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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 21 — Oct. 21, 2013
  • pp: 24799–24808
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Correction of rotational inaccuracy in lateral shearing interferometry for freeform measurement

Hyug-Gyo Rhee, Young-Sik Ghim, Joohyung Lee, Ho-Soon Yang, and Yun-Woo Lee  »View Author Affiliations


Optics Express, Vol. 21, Issue 21, pp. 24799-24808 (2013)
http://dx.doi.org/10.1364/OE.21.024799


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Abstract

A lateral shearing interferometer has an advantage over previous wavefront measuring interferometers since it requires no reference. Therefore the lateral shearing interferometer can be a powerful solution to measure a freeform surface. It, however, has some issues to be resolved before it can be implemented. One of them is the orthogonality problem between two shearing directions in LSI. Previous wavefront reconstruction algorithms assume that the shearing directions are perfectly orthogonal to each other and lateral shear is obtained simultaneously in the sagittal and tangential directions. For practical LSI, however, there is no way to guarantee perfect orthogonality between two shearing directions. Motivated by this, we propose a new algorithm that is able to compensate the rotational inaccuracy. The mathematical model is derived in this paper. Computer simulations and experiments are also displayed to verify our algorithm.

© 2013 Optical Society of America

1. Introduction

In wavefront measuring interferometers, such as a Fizeau [1

1. M. V. Mantravadi, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, 2nd ed., (Wiley, 1992), Chap.1.

], it is relatively easy to generate a reference wavefront for a flat or spherical test surface. To realize an aspheric reference is not quite as easy, but it is possible with the help of a null optics including a computer generated hologram [2

2. A. G. Poleshchuk, E. G. Churin, V. P. Koronkevich, V. P. Korolkov, A. A. Kharissov, V. V. Cherkashin, V. P. Kiryanov, A. V. Kiryanov, S. A. Kokarev, and A. G. Verhoglyad, “Polar coordinate laser pattern generator for fabrication of diffractive optical elements with arbitrary structure,” Appl. Opt. 38(8), 1295–1301 (1999). [CrossRef] [PubMed]

4

4. H.-G. Rhee and Y.-W. Lee, “Improvement of linewidth in laser beam lithographed computer generated hologram,” Opt. Express 18(2), 1734–1740 (2010). [CrossRef] [PubMed]

]. However, it is difficult to generate a proper reference wavefront for freeform optics [5

5. H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19(3), 590–595 (2002). [CrossRef] [PubMed]

, 6

6. R. Hu, X. Luo, H. Zheng, Z. Qin, Z. Gan, B. Wu, and S. Liu, “Design of a novel freeform lens for LED uniform illumination and conformal phosphor coating,” Opt. Express 20(13), 13727–13737 (2012). [CrossRef] [PubMed]

] in existing wavefront measuring interferometers. The lateral shearing interferometer (LSI) has an advantage over these wavefront measuring interferometers, since the test wavefront of an LSI is made to interfere with the sheared version of itself, i.e., the LSI requires no reference wavefront [7

7. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. 69(7), 972–977 (1978). [CrossRef]

25

25. F. Dai, F. Tang, X. Wang, and O. Sasaki, “Generalized zonal wavefront reconstruction for high spatial resolution in lateral shearing interferometry,” J. Opt. Soc. Am. A 29(9), 2038–2047 (2012). [CrossRef]

]. Another known advantage is that environmental effects, such as mechanical vibration or air turbulence, cause less problems with an LSI than with other wavefront measuring interferometers.

The other issue of LSI is the orthogonality problem between two shearing directions, as illustrated in Fig. 1
Fig. 1 (a) Type 1: Wavefront slopes in x- and y-directions are obtained in order by part rotation. (b) Type 2: Wavefront slopes in x- and y-directions are obtained at the same time. (c) Orthogonality problem between x- and y-shearing-directions.
. Most reconstruction algorithms assume that the shearing directions are perfectly orthogonal to each other and lateral shear is obtained simultaneously in the sagittal and tangential directions. For practical LSI, however, there is no way to guarantee perfect orthogonality between two shearing directions. Motivated by this, we propose a novel method that adopts an nonlinear least square optimization technique [26

26. H.-G. Rhee, Y.-W. Lee, and S.-W. Kim, “Azimuthal position error correction algorithm for absolute test of large optical surfaces,” Opt. Express 14(20), 9169–9177 (2006). [CrossRef] [PubMed]

] to compensate for lack of orthogonality between two shearing directions. The mathematical model is derived in Section II. The simulation and experimental results are shown in Section III.

