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Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm
Optics and Lasers in Engineering 84 (2016) 89–95
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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm Yuanyuan Xu a, Yawei Wang a,b,n, Ying Ji b, Hao Han b, Weifeng Jin a a b
School of Mechanical Engineering, Jiangsu University, Zhenjiang 212013, China Faculty of Science, Jiangsu University, Zhenjiang 212013, China
art ic l e i nf o
a b s t r a c t
Article history: Received 31 October 2015 Received in revised form 8 April 2016 Accepted 11 April 2016 Available online 19 April 2016
Generalized phase-shifting interferometry (GPSI) is one of the most effective techniques in imaging of a phase object, in which phase retrieval is an essential and important procedure. In this paper, a simple and rapid algorithm for retrieval of the unknown phase shifts in three-frame GPSI is proposed. Using this algorithm, the value of phase shift can be calculated by a determinate formula consisting of three different Euclidean matrix norms of the intensity difference between two phase shifted interferograms, and then the phase can be retrieved easily. The algorithm has the advantages of freeing from the background elimination and less computation, since it only needs three phase-shifted interferograms without no extra measurements, the iterative procedure or the integral transformation. The reliability and accuracy of this algorithm were demonstrated by simulation and experimental results. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Phase-shifting interferometry Phase retrieval Euclidean matrix norm
1. Introduction Phase-shifting interferometry (PSI) is an important technique widely used in optical measurement and microscopy [1–4]. In traditional PSI, a special constant phase shift, 2π /N with the integer N ≥ 3 between two adjacent frames, is often assumed. Subsequently, Greivenkamp [5] and Stoilov [6] proposed general methods to deal with arbitrary phase shifts. Although the phase shifts do not have to be special values, they are either known precisely or equal in these methods. In fact, it is still a strict requirement on the precision of the phase shifter. In order to remove this requirement, many methods have been reported [7–18], among which the generalized phase shift exaction techniques [10– 18] provide a possible method for extracting arbitrary and unknown phase shifts from holograms directly. In general, the phase shift extraction methods can be classified into two categories: iterative and noniterative. The iterative methods [10,11] are greatly time-consuming since the procedures are repeated many times to achieve acceptable accuracy. For example, an advanced iterative algorithm based on the least squares method is proposed to determine the phase shift and the phase distribution simultaneously [10]. For this reason, noniterative methods [12–21] have been favored for their speed and less computational loads. Cai and other researchers proposed a class of n Corresponding author at: Faculty of Science, Jiangsu University, Zhenjiang 212013, China. E-mail address: [email protected] (Y. Wang).
http://dx.doi.org/10.1016/j.optlaseng.2016.04.011 0143-8166/& 2016 Elsevier Ltd. All rights reserved.
statistical algorithms to calculate the unknown phase shift directly and reconstruct the wavefront from two or more holograms [12– 15]. These algorithms are based on the assumption that the measured object has a random phase in [0, 2π ] over the whole interferogram. To relax this assumption, another method has been proposed to extract the phase shift accurately using the histogram of phase difference between two adjacent frames [16]. However, it requires determining both the background intensity and the modulation amplitude by searching for the maximum and the minimum intensity in each pixel of interferograms, which still takes considerable time. Later, a self-tuning approach is proposed to retrieve the phase shift by looking for the minimum of a merit function [17], where the accuracy of phase shift decreases when the phase shift is far from π/2 and the interferograms are required to be normalized beforehand. In Ref. [18], an accurate phase shift extraction algorithm is proposed by using the maximum and the minimum values of the interference term. In Ref. [19], the GramSchmidt (GS) orthonormalization algorithm is employed to extract the phase with high precision and rapid speed. Both of these twostep demodulation methods need a precondition of filtering out the background term by a high-pass filter in advance. In Refs. [20,21], a new kind of phase-shifting demodulation method based on the use of principal component analysis (PCA) algorithm is proposed. It is worth noting that this method does not require the extraction of the phase shift to retrieve the modulating phase. Although the PCA method is fast, it still requires the elimination of the background term by a temporal average in advance. However, the filtering and averaging algorithms do not work well for the background elimination when the interferograms are with rapid
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background variation [22]. New methods that are robust to background vibration have been also proposed [23–25]. In Ref. [25], a combination of the GS orthonormalization process and the twodimensional continuous wavelet transform (2D-CWT) algorithm is used to analyze two-step arbitrarily phase-shifted interferograms. The method works well when the interferograms contain complex fringes, large fringe-frequency variations, noise or defect fringes. It is still time-consuming because the 2D-CWT algorithm refers to several times Fourier transforms. In this paper, we propose a Euclidean matrix norm (EMN) algorithm to extract the unknown phase shifts from only three interferograms. After substituting the phase shifts to the three-step phase-shifting algorithm, the phase can be retrieved easily. This algorithm is faster since it does not use the iterative procedure or the integral transformation. Moreover, it is easy to implement without the background elimination or the measurements of other parameters excepting the interferograms in the entire retrieval process. The only requirement is the fringe limit condition, which is easy to meet in real cases.
Epq ≈
⎧ M N ⎫1/2 ⎛ δq − δ p ⎞ ⎪ ⎪ ⎛ δq − δ p ⎞ 2 ⎬ ⎨ ∑ ∑ bmn ⎟ ⎪ ⎟ = C × sin ⎜ 2 sin ⎜ ⎪ ⎝ 2 ⎠ ⎩ ⎝ 2 ⎠ ⎭ m= 1 n= 1
with C =
M N 2 1/2 2 ⋅{∑m = 1 ∑n = 1 bmn } .
