Correction of phase extraction error in phase-shifting interferometry based on Lissajous figure and ellipse fitting technology

Fengwei Liu; Yongqian Wu; Fan Wu

doi:10.1364/OE.23.010794

1. Introduction

The phase-shifting interferometry (PSI) is widely used in high precision optical surface testing domain [1], especially for large-aperture mirrors’ surface testing in astronomical telescopes. The conventional PSI [2–5] requires a strict precondition that the phase shift between each interferogram should be a special constant (e.g. $π / 2$ ). Therefore, the accuracy of PSI will be impaired if the phase-shift value is not the required one. Due to the large size of the measured surface the optical path often reaches several meters in practice, so the mechanical vibration and air turbulence will lead to some unavoidable phase-shift errors [6,7]. Such errors can further cause substantial errors of the phase distribution.

To deal with those problems plentiful of efforts have been done, which are mainly divided into two classes: (1) Optimizing the configuration of the interferometer, represented by simultaneously phase shifting interferometry (SPSI). It can freeze the vibration by collecting three or four interferograms instantaneously [8–10]. However, SPSI system is complicate in the configuration and except for the hardware expenditure most of them are polarization based system, which may introduce systematic polarization aberrations. (2) Optimizing the phase-shifting algorithm (PSA), many algorithms aimed to diminish the random phase shift errors have been proposed in recent years. In 1991, Okada et al. [11] proposed an iterative algorithm based on the least-square method to cope with the uncertainties in phase-shift amounts. Since then, several similar self-calibrating PSAs have been proposed to correct the phase-shift errors [12–15], whereas all of these approaches are time consuming because of the iterative process. Meanwhile, some non-iterative methods have also been published. The Lissajous elliptical curve fitting is an attractive approach to calibrate the phase-shift errors for its simple principle and high accuracy [16–21]. In 1988, Kinnstaetter et al. [16] first discussed the systematic errors in PSI by using of the Lissajous display technique. But the correction of the inaccuracies of phase shift is still using an iterative process in their paper and the Lissajous ellipse is created by choosing two points, roughly in phase quadrature, on the fringe pattern which determines more than 5 interferograms are needed. In 1992, Farrell and Player [17,18] utilized Lissajous figures and ellipse fitting technology to calculate the phase shift and the intensity bias between two interferograms, they also shared the idea of correcting the measured intensity field in traditional PSA with the calculated bias, offset and phase shift error. However, since the points that created the Lissajous ellipse are from the whole fringe pattern the illumination is then assumed to be uniform, which is too strong to be met exactly in practical experiments. In 2003, K. S. Moon and Y. D. Wang [19] proposed a similar method to estimate the arbitrary phase shift between each interferogram using Lissajous figure technique, a scan line points on two intensity profiles are used to create the Lissajous ellipse.

In one word, not only the iterative method but also the Lissajous elliptical curve fitting method mentioned above are all aimed at suppressing the phase shift error by calculating the comparatively true phase-shift amounts between each interferogram first and then using the general PSA to extract the error free phase [11–15, 19] or correcting the intensity fringe field in traditional PSA [17]. Considering the fact that the numerator, N, and the denominator, D, terms of the phase calculation function in all kinds of PSAs are in phase quadrature, the inaccurate phase shifts of each interferogram can be ignored first and then they can be directly corrected by calibrating the phase extraction error from the phase calculation function (i.e. arctangent function) in PSA .

Recently, B. Kimbrough [21] proposed a method to correct the systematic errors of polarization based SPSI system by Lissajous figure technique. The offset, bias and phase shift error are calculated by fitting a series of points to a Lissajous ellipse. It can only correct the static systematic error such as the phase dependent error caused by mechanisms, and what’s more the Lissajous ellipse is formed by taking M (>5) measurements. In this paper, we propose a general method to correct the phase extraction error caused by inaccurate phase shifts in PSI. Ostensibly, the process of correcting the Lissajous ellipse to a unit circle is similar with B. Kimbrough’s method, but the Lissajous ellipse does have an absolutely different meaning, thus the meaning of correction is totally different. Proposed method can correct the dynamic random phase-shift errors in PSI that B. Kimbrough’s method cannot. More importantly, we assume the background intensity and the intensity contrast of the interferogram are circular symmetric in practice, proposed method then can correct the phase extraction error accurately from interferograms even with excessively large variations of background intensity and modulation amplitude. This approach is non-iterated, adapts to all PSAs, and has high accuracy.

