Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Two-step orthogonalization phase demodulation method based on a single differential interferogram

Jiaosheng Li and Qinnan Zhang

Open Access Open Access

Abstract

To reduce the acquisition time of interferogram and provide a dynamic phase retrieval method with arbitrary phase shift using a dual-channel simultaneous polarization phase-shifting system, a two-step orthogonalization phase demodulation method (TOPD) based on a single differential interferogram is proposed in this paper. In this method, the differential interferogram obtained by subtracting two phase-shifting interferograms and one of the Gaussian filtered based-interferograms are used to normalize and orthogonalize, and then the phase related parameters are solved by the Lissajous ellipse fitting method. Finally, the measured phase is obtained with high accuracy. The proposed method further reduces the deviation caused by the filtering operation performed in the two-step phase demodulation method. At the same time, combined it with the Lissajou ellipse fitting method reduces the limitation associated with the approximation conditions of the orthogonalization and normalization method. The experimental and simulation results demonstrate that this method provides a solution with high accuracy, high stability, strong practicability, and few restrictions for phase extraction in quantitative phase imaging.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Phase-shifting interferometry is often applied to extract phase information in the fields of quantitative phase imaging, fluid measurement, and cell detection. It is fast, highly accurate, non-destructive, and contactless [1–4]. These methods typically require more than three phase-shifting interferograms that are obtained with the help of phase-shifting tools or other means to achieve high-accuracy phase extraction. Therefore, such methods are often limited by the accuracy and stability of the phase shift device, and it is difficult to achieve fast and high-accuracy dynamic phase extraction owing to the time-consuming interferogram acquisition process [5–7]. Large linear carrier frequency modulation-based single interferogram can be used to realize dynamic phase extraction [8,9], but it ignores the use of spatial bandwidth product, and the calculation accuracy is not high. The end-to-end deep learning-based phase recovery method is limited by the generalization capability of deep learning, and it is difficult to achieve high-accuracy measurement of a large number of different samples at the same time [10,11].

In contrast, two-step phase demodulation method requires less sampling time and is suitable for realizing fast and high-accuracy phase extraction compared with multi-step phase shifting phase demodulation methods [12–16]. It can be combined with the dual-channel phase-shifting system proposed in [17] to realize dynamic phase measurement. In the reported two-step phase demodulation methods, demodulation is often conducted using preprocessing operations such as filtering or normalization of an interferogram. Most methods require Gaussian filtering to remove the background intensity in the interferogram. Gaussian filtering applied to a phase-shifting interferogram will have adverse effects on the measurement accuracy owing to the uneven fringe distribution and background intensity distribution of interferograms [12–14]. For the two-step phase demodulation method with the help of normalization, its calculation accuracy is affected by the fringe number or the fringe shape in the interferogram to some extent [12,18]. Recently, many hybrid technologies have emerged [19–21] with the intent of providing a more effective means for phase extraction using two interferograms, but these methods have problems similar to the two-step demodulation method using orthogonalization.

Based on this, to realize fast and accurate phase extraction for dynamic process combined with a dual-channel polarization phase-shifting system, we propose a two-step orthogonalization phase demodulation method (TOPD) using single-frame differential interferogram in this paper. In contrast to the two-step phase extraction method that requires pre-filtering, the proposed method only needs to filter once, which reduces the impact of filtering operations on accuracy; Compared with the traditional method requiring normalization, this method reduces the associated limitation of the approximation conditions in the normalization method through the introduction of the Lissajou ellipse fitting method. Moreover, the simulation and experimental data and performance analysis prove that the proposed approach can achieve accurate phase recovery with better stability and anti-noise performance.

