Phase shifting interferometry from two normalized interferograms with random tilt phase-shift

Fengwei Liu; Yongqian Wu; Fan Wu

doi:10.1364/OE.23.019932

1. Introduction

Phase-shifting techniques are regarded as a powerful toolbox for extracting the wrapped phase from a series of phase shifted fringe patterns [1], and the phase-shifting interferometry (PSI) is an important technique for phase extraction in high accuracy optical surface measurement domain [2]. Traditional PSI presumes that the phase shift between adjacent frames is a constant (e.g. $π / 2$ ). However, several practical influence factors may limit the accuracy of PSI [3], such as errors related to the phase shift fidelity which may caused by the nonlinearity of phase shifter and the environmental vibration, intensity errors caused by nonlinearity of detector and noise. Among these errors, inaccurate phase shift is the dominate one and should be considered firstly.

The phase shift value is expected to be uniform at all pixels in an interferogram, typically named as piston phase shift. Correspondingly, plentiful efforts have been marshaled against the random piston phase shift errors of PSI in recent decades [4–9,19]. Unfortunately, vibrations may change the relative orientation between the reference and the test wavefronts during the temporal phase shifting process, thus the phase-shift plane between each frame may be unparallel and then the tilt-shift errors will occur in PSI as a consequence. In this case, the methods mentioned above will fail to completely compensate the inaccurate phase shifts caused by vibration since tilt-shift error is included. It can be demonstrated by our previous work [9]. In [9], the so proposed ETC method can successfully correct the artificial random piston phase-shift errors in experiment 4.1, but for the inaccurate phase-shift that caused by vibration in experiment 4.2 it can only compensate the piston phase-shift error. Thus, the residual surface error of ETC still contains a minor “ripple” error which may caused by the uncompensated tilt-shift error.

This paper is devoted to reconstruct the phase distribution of the measured object when random tilt phase-shift error is introduced in PSI. Although many methods have already been presented to cope with the tilt-shift error in PSI these years [10–16], most of them have inherent limitations. In 2000, Chen et al. [10] proposed an iterative method with first-order Taylor series expansion to compensate the random tilt-shift errors. Their algorithm could only cope with small magnitude of tilts, which is 0.00125 wave over the entire pupil. In 2008, Xu et al. [11] modified the advanced iterative algorithm (AIA) [5] to compensate the tilt-shift error by subdividing the interferograms into small regions. The phase steps in each region are assumed to be a constant and calculated by AIA, and then the global tilted phase-shift plane is estimated by least-squares fitting method. However, an inherent error is inevitable due to the assumption, besides their method can only deal with small tilts up to 0.1 wave over the entire pupil. In [12], the tilt-shift error is detected by spatial and time-domain process. Since Fourier Transform (FT) method is utilized, it can only deal with interferograms with adequate carrier frequency and the fringes cannot be closed. In 2013, Liu et al. [13] proposed a three step iterative method to calculate the tilt parameters by separating the tilt phase-shift into two independent parts, namely x- and y- directional tilt components. In each circle the wavefront phase and tilt-shift in x- and y- directions are calculated by linear least-squares method. Their method is claimed to be high accuracy and could deal with large tilts up to 1.7 wavelengths over the entire pupil, but the calculation process is time consuming and suffers a low efficiency. Additionally, the various background intensity and modulation amplitude will lead to a “coupling” effect when performing the iteration, which will definitely limits the accuracy of tilt calculation. A little earlier before [13], Zeng et al [14] proposed a method by extending the regularized optical flow method [17] with the spatial image processing and frequency estimation technology. The method can deal with tilts with amount up to several wavelengths. However, an unwrapping process of phase shift is necessary before the estimation of tilt parameters, and the adopted spatial image process may be complex and time consuming. To deal with tilt-shift errors that more than 1 wave in presence, Soloviev and Vdovin [15] presented an interesting method to calculate the tilt phase-shift by analyzing zero-crossing points’ distribution of the interferogram difference. In 2013, Li et al. [16] adopted the idea of line detection and proposed phase-tilting interferometry (PTI), in which the costly and accurate phase shifter is claimed to be useless. However, both methods can only deal with tilt phase-shift higher than 1 wave over entire pupil, and since the calculation of tilt parameters is based on line detection, they are sensitive to the orientation of fringe.

Actually, since the phase-shift and measured objects, e.g. surface height, are both encoded as phase items which are dynamic and static respectively in a cosinusoidal way as represented in Eq. (1), thus the dynamic phase-shift item could be extracted from the phase difference of two interferograms provided that they can be normalized first.