2. Basic theory

The wavefront under test, W, can be expanded by polar coordinate polynomials in the following form [27

27. D. Malacara and S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, 2nd ed., (Wiley, 1992), Chap.13.

]:
W(r,θ)=l,kRlk(r)[clkcos(kθ)+dlksin(kθ)]
(1)
where r and θ are the normalized radial and angular coordinates; Rkl(r) the radial polynomials; clk and dlk the coefficients. In Eq. (1), if the test surface is rotated along the optical axis by the amount of α, the rotated wavefront W(r, θ - α) can be described as

W(r,θα)=l,kRlk(r)[clkcos(kθkα)+dlksin(kθkα)]=l,kRlk(r)[(clkcos(kα)dlksin(kα))cos(kθ)+(dlkcos(kα)+clksin(kα))sin(kθ)]=l,kRlk(r)[clk'cos(kθ)+dlk'sin(kθ)]
(2)

This equation indicates that the coefficients clk and dlk comply with the well-known transformation rule of vector rotation when they are regarded as the two orthogonal magnitude components of a two-dimensional vector,
[clk'dlk']=[cosαsinαsinαcosα][clkdlk]=A[clkdlk]fork=1,and[clk'dlk']=[cos(kα)sin(kα)sin(kα)cos(kα)][clkdlk]=Ak[clkdlk]forarbitraryk
(3)
where A is the well-known counterclockwise rotation matrix. From Eq. (3), Eq. (2) is rewritten as

W(r,θα)=l,kRlk(r)[clkcos(kθkα)+dlksin(kθkα)]=l,kRlk(r)[clkcos(kθ)+dlksin(kθ)]cos(kα)+l,kRlk(r)[dlkcos(kθ)+clksin(kθ)]sin(kα)=k[Wk(r,θ)cos(kα)+Wk(r,θ900)sin(kα)]=k[Wk(r,θ)cos(kα)+W˜k(r,θ)sin(kα)]
(4)

[clk'dlk']=[cos90osin90osin90ocos90o][clkdlk]=[0110][clkdlk]=[dlkclk]
(5)

LSI measures the x-directional derivative of the original wavefront, ∂W/∂x, or the y-directional derivative of the original wavefront, ∂W/∂y, or both at a particular location at the same time. Equations (2), (3), and (4) indicate that W˜/x is theoretically the same as ∂W/∂y, [also see Fig. 1(a) and (b)] and the orthogonal error between two shearing directions are eliminated when α is exactly 90°, or when we know the actual value of α.

The main idea of our method is that clk, dlk and α are treated as unknowns at the same time. Then, as shown in Fig. 2
Fig. 2 Four measurements scheme to estimate exact αj.
, two more measurements are added to solve the unknowns by using a nonlinear optimization technique. According to the Nyquist theory [28

28. J. W. Goodman, “Analog optical information processing,” in Introduction to Fourier optics, 2nd ed., (McGraw-Hill, 1996), Chap. 8.

], four or more measurements are required to estimate the angle αj in a periodic wave. The subscript j indicates the rotation index ranging from 0 to N-1. The parameter N is the total number of rotations. For convenience of explanation, let the wavefront be decomposed in terms of the angular order in k. Letting L be the maximum radial order to be considered for each k, the partial sum is made up in detail such as [18

18. 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). [CrossRef] [PubMed]

]

Wk(r,θ)=lLRlk(r)[clkcos(kθ)+dlksin(kθ)]=lLβlkZlk(x,y),whereZlk(x,y)Rlk(r){cossin}(kθ)
(6)