2. Principle derivation

The interferogram intensity distribution obtained by phase-shifting interferometry is described as a k-dimensional column vector, and the intensity of each pixel is the value of the elements in the column vector.

$${I_{n,k}} = {a_k} + {b_k}\cos ({\phi _k} + {\theta _n})$$

Here, $n$ and $k$ respectively represent the sequence number and pixel position in this series of interferograms, $n = 1,2$ and $k = 1,2,\ldots ,K$, where K is the total number of pixels in the interferogram. ${a_k}$ is defined as the background intensity, ${b_k}$ is set as the modulation intensity and ${\phi _k}$ is defined as the phase distribution. ${\theta _n}$ represents the phase shift between the interferograms, and let ${\theta _1} = 0$ and ${\theta _2} = \theta $. Then, the above interferograms with the phase shift of $\theta $ are respectively processed as follows.

First, a Gaussian filtering operation [22] is performed to remove the background of the first interferogram defined by

$${\bar{I}_{1,k}} = {b_k}\cos ({\phi _k}).$$

At the same time, the two interferograms are subtracted to obtain the differential interferogram:

$$\begin{aligned} {{\bar{I}}_{2,k}}& = {I_{2,k}} - {I_{1,k}} = {b_k}\cos ({\phi _k} + \theta ) - {b_k}\cos ({\phi _k})\\& ={-} 2{b_k}\sin \left( {{\phi_k} + \frac{\theta }{2}} \right)\sin \left( {\frac{\theta }{2}} \right) \end{aligned}$$

Next, the basic concepts of normalization and orthogonalization are used for the existing vectors. The vector ${\bar{I}_{1,k}}$ is normalized first.

$${S_{1,k}} = \frac{{{{\bar{I}}_{1,k}}}}{{\sqrt {\left\langle {{{\bar{I}}_{1,k}},{{\bar{I}}_{1,k}}} \right\rangle } }} = \frac{{{{\bar{I}}_{1,k}}}}{{||{{{\bar{I}}_{1,k}}} ||}} = \frac{{\cos ({\phi _k})}}{{{m_1}}}$$

For interferograms, the inner product operation can be expressed as

$$\left\langle {{{\bar{I}}_{1,k}},{{\bar{I}}_{1,k}}} \right\rangle = \sum\limits_{k = 1}^K {{{\bar{I}}_{1,k}} \cdot {{\bar{I}}_{1,k}}} .$$

The second vector is further normalized as

$${S_{2,k}} = \frac{{{{\bar{I}}_{2,k}}}}{{\sqrt {\left\langle {{{\bar{I}}_{2,k}},{{\bar{I}}_{2,k}}} \right\rangle } }} = \frac{{{{\bar{I}}_{2,k}}}}{{||{{{\bar{I}}_{2,k}}} ||}} = \frac{{\sin ({\phi _k} + \frac{\theta }{2})}}{{{m_2}}}.$$

Here, ${m_1} = \sqrt {\sum\limits_k^K {{{\cos }^2}} ({\phi _k})}$ and ${m_2} = \sqrt {\sum\limits_k^K {{{\sin }^2}} ({\phi _k} + \frac{\theta }{2})}$. Furthermore, the two normalized vectors are orthogonalized to obtain

$$\begin{aligned} {S_{add}}& = {S_{1,k}} + {S_{2,k}} = \frac{{{m_2}\cos ({\phi _k}) + {m_1}\sin ({\phi _k} + \frac{\theta }{2})}}{{{m_1}{m_2}}}\\& = \left[ {{m_2} + {m_1}\sin (\frac{\theta }{2})} \right]cos({\phi _k}) + {m_1}\cos (\frac{\theta }{2})\sin ({\phi _k})\\& = {\eta _1}cos({\phi _k}) + {\mu _1} \end{aligned}$$
$$\begin{aligned} {S_{sub}} &= {S_{1,k}} - {S_{2,k}} = \frac{{{m_2}\cos ({\phi _k}) - {m_1}\sin ({\phi _k} + \frac{\theta }{2})}}{{{m_1}{m_2}}}\\& = \left[ {{m_2} - {m_1}\sin (\frac{\theta }{2})} \right]cos({\phi _k}) - {m_1}\cos (\frac{\theta }{2})\sin ({\phi _k})\\& = {\eta _2}sin({\phi _k}) + {\mu _2} \end{aligned}$$
where
$$\begin{aligned} {\eta _1}& = {m_2} + {m_1}\sin (\frac{\theta }{2})\\ {\mu _1}& = {m_1}\cos (\frac{\theta }{2})\sin ({\phi _k})\\ {\eta _2}& ={-} {m_1}\cos (\frac{\theta }{2})\\ {\mu _2}& = \left[ {{m_2} - {m_1}\sin (\frac{\theta }{2})} \right]cos({\phi _k}) \end{aligned}$$