In 2010, Xu et al. [18] proposed a method of extracting the phase-shift by histogram of the phase difference of two normalized interferograms. Their method is non-iterative and of high accuracy, but tilt phase-shift is not considered. Juarez-Salazar et al. [1,19] proposed a new method for normalization based on polynomials surface fitting and parameters estimation method. In [19] the phase shift value is homogeneous phase step, whereas in [1] the phase shift between each adjacent frame can even be inhomogeneous surface where tilt phase-shift is just a special situation. Their work is definitely important however, the results are obtained in an approximately way and the normalization process is accurate only if there are many fringes in the interferogram. In this paper, we will calculate the random translational phase-shift and tilt gradients in x- and y- directions by analyzing phase difference distribution between two normalized interferograms. By selecting and fitting some points with small gradients (related to small tilts) or zero crossing points (related to large tilts) from the phase difference to a tilted phase-shift plane, the parameters of tilt phase-shift can be determined. Proposed method is non-iterative and of high accuracy, both closed and open fringes can be analyzed, more importantly it cannot only adapt to small tilt phase-shift (less than 0.5 wave over the entire pupil) but also calculate large tilt phase-shift (more than 1 wave over the entire pupil but not as large as in FT-based methods [15]).

2. Principles

2.1 The normalization of background intensity and modulation amplitude

The intensity of an arbitrary pixel $(x, y)$ in the kth interferogram can be expressed mathematically as,

I_{k} (x, y) = A (x, y) + B (x, y) \cdot \cos [ϕ (x, y) + δ_{k} (x, y)] \begin{matrix} k = 1, 2, 3 \dots M \end{matrix} .

where k is the frame index and M is the number of frames.

A (x, y), B (x, y), φ (x, y), δ_{k} (x, y)

represent the background intensity, the modulation amplitude, the phase to be measured and the random phase-shift at pixel

(x, y)

respectively.

A (x, y), B (x, y), φ (x, y)

always stay the same during the temporal phase shifting process, but

δ_{k} (x, y)

usually changes.

There are too many approaches to calculate the background intensity and modulation amplitude in literature, such as polynomials surface fitting method [19], direct measurement [20], low-pass filtering method [21], and maximum and minimum values method [22] and so on. Each method has its pro and con, the appropriate one is the best. Since it is not the theme of this paper, the material normalization process is out of our consideration and the maximum and minimum values method is selected for its simplicity and high accuracy in our work.

As long as the phase-shift between two adjacent frames is small enough and M is large enough [22], the intensity at $(x, y)$ can be ergodic over M frames. The maximum and the minimum intensity at pixel $(x, y)$ , denoted as $I_{\max} (x, y), I_{\min} (x, y)$ can be easily found from the M frames. Then $A (x, y), B (x, y)$ can be derived from

A (x, y) = \frac{I_{\max} + I_{\min}}{2}, B (x, y) = \frac{I_{\max} - I_{\min}}{2} .

A (x, y)

and

B (x, y)

are assumed as only functions of pixels and have no frame-to-frame variation, which can be measured and used as the system parameters before phase measurement. The intensity of normalized interferogram then can be expressed as,

\bar{I_{k}} (x, y) = \frac{I_{k} (x, y) - A (x, y)}{B (x, y)} .

2.2 The phase difference of two normalized interferograms

Since the interferograms have already been normalized, then the phase of the kth frame, denoted as $ϕ (x, y, k) = φ (x, y) + δ_{k} (x, y)$ , can be calculated by the arccosine function, namely,

ϕ_{k}^{[0, π]} (x, y) = arc \cos (\bar{I_{k}} (x, y)) \begin{matrix} k = 1, 2, ... M \end{matrix} .

It should be noted that the range of the calculated phase is

[0, π]

, Xu et al. [23] proposed a method to recover

ϕ_{k}^{[0, π]} (x, y)

to its principal phase

ϕ_{k}^{[- π, π]} (x, y)

and then extracted the phase distribution by unwrapping

ϕ_{k}^{[- π, π]} (x, y)

and removing the tilt. However, the process is usually time consuming and may cause phase error when

ϕ_{k}^{[0, π]} (x, y)

approaches zero or

π

. It has been demonstrated [18] that the phase shift

Δ δ_{k} (x, y)

, denoted as

Δ δ_{k} = δ_{k} - δ_{k - 1}

, can be extracted from the phase difference, denoted as

Δ ϕ_{k}^{[0, π]} = ϕ_{k}^{[0, π]} - ϕ_{k - 1}^{[0, π]}

. Once the phase shift is accurately achieved, the principal phase distribution can be extracted from,

ϕ_{k}^{[- π, π]} = \tan^{- 1} [\cot (Δ δ_{k}) - \frac{\bar{I_{k}}}{\sin (Δ δ_{k}) \bar{I_{k - 1}}}] .