(5)
From Eq. (5), it is clear that Epq is
proportional to sin [(δq − δp ) /2] , and there are only three unknown quantities, namely δ2, δ3 and C . To determine these quantities, at least three equations are required. According to Eq. (5), there are three quantities as follows:
E12 = C × sin ( δ2/2) , E13 = C × sin ( δ3/2) , E23 = C × sin ⎡⎣ ( δ3 − δ2 )/2⎤⎦
(6)
In order to avoid the uncertainty of the sign, the phase shift δ is normally constrained within the range of [0, π ]. Therefore, δ/2 ranges from 0 to π/2, and the function of sin (δ /2) is monotonically sin (δ2/2) = sin (δ2/2) and increasing and positive. So sin (δ3/2) = sin (δ3/2). The sign of sin [(δ3 − δ2 ) /2] can be determined by comparing the values of E12 and E13. For example, if E13 > E12, it can be obtained that sin (δ3/2) > sin (δ2/2)andδ3/2 > δ2/2, and then sin [(δ3 − δ2 )/2] = sin [(δ3 − δ2 ) /2]. Thus, Eq. (6) can be rewritten as
E12 = C × sin ( δ2/2), E13 = C × sin ( δ3/2), E23
2. Method
= C × sin ⎡⎣ ( δ3 − δ2 )/2⎤⎦
(7)
For the three-frame generalized phase-shifting interferometry, the distribution of intensity for each interferogram can be given in the following form:
Since E12 , E13 and E23 can be determined, we get the following relationship which relates E12 etc., with the parameter C ,
Ikmn = amn + bmn cos [φmn + δk ],
E13 = E12 ×
(k = 1, 2, 3)
(1)
where m and n denote the pixel position of rows and columns of interferograms respectively. amn , bmn and φmn represent the background intensity, the modulation amplitude and the measured phase, respectively. The phase shift related to the kth interferogram, δk , is usually assumed to be zero when k ¼1. With the measured intensities, the difference between the pth and qth interferograms can be expressed as
(2)
Here, we consider the Euclidean matrix norm (EMN) of the intensity difference. In general, for a matrix T = [tmn ], with M × N M N [∑m = 1 ∑n = 1 (tmn )2]1/2,
order, its EMN is defined as ‖T‖2 = in which the sign ‖‖2 is the EMN operator. Therefore, the EMN of ΔIpq can be expressed as
Epq = ΔIpq
2
M N ⎡ ⎧ ⎫1/2 ⎪ ⎛ δ + δ ⎞ ⎛ δ − δ ⎞ ⎤2 ⎪ =⎨ ∑ ∑ ⎢ 2bmn sin ⎜⎝ ϕmn + q p ⎟⎠ sin ⎜⎝ q p ⎟⎠ ⎥ ⎪⎬ ⎪ ⎦ ⎭ 2 2 ⎩ m= 1 n= 1 ⎣
⎛ δq − δ p ⎞ ⎟ 2 sin ⎜ ⎝ 2 ⎠
=
⎧ M N ⎫1/2 2 ⎡ ⎤⎦ ⎪ ⎨ ⎬ 1 cos 2 − ϕ + δ + δ b ∑ ∑ ( ) ⎣ q p mn mn ⎪ ⎪ ⎩ m= 1 n= 1 ⎭ ⎪
(3)
If the fringe number in each interferogram is more than one, then the measured phase varies more than 2π (rad) in the observed area. As a result, the following approximation can be applied, M
N
M
N
2 2 > > ∑ ∑ bmn cos ( 2φmn + δq + δ p ) ∑ ∑ bmn m= 1 n= 1
m= 1 n= 1
Thus, Eq. (3) can be simplified as
2
1 − ( E12/C )
(8)
After solving for C , δ2 and δ3 can be directly calculated from E12 and E13. Once the phase shifts δk (k = 2, 3) are known, the wrapped phase φ can be solved with following expression:
φ = arctan
I3 − I2 + (I1 − I3 ) cos δ2 + (I2 − I1) cos δ3 (I1 − I3 ) sin δ2 + (I2 − I1) sin δ3
(9)
Here, the pixel coordinates have been omitted for simplicity.
⎡ δq + δ p ⎤ ⎡ δq − δ p ⎤ ΔIpq = Ipmn − Iqmn = 2bmn sin ⎢ ϕmn + ⎥ sin ⎢ ⎥ ⎣ 2 ⎦ ⎣ 2 ⎦ (p, q = 1, 2, 3)
2
1 − ( E23/C ) + E23 ×
(4)
3. Numerical simulations A series of numerical simulations of three-frame GPSI have been carried out to verify the effectiveness of the method proposed above. First, we tested the method with two kinds of fringe patterns: the closed ring fringe pattern and the open straight fringe pattern. For the ring fringe pattern, the background intensity and the modulation amplitude are set as amn = 120 × exp { − [m2 + n2] /5002} and bmn = 100 × exp { − [m2 + n2] /5002} respectively. The measured phase is set as φmn = − π × Nf × [m2 + n2] /2002, in which Nf = 2 is the fringe number in the fringe pattern, m, n¼ 500, 499, …, 500. The phase shift values of the 1th, 2th and 3th fringe patterns are preset as δ1 = 0 rad, δ2 = 0.1 rad and δ3 = 0.2 rad, respectively. Moreover, Gaussian noise with a signal-to-noise ratio (SNR) of 30 dB is added to the fringe pattern. With above parameter setting, three simulated patterns with the size of 1000 1000 pixels can be generated with Eq. (1), as shown in Fig. 1(a–c). By means of numerical calculation, δ2 and δ3 are determined as 0.0960 rad and 0.1907 rad with the associated errors of 0.0040 rad and 0.0093 rad, respectively. Then, the wrapped and unwrapped phase maps can be obtained easily after the determination of the phase shift, which are illustrated in Fig. 1(d) and (e), respectively. Fig. 1(f) is the theoretical phase. From Fig. 1(e) and (f), no significant difference can be observed.
