Improved phase retrieval method of dual-wavelength interferometry based on a shorter synthetic-wavelength

Jiaxiang Xiong; Liyun Zhong; Shengde Liu; Xiang Qiu; Yunfei Zhou; Jindong Tian; Xiaoxu Lu

doi:10.1364/OE.25.007181

1. Introduction

Optical interferometry has been widely utilized in surface micro-topography [1] and biological cell imaging [2, 3] due to its high accuracy, rapid speed, full-field and non-intervention. In recent years, a lot of methods, such as single-wavelength interferometry (SWI) and dual-wavelength interferometry (DWI), have been proposed [4–18]. In SWI, if the height variation of measured sample between adjacent sampling points is more than half of illumination wavelength, the problem of phase ambiguity will appear, so the corresponding measurement range is restricted. To solve this problem, DWI [7–18] is introduced, in which the value of synthetic-wavelength is inversely proportional to the reciprocal difference between two wavelengths: the smaller the difference between two wavelengths, the larger the measurement range. In [10], a monochrome camera is utilized to successively capture the interferograms of two wavelengths generated from two individual lasers, then the wrapped phases of single-wavelength are calculated, respectively, this method achieves high accuracy because the interferograms of two wavelengths are captured separately without crosstalk. However, it is time-consuming in both interferogram acquisition and phase calculation. In [11], by simultaneously capturing interferograms of two wavelengths through using different color channels of a color camera, the acquisition time can be decreased to a half relative to the above method, but the crosstalk and pixel deviation between different channels will lead to additional error. In [12–18], the hybrid interferograms of two wavelengths are captured by a monochrome camera, and then the wrapped phases of single-wavelength can be separated. Compared with the above two methods, this method greatly simplify both interferogram acquisition and phase calculation, but the accuracy will be decreased.

Though the phase ambiguity problem in SWI can be solved through using DWI, the corresponding error of phase retrieval will be amplified due to the enlargement of synthetic- wavelength, so the accuracy of phase retrieval is greatly decreased. Actually, the phase variation of measured sample possibly reveals rapid in some areas but becomes slow in another areas, thus it is needed to achieve phase retrieval with high accuracy and large measurement range. In a word, the accuracy improvement of phase retrieval is still an important research in DWI. To address this, an immune algorithm of phase ambiguity is proposed [19], in which the height achieved by the longer synthetic-wavelength is utilized to perform the phase unwrapping of single- wavelength, measurement range extension and noise reduction. According to this idea, several similar methods are also developed. In [20], the linear regression equation is introduced to perform the phase unwrapping, but the corresponding calculation procedure is complicated. In [21, 22], by selecting two closer wavelengths, the measurement range can be further expanded, but the corresponding noise was also increased. To solve this problem, the third wavelength which is far away from the two above wavelengths is introduced to perform the noise reduction. However, this DWI method requires an additional wavelength, so the corresponding experimental process becomes more complicated. After that, by using the shorter synthetic-wavelength of DWI, a high accuracy phase retrieval method of DWI is developed [23, 24], but the corresponding measurement range is decreased due to the phase ambiguity. In this study, by combining the advantage of the shorter synthetic-wavelength and immune algorithm of phase ambiguity [19], we propose an improved DWI to achieve phase retrieval with high accuracy and large measurement range without the additional wavelength. Following, we will introduce the principle of proposed method, and present the corresponding simulation and experimental results.

2. Principle

In SWI, the height of measured sample can be calculated by

h_{λ_{1}} (x, y) = [n_{λ_{1}} (x, y) + \frac{φ_{1} (x, y)}{2 π}] λ_{1} .

Where

h_{λ_{1}}

represents the height at single-wavelength

λ_{1}

,

φ_{1} (0 \leq φ_{1} \leq 2 π)

denotes the corresponding wrapped phase achieved from interferogram;

n_{λ_{1}}

is an integer employed for phase unwrapping operation. For convenience, we omit the cartesian coordinates in following derivation. As we know, if the optical thickness variation of measured sample between two adjacent pixels is larger than

λ_{1} / 2

, it is impossible to achieve accurate

n_{λ_{1}}

through using SWI method. To address this, DWI is introduced, in which a longer synthetic-wavelength relative to the illumination wavelengths of

λ_{1}

or

λ_{2} (λ_{1} > λ_{2})

is introduced

Λ_{s u b} = \frac{λ_{1} λ_{2}}{λ_{1} - λ_{2}} .