Therefore, the key problem is to extract the phase-shift plane from the phase difference. Considering environmental vibration, the phase-shift plane may be tilted. In this case the phase-shift plane can be expressed as $δ_{k} (x, y) = a x + b y + c$ and the absolute phase difference of two normalized interferograms, denoted as $| Δ ϕ_{k}^{[0, π]} |$ may has three different kinds of distribution according to different magnitudes of tilts, which can be divided by the number of straight lines in the binary image of $| Δ ϕ_{k}^{[0, π]} |$ .

a) The magnitude of tilts is less than 0.5 wave over the entire pupil and there is no straight lines in the binary image of $| Δ ϕ_{k}^{[0, π]} |$ .
b) The magnitude of tilts is less than 1 wave but higher than 0.5 wave over the entire pupil, there is one straight line in the binary image of $| Δ ϕ_{k}^{[0, π]} |$ .
c) The magnitude of tilts is higher than 1 wave over the entire pupil and there are more than two straight lines in the binary image of $| Δ ϕ_{k}^{[0, π]} |$ .

Actually, the ambit of the magnitude of tilts in the division is not that rigorous, especially when the translational phase shift, i.e., $c$ is close to 0 or $π$ . It should also be noted in cases that the fringe themselves are straight ones then the division of magnitude of tilts above will lose its significance, but what the authors want to express is that the division above is just for the convenient view to explain the following different methods that adopted to extract the tilted phase-shift plane in this proposal.

2.3 The extraction of tilted phase-shift plane from phase difference and examples

Since different magnitudes of tilts will generate different distributions of phase difference, the approaches of phase-shift plane extraction are not the same but based on the same principle, namely the phase shift value can be extracted from the phase difference.

For simplicity and without loss of generality, we will give some examples to explicate the methods of tilted phase-shift plane extraction according to different cases.

a) For small tilts, the absolute phase difference distribution $| Δ ϕ_{k}^{[0, π]} |$ is shown in Fig. 1(a) and the corresponding binary image is shown in Fig. 1(b). It can be seen from Fig. 1(b) there is no straight lines. In this case, the tilted phase-shift plane can be extracted by fitting the points with small gradients in $| Δ ϕ_{}^{[0, π]} |$ . Provided that the phase shift plane has no tilt, the points with gradient of zero will be selected to fit the phase-shift plane, which correspond to the points with the bin of highest frequency in the histogram of phase difference in [18]. Since the magnitude of tilts is usually less than 0.5 wave over the entire pupil, the gradients of the points that should be selected are typically less than $π / m$ rad/pixel, where m is the number of pixels in the row or column of the interferogram. To determine the tilted phase-shift plane, one can separate the tilted phase-shift plane into x- and y- directional tilt components which are independent, namely $δ_{k x} = a x + c_{x}$ $δ_{k y} = b y + c_{y}$ $c = c_{x} + c_{y}$

At appointed row and column one can estimate the expressions of oblique lines by fitting the points with small gradients, as Figs. 1(d) and 1(f) shows, and then the translational phase-shift c can be determined. To improve the robustness of this method against noise, the process can be repeated at different rows and columns and parameters of the final tilted phase-shift plane are determined by use of average operation.

Fig. 1 Extraction of the tilted phase-shift plane: (a) the distribution of phase difference; (b) the binary image of (a); (c) the x- directional component of phase difference; (d) the selected points and the x- directional fitted line; (e) the y- directional component of phase difference; (f) the selected points and the y- directional fitted line.

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b) If the magnitude of tilts is higher than 0.5 wave and less than 1 wave over the entire pupil, there will be three special distributions of $| Δ ϕ_{k}^{[0, π]} |$ . (1) The tilted phase-shift plane over the entire pupil is in the range of $(- π, π)$ , just as Fig. 2(a) shows. (2) The tilted phase-shift plane over the entire pupil is in the range of $(0, 2 π)$ , just as Fig. 2(c) shows. (3) The tilted phase-shift plane over the entire pupil is in the range of $(- π / 2, 3 π / 2)$ , just as Fig. 2(e) shows. In situation (1), two tilted phase-shift plane with the opposite sign are intersected at the position of phase shift value equals zero. Firstly, one can select the points with small gradients, besides the phase shift values of them should approach zero in $| Δ ϕ_{k}^{[0, π]} |$ then these selected points can be fitted to a straight line. Finally, the points with small gradients that located at the left (or right) of the straight line can be fitted to a plane, which is the tilted phase-shift plane just as Fig. 2(b) shows. Noted that the fitted plane on the left of the detected straight line will has an opposite sign compared with that on the right of the line, thus the extracted phase-shift plane may has a sign ambiguity. In situation (2), two tilted phase-shift plane with the opposite sign are intersected at the position of phase-shift value equals $π$ . Similarly, the tilted phase-shift plane can be extracted as Fig. 2(d) shows. In situation (3), two straight lines can be fitted by selecting the points with small gradients approach zero and $π$ . The tilted phase-shift plane then can be extracted by fitting the points with small gradients between the two detected straight lines, which is shown in Fig. 2(f). The “small” gradient values in three situations of this case are typically higher than $π / m$ but less than $2 π / m$ rad/pixel.

