Dynamic phase measurement based on spatial carrier-frequency phase-shifting method

Linbo Huang; Xiaoxu Lu; Jiaosheng Li; Yunfei Zhou; Jiaxiang Xiong; Jindong Tian; Liyun Zhong

doi:10.1364/OE.24.013744

1. Introduction

Dynamic phase measurement is an important content of optical interferometry [1–3]. In general, the phase retrieval of optical interferometry are classified into the temporal phase-shifting (TPS) [4, 5] and spatial Fourier transform(SFT) [6, 7] methods. Though TPS based interferometry reveals high accuracy, but its measuring process is time-consuming, complicated and sensitive to the environmental disturbance due to at least three-frame phase-shifting interferograms are required. Therefore, TPS based interferometry is not suitable for dynamic phase measurement. Though SFT based interferometry is a well-known method of dynamic phase measurement due to the wrapped phase can be retrieved from one-frame interferogram through Fourier transform, but its accuracy is greatly influenced by the filtering window, noise and Gibbs effects [8]. To address this, several methods of dynamic phase measurement are reported and developed [9, 10]. For example, Kakue et. al. [9] propose a high-speed phase imaging through using the parallel phase-shifting digital holography, in which two-frame interferograms with the phase-shift of /2 are captured in single-shot by a special polarization array image sensor, then the wrapped phase is retrieved by two-step phase demodulation algorithm [11]. However, two-frame interferograms with phase-shift of /2 generated by the interpolation method will reduce the spatial bandwidth of system. Moreover, the polarization array image sensor is specially customized and expensive, greatly limiting its application. Recently, our research group [10] presents a dynamic phase measurement method by dual-channel simultaneous phase-shifting interferometry, in which two image sensors are used to simultaneously capture two-frame interferograms with phase-shift of /2 introduced by the polarization element and wave plate; then the wrapped phase can be obtained by two-step phase demodulation algorithm [11–13]. However, both the pixel matching of two image sensors and two-frame interferograms simultaneously captured are very difficult. In addition, there is similar problem in three-channel or four-channel simultaneous phase-shifting interferometry method [14, 15]. To address this, Ichiok et. al. [16] firstly propose a novel phase retrieval method based on the spatial carrier-frequency phase-shifting(SCPS) technique, in which the phase-shift is obtained by numerical simulation instead of conventional phase-shifting method, thus the phase retrieval can be performed with the conventional temporal phase-shifting algorithm. Clearly, using this SCPS method, the wrapped phase can be retrieved from only one-frame spatial carrier-frequency interferogram (SCFI), so it should be a good candidate for dynamic phase measurement. However, the existing SCPS methods need prior knowledge about the value of carrier-frequency and the phase-shift between adjacent pixels [17–19]. For example, Debnath et al [19] proposes a real-time quantitative phase imaging method based on SCPS technique, in which three-frame phase-shifting interferograms are constructed from SCFI by SCPS technique, then the wrapped phase can be obtained with three-step demodulation algorithm [20]. However, this method needs to calibrate the carry-frequency and the phase-shift between adjacent pixels by an optical grating in advance. After that, Xu et al [21] proposes a SCPS method based on the least-square iteration operation. Though this method possesses the advantages of high accuracy and good stability, but it is very time-consuming due to the least-square iteration operation. Besides, Du et al [22] proposes a principal component analysis(PCA) based SCPS method, which does not need to calibrate the phase-shifts and the carrier-frequency. Using this PCA based SCPS method, the processing time of this approach is shorter than Xu’ method, but its accuracy of phase retrieval is lower than Xu’s least-square iteration method.

In this study, combining SCPS technique and Fourier-transform method, from one-frame SCFI, we propose a novel non-iterative phase retrieval method to achieve dynamic phase measurement with high accuracy. Following, we will introduce the principle of the proposed method, and then present the simulation and the experimental results.