Y. Xu et al. / Optics and Lasers in Engineering 84 (2016) 89–95
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Fig. 1. The simulated results in case of the ring fringe pattern: (a–c) the 1th, 2th and 3th interferograms, (d and e) the reconstructed wrappped phase and unwrapped phase, (f) the theoretical phase.
Fig. 2. The corresponding simulated results in case of the straight fringe pattern.
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Fig. 3. (a) The difference between Fig. 1(e) and 1(f), (c) the difference between Fig. 2(e) and 2(f), (b) and (d) are the transverse profiles corresponding to Fig. 3(a) and (c), respectively.
For the straight fringe pattern, the theoretical phase is set as φmn = π × Nf × n/200, in which Nf = 2. Other parameters are set as the same as previous. With such parameter setting, δ2 and δ3 are calculated as 0.0998 rad and 0.1997 rad with the associated errors of 0.0002 rad and 0.0003 rad, respectively. It indicates that the phase shifts can be extracted with high precision. Fig. 2(a–c) are the simulated phase-shifted straight fringe patterns, Fig. 2(d) is the wrapped phase, Fig. 2(e) and (f) are the measured phase and the theoretical phase, respectively. Similar to the case of ring pattern demonstrated above, the measured phase and the theoretical phase show a good agreement. In order to discuss the accuracy of the reconstructed phase with the method presented above, we calculated the difference between the theoretical phase and the reconstructed one. Fig. 3 (a) and (c) are the results of the ring and straight fringe patterns, respectively. Fig. 3(b) and (d) are the transverse profiles corresponding to Fig. 3(a) and (c), respectively. It could be found that most of the values are less than 0.4 rad in Fig. 3(b) and the absolute values are less than 0.4 rad in Fig. 3(d). In addition, we also calculated the root-mean-square error (RMSE) of the difference. The related results are 0.1716 rad and 0.1354 rad for the cases of ring and straight fringe patterns, respectively. Next, we changed the fringe number Nf in the pattern and discussed the variation of the extracted phase shift with Nf . The ring fringe pattern is taken as an example and the associated results are listed in Table 1 when the preset values of δ2 and δ3 are π/8 and π/4 , respectively. From Table 1, it could be found that with the increase of the fringe number, the calculated value of δ2 decreases, while the calculated value of δ3 increases except for the case of Nf = 1, but the absolute values of their corresponding errors both decrease. Meanwhile, the RMSE of phase difference also decreases with the increase of the fringe number. It indicates that the
Table 1 Phase shift and phase retrieval in different fringe number.
Nf
Calculated δ2 and its error (rad)
Calculated δ3 and its error (rad)
1 1.5 2 2.5 3 3.5 4 4.5 5
0.4031 0.3989 0.3969 0.3958 0.3951 0.3950 0.3948 0.3946 0.3944
0.7714 0.7697 0.7699 0.7703 0.7711 0.7721 0.7729 0.7733 0.7736
0.0104 0.0062 0.0042 0.0031 0.0025 0.0023 0.0021 0.0019 0.0017
0.0140 0.0157 0.0155 0.0151 0.0143 0.0133 0.0125 0.0121 0.0118
RMSE (rad)
0.0849 0.0692 0.0585 0.0535 0.0490 0.0457 0.0431 0.0411 0.0394
larger fringe number, the calculated value of the phase shift is more close to its theoretical value, and the accuracy of phase retrieval is higher.
4. Experimental results In order to further verify the feasibility of the proposed method, a Mach–Zehnder interferometer (see Fig. 4) was employed to perform optical experiments. A He–Ne laser with the wavelength λ = 632.8 nm is used as a light source. In the arms of the interferometer, different components are configured for different experimental purposes. The details of this part are described in the following paragraphs. A quarter-wave plate or a thin glass plate is employed as a phase shifter, which is placed in the reference arm. Among which, the thickness and the refractive index of the glass plate are d = 0. 28 mm and n = 1.55, respectively.
Y. Xu et al. / Optics and Lasers in Engineering 84 (2016) 89–95
Fig. 4. Experimental setup for phase-shifting interferometry. BE: beam expander, BS1, BS2: beam splitter, M1, M2: mirror, L1, L2, L3: lens, MO1, MO2: micro objective, PS: phase shifter.