Fig. 2 Extraction of the tilted phase-shift plane: (a) the distribution of phase difference in situation (1); (b) the selected points and the fitted phase-shift plane in situation (1); (c) the distribution of phase difference in situation (2); (d) the selected points and the fitted phase-shift plane in situation (2); (e) the distribution of phase difference in situation (3); (f) the selected points and the fitted phase-shift plane in situation (3).

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c) For large tilts, the magnitude of tilts is higher than 1 wave over the entire pupil that is to say the number of straight lines in the binary image of $| Δ ϕ_{k}^{[0, π]} |$ is more than two. In this case, it is difficult to extract the tilted phase shift plane by using points fitting method just as a) and b) cases represent, because the gradient of points that in the phase shift plane may even be larger than other points. So it is necessary to find a new way to extract the tilted phase-shift plane. Considering the fact that the straight lines in the binary image of $| Δ ϕ_{k}^{[0, π]} |$ are generated by two tilted plane with opposite sign, namely $δ_{k 1} (x, y) = a x + b y + c$ $δ_{k 2} (x, y) = - a x - b y - c$
then we can deduce from Eq. (7) that the straight parallel lines are:
$a x + b y + c + (n - 1) \cdot 2 π = 0, n = 1, 2, 3...$

Therefore, to estimate parameters of the tilted phase-shift plane, i.e. a, b, c, we just need to determine the expression of the first straight line of these parallel lines in the binary image of $| Δ ϕ_{k}^{[0, π]} |$ . Namely, we can extract the tilted phase-shift plane by detecting the straight parallel lines in the binary images of $| Δ ϕ_{k}^{[0, π]} |$ . Hough transform is a powerful method in lines detection and was first utilized by Soloviev and Vdovin [15] to detect lines in the binary image of inteferogram difference. Li et al. [16] also demonstrated the feasibility of lines detection in the binary image of inteferogram difference to extract the tilted phase-shift plane and proposed PTI. Figure 3(a) is the distribution of $| Δ ϕ_{k}^{[0, π]} |$ and the platform of Fig. 3(a) is shown in Fig. 3(b). It can be seen, the phase difference is wrapped in the range of $[0, π]$ . The binary image of $| Δ ϕ_{k}^{[0, π]} |$ is shown in Fig. 3(c), there are totally about 7 straight lines. Figure 3(d) exhibits the Hough Transform process. By detecting the peaks in Hough domain, the straight parallel lines in the binary image of $| Δ ϕ_{k}^{[0, π]} |$ can be detected. The white squares in Fig. 3(e) represent the straight parallel lines in Fig. 3(c).

Fig. 3 Extraction of the tilted phase-shift plane: (a) the distribution of phase difference; (b) the platform of (a); (c) the binary image of (b); (d) the Hough transform process; (e) the peak points in the Hough domain.

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With the achieved first peak in Hough domain, denoted as $(ρ, θ)$ , and the interval of two adjacent peaks, denoted as $Δ ρ$ , one can calculate the parameters of the tilted phase-shift plane from the following equations [16],

a = 2 π \cdot \cos (ϑ) / Δ ρ

b = 2 π \cdot \sin (ϑ) / Δ ρ

c = - b \cdot ρ \cdot \sin (ϑ) - a \cdot ρ \cdot \cos (ϑ) \begin{matrix} (ϑ = 90^{\circ} - θ) \end{matrix}

it should be noted that in case c), i.e. the magnitude of tilts is larger than 1 wave, since the parameters of tilts are calculated by line detection of the binary image of phase difference, proposed method will also meets the disadvantage that [15,16] have, namely the orientation of fringe should not be straight parallel ones. However, in case 1) and case 2), proposed method is insensitive to the orientation of fringe.