2. Principle

In general, the intensity distribution of interferogram with a linear carrier-frequency can be expressed as [23]

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

where x and y respectively denote the pixel coordinate(

1 \leq x \leq X

and

1 \leq y \leq Y

);

X

and

Y

are the number of row and column of SCFI, respectively;

a (x, y)

,

b (x, y)

and

φ (x, y)

respectively represent the background, the modulation amplitude and the measured phase;

f_{0 x}

and

f_{0 y}

are the spatial carrier-frequency along x and y directions, respectively.

Typically, the background and modulation amplitude are assumed to be the slowly varying signals in the interferogram, and the measured phase is modulated by a much larger carrier-frequency phase. That is to say, the modulating phase is much smoother than the carrier-frequency phase $2 π f_{0 x} x + 2 π f_{0 y} y$ , so the phase-shifts between adjacent pixels in the interferogram are approximately the same. Based on this, four phase-shifting sub-interferograms can be constructed from four consecutive and adjacent pixels of the SCFI, and the detailed process is following: From one-frame SCFI, we first intercept an area with size of $X^{'} \times Y^{'}$ as the first sub-interferogram $I_{1}$ , in which the measured phase is included; then, the intercepted area is moved one pixel along x and y directions, respectively and simultaneously, corresponding sub-interferograms $I_{2}$ , $I_{3}$ , $I_{4}$ can be obtained as

I_{1} (x^{'}, y^{'}) = I (x, y) = a (x^{'}, y^{'}) + b (x^{'}, y^{'}) \cos [2 π (f_{0 x} x^{'} + f_{0 y} y^{'}) + φ (x^{'}, y^{'}) + θ_{1}],

I_{2} (x^{'}, y^{'}) = I (x + 1, y) = a (x^{'}, y^{'}) + b (x^{'}, y^{'}) \cos [2 π (f_{0 x} x^{'} + f_{0 y} y^{'}) + φ (x^{'}, y^{'}) + θ_{2}],

I_{3} (x^{'}, y^{'}) = I (x, y + 1) = a (x^{'}, y^{'}) + b (x^{'}, y^{'}) \cos [2 π (f_{0 x} x^{'} + f_{0 y} y^{'}) + φ (x^{'}, y^{'}) + θ_{3}],

I_{4} (x^{'}, y^{'}) = I (x + 1, y + 1) = a (x^{'}, y^{'}) + b (x^{'}, y^{'}) \cos [2 π (f_{0 x} x^{'} + f_{0 y} y^{'}) + φ (x^{'}, y^{'}) + θ_{4}],

In Eqs. (2)-(5),

x^{'}

and

y^{'}

respectively denote the pixel coordinates in the sub- interferogram(

1 \leq x^{'} \leq X^{'}

and

1 \leq y^{'} \leq Y^{'}

);

X^{'}

and

Y^{'}

are the number of row and column of reconstructed sub-interferograms;

θ_{n} (n = 1, 2, 3, 4)

denotes the phase-shifts of these four sub-interferograms, which are equal to 0,

2 π f_{0 x}

,

2 π f_{0 y}

and

2 π f_{0 x} + 2 π f_{0 y}

, respectively. Similarly, we also can construct nine or more sub-interferograms from nine or more consecutive and adjacent pixels of the SCFI. In general, the noise tolerance of phase retrieval is better if the more sub-interferograms are constructed. Since the accuracy of phase retrieval is high enough when four sub-interferograms are constructed, and considering time consumption and convenience in dynamic measurement, we select the way of constructing four sub-interferograms to perform phase retrieval. Besides, for convenience of the following derivation, we rewrite Eqs. (2)-(5) as

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

Because there is no confusedness, we still write the coordinates of phase-shifting sub-interferograms as (x, y).