2. Principles

2.1 The principle of error compensating

Regardless of the technique, almost all of the PSAs ultimately recur to an arctangent function to calculate a phase value, which is wrapped in the range of $(- π, π)$ . With the help of unwrapping algorithm, the true phase distribution is finally established.

As we all know the arctangent function is a ratio of two values, which can be referred to as the numerator, N, and the denominator, D, terms here, shown in Eq. (1),

ϕ = \tan^{- 1} (\frac{N}{D}) .

where

ϕ

is the phase that calculated from the phase extraction function and both N and D are functions of the real phase

φ

, a general expression [21] for them are given in Eq. (2),

N (φ) = x_{0} + a_{x} \cdot \sin (φ)

D_{ε} (φ) = y_{0} + a_{y} \cdot \cos (φ + ε)

which are coincident with the parametric expression of an ellipse centered at

(x_{0}, y_{0})

and constrained to lie within a rectangle of length

a_{x}

by

a_{y}

. Pairs of points

{N (x, y), D (x, y)}

are located at that ellipse just like Fig. 1(a) shows.

Fig. 1 Lissajous figures by plotting N against D: (a) an ellipse (with errors); (b) a unit circle (without errors).

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Provided that both N and D terms have no offset, the amplitudes are equal and their phases are in quadrature, namely,

x_{0} = y_{0} = 0; a_{x} = a_{y}; ε = 0.

Then Eq. (1) can be rewritten as

ϕ = \tan^{-1} [\frac{\sin (φ)}{\cos (φ)}]

Therefore, the phase value $ϕ$ that calculated from Eq. (3) is exactly the real phase $φ$ . Pairs of points ${N (x, y), D (x, y)}$ are transformed to lie on a perfect circle centered at the origin as Fig. 1(b) shows. Consequently, the phase extraction error, which usually is a non-liner combination of offset error, amplitude error and phase error, has already been corrected. Thus, to correct the phase extraction error we should transform the ellipse to a unit circle.

By expanding Eq. (2b) and substituting for $\cos (φ)$ and $\sin (φ)$ , we get

D_{ε} (φ) = y {}_{0}+ (D - y_{0}) \cdot \cos (ε) - (a_{y} / a_{x}) (N - x_{0}) \cdot \sin (ε)

(in which,

N (φ) = x_{0} + a_{x} \cdot \sin (φ) S

and

D (φ) = y_{0} + a_{y} \cdot \cos (φ)

)

Assuming N is accurate and rearranging Eq. (4), then the ideal value of D can be calculated as follow,

D = [(D_{ε} - y_{0}) + (a_{y} / a_{x}) \cdot (N - x {}_{0}) \cdot \sin (ε)] / \cos (ε) + y_{0}

Once D is established, the less of the quadrature

ε

(typically,

| ε | < < π / 2

) caused by the inaccurate phase shifts has been compensated. However, if we want to get the Lissajous circle as Fig. 1(b) shows, we have to correct the offset error (i.e.,

x_{0} \neq 0, y_{0} \neq 0

) and the amplitude bias (i.e.,

a_{x} \neq a_{y}

) using the following equation,

\bar{N} = (N - x_{0}) / a_{x}; \bar{D} = (D - y_{0}) / a_{y} .

Then the calibrated error free phase is given in Eq. (7),

φ = ϕ = arc \tan (\frac{\bar{N}}{\bar{D}}) = arc \tan (\frac{1}{\frac{a_{x}}{a_{y}} (\frac{D_{ε} - y_{0}}{N - x_{0}}) \frac{1}{\cos ε} + \tan ε})

Finally, the phase value that calculated from Eq. (7) is correct. Together, Eqs. (5)-(7) are referred to as ETC process for brevity in the following discussions.

2.2 Analysis and example of the phase extraction error caused by random phase shift errors

The two-beam interference pattern can be written algebraically in Eq. (8)

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

Where

A (x, y), B (x, y), φ (x, y)

represent the background intensity, the modulation amplitude and the real phase at the location of

(x, y)

respectively, and

β_{k}

represents the

k

th phase shift value. The general expression of PSA [22] can be written in Eq. (9)

ϕ = arc \tan \frac{\sum_{k = 0}^{m - 1} [(I_{\mod (k + 1, m)} - I_{\mod (m + k - 1, m)}) \cdot \cos (β_{k})]}{\sum_{k = 0}^{m - 1} [(I_{\mod (k + 1, m)} - I_{\mod (m + k - 1, m)}) \cdot \sin (β_{k})]}

Where mod

(i, m)

represents the remainder when

i

is divided by

m

and the location

(x, y)

is omitted for brevity in Eq. (9).

