Abstract

In dual-wavelength interferometry (DWI), by combing the advantage of the shorter synthetic-wavelength and the immune algorithm of phase ambiguity, we propose an improved phase retrieval method with both high accuracy and large measurement range, which is a pair of contradiction in the reported DWI method. First, we calculate the height of measured object at longer synthetic-wavelength through using the wrapped phases of two single-wavelengths. Second, by combining the immune algorithm of phase ambiguity and the height of measured object at longer synthetic-wavelength, we can perform the phase unwrapping of the larger one of the two single-wavelengths, then achieve 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 at shorter synthetic-wavelength can be achieved. 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 procedures is greatly simplified. Simulation and experimental results demonstrate the performance of proposed method.

© 2017 Optical Society of America

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

hλ1(x,y)=[nλ1(x,y)+φ1(x,y)2π]λ1.
Where hλ1 represents the height at single-wavelength λ1, φ1(0φ12π)denotes the corresponding wrapped phase achieved from interferogram; nλ1 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 λ1/2, it is impossible to achieve accurate nλ1 through using SWI method. To address this, DWI is introduced, in which a longer synthetic-wavelength relative to the illumination wavelengths of λ1 or λ2(λ1>λ2) is introduced
Λsub=λ1λ2λ1λ2.
And the phase of measured sample can be calculated by
φsub=φ2φ1.
The corresponding height is
hsub=φsubΛsub2π.
Where φ1 and φ2 denote the wrapped phases of single-wavelength at λ1 and λ2, respectively; and hsub is the height at Λsub. As we know, the height of measured sample at the longer synthetic-wavelength has the magnification error relative to the single-wavelength, then hsub can be described as
hsub=hλ1+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 nλ1 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 nλ1 by rounding operation, we can achieve an integer r1, which is very close to the integer nλ1,
r1=round(hsubλ1/2λ1).
According to Eqs. (1) and (5), we have that
hsubλ1/2=nλ1λ1+φ12πλ1+Δλ1/2.
If the condition that |Δ|<λ1/2 can be satisfied, and due to 0φ12π, we have that
λ1<Δ+φ12πλ1λ1/2<λ1.
Thus the possible values of r1 can be described as
r1={nλ11λ1/2<Δ+φ12πλ1<0nλ10<Δ+φ12πλ1<λ1nλ1+1λ1<Δ+φ12πλ1<3λ1/2.
By replacing nλ1 with r1, we achieve the hλ1' with some sharp peaks named as the burring height,
hλ1'=[r1+φ12π]λ1.
To eliminate those sharp peaks, we introduce a difference parameter D,
D=hsubhλ1'={Δ+λ1r1=nλ11Δr1=nλ1Δλ1r1=nλ1+1.
Because |Δ|<λ1/2, we can achieve a correction factor
c1=round(Dλ1)={1r1=nλ110r1=nλ11r1=nλ1+1.
Obviously, it is found that nλ1=r1+c1, and then the accurate height at single-wavelength λ1 can be achieved by
hλ1=[r1+c1+φλ1(x,y)2π]λ1.
Considering a more general situation, if mλ1/2<Δ<(m+1)λ1/2(m is an integer), there will be the following relationship
r1+c1={nλ1+m+12(m=odd)nλ1+m+12(m=odd)nλ1+m2(m=ever)nλ1+m2(m=ever).
Clearly, only when m=0,1, |Δ|<λ1/2, we have that r1+c1=nλ1, 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 hsub and hλ1 denote the heights of measured sample at Λsub and λ1, respectively; hλ1' 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

Λadd=λ1λ2λ1+λ2.
The wrapped phase of shorter synthetic-wavelength can be achieved by
φadd=φ1+φ2.
Like Eq. (1), the corresponding height can be expressed as
hadd(x,y)=[nadd(x,y)+φadd(x,y)2π]Λadd.
Moreover, to satisfy the condition0φadd2π, we perform the transformation
φadd={φadd+2πφadd<0φadd2πφadd>2π.
Here, we define hλ1 calculated from Eq. (13) as the transition height. By respectively replacing hsub and λ1 in Eq. (6) with hλ1 and Λadd, and then performing Eq. (6)-(13) again, we can achieve the accuracy improvement of height hadd by

hadd=[radd+cadd+φadd(x,y)2π]Λadd.