According to the conventional phase-shifting method, the measured phase can be obtained in the case that the phase-shift is known. Resulting from the introduced spatial carrier-frequency, Fourier-transform method should be an excellent candidate to extract the phase-shift [24]. From this view, we rewrite Eq. (6) as

\begin{array}{l} I_{n} (x, y) = a (x, y) + \exp (i θ_{n}) q (x, y) \exp [i (2 π f_{0 x} x + 2 π f_{0 y} y)] \\ + \exp (- i θ_{n}) q^{*} (x, y) \exp [- i (2 π f_{0 x} x + 2 π f_{0 y} y)], \end{array}

where

q (x, y) = \frac{1}{2} b (x, y) \exp [i φ (x, y)]

; * denotes the complex conjugate operation. The Fourier transform of

I_{n} (x, y)

can be expressed as

i_{n} (f_{x}, f_{y})

i_{n} (f_{x}, f_{y}) = A (f_{x}, f_{y}) + \exp (i θ_{n}) Q (f_{x} - f_{0 x}, f_{y} - f_{0 y}) + \exp (- i θ_{n}) Q^{*} (f_{x} + f_{0 x}, f_{y} + f_{0 y}),

where

A (f_{x}, f_{y})

and

Q (f_{x}, f_{y})

denote the Fourier transforms of

a (x, y)

and

q (x, y)

, respectively;

f_{x}

and

f_{y}

are the spatial frequency along x and y directions, respectively. Since the variations of

a (x, y)

,

b (x, y)

and

φ (x, y)

are slow relative to the spatial frequency

f_{x}

and

f_{y}

, so the zero frequency, positive and negative level spectrum can be completely separated each other. Using a rectangular filter, the spectral component

\exp (i θ_{n}) Q (f_{x} - f_{0 x}, f_{y} - f_{0 y})

can be extracted easily, and then the phase-shift can be obtained by retrieving the peak value of filtered spectral component. Concretely,

\exp (i θ_{n}) Q (f_{x} - f_{0 x}, f_{y} - f_{0 y})

reaches its peak value when

f_{x} = f_{0 x}

and

f_{y} = f_{0 y}

. Then

\begin{array}{l} i_{n} (f_{0 x}, f_{0 y}) = A (f_{0 x}, f_{0 y}) + \exp (i θ_{n}) Q (0, 0) + \exp (- i θ_{n}) Q^{*} (2 f_{0 x}, 2 f_{0 y}) \\ \approx \exp (i θ_{n}) Q (0, 0), \end{array}

where

Q (0, 0)

denotes a complex constant. Therefore, aparting from a constant phase offset, the phase-shift can be expressed as

θ_{n} \approx \tan^{- 1} [i_{n} (f_{0 x}, f_{0 y})],

Based on the above analysis, it can be found out that the phase-shift $θ_{n}$ can be obtained by Fourier-transform, and simultaneously, the spatial frequency $f_{x}$ and $f_{y}$ also can be determined. Following, we substitute the obtained phase-shift into the least-square equation and retrieve the wrapped phase, and then Eq. (6) is rewritten as

I_{n} (x, y) = a^{'} (x, y) + b^{'} (x, y) \cos θ_{n} + c^{'} (x, y) \sin θ_{n}

where

a^{'} (x, y) = a (x, y)

,

b^{'} (x, y) = b (x, y) \cos [Φ (x, y)]

,

c^{'} (x, y) = - b (x, y) \sin [Φ (x, y)]

;

Φ (x, y) = 2 π (f_{0 x} x + f_{0 y} y) + φ (x, y)

denotes the sum of spatial carrier-frequency phase and measured phase. According to the principle of least-square algorithm, in order to calculate the wrapped phase, it is required that the deviation square sum

S (x, y)

, accumulated from the same pixel of all phase-shifting sub-interferograms in Eq. (11), should reach the minimum

S (x, y) = \sum_{n = 1}^{N} {[I_{n}^{e} (x, y) - I_{n} (x, y)]}^{2},

where

I_{n}^{e} (x, y)

represents the experimentally captured intensity of interferogram. N denotes the total number of interferograms. To make Eq. (12) reach the minimum, the following extreme value condition should be satisfied