For simplicity, the traditional 4 buck PSA is selected as an example to analysis the phase extraction error that caused by the inaccurate phase shifts of each interferogram.

Provided $β_{1} = 0, β_{2} = π / 2, β_{3} = π, β_{4} = 3 π / 2$ then we can easily obtain the expression of 4 buck PSA from Eq. (9),

ϕ = arc \tan \frac{I_{4} - I_{2}}{I_{1} - I_{3}}

which is the same with that represented in [23]. However, if the phase-shift value of each interferogram contains a random error. For instance,

β_{1} = 0, β_{2} = π / 2 + δ_{1}, β_{3} = π + δ_{2}, β_{4} = 3 π / 2 + δ_{3}

, then substituting Eq. (8) in Eq. (10) the following deduction will happens,

\begin{array}{l} ϕ = arc \tan \frac{N}{D} =arc \tan \frac{I_{4} - I_{2}}{I_{1} - I_{3}} \\ = arc \tan \frac{B [\sin (φ + δ_{3}) + \sin (φ + δ_{1})]}{B [\cos φ + \cos (φ + δ_{2})]} \\ = arc \tan \frac{B [\sin (φ + \frac{δ_{3} + δ_{1}}{2}) \cdot \cos (\frac{δ_{3} - δ_{1}}{2})]}{B [\cos (φ + \frac{δ_{2}}{2}) \cdot \cos (\frac{δ_{2}}{2})]} \\ = arc \tan \frac{b_{1} \sin (φ + \frac{δ_{3} + δ_{1}}{2})}{b_{2} \cos (φ + \frac{δ_{2}}{2})} \\ = arc \tan \frac{b_{1} \sin (φ)}{b_{2} \cos (φ + ε)} \end{array}

where,

b_{1} = B \cos (\frac{δ_{3} - δ_{1}}{2}), b_{2} = B \cos (\frac{δ_{2}}{2}), ε = \frac{δ_{2} - δ_{1} - δ_{3}}{2}

, comparing Eq. (11) with Eq. (2) it can be found that

x_{0} = 0, y_{0} = 0; a_{x} = b_{1}, a_{y} = b_{2}; ε = ε

.

From the deduction above one can conclude that Eq. (2) is a reasonable expression of the phase extraction function when the phase shift of each interferogram contains a random error. It is based on Eq. (2) that proposed ETC method can calibrate the phase extraction error by calculating the parameters that Eq. (7) needs (i.e., $x_{0}, y_{0}, a_{x} / a_{y}, ε$ ) from a single Lissajous ellipse.

2.3 The method of producing the Lissajous ellipse

Principle of 2.1 tells that if we know the parameters of the ellipse the phase extraction error can be compensated. Actually, the way of producing the Lissajous ellipse not only affects the accuracy of the calculated parameters but also determines the meaning of the Lissajous ellipse. Hence, a proper way to create the Lissajous ellipse is essential to proposed algorithm. As far as we know, there are two feasible approaches to produce a Lissajous ellipse in literature.

A). Taking a series of (M>5) phase measurements, where a phase offset is introduced between each measurement. The average phase offset should ensure that the six measurements, for example, can form an orbicular phase shift period (i.e., $2 π$ ). Firstly, choosing a field point (pixel) in the numerator, designated ${N_{i} (x, y)} (0 \leq i \leq 5)$ , and then plotting the corresponding point in the denominator, designated ${D_{i} (x, y)} (0 \leq i \leq 5)$ against it in a Cartesian coordinate. Finally, a Lissajous ellipse is created by fitting the six scattered points, i.e. $(N_{i} (x, y), D_{i} (x, y)) (0 \leq i \leq 5)$ . By repeating the process to all pixels in the field, each pixel will has a corresponding ellipse.

B). The other one needs only one time measurement and returns two data sets ${N (x, y)}, {D (x, y)}$ . The Lissajous figure is created by plotting ${D (x, :)}$ against ${N (x, :)}$ , or ${D (:, y)}$ against ${N (:, y)}$ . In [17], C.T Farrell first utilized this mode to calculate the phase step and intensity bias between two interferograms. It should be noted that this approach requires at least one fringe across each field, and it performs best when the illumination of the fringe field is uniform.