And the phase of measured sample can be calculated by

φ_{s u b} = φ_{2} - φ_{1} .

The corresponding height is

h_{s u b} = \frac{φ_{s u b} Λ_{s u b}}{2 π} .

Where

φ_{1}

and

φ_{2}

denote the wrapped phases of single-wavelength at

λ_{1}

and

λ_{2}

, respectively; and

h_{s u b}

is the height at

Λ_{s u b}

. As we know, the height of measured sample at the longer synthetic-wavelength has the magnification error relative to the single-wavelength, then

h_{s u b}

can be described as

h_{s u b} = h_{λ_{1}} + Δ .

Where

Δ

denotes the magnification error. To reduce this error, we introduce an immune algorithm of phase ambiguity [21], in which the height achieved from the longer synthetic- wavelength is utilized to perform phase unwrapping of single-wavelength, and then the accurate integer

n_{λ_{1}}

can be achieved. As a result, the correct height at single-wavelength can be determined in the presence of phase ambiguity, as shown in Fig. 1. Following, we first need to determine the integer

n_{λ_{1}}

by rounding operation, we can achieve an integer

r_{1}

, which is very close to the integer

n_{λ_{1}}

,

r_{1} = r o u n d (\frac{h_{s u b} - λ_{1} / 2}{λ_{1}}) .

According to Eqs. (1) and (5), we have that

h_{s u b} - λ_{1} / 2 = n_{λ_{1}} λ_{1} + \frac{φ_{1}}{2 π} λ_{1} + Δ - λ_{1} / 2.

If the condition that

| Δ | < λ_{1} / 2

can be satisfied, and due to

0 \leq φ_{1} \leq 2 π

, we have that

- λ_{1} < Δ + \frac{φ_{1}}{2 π} λ_{1} - λ_{1} / 2 < λ_{1}

Thus the possible values of

r_{1}

can be described as

r_{1} = {\begin{cases} n_{λ_{1}} - 1 - λ_{1} / 2 < Δ + \frac{φ_{1}}{2 π} λ_{1} < 0 \\ n_{λ_{1}} 0 < Δ + \frac{φ_{1}}{2 π} λ_{1} < λ_{1} \\ n_{λ_{1}} + 1 λ_{1} < Δ + \frac{φ_{1}}{2 π} λ_{1} < 3 λ_{1} / 2 \end{cases} .

By replacing

n_{λ_{1}}

with

r_{1}

, we achieve the

h_{λ_{1}}'

with some sharp peaks named as the burring height,

h_{λ_{1}}' = [r_{1} + \frac{φ_{1}}{2 π}] λ_{1} .

To eliminate those sharp peaks, we introduce a difference parameter

D

,

D = h_{s u b} - h_{λ_{1}}' = {\begin{cases} Δ + λ_{1} r_{1} = n_{λ_{1}} - 1 \\ Δ r_{1} = n_{λ_{1}} \\ Δ - λ_{1} r_{1} = n_{λ_{1}} + 1 \end{cases} .

Because

| Δ | < λ_{1} / 2

, we can achieve a correction factor

c_{1} = r o u n d (\frac{D}{λ_{1}}) = {\begin{cases} 1 r_{1} = n_{λ_{1}} - 1 \\ 0 r_{1} = n_{λ_{1}} \\ - 1 r_{1} = n_{λ_{1}} + 1 \end{cases} .

Obviously, it is found that

n_{λ_{1}} = r_{1} + c_{1}

, and then the accurate height at single-wavelength

λ_{1}

can be achieved by

h_{λ_{1}} = [r_{1} + c_{1} + \frac{φ_{λ_{1}} (x, y)}{2 π}] λ_{1} .

Considering a more general situation, if

m λ_{1} / 2 < Δ < (m + 1) λ_{1} / 2

(

m

is an integer), there will be the following relationship

r_{1} + c_{1} = {\begin{cases} n_{λ_{1}} + \frac{m + 1}{2} (m = o d d) \\ n_{λ_{1}} + \frac{m + 1}{2} (m = o d d) \\ n_{λ_{1}} + \frac{m}{2} (m = e v e r) \\ n_{λ_{1}} + \frac{m}{2} (m = e v e r) \end{cases} .