3. Simulations

Numerical simulations are carried out to verify the performance of proposed method. It is assumed that the background intensity and the modulation amplitude are both obey to Gaussian distribution, which are

A (x, y) = 145 \exp (- 1.8 (x^{2} + y^{2})), B (x, y) = 100 \exp (- 0.2 (x^{2} + y^{2})); (- 1 \leq x, y \leq 1)

respectively, and the number of pixels is 200 in both x and y directions. The phase distribution is

φ (x, y) = π (0.3 (x^{2} + y^{2}) - 0.2) (- 1 \leq x, y \leq 1)

and the corresponding surface distribution is shown in Fig. 4(c). The random phase-shift at pixel

(x, y)

is

δ_{k} (x, y) = a x + b y + c

. Tilt gradients of x- and y- directions, i.e., a and b, are in the range of 1e-4 to 1e-1 with the unit of rad/pixel, which means the largest tilt phase-shift is 200 × 0.1 = 20 rad and the smallest tilted phase shift is 200 × 0.0001 = 0.02 rad over the entire pupil. The translational phase-shift c is in the range of

(0, 2 π)

. In addition, Gaussian white noise with the Signal-to-Noise Ratio (SNR) of 40dB is added to the interferograms. 30 randomly phase shifted interferograms are generated according to Eq. (1).

Fig. 4 Simulation results1: (a) and (b) the two normalized interferograms; (c) the surface distribution of the phase to be measured; (d) the phase difference of the two normalized interferograms; (e) the histogram of phase shift; (f) and (g) the retrieved surface and the residual surface error of Xu’s method; (h) the extracted tilted phase-shift plane; (i) and (j) the retrieved surface and the residual surface error of proposed method.

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From these 30 interferograms, the background intensity and the modulation amplitude can be calculated. Then two inteferograms are randomly selected and normalized, Figs. 4(a) and 4(b) is two of the normalized interferograms. Figure 4(d) is the distribution of phase difference between Figs. 4(a) and 4(b). Both Xu’s algorithm mentioned in [18] and proposed method are used to retrieve the phase distribution from the two normalized interferograms. On the one hand, from the histogram of the phase difference distribution, just as Fig. 4(e) shows, the phase-shift range can be achieved by extracting the bin with highest frequency which is [1.365, 1.547]. To extract the more accurate phase shift value, the interval of [1.365, 1.547] is set to 0.02 rad and the ultimate phase shift is achieved, i.e. 1.419 rad. Then with the help of Eq. (5) the retrieved surface distribution of Xu’s method is shown in Fig. 4(f), and the distribution of the residual surface error is exhibited in Fig. 4(g). From which we can see the root-mean-square (rms) value of the residual surface error is as high as 3.261nm since the tilt phase-shift is not in the consideration of Xu’s method. On the other hand, proposed method is also used to extract the phase-shift, by selecting the points with small gradients in phase difference, i.e. Figure 4(d) the tilted phase-shift plane can be fitted by the least-squares method, which is expressed as $δ (x, y) = - 0.0026 x + 0.0017 y + 1.5908$ . The tilted phase-shift plane is appreciated in Fig. 4(h). The reconstructed surface of proposed method is shown in Fig. 4(i) and the residual surface error of proposed method, shown in Fig. 4(j), is 0.366nm (rms), which is mainly caused by high frequency noise.

To demonstrate the performance of proposed method, we make a comparison between Liu’s three step iterative method [13], since Liu’s method is confirmed to be more flexible and accurate than Xu’s iterative method [11], with proposed method. The interferograms used for comparison are also generated according to Eq. (1), and the SNR of the noise is 40dB. The normalization of frames is performed with 30 randomly phase shifted interferograms. It should be noted firstly that the nonuniform background intensity and modulation amplitude will cause a “coupling” effect when calculating the tilt parameters through the iterative process. Therefore, we also verified the ability of Liu’s three step iterative method with normalized interferograms as reference. Since the iterative method needs at least three interferograms to calculate the tilt phase-shift and phase distribution, whereas only two normalized frames are needed in proposed method. For the equality view, five raw frames are dealt with by Liu’s iterative method first and then the corresponding five normalized frames are dealt with by both Liu’s iterative method and proposed method. With all the calculated coefficients of tilt phase-shift between these five frames, the phase distribution can be extracted by the least-squares method (LSM) [11,14], which will be served as the final reconstructed surface of proposed method.