The first method can correct the phase extraction error pixel by pixel, thus it is less sensitive to the quality of interferograms and does not need the illumination to be uniform. It seems like a perfect way to calibrate the phase extraction error caused by inaccurate phase shifts. Unfortunately, a series of measurements always require a dual phase shift arrangement, so the scheme is complicated and expensive. More importantly, it can only calibrate the static systematic error, which is improper to analysis the random phase shift errors in PSI. On the contrary, the second one needs only one time measurement and ETC can be easily used in all kinds of PSAs to correct the phase extraction error. Therefore, it is more convenient and practical in optical experiment. However, the illumination is not uniform as C.T. Farrell assumes in practice, the parameters calculated from the Lissajous ellipse created by a scan line points is inaccurate when the illumination has excessively large variations. Thus, it is necessary to find a new way to produce the Lissajous ellipse in the case that the illumination is nonuniform. The key is to find out the points with uniform background and modulation amplitude on the fringe pattern to create the Lissajous ellipse. It seems like impossible since the illumination of the interferogram is in the dark, but the background intensity and modulation amplitude are most likely to be a Gaussian symmetric distribution experientially so we assume the illumination is circular symmetric.

For different PSAs, the numerators, N, and the denominators, D, of the phase extraction function are regarded as two interferograms with the phase-shift amount of $π / 2$ approximately in our algorithm. We directly correct the phase extraction error by calculating the parameters of a Lissajous ellipse created by plotting ${N (i, j)}$ against ${D (i, j)}$ , where $(i, j)$ is the point laid on a special circle. The circle should crosses a whole fringe field and the illumination on the circle is assumed to be uniform. We refer this way as the “circular mode”. Correspondingly, the scan line way is the “linear mode”.

Actually, choosing one special circle to create the Lissajous ellipse is enough even though different circles in the fringe pattern will lead to different Lissajous ellipses. Because from Eq. (11) and the following Eq. (15) we know, the parameter $ε$ is the same value for the whole fringe pattern, and though the parameters $a_{x}, a_{y}$ are definitely different corresponding to different ellipses, the value of $a_{x} / a_{y}$ , however, is a constant. For the offset, $x_{0}, y_{0}$ are approximately to zero. Consequently, the corrected function that Eq. (7) represents is almost the same to different ellipses.

It should be noted that in the noisy circumstance, we have two methods to improve the robustness of proposed algorithm: Firstly, we can control the width of the circle line, which determines the number of the points creating the Lissajous ellipse. Secondly, an average process should be taken by repeated plotting and fitting across the full fringe pattern.

2.4 The ellipse fitting and calculation of coefficients

The points creating the Lissajous figure are always scattered because of the noisy interferogram. If those points are located over one period of the intensity, a conic can be fitted to the points by using the least square fitting method [24].

The general expression of a conic curve is [17,18]

A \cdot I_{1}^{2} + 2 B \cdot I_{1} I_{2} + C \cdot I_{2}^{2} + 2 D \cdot I_{1} + 2 E \cdot I_{2} + F = 0

If we write the parametric Eq. (2) in a general ellipse form, it is found to be

\frac{{(I_{1} - x_{0})}^{2}}{a_{x}^{2}} + \frac{2 \sin ε (I_{1} - x {}_{0}) (I_{2} - y {}_{0})}{a_{x} a_{y}} + \frac{{(I_{2} - y_{0})}^{2}}{a_{y}^{2}} = \cos^{2} ε

Where

I_{1} = N, I_{2} = D {}_{ε}

, by rewriting Eq. (13) in the general conic curve form (i.e. Equation (12)) and comparing with each other it yields the parametric-to-conic transformation coefficients as follow,

\begin{array}{l} A = \frac{1}{a_{x}^{2}} \\ B = \frac{\sin ε}{a_{x} a_{y}} \\ C = \frac{1}{a_{y}^{2}} \\ D = - \frac{x_{0}}{a_{x}^{2}} - \frac{y_{0} \sin ε}{a_{x} a_{y}} \\ E = - \frac{y_{0}}{a_{y}^{2}} - \frac{x_{0} \sin ε}{a_{x} a_{y}} \\ F = \frac{x_{0}^{2}}{a_{x}^{2}} + \frac{y_{0}^{2}}{a_{y}^{2}} + \frac{2 x_{0} y_{0} \sin ε}{a_{x} a_{y}} - \cos^{2} ε \end{array}

A convention is sometimes followed in which $F$ is set to −1 [18], this, however, precludes the fitting of conics which pass through the origin, and should be avoided.

