Spatial dual-orthogonal (SDO) phase-shifting algorithm by pre-recomposing the interference fringe

Yi Wang; Bingbo Li; Liyun Zhong; Jindong Tian; Xiaoxu Lu

doi:10.1364/OE.25.017446

1. Introduction

Phase-shifting interferometry (PSI) is a powerful tool for high accuracy optical metrology [1], in which the phase distortion induced by the tested sample is encoded into a sequence of phase-shifting interferograms. To demodulate the encoded phase quantitatively and accurately, it is needed to choose a proper phase retrieval algorithm. Early algorithms can work only for the phase-shifting interferograms with known phase shifts [2] or the phase shifts are linearly distributed in the integer periods of $2 π$ , which makes them be similar in their time-frequency property [3, 4]. If the above conditions are not satisfied, the retrieved phase will suffer from an apparent error [5]. And this inspires the born of self–calibrating algorithm without the prior knowledge requirement of phase shifts [6–8].

Actually, we still seek for more accurate phase retrieval algorithm from the minimum number of phase-shifting interferograms. Although some self-calibrating algorithms can work with three or less phase-shifting interferograms, it is needed to set a reasonable assumption about the fringe patterns [6–13], meanwhile the assumption should be reliable and simple. Generally speaking, the reliability and simplicity of the assumption are respectively related with the accuracy and speed of phase retrieval. The slow variation rates of background/phase modulation amplitude have been assumed in the previous reports [6–9, 12], in which the advanced iterative algorithm (AIA) [9] is known as its high accuracy while its speed is low due to the iterative calculation. However, in practical application, it is impossible to ensure that the background/modulation amplitude are the absolute constants. Another assumptions, just like minimizing the phase-related correlation function are also proposed [10, 14]. In 2011, the principal components analysis (PCA) algorithm is introduced into PSI [11], which has been widely used and developed till now [15–23]. By utilizing the improved PCA algorithm, we can achieve the tested phase rapidly and accurately from three or more interferograms with arbitrary phase shifts while the original PCA algorithm requires both the spatial uniformity of encoded phase and temporal uniformity of phase shifts [21]. The corresponding error analysis indicates that in PCA algorithm, the temporal distribution uniformity of phase shifts plays an especially important role [22]. Till now, the temporal requirement can be cancelled by using the spatial statistical property of interferograms [19] when the fringes number is large, and the spatial requirement also can be cancelled by combining PCA algorithm with other methods [23], or using the temporal property of phase shifts when the number of interferograms is large [24]. When the number of interferograms is not sufficiently large, it is needed to choose the former strategy based on the global spatial uniformity of encoded phase. The nonuniform phase distribution usually appears in the objective or lens matching testing system for aberration phase elimination [25]. In this system, the fringe number in interferogram is greatly reduced, as a result, the spatial bandwidth of CCD can be used effectively. In this situation, even if the phase distribution of fringe pattern is uniform, it is still difficult to achieve accurate result with PCA algorithm. The removal of quadratic aberration induced phase will lead to both the reduction of fringes number and the extreme unbalanced weight factors of different phase values, thus the assumption of PCA algorithm will become invalid. Except this, in the case that a partially coherent source is used, the fringe number is also limited to satisfy the requirement of PCA algorithm [26, 27].

In this manuscript, we propose a spatial dual-orthogonal (SDO) phase-shifting algorithm. First, we recompose the spatial phase distribution to construct a new sequence of fringe patterns. Second, using these new-constructed fringe patterns, the optimum combination coefficients (CCs) corresponding to each original interferograms can be determined. Finally, by combining the CCs and the original interferograms, we can achieve the modulated cosine and sine terms of tested phase, and then the accurate phase can be retrieved. Following, we will introduce the proposed method in details.

2. Principle

In PSI, the intensity of the nth-frame phase-shifting interferogram can be described as

I_{n} (x, y) = A (x, y) + B (x, y) \cos [ϕ (x, y) + θ_{n}]

where x and y denote the transverse and longitude coordinates, respectively; A(x, y) and B(x, y) represent the background intensity and modulation amplitude, respectively;

ϕ (x, y)

is the tested phase;

θ_{n}

represents the phase shifts and

θ_{0} = 0

,

n = 1, 2, 3 \dots N

are the index numbers of phase-shifting interferograms.