The parameters in Eq. (9) can be solved using the Lissajou ellipse fitting method [23], where

$$cos({\phi _k}) = \frac{{{S_{add}} - {\mu _1}}}{{{\eta _1}}}$$
$$\sin ({\phi _k}) = \frac{{{S_{sub}} - {\mu _2}}}{{{\eta _2}}}.$$

The two equations can be manipulated to define the ellipse $\frac{{{{({{S_{add}} - {\mu_1}} )}^2}}}{{\eta _1^2}} + \frac{{{{({{S_{sub}} - {\mu_2}} )}^2}}}{{\eta _2^2}} = 1$. The center of the ellipse is $({\mu _1},{\mu _2})$, and the major axis and minor axis are ${\eta _1}$ and ${\eta _2}$, respectively. Therefore, the measured phase can be obtained after achieving the above parameters:

$${\phi _k}(x,y) = \arctan [({S_{sub}} - {\mu _2}){\eta _1}/({S_{add}} - {\mu _1}){\eta _2}]$$

3. Simulation results and performance analysis

3.1 Simulation verification

To verify the feasibility of the method, two groups of simulated circular fringe interferograms with different backgrounds and modulations are used to verify the applicable conditions of the proposed TOPD method. The background and modulation items in the first group are set as

$${a_1}(x,y) = {a_2}(x,y) = 100 + 20\exp [{ - 0.025({x^2} + {y^2})} ]$$
$${b_1}(x,y) = {b_2}(x,y) = 60 + 10\exp [{ - 0.025({x^2} + {y^2})} ].$$

In the second group, the background and modulation terms are set as

$${a_1}(x,y) = 100 + 20\exp [{ - 0.025({x^2} + {y^2})} ]$$
$${b_1}(x,y) = 60 + 10\exp [{ - 0.025({x^2} + {y^2})} ]$$
$${a_2}(x,y) = 80 + 20\exp [{ - 0.025({x^2} + {y^2})} ]$$
$${b_2}(x,y) = 40 + 10\exp [{ - 0.025({x^2} + {y^2})} ].$$

The phases of these two groups of interferograms are preset to $\varphi (x,y) = 2\pi ({1/2.56/2.56} )({{x^2} + {y^2}} )+ ({x + y} )$, and the preset height is 18.2 rad. The interferogram size is set to 230 × 230 pixels. To consider a more realistic scenario, Gaussian white noise with a signal-to-noise ratio (SNR) of 35 dB is added to the above interferograms. The two phase-shifting interferograms with the phase shift of 1.57 rad and the preset phase distribution in the first group are shown in Fig. 1. To prove the feasibility and superiority over other commonly used two-step algorithms, the proposed method (TOPD), Gram-Schmidt orthonormalization (GS) [12], Diamond diagonal vectors (DDV) [13] and fringe spatial extremum features based-phase demodulation method, which is referred to as the extreme value of interference (EVI) [14], are used to calculate the above two groups of fringes. To prove the advantages of applying the simultaneous difference and filtering in this method, we also use this method to calculate two unfiltered interferograms, and analyze the calculation accuracy and processing time. For the sake of comparison, we refer to this method as TOPD1. Figure 2 shows the phase distributions calculated by the different methods on two groups of different interferograms (Situation 1 and Situation 2). To show the deviation calculated using each methods, we subtract the obtained phase distributions from the reference phases to obtain the phase deviation distributions, which are also shown in Fig. 2.

 figure: Fig. 1.