According to the relationship between the phase shift δ and the tilt angle θ of the glass plate, that is, δ ≈ πd (n − 1) θ 2/(nλ ), the introduced phase shift is about 2.4 rad when the glass plate is rotated by 4° from its normal direction. In order to achieve three-
93
frame phase-shifting interferometry, we record an interferogram without the phase-shifter as the 1th interferogram, and record two interferograms with the phase shifts introduced by the quarterwave plate and the glass plate as the 2th and 3th interferograms, respectively. Thus, the phase shifts of 1th, 2th and 3th frames are respectively 0 rad, π/2 rad and 2.4 rad theoretically. In the following experiments, the interferogram is captured by a complementary metal oxide semiconductor (CMOS) with size of 1500 1500 pixels (6 mm 6 mm). Similar to the previous simulation results, both the ring and straight-type interferograms were discussed in our experiments. In the case of ring-type interferogram, two lenses, L1 and L2 with different curvature radiuses are positioned in the reference and sample arms, respectively. Consequently, the difference between two spherical wavefronts introduced by lenses L1 and L2 is introduced in inline interferometer. By adjusting the phase shifter according to the method described above, three interferograms of ring-type with different phase shift values are obtained, as shown in Fig. 5(a–c). With these three interferograms, three different EMNs of the intensity difference, namely E12 , E13 and E23, can be calculated. Then the phase shift values in the 2th and 3th frames ( δ2 and δ3) can be determined with Eqs. (7) and (8). The measured
Fig. 5. The experimental results of the ring interferogram: (a–c) the 1th, 2th and 3th interferograms, (d) the reconstructed wrappped phase, (e) and (f) are the 2D display and 3D display of the unwrapped phase respectively, (g) the diagram of optical path difference induced by two lenses, (h) the transverse profiles of the reconstructed and the theoretical phases. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
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Fig. 6. The experimental results of the straight interferogram: (a–c) the 1th, 2th and 3th interferograms, (d) the wrappped phase, (e) and (f) are the 2D display and 3D display of the unwrapped phase, respectively.
results of δ2 and δ3 are 1.623 rad and 2.339 rad with the associated errors of 0.052 rad and 0.061 rad compared to their theoretical values, respectively. Lastly, the wrapped and unwrapped phases can be reconstructed according to Eq. (9), as shown in Fig. 5(d) and (e). Fig. 5(f) is the 3D display of the unwrapped phase. In order to discuss the actual accuracy of reconstructed phase, the reconstructed and theoretical phases at the horizontal central section were analyzed. Fig. 5(g) is the diagram of optical path difference induced by two lenses. D1 and D2 are the radiuses of lens L1 and lens L2, respectively, which both are equal. R0 and R are the radiuses of the beam and the experimental screenshoot. In addition, two lenses are thin lenses and their thicknesses are equal at the R position. According to the geometry calculation and the approximation of Taylor series, the optical path difference introduced by lenses L1 and L2 can be deduced as the following expression when the relationship, r ≤ r1, r2, R ≤ r1, r2 is met
⎡ Δ(r ) ≈ 2 ⎢⎣ r2 − r1 −
(
r22 − R2 −
⎤ r − r1 r12 − R2 ⎥⎦ − 2 × r2 r2 × r1
)
(10)
where r1 and r2 are the curvature radiuses of lens L1 and lens L2, respectively, r is the vertical distance from an arbitrary point in the lens to optical axis. Thus the corresponding phase difference is 2π (n2 − n1)Δ(r ) λ ⎧ ⎡ 2π = (n2 − n1) ⎨ 2 ⎢ r2 − r1 − ⎩ ⎣ λ
5. Conclusions
ϕ (r ) =
(
r22 − R2 −
⎫ ⎤ r − r1 × r 2⎬ r12 − R2 ⎥ − 2 ⎦ r2 × r1 ⎭
)
(11)
where n1 and n2 are the refractive indices of the air and the lenses, respectively. In our experiment, r1 = 110 mm , r2 = 120 mm , n1 = 1.0, n2 = 1.5 and R = 3 mm , thus such a relationship, namely, r ≤ R, R < r1, R < r2 is existed inside the range of experimental screenshoot. Consequently, Eq. (11) can be approximated as
φ (r ) ≈ −
with its opening downward. After substituting related parameters, the profile of the phase φ can be made (see red profile in Fig. 5(h)). The blue profile in Fig. 5(h) is the transverse profile of the reconstructed phase (Fig. 5(e)). We found that these two profiles almost overlap. Via the calculation, the maximum deviation between them is 2.6344 rad and the RMSE of the deviation is 0.6913 rad. In case of straight-type interferogram, two lenses above are replaced by two identical microscope objectives with the magnification of 20 in the reference and sample arms, other optical devices are the same as those of the above experiment. By slightly adjusting the angle of the mirror M1, the reference field is tilted with respect to the sample beam to create a uniform off-axis interferogram. Fig. 6(a–c) are the 1th, 2th and 3th interferograms, respectively. Applying the algorithm proposed above, the associated results are achieved, as shown in Fig. 6(d–f). Especially, Fig. 6(f) shows a phase distribution of a plane wave well. The measured results of δ2 and δ3 are about 1.532 rad and 2.344 rad with the associated errors of 0.039 rad and 0.056 rad, respectively.
2π r − r1 × (n2 − n1) × 2 × r 2. λ r2 × r1
(12)
It can be found that Eq. (12) is a standard parabolic equation
In summary, by analyzing the character of EMN of the intensity difference between two phase shifted interferograms, a simple phase shift extraction algorithm has been proposed in the GPSI from only three interferograms. The phase shift can be any value between [0, π ]. The phase can be extracted after determining the phase shifts. Neither the iterative calculations nor extra measurements on some other parameters are needed in this algorithm. It makes the algorithm very fast and easy to implement. The algorithm was tested with the simulated and experimental interferograms. Results show that the algorithm can be applied for any kind of fringe pattern and it is suitable for blind PSI.
Y. Xu et al. / Optics and Lasers in Engineering 84 (2016) 89–95
Acknowledgments This work was supported by National Natural Science Foundation of China (Nos. 11374130, and 11474134), Natural Science Foundation of Jiangsu Province (Nos. SBK2015042429 and BK20141296), Innovation Program for College Graduates of Jiangsu Province (No. KYLX_1017), Postdoctoral Science Fund of China (No. 2014M561574), and Foundation of Six Kinds of High Talented People of Jiangsu Province (No. 2015-DZXX-023).
