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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 14 — Jul. 2, 2012
  • pp: 15723–15733
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Two-dimensional wavefront reconstruction from lateral multi-shear interferograms

Yun-feng Guo, Hao Chen, Ji Xu, and Jianping Ding  »View Author Affiliations


Optics Express, Vol. 20, Issue 14, pp. 15723-15733 (2012)
http://dx.doi.org/10.1364/OE.20.015723


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Abstract

We propose and demonstrate multiple shearing interferometry for measuring two-dimensional phase object. Multi-shear interference can effectively eliminate the problem of spectral leakage that results from the single-shear interference. The Fourier coefficients of a two-dimensional wavefront are computed from phase differences obtained from multiple shearing interferograms, which are acquired by a shearing interferometer, and the desired phase is then reconstructed. Numerical and optical tests have confirmed that the multiple shearing interferometry has a higher recovery accuracy than single-shear interferometry and the reconstruction precision increases as the number of shear steps increases.

© 2012 OSA

1. Introduction

In this paper, we propose a new, to the best of our knowledge, lateral multiple-shearing scheme to resolve the spectral leakage problem. Multi-shear interference is accomplished in a three-wave shearing interferometer based on a spatial light modulator (SLM) [22

22. S. H. Zhai, J. Ding, J. Chen, Y. X. Fan, and H. T. Wang, “Three-wave shearing interferometer based on spatial light modulator,” Opt. Express 17(2), 970–977 (2009). [CrossRef] [PubMed]

]. The influence of the amount of shear on the reconstruction accuracy is tested, and robustness against noise is also evaluated. Numerical and optical tests have confirmed that this multiple shearing interferometry method results in higher recovery accuracy than single-shear interference, especially for objects with discontinuous phase profiles (e.g., step-like optic surface)

2. Principle

A wavefront φ(m,n) can be expanded in a set of basis functions zpq(m,n) as follows:
φ(m,n)=p=0N1q=0N1αpqZpq(m,n),m,n=0,1,2,...,N1,
(1)
where N denotes the number of sampling points along the x or y direction. The original phase distribution φ(m,n) can be reconstructed if the coefficients αpq are determined from optical interferometry. If we choose the following complex exponential as the basis function:
Zpq(m,n)=1Nexp[i2πN(pm+qn)],p,q=0,1,2...,N1,
(2)
then Eq. (1) represents a two-dimensional discrete Fourier transform; thus, the coefficients αpq are the Fourier expansion coefficients of the desired wavefront.

Dx(m,n)=p=0N1q=0N1αpqZpq(m,n)[1exp(i2πpsN)], (4.a)
Dy(m,n)=p=0N1q=0N1αpqZpq(m,n)[1exp(i2πqsN)]. (4.b)

The measured phase differences in the x and y directions are denoted by Dx'(m,n)and Dy'(m,n), respectively. The squared sum of the deviation between the measured and original phase differences can be defined as follows:

F=m=0N1n=0N1{[Dx(m,n)Dx(m,n)]2+[Dy(m,n)Dy(m,n)]2}.
(5)

The minimization of this residual error yields the optimal phase estimate. By performing a least-squares procedure, we derive the following condition:
Fαpq=0.
(6)
Substituting Eq. (4) into Eq. (5) and solving Eq. (6), we get the Fourier coefficients of the wavefront as follows:
αpq=14[sin2(πps/N)+sin2(πqs/N)]{[1exp(i2πpsN)]FTpq{Dx(m,n)}}+{[1exp(i2πqsN)]FTpq{Dy(m,n)}},
(7)
where FTpq represents the 2-D Fourier transform and p and q are the corresponding spatial frequencies. If p and q simultaneously reach a multiple of N/s, the denominator of Eq. (7) becomes zero, which leads to undetermined results; i.e., the Fourier spectral components of the wavefront at these points are lost. To circumvent this problem, one can replace the frequency components at these leakage points with the average value of the adjacent points [25

25. T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A 14(10), 2692–2701 (1997). [CrossRef]

]. However, such processing may result in reconstruction errors. Our interferometry experience thus far has shown that replacing the leaking frequency components with the average value may severely impair the reconstruction accuracy especially for object wavefronts with a discontinuous phase profile.

