theta (n)$. In the other parts of the interferogram, $\left|\Delta {\Phi }_{0,\pi }(n)\right|$varies nearly uniform from zero to$\Delta \theta (n)$. Thus, the frequency of$\left|\Delta {\Phi }_{0,\pi }(n)\right|=\Delta \theta (n)$is much larger than that of $\left|\Delta {\Phi }_{0,\pi }(n)\right|$ equaling any other values. Therefore, if the air turbulence and the calculation error are considered, then the relation between $\left|\Delta {\Phi }_{0,\pi }(n)\right|$ and $\Delta \theta (n)$can be approximately expressed as

In order to determine the phase shift$\Delta \theta (n)$from the phase difference$\left|\Delta {\Phi }_{0,\pi }(n)\right|$, we divide $\left|\Delta {\Phi }_{0,\pi }(n)\right|$ into many groups, named as bins in the context of histograms. By setting the bin width ε, we can calculate the histogram of $\left|\Delta {\Phi }_{0,\pi }(n)\right|$. 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., $(m-0.5)\epsilon <\left|\Delta {\Phi }_{0,\pi }(n)\right|\le (m+0.5)\epsilon$, 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 ${\Phi }_{0,\pi }(n)$ is noisy, then ε should be chosen larger, such as ε = 0.01 rad. According to the property of histogram of$\left|\Delta {\Phi }_{0,\pi }(n)\right|$, we know that the $\Delta \theta (n)$ would be located in the bin with the highest frequency. So the phase shift$\Delta \theta (n)$can be estimated as the average of $\left|\Delta {\Phi }_{0,\pi }(n)\right|$ 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. $A=130\exp [-0.02(x^2+y^2)]$ , $B=120\exp [-0.02(x^2+y^2)]$, and $\phi =2\pi \cos [(x^2+y^2)+3x]$ where$-1\le x\le 1$ and $-1\le y\le 1$ . ${\theta }_1(n)=n\pi /4$ and ${\theta }_2(n)$ 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) .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.

 

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.

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3.2 Influence of quantization error

The fringe-intensity error caused by quantization is$\Delta I_n=I_n-INT(I_n)$, where INT indicates the nearest integer representation. We assume that the maximum of $INT(I_n)$to be 2^t-1, where t means the quantization bit of CCD. Figure 3(a) 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 $INT(I_n)$close to 2^t-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.

 

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.

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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 obtained$A(x,y)$and$B(x,y)$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) 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 ${\theta }_1(n)=n\pi /4$ and the random phase shift${\theta }_2(n)$<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 than$2\pi /{\theta }_1(1)$ for small random phase shift. In practical phase measurement, due to mechanical vibration and air turbulence, the random phase shift${\theta }_2(n)$may be larger than 0.3 rad, and so N should be about 2~5 times as large as$2\pi /{\theta }_1(1)$.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.

 

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.

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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 shift$\Delta \theta (n)$is estimated as the average of $\left|\Delta {\Phi }_{0,\pi }(n)\right|$ 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 . 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.

 

Fig. 5 The relation between the average phase-shift extraction error and the bin width.

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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], Xu et al [22,23] and Meng et al [24].We make a comparison between our method and Xu’s method [23] 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) . The SNR of the interferogram is 60dB and the bit of CCD is 8. The real phase shift ${\theta }_1(n)+{\theta }_2(n)$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 ($\phi (x,y)+\theta (n)$) 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 [21–24]. 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.

 

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.

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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 is$n\pi /4$, are captured at a frame frequency of 20 fps. Two typical adjacent interferograms are shown in Figs. 7(a) 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.

 

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).

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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.