The spatial approximation of PCA algorithm can be described as [11]

\sum_{x, y} B^{2} (x, y) \cos^{2} ϕ (x, y) \approx \sum_{x, y} B^{2} (x, y) \sin^{2} ϕ (x, y)

\sum_{x, y} B^{2} (x, y) \cos ϕ (x, y) \sin ϕ (x, y) < < \sum_{x, y} B^{2} (x, y) \sin^{2} ϕ (x, y)

In this study, we name Eqs. (2) as the second order orthogonal property of interferogram. If the phase distribution of interferogram satisfy Eqs. (2), the interferograms are considered as uniform. In the case that fringe number in interferogram is only one, Eqs. (2) will probably deviate from the practical situation. However, if the fringe number in interferogram is large enough, the deviations in every fringe periods can be mutually compensated, and then the quasi-uniform global property of interferogram can be satisfied. That is to say, the increasing fringes number is helpful to the satisfaction of Eqs. (2). But if the nonuniformity of interferograms is too large to be compensated by the fringes number, the orthogonal condition will become invalid. In this situation, it is needed to reconstruct a new sequence of fringe patterns with good uniformity.

Accordingly, a guideline for the recomposing of original phase-shifting interferograms is needed. First, we set a reasonable hypothesis as following: If B(x, y) and $ϕ (x, y)$ can satisfy following condition

\sum_{x, y} B (x, y) \cos ϕ (x, y) = 0

\sum_{x, y} B (x, y) \sin ϕ (x, y) = 0

Then, they also satisfy Eqs. (2), so we define the condition in Eqs. (3) as the first order orthogonal property of interferograms. Actually, Eqs. (2) and Eqs. (3) are physically equivalent, revealing two different descriptions about the uniformity of fringe distribution. That is to say, Eqs. (2) and Eqs. (3) should be satisfied at the same time. This implies that we can make the interferograms consistent with Eqs. (2) by forcing them to satisfy Eqs. (3).

If a high-pass filter $H (f_{x})$ is utilized to modulate the spatial spectrum $I_{n} (f_{x})$ of $I_{n} (x)$ , the corresponding result ${\tilde{I}}_{n} (f_{x})$ represents the spatial spectrum of a new background-free interferogram $B' (x) \cos [ϕ' (x) + δ_{n}]$ , in which the phase distribution $ϕ' (x)$ denotes the spatial recomposing of $ϕ (x)$ . For convenience, here, we neglect the y component, and the corresponding result can be expressed as

\begin{array}{l} {\tilde{I}}_{n} (f_{x}) = H (f_{x}) I_{n} (f_{x}) \\ = H (f_{x}) (A (f_{x}) + ξ {B (x) \cos [ϕ (x) + δ_{n}]}) \\ = H (f_{x}) ξ {B (x) \cos [ϕ (x) + δ_{n}]} \end{array}

where

ξ

denotes the Fourier transform operator. It is observed that A(x) has been eliminated but the signal term

B (x) \cos [ϕ (x) + δ_{n}]

is modulated. Then, by performing the inverse Fourier transform

ξ^{- 1}

for Eq. (4), we have that

H (x) = ξ^{- 1} {H (f_{x})}

and

\begin{array}{l} {\tilde{I}}_{n} (x) = H (x) \otimes B (x) \cos [ϕ (x) + δ_{n}] = \sum_{τ} H (τ) B (x - τ) \cos [ϕ (x - τ) + δ_{n}] \\ = \sum_{τ} H (τ) B (x - τ) \cos [ϕ (x) + f (x, τ) + δ_{n}] \\ = \cos [ϕ (x) + δ_{n}] \sum_{τ} H (τ) B (x - τ) \cos f (x, τ) - \sin [ϕ (x) + δ_{n}] \sum_{τ} H (τ) B (x - τ) \sin f (x, τ) \\ = \sqrt{T (x)} \cos [ϕ (x) + δ_{n} + η (x)] = \sqrt{T (x)} \cos [ϕ' (x) + δ_{n}] \end{array}

in which

f (x, τ) = ϕ (x - τ) - ϕ (x)

T (x) = {[\sum_{τ} H (τ) B (x - τ) \sin f (x, τ)]}^{2} + {[\sum_{τ} H (τ) B (x - τ) \cos f (x, τ)]}^{2}

η (x) = arc \cos (\frac{\sum_{τ} H (τ) B (x - τ) \cos f (x, τ)}{\sqrt{T (x)}}) .