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Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm Yuanyuan Xu a, Yawei Wang a,b,n, Ying Ji b, Hao Han b, Weifeng Jin a a b
School of Mechanical Engineering, Jiangsu University, Zhenjiang 212013, China Faculty of Science, Jiangsu University, Zhenjiang 212013, China
art ic l e i nf o
a b s t r a c t
Article history: Received 31 October 2015 Received in revised form 8 April 2016 Accepted 11 April 2016 Available online 19 April 2016
Generalized phase-shifting interferometry (GPSI) is one of the most effective techniques in imaging of a phase object, in which phase retrieval is an essential and important procedure. In this paper, a simple and rapid algorithm for retrieval of the unknown phase shifts in three-frame GPSI is proposed. Using this algorithm, the value of phase shift can be calculated by a determinate formula consisting of three different Euclidean matrix norms of the intensity difference between two phase shifted interferograms, and then the phase can be retrieved easily. The algorithm has the advantages of freeing from the background elimination and less computation, since it only needs three phase-shifted interferograms without no extra measurements, the iterative procedure or the integral transformation. The reliability and accuracy of this algorithm were demonstrated by simulation and experimental results. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Phase-shifting interferometry Phase retrieval Euclidean matrix norm
1. Introduction Phase-shifting interferometry (PSI) is an important technique widely used in optical measurement and microscopy [1–4]. In traditional PSI, a special constant phase shift, 2π /N with the integer N ≥ 3 between two adjacent frames, is often assumed. Subsequently, Greivenkamp [5] and Stoilov [6] proposed general methods to deal with arbitrary phase shifts. Although the phase shifts do not have to be special values, they are either known precisely or equal in these methods. In fact, it is still a strict requirement on the precision of the phase shifter. In order to remove this requirement, many methods have been reported [7–18], among which the generalized phase shift exaction techniques [10– 18] provide a possible method for extracting arbitrary and unknown phase shifts from holograms directly. In general, the phase shift extraction methods can be classified into two categories: iterative and noniterative. The iterative methods [10,11] are greatly time-consuming since the procedures are repeated many times to achieve acceptable accuracy. For example, an advanced iterative algorithm based on the least squares method is proposed to determine the phase shift and the phase distribution simultaneously [10]. For this reason, noniterative methods [12–21] have been favored for their speed and less computational loads. Cai and other researchers proposed a class of n Corresponding author at: Faculty of Science, Jiangsu University, Zhenjiang 212013, China. E-mail address: [email protected] (Y. Wang).
http://dx.doi.org/10.1016/j.optlaseng.2016.04.011 0143-8166/& 2016 Elsevier Ltd. All rights reserved.
statistical algorithms to calculate the unknown phase shift directly and reconstruct the wavefront from two or more holograms [12– 15]. These algorithms are based on the assumption that the measured object has a random phase in [0, 2π ] over the whole interferogram. To relax this assumption, another method has been proposed to extract the phase shift accurately using the histogram of phase difference between two adjacent frames [16]. However, it requires determining both the background intensity and the modulation amplitude by searching for the maximum and the minimum intensity in each pixel of interferograms, which still takes considerable time. Later, a self-tuning approach is proposed to retrieve the phase shift by looking for the minimum of a merit function [17], where the accuracy of phase shift decreases when the phase shift is far from π/2 and the interferograms are required to be normalized beforehand. In Ref. [18], an accurate phase shift extraction algorithm is proposed by using the maximum and the minimum values of the interference term. In Ref. [19], the GramSchmidt (GS) orthonormalization algorithm is employed to extract the phase with high precision and rapid speed. Both of these twostep demodulation methods need a precondition of filtering out the background term by a high-pass filter in advance. In Refs. [20,21], a new kind of phase-shifting demodulation method based on the use of principal component analysis (PCA) algorithm is proposed. It is worth noting that this method does not require the extraction of the phase shift to retrieve the modulating phase. Although the PCA method is fast, it still requires the elimination of the background term by a temporal average in advance. However, the filtering and averaging algorithms do not work well for the background elimination when the interferograms are with rapid
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background variation [22]. New methods that are robust to background vibration have been also proposed [23–25]. In Ref. [25], a combination of the GS orthonormalization process and the twodimensional continuous wavelet transform (2D-CWT) algorithm is used to analyze two-step arbitrarily phase-shifted interferograms. The method works well when the interferograms contain complex fringes, large fringe-frequency variations, noise or defect fringes. It is still time-consuming because the 2D-CWT algorithm refers to several times Fourier transforms. In this paper, we propose a Euclidean matrix norm (EMN) algorithm to extract the unknown phase shifts from only three interferograms. After substituting the phase shifts to the three-step phase-shifting algorithm, the phase can be retrieved easily. This algorithm is faster since it does not use the iterative procedure or the integral transformation. Moreover, it is easy to implement without the background elimination or the measurements of other parameters excepting the interferograms in the entire retrieval process. The only requirement is the fringe limit condition, which is easy to meet in real cases.
Epq ≈
⎧ M N ⎫1/2 ⎛ δq − δ p ⎞ ⎪ ⎪ ⎛ δq − δ p ⎞ 2 ⎬ ⎨ ∑ ∑ bmn ⎟ ⎪ ⎟ = C × sin ⎜ 2 sin ⎜ ⎪ ⎝ 2 ⎠ ⎩ ⎝ 2 ⎠ ⎭ m= 1 n= 1
with C =
M N 2 1/2 2 ⋅{∑m = 1 ∑n = 1 bmn } .