The inverse, conic-to-parametric transformation reads:

ε = arc \sin (\frac{B}{\sqrt{A C}}), x_{0} = \frac{B E - C D}{δ}, y_{0} = \frac{B D - A E}{δ}, a_{x} = \frac{\sqrt{- C Δ}}{δ}, a_{y} = \frac{\sqrt{- A Δ}}{δ} .

where

Δ

,

δ

are defined as,

\bar{Δ} = [\begin{matrix} A & B & D \\ B & C & E \\ D & E & F \end{matrix}], Δ = \det \bar{Δ}, δ = | \begin{matrix} A & B \\ B & C \end{matrix} |

With the established coefficients, we can calculate the error free phase using Eq. (7).

3. Simulations and discussions

Numerical simulations are carried out to test the performance of proposed method. We generate 4 interferograms according to Eq. (8). One of them is shown in Fig. 2(a). Here, $A (x, y) = 145 \exp (- 1.8 (x^{2} + y^{2})); B (x, y) = 100 \exp (- 0.2 (x^{2} + y^{2})); (- 1 \leq x, y \leq 1)$ the number of pixels is 200 in both $x$ and $y$ directions. The phase distribution $φ (x, y)$ is produced by the Zernike polynomials [25] with the first five coefficients setting to 0.1, 2, 0.5, 0.2, 0.1 respectively, namely

ϕ (x, y) = π (0.1 + 2 x + 0.5 y + 0.2 (x^{2} + y^{2} - 1) + 0.1 (y^{2} - x^{2})) (- 1 \leq x, y \leq 1)

the corresponding surface is shown in Fig. 2(b), which will be regarded as the reference surface. The phase shift of each interferogram is

β_{k} = (π / 2) \cdot (k - 1) + δ (k - 1); (1 \leq k \leq 4)

where

δ (k - 1)

is a random value ranging from −1rad to 1rad. Since the phase shift between each interferogram is not exactly

π / 2

, the phase distribution that extracted by the traditional equal step PSA will have a double frequency “fringe-print-through” error [21]. Therefore, we have to correct this error.

Fig. 2 Simulation results, (a) the fringe; (b) the given measured surface; (c) and (d) the extracted surface and the residual error of 4-buck PSA; (e) and (f) the compensated surface and the residual error of using proposed method in 4-buck PSA method.

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As a most common PSA, 4-buck PSA that Eq. (10) represents is used to calculate the phase in the case that the random phase-shift errors are set to: $δ_{0} = 0, δ_{1} = - 0.1924, δ_{2} = 0.7 278, δ_{3} = - 0.2 417$ (created by the rand function in MATLAB) respectively. Figure 2(c) shows the final surface that calculated by 4-buck PSA with the inaccurate phase-shift. Comparing with the reference, we can see the double frequency “fringe-print-through” residual surface error is as high as 10.8553nm (RMS), which is shown in Fig. 2(d). In this case, almost all of the self-calibrating algorithms [11–19] are dedicated to calculate the phase shift errors of each interferogram and then compensate them by GPSA. Proposed method, however, ignores the random phase shift errors and calculates the offset, bias, and less of quadrature of the numerator and the denominator of the arctangent function in PSA directly. Then the phase extraction error can be corrected using Eq. (7).

The ETC process is shown in Figs. 3(a)-3(f). We choose 6 different circles with the radius interval of 10 pixels at the whole fringe pattern, the corresponding Lissajous ellipses are shown as Fig. 3(a)-3(c), and Fig. 3(d)-(f) illustrate the offset, bias and less of quadrature of each ellipse. Evidently, Fig. 3(f) manifests the less of the quadrature is a constant to each ellipse, and the bias of each ellipse is different but $a_{x} / a_{y}$ is a constant as Fig. 3(e) shows, the offset is very close to zero, which is negligible in our calculation. More importantly, the value of less of quadrature