ϕ' (x) = ϕ (x) + η (x) .

Equations (6) mathematically present the phase recomposing procedure, in which

ϕ (x)

is changed into

ϕ' (x)

while the phase shifts

δ_{n}

remains unchanged. Thus, we have that

{\tilde{I}}_{n} {(f_{x})}_{f_{x} = 0} = \sum_{x, y} \sqrt{T (x, y)} \cos [ϕ' (x, y) + δ_{n}] = 0, n = 1...... N .

Clearly, Eq. (7) can guarantee that all

{\tilde{I}}_{n} (x)

can satisfy Eqs. (3). According to the physical equivalence relationship of Eqs. (3) and Eqs. (2) we assumed,

{\tilde{I}}_{n} (x)

will satisfy Eqs. (2) better than

I_{n} (x)

. From this result, it is presented that the second order orthogonal property of constructed interference pattern can be changed by adjusting the first order orthogonal property of original interferogram, so we name the proposed algorithm as the spatial dual-orthogonal (SDO) method.

Because the phase distribution $ϕ' (x)$ of ${\tilde{I}}_{n} (x)$ is different from the original phase $ϕ (x)$ of $I_{n} (x)$ while the phase shifts $δ_{n}$ remains unchanged, so we will not directly use ${\tilde{I}}_{n} (x)$ to perform phase retrieval but to determine the corresponding CCs. Actually, this deviation is also the main error source of 2-step PSI algorithm [13, 15, 28], in which the background term A(x) is removed by a spatial convolution operation. Therefore, Eqs. (5) also quantitatively present the filter-induced phase distortion in 2-step PSI algorithm. What’s more, they also proves that $δ_{n}$ will not change after the spatial convolution operation. This is a hypothesis in 2-step PSI algorithms but has not been strictly proved till now.

Finally, we construct a matrix A with N × N elements as

{\begin{cases} A_{m n} = \sum_{x} {\tilde{I}}_{m} (x) {\tilde{I}}_{n} (x) \\ m = 1... N, n = 1... N \end{cases}

By performing the singular values decomposition (SVD) operation, we can achieve the first eigenvalue

λ_{1}

and eigenvector

P_{1}

, as well as the second eigenvalue

λ_{2}

and eigenvector

P_{2}

, respectively. The meaning of SVD procedure can be described as following

A = I_{p} \cdot I_{p}^{T} .

P A P^{T} = P I_{p} I_{p}^{T} P^{T} = P I_{p} {(P I_{p})}^{T} = O .

in which

O = (\begin{matrix} λ_{1} & 0 & 0 & 0 \\ 0 & λ_{2} & 0 & 0 \\ 0 & 0 & ...... & 0 \\ 0 & 0 & 0 & λ_{N} \end{matrix}) .

I_{p} = (\begin{matrix} I_{1} \\ I_{2} \\ ...... \\ I_{N} \end{matrix}) .

P = (\begin{matrix} P_{1} (1) ...... P_{1} (N) \\ P_{2} (2) ...... P_{2} (N) \\ ...... \\ P_{N} (2) ...... P_{N} (N) \end{matrix}) .

{[\sum_{n}^{N} P_{1} (n) I_{n} (x)]}^{2} = C_{1} \sum_{x, y} {[\sqrt{T (x, y)} \cos ϕ (x, y)]}^{2} = λ_{1} .

{[\sum_{n}^{N} P_{2} (n) I_{n} (x)]}^{2} = C_{2} \sum_{x, y} {[\sqrt{T (x, y)} \sin ϕ (x, y)]}^{2} = λ_{2} .

