## Correction of rotational inaccuracy in lateral shearing interferometry for freeform measurement |

Optics Express, Vol. 21, Issue 21, pp. 24799-24808 (2013)

http://dx.doi.org/10.1364/OE.21.024799

Acrobat PDF (1694 KB)

### 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

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. 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]

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]

*∂W/∂x*and

*∂W/∂y*, rather than the wavefront under test itself, addition data processing is necessary to reconstruct the original wavefront

*W*. Many studies have discussed how to reconstruct

*W*, and the reconstruction algorithms are basically categorized into either modal [7

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

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]

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]

*W*is assumed to be decomposed into a set of complete basis functions such as the Zernike circular polynomials [12

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]

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]

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]

*∂W/∂x*and

*∂W/∂y*. In zonal methods,

*W*is directly estimated in a local zone from the measured wavefront slopes. The modal method generally shows better performance than the zonal method under noisy conditions, but the zonal method is more appropriate when the test surface has a freeform shape. The zonal method is also better for retaining high spatial frequency in the wavefront.

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]

## 2. Basic theory

*W*, can be expanded by polar coordinate polynomials in the following form [27]:where

*r*and

*θ*are the normalized radial and angular coordinates;

*R*the radial polynomials;

^{k}_{l}(r)*c*and

_{lk}*d*the coefficients. In Eq. (1), if the test surface is rotated along the optical axis by the amount of α, the rotated wavefront

_{lk}*W(r, θ - α)*can be described as

*c*and

_{lk}*d*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,

_{lk}*A*is the well-known counterclockwise rotation matrix. From Eq. (3), Eq. (2) is rewritten as

*∂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/∂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*.

*c*,

_{lk}*d*and α are treated as unknowns at the same time. Then, as shown in Fig. 2, two more measurements are added to solve the unknowns by using a nonlinear optimization technique. According to the Nyquist theory [28], four or more measurements are required to estimate the angle

_{lk}*α*in a periodic wave. The subscript

_{j}*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]

*β*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

^{k}_{l}*α*is exactly 0°, but the other angles,

_{0}*α*, and

_{1}α_{2}*α*have the inevitable errors along the azimuthal direction, as described in Fig. 2. If the values of

_{3},*α*are not correctly estimated, the actual measured values of

_{j}*∂W*. Now, with the intention of optimization, we define the difference wavefront slope

_{j}/∂x*D*as

_{j}*F*represents the partial sum of errors induced in the radial coefficients of Zernike fitting by the inaccurate estimation of rotation angles

_{l}^{k}*α*. On the other hand,

_{j}*F*means the partial sum of errors resulting in the

_{j}^{k}*j*-th slope. These functions have to be minimized with respect to the unknowns, so

*α*= 0°,

_{0}*α*= 90°,

_{1}*α*= 180°, and

_{2}*α*= 270° (the no rotational error case). Equations (12) and (14) are nonlinear, so we adopted an iterative scheme. For each

_{3}*k*, an initial guess is made for each

*α*to estimate

_{j}*α*= 0°,

_{0}*α*= 90°,

_{1}*α*= 180°, and

_{2}*α*= 270° as the initial guess values. Then by using the estimated values of

_{3}*α*are updated from Eq. (14). Next, we go back to Eq. (12) with the new values of

_{j}*α*and repeat the computation of

_{j}*α*converges into a predefined small value. Most of case, the number of iteration was 4~7 when the threshold of the change of

_{j}*α*was 0.001°. The total computational time is influenced by the number of terms of Zernike polynomials in consideration (maximum

_{j}*l*and

*k*), the number of rotation

*N*, and the precision of the initial guess of

*α*. Finally, the original wavefront

_{j}*W*can be reconstructed by using Eqs. (4) and (6) with the converged value of Zernike coefficient

*D*in Eq. (8) is used to prevent the determinant of matrix M in Eq. (11) going to 0.

_{j}## 3. Simulation and experimental results

### 3.1 Modal-based results

*l*= 0 to 4 and

*k*= 0 to 4 (total 15 terms [27]) as shown in Fig. 3(a). Next, two wavefront slopes,

_{2}and α

_{3}are possible. However we used 0°, 90°, 180°, and 270° for convenience. In Fig. 4, we applied the same amount of the rotational error. For α

_{1}= 90.5° case, α

_{2}and α

_{3}were 180.5° and 270.5°, respectively ( + 0.5° error). For α

_{1}= 89.5° case, α

_{2}and α

_{3}are 179.5° and 269.5°, respectively (−0.5° error).