Clearly, only when

m = 0, - 1

,

| Δ | < λ_{1} / 2

, we have that

r_{1} + c_{1} = n_{λ_{1}}

, otherwise, the sharp peaks in Eq. (10) cannot be eliminated. To solve this problem, the third wavelength is introduced into DWI [21,22].

Fig. 1 Schematic of immune algorithm of phase ambiguity, in which $h_{s u b}$ and $h_{λ_{1}}$ denote the heights of measured sample at $Λ_{s u b}$ and $λ_{1}$ , respectively; $h_{λ_{1}}'$ is the corresponding burring height.

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Since the error of phase retrieval induced by the shorter synthetic-wavelength [23,24] is smaller than single-wavelength. In this study, in order to fully utilize the existing results in DWI, by combining the advantage of the shorter synthetic-wavelength DWI and the immune algorithm of phase ambiguity, we intend to achieve the phase retrieval with improved high accuracy and large measurement range. As described in [23,24], the shorter synthetic-wavelength can be expressed as

Λ_{a d d} = \frac{λ_{1} λ_{2}}{λ_{1} + λ_{2}} .

The wrapped phase of shorter synthetic-wavelength can be achieved by

φ_{a d d} = φ_{1} + φ_{2} .

Like Eq. (1), the corresponding height can be expressed as

h_{a d d} (x, y) = [n_{a d d} (x, y) + \frac{φ_{a d d} (x, y)}{2 π}] Λ_{a d d} .

Moreover, to satisfy the condition

0 \leq φ_{a d d} \leq 2 π

, we perform the transformation

φ_{a d d} = {\begin{cases} φ_{a d d} + 2 π φ_{a d d} < 0 \\ φ_{a d d} - 2 π φ_{a d d} > 2 π \end{cases} .

Here, we define

h_{λ_{1}}

calculated from Eq. (13) as the transition height. By respectively replacing

h_{s u b}

and

λ_{1}

in Eq. (6) with

h_{λ_{1}}

and

Λ_{a d d}

, and then performing Eq. (6)-(13) again, we can achieve the accuracy improvement of height

h_{a d d}

by

h_{a d d} = [r_{a d d} + c_{a d d} + \frac{φ_{a d d} (x, y)}{2 π}] Λ_{a d d} .

As mentioned above, $| Δ | < λ_{1} / 2$ is the precondition of proposed method, corresponding to that the difference of two single-wavelengths cannot be too small. Otherwise, the amplification error will lead to the invalid of precondition, and the third wavelength is needed to be introduced [21,22]. In addition, to ensure the larger range of $Δ$ , it is required that $λ_{1}$ should be set as the larger one of two single-wavelengths. In summary, in the proposed method, there are three steps: (1) like the conventional DWI method, it is needed to calculate the height at longer synthetic-wavelength through using the two wrapped phases of single-wavelength; (2) by using the immune algorithm of phase ambiguity, we perform phase unwrapping at single-wavelength, and then achieve the accurate height of single-wavelength, defined as the transition height; (3) we perform phase unwrapping of shorter synthetic-wavelength through using the immune algorithm of phase ambiguity and the transition height, and then the high accuracy height at shorter synthetic-wavelength can be achieved.