Here,

C_{1}

and

λ_{1}

,

C_{2}

and

λ_{2}

may not be the same value due to the phase shifts

δ_{n}

may not linearly distributed in the integral periods of

2 π

. Due to

\sum_{x, y} {[\sqrt{T (x, y)} \cos ϕ (x, y)]}^{2} \approx \sum_{x, y} {[\sqrt{T (x, y)} \sin ϕ (x, y)]}^{2} .

because the

{\tilde{I}}_{n}

satisfy Eqs. (2). Then, we have that

{\tilde{P}}_{1} = \frac{P_{1}}{\sqrt{λ_{1}}}, {\tilde{P}}_{2} = \frac{P_{2}}{\sqrt{λ_{2}}} .

where

{\tilde{P}}_{1}

and

{\tilde{P}}_{2}

denote one-dimension vectors with length of N, and their elements are corresponding to the CCs of N-frame constructed interference patterns, respectively. The role of Eq. (12) is to eliminate the effect of the difference between

C_{1}

and

C_{2}

. After that,

B (x) \cos ϕ (x)

and

B (x) \sin ϕ (x)

can be achieved by combining the original interferograms

I_{n} (x)

with their corresponding CCs

U_{1} (x) = B (x) \cos ϕ (x) = \sum_{n}^{N} {\tilde{P}}_{1} (n) I_{n} (x) .

U_{2} (x) = B (x) \sin ϕ (x) = \sum_{m}^{N} {\tilde{P}}_{2} (m) I_{m} (x) .

ϕ (x) = arc \tan (\frac{U_{2} (x)}{U_{1} (x)}) .

3. Numerical simulation

Following, we will quantitatively discuss the property of proposed SDO algorithm by simulation. A Gaussian phase distribution with quadratic aberration is chosen as the first tested model and a sloping surface is employed the second one, in which the function for constructing the first model is set as $ϕ_{1} = 15exp[-0 .25(x^{2} + y^{2})]+0 .7 π (x^{2} + y^{2})$ with pixel size of 10μm × 10μm. In the second model, the slope part occupies half of the interferogram with the phase gradient of 0.06rad/pixels and the other part is plane. Three-frame phase-shifting interferograms with the phase shifts of 0rad, 0.4rad and 0.8rad and the size of 500 × 500 pixels are generated. Random noise of 45db is added to the interferogram. The background and modulation amplitude are set as A(x) = 50exp[-0.15(x² + y²)] + 50 and B(x) = 50exp[-0.15(x² + y²)], respectively. The high-pass filter is $H (f_{x}, f_{y}) = 1 - \exp [- \frac{f_{x}^{2} + f_{y}^{2}}{2 σ^{2}}], σ = 3$ .To quantitatively describe the “uniformity” of fringe patterns, we define two parameters as

T_{1} = \frac{\sum_{x, y} B {(x, y)}^{2} \cos^{2} ϕ (x, y)}{\sum_{x, y} B {(x, y)}^{2} \sin^{2} ϕ (x, y)} .

T_{2} = \frac{\sum_{x, y} B {(x, y)}^{2} \cos ϕ (x, y) \sin ϕ (x, y)}{\sum_{x, y} B {(x, y)}^{2}} .

In the ideal situation,

T_{1} = 1, T_{2} = 0

, so it is not needed to perform the SDO phase re-composing procedure. However, in most cases, the above condition cannot be satisfied, for example, as we will see in Fig. 1, even if the fringe number in interferogram is not small, the interferograms are still nonuniform. In these situation, we need to use the SDO algorithm to adjust the values of

T_{1}, T_{2}

, and the corresponding procedure can be described as

K_{1} = H (x, y) \otimes B (x, y) \cos ϕ (x, y) .

K_{2} = H (x, y) \otimes B (x, y) \sin ϕ (x, y) .

T_{1}' = \frac{\sum_{x, y} K_{1}^{2}}{\sum_{x, y} K_{2}^{2}} .

T_{2}' = \frac{\sum_{x, y} K_{1} \cdot K_{2}}{\sum_{x, y} K_{1}^{2} + K_{2}^{2}} .

After the SDO processing, we will have that

T_{1}' \approx 1, T_{2}' \approx 0

, then the uniformity of these fringe patterns will be improved. Figure 1 shows the results achieved with the PCA, AIA and SDO algorithms, respectively.