Here the wavefront is presented in rectangular coordinates rather than polar coordinates, to accommodate the conventional wavefront sensors (CCD) that are directly describable with Cartesian coordinates, rather than polar coordinates. In Eq. (6), βkl represents another form of Zernike coefficients that should be chosen to minimize the error between the estimates and the actual measurements. From Eqs. (4) and (6), the actual wavefront slopes with the rotational errors are given by
W^jkx|j=0=W^k(r,θ)x=lLβlkZlk(x,y)xcos(kα0)+lLβ˜lkZlk(x,y)xsin(kα0),W^jkx|j=1=W^k(r,θα1)x=lLβlkZlk(x,y)xcos(kα1)+lLβ˜lkZlk(x,y)xsin(kα1),W^jkx|j=2=W^k(r,θα2)x=lLβlkZlk(x,y)xcos(kα2)+lLβ˜lkZlk(x,y)xsin(kα2),W^jkx|j=3=W^k(r,θα3)x=lLβlkZlk(x,y)xcos(kα3)+lLβ˜lkZlk(x,y)xsin(kα3)
(7)
where α0 is exactly 0°, but the other angles, α1 α2, and α3, have the inevitable errors along the azimuthal direction, as described in Fig. 2. If the values of αj are not correctly estimated, the actual measured values of W^j/xnever equal the true values of ∂Wj/∂x. Now, with the intention of optimization, we define the difference wavefront slope Dj as

D^jk=W^jk(r,θ)xW^0k(r,θ)x=lKβlkZlk(x,y)x[cos(kαj)1]+lKβ˜lkZlk(x,y)xsin(kαj).
(8)

From Eq. (8), the least square optimization functions are given by

Flk=j=0N1{DljkD^ljk}2=j=0N1{βlkZlk(x,y)x[cos(kαj)1]+β˜lkZlk(x,y)xsin(kαj)D^ljk}2,andFjk=lL{DljkD^ljk}2=lL{βlkZlk(x,y)x[cos(kαj)1]+β˜lkZlk(x,y)xsin(kαj)D^ljk}2.
(9)

The function Flk represents the partial sum of errors induced in the radial coefficients of Zernike fitting by the inaccurate estimation of rotation angles αj. On the other hand, Fjk means the partial sum of errors resulting in the j-th slope. These functions have to be minimized with respect to the unknowns, so

Flkβlk=0,Flkβ˜lk=0,Fjkcos(kαj)=0,andFjksin(kαj)=0.
(10)

This results in the following form of matrix equations,

M[βlkβ˜lk]=N,whereM=[j=0N1[cos(kαj)1]2Zlk(x,y)xj=0N1sin(kαj)[cos(kαj)1]Zlk(x,y)xj=0N1sin(kαj)[cos(kαj)1]Zlk(x,y)xj=0N1sin2(kαj)Zlk(x,y)x],N=[j=0N1D^ljk[cos(kαj)1]j=0N1D^ljksin(kαj)],and
(11)
[βlkβ˜lk]=M1N.
(12)
P[cos(kαj)sin(kαj)]=Q,whereP=[lL[βlkZlk(x,y)x]2lLβlkβ˜lk[Zlk(x,y)x]2lLβlkβ˜lk[Zlk(x,y)x]2lL[β˜lkZlk(x,y)x]2],Q=[lL{D^ljkβlkZlk(x,y)x+[βlkZlk(x,y)x]2}lL{D^ljkβ˜lkZlk(x,y)x+βlkβ˜lk[Zlk(x,y)x]2}],and
(13)
[cos(kαj)sin(kαj)]=P1Q.
(14)

3. Simulation and experimental results

3.1 Modal-based results

To check our algorithm by experiment, we tested a DVD pickup lens (an aspheric optics), as shown in Fig. 5
Fig. 5 (a) Photographic view of a DVD pickup lens. (b) Measured wavefront of the lens in LSI without (PV: 209.6 nm, rms: 30.4 nm), and (c) with application of the proposed algorithm (PV: 172.8 nm, rms: 30.1 nm). α1, α2, and α3 were estimated to be 90.33°, 179.74° and 270.40 o, respectively. (d) Wavefront error due to the rotational inaccuracy (PV: 8.1 nm, rms: 1.27 nm). (e) Measured Zernike coefficients obtained by the LSI and a commercial Fizeau interferometer.
. After applying our algorithm, α1 was estimated to be 90.33°. As shown in Fig. 5(e), the wavefront obtained by the Fizeau interferometer showed about 181.8 nm in PV value and 29.7 nm in rms. This is close to the result of the compensated LSI.

3.2 Zonal-based results

4. Conclusions

Acknowledgments

This work was supported by Korea Research Council of Fundamental Science & Technology – Grant funded by the Korean Government (KRCF-2013-CAP-1345194477).

References and links

1.

M. V. Mantravadi, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, 2nd ed., (Wiley, 1992), Chap.1.

2.