(5)
From Eq. (5), it is clear that Epq is
proportional to sin [(δq − δp ) /2] , and there are only three unknown quantities, namely δ2, δ3 and C . To determine these quantities, at least three equations are required. According to Eq. (5), there are three quantities as follows:
E12 = C × sin ( δ2/2) , E13 = C × sin ( δ3/2) , E23 = C × sin ⎡⎣ ( δ3 − δ2 )/2⎤⎦
(6)
In order to avoid the uncertainty of the sign, the phase shift δ is normally constrained within the range of [0, π ]. Therefore, δ/2 ranges from 0 to π/2, and the function of sin (δ /2) is monotonically sin (δ2/2) = sin (δ2/2) and increasing and positive. So sin (δ3/2) = sin (δ3/2). The sign of sin [(δ3 − δ2 ) /2] can be determined by comparing the values of E12 and E13. For example, if E13 > E12, it can be obtained that sin (δ3/2) > sin (δ2/2)andδ3/2 > δ2/2, and then sin [(δ3 − δ2 )/2] = sin [(δ3 − δ2 ) /2]. Thus, Eq. (6) can be rewritten as
E12 = C × sin ( δ2/2), E13 = C × sin ( δ3/2), E23
2. Method
= C × sin ⎡⎣ ( δ3 − δ2 )/2⎤⎦
(7)
For the three-frame generalized phase-shifting interferometry, the distribution of intensity for each interferogram can be given in the following form:
Since E12 , E13 and E23 can be determined, we get the following relationship which relates E12 etc., with the parameter C ,
Ikmn = amn + bmn cos [φmn + δk ],
E13 = E12 ×
(k = 1, 2, 3)
(1)
where m and n denote the pixel position of rows and columns of interferograms respectively. amn , bmn and φmn represent the background intensity, the modulation amplitude and the measured phase, respectively. The phase shift related to the kth interferogram, δk , is usually assumed to be zero when k ¼1. With the measured intensities, the difference between the pth and qth interferograms can be expressed as
(2)
Here, we consider the Euclidean matrix norm (EMN) of the intensity difference. In general, for a matrix T = [tmn ], with M × N M N [∑m = 1 ∑n = 1 (tmn )2]1/2,
order, its EMN is defined as ‖T‖2 = in which the sign ‖‖2 is the EMN operator. Therefore, the EMN of ΔIpq can be expressed as
Epq = ΔIpq
2
M N ⎡ ⎧ ⎫1/2 ⎪ ⎛ δ + δ ⎞ ⎛ δ − δ ⎞ ⎤2 ⎪ =⎨ ∑ ∑ ⎢ 2bmn sin ⎜⎝ ϕmn + q p ⎟⎠ sin ⎜⎝ q p ⎟⎠ ⎥ ⎪⎬ ⎪ ⎦ ⎭ 2 2 ⎩ m= 1 n= 1 ⎣
⎛ δq − δ p ⎞ ⎟ 2 sin ⎜ ⎝ 2 ⎠
=
⎧ M N ⎫1/2 2 ⎡ ⎤⎦ ⎪ ⎨ ⎬ 1 cos 2 − ϕ + δ + δ b ∑ ∑ ( ) ⎣ q p mn mn ⎪ ⎪ ⎩ m= 1 n= 1 ⎭ ⎪
(3)
If the fringe number in each interferogram is more than one, then the measured phase varies more than 2π (rad) in the observed area. As a result, the following approximation can be applied, M
N
M
N
2 2 > > ∑ ∑ bmn cos ( 2φmn + δq + δ p ) ∑ ∑ bmn m= 1 n= 1
m= 1 n= 1
Thus, Eq. (3) can be simplified as
2
1 − ( E12/C )
(8)
After solving for C , δ2 and δ3 can be directly calculated from E12 and E13. Once the phase shifts δk (k = 2, 3) are known, the wrapped phase φ can be solved with following expression:
φ = arctan
I3 − I2 + (I1 − I3 ) cos δ2 + (I2 − I1) cos δ3 (I1 − I3 ) sin δ2 + (I2 − I1) sin δ3
(9)
Here, the pixel coordinates have been omitted for simplicity.
⎡ δq + δ p ⎤ ⎡ δq − δ p ⎤ ΔIpq = Ipmn − Iqmn = 2bmn sin ⎢ ϕmn + ⎥ sin ⎢ ⎥ ⎣ 2 ⎦ ⎣ 2 ⎦ (p, q = 1, 2, 3)
2
1 − ( E23/C ) + E23 ×
(4)
3. Numerical simulations A series of numerical simulations of three-frame GPSI have been carried out to verify the effectiveness of the method proposed above. First, we tested the method with two kinds of fringe patterns: the closed ring fringe pattern and the open straight fringe pattern. For the ring fringe pattern, the background intensity and the modulation amplitude are set as amn = 120 × exp { − [m2 + n2] /5002} and bmn = 100 × exp { − [m2 + n2] /5002} respectively. The measured phase is set as φmn = − π × Nf × [m2 + n2] /2002, in which Nf = 2 is the fringe number in the fringe pattern, m, n¼ 500, 499, …, 500. The phase shift values of the 1th, 2th and 3th fringe patterns are preset as δ1 = 0 rad, δ2 = 0.1 rad and δ3 = 0.2 rad, respectively. Moreover, Gaussian noise with a signal-to-noise ratio (SNR) of 30 dB is added to the fringe pattern. With above parameter setting, three simulated patterns with the size of 1000 1000 pixels can be generated with Eq. (1), as shown in Fig. 1(a–c). By means of numerical calculation, δ2 and δ3 are determined as 0.0960 rad and 0.1907 rad with the associated errors of 0.0040 rad and 0.0093 rad, respectively. Then, the wrapped and unwrapped phase maps can be obtained easily after the determination of the phase shift, which are illustrated in Fig. 1(d) and (e), respectively. Fig. 1(f) is the theoretical phase. From Fig. 1(e) and (f), no significant difference can be observed.