Fig. 1 Simulation results of a Gauss phase distribution sample with quadratic background and a sloping surface. (a1)(a2) three-frame simulated phase-shifting interference patterns; (b1)(b2) theoretical phase of (a1) and (a2), respectively ; the phases achieved with different algorithms (c1)(c2) PCA; (d1)(d2) SDO; (e1)(e2) AIA.

Download Full Size | PDF

It is observed that the apparent error appears in the results achieved with the AIA and PCA algorithms. This is because the fringes number are too small to compensate the unbalanced weight factors of different phase values induced by the wide-range of plane phase and the nonconstant background/modulation amplitude, thus the hypothesis in both AIA and PCA algorithms will no longer be valid. Clearly, these simulation results are greatly accordant with the above theoretical analysis.

As we know, there are two significant disadvantages in the phase-shifting algorithms: (1) the accuracy is changed with the SNR (signal to noise ratio); (2) the accuracy is sensitive to the variation of fringe pattern. Following, based on the AIA, PCA and SDO algorithms, we perform the comparison of these two disadvantages, as shown in Fig. 2. For the circular phase distribution sample without quadratic background component, the SNR is changed from 30db to 50db; for the $K \cdot p e a k s (500)$ function, the K value is changed from 0.3 to 3.3, in which the phase shifts of three-frame interference patterns are set as 0rad, 1.4rad and 2.6rad, respectively.

Fig. 2 Phase distributions achieved with the proposed SDO algorithm in different SNR (a) 30db; (c) 50db; (b) variation curves of RMSE with the SNR; the achieved fringe patterns in different K values (d) 0.3; (f) 3.3 ; (e) variation curves of RMSE with the K value.

Download Full Size | PDF

From Fig. 2, we can see that the proposed SDO algorithm still remains high accuracy in different noise levels, moreover, the accuracy is insensitive to the variation of fringe pattern, indicating the obvious advantage of proposed SDO algorithm in accuracy stability of phase demodulation. In addition, in the proposed SDO algorithm, a high-pass filter $H (f_{x}, f_{y})$ is employed to ensure the satisfaction of Eqs. (3), and then Eqs. (2) also can be satisfied. Apparently, the choice of filtering window will affect the accuracy of proposed SDO algorithm. In this simulation, we choose $H (f_{x}, f_{y}) = 1 - \exp (\frac{- f_{x}^{2} - f_{y}^{2}}{2 σ^{2}}), σ = 3$ , in which a small part of spatial spectrum near zero frequency is eliminated, but not only zero frequency itself. This is due to considering of slight spectrum extension of $B (x, y)$ , which will make the original zero frequency components of $\cos ϕ (x, y)$ and $\sin ϕ (x, y)$ have the spectrum extension by multiplying $B (x, y)$ , i.e. $B (x, y) \cos ϕ (x, y)$ and $B (x, y) \sin ϕ (x, y)$ . Next, we will present the influence of window filtering function $H (f_{x}, f_{y})$ on the accuracy, as shown in Fig. 3, in which the phase model is generated by Matlab function $p e a k s (500)$ , and the size of pixels is set as 0.01mm, A(x) = 50exp[-0.05(x² + y²)] + 50, B(x) = 50exp[-0.05(x² + y²)]; the phase shifts is 0rad, 0.8rad, 1.5rad, respectively. The SNR of fringe pattern is set as 55db.

Fig. 3 RMSE variation curve of SDO algorithm with the value of filtering window $σ$ .

Download Full Size | PDF

From Fig. 3, we can see that the accuracy of SDO algorithm is changed in the range of 0.023rad to 0.055rad while the $σ$ is changed from 0.2 to 6. In the case that $σ = 0.2$ , the effect of $H (f_{x}, f_{y})$ is very close to only eliminating zero frequency point, and the corresponding RMSE is 0.055rad, which is smaller than 0.068rad of PCA algorithm and 0.471rad of AIA algorithm, but larger than 0.023rad in the case that $σ = 1.4$ . From these results, we can find that so long as in all the reasonable range of $σ$ , the SDO processing can greatly improve the accuracy of phase retrieval relative to the PCA and AIA algorithms.

