Phase extraction from two phase-shifting fringe patterns using spatial-temporal fringes method

Ronggang Zhu; Bo Li; Rihong Zhu; Yong He; Jianxin Li

doi:10.1364/OE.24.006814

1. Introduction

To demodulate the measured phase from interferograms, the phase shift methods [1–6]and the Fourier transform methods are most commonly used [7–10]. Generally the phase shift algorithms are fast and accurate, but their performance is usually effected by the inaccurate phase shift value arising from the detuning of the phase shifter [6,11]and the environment disturbances such as the vibration and the air flow [12]. To solve this problem, some algorithms assume the phase shift values as the unknown quantities, and extract the measured phase by the iterative process [13–16]. On the other hand, Fourier transform methods can demodulate the phase with only one frame of fringes, therefore it is immune to the phase shift error. However, large linear carrier is usually necessary to ensure that the signal lobe is far enough from the background lobe in the frequency domain so that it can be extracted to retrieve the phase, which will result in the extra retrace error in the measurement.

Different from the above two kinds of methods, the Spatial-Temporal Fringes (STF) method can be seen as the fusion of them, which was firstly proposed by M. Servin [17]with the name of “Squeezing Interferometry”. In this method, at least three phase shift fringes are combined into one STF image which contains both the measured phase in the spatial domain and the phase shift information in the temporal domain. Due to the phase shift, the background lobe and the signal lobe in the frequency domain of the STF image are so far away from each other, therefore the phase can be extracted easily. But only the linear detuning error of the phase shift can be suppressed in this method, which is not actual in many cases such as the vibration environment. Then Bo Li et al proposed that if there is suitable linear carrier in the original phase shift fringes, arbitrary form of small phase shift errors can be suppressed in the STF method [18,19]. But their research is still based on the framework with at least three phase shift fringes. In addition, because of the one order approximation in the deduction, their theory is not suitable for the case of relative large phase shift error (such as the value larger than 20°).

In this paper we propose the advanced STF method that uses only two frames of phase shift fringes, and even the large phase shift error can be acceptable, which means it can be used to solve the nearly “random” phase shift problem. Generally, capturing two frames of phase shift fringes is more benefit than three or more frames since it means less capturing time for the temporal phase shift interferometers or less hardware for the spatial phase shift interferometers. Unfortunately, the condition of accurate phase step and approximately uniform background (or background suppressed by algorithm) is required in traditional two-step phase shift algorithms [20–23]. On the other hand, compared with the Fourier transform methods using only one interferogram, the necessary linear carrier to separate the background lobe and the signal lobe is significantly lower, which means less retrace error in the measurement system.

2. Theory

Assuming the measured surface in the interferometry system is $w (x, y)$ , a group of two-step fringes can be expressed by:

{\begin{cases} I_{0} (x, y) = a (x, y) + b (x, y) \cos {\frac{4 π}{λ} [x \tan (θ) + w (x, y)]} \\ I_{1} (x, y) = a (x, y) + b (x, y) \cos {\frac{4 π}{λ} [x \tan (θ) + w (x, y)] + ϕ} \end{cases}

where θ is the angle between the measured surface and the reference one, λ is the wavelength,

a (x, y)

is the background,

b (x, y)

is the modulation and ϕ is the phase shift value.

To give an example, a simulated surface and its phase shift fringes are shown in Fig. 1, where $ϕ = π$ .

Fig. 1 The simulated surface (a) and its fringes (b).

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For the deduction simplicity, we re-write Eq. (1) as:

I_{i} (x, y) = a (x, y) + b (x, y) \cos [2 π f_{0} x + α (x, y) + ϕ], (i = 0, 1)

where

f_{0} = 2 \tan (θ) / λ

is the original spatial frequency in the fringes, and

α (x, y) = 4 π w (x, y) / λ

is the measured phase.

The phase shift fringes can be combined into the STF image by row or by column and there is almost no difference [10–12]. However, in the earlier references the STF theory is based on at least three phase shift fringes. Assuming the image size of the fringes is M × N, the STF image with the size of M × 2N can be obtained by the column re-arrangement, which means the odd columns in the STF comes from the first phase shift fringes while the even column comes from the second one. In other words, the 1st, 3rd, 5th, … columns of the combined fringes are the 1st, 2nd, 3rd, … columns of the first fringes and its 2nd, 4th, 6th, … columns are the 1st, 2nd, 3rd, ... columns of the second fringes. As shown in Fig. 2, the STF image is extended in x-direction by 2 times compared with the original phase shift fringes, so we can mark the new coordinate by x' and express the STF image as:

I' (x', y) = a (\frac{x'}{2}, y) + b (\frac{x'}{2}, y) \cos [2 π f_{0}' x' + α (\frac{x'}{2}, y) + ε_{ϕ}]

where

f_{0}' = f_{0} / 2

since the spatial frequency is reduced by half caused by the coordinate extension, and the period order

ε_{ϕ}

is introduced by the phase shift ϕ, which is defined by:

Fig. 2 The STF image rearranged by the two phase shift fringes.