Y. Xu et al. / Optics and Lasers in Engineering 84 (2016) 89–95
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Fig. 1. The simulated results in case of the ring fringe pattern: (a–c) the 1th, 2th and 3th interferograms, (d and e) the reconstructed wrappped phase and unwrapped phase, (f) the theoretical phase.
Fig. 2. The corresponding simulated results in case of the straight fringe pattern.
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Fig. 3. (a) The difference between Fig. 1(e) and 1(f), (c) the difference between Fig. 2(e) and 2(f), (b) and (d) are the transverse profiles corresponding to Fig. 3(a) and (c), respectively.
For the straight fringe pattern, the theoretical phase is set as φmn = π × Nf × n/200, in which Nf = 2. Other parameters are set as the same as previous. With such parameter setting, δ2 and δ3 are calculated as 0.0998 rad and 0.1997 rad with the associated errors of 0.0002 rad and 0.0003 rad, respectively. It indicates that the phase shifts can be extracted with high precision. Fig. 2(a–c) are the simulated phase-shifted straight fringe patterns, Fig. 2(d) is the wrapped phase, Fig. 2(e) and (f) are the measured phase and the theoretical phase, respectively. Similar to the case of ring pattern demonstrated above, the measured phase and the theoretical phase show a good agreement. In order to discuss the accuracy of the reconstructed phase with the method presented above, we calculated the difference between the theoretical phase and the reconstructed one. Fig. 3 (a) and (c) are the results of the ring and straight fringe patterns, respectively. Fig. 3(b) and (d) are the transverse profiles corresponding to Fig. 3(a) and (c), respectively. It could be found that most of the values are less than 0.4 rad in Fig. 3(b) and the absolute values are less than 0.4 rad in Fig. 3(d). In addition, we also calculated the root-mean-square error (RMSE) of the difference. The related results are 0.1716 rad and 0.1354 rad for the cases of ring and straight fringe patterns, respectively. Next, we changed the fringe number Nf in the pattern and discussed the variation of the extracted phase shift with Nf . The ring fringe pattern is taken as an example and the associated results are listed in Table 1 when the preset values of δ2 and δ3 are π/8 and π/4 , respectively. From Table 1, it could be found that with the increase of the fringe number, the calculated value of δ2 decreases, while the calculated value of δ3 increases except for the case of Nf = 1, but the absolute values of their corresponding errors both decrease. Meanwhile, the RMSE of phase difference also decreases with the increase of the fringe number. It indicates that the
Table 1 Phase shift and phase retrieval in different fringe number.
Nf
Calculated δ2 and its error (rad)
Calculated δ3 and its error (rad)
1 1.5 2 2.5 3 3.5 4 4.5 5
0.4031 0.3989 0.3969 0.3958 0.3951 0.3950 0.3948 0.3946 0.3944
0.7714 0.7697 0.7699 0.7703 0.7711 0.7721 0.7729 0.7733 0.7736
0.0104 0.0062 0.0042 0.0031 0.0025 0.0023 0.0021 0.0019 0.0017
0.0140 0.0157 0.0155 0.0151 0.0143 0.0133 0.0125 0.0121 0.0118
RMSE (rad)
0.0849 0.0692 0.0585 0.0535 0.0490 0.0457 0.0431 0.0411 0.0394
larger fringe number, the calculated value of the phase shift is more close to its theoretical value, and the accuracy of phase retrieval is higher.
4. Experimental results In order to further verify the feasibility of the proposed method, a Mach–Zehnder interferometer (see Fig. 4) was employed to perform optical experiments. A He–Ne laser with the wavelength λ = 632.8 nm is used as a light source. In the arms of the interferometer, different components are configured for different experimental purposes. The details of this part are described in the following paragraphs. A quarter-wave plate or a thin glass plate is employed as a phase shifter, which is placed in the reference arm. Among which, the thickness and the refractive index of the glass plate are d = 0. 28 mm and n = 1.55, respectively.
Y. Xu et al. / Optics and Lasers in Engineering 84 (2016) 89–95
Fig. 4. Experimental setup for phase-shifting interferometry. BE: beam expander, BS1, BS2: beam splitter, M1, M2: mirror, L1, L2, L3: lens, MO1, MO2: micro objective, PS: phase shifter.