ε = 0.581 \approx 0.5809 = \frac{0.7278 + 0.1924 + 0.2417}{2} = \frac{δ_{2} - δ_{1} - δ_{3}}{2}

and

\frac{a_{x}}{a_{y}} = \frac{195.1}{182.4} = 1 .0696 \approx 1.0698 = \frac{\cos (\frac{- 0.2417 + 0.1924}{2})}{\cos (\frac{0.7278}{2})} = \frac{\cos (\frac{δ_{3} - δ_{1}}{2})}{\cos (\frac{δ_{2}}{2})}

coincide very well with the analysis of principle 2.2, which validate the correctness of the theory. Hence, only one special circle is enough to achieve the corrected parameters of Lissajous ellipse when no noise is introduced into the interferogram. We use the Lissajous ellipse that created at circle1 to correct the phase extraction error. The calibrated surface and the residual error are shown in Fig. 2(e)-2(f) respectively. It can be seen the double frequency “fringe-print-through” error is fully removed.

Fig. 3 Proposed circular mode: (a) Lissajous ellipse at circle1; (b) Lissajous ellipse at circle2; (c) Lissajous ellipse at circle6; (d) the calculated offset of the Lissajous ellipse at different circles; (e) the calculated bias of the Lissajous ellipse at different circles; (f) the less of quadrature of the Lissajous ellipse at different circles;

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In the simulations we find several factors may affect the performance of proposed method. Such as the number of interferograms, random noise in the interferogram and amplitude of the random phase shift error. Those factors will be analyzed and discussed as follows.

We will compare proposed scan circle mode with C.T. Farrell’s or K. S. Moon’s scan line mode in creating the Lissajous ellipse under the same influence factor. (Here, though C.T. Farrell simply corrected the measured fringe field in traditional PSA, and K. S. Moon simply used this mode to calculate the phase shift between each two interferograms, it is convenient to consider the scan line mode in directly correcting the phase extraction error of N and D terms in PSA as their method for comparison)

Firstly, we assume the SNR of the white Gaussian noise is 35dB, the random phase shift error of each interferogram is between $(- π / 9, π / 9)$ and let the number of the frames varies from 3 to 20. Then proposed method and C.T. Farrell’s method are used separately to analyze these interferograms, the residual errors are shown in Fig. 4(a). It can be seen that with the number of frames increasing from 3 to 20, both methods will have a better performance, but proposed method with “circular mode” always has a higher accuracy than Farrell’s “linear mode” at a given number of frames.

Fig. 4 Discussions of the performance of proposed method and C.T. Farrell’s method with different influence factors: (a) the number of the frames; (b) the SNR of the interferogram; (c) the amplitude of the phase shift error; (d) the difference between proposed method and AIA at the different amplitudes of phase shift error.

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Secondly, we assume the number of the frames is 4, the random phase shift error of each frame is between $(- π / 9, π / 9)$ and let the SNR of the noise varies from 20dB to 80dB, the interval is 5dB. Then proposed method is used to calibrate the phase extraction error, root-mean-square (RMS) value of the residual surface error decreases from 6.4127nm to 0.002nm with the SNR of the interferogram increasing from 20dB to 80dB. Meanwhile, Farrell’s scan line mode is also used for comparison, which has the same trend with proposed method. However, it can be seen from the red line in Fig. 4(b), when the SNR is higher than 50dB, the RMS of the residual surface error is convergent to 0.28nm and no longer decreases, which means the scan line mode has a theory error when the illumination is variational, whereas proposed scan circle mode has not.

Thirdly, we assume the number of the frames is 4, the SNR of the noise is 35dB, and let the amplitude of the random phase shift errors varies from 0.1rad to1.5rad, the interval is 0.1rad. Figure 4(c) manifests that, both Farrell’s scan line mode and proposed scan circle mode have a similar response to the amplitude of the random errors. Namely, the RMS of the residual surface error of Lissajous calibration method (ETC) is increasing from 0.9nm to 1.5nm as the amplitude of the random phase shift error increases from 0.1rad to 1.5rad, but ETC process with proposed “circular mode” has a better performance than Farrell’s “linear mode” in most general case.