Figures 5(a)–5(c) appreciate the three coefficients of tilt phase-shift, i.e. a, b, c of the five frames. It is not lose of generality, the three coefficients of the first interferogram are set to zero and this frame is regarded as reference. From Figs. 5(a)–5(c) it can be seen clearly that Liu’s three step iterative method with normalized interferograms has the most accurate calculations, the calculated results of which are almost equal the real data since the “coupling” effect has been suppressed by the normalization process. Proposed method is of high accuracy either, however Liu’s iterative method will has significant errors when dealing with interferograms with the background intensity and modulation amplitude equal $A (x, y)$ and $B (x, y)$ in this simulation. It also can be seen the maximum error of Liu’s method in tilt gradients extraction and translational phase-shift extraction are $6.83 \times 10^{- 4}$ rad/pixel and 0.092 rad respectively, which coincide well with [13]. Figure 5(c) represents the computing time and residual surface error (rms) of the three methods, i.e. Liu’s iterative method with normalized inerferogram, Liu’s iterative method and proposed method, which are substitute for A, B and C in Fig. 5(c) respectively. It indicates that proposed method needs 0.891s to achieve the residual error of 0.256 nm (rms), whereas Liu’s iterative method with normalized inerferogram costs 8.6s to achieve the residual error of 0.244 nm (rms) and Liu’s three step iterative method however, suffers a low efficiency and low accuracy correspondingly. It should be mentiond the previous normalization process costs 0.188s which is not included in the above list, but obviously it is reasonable to draw a conclusion that proposed method is of high accuracy and speed compared with Liu’s iterative method. Noted that all the methods are performed on an Intel i5 processor based on Matlab computing tool.

Fig. 5 Simulation results2: (a)-(c) the calculated tilt gradients, i.e. a, b and translational phase shift, i.e. c of the 5 interferograms with three different methods; (d) the computing time and residual surface error (rms) of three different methods .

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To demonstrate the validity of proposed method when dealing with different magnitudes of tilts, a total of 300 simulations (100 sets for each case with magnitude of tilts up to 0.5 wave, 1 wave, and larger than 1 wave over the entire pupil) are carried out. In the simulations, the background intensity and modulation amplitude of the interferograms are calculated from 30 frames, the SNR of the noise is 45dB and parameters of the tilted phase-shift plane are totally random.

Table 1 summarizes the results of 12 representative cases, in which the average phase shift extraction error is calculated by the following equation,

Tabel 1. Comparison of the tilt phase shift coefficients according to different magnitudes of tilts

View Table

e = {[| (a' - a) | + | (b' - b) |] \times 200 + | (c' - c) |} / 3

It can be seen proposed method can easily extract the tilted phase-shift plane with high accuracy especially when the magnitude of tilts is less than 1 wave. The phase shift extraction error is typically less than 0.02 rad, but if the tilt is higher than 1 wave the phase shift extraction error may increases to 0.17 rad and the extraction errors of tilt gradients, i.e. a, b are typically less than 0.001 rad/pixel which coincide well with [16]. The main error is the translational phase shift extraction error, i.e. c. From Eq. (9c) we know three variables, namely the distance $ρ$ , the angle $θ$ , the interval $Δ ρ$ in Hough domain will affect the accuracy of c. However, since both $θ$ and $Δ ρ$ are calculated by applying the average operation to all peaks in Hough domain, they are taken for accurate values (that’s why the extraction errors of tilt gradients, i.e. a, b are accurate). Thus, it is $ρ$ of the first peak in Hough domain that effect the accuracy of c. To improve the accuracy of c, we could calculate the three coefficients of all the parallel lines (related to all peaks in Hough domain), i.e. Equation (8). Finally, the translational phase shift, i.e. c can be achieved by use of average operation.

Unfortunately, it should be noted that the calculated parameters of tilted phase-shift plane may have incorrect global sign. The phase shift with incorrect sign will lead to wrong global sign of phase distribution, so it is necessary to analyze the relation between the sign of extracted phase-shift plane and the initial phase shifts according to different magnitudes of tilts. From the simulations we also find some factors may influence the accuracy of proposed method and they will be analyzed in the following discussion.

4. Discussions

4.1 The number of frames used

Since the adopted normalization process is based on the calculation of background intensity and modulation amplitude, and the number of frames (M) is assumed to be large enough to ensure the intensity of each pixel is ergodic over M frames, so if M is not large enough the calculated A(x,y) and B(x,y) are not accurate, consequently the normalization will contains inevitable errors. Thus, it is crucial to determine the number of frames that should be used, a simulation has been done to address this problem correspondingly. In the simulation, the SNR of the noise is set to 40dB, and the tilted phase-shift plane of the two measured interferograms is set to a random expression, e.g, $δ (x, y) = - 0.0026 x + 0.0017 y + 1.5908$ , and stays the same, whereas the tilt gradients of other frames that used to calculate A(x,y) and B(x,y) are set to random values varying from 1e-4 to 1e-1, and the translational random phase shift is in the range of $(0, 2 π)$ . Then we let M vary from 10 to 80 with the interval of 5. Figure 6 indicates the average extraction error of the phase shift according to different number of frames that used for normalization. The error is calculated by Eq. (10). It can be seen when M is smaller than 15 the average extraction error of phase shift can be reduced significantly with the increasing of M, but when it is higher than 15 the average extraction error of phase shift are reduced gently as M increases. Thus, 30 frames that used to normalize the interferogram in the simulation above is reliable and efficient. Considering the air turbulence and mechanical vibration in practical, more than 30 (2 times as large as 15 [18]) frames should be used for normalization.