As mentioned above, |Δ|<λ1/2 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 λ1 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 λ1=632.8nm and λ2=532nmare utilized, so the corresponding longer and shorter synthetic-wavelength are equal to Λsub=3334.2nm and Λadd=289.1nm, respectively. A sequence of 5-frame simulated phase-shifting interference patterns with size of 150×150pixels are generated, respectively; and the phase shifts is distributed in one period with random error form 0.2rad to 0.2rad. The backgrounds at λ1 and λ2 are set as A1=120exp[0.05(u2+v2)], A2=110exp[0.05(u2+v2)](1.5u,v1.5) and the corresponding modulation amplitudes are set asB1=80exp[0.04(u2+v2)], B2=70exp[0.04(u2+v2)], 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 hsub of longer synthetic-wavelength Λsub through using Eq. (4), as shown in Fig. 2(c). It can be seen that the surface of hsub is very rough due to the magnification error. Second, by employing the immune algorithm of phase ambiguity, we achieve the height hλ1 of single-wavelength (Fig. 2(d)). It is found that even if the step height is more than half of λ1, 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 Λadd, and then performing the operations as described in Eqs. (6)-(13), we can achieve the accurate hadd of shorter synthetic-wavelength, as shown in Fig. 2(e). For comparison, Fig. 2(f) also shows the height map of hadd directly achieved through the immune algorithm of phase ambiguity and the wrapped phase of φadd. It can be seen that a lot of sharp peaks appear in the surface, indicating that the condition |hsubhadd|<Λadd/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 hadd and hλ1 is less than λ1/2 but greatly larger than Λadd/2, while the difference between hλ1 and hadd is small, the condition |hλ1hadd|<Λadd/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 PVsub=1026.41nm, PVλ1=827.98nm and PVadd=813.78nm, and the height distribution curves of the 40th row are also shown (Fig. 3). Obviously, hadd is closer to the theoretical height relative to hλ1 and hsub, 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|<λ1/2can 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 hsub, hλ1 and hadd, respectively; (f) height map that achieved with the immune algorithm of phase ambiguity and φadd 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 40th 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 632.8nm and 532nm are utilized. For each wavelength, 30-frame phase–shifting interferograms with size of 200×200 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 hsub 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 hλ1 through combining the immune algorithm of phase ambiguity and hsub 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 hλ1, and then the super high accuracy of hadd can be achieved, as shown in Fig. 7(c). Quantitatively, it is found that the values of peak-valley (PV) in hsub, hλ1 and hadd are PVsub=867.91nm, PVλ1=617.93nm and PVadd=609.07nm, respectively. In addition, Fig. 8 also shows the height distribution curves of the 50th row in hsub, hλ1 and hadd, 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 hsub achieved with the wrapped phases of single-wavelength; (b) height map of hλ1 achieved through combining the immune algorithm of phase ambiguity and hsub; (c) height map of hadd achieved through combining the immune algorithm of phase ambiguity and hλ1.

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Fig. 8 Height distribution curves of the 50th 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).

References and links

1. A. Safrani and I. Abdulhalim, “Real-time phase shift interference microscopy,” Opt. Lett. 39(17), 5220–5223 (2014). [CrossRef]   [PubMed]  

2. N. T. Shaked, “Quantitative phase microscopy of biological samples using a portable interferometer,” Opt. Lett. 37(11), 2016–2018 (2012). [CrossRef]   [PubMed]  

3. G. Popescu, Y. Park, W. Choi, R. R. Dasari, M. S. Feld, and K. Badizadegan, “Imaging red blood cell dynamics by quantitative phase microscopy,” Blood Cells Mol. Dis. 41(1), 10–16 (2008). [CrossRef]   [PubMed]  

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

5. J. Vargas, J. A. Quiroga, and T. Belenguer, “Analysis of the principal component algorithm in phase-shifting interferometry,” Opt. Lett. 36(12), 2215–2217 (2011). [CrossRef]   [PubMed]  