\frac{\partial S (x, y)}{\partial a^{'} (x, y)} = 0, \frac{\partial S (x, y)}{\partial b^{'} (x, y)} = 0, \frac{\partial S (x, y)}{\partial c^{'} (x, y)} = 0,

Next, we rewrite Eq. (13) as (the spatial coordinates x, y have been omitted)

[\begin{matrix} a^{'} \\ b^{'} \\ c^{'} \end{matrix}] = {| \begin{array}{l} N & \sum_{n = 1}^{N} \cos θ_{n} & \sum_{n = 1}^{N} \sin θ_{n} \\ \sum_{n = 1}^{N} \cos θ_{n} & \sum_{n = 1}^{N} \cos^{2} θ_{n} & \sum_{n = 1}^{N} \sin θ_{n} \cos θ_{n} \\ \sum_{n = 1}^{N} \sin θ_{n} & \sum_{n = 1}^{N} \sin θ_{n} \cos θ_{n} & \sum_{n = 1}^{N} \sin^{2} θ_{n} \end{array} |}^{- 1} | \begin{array}{l} \sum_{n = 1}^{N} I_{n}^{e} \\ \sum_{n = 1}^{N} I_{n}^{e} \cos θ_{n} \\ \sum_{n = 1}^{N} I_{n}^{e} \sin θ_{n} \end{array} |

From above equation, it can be seen that the unknowns

a^{'} (x, y)

,

b^{'} (x, y)

and

c^{'} (x, y)

can be determined when the phase-shift

θ_{n}

is obtained by Fourier-transform in advance. Then the wrapped phase

Φ (x, y)

can be determined by

Φ (x, y) = arc \tan [- c^{'} (x, y) / b^{'} (x, y)],

Thus, the measured phase

φ (x, y)

can be obtained easily after subtracting the linear carrier-frequency phase

2 π (f_{0 x} x + f_{0 y} y)

.

3. Simulation

In order to verify the validity of the proposed method, one-frame SCFI with size of 512 $\times$ 512 pixels is generated according to Eq. (1), as shown in Fig. 1(a), in which the background is set as $a (x, y) = 45 \exp [- 0.125 (x^{2} + y^{2})]$ ; and the modulation amplitude and measured phase are respectively set as $b (x, y) = 100 \exp [- 0.125 (x^{2} + y^{2})]$ and $φ (x, y) = 2 (x^{2} + y^{2})$ , where $(x, y) \in [- 2, 2]$ . The spatial carrier-frequency along x and y directions are respectively set as $f_{0 x} = 0.156$ rad/pixel and $f_{0 y} = 0.195$ rad/pixel.

Fig. 1 Simulation results. (a) one-frame SCFI; (b) the preset phase (reference phase); (c)the phase retrieved with the proposed method; (d)the difference of phase retrieval between (b) and (c).

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For simplicity, in this study, the preset wrapped phase is used as the reference phase, as shown in Fig. 1(b). Next, we will retrieve the measured phase from one-frame SCFI with the proposed method. First, four phase-shifting sub-interferograms are constructed by the SCPS technique, and then the phase-shifts of these four sub-interferograms extracted by Fourier-transform method are shown in Table 1. For comparison, we also give the corresponding theoretical phase-shifts. It can be seen that the error of phase-shift between the theoretical value and the obtained result is very small, indicating that Fourier-transform method is an accurate method of phase-shift extraction. Importantly, due to there is no requirement for the calibration of carrier-frequency or phase-shift in advance, usually needed in the conventional SCPS method, the proposed method will greatly improve the processing speed of phase retrieval. Finally, the phase retrieved with the least-square algorithm through using the above phase-shifts is shown in Fig. 1(c). And Fig. 1(d) presents the phase difference between the preset and the retrieved phase. In contrast, we also present the root mean square error (RMSE), peak to valley error (PVE) and processing time obtained with different methods, as shown in Table 2. It is found that though the accuracy of phase retrieval with the proposed method is nearly the same with Xu’s least-square iteration algorithm and higher than SFT method, but its processing time is an order magnitude lower than Xu’s method, and this will greatly facilitate its application in dynamic phase measurement.