4. Experimental result

Next, the experimental research is carried to verify the advantage of proposed SDO algorithm. A He-Ne stabilized frequency laser with the wavelength of 632.8nm is employed as the light source and the pixel size of CCD camera is 10μm × 10μm. A sequence of phase-shifting interferograms are captured, in which the quadratic background phase is eliminated through the objective matching in the reference light, and 299-frame phase-shifting interferograms are captured to achieve the reference phase by using Fourier Transform algorithm [29], which has high accuracy when a large number of phase-shifting interferograms are utilized and the phase-shifts are quasi-linear distribution. To address this, in our experiment, a sequence of phase-shifting interferograms are captured only after the driving voltage of PZT-based phase-shifting system reaches to 70V, and the total phase shifts of 299-frame interferograms is about 63rad. And three-frame phase-shifting interferograms with arbitrary phase shifts are chosen to perform phase retrieval with the AIA, PCA and SDO algorithms, respectively. The achieved results are shown in Fig. 4 and Table. 1.

Fig. 4 (a) (b) One-frame interferogram with size of 1300 × 1300 pixels and the intercepted area marked with red square and size of 480 × 520 pixels, respectively; (c) the reference phase distribution achieved with Fourier Transform algorithm; the error distributions achieved with different algorithms (d) PCA; (e) AIA (f) SDO; (g) variation curves (the 250th row) of the phase difference between the reference phase and the phases achieved with different algorithms.

Download Full Size | PDF

Table 1. PVE, RMSE and Processing time of phase retrieval achieved with different algorithms

View Table

Like the simulation result, we can see that even if a few phase-shifting interferograms with nonuniform phase distribution, nonconstant background/modulation amplitude and arbitrary phase shifts are used, the RMSE, PVE and processing time of phase retrieval achieved with the proposed SDO algorithm reveal obvious advantage compared to the AIA, PCA algorithms. These results further demonstrate that the proposed SDO algorithm should be a better candidate for phase retrieval with high accuracy and rapid speed.

5. Summary

In this study, from three or more phase-shifting interferograms with unknown phase-shifts, we propose an accurate and rapid SDO algorithm to perform phase retrieval, in which a new sequence of interference patterns are constructed by recomposing the spatial phase distribution of original interferograms to determine their corresponding optimum CCs, which are closely related with the phase shifts. The achieved results present that the proposed SDO algorithm reveals obvious accuracy advantage relative to current self-calibration algorithms. Specially, it is found that in the proposed SDO algorithm, the accuracy of phase retrieval is insensitive to the variation of fringe pattern, and this will supply a guarantee for high accuracy phase measurement and then facilitate its application, especially in the case that the quadratic phase is eliminated or the fringe number in interferogram is less.

Funding

This work is supported by National Natural Science Foundation of China grants (61475048, 61575069 and 61275015).

References and links

1. M. Servín, J. A. Quiroga, and J. M. Padilla, Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications (Wiley-VCH, 2014).

2. C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7(8), 368–370 (1982). [CrossRef] [PubMed]

3. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. 7(4), 542–551 (1990). [CrossRef]

4. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009). [CrossRef] [PubMed]

5. K. Creath, “V Phase-Measurement Interferometry Techniques,” Prog. Opt. 26, 349–393 (1988). [CrossRef]

6. X. F. Xu, L. Z. Cai, X. F. Meng, G. Y. Dong, and X. X. Shen, “Fast blind extraction of arbitrary unknown phase shifts by an iterative tangent approach in generalized phase-shifting interferometry,” Opt. Lett. 31(13), 1966–1968 (2006). [CrossRef] [PubMed]

7. P. Gao, B. Yao, N. Lindlein, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett. 34(22), 3553–3555 (2009). [CrossRef] [PubMed]

8. C.-S. Guo, B. Sha, Y.-Y. Xie, and X.-J. Zhang, “Zero difference algorithm for phase shift extraction in blind phase-shifting holography,” Opt. Lett. 39(4), 813–816 (2014). [CrossRef] [PubMed]

9. 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]

10. H. Guo, “Blind self-calibrating algorithm for phase-shifting interferometry by use of cross-bispectrum,” Opt. Express 19(8), 7807–7815 (2011). [CrossRef] [PubMed]