With the aim of resolving the abovementioned frequency-leaking problem in shearing interferometry, we propose a multi-shear interference scheme based on Fourier expansion. First, we obtain a series of phase differences from the multiple shearing interferograms with different shears. Denoting the number of shears by K (i.e., K groups of phase differences), we reconstitute the error function as follows:

ε=m=0N1n=0N1{j=1K[Dx'j(m,n)Dxj(m,n)]2+j=1K[Dy'j(m,n)Dyj(m,n)]2},j=1,2...k.
(8)

We then derive the following optimal solution of the Fourier coefficients after carrying out some mathematical manipulations based on least-squares fitting:
α(p,q)=j=1K(1ei2πpsj/N)FTpq{Dx'j(m,n)}+j=1K(1ei2πqsj/N)FTpq{Dy'j(m,n)}j=1K[4sin2(πpsjN)+4sin2(πqsjN)],
(9)
where sj represents the j-th shear amount in the x or y direction. From Eq. (9), it follows that the denominator cannot vanish so long as any two of the shear amounts have no common divisors, and thus the spectral leaking problem can be removed. The wavefront phase φ(m,n) can be reconstructed exactly over all frequencies by performing the inverse Fourier transform of the Fourier coefficients αpq.

3. Error propagation

(bpqyj)=1exp(i2πqsjN)j=1K{4[sin2(πpsj/N)+sin2(πqsj/N)]}.
(12)

For the following analysis it is assumed that the measured phase difference are perturbed by the uncorrelated zero-mean additive random noises, nxj(m,n) and nyj(m,n), which have the following statistics,
nui(k,l)nvj(m,n)=σn2δ(u,v)δ(i,j)δ(k,m)δ(l,n),
(13)
where ﹤﹒﹥ stands for the ensemble average. From Eq. (10) and using the Fourier transform and autocorrelation theorem, we get the mean variance of the error in the coefficients a(p, q)
σa2(p,q)=j=1K(|bpqxj|2+|bpqyj|2)σn2.
(14)
Since a(p, q) is the Fourier coefficient of the desired phase, the mean variance of the error nφ(m,n) in the reconstructed phase φ(m, n) can be derived as follows,
σφ2=1N2m,n|nφ(m,n)|2=1N2m,nσa2(p,q).
(15)
where the second equal sign is due to Parseval's theorem. Finally, we obtain the noise coefficient:
C=σφ2σn2=1N2p,qj=1K(|bpqxj|2+|bpqyj|2),
(16)
Substituting Eqs. (11) and (12) into Eq. (16) gives a simple form of the noise coefficient:

C=1N2p,q1j=1K[4sin2(πpsjN)+4sin2(πqsjN)].
(17)

Figure 1
Fig. 1 Plot of the noise coefficient for different shear amounts versus the number of sample points.
shows the plots of the noise coefficients for different shear amounts and various numbers of sample points. The data on the noise coefficient can be fitted to a logarithmic curve. The result is summarized in Table 1

Table 1. Noise coefficient versus the shear amount s

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. It may be questioned that, since the number of shear steps contributes to both the numerator and the denominator in Eq. (9) at the same time, the noise coefficient should not change as dramatically as in the plots in Fig. 1. Our answer is that the denominator is a sum of absolute values while the numerator consists of a sum of complex terms, and accordingly, the denominator increases faster with the number of shear steps than the numerator when more shearing data are added. The simulations and experiments presented in the following sections will reaffirm our argument that the noise coefficient decreases significantly as the number of shear steps increases.

4. Computer simulation

To evaluate the reliability of the proposed algorithm, we use the following phase function φ(x,y) for a reconstruction simulation:

φ(x,y)=1.014×[(x23×y2)x+y+x2y2].
(18)

This function is a combination of Zernike polynomials and is sampled across a 420 × 420 grid. The Zernike function is selected because it is most often used in imaging system evaluations, and thus represents a typical phase distribution. The phase distribution is plotted in Fig. 2(a)
Fig. 2 Computer simulation of wavefront reconstruction from multiple phase differences in orthogonal directions: (a) original phase function, (b) x-directional phase difference (4-pixel shear), (c) y-directional phase difference (4-pixel shear), and (d) reconstructed distribution using multi shears of 4, 5, and 6 pixels.
. The multi-shears sj are chosen to be 4, 5, and 6 pixels in this case. The phase differences with a shear of 4 pixels in two orthogonal directions are shown in Fig. 2(b) and 2(c). Figure 2(d) shows the reconstructed phase for the multi-shears sj of 4, 5, and 6 pixels. The deviation of the reconstructed phase from the original phase is computed using the root mean square (RMS) and peak-to-valley (PV) value of the deviations as error measures. The unit of the numerical errors is converted to wavelength, λ, by considering a phase of 2π as being equivalent to an optical length of λ. In this simulation, the RMS is 4.59 × 10−9 λ and the PV value is 9.54 × 10−8 λ. The relative error is expressed as a ratio of the PV or RMS value to the average value of the original phase. The relative RMS value is 3.06 × 10-6 %, and the PV value is 6.36 × 10-5 %.