Table 1. Phase-shift extracted by Fourier-transform method and its difference with the theoretical value.

View Table | View all tables in this article

Table 2. RMSE, PVE and Processing time of phase retrieval with different methods.

View Table | View all tables in this article

4. Experimental research

A Mach-Zehnder interferometer based experimental setup system, as shown in Fig. 2, is built up to further verify the performance of the proposed method, in which the illumination source is a He-Ne laser with the wavelength of 632.8 nm. First, the intensity of laser beam is attenuated by a neutral density filter (ND) and then separated into two beams by the beam splitter (BS1). Second, a light beam is reflected by the mirror M1 and then passes the micro-objective (MO1) as the reference beam. The other light beam is reflected by a mirror M2, and illuminates the sample then passes the micro-objective (MO2) as the object beam. Third, two beams are combined by a beam splitter BS2 and interference on the CCD plane.

Fig. 2 Experimental setup for recording SCFI. ND: neutral density filter; BS1-BS2: beam splitter; M1-M2: mirror; MO1-MO2: micro-objective.

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To present the accuracy of the proposed method, we first perform the phase retrieval of a static transparent light-guide plate of cell phone screen, which is made of Yakeli board and full of conical groove with depth of 2 ~3μm, is chosen as measured object. The magnifications of micro-objective MO1 and MO2 are 25, and their numerical apertures (N.A.) are equal to 0.4. The CCD is a 8-bit monochrome camera with size of 1280 $\times$ 1024 pixels and pixel size of $5 .2μm \times 5 .2μm$ . Figure 3(a) gives one-frame 800 $\times$ 800 pixels SCFI of light-guide plate captured by CCD in one shot, and Fig. 3(b) presents the corresponding phase obtained with the proposed method. For comparison, the phase retrieved with high accuracy TPS method is shown in Fig. 3(c), in which 50-frame phase-shifting interferograms introduced by a PZT are employed to perform phase retrieval with advanced iterative algorithm (AIA) [25]. Figure 3(d) gives the phase distributions of 646th row in Fig. 3 (b) and 3(c) and their difference. It is found that the RMSE of phase retrieval between the proposed method and TPS method is only 0.3005rad, indicating the proposed method possesses high accuracy.

Fig. 3 Experimental results. (a) one-frame SCFI of light-guide plate; the phases retrieved with (b) the proposed method; (c)TPS method; (d)the phase distributions of the 646th row in (b) and (c) and their difference.

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Following, to illustrate the feasibility of dynamic phase measurement with the proposed method, the evaporation process of a tiny water droplet is chosen as the measured object. With the exception of the magnification and N.A of micro-objectives MO1 and MO2 are respectively changed as 4 $\times$ and 0.1, and a 8-bit monochrome CCD with size of 768576 pixels (7.68mm5.76mm) is used to capture interferogram; the other experimental setup is the same with the above static measurement. The experiment is carried out in a closed environment with temperature of 20^。C and humidity of 67%. The sampling interval is set as 50 ms, and the total number of captured interferogram are 200 during the evaporation process. LabVIEW software is used to control the image acquisition. Figure 4 shows the experimental result, in which Figs. 4(a)-4(c) present the captured interferograms (430 $\times$ 400 pixels) of a droplet evaporation at the moments of t = 0s, 5s, 10s, respectively; Figs. 4(d)-4(f) are the corresponding phases retrieved with the proposed method. For directly visual exhibition, Visualization 1 give the phase variation of a droplet evaporation process.

Fig. 4 Interferograms of a water droplet at different moments: (a) t = 0s, (b) t = 5s, (c) t = 10s and the corresponding phases (d-f) retrieved by the proposed method. The whole dynamic process of phase variation can be seen in Visualization 1.