At last, to demonstrate the performance of proposed ETC method we make a comparison with Wang’s AIA method [13] published in 2004. AIA is an advanced iterative algorithm which provides stable convergence and accurate phase shift extraction and has became a powerful tool to cope with the completely random phase shift interferograms since it has been published. We assume the number of frames is 4, the SNR of the noise is 40dB and let the amplitude of the random phase shift errors varies from 0.1rad to 3rad, the interval is 0.2rad. Since the background intensity and modulation amplitude is variational, we also select a small portion of the interferogram to perform the iterative process in AIA for comparison. Figure 4(d) indicates when the amplitude of the random phase shift error is below 1.5rad, ETC has a comparatively higher precision than AIA. It should be noted that, though the average phase shift extraction error is reduced from 0. 035rad to 0.015rad by selecting part of the image in AIA, the residual surface error of AIA does not seem to have a significant decrease. Figure 4(d) also indicates when the random phase shift error is higher than 1.5rad proposed ETC method may results in large errors, however, AIA has a stable performance. Actually, if the amplitude of the random phase shift error is higher than 1.5rad ( $π / 2$ ), then the phase step between each interferogram may be completely random, which means the traditional equal step PSA can no more be used to calculate the phase distribution. In this case, we can combine ETC with AIA to compensate the phase extraction error after the first step in each iterative cycle of AIA, we believe that the iterations will have a considerable decrease. Hence, the rapidity of AIA will has a notable rise. Considering the limit of this paper, we will discuss the hybrid algorithm of retrieving the phase from a series of (>3) totally random phase-shifting interferograms in another paper.

4. Experiments

To illustrate the application of proposed method, we apply it to the practical thirteen phase-shifted interferograms, which are captured by a Zygo GPI Interferometer under two different circumstances.

4.1 The random phase shift errors of each practical interferogram are artificial

An optical flat with aperture of 100mm is measured with the active vibration isolation workstation turned on. A 260 × 260 square mask is used to select only part of the interferogram. One of the frames is shown in Fig. 5(a). The measured surface that Zygo gives is shown in Fig. 5(b), with the PV of 28.5135nm and RMS of 3.2071nm.

Fig. 5 Experimental results:(a) the fringe; (b) the measured surface Zygo gives; (c) and (d) the extracted surface and the residual error of LSM (with artificial phase shift errors); (e) and (f) the extracted surface and the residual error of AIA; (g) and (h) the extracted surface and the residual error of proposed method with the “circular mode”; (i) the residual error of proposed method with the “linear mode”.

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As we know, the phase step between the 13 interferograms is constantly $π / 4$ and the phase shift error is negligible, so Fig. 5(b) can serves as the reference surface in the experiment. However, to verify the performance of proposed method, we have to make some artificial phase shift errors to these interferograms. We take 4 frames, $I_{1}, I_{3}, I_{5}, I_{7}$ for example. The phase shifts between the 4 interferograms are assumed to $δ_{1} = (0 - 0.01) r a d, δ_{2} = (π / 2 + 0.1) r a d, δ_{3} = (π - 0.2) r a d, δ_{4} = (3 π / 2 + 0.02) r a d$ respectively, which should be $δ_{1} = 0, δ_{2} = π / 2, δ_{3} = π, δ_{4} = 3 π / 2$ . It means an artificial phase shift error has been successfully added to each interferogram. Since the assumed phase shifts are regarded as the real one, the phase distribution can be extracted by using of the least squares method (LSM) [26], the calculated surface is shown in Fig. 5(c). The residual surface error that caused by the artificial phase shift errors is shown in Fig. 5(d). It can be seen a distinct double frequency “fringe-print-through” error with the RMS of 2.7951nm.

To correct the double frequency “fringe-print-through” error, Wang’s AIA method is firstly used to extract the error free phase for comparison. In this method, the assumed phase shifts are regarded as the initial phase shifts and 14 iterations later the true phase shifts, i.e.

δ_{1} = 0 r a d, δ_{2} = 1.5778 r a d, δ_{3} = 3.1838 r a d, δ_{4} = 4.7513 r a d

and the calculated phase distribution, shown in Fig. 5(e), are acquired simultaneously. The whole process takes 7.6602s using MATLAB with an Intel i5 processor. The residual surface error is represented in Fig. 5(f). Proposed ETC method is secondly used to correct the double frequency “fringe-print-through” error. It does not have to calculate the true phase shifts of each interferogram, which directly calibrate and compensate the phase extraction error caused by the inaccurate phase shifts from the arctangent function in LSM algorithm. The compensated surface and the residual error are shown in Figs. 5(g) and 5(h). The whole progress costs only 0.73s. In this approach the “linear mode” that Lissajous ellipse created is also applied for comparison, Fig. 5(i) represents the corresponding residual surface error.