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ε_{ϕ} = {\begin{matrix} 0 & , \begin{matrix} while & x' & is & odd \end{matrix} \\ ϕ & , \begin{matrix} while & x' & is & even \end{matrix} \end{matrix}

Using the exponential form, Eq. (3) can be re-written as:

\begin{matrix} I' (x', y) = a (\frac{x'}{2}, y) + \frac{1}{2} b (\frac{x'}{2}, y) \exp {j [2 π f_{0}' x' + α (\frac{x'}{2}, y)]} \exp (j ε_{ϕ}) \\ + \frac{1}{2} b (\frac{x'}{2}, y) \exp {- j [2 π f_{0}' x' + α (\frac{x'}{2}, y)]} \exp (- j ε_{ϕ}) \end{matrix}

If the period order with finite length $\exp (\pm j ε_{ϕ})$ is seen as the infinite one approximately, its Fourier transform is:

\frac{1 - \exp (\pm j ϕ)}{2} δ (f + f_{c}) + \frac{1 + \exp (\pm j ϕ)}{2} δ (f) + \frac{1 - \exp (\pm j ϕ)}{2} δ (f - f_{c})

where

f_{c} = 1 / 2

in the normalized frequency coordinate, which is called the phase shift modulated frequency here because it is generated by the phase shift.

Then the Fourier transform of Eq. (5) is:

ℱ [I' (x', y)] = A (f) + \frac{1}{4 π} B (f) * S (f)

where

A (f) = ℱ [a (x' / 2, y)]

is the background lobe in the spectrum,

B (f) = ℱ [b (x' / 2, y)]

is the Fourier transform of modulation,

S (f) = S_{+} (f) + S_{-} (f)

is the signal spectrum and

S_{\pm} (f) = ℱ (\exp {\pm j [2 π f_{0}' x' + α (x' / 2, y)]} \exp (\pm j ε_{ϕ}))

.

By Eq. (6) we can express S(f) as:

\begin{array}{r} S (f) = \frac{1}{4 π^{2}} Θ (f) * [\begin{array}{l} \frac{1 - \exp (j ϕ)}{2} δ (f + f_{c} - f_{0}') + \frac{1 + \exp (j ϕ)}{2} δ (f - f_{0}') \\ + \frac{1 - \exp (j ϕ)}{2} δ (f - f_{c} - f_{0}') \end{array}] \\ + \frac{1}{4 π^{2}} Θ^{*} (f) * [\begin{array}{l} \frac{1 + \exp (- j ϕ)}{2} δ (f + f_{c} + f_{0}') + \frac{1 - \exp (- j ϕ)}{2} δ (f + f_{0}') \\ + \frac{1 + \exp (- j ϕ)}{2} δ (f - f_{c} + f_{0}') \end{array}] \end{array}

where

Θ (f) = ℱ {\exp [j α (x' / 2, y)]}

and

Θ^{*} (f) = ℱ {\exp [- j α (x' / 2, y)]}

are a pair of conjugate phase spectrum.

From Eq. (8) we can know that the lobes of $Θ (f)$ are shifted to the locations of $- (f_{c} - f_{0}')$ , $f_{0}'$ and $f_{c} + f_{0}'$ in the frequency domain, and the lobes of $Θ^{*} (f)$ are at $- (f_{c} + f_{0}')$ , $- f_{0}'$ and $f_{c} - f_{0}'$ respectively. In these six lobes, the two lobes at $\pm (f_{c} + f_{0}')$ can be ignored since they exceed the normalized frequency limit $[- 1 / 2, 1 / 2]$ , and the other four lobes all contain the signal spectrum that can be used to retrieve the phase. However, the two lobes at $\pm f_{0}'$ are easy to overlap with background lobe $A (f)$ if the original carrier frequency $f_{0}$ is not large enough. Therefore, the two lobes at $\pm (f_{c} - f_{0}')$ are the best choices to obtain the measured phase. If we extract lobe at $f_{c} - f_{0}'$ by filtering in the frequency domain we can get:

ℱ {[I' (x', y)]}_{f_{c} - f_{0}'} = \frac{1}{4 π} ℱ [b (\frac{x'}{2}, y)] * \frac{1}{4 π^{2}} Θ^{*} (f) * [\frac{1 + \exp (- j ϕ)}{2} δ (f - f_{c} + f_{0}')]

And the inverse Fourier transform is executed on Eq. (9) to obtain:

I " (x', y) = \frac{1}{4 π} [\frac{1 + \exp (- j ϕ)}{2}] b (\frac{x'}{2}, y) \exp {j [2 π (f_{c} - f_{0}') x' - α (\frac{x'}{2}, y)]}

And then the phase can be obtained by:

α (\frac{x'}{2}, y) = - \tan^{- 1} {\frac{i m [I " (x', y)]}{r e a l [I " (x', y)]}} + 2 π (f_{c} - f_{0}') x'

where the last component

2 π (f_{c} - f_{0}') x'

can be omitted in many occasions because the tilt term is usually not concerned in the surface information.