A. G. Poleshchuk, E. G. Churin, V. P. Koronkevich, V. P. Korolkov, A. A. Kharissov, V. V. Cherkashin, V. P. Kiryanov, A. V. Kiryanov, S. A. Kokarev, and A. G. Verhoglyad, “Polar coordinate laser pattern generator for fabrication of diffractive optical elements with arbitrary structure,” Appl. Opt. 38(8), 1295–1301 (1999). [CrossRef] [PubMed]

3.

P. Zhou and J. H. Burge, “Coupling of surface roughness to the performance of computer-generated holograms,” Appl. Opt. 46(26), 6572–6576 (2007). [CrossRef] [PubMed]

4.

H.-G. Rhee and Y.-W. Lee, “Improvement of linewidth in laser beam lithographed computer generated hologram,” Opt. Express 18(2), 1734–1740 (2010). [CrossRef] [PubMed]

5.

H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19(3), 590–595 (2002). [CrossRef] [PubMed]

6.

R. Hu, X. Luo, H. Zheng, Z. Qin, Z. Gan, B. Wu, and S. Liu, “Design of a novel freeform lens for LED uniform illumination and conformal phosphor coating,” Opt. Express 20(13), 13727–13737 (2012). [CrossRef] [PubMed]

7.

R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. 69(7), 972–977 (1978). [CrossRef]

8.

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

9.

J. Herrmann, “Cross coupling and aliasing in modal wave-front estimation,” J. Opt. Soc. Am. 71(8), 989–992 (1981). [CrossRef]

10.

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]

11.

G. W. R. Leibbrandt, G. Harbers, and P. J. Kunst, “Wave-front analysis with high accuracy by use of a double-grating lateral shearing interferometer,” Appl. Opt. 35(31), 6151–6161 (1996).

12.

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]

13.

W. Shen, M. Chang, and D. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng. 36(3), 905–913 (1997). [CrossRef]

14.

H. van Brug, “Zernike polynomials as a basis for wave-front fitting in lateral shearing interferometry,” Appl. Opt. 36(13), 2788–2790 (1997). [CrossRef] [PubMed]

15.

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]

16.

F. Dai, F. Tang, X. Wang, P. Feng, and O. Sasaki, “Use of numerical orthogonal transformation for the Zernike analysis of lateral shearing interferograms,” Opt. Express 20(2), 1530–1544 (2012). [CrossRef] [PubMed]

17.

F. Dai, F. Tang, X. Wang, O. Sasaki, and P. Feng, “Modal wavefront reconstruction based on Zernike polynomials for lateral shearing interferometry: comparisons of existing algorithms,” Appl. Opt. 51(21), 5028–5037 (2012). [CrossRef] [PubMed]

18.

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). [CrossRef] [PubMed]

19.

D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67(3), 370–374 (1977). [CrossRef]

20.

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

21.

R. H. Hudgin, “Optimal wave-front estimation,” J. Opt. Soc. Am. 67(3), 378–382 (1977). [CrossRef]

22.

B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69(3), 393–399 (1979). [CrossRef]

23.

J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am. 70(1), 28–35 (1980). [CrossRef]

24.

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). [CrossRef] [PubMed]

25.

F. Dai, F. Tang, X. Wang, and O. Sasaki, “Generalized zonal wavefront reconstruction for high spatial resolution in lateral shearing interferometry,” J. Opt. Soc. Am. A 29(9), 2038–2047 (2012). [CrossRef]

26.

H.-G. Rhee, Y.-W. Lee, and S.-W. Kim, “Azimuthal position error correction algorithm for absolute test of large optical surfaces,” Opt. Express 14(20), 9169–9177 (2006). [CrossRef] [PubMed]

27.

D. Malacara and S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, 2nd ed., (Wiley, 1992), Chap.13.

28.

J. W. Goodman, “Analog optical information processing,” in Introduction to Fourier optics, 2nd ed., (McGraw-Hill, 1996), Chap. 8.