Fig. 1. (a) (b) Two simulated circular fringe interferograms, and (c) the preset phase distribution.

Download Full Size | PDF

 figure: Fig. 2.

Fig. 2. Phase distribution and phase deviation distribution results calculated for the different methods in situation 1 and situation 2.

Download Full Size | PDF

For further quantitative analysis, the root mean square error (RMSE) of the different methods under the two situations is calculated, and the values are recorded in Table 1. The calculation is implemented by MATLAB R2017a version on a PC with Intel Core i7-8550U CPU. To compare computational efficiency, the calculation time required by each of the different methods is also recorded in Table 1. Combined with the phase deviation distribution results shown in Fig. 2, it can be concluded that the proposed TOPD method achieves the highest accuracy. In terms of computational efficiency, the calculation time is comparable to that of GS in all situations. Moreover, compared with the calculation results using TOPD1, the proposed TOPD method is more efficient and accurate.

Tables Icon

Table 1. Performance of the different methods when calculating the interferograms under three situations (Simulation)

View Table | View all tables in this article

To eliminate the influence of the filtering algorithm on Situation 1, the performance of the the methods is also analyzed (Situation 3). The same conditions apply, but let a1 =a2 = 0 in Eq. (1). Similarly, phase distribution and phase deviation distribution results calculated using the different methods are shown in Fig. 3, and the quantitative analysis results are provided in Table 1. The results show that the accuracy of all the methods is improved when the influence of filtering is eliminated, which proves that the filtering algorithm has great influence on the two-step phase-shifting methods. Nevertheless, the proposed TOPD method is still outperforms GS, DDV, and EVI methods regarding calculation accuracy.

 figure: Fig. 3.

Fig. 3. Phase distribution and phase deviation distribution results calculated by different methods in situation 3 (no filtering).

Download Full Size | PDF

3.2 Influence of phase shift on the performance of the proposed TOPD method

The distribution of the phase shift and noise will have a significant impact on the phase shifting-based phase extraction methods. In this section, the influence of the phase shift distribution on the accuracy of phase demodulation is discussed. According to the simulation conditions under the first situation, the phase shift setting is changed from 0.1 rad to 3 rad in intervals of 0.1 rad. The TOPD, TOPD1, GS, DDV, and EVI algorithms are used for comparison purposes. The curve distribution is shown in Fig. 4, and the result shows that the variation trend of the accuracy calculated by the five methods with respect to the phase shift is basically the same, except that the accuracy fluctuation of EVI is relatively large when the phase shift is less than 1 rad. Moreover, the accuracy of the TOPD method is better than that of GS, DDV, and EVI under different phase shifts, and it can be concluded from the accuracy distribution that the accuracy of TOPD is relatively high in the 0.1 to 3 rad phase shift range.

 figure: Fig. 4.

Fig. 4. RMSE distributions calculated with respect to phase shift for the different methods.

Download Full Size | PDF

3.3 Noise resistance analysis

The influence of the noise level on the calculation accuracy using the different methods is discussed. We add different levels of white Gaussian noise to the interferograms, where the operation of “y = awgn (x, SNR, ‘measured’) “ is used for generating the noise, in which white Gaussian noise is added to the vector signal x. The scalar SNR specifies the signal to noise ratio at each sampling point, in dB. And ‘measured’ means that the energy of x is measured before adding noise. The SNR range is set from 20 dB to 46 dB in intervals of 2 dB. The results calculated using the different methods are shown in Fig. 5. The results reflect that except for GS and DDV, when the SNR of the interferogram increases, the accuracy of the other three methods increases. Because GS and DDV use the pre-filtering operation on the interferograms, the noise level has little effect on them. The EVI method based on fringe spatial extremum features has the most unstable accuracy and the worst noise resistance among all of the two-step methods compared. The main reason is that EVI needs to find the spatial extremum of interference fringes in the calculation process, and the influence of random noise on the spatial extremum of an interferogram is random, so its accuracy often exhibits randomness, and the stability of the algorithm is poor. Finally, compared with its unfiltered version (TOPD1), the proposed method (TOPD) is less affected by noise.

 figure: Fig. 5.