11. J. Vargas, J. A. Quiroga, and T. Belenguer, “Phase-shifting interferometry based on principal component analysis,” Opt. Lett. 36(8), 1326–1328 (2011). [CrossRef] [PubMed]

12. F. Liu, Y. Wu, and F. Wu, “Correction of phase extraction error in phase-shifting interferometry based on Lissajous figure and ellipse fitting technology,” Opt. Express 23(8), 10794–10807 (2015). [CrossRef] [PubMed]

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

14. H. Guo, Y. Yu, and M. Chen, “Blind phase shift estimation in phase-shifting interferometry,” J. Opt. Soc. Am. A 24(1), 25–33 (2007). [CrossRef] [PubMed]

15. 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] [PubMed]

16. Y. Du, G. Feng, H. Li, J. Vargas, and S. Zhou, “Spatial carrier phase-shifting algorithm based on principal component analysis method,” Opt. Express 20(15), 16471–16479 (2012). [CrossRef]

17. H. Wang, C. Luo, L. Zhong, S. Ma, and X. Lu, “Phase retrieval approach based on the normalized difference maps induced by three interferograms with unknown phase shifts,” Opt. Express 22(5), 5147–5154 (2014). [CrossRef] [PubMed]

18. J. Xu, W. Jin, L. Chai, and Q. Xu, “Phase extraction from randomly phase-shifted interferograms by combining principal component analysis and least squares method,” Opt. Express 19(21), 20483–20492 (2011). [CrossRef] [PubMed]

19. J. Deng, K. Wang, D. Wu, X. Lv, C. Li, J. Hao, J. Qin, and W. Chen, “Advanced principal component analysis method for phase reconstruction,” Opt. Express 23(9), 12222–12231 (2015). [CrossRef] [PubMed]

20. K. Yatabe, K. Ishikawa, and Y. Oikawa, “Improving principal component analysis based phase extraction method for phase-shifting interferometry by integrating spatial information,” Opt. Express 24(20), 22881–22891 (2016). [CrossRef] [PubMed]

21. 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]

22. J. Vargas, J. M. Carazo, and C. O. S. Sorzano, “Error analysis of the principal component analysis demodulation algorithm,” Appl. Pys. B B 115(3), 355–364 (2014). [CrossRef]

23. J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the Principal Component Analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013). [CrossRef]

24. J. Vargas and C. O. S. Sorzano, “Quadrature Component Analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013). [CrossRef]

25. Q. Weijuan, Y. Yingjie, C. O. Choo, and A. Asundi, “Digital holographic microscopy with physical phase compensation,” Opt. Lett. 34(8), 1276–1278 (2009). [CrossRef] [PubMed]

26. M. Roy, J. Schmit, and P. Hariharan, “White-light interference microscopy: minimization of spurious diffraction effects by geometric phase-shifting,” Opt. Express 17(6), 4495–4499 (2009). [CrossRef] [PubMed]

27. Z. Wang, L. Millet, M. Mir, H. Ding, S. Unarunotai, J. Rogers, M. U. Gillette, and G. Popescu, “Spatial light interference microscopy (SLIM),” Opt. Express 19(2), 1016–1026 (2011). [CrossRef] [PubMed]

28. J. C. Estrada, M. Servin, and J. A. Quiroga, “A self-tuning phase-shifting algorithm for interferometry,” Opt. Express 18(3), 2632–2638 (2010). [CrossRef] [PubMed]

29. Y. Kikuchi, D. Barada, T. Kiire, and T. Yatagai, “Doppler phase-shifting digital holography and its application to surface shape measurement,” Opt. Lett. 35(10), 1548–1550 (2010). [CrossRef] [PubMed]

Algorithms	AIA	PCA	SDO
PVE(rad)	0.869	1.139	0.822
RMSE(rad)	0.067	0.201	0.046
Time(s)	1.980	0.039	0.052

Spatial dual-orthogonal (SDO) phase-shifting algorithm by pre-recomposing the interference fringe

Abstract

1. Introduction

2. Principle

3. Numerical simulation

4. Experimental result

5. Summary

Funding

References and links

Cited By

Figures (4)

Tables (1)

Equations (32)

Optics Express