OCIS Codes
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.3940) Instrumentation, measurement, and metrology : Metrology

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: August 14, 2013
Revised Manuscript: September 30, 2013
Manuscript Accepted: October 2, 2013
Published: October 9, 2013

Citation
Hyug-Gyo Rhee, Young-Sik Ghim, Joohyung Lee, Ho-Soon Yang, and Yun-Woo Lee, "Correction of rotational inaccuracy in lateral shearing interferometry for freeform measurement," Opt. Express 21, 24799-24808 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-21-24799


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References

  1. M. V. Mantravadi, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, 2nd ed., (Wiley, 1992), Chap.1.
  2. A. G. Poleshchuk, E. G. Churin, V. P. Koronkevich, V. P. Korolkov, A. A. Kharissov, V. V. Cherkashin, V. P. Kiryanov, A. V. Kiryanov, S. A. Kokarev, and A. G. Verhoglyad, “Polar coordinate laser pattern generator for fabrication of diffractive optical elements with arbitrary structure,” Appl. Opt.38(8), 1295–1301 (1999). [CrossRef] [PubMed]
  3. P. Zhou and J. H. Burge, “Coupling of surface roughness to the performance of computer-generated holograms,” Appl. Opt.46(26), 6572–6576 (2007). [CrossRef] [PubMed]
  4. H.-G. Rhee and Y.-W. Lee, “Improvement of linewidth in laser beam lithographed computer generated hologram,” Opt. Express18(2), 1734–1740 (2010). [CrossRef] [PubMed]
  5. H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A19(3), 590–595 (2002). [CrossRef] [PubMed]
  6. R. Hu, X. Luo, H. Zheng, Z. Qin, Z. Gan, B. Wu, and S. Liu, “Design of a novel freeform lens for LED uniform illumination and conformal phosphor coating,” Opt. Express20(13), 13727–13737 (2012). [CrossRef] [PubMed]
  7. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am.69(7), 972–977 (1978). [CrossRef]
  8. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am.70(8), 998–1006 (1980). [CrossRef]
  9. J. Herrmann, “Cross coupling and aliasing in modal wave-front estimation,” J. Opt. Soc. Am.71(8), 989–992 (1981). [CrossRef]
  10. 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]
  11. G. W. R. Leibbrandt, G. Harbers, and P. J. Kunst, “Wave-front analysis with high accuracy by use of a double-grating lateral shearing interferometer,” Appl. Opt.35(31), 6151–6161 (1996).
  12. 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]
  13. W. Shen, M. Chang, and D. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng.36(3), 905–913 (1997). [CrossRef]
  14. H. van Brug, “Zernike polynomials as a basis for wave-front fitting in lateral shearing interferometry,” Appl. Opt.36(13), 2788–2790 (1997). [CrossRef] [PubMed]
  15. 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]
  16. F. Dai, F. Tang, X. Wang, P. Feng, and O. Sasaki, “Use of numerical orthogonal transformation for the Zernike analysis of lateral shearing interferograms,” Opt. Express20(2), 1530–1544 (2012). [CrossRef] [PubMed]
  17. F. Dai, F. Tang, X. Wang, O. Sasaki, and P. Feng, “Modal wavefront reconstruction based on Zernike polynomials for lateral shearing interferometry: comparisons of existing algorithms,” Appl. Opt.51(21), 5028–5037 (2012). [CrossRef] [PubMed]
  18. 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). [CrossRef] [PubMed]
  19. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am.67(3), 370–374 (1977). [CrossRef]
  20. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am.67(3), 375–378 (1977). [CrossRef]
  21. R. H. Hudgin, “Optimal wave-front estimation,” J. Opt. Soc. Am.67(3), 378–382 (1977). [CrossRef]
  22. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am.69(3), 393–399 (1979). [CrossRef]
  23. J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am.70(1), 28–35 (1980). [CrossRef]
  24. 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). [CrossRef] [PubMed]
  25. F. Dai, F. Tang, X. Wang, and O. Sasaki, “Generalized zonal wavefront reconstruction for high spatial resolution in lateral shearing interferometry,” J. Opt. Soc. Am. A29(9), 2038–2047 (2012). [CrossRef]
  26. H.-G. Rhee, Y.-W. Lee, and S.-W. Kim, “Azimuthal position error correction algorithm for absolute test of large optical surfaces,” Opt. Express14(20), 9169–9177 (2006). [CrossRef] [PubMed]
  27. D. Malacara and S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, 2nd ed., (Wiley, 1992), Chap.13.
  28. J. W. Goodman, “Analog optical information processing,” in Introduction to Fourier optics, 2nd ed., (McGraw-Hill, 1996), Chap. 8.

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