6. K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40(17), 2886–2894 (2001). [CrossRef]   [PubMed]  

7. P. S. Lam, J. D. Gaskill, and J. C. Wyant, “Two-wavelength holographic interferometer,” Appl. Opt. 23(18), 3079–3081 (1984). [CrossRef]   [PubMed]  

8. K. Creath, “Step height measurement using two-wavelength phase-shifting interferometry,” Appl. Opt. 26(14), 2810–2816 (1987). [CrossRef]   [PubMed]  

9. Y. Ishii and R. Onodera, “Two-wavelength laser-diode interferometry that uses phase-shifting techniques,” Opt. Lett. 16(19), 1523–1525 (1991). [CrossRef]   [PubMed]  

10. D. G. Abdelsalam and D. Kim, “Two-wavelength in-line phase-shifting interferometry based on polarizing separation for accurate surface profiling,” Appl. Opt. 50(33), 6153–6161 (2011). [CrossRef]   [PubMed]  

11. A. Pförtner and J. Schwider, “Red-green-blue interferometer for the metrology of discontinuous structures,” Appl. Opt. 42(4), 667–673 (2003). [CrossRef]   [PubMed]  

12. L. Fei, X. Lu, H. Wang, W. Zhang, J. Tian, and L. Zhong, “Single-wavelength phase retrieval method from simultaneous multi-wavelength in-line phase-shifting interferograms,” Opt. Express 22(25), 30910–30923 (2014). [CrossRef]   [PubMed]  

13. W. Zhang, X. Lu, L. Fei, H. Zhao, H. Wang, and L. Zhong, “Simultaneous phase-shifting dual-wavelength interferometry based on two-step demodulation algorithm,” Opt. Lett. 39(18), 5375–5378 (2014). [CrossRef]   [PubMed]  

14. W. Zhang, X. Lu, C. Luo, L. Zhong, and J. Vargas, “Principal component analysis based simultaneous dual-wavelength phase-shifting interferometry,” Opt. Commun. 341, 276–283 (2015). [CrossRef]  

15. X. Qiu, L. Zhong, J. Xiong, Y. Zhou, J. Tian, D. Li, and X. Lu, “Phase retrieval based on temporal and spatial hybrid matching in simultaneous phase-shifting dual-wavelength interferometry,” Opt. Express 24(12), 12776–12787 (2016). [CrossRef]   [PubMed]  

16. R. Onodera and Y. Ishii, “Two-wavelength interferometry that uses a fourier-transform method,” Appl. Opt. 37(34), 7988–7994 (1998). [CrossRef]   [PubMed]  

17. D. G. Abdelsalam, R. Magnusson, and D. Kim, “Single-shot, dual-wavelength digital holography based on polarizing separation,” Appl. Opt. 50(19), 3360–3368 (2011). [CrossRef]   [PubMed]  

18. L. Huang, X. Lu, Y. Zhou, J. Tian, and L. Zhong, “Dual-wavelength interferometry based on the spatial carrier-frequency phase-shifting method,” Appl. Opt. 55(9), 2363–2369 (2016). [CrossRef]   [PubMed]  

19. J. Gass, A. Dakoff, and M. K. Kim, “Phase imaging without 2π ambiguity by multiwavelength digital holography,” Opt. Lett. 28(13), 1141–1143 (2003). [CrossRef]   [PubMed]  

20. A. Khmaladze, R. L. Matz, C. Zhang, T. Wang, M. M. Banaszak Holl, and Z. Chen, “Dual-wavelength linear regression phase unwrapping in three-dimensional microscopic images of cancer cells,” Opt. Lett. 36(6), 912–914 (2011). [CrossRef]   [PubMed]  

21. D. Parshall and M. K. Kim, “Digital holographic microscopy with dual-wavelength phase unwrapping,” Appl. Opt. 45(3), 451–459 (2006). [CrossRef]   [PubMed]  

22. C. J. Mann, P. R. Bingham, V. C. Paquit, and K. W. Tobin, “Quantitative phase imaging by three-wavelength digital holography,” Opt. Express 16(13), 9753–9764 (2008). [CrossRef]   [PubMed]  

23. J. Di, W. Qu, B. Wu, X. Chen, J. Zhao, and A. Asundi, “Dual wavelength digital holography for improving the measurement accuracy,” Proc. SPIE 8769(19), 9149–9427 (2013).