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Finally, we also present the phase variation of a Jurkat cell apoptosis induced by a drug of daunorubicin (DNR), in which the apoptosis of Jurkat cell treated with DNR are according to our previous report [26]. With the exception of the magnification and N.A of MO1 and MO2 are respectively changed as 25 $\times$ and 0.4, and a 8-bit monochrome CCD with size of 768 $\times$ 576 pixels (7.68mm $\times$ 5.76mm) is used, the other experimental setup is the same with the above static measurement. The sampling interval is set as 10s, and the total number of captured interferogram are 25 during the apoptosis process. Figure 5 shows the experimental results, in which Figs. 5(a)-5(c) are the captured interferograms (210 $\times$ 220 pixels) of Jurkat cell treated with DNR at the moments of t = 0s,160s, 240s after 37 hours 31 minutes, respectively; Figs. 5(d)-5(f) show the corresponding phases retrieved with the proposed method. For directly visual exhibition, Visualization 2 gives the phase variation of a Jurkat cell apoptosis treated with DNR. From above results, we can conclude that the proposed method is greatly suitable for dynamic phase measurement.

Fig. 5 Interferograms (a-c) of a Jurkat cell Jurkat cell treated with DNR at different moments after 37 hours 31 minutes (a) 0s, (b) 160s, (c) 240s and the corresponding phases (d-f) retrieved by the proposed method. The whole dynamic process of phase variation can be seen in Visualization 2.

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5. Conclusion

In this study, combining the SCPS technique and Fourier transform method, a novel phase retrieval method from one-frame SCFI is proposed and applied to dynamic phase measurement. First, by introducing SCPS technique, four-frame phase-shifting sub-interferograms can be constructed from one-frame SCFI. Second, by using Fourier transform method, the accurate phase-shifts of above sub-interferograms can be extracted easily and rapidly. Third, based on the obtained phase-shifts, the wrapped phase can be retrieved with the least-squares algorithm. Finally, the practical measured phase can be obtained after a phase unwrapping operation. Compared with the existing SCPS methods, in addition to maintaining high accuracy of phase-shifting method, due to there is no requirement of calibration for carrier-frequency and phase-shift in advance, the proposed method reveals more rapid processing speed of phase retrieval. Furthermore, two practical examples of dynamic phase variation, a tiny water droplet evaporation and a Jurkat cell apoptosis induced by DNR, are presented with the proposed method. That is to say, for each time point of dynamic process, only one-frame interferogram is enough for the phase retrieval, the proposed method is suitable for dynamic measurement while it is still difficult for real-time dynamic measurement due to the process of phase retrieval are performed in an off-line mode.

Acknowledgments

This work is supported by National Nature Science Foundation of China grants (61275015, 61475048 and 61177005).

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Name	Description
Visualization 1: MP4 (7209 KB)	The phase variation of a droplet evaporation process
Visualization 2: MP4 (866 KB)	The phase variation of a Jurkat cell apoptosis treated with DNR

Frame index	1	2	3	4
Phase-shift (rad)	0	0.9801	1.2252	2.2053
Theoretical value(rad)	0	1.0053	1.2068	2.2122
Difference(rad)	–	0.0252	−0.0184	0.0069

Methods	Proposed method	Xu’s method	SFT
RMSE(rad)	0.0273	0.0272	0.0531
PVE(rad)	0.1970	0.1972	0.4782
Time(s)	2.3324	21.3094	0.0898

Frame index	1	2	3	4
Phase-shift (rad)	0	0.9801	1.2252	2.2053
Theoretical value(rad)	0	1.0053	1.2068	2.2122
Difference(rad)	–	0.0252	−0.0184	0.0069

Methods	Proposed method	Xu’s method	SFT
RMSE(rad)	0.0273	0.0272	0.0531
PVE(rad)	0.1970	0.1972	0.4782
Time(s)	2.3324	21.3094	0.0898

Dynamic phase measurement based on spatial carrier-frequency phase-shifting method

Abstract

1. Introduction

2. Principle

3. Simulation

4. Experimental research

5. Conclusion

Acknowledgments

References and links

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Optics Express