Fig. 6 The average phase shift extraction error for different numbers of frames that used for normalization.

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In the latest literature [1,19], it has been demonstrated that the interferogram can be normalized from only one single frame as long as the interferogram has many fringes (to ensure the least-squares estimation method is accurate). Without doubt, it is good news to this proposal since it is possible to extract the tilted phase-shift plane by proposed method with two interferograms really and truly if the assumption has been met.

4.2 The random noise of the interferogram

The noise invariably affect the accuracy of proposed method, since the parameters of tilted phase-shift plane are achieved by selecting special points in phase difference. Thus it is necessary to study the robustness of proposed method in presence of noise.

As an example we perform numerical simulations at different SNRs of noise to obtain the residual surface error of proposed method in case a), i.e. the magnitude of tilts is less than 0.5 wave over the entire pupil. The noise is added to the interferogram intensity by the awgn function in matlab with ‘measured’ mode. Figure 7 shows the relation between the residual surface error and SNR. It shows that proposed method is sensitive to noise, and when SNR of the additive noise is less than 30dB proposed method may leads to results with significant errors. When SNR of the additive noise is higher than 50dB, the residual surface error is less than 0.1nm. Therefore, it is advantageous to filter out the noise of interferograms before performing phase measurement. In the simulation above and following experiments the noise is filtered by medfilt2 function in Matlab with a $7 \times 7$ window.

Fig. 7 The relation between the residual surface error and SNR.

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4.3 Tilt range and ambiguity in the detection of phase shift sign

The ambiguity of phase shift sign directly leads to opposite phase distribution, hence it is important to analyze the relation between extracted phase-shift sign with initial phase-shift according to different tilt amplitudes to ensure the sign is correct. The 300 simulations above indicate, in case 1) i.e., magnitude of tilts over entire pupil is less than 0.5 wave, the extracted phase shift sign is correct if the initial translational phase-shift is in the range of $(0, π)$ , else if the initial translational phase-shift is in the range of $(π, 2 π)$ the sign of the extracted tilt phase-shift will be opposite. Therefore, the translational phase-shift between the two interferograms should be controlled in range of $(0, π)$ to ensure the extracted phase shift sign is correct in experiment, which is feasible in practical measurement [8]. In case 2) and case 3) i.e., magnitude of tilts over the entire pupil is higher than 0.5 wave, since one straight line in the binary image of $| Δ ϕ_{k}^{[0, π]} |$ always has two opposite expressions, it is difficult to detect the positive one. Some additional information is always needed to achieve the correct sign, in practical we can limit all possible tilts only to those increasing in y-direction [15,16].

5. Experiments

To illustrate the application of proposed method, optical experiments have been carried out on a Zygo GPI interferometer. An optical surface, with PV value of 72.39nm and RMS value of 14.15nm, is first measured on a vibration-isolating platform as reference, which is shown in Fig. 8(e). Subsequently, about 50 frames, whose preset piston phase-shift is $k π / 4$ and the tilt-shift error is introduced by pressing on the object holder slightly with a finger, are collected at a frame frequency of 20 fps. The so calculated phase distribution by Zygo’s 13-step phase shifting algorithm (PSA) is shown in Fig. 8(f), which indicates a distinct “ripple” error that caused by introduced tilt-shift error. By comparing with the reference surface i.e. Figure 8(e), we can obtain the distribution of the “ripple” error, which is appreciated in Fig. 8(g) and the RMS value of the “ripple” error is 9.416nm. To validate the performance of proposed method two interferograms are selected from these 50 frames, as appreciated in Figs. 8(a) and 8(b). By use of principle 2.1 the background intensity and modulation amplitude can be calculated from these 50 frames and then the two selected interferograms are normalized which are shown in Figs. 8(c) and 8(d). The phase difference of the two normalized interferograms is calculated as Fig. 8(h) shows, and the tilted phase-shift plane is estimated as $δ (x, y) = - 0.0012 x - 0.0061 y + 0.3226$ . With the help of Eq. (5), the extracted phase distribution is appreciated in Fig. 8(i). It can be seen that the surface error that caused by tilt phase-shift error is reduced effectively and the “ripple” error has almost been wiped off compared with Fig. 8(f). Likewise, we achieved the distribution of residual surface error of proposed method by subtracting Fig. 8(i) from Fig. 8(e), the residual surface error with RMS value of 4.83nm is shown in Fig. 8(j) and from which we can see the error is mainly caused by air turbulence. Therefore, the calculated tilt phase-shift of proposed method is accurate and reliable.