24. J. Di, J. Zhang, T. Xi, C. Ma, and J. Zhao, “Improvement of measurement accuracy in digital holographic microscopy by using dual-wavelength technique,” J. Micro. Nanolithogr. MEMS MOEMS 14(4), 041313 (2015). [CrossRef]  

References

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  1. A. Safrani and I. Abdulhalim, “Real-time phase shift interference microscopy,” Opt. Lett. 39(17), 5220–5223 (2014).
    [Crossref] [PubMed]
  2. N. T. Shaked, “Quantitative phase microscopy of biological samples using a portable interferometer,” Opt. Lett. 37(11), 2016–2018 (2012).
    [Crossref] [PubMed]
  3. G. Popescu, Y. Park, W. Choi, R. R. Dasari, M. S. Feld, and K. Badizadegan, “Imaging red blood cell dynamics by quantitative phase microscopy,” Blood Cells Mol. Dis. 41(1), 10–16 (2008).
    [Crossref] [PubMed]
  4. Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004).
    [Crossref] [PubMed]
  5. J. Vargas, J. A. Quiroga, and T. Belenguer, “Analysis of the principal component algorithm in phase-shifting interferometry,” Opt. Lett. 36(12), 2215–2217 (2011).
    [Crossref] [PubMed]
  6. K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40(17), 2886–2894 (2001).
    [Crossref] [PubMed]
  7. P. S. Lam, J. D. Gaskill, and J. C. Wyant, “Two-wavelength holographic interferometer,” Appl. Opt. 23(18), 3079–3081 (1984).
    [Crossref] [PubMed]
  8. K. Creath, “Step height measurement using two-wavelength phase-shifting interferometry,” Appl. Opt. 26(14), 2810–2816 (1987).
    [Crossref] [PubMed]
  9. Y. Ishii and R. Onodera, “Two-wavelength laser-diode interferometry that uses phase-shifting techniques,” Opt. Lett. 16(19), 1523–1525 (1991).
    [Crossref] [PubMed]
  10. D. G. Abdelsalam and D. Kim, “Two-wavelength in-line phase-shifting interferometry based on polarizing separation for accurate surface profiling,” Appl. Opt. 50(33), 6153–6161 (2011).
    [Crossref] [PubMed]
  11. A. Pförtner and J. Schwider, “Red-green-blue interferometer for the metrology of discontinuous structures,” Appl. Opt. 42(4), 667–673 (2003).
    [Crossref] [PubMed]
  12. L. Fei, X. Lu, H. Wang, W. Zhang, J. Tian, and L. Zhong, “Single-wavelength phase retrieval method from simultaneous multi-wavelength in-line phase-shifting interferograms,” Opt. Express 22(25), 30910–30923 (2014).
    [Crossref] [PubMed]
  13. W. Zhang, X. Lu, L. Fei, H. Zhao, H. Wang, and L. Zhong, “Simultaneous phase-shifting dual-wavelength interferometry based on two-step demodulation algorithm,” Opt. Lett. 39(18), 5375–5378 (2014).
    [Crossref] [PubMed]
  14. W. Zhang, X. Lu, C. Luo, L. Zhong, and J. Vargas, “Principal component analysis based simultaneous dual-wavelength phase-shifting interferometry,” Opt. Commun. 341, 276–283 (2015).
    [Crossref]
  15. X. Qiu, L. Zhong, J. Xiong, Y. Zhou, J. Tian, D. Li, and X. Lu, “Phase retrieval based on temporal and spatial hybrid matching in simultaneous phase-shifting dual-wavelength interferometry,” Opt. Express 24(12), 12776–12787 (2016).
    [Crossref] [PubMed]
  16. R. Onodera and Y. Ishii, “Two-wavelength interferometry that uses a fourier-transform method,” Appl. Opt. 37(34), 7988–7994 (1998).
    [Crossref] [PubMed]
  17. D. G. Abdelsalam, R. Magnusson, and D. Kim, “Single-shot, dual-wavelength digital holography based on polarizing separation,” Appl. Opt. 50(19), 3360–3368 (2011).
    [Crossref] [PubMed]
  18. L. Huang, X. Lu, Y. Zhou, J. Tian, and L. Zhong, “Dual-wavelength interferometry based on the spatial carrier-frequency phase-shifting method,” Appl. Opt. 55(9), 2363–2369 (2016).
    [Crossref] [PubMed]
  19. J. Gass, A. Dakoff, and M. K. Kim, “Phase imaging without 2π ambiguity by multiwavelength digital holography,” Opt. Lett. 28(13), 1141–1143 (2003).
    [Crossref] [PubMed]
  20. A. Khmaladze, R. L. Matz, C. Zhang, T. Wang, M. M. Banaszak Holl, and Z. Chen, “Dual-wavelength linear regression phase unwrapping in three-dimensional microscopic images of cancer cells,” Opt. Lett. 36(6), 912–914 (2011).
    [Crossref] [PubMed]
  21. D. Parshall and M. K. Kim, “Digital holographic microscopy with dual-wavelength phase unwrapping,” Appl. Opt. 45(3), 451–459 (2006).
    [Crossref] [PubMed]
  22. C. J. Mann, P. R. Bingham, V. C. Paquit, and K. W. Tobin, “Quantitative phase imaging by three-wavelength digital holography,” Opt. Express 16(13), 9753–9764 (2008).
    [Crossref] [PubMed]
  23. J. Di, W. Qu, B. Wu, X. Chen, J. Zhao, and A. Asundi, “Dual wavelength digital holography for improving the measurement accuracy,” Proc. SPIE 8769(19), 9149–9427 (2013).
  24. J. Di, J. Zhang, T. Xi, C. Ma, and J. Zhao, “Improvement of measurement accuracy in digital holographic microscopy by using dual-wavelength technique,” J. Micro. Nanolithogr. MEMS MOEMS 14(4), 041313 (2015).
    [Crossref]