According to the relationship between the phase shift δ and the tilt angle θ of the glass plate, that is, δ ≈ πd (n − 1) θ 2/(nλ ), the introduced phase shift is about 2.4 rad when the glass plate is rotated by 4° from its normal direction. In order to achieve three-
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frame phase-shifting interferometry, we record an interferogram without the phase-shifter as the 1th interferogram, and record two interferograms with the phase shifts introduced by the quarterwave plate and the glass plate as the 2th and 3th interferograms, respectively. Thus, the phase shifts of 1th, 2th and 3th frames are respectively 0 rad, π/2 rad and 2.4 rad theoretically. In the following experiments, the interferogram is captured by a complementary metal oxide semiconductor (CMOS) with size of 1500 1500 pixels (6 mm 6 mm). Similar to the previous simulation results, both the ring and straight-type interferograms were discussed in our experiments. In the case of ring-type interferogram, two lenses, L1 and L2 with different curvature radiuses are positioned in the reference and sample arms, respectively. Consequently, the difference between two spherical wavefronts introduced by lenses L1 and L2 is introduced in inline interferometer. By adjusting the phase shifter according to the method described above, three interferograms of ring-type with different phase shift values are obtained, as shown in Fig. 5(a–c). With these three interferograms, three different EMNs of the intensity difference, namely E12 , E13 and E23, can be calculated. Then the phase shift values in the 2th and 3th frames ( δ2 and δ3) can be determined with Eqs. (7) and (8). The measured
Fig. 5. The experimental results of the ring interferogram: (a–c) the 1th, 2th and 3th interferograms, (d) the reconstructed wrappped phase, (e) and (f) are the 2D display and 3D display of the unwrapped phase respectively, (g) the diagram of optical path difference induced by two lenses, (h) the transverse profiles of the reconstructed and the theoretical phases. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
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Fig. 6. The experimental results of the straight interferogram: (a–c) the 1th, 2th and 3th interferograms, (d) the wrappped phase, (e) and (f) are the 2D display and 3D display of the unwrapped phase, respectively.
results of δ2 and δ3 are 1.623 rad and 2.339 rad with the associated errors of 0.052 rad and 0.061 rad compared to their theoretical values, respectively. Lastly, the wrapped and unwrapped phases can be reconstructed according to Eq. (9), as shown in Fig. 5(d) and (e). Fig. 5(f) is the 3D display of the unwrapped phase. In order to discuss the actual accuracy of reconstructed phase, the reconstructed and theoretical phases at the horizontal central section were analyzed. Fig. 5(g) is the diagram of optical path difference induced by two lenses. D1 and D2 are the radiuses of lens L1 and lens L2, respectively, which both are equal. R0 and R are the radiuses of the beam and the experimental screenshoot. In addition, two lenses are thin lenses and their thicknesses are equal at the R position. According to the geometry calculation and the approximation of Taylor series, the optical path difference introduced by lenses L1 and L2 can be deduced as the following expression when the relationship, r ≤ r1, r2, R ≤ r1, r2 is met
⎡ Δ(r ) ≈ 2 ⎢⎣ r2 − r1 −
(
r22 − R2 −
⎤ r − r1 r12 − R2 ⎥⎦ − 2 × r2 r2 × r1
)
(10)
where r1 and r2 are the curvature radiuses of lens L1 and lens L2, respectively, r is the vertical distance from an arbitrary point in the lens to optical axis. Thus the corresponding phase difference is 2π (n2 − n1)Δ(r ) λ ⎧ ⎡ 2π = (n2 − n1) ⎨ 2 ⎢ r2 − r1 − ⎩ ⎣ λ
5. Conclusions
ϕ (r ) =
(
r22 − R2 −
⎫ ⎤ r − r1 × r 2⎬ r12 − R2 ⎥ − 2 ⎦ r2 × r1 ⎭
)
(11)
where n1 and n2 are the refractive indices of the air and the lenses, respectively. In our experiment, r1 = 110 mm , r2 = 120 mm , n1 = 1.0, n2 = 1.5 and R = 3 mm , thus such a relationship, namely, r ≤ R, R < r1, R < r2 is existed inside the range of experimental screenshoot. Consequently, Eq. (11) can be approximated as
φ (r ) ≈ −
with its opening downward. After substituting related parameters, the profile of the phase φ can be made (see red profile in Fig. 5(h)). The blue profile in Fig. 5(h) is the transverse profile of the reconstructed phase (Fig. 5(e)). We found that these two profiles almost overlap. Via the calculation, the maximum deviation between them is 2.6344 rad and the RMSE of the deviation is 0.6913 rad. In case of straight-type interferogram, two lenses above are replaced by two identical microscope objectives with the magnification of 20 in the reference and sample arms, other optical devices are the same as those of the above experiment. By slightly adjusting the angle of the mirror M1, the reference field is tilted with respect to the sample beam to create a uniform off-axis interferogram. Fig. 6(a–c) are the 1th, 2th and 3th interferograms, respectively. Applying the algorithm proposed above, the associated results are achieved, as shown in Fig. 6(d–f). Especially, Fig. 6(f) shows a phase distribution of a plane wave well. The measured results of δ2 and δ3 are about 1.532 rad and 2.344 rad with the associated errors of 0.039 rad and 0.056 rad, respectively.
2π r − r1 × (n2 − n1) × 2 × r 2. λ r2 × r1
(12)
It can be found that Eq. (12) is a standard parabolic equation
In summary, by analyzing the character of EMN of the intensity difference between two phase shifted interferograms, a simple phase shift extraction algorithm has been proposed in the GPSI from only three interferograms. The phase shift can be any value between [0, π ]. The phase can be extracted after determining the phase shifts. Neither the iterative calculations nor extra measurements on some other parameters are needed in this algorithm. It makes the algorithm very fast and easy to implement. The algorithm was tested with the simulated and experimental interferograms. Results show that the algorithm can be applied for any kind of fringe pattern and it is suitable for blind PSI.
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Acknowledgments This work was supported by National Natural Science Foundation of China (Nos. 11374130, and 11474134), Natural Science Foundation of Jiangsu Province (Nos. SBK2015042429 and BK20141296), Innovation Program for College Graduates of Jiangsu Province (No. KYLX_1017), Postdoctoral Science Fund of China (No. 2014M561574), and Foundation of Six Kinds of High Talented People of Jiangsu Province (No. 2015-DZXX-023).
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