To study the reconstruction accuracy of the multi-shear method and the influence of the number of shears on the accuracy, we perform simulations with different number of shears, in which the reconstruction accuracies of single-shear and multi-shear interferences are compared. The numerical results for the two methods are presented in Tables 2

Table 2. Deviation of reconstructed phase from original phase using singe-shear versus shear amount s (Units of relative RMS and PV are in percent).

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and 3

Table 3. Deviation of the reconstructed phase from the original phase using multi-shear versus shear amount s (Units of relative RMS and PV are in percent)..

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, respectively. Table 2 shows the dependence of reconstruction precision of singe-shear interferometry on the shear amount s. Table 3 displays the dependence of reconstruction precision on the number of shearing steps. From Tables 2 and 3 we can see that the reconstruction accuracy of the multi-shear method is higher than that of the single-shear method, especially for larger shear amounts. In addition, from Table 3, we know that both the RMS and PV decrease as the number of shears increases, which agrees with the theoretical analysis presented in the last section.

Objects with discontinuous phase distributions are also tested. Owing to its step-like phase map, we choose the letter B for use in reconstruction simulations with 420 × 420 grid points. The original phase is plotted in Fig. 3(a)
Fig. 3 Comparison between reconstruction accuracy of single-shear and multi-shear methods: (a) original phase map, (b) difference between original and reconstructed phases obtained using single-shear algorithm, and (c) difference between original and reconstructed phases obtained using multi-shear algorithm.
. The obtained RMS of the reconstruction phases are 1.3 × 10−3 λ for the single-shearing interferograms with shear of 4 pixels and 3.1 × 10−14 λ for multiple shears of 4, 5, and 6 pixels, respectively. The multi-shear algorithm demonstrates superior performance in recovering the discontinuous phase distribution. To visualize the performance of the two algorithms from the plot, we use the difference between the original and reconstructed phases. The difference between the original and reconstructed phases using the single-shear algorithm is plotted in Fig. 3(b), and the difference using the multi-shear algorithm is shown in Fig. 3(c). Figures 3(b) and 3(c) show that the multi-shear algorithm has a much higher reconstruction precision than the singe-shear algorithm. In addition, we use different numbers of shears in the simulation and find that the precision increases as the number of shear amounts increase, which is similar to the trend observed in Table 3.

We believe that the superior performance of the multi-shear algorithm in recovering the discontinuous phase is due to the fact that a discontinuous function has a broader Fourier frequency range than does a smooth function. A step-like phase, which has more high-frequency components in the Fourier domain, is more likely susceptible to frequency-leaking damage, which occurs when the single-shear algorithm is used but can be avoided completely by using the multi-shear algorithm. To validate our viewpoint, we compare the recovery accuracy of the above two phase (one with continuous distribution and the other with discontinuous distribution) as they suffer from the spectral leaking. For the two phases, their Fourier spectra at the period of N/s are blocked (set to zero), and then inversely Fourier transformed to an erroneous phase. The recovery error due to the spectral leaking is listed in Table 4

Table 4. Deviation of the recovered phase from the original phase under different spectral leaking(case 1- the phase φ(x, y) in Eq. (18), case 2- the phase “B” in Fig. 4(a))

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, which clearly suggests that a discontinuous phase is more susceptible to frequency-leaking than a discontinuous phase is. Consequently, our multi-shear interferometry can achieve a significant improvement over a single shear in the reconstruction of the discontinuous phase.