Fig. 8 Experimental results: (a) and (b) the two selected interferograms with tilt phase shift error; (c) and (d) the two normalized interferograms; (e) the reference surface calculated without tilt phase-shift error; (f) the surface calculated by Zygo’s 13-step PSA when tilt phase shift error is introduced; (g) the residual surface error between (e) and (f); (h) the phase difference of the two normalized interferograms; (i) the retrieved surface of proposed method; (j) the residual surface error between (e) and (i).

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The experimental results indicate proposed method can relaxes the requirements on the accuracy of phase shifter and if the amplitude of vibration is high enough, in which the tilt-shift can be regarded as phase-shift, proposed method can even be used to extract the phase distribution without a costly phase shifter.

6. Conclusions

We have presented a novel method to extract the phase distribution from two normalized interferograms with random phase shift and tilt. The proposed method determines the parameters of the tilt phase-shift by extracting the tilted phase-shift plane from the phase difference of two normalized interferograms. According to different magnitudes of tilts, the phase difference will have three different kinds of distributions, and correspondingly we proposed three different methods to extract the tilted phase-shift plane. Proposed method cannot only deal with small tilt-shift errors (typically less than 0.5 wave over entire pupil) in PSI but also detect the tilt phase-shift that is larger than 1 wave over entire pupil in PTI. What’s more, both closed and open fringes can be dealt with by proposed method. Additionally, it is far faster than the iterative method and has a higher accuracy. It should be noted that to ensure the correctness of the tilt phase-shift sign, the translational phase-shift between two adjacent interferograms should be controlled in the range of $(0, π)$ in practical measurements and when the tilt phase-shift is higher than 0.5 wave over the entire pupil other additional information should be known previously. Experimental results indicate that proposed method can reduce the “ripple” error that caused by tilt-shift error efficiently, which demonstrates the extracted tilted-phase shift plane of proposed method are accurate.

Acknowledgments

The research is supported by the National Natural Science Foundation of China (61108045) and the Creative Foundation of Institute of Optics and Electronics, CAS. Furthermore, we would like to express our gratitude to the anonymous reviewers for their invaluable help and useful recommendations to improve this work.

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Number Magnitude		The real parameters of tilts (rad/pixel)				The calculated parameters of tilts (rad/pixel)			Error (rad)
Number Magnitude		$a$	$b$	$c$		$a'$	$b'$	$c'$	$e$
Less than 0.5-wave	1	−1.5192e-4	4.6194e-4	5.8009		−1.5116e-4	4.6348e-4	5.7391	0.0208
	2	−3.1474e-5	1.1718e-4	1.2826		−3.1225e-5	1.1938e-4	1.2653	0.0059
	3	0.0011	0.0014	1.4582		0.0011	0.0014	1.4608	0.0029
	4	0.0013	−0.0035	4.0390		0.0013	−0.0035	4.0373	0.0026

Less than 1-wave	5	0.0087	0.0086		1.3968	0.0087	0.0086	1.4270	0.0127
	6	0.0108	0.0031		1.2028	0.0108	0.0031	1.1886	0.0074
	7	0.0253	−0.0111		2.6704	0.0253	−0.0111	2.6858	0.0078
	8	0.0147	−0.0057		4.6956	0.0147	−0.0057	4.7053	0.0046

Higher than 1-wave	9	0.1614	0.2663		2.8185	0.1607	0.2669	2.9114	0.1190
	10	0.1576	−0.1913		3.1668	0.1578	−0.1914	3.1003	0.0442
	11	0.3557	−0.2385		5.4987	0.3552	−0.2396	5.7120	0.1798
	12	−0.1826	0.0971		2.7628	−0.1826	0.0973	2.6632	0.0472

Phase shifting interferometry from two normalized interferograms with random tilt phase-shift

Abstract

1. Introduction

2. Principles

2.1 The normalization of background intensity and modulation amplitude

2.2 The phase difference of two normalized interferograms

2.3 The extraction of tilted phase-shift plane from phase difference and examples

3. Simulations

4. Discussions

4.1 The number of frames used

4.2 The random noise of the interferogram

4.3 Tilt range and ambiguity in the detection of phase shift sign

5. Experiments

6. Conclusions

Acknowledgments

References and links

Cited By

Figures (8)

Tables (1)

Equations (16)

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