2016 (2)

2015 (2)

J. Di, J. Zhang, T. Xi, C. Ma, and J. Zhao, “Improvement of measurement accuracy in digital holographic microscopy by using dual-wavelength technique,” J. Micro. Nanolithogr. MEMS MOEMS 14(4), 041313 (2015).
[Crossref]

W. Zhang, X. Lu, C. Luo, L. Zhong, and J. Vargas, “Principal component analysis based simultaneous dual-wavelength phase-shifting interferometry,” Opt. Commun. 341, 276–283 (2015).
[Crossref]

2014 (3)

2013 (1)

J. Di, W. Qu, B. Wu, X. Chen, J. Zhao, and A. Asundi, “Dual wavelength digital holography for improving the measurement accuracy,” Proc. SPIE 8769(19), 9149–9427 (2013).

2012 (1)

2011 (4)

2008 (2)

C. J. Mann, P. R. Bingham, V. C. Paquit, and K. W. Tobin, “Quantitative phase imaging by three-wavelength digital holography,” Opt. Express 16(13), 9753–9764 (2008).
[Crossref] [PubMed]

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J. Di, W. Qu, B. Wu, X. Chen, J. Zhao, and A. Asundi, “Dual wavelength digital holography for improving the measurement accuracy,” Proc. SPIE 8769(19), 9149–9427 (2013).

Badizadegan, K.

G. Popescu, Y. Park, W. Choi, R. R. Dasari, M. S. Feld, and K. Badizadegan, “Imaging red blood cell dynamics by quantitative phase microscopy,” Blood Cells Mol. Dis. 41(1), 10–16 (2008).
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J. Di, W. Qu, B. Wu, X. Chen, J. Zhao, and A. Asundi, “Dual wavelength digital holography for improving the measurement accuracy,” Proc. SPIE 8769(19), 9149–9427 (2013).

Chen, Z.

Choi, W.