Fig. 5. RMSE distributions calculated for the different methods with respect to SNR.

Download Full Size | PDF

3.4 Stability analysis of the proposed TOPD method

To explore the stability of the TOPD method, 110 groups of different surfaces are randomly transformed by the Gaussian function and are used as the measured samples, and different backgrounds and modulation terms are randomly superimposed [24]. In the above combination, the variation range of the phase shift is set to [0.2 rad 1.57 rad], and the preset phase height varies from 3 to 60 rad. Examples of some simulated interferograms are shown in Fig. 6. The 110 groups of interferograms are calculated using the proposed method and other two-step phase-shifting demodulation methods, and the calculated accuracy distribution results are shown in Fig. 7. In general, the RMSE value of the proposed TOPD method remains below 0.2 rad, and the accuracy is stably distributed. The TOPD1 method has a similar trend to the TOPD method, but the overall accuracy is slightly lower than that of the TOPD method. The RMSE distributions reflect that the proposed TOPD method has obvious advantages in accuracy when random fringe shapes are considered. In contrast, the other three methods will fail in some cases, further proving that TOPD is a phase demodulation method exhibiting high accuracy and high stability.

 figure: Fig. 6.

Fig. 6. Examples of randomly generated interferograms.

Download Full Size | PDF

 figure: Fig. 7.

Fig. 7. RMSE values calculated for different methods using 110 groups of interferograms with different distribution combinations.

Download Full Size | PDF

4. Experimental results and analysis

In this section, the feasibility of the method will be verified experimentally. A dual-channel simultaneous polarization phase-shifting system (Fig. 8) is applied to obtain experimental interferograms. The linearly polarized light emitted by the laser with a wavelength of 632.8 nm enters the half-wave plate (HWP) after passing through a variable neutral density (ND) filter, and is split by the first polarization beam splitter (PBS) to obtain s-polarized light and p-polarized light. The s-polarized light becomes a circular polarized light after passing through a quarter-wave plate (QWP) and is used as the reference light. The p-polarized light is irradiated on the HWP and then reflected by the piezoelectric ceramic transducer (PZT) through the sample and imaged on the two camera targets. After the two beams are combined by the beam splitter (BS), they then pass through the other PBS. At this time, p-polarized light and s-polarized light form interference patterns on the two channels. Finally, two phase-shifting interferograms with phase shift $\theta$ are obtained simultaneously by using the two cameras.

 figure: Fig. 8.

Fig. 8. The principle sketch of the dual-channel simultaneous polarization phase-shifting system.

Download Full Size | PDF

A group of experimental circular fringes similar to the simulated interferograms are collected for calculation. The two interferograms collected for calculation are shown in Fig. 9(a) and 9(b). The image size used for calculations is 511 × 633 pixels, and the phase shift is 1.25 rad. To obtain the reference phase, 180 frames of the phase-shifted interferograms are calculated using the advanced iterative algorithm (AIA) [7], and the calculated result is used as the reference phase shown in Fig. 9(c). Similarly, the five methods previously compared are used to calculate the two interferograms, and the calculated results are given in Fig. 10. The phase deviation results indicate that the proposed method achieves the smallest deviation, and the phase deviation distribution of the other three methods is similar. The performances of the different methods are quantitatively analyzed in Table 2. The results of quantitative analysis indicate that the accuracy of the proposed TOPD method is twice that of the other three methods, and it is also higher than that without filtering (TOPD1). This further verifies the experimental feasibility and superiority of the proposed method.

 figure: Fig. 9.