From Figs. 5(f), 5(h) and 5(i) we can conclude that both the AIA and the ETC method of two modes can remove the double frequency “fringe-print-through” error, and the ETC method of the “circular mode” (RMS = 0.681nm) has a comparative precision with AIA (RMS = 0.688nm). But the ETC method with the “linear mode” may leads to a minor single frequency “fringe-print-through” error (RMS = 0.807nm) which may caused by the inaccurate calculation of the offset [21]. It should be noted that proposed ETC method does not need the iterative process to calculate the true phase shifts, thus it is far faster than Wang’s iterative method.

4.2 The random phase shift errors of each practical interferogram are caused by vibration

Among numerous factors that may affect the accuracy of phase shift value in PSI, the mechanical vibration is the dominate one [6]. Thus, we measured another optical flat with aperture of 130mm by a Zygo GPI Interferometer with the active vibration isolation workstation turned on/off in workshop. For comparison, the flat was firstly measured with the active vibration isolation workstation turned on. The calculated surface is shown in Fig. 6(b), which can be regarded as the reference. Then the vibration-isolating platform was turned off, and the flat is measured once again. Figure 6(a) is one of the 13 interferograms captured by Zygo GPI Interferometer in vibrated flat cavity, and the corresponding measured surface is shown in Fig. 6(c). It can be seen a distinct “ripple” error from Fig. 6(c). By using of proposed ETC method the phase extraction error caused by inaccurate phase shift can be corrected. The compensated surface is shown in Fig. 6(d), we can see the “ripple” error has been removed successfully and the compensated profile coincides very well with Fig. 6(b). Comparing Fig. 6(c) and Fig. 6(d) with Fig. 6(b), we can achieve the phase extraction error of 13-frame PSA in vibrated circumstance and that calibrated by proposed ETC method. The distributions of the residual surface error are shown in Figs. 6(e) and 6(f) respectively, from Fig. 6(e) we can see the double frequency “fringe print through” error caused by vibration is 4.23nm (RMS), and by using of proposed ETC method the double frequency “fringe print through” error has almost been eliminated, shown in Fig. 6(f), the main residual error may caused by inaccurate calculation of offset. By improving the SNR of interferograms and repeating the ETC process at different circles in the fringe field the minor single frequency “fringe print through” error can be reduced. It should be noted the vibration in this experiment is not induced but occurred naturally from the environment of the workshop.

Fig. 6 Experimental results: (a) the fringe in vibration; (b) the measured surface with no vibration; (c) the surface measured by 13-frame PSA in vibration; (d) the compensated surface by proposed ETC method; (e) the residual surface error caused by vibration; (f) the residual surface error of compensated surface.

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The two experimental results indicate that proposed method can successfully correct the random phase shift errors in PSI with high accuracy and speed, and it can be used in workshop with no vibration-isolating platform to compensate the phase extraction error caused by vibration.

5. Conclusions

We have presented a general method to calibrate and compensate the phase extraction error in PSA. The principles of this method are elaborated first, and then a series of simulations have been done. Through the discussions we can conclude, (1). Proposed method can successfully compensate the phase extraction error without calculating the true phase shift values of each interferogram when the random phase shift errors are less than 1.5rad; (2). Proposed method is insensitive to the variation of the background and modulation amplitude when they are circular symmetric; (3). The experimental results manifest proposed method is far faster than Wang’s AIA method and has a comparative precision when dealing with interferograms with random phase shift errors. (4). Proposed method can be used in workshop with no vibration-isolating platform to compensate the phase extraction error caused by vibration. Further investigation is being made for completely random phase-shifted interferograms, proposed method will be combined with AIA to improve the calculation precision and the rapidity simultaneously, and it should be an attractive method for dynamic measurement.

Acknowledgments

The authors acknowledge the support by the National Natural Science Foundation of China (61108045)

References and links

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Correction of phase extraction error in phase-shifting interferometry based on Lissajous figure and ellipse fitting technology

Abstract

1. Introduction

2. Principles

2.1 The principle of error compensating

2.2 Analysis and example of the phase extraction error caused by random phase shift errors

2.3 The method of producing the Lissajous ellipse

2.4 The ellipse fitting and calculation of coefficients

3. Simulations and discussions

4. Experiments

4.1 The random phase shift errors of each practical interferogram are artificial

4.2 The random phase shift errors of each practical interferogram are caused by vibration

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (6)

Equations (22)

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