G. Popescu, Y. Park, W. Choi, R. R. Dasari, M. S. Feld, and K. Badizadegan, “Imaging red blood cell dynamics by quantitative phase microscopy,” Blood Cells Mol. Dis. 41(1), 10–16 (2008).
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G. Popescu, Y. Park, W. Choi, R. R. Dasari, M. S. Feld, and K. Badizadegan, “Imaging red blood cell dynamics by quantitative phase microscopy,” Blood Cells Mol. Dis. 41(1), 10–16 (2008).
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J. Di, J. Zhang, T. Xi, C. Ma, and J. Zhao, “Improvement of measurement accuracy in digital holographic microscopy by using dual-wavelength technique,” J. Micro. Nanolithogr. MEMS MOEMS 14(4), 041313 (2015).
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J. Di, W. Qu, B. Wu, X. Chen, J. Zhao, and A. Asundi, “Dual wavelength digital holography for improving the measurement accuracy,” Proc. SPIE 8769(19), 9149–9427 (2013).

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Feld, M. S.

G. Popescu, Y. Park, W. Choi, R. R. Dasari, M. S. Feld, and K. Badizadegan, “Imaging red blood cell dynamics by quantitative phase microscopy,” Blood Cells Mol. Dis. 41(1), 10–16 (2008).
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W. Zhang, X. Lu, C. Luo, L. Zhong, and J. Vargas, “Principal component analysis based simultaneous dual-wavelength phase-shifting interferometry,” Opt. Commun. 341, 276–283 (2015).
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J. Di, J. Zhang, T. Xi, C. Ma, and J. Zhao, “Improvement of measurement accuracy in digital holographic microscopy by using dual-wavelength technique,” J. Micro. Nanolithogr. MEMS MOEMS 14(4), 041313 (2015).
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G. Popescu, Y. Park, W. Choi, R. R. Dasari, M. S. Feld, and K. Badizadegan, “Imaging red blood cell dynamics by quantitative phase microscopy,” Blood Cells Mol. Dis. 41(1), 10–16 (2008).
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G. Popescu, Y. Park, W. Choi, R. R. Dasari, M. S. Feld, and K. Badizadegan, “Imaging red blood cell dynamics by quantitative phase microscopy,” Blood Cells Mol. Dis. 41(1), 10–16 (2008).
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Qu, W.

J. Di, W. Qu, B. Wu, X. Chen, J. Zhao, and A. Asundi, “Dual wavelength digital holography for improving the measurement accuracy,” Proc. SPIE 8769(19), 9149–9427 (2013).

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W. Zhang, X. Lu, C. Luo, L. Zhong, and J. Vargas, “Principal component analysis based simultaneous dual-wavelength phase-shifting interferometry,” Opt. Commun. 341, 276–283 (2015).
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J. Di, W. Qu, B. Wu, X. Chen, J. Zhao, and A. Asundi, “Dual wavelength digital holography for improving the measurement accuracy,” Proc. SPIE 8769(19), 9149–9427 (2013).

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Xi, T.

J. Di, J. Zhang, T. Xi, C. Ma, and J. Zhao, “Improvement of measurement accuracy in digital holographic microscopy by using dual-wavelength technique,” J. Micro. Nanolithogr. MEMS MOEMS 14(4), 041313 (2015).
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J. Di, J. Zhang, T. Xi, C. Ma, and J. Zhao, “Improvement of measurement accuracy in digital holographic microscopy by using dual-wavelength technique,” J. Micro. Nanolithogr. MEMS MOEMS 14(4), 041313 (2015).
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J. Di, J. Zhang, T. Xi, C. Ma, and J. Zhao, “Improvement of measurement accuracy in digital holographic microscopy by using dual-wavelength technique,” J. Micro. Nanolithogr. MEMS MOEMS 14(4), 041313 (2015).
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J. Di, W. Qu, B. Wu, X. Chen, J. Zhao, and A. Asundi, “Dual wavelength digital holography for improving the measurement accuracy,” Proc. SPIE 8769(19), 9149–9427 (2013).