It is notable that the amplitudes of the four lobes are decided by the phase shift ϕ, deduced by their coefficients $1 \pm \exp (\pm j ϕ)$ in Eq. (8). Evidently the extracted lobes will get the maximum amplitude when $ϕ = π$ , which means the best Signal-to-Noise Ratio (SNR), so it is the optimal phase shift value. To give the example, the STF spectrums with the phase shift values $ϕ = π / 2$ and $ϕ = π$ are shown in Figs. 3(a) and 3(b) respectively, where the amplitude difference of the extracted lobes can be seen clearly.

Fig. 3 The spectrums of STF when phase shift values are (a) ϕ = π/2 and (b) ϕ = π.

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The final step is to transform the phase of STF $α (x' / 2, y)$ to the original measured phase $α (x, y)$ , which can be completed easily by extracting the odd columns and combine them. The measured surface can be obtained from the phase directly by $w (x, y) = λ α (x, y) / 4 π$ and the retrieved surface by our method with $ϕ = π$ is shown in Fig. 4(a). From the calculated error given in Fig. 4(b) we can see that the deviation is less than 5 × 10⁻³ λ and its Error Root Mean Square (ERMS) is 1.18 × 10⁻³ λ.

Fig. 4 (a) The retrieved surface and (b) its error (ERMS value: 1.18 × 10⁻³ λ).

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3. Experiments

To verify the algorithm, a flat mirror with 100mm diameter is tested on a Zygo GPI^TM phase shift interferometer, whose work wavelength is 632.8nm and the camera resolution is 256 × 256. The phase shift value is set to π/2, but it is potential to be influenced by the vibration. A series of phase shift fringes are captured continuously where the first and third frames of them are shown in Figs. 5(a) and 5(b), which means, the phase step between them is π. The fused STF image of these two fringes generated by column method is shown in Fig. 5(c) whose spectrum is mainly composed of the background lobe and the two signal lobes. The un-uniform background intensity of the fringes present approximately the shape of the 2D Gauss Function, which makes its bandwidth wide. Since this phenomena is very common in the interferometry, the necessary linear carrier should be large enough to separate the signal lobes and the background one, which means the large tilt of the reference mirror and corresponding retrace error. Usually the tilt of reference mirror should be large enough to make sure there are 40 or more fringes in the interferogram when using the Fourier transform method. But in our method, the signal lobes locate at the two edges of the frequency domain which are very far from the background lobe. Therefore the necessary linear carrier is significantly smaller in our method. The detailed comparison will be given with simulations in the next section. To be contrast, the fused STF by the first and the second phase shift fringes is also generated where the phase step is π/2, and its spectrum is shown in Fig. 6(b).

Fig. 5 The phase shift fringes captured in the experiment with the phase shift values of (a) 0 and (b) π, and the fused STF image (c).

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Fig. 6 The spectrums of the STF in the experiment when phase-step are (a) π and (b) π/2.

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Figures 7(a) and 7(b) illustrate the surface results retrieved from these two STF images, where the deviation between them is shown in Fig. 7(d). The PV and RMS values of the deviation are 1.8 × 10⁻³ λ and 3.3 × 10⁻⁴ λ respectively, therefore it can be deduced that our method can give the reliable result from two phase shift fringes even if the phase step changes so greatly. The surface obtained by traditional phase shift algorithm is given in Fig. 7(c) which contains the significant “ripple” surface error with the spatial frequency twice of the original fringes. This ripple error is potentially caused by the phase shift error and the 2nd order harmonics of the fringes, but they can be both suppressed in our method. Actually, the harmonics rejection is the inherent ability of all kinds of Squeezing Interferometry methods [17]. It is the typical surface error caused by the vibration, which can be suppressed well in our method. The deviation between Figs. 7(a) and 7(c) is shown in Fig. 7(e) and its RMS value is 2.6 × 10⁻³ λ.

Fig. 7 The retrieved surface by our algorithm with phase-step of (a) π and (b) π/2. (c) The surface obtained by phase shift method. (d) The deviation between (a) and (b). (e) The deviation between (a) and (c).