5. Optical experiment

In most shearing interferometers, mechanically moving devices are required to adjust the phase shifts and wavefront shears. The accuracy and adaptability of shearing interferometers are therefore limited by mechanical elements. Because our proposed algorithm requires multiple shears, and thus both the measured phase differences and reconstruction precision are easily affected by any error in the moving parts. Therefore, we carry out optical experiments in a three-wave lateral shearing interferometer based on a SLM [22

22. S. H. Zhai, J. Ding, J. Chen, Y. X. Fan, and H. T. Wang, “Three-wave shearing interferometer based on spatial light modulator,” Opt. Express 17(2), 970–977 (2009). [CrossRef] [PubMed]

], as schematically shown in Fig. 4
Fig. 4 Three-wave lateral shearing interferometer based on SLM with four lenses (L1, L2, L3 and L4) and a rotating ground glass that can lower the spatial coherence of light in order to reduce speckle noise.
. A He-Ne laser provides light at a wavelength of 632.8 nm. A cosine grating is displayed on the SLM with 1024 × 768 pixels (each pixel is 18 μm × 18 μm in size) controlled by a computer. The cosine grating is used to diffract the incident light into the + 1, −1 and zero diffraction orders so that an object wave is split into three sheared waves. The common path set-up and zero optical path difference between the interfering diffraction orders makes this device easy to align. The flexibility of the SLM can be fully utilized in the sense that the phase shift, as well as the direction and amount of shear can be dynamically controlled. The interferograms are captured using a CCD camera with a cell size of 8.6 μm × 8.6 μm. The eight interferograms required for an 8-step phase-shifting algorithm are recorded [22

22. S. H. Zhai, J. Ding, J. Chen, Y. X. Fan, and H. T. Wang, “Three-wave shearing interferometer based on spatial light modulator,” Opt. Express 17(2), 970–977 (2009). [CrossRef] [PubMed]

], and then the phase differences are extracted and unwrapped. The phase difference in the perpendicular direction is obtained by rotating the grating through 90 degrees. Multiple phase differences are acquired for different shear amounts, and are then used to reconstruct the desired phase.

6. Conclusion

We have developed a multiple shearing interferometry for measuring two-dimensional object phase. Multiple phase differences are acquired using different shears in the shearing interferometer. The algorithm can perform a fast phase reconstruction based on least-square estimate and the two-dimensional Fourier transform of the multiple phase differences. The multiple-shearing scheme demonstrates higher phase recovery accuracy than single-shear interference, especially for discontinuous phase distributions, because it is not subject to spectral leakage. The influence of shear steps and noise levels on reconstruction accuracy is evaluated. Multi-shear interference measurements are accomplished in a three-wave shearing interferometer based on SLM, showing a good stability and reliability in wavefront phase reconstruction.

Acknowledgments

This work is supported in part by the National Natural Science Foundation of China (Grant Nos. 11074116, 10934003 and 10874078), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20090091110012). We thank the reviewers, whose comments made possible a substantial improvement in the presentation of this work.

References and links

1.

W. J. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc. 59(6), 940–950-2 (1947). [CrossRef]

2.

M. P. Rimmer and J. C. Wyant, “Evaluation of large aberrations using a lateral-shear interferometer having variable shear,” Appl. Opt. 14(1), 142–150 (1975). [PubMed]

3.

R. S. Kasana and K.-J. Rosenbruch, “Determination of the refractive index of a lens using the Murty shearing interferometer,” Appl. Opt. 22(22), 3526–3531 (1983). [CrossRef] [PubMed]

4.

T. Nomura, K. Kamiya, H. Miyashiro, S. Okuda, H. Tashiro, and K. Yoshikawa, “Shape measurements of mirror surfaces with a lateral-shearing interferometer during machine running,” Precis. Eng. 22(4), 185–189 (1998). [CrossRef]

5.

M. V. Mantravadi, “Lateral shearing interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992), 123–172.

6.

P. Bon, G. Maucort, B. Wattellier, and S. Monneret, “Quadriwave lateral shearing interferometry for quantitative phase microscopy of living cells,” Opt. Express 17(15), 13080–13094 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-15-13080. [CrossRef] [PubMed]

7.

J. Rizzi, T. Weitkamp, N. Guérineau, M. Idir, P. Mercère, G. Druart, G. Vincent, P. da Silva, and J. Primot, “Quadriwave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett. 36(8), 1398–1400 (2011). [CrossRef] [PubMed]

8.

K. R. Freischlad and C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A 3(11), 1852–1861 (1986). [CrossRef]

9.

A. Dubra, C. Paterson, and C. Dainty, “Wave-front reconstruction from shear phase maps by use of the discrete Fourier transform,” Appl. Opt. 43(5), 1108–1113 (2004). [CrossRef] [PubMed]

10.

X. Tian, M. Itoh, and T. Yatagai, “Simple algorithm for large-grid phase reconstruction of lateral-shearing interferometry,” Appl. Opt. 34(31), 7213–7220 (1995). [CrossRef] [PubMed]

11.