3. Simulation

Numerical simulation is carried out to verify the effectiveness of proposed method, in which two single-wavelengths with $λ_{1} = 632.8 n m$ and $λ_{2} = 532 n m$ are utilized, so the corresponding longer and shorter synthetic-wavelength are equal to $Λ_{s u b} = 3334.2 n m$ and $Λ_{a d d} = 289.1 n m$ , respectively. A sequence of 5-frame simulated phase-shifting interference patterns with size of $150 \times 150$ pixels are generated, respectively; and the phase shifts is distributed in one period with random error form $- 0.2 r a d$ to $0.2 r a d$ . The backgrounds at $λ_{1}$ and $λ_{2}$ are set as $A_{1} = 120 \exp [- 0.05 (u^{2} + v^{2})]$ , $A_{2} = 110 \exp [- 0.05 (u^{2} + v^{2})]$ $(- 1.5 \leq u, v \leq 1.5)$ and the corresponding modulation amplitudes are set as $B_{1} = 80 \exp [- 0.04 (u^{2} + v^{2})]$ , $B_{2} = 70 \exp [- 0.04 (u^{2} + v^{2})]$ , respectively. The sample is a simulated vortex phase plate with a step of 800nm. In addition, the zero-mean Gaussian white noise with standard deviation $σ = 8$ is added to the interference pattern as shown in Figs. 2(a) and 2(b). The wrapped phases of each single-wavelength are achieved by the advanced iterative algorithm(AIA) [4] due to its high accuracy. First, we achieve the height $h_{s u b}$ of longer synthetic-wavelength $Λ_{s u b}$ through using Eq. (4), as shown in Fig. 2(c). It can be seen that the surface of $h_{s u b}$ is very rough due to the magnification error. Second, by employing the immune algorithm of phase ambiguity, we achieve the height $h_{λ_{1}}$ of single-wavelength (Fig. 2(d)). It is found that even if the step height is more than half of $λ_{1}$ , the proposed method still can work well in the case that the location information of height jump is lack. Third, by introducing a shorter synthetic-wavelength $Λ_{a d d}$ , and then performing the operations as described in Eqs. (6)-(13), we can achieve the accurate $h_{a d d}$ of shorter synthetic-wavelength, as shown in Fig. 2(e). For comparison, Fig. 2(f) also shows the height map of $h_{a d d}$ directly achieved through the immune algorithm of phase ambiguity and the wrapped phase of $φ_{a d d}$ . It can be seen that a lot of sharp peaks appear in the surface, indicating that the condition $| h_{s u b} - h_{a d d} | < Λ_{a d d} / 2$ is not well satisfied. Moreover, Figs. 2(g) and 2(h) give the differences between Figs. 2(c) and 2(d), as well as Figs. 2(d) and 2(e), respectively. It is found that the difference between $h_{a d d}$ and $h_{λ_{1}}$ is less than $λ_{1} / 2$ but greatly larger than $Λ_{a d d} / 2$ , while the difference between $h_{λ_{1}}$ and $h_{a d d}$ is small, the condition $| h_{λ_{1}} - h_{a d d} | < Λ_{a d d} / 2$ can be satisfied well. For clarity, we also calculate the corresponding values of peak-valley (PV) of Figs. 2(c)-2(e) it is found that $P V_{s u b} = 1026.41 n m$ , $P V_{λ_{1}} = 827.98 n m$ and $P V_{a d d} = 813 .78nm$ , and the height distribution curves of the $40^{t h}$ row are also shown (Fig. 3). Obviously, $h_{a d d}$ is closer to the theoretical height relative to $h_{λ_{1}}$ and $h_{s u b}$ , indicating the height accuracy of measured sample is improved has through using the proposed method. At last, Fig. 4 presents the variation of root mean square error (RMSE) of achieved height in different standard deviations of zero-mean Gaussian white noise. It is observed that the accuracy of phase retrieval with the proposed method is higher than other two methods. That is to say, so long as the condition $| Δ | < λ_{1} / 2$ can be satisfied, the proposed DWI method can work well, the larger the noise, the better accuracy improvement.

Fig. 2 Simulated result of a vortex phase plate through using the proposed method with the zero-mean Gaussian white noise with standard deviation $σ = 8$ , one-frame interferogram of single-wavelength at (a) 532nm and (b) 632.8nm; (c)-(e) height maps of $h_{s u b}$ , $h_{λ 1}$ and $h_{a d d}$ , respectively; (f) height map that achieved with the immune algorithm of phase ambiguity and $φ_{a d d}$ directly; (g) the difference maps between (c) and (d); (h) the difference maps between (e) and (d).

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Fig. 3 Height distributions of the $40^{t h}$ row retrieved from Figs. 2(c)-2(e), respectively.

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Fig. 4 RMSEs of achieved height in different standard deviations of zero-mean Gaussian white noise.