Fig. 9. (a) (b) Interferograms collected for the experiment and (c) the reference phase distribution.

Download Full Size | PDF

 figure: Fig. 10.

Fig. 10. Experimental calculation results, including phase distribution results (top row) and phase deviation distribution results (bottom row).

Download Full Size | PDF

Tables Icon

Table 2. Performance of different methods in calculating the experimental interferograms

View Table | View all tables in this article

5. Conclusion

In this paper, we propose a two-step orthogonalization phase demodulation method (TOPD) based on single differential interferogram. This method can be combined with a dual-channel phase-shifting system to achieve dynamic phase extraction with high accuracy. This method only needs one filtering step, which reduces the impact of pre-filtering on the calculation accuracy. At the same time, combined with the Lissajou ellipse fitting method, it reduces the the limitation associated with the approximation conditions in the two-step normalization method, enhancing both accuracy and stability. The proposed method is verified by both simulation and experimental results. The effects of the phase shift and noise on the performance of the method are discussed, and the stability is also analyzed. The above results prove that the proposed method can provide an effective means in the phase extraction related applications.

Funding

National Natural Science Foundation of China (62205059, 61805086); Basic and Applied Basic Research Foundation of Guangdong Province (2021A1515110664); Basic and Applied Basic Research Foundation of Guangzhou (202201011277); Start-Up Funding of Guangdong Polytechnic Normal University (2022SDKYA008); Key-Area Research and Development Program of Guangdong Province (2020B090922005).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. C. Zuo, S. Feng, L. Huang, T. Y. Tao, W. Yin, and Q. Chen, “Phase shifting algorithms for fringe projection profilometry: A review,” Opt. Laser. Eng. 109, 23–59 (2018). [CrossRef]  

2. Y. Park, C. Depeursinge, and G. Popescu, “Quantitative phase imaging in biomedicine,” Nat. Photonics 12(10), 578–589 (2018). [CrossRef]  

3. J. S. Li, Q. N. Zhang, L. Y. Zhong, J. D. Tian, G. Pedrini, and X. X. Lu, “Quantitative phase imaging in dual-wavelength interferometry using a single wavelength illumination and deep learning,” Opt. Express 28(19), 28140–28153 (2020). [CrossRef]  

4. H. Zhu and H. Guo, “Anti-aliasing phase reconstruction via a non-uniform phase-shifting technique,” Opt. Express 30(3), 3835–3853 (2022). [CrossRef]  

5. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22(21), 3421–3432 (1983). [CrossRef]  

6. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987). [CrossRef]  

7. Z. Y. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004). [CrossRef]  

8. H. Zou, L. Zhong, J. S. Li, Z. D. Sun, Y. C. Liu, Q. W. Ning, and X. X. Lu, “Phase extraction algorithm based on the spatial carrier-frequency phase-shifting gradien,” IEEE Photonics J. 11(6), 1–7 (2019). [CrossRef]  

9. C. H. Hu, L. Chen, D. H. Zheng, Y. T. Wang, Z. Y. Ma, Z. Zhang, and J. X. Li, “Parameter mismatch phase extraction method for spatial phase-shifting interferograms,” Opt. Laser. Eng. 154, 107002 (2022). [CrossRef]  

10. Z. Zhao, B. Li, J. S. Lu, X. Q. Kang, and T. K. Liu, “One-shot phase retrieval method for interferometry using a hypercolumns convolutional neural network,” Opt. Express 29(11), 16406–16421 (2021). [CrossRef]  

11. J. S. Li, Q. N. Zhang, L. Y. Zhong, and X. X. Lu, “Hybrid-net: a two-to-one deep learning framework for three-wavelength phase-shifting interferometry,” Opt. Express 29(21), 34656–34670 (2021). [CrossRef]  

12. J. Vargas, J. A. Quiroga, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Two-step demodulation based on the Gram-Schmidt orthonormalization method,” Opt. Lett. 37(3), 443–445 (2012). [CrossRef]  