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4. Analysis and discussions

4.1 Compared with the Fourier transform method

As explained before, the linear carrier is necessary in both our method and the Fourier transform one, but the most significant difference is that the former requirement is much lower. To give an example, the surface obtained by the Fourier transform method from the same fringes used in Fig. 1 is shown in Fig. 8(a), as well as the retrieved error in Fig. 8(b). The filter window, which is used to extract the signal lobe in the frequency domain, is a gauss function with the optimal bandwidth. It can be seen that the retrieved error is much larger than our method shown in Fig. 4(b) due to the overlapping between the signal lobe and the background one. That means, although the linear carrier in the fringes of Fig. 1 is suitable in our method, it is not large enough to obtain accurate surface by Fourier transform method.

Fig. 8 (a) The retrieved surface by Fourier transform method and (b) its error (ERMS value: 3.66 × 10⁻³ λ).

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To give a comprehensive comparison between our method and the Fourier transform method, we do the simulations with different spatial frequencies in Fig. 9. The retrieved results are merited by the ERMS values, which are the RMS values of the deviations between the retrieved surfaces and the original ones. Since the spatial frequency f₀ of the phase shift fringes is the function about the tilt angle θ between the reference mirror and the measured one, here we prefer to use θ as the x-axis while not f₀ so that the physical meaning is more clear. In this Fig. it can be seen that the performance of our method is significantly more accurate than the Fourier transform method with small tilt (i.e. the low linear carrier) conditions. But the ERMS values of the Fourier transform method trend to decrease when the tilt increases, and obtain the same accuracy as our method at last.

Fig. 9 ERMS values of the retrieved surfaces with different tilt by our method and the Fourier transform method.

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4.2 The carrier requirement

Although the carrier requirement of our method is lower than the Fourier transform method, it can’t be too small. From Eq. (9) we can know that the distance between the extracted signal lobe and the edge of the frequency domain is f₀, therefore the signal lobe will exceed the maximum frequency when the carrier f₀ is very small. Furthermore, the frequency edge is actually a rectangle filter window which will result in the Gibbs effect if the signal lobe is too close to it.

In Fig. 10 the retrieved errors of our method with different carrier are given, where the tilt angles are within the range of 0.003”~0.027”. Evidently the accuracy also improves with the increment of the tilt angle while the accuracy is always better than 1.2 × 10⁻³ λ as if the tilt angle is greater than 0.017”, which can be satisfied in most cases.

Fig. 10 ERMS values of the retrieved surfaces with a series of small tilt values.

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4.3 The effect of different phase shift values

As shown in Eq. (9) and Eq. (10), the amplitude of the extracted signal lobe in our method is the function of the phase shift value ϕ, which will obtain the maximum value when $ϕ = π$ . The lager amplitude is more benefit to retrieve the surface since it means better SNR. On the opposite, if the phase shift is close to zero, the SNR will be too low to obtain the measured phase. Unfortunately, the phase shift value is usually unable to be controlled perfectly because of various of systematical and random errors. So it is necessary to know the performance of our method when the phase shift value changes.

We do the simulations with phase shift from 15° to 180° and the retrieved errors are given in Fig. 11, where the random noise with 0.15 amplitude of the signal is added into the phase shift fringes. It is notable that this range actually represents the cases of phase shift 15°~345° since the period of phase shift is 360°. From this Fig. it can be seen that the retrieved accuracy will be better if the phase shift is closer to 180°, which accords with the theory. However, our method can obtain the accuracy better than 2.1 × 10⁻³ λ if the phase shift is within the range of 180 ± 90°. Even when the range is extended to 180 ± 165° the RMS value of the error is still less than 11.2 × 10⁻³ λ. It is a very loose tolerance about the phase shift so it can be concluded that our method can be used to solve the “random phase shift” problem.

Fig. 11 The retrieved surface errors of our method with different phase shift value.

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5. Conclusion

In this paper we propose a method to demodulate the phase from two phase shift fringes with linear carrier and almost arbitrary unknown phase step. It is deduced by extending the traditional STF theory using at least three phase shift fringes to the two fringes case. Its loose requirements on the phase shift value and fringes frame amount give it wider application area than the traditional phase shift methods, such as the measurement cases with inaccurate phase shift calibration and environment vibrations, or there is the demand of high speed capturing. On the other hand, the necessary carrier is lower than the Fourier transform methods that uses only one frame of fringes, which means the less tilt of reference mirror during the interferometry measurement and correspondingly less retrace error.

Acknowledgments

The work of this paper is supported by the National Natural Science Foundation of China (NSFC) under grant No. A030801, No. KT04081 and No.11303068 and the Natural Science Foundation of Jiangsu Province under grant No. BK20131060.

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Phase extraction from two phase-shifting fringe patterns using spatial-temporal fringes method

Abstract

1. Introduction

2. Theory

3. Experiments

4. Analysis and discussions

4.1 Compared with the Fourier transform method

4.2 The carrier requirement

4.3 The effect of different phase shift values

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (11)

Equations (11)

Optics Express