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Blood Cells Mol. Dis. (1)

G. Popescu, Y. Park, W. Choi, R. R. Dasari, M. S. Feld, and K. Badizadegan, “Imaging red blood cell dynamics by quantitative phase microscopy,” Blood Cells Mol. Dis. 41(1), 10–16 (2008).
[Crossref] [PubMed]

J. Micro. Nanolithogr. MEMS MOEMS (1)

J. Di, J. Zhang, T. Xi, C. Ma, and J. Zhao, “Improvement of measurement accuracy in digital holographic microscopy by using dual-wavelength technique,” J. Micro. Nanolithogr. MEMS MOEMS 14(4), 041313 (2015).
[Crossref]

Opt. Commun. (1)

W. Zhang, X. Lu, C. Luo, L. Zhong, and J. Vargas, “Principal component analysis based simultaneous dual-wavelength phase-shifting interferometry,” Opt. Commun. 341, 276–283 (2015).
[Crossref]

Opt. Express (3)

Opt. Lett. (8)

Proc. SPIE (1)

J. Di, W. Qu, B. Wu, X. Chen, J. Zhao, and A. Asundi, “Dual wavelength digital holography for improving the measurement accuracy,” Proc. SPIE 8769(19), 9149–9427 (2013).

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Figures (8)

Fig. 1
Fig. 1 Schematic of immune algorithm of phase ambiguity, in which h sub and h λ 1 denote the heights of measured sample at Λ sub and λ 1 , respectively; h λ 1 ' is the corresponding burring height.
Fig. 2
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 sub , h λ1 and h add , respectively; (f) height map that achieved with the immune algorithm of phase ambiguity and φ add directly; (g) the difference maps between (c) and (d); (h) the difference maps between (e) and (d).
Fig. 3
Fig. 3 Height distributions of the 40 th row retrieved from Figs. 2(c)-2(e), respectively.
Fig. 4
Fig. 4 RMSEs of achieved height in different standard deviations of zero-mean Gaussian white noise.
Fig. 5
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.
Fig. 6
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.
Fig. 7
Fig. 7 (a) Height map of h sub 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 sub ; (c) height map of h add achieved through combining the immune algorithm of phase ambiguity and h λ 1 .
Fig. 8
Fig. 8 Height distribution curves of the 50 th row extracted from Fig. 7(a)-7(c), respectively.

Equations (19)

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h λ 1 (x,y)=[ n λ 1 (x,y)+ φ 1 (x,y) 2π ] λ 1 .
Λ sub = λ 1 λ 2 λ 1 λ 2 .
φ sub = φ 2 φ 1 .
h sub = φ sub Λ sub 2π .
h sub = h λ 1 +Δ.
r 1 =round( h sub λ 1 /2 λ 1 ).
h sub λ 1 /2= n λ 1 λ 1 + φ 1 2π λ 1 +Δ λ 1 /2.
λ 1 <Δ+ φ 1 2π λ 1 λ 1 /2< λ 1
r 1 ={ n λ 1 1 λ 1 /2<Δ+ φ 1 2π λ 1 <0 n λ 1 0<Δ+ φ 1 2π λ 1 < λ 1 n λ 1 +1 λ 1 <Δ+ φ 1 2π λ 1 <3 λ 1 /2 .
h λ 1 '=[ r 1 + φ 1 2π ] λ 1 .
D= h sub h λ 1 '={ Δ+ λ 1 r 1 = n λ 1 1 Δ r 1 = n λ 1 Δ λ 1 r 1 = n λ 1 +1 .
c 1 =round( D λ 1 )={ 1 r 1 = n λ 1 1 0 r 1 = n λ 1 1 r 1 = n λ 1 +1 .
h λ 1 =[ r 1 + c 1 + φ λ 1 (x,y) 2π ] λ 1 .
r 1 + c 1 ={ n λ 1 + m+1 2 (m=odd) n λ 1 + m+1 2 (m=odd) n λ 1 + m 2 (m=ever) n λ 1 + m 2 (m=ever) .
Λ add = λ 1 λ 2 λ 1 + λ 2 .
φ add = φ 1 + φ 2 .
h add (x,y)=[ n add (x,y)+ φ add (x,y) 2π ] Λ add .
φ add ={ φ add +2π φ add <0 φ add 2π φ add >2π .
h add =[ r add + c add + φ add (x,y) 2π ] Λ add .

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