G. Harbers, P. J. Kunst, and G. W. R. Leibbrandt, “Analysis of lateral shearing interferograms by use of Zernike polynomials,” Appl. Opt. 35(31), 6162–6172 (1996). [CrossRef] [PubMed]

12.

W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70(8), 998–1006 (1980). [CrossRef]

13.

A. Talmi and E. N. Ribak, “Wavefront reconstruction from its gradients,” J. Opt. Soc. Am. A 23(2), 288–297 (2006). [CrossRef] [PubMed]

14.

S. Okuda, T. Nomura, K. Kamiya, H. Miyashiro, K. Yoshikawa, and H. Tashiro, “High-precision analysis of a lateral shearing interferogram by use of the integration method and polynomials,” Appl. Opt. 39(28), 5179–5186 (2000). [CrossRef] [PubMed]

15.

M. Servin, D. Malacara, and J. L. Marroquin, “Wave-front recovery from two orthogonal sheared interferograms,” Appl. Opt. 35(22), 4343–4348 (1996). [CrossRef] [PubMed]

16.

C. Elster and I. Weingärtner, “Exact wave-front reconstruction from two lateral shearing interferograms,” J. Opt. Soc. Am. A 16(9), 2281–2285 (1999). [CrossRef]

17.

C. Elster, “Exact two-dimensional wave-front reconstruction from lateral shearing interferograms with large shears,” Appl. Opt. 39(29), 5353–5359 (2000). [CrossRef] [PubMed]

18.

C. Falldorf, Y. Heimbach, C. von Kopylow, and W. Jüptner, “Efficient reconstruction of spatially limited phase distributions from their sheared representation,” Appl. Opt. 46(22), 5038–5043 (2007). [CrossRef] [PubMed]

19.

R. Legarda-Sáenz, M. Rivera, R. Rodríguez-Vera, and G. Trujillo-Schiaffino, “Robust wave-front estimation from multiple directional derivatives,” Opt. Lett. 25(15), 1089–1091 (2000). [CrossRef] [PubMed]

20.

S. Velghe, J. Primot, N. Guérineau, M. Cohen, and B. Wattellier, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett. 30(3), 245–247 (2005). [CrossRef] [PubMed]

21.

P. Liang, J. Ding, Z. Jin, C. S. Guo, and H. T. Wang, “Two-dimensional wave-front reconstruction from lateral shearing interferograms,” Opt. Express 14(2), 625–634 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-2-625. [CrossRef] [PubMed]

22.

S. H. Zhai, J. Ding, J. Chen, Y. X. Fan, and H. T. Wang, “Three-wave shearing interferometer based on spatial light modulator,” Opt. Express 17(2), 970–977 (2009). [CrossRef] [PubMed]

23.

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

24.

M. H. Takeda, H. Ina, and S. Kobayashi, “Fourier transforms method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]

25.

T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A 14(10), 2692–2701 (1997). [CrossRef]

26.

G. Fornaro, G. Franceschetti, R. Lanari, and E. Sansosti, “Robust phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 13(12), 2355–2366 (1996). [CrossRef]

27.

S. W. Bahk, “Highly accurate wavefront reconstruction algorithms over broad spatial-frequency bandwidth,” Opt. Express 19(20), 18997–19014 (2011). [CrossRef] [PubMed]

28.

R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67(3), 375–378 (1977). [CrossRef]

OCIS Codes
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.5050) Instrumentation, measurement, and metrology : Phase measurement
(070.2465) Fourier optics and signal processing : Finite analogs of Fourier transforms

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: February 9, 2012
Revised Manuscript: April 13, 2012
Manuscript Accepted: June 19, 2012
Published: June 27, 2012

Citation
Yun-feng Guo, Hao Chen, Ji Xu, and Jianping Ding, "Two-dimensional wavefront reconstruction from lateral multi-shear interferograms," Opt. Express 20, 15723-15733 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15723