13. C. S. Luo, L. Y. Zhong, P. Sun, H. L. Wang, J. D. Tian, and X. X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119(2), 387–391 (2015). [CrossRef]  

14. J. Deng, H. L. Wang, F. Zhang, D. Zhang, L. Y. Zhong, and X. X. Lu, “Two-step phase demodulation algorithm based on the extreme value of interference,” Opt. Lett. 37(22), 4669–4671 (2012). [CrossRef]  

15. M. Trusiak, “Fringe analysis: single-shot or two-frames? Quantitative phase imaging answers,” Opt. Express 29(12), 18192–18211 (2021). [CrossRef]  

16. T. Tahara, Y. Kozawa, A. Ishii, K. Wakunami, Y. Ichihashi, and R. Oi, “Two-step phase-shifting interferometry for self-interference digital holography,” Opt. Lett. 46(3), 669–672 (2021). [CrossRef]  

17. P. Sun, L. Y. Zhong, C. S. Luo, W. H. Niu, and X. X. Lu, “Visual measurement of the evaporation process of a sessile droplet by dual-channel simultaneous phase-shifting interferometry,” Sci. Rep. 5(1), 12053–10 (2015). [CrossRef]  

18. H. Y. Zhang, F. Yang, and L. C. Cao, “Phase recovery technology of a dual-frame phase-shifting interferogram based on first-order norm vector normalization,” Appl. Opt. 61(5), B200–B205 (2022). [CrossRef]  

19. Y. Zhang, X. Tian, and R. Liang, “Random two-step phase shifting interferometry based on Lissajous ellipse fitting and least squares technologies,” Opt. Express 26(12), 15059–15071 (2018). [CrossRef]  

20. H. Y. Zhang, H. Zhao, Z. X. Zhao, Y. Y. Zhuang, and C. Fan, “Two-frame fringe pattern phase demodulation using Gram-Schmidt orthonormalization with least squares method,” Opt. Express 27(8), 10495–10508 (2019). [CrossRef]  

21. Y. Zhang, X. B. Tian, and R. G. Liang, “Two-step random phase retrieval approach based on Gram-Schmidt orthonormalization and Lissajous ellipse fitting method,” Opt. Express 27(3), 2575–2588 (2019). [CrossRef]  

22. J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003). [CrossRef]  

23. C. Farrell and M. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992). [CrossRef]  

24. Q. N. Zhang, S. Y. Lu, J. S. Li, D. Li, X. X. Lu, L. Y. Zhong, and J. D. Tian, “Phase-shifting interferometry from single frame in-line interferogram using deep learning phase-shifting technology,” Opt. Commun. 498, 127226 (2021). [CrossRef]  

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)
Fig. 1.
Fig. 1. (a) (b) Two simulated circular fringe interferograms, and (c) the preset phase distribution.

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. Phase distribution and phase deviation distribution results calculated for the different methods in situation 1 and situation 2.

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. Phase distribution and phase deviation distribution results calculated by different methods in situation 3 (no filtering).

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. RMSE distributions calculated with respect to phase shift for the different methods.

View in Article | Download Full Size | PDF

Fig. 5.
Fig. 5. RMSE distributions calculated for the different methods with respect to SNR.

View in Article | Download Full Size | PDF

Fig. 6.
Fig. 6. Examples of randomly generated interferograms.

View in Article | Download Full Size | PDF

Fig. 7.
Fig. 7. RMSE values calculated for different methods using 110 groups of interferograms with different distribution combinations.

View in Article | Download Full Size | PDF

Fig. 8.
Fig. 8. The principle sketch of the dual-channel simultaneous polarization phase-shifting system.

View in Article | Download Full Size | PDF

Fig. 9.
Fig. 9. (a) (b) Interferograms collected for the experiment and (c) the reference phase distribution.