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4. Experiment

In experiment, we build an in-line Mach-Zehnder dual-wavelength interference system to further verify the flexibility of proposed method, as shown in Fig. 5. The sample is a vortex phase plate (RPC Photonics Co.VPP-1c) with height of 607nm; two lasers with wavelength of $632.8 n m$ and $532 n m$ are utilized. For each wavelength, 30-frame phase–shifting interferograms with size of $200 \times 200$ pixel are captured, and the phase shifts is distributed in one period, as shown in Fig. 6. The wrapped phases of single-wavelength are achieved by the AIA method, and then $h_{s u b}$ can be calculated with Eqs. (3) and (4), as shown in Fig. 7(a). Like the simulation result, we can see that the surface of sample is rough due to the magnification error. Moreover, Fig. 7(b) gives the height map of $h_{λ_{1}}$ through combining the immune algorithm of phase ambiguity and $h_{s u b}$ for phase unwrapping of single-wavelength. It is observed that the noise of phase retrieval is effectively decreased and the step position is properly unwrapped. Further, we perform phase unwrapping of shorter synthetic-wavelength through using the immune algorithm of phase ambiguity and $h_{λ_{1}}$ , and then the super high accuracy of $h_{a d d}$ can be achieved, as shown in Fig. 7(c). Quantitatively, it is found that the values of peak-valley (PV) in $h_{s u b}$ , $h_{λ_{1}}$ and $h_{a d d}$ are ${PV}_{s u b} =867 .91nm$ , ${PV}_{λ_{1}} =617 .93nm$ and ${PV}_{a d d} =609 .07nm$ , respectively. In addition, Fig. 8 also shows the height distribution curves of the $50^{t h}$ row in $h_{s u b}$ , $h_{λ_{1}}$ and $h_{a d d}$ , respectively. These results further demonstrates that the outstanding advantage of the proposed method in accuracy improvement of phase retrieval.

Fig. 5 Mach-Zehnder interferometer based dual-wavelength phase-shifting interferometry system, CLBE: collimating laser beam expander, ND: neutral density filter, BS1 and BS2:beam splitter, PZT: piezoelectric transducer, M1 and M2:mirror.

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Fig. 6 Experimental phase-shifting interferograms of a vortex phase plate of single-wavelength at (a) 532nm; (b) 632.8nm; (c),(d) the corresponding background of (a) and(b), respectively.

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Fig. 7 (a) Height map of $h_{s u b}$ achieved with the wrapped phases of single-wavelength; (b) height map of $h_{λ_{1}}$ achieved through combining the immune algorithm of phase ambiguity and $h_{s u b}$ ; (c) height map of $h_{a d d}$ achieved through combining the immune algorithm of phase ambiguity and $h_{λ_{1}}$ .

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Fig. 8 Height distribution curves of the $50^{t h}$ row extracted from Fig. 7(a)-7(c), respectively.

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5. Summary

In this study, by combining the advantage of the shorter synthetic-wavelength DWI and immune algorithm of phase ambiguity, we propose an improved phase retrieval method of DWI with high accuracy and large measurement range. First, like the conventional DWI method, we calculate the height at longer synthetic-wavelength through using the wrapped phases of two single-wavelengths. Subsequently, by combining the immune algorithm of phase ambiguity and the height at longer synthetic-wavelength, we perform phase unwrapping of single-wavelength, and then achieve the accurate height at single-wavelength named as the transition height. Finally, we perform phase unwrapping of shorter synthetic-wavelength through using the immune algorithm of phase ambiguity and the transition height, and then the height accuracy of shorter synthetic-wavelength can be further improved as long as the condition requirement of noise can be satisfied. Compared with the reported method, in addition to maintaining the advantage of high accuracy, the proposed method does not need the additional wavelength, so the corresponding measurement and calculation procedures is greatly simplified. And this will facilitate the application of DWI of in optical phase measurement.

Funding

National Natural Science Foundation of China (NSFC) (61475048, 61275015, 61177005).

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Improved phase retrieval method of dual-wavelength interferometry based on a shorter synthetic-wavelength

Abstract

1. Introduction

2. Principle

3. Simulation

4. Experiment

5. Summary

Funding

References and links

Cited By

Figures (8)

Equations (19)

Optics Express