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References

  1. W. J. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc.59(6), 940–950-2 (1947). [CrossRef]
  2. M. P. Rimmer and J. C. Wyant, “Evaluation of large aberrations using a lateral-shear interferometer having variable shear,” Appl. Opt.14(1), 142–150 (1975). [PubMed]
  3. R. S. Kasana and K.-J. Rosenbruch, “Determination of the refractive index of a lens using the Murty shearing interferometer,” Appl. Opt.22(22), 3526–3531 (1983). [CrossRef] [PubMed]
  4. T. Nomura, K. Kamiya, H. Miyashiro, S. Okuda, H. Tashiro, and K. Yoshikawa, “Shape measurements of mirror surfaces with a lateral-shearing interferometer during machine running,” Precis. Eng.22(4), 185–189 (1998). [CrossRef]
  5. M. V. Mantravadi, “Lateral shearing interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992), 123–172.
  6. P. Bon, G. Maucort, B. Wattellier, and S. Monneret, “Quadriwave lateral shearing interferometry for quantitative phase microscopy of living cells,” Opt. Express17(15), 13080–13094 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-15-13080 . [CrossRef] [PubMed]
  7. J. Rizzi, T. Weitkamp, N. Guérineau, M. Idir, P. Mercère, G. Druart, G. Vincent, P. da Silva, and J. Primot, “Quadriwave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett.36(8), 1398–1400 (2011). [CrossRef] [PubMed]
  8. K. R. Freischlad and C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A3(11), 1852–1861 (1986). [CrossRef]
  9. A. Dubra, C. Paterson, and C. Dainty, “Wave-front reconstruction from shear phase maps by use of the discrete Fourier transform,” Appl. Opt.43(5), 1108–1113 (2004). [CrossRef] [PubMed]
  10. X. Tian, M. Itoh, and T. Yatagai, “Simple algorithm for large-grid phase reconstruction of lateral-shearing interferometry,” Appl. Opt.34(31), 7213–7220 (1995). [CrossRef] [PubMed]
  11. G. Harbers, P. J. Kunst, and G. W. R. Leibbrandt, “Analysis of lateral shearing interferograms by use of Zernike polynomials,” Appl. Opt.35(31), 6162–6172 (1996). [CrossRef] [PubMed]
  12. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am.70(8), 998–1006 (1980). [CrossRef]
  13. A. Talmi and E. N. Ribak, “Wavefront reconstruction from its gradients,” J. Opt. Soc. Am. A23(2), 288–297 (2006). [CrossRef] [PubMed]
  14. S. Okuda, T. Nomura, K. Kamiya, H. Miyashiro, K. Yoshikawa, and H. Tashiro, “High-precision analysis of a lateral shearing interferogram by use of the integration method and polynomials,” Appl. Opt.39(28), 5179–5186 (2000). [CrossRef] [PubMed]
  15. M. Servin, D. Malacara, and J. L. Marroquin, “Wave-front recovery from two orthogonal sheared interferograms,” Appl. Opt.35(22), 4343–4348 (1996). [CrossRef] [PubMed]
  16. C. Elster and I. Weingärtner, “Exact wave-front reconstruction from two lateral shearing interferograms,” J. Opt. Soc. Am. A16(9), 2281–2285 (1999). [CrossRef]
  17. C. Elster, “Exact two-dimensional wave-front reconstruction from lateral shearing interferograms with large shears,” Appl. Opt.39(29), 5353–5359 (2000). [CrossRef] [PubMed]
  18. C. Falldorf, Y. Heimbach, C. von Kopylow, and W. Jüptner, “Efficient reconstruction of spatially limited phase distributions from their sheared representation,” Appl. Opt.46(22), 5038–5043 (2007). [CrossRef] [PubMed]
  19. R. Legarda-Sáenz, M. Rivera, R. Rodríguez-Vera, and G. Trujillo-Schiaffino, “Robust wave-front estimation from multiple directional derivatives,” Opt. Lett.25(15), 1089–1091 (2000). [CrossRef] [PubMed]
  20. S. Velghe, J. Primot, N. Guérineau, M. Cohen, and B. Wattellier, “Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers,” Opt. Lett.30(3), 245–247 (2005). [CrossRef] [PubMed]
  21. P. Liang, J. Ding, Z. Jin, C. S. Guo, and H. T. Wang, “Two-dimensional wave-front reconstruction from lateral shearing interferograms,” Opt. Express14(2), 625–634 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-2-625 . [CrossRef] [PubMed]
  22. S. H. Zhai, J. Ding, J. Chen, Y. X. Fan, and H. T. Wang, “Three-wave shearing interferometer based on spatial light modulator,” Opt. Express17(2), 970–977 (2009). [CrossRef] [PubMed]
  23. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt.26(13), 2504–2506 (1987). [CrossRef] [PubMed]
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