View in Article | Download Full Size | PDF

Fig. 10.
Fig. 10. Experimental calculation results, including phase distribution results (top row) and phase deviation distribution results (bottom row).

View in Article | Download Full Size | PDF

Tables (2)

Tables Icon

Table 1. Performance of the different methods when calculating the interferograms under three situations (Simulation)

View Table | View all tables in this article
Tables Icon

Table 2. Performance of different methods in calculating the experimental interferograms

View Table | View all tables in this article

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

I n , k = a k + b k cos ⁡ ( ϕ k + θ n )
I ¯ 1 , k = b k cos ⁡ ( ϕ k ) .
I ¯ 2 , k = I 2 , k − I 1 , k = b k cos ⁡ ( ϕ k + θ ) − b k cos ⁡ ( ϕ k ) = − 2 b k sin ⁡ ( ϕ k + θ 2 ) sin ⁡ ( θ 2 )
S 1 , k = I ¯ 1 , k ⟨ I ¯ 1 , k , I ¯ 1 , k ⟩ = I ¯ 1 , k | | I ¯ 1 , k | | = cos ⁡ ( ϕ k ) m 1
⟨ I ¯ 1 , k , I ¯ 1 , k ⟩ = ∑ k = 1 K I ¯ 1 , k ⋅ I ¯ 1 , k .
S 2 , k = I ¯ 2 , k ⟨ I ¯ 2 , k , I ¯ 2 , k ⟩ = I ¯ 2 , k | | I ¯ 2 , k | | = sin ⁡ ( ϕ k + θ 2 ) m 2 .
S a d d = S 1 , k + S 2 , k = m 2 cos ⁡ ( ϕ k ) + m 1 sin ⁡ ( ϕ k + θ 2 ) m 1 m 2 = [ m 2 + m 1 sin ⁡ ( θ 2 ) ] c o s ( ϕ k ) + m 1 cos ⁡ ( θ 2 ) sin ⁡ ( ϕ k ) = η 1 c o s ( ϕ k ) + μ 1
S s u b = S 1 , k − S 2 , k = m 2 cos ⁡ ( ϕ k ) − m 1 sin ⁡ ( ϕ k + θ 2 ) m 1 m 2 = [ m 2 − m 1 sin ⁡ ( θ 2 ) ] c o s ( ϕ k ) − m 1 cos ⁡ ( θ 2 ) sin ⁡ ( ϕ k ) = η 2 s i n ( ϕ k ) + μ 2
η 1 = m 2 + m 1 sin ⁡ ( θ 2 ) μ 1 = m 1 cos ⁡ ( θ 2 ) sin ⁡ ( ϕ k ) η 2 = − m 1 cos ⁡ ( θ 2 ) μ 2 = [ m 2 − m 1 sin ⁡ ( θ 2 ) ] c o s ( ϕ k )
c o s ( ϕ k ) = S a d d − μ 1 η 1
sin ⁡ ( ϕ k ) = S s u b − μ 2 η 2 .
ϕ k ( x , y ) = arctan ⁡ [ ( S s u b − μ 2 ) η 1 / ( S a d d − μ 1 ) η 2 ]
a 1 ( x , y ) = a 2 ( x , y ) = 100 + 20 exp ⁡ [ − 0.025 ( x 2 + y 2 ) ]
b 1 ( x , y ) = b 2 ( x , y ) = 60 + 10 exp ⁡ [ − 0.025 ( x 2 + y 2 ) ] .
a 1 ( x , y ) = 100 + 20 exp ⁡ [ − 0.025 ( x 2 + y 2 ) ]
b 1 ( x , y ) = 60 + 10 exp ⁡ [ − 0.025 ( x 2 + y 2 ) ]
a 2 ( x , y ) = 80 + 20 exp ⁡ [ − 0.025 ( x 2 + y 2 ) ]
b 2 ( x , y ) = 40 + 10 exp ⁡ [ − 0.025 ( x 2 + y 2 ) ] .
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use