*α*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.

_{1}### 3.2 Zonal-based results

*K*= 0,

*C*= 1/62.65988,

_{1}*C*= −0.10862,

_{2}*A*= 0.180600,

_{3}*A*= −0.198581,

_{4}*A*= 0.116195,

_{5}*A*= −0.37086,

_{6}*B*= −0.247026,

_{3}*B*= −0.435904,

_{4}*B*= 0.247818, and

_{5}*B*= −0.644216. For a freeform surface, it is hard to apply our algorithm directly since the Zernike polynomials are not suitable to this kind of shape. Therefore another compensation procedure was performed as follows: (1) first, we set up a standard mirror whose wavefront was already known on the LSI [Type 2 in Fig. 1(b)], and measured

_{6}*α*-

_{3}*α*. (4) We replaced the standard mirror by a freeform target. (4) We obtained

_{2}*∂W/∂x*and

*∂W/∂y*from the freeform target. (5) We applied the rotational error that is estimated in Step 3 and corrected

*∂W/∂y*. (6) Finally, we applied the zonal technique to

*∂W/∂x*and

*∂W/∂y,*and reconstructed

*W*for the freeform target. The result of this procedure is displayed in Fig. 7. Before the compensation, the maximum form error was appeared at

*α*= 88° and 92° with 64 µmin PV value, but it was reduced to 2.11 µm after applying the proposed method.

_{1}## 4. Conclusions

## Acknowledgments

## References and links

1. | M. V. Mantravadi, “Newton, Fizeau, and Haidinger interferometers,” in |

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. |

3. | P. Zhou and J. H. Burge, “Coupling of surface roughness to the performance of computer-generated holograms,” Appl. Opt. |

4. | H.-G. Rhee and Y.-W. Lee, “Improvement of linewidth in laser beam lithographed computer generated hologram,” Opt. Express |

5. | H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A |

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 |

7. | R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. |

8. | W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. |

9. | J. Herrmann, “Cross coupling and aliasing in modal wave-front estimation,” J. Opt. Soc. Am. |

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 |

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. |

12. | G. Harbers, P. J. Kunst, and G. W. R. Leibbrandt, “Analysis of lateral shearing interferograms by use of Zernike polynomials,” Appl. Opt. |

13. | W. Shen, M. Chang, and D. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng. |

14. | H. van Brug, “Zernike polynomials as a basis for wave-front fitting in lateral shearing interferometry,” Appl. Opt. |

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. |

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 |

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. |

18. | M. P. Rimmer and J. C. Wyant, “Evaluation of large aberrations using a lateral-shear interferometer having variable shear,” Appl. Opt. |

19. | D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. |

20. | R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. |

21. | R. H. Hudgin, “Optimal wave-front estimation,” J. Opt. Soc. Am. |

22. | B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. |

23. | J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am. |

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 |

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 |

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 |

27. | D. Malacara and S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in |

28. | J. W. Goodman, “Analog optical information processing,” in |

**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

Sort: Year | Journal | Reset

### References

- M. V. Mantravadi, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, 2nd ed., (Wiley, 1992), Chap.1.
- 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]
- 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]
- 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]
- H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A19(3), 590–595 (2002). [CrossRef] [PubMed]
- 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]
- R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am.69(7), 972–977 (1978). [CrossRef]
- W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am.70(8), 998–1006 (1980). [CrossRef]
- J. Herrmann, “Cross coupling and aliasing in modal wave-front estimation,” J. Opt. Soc. Am.71(8), 989–992 (1981). [CrossRef]
- 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]
- 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).
- 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]
- W. Shen, M. Chang, and D. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng.36(3), 905–913 (1997). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am.67(3), 375–378 (1977). [CrossRef]
- R. H. Hudgin, “Optimal wave-front estimation,” J. Opt. Soc. Am.67(3), 378–382 (1977). [CrossRef]
- B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am.69(3), 393–399 (1979). [CrossRef]
- J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am.70(1), 28–35 (1980). [CrossRef]
- 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]
- 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]
- 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]
- D. Malacara and S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, 2nd ed., (Wiley, 1992), Chap.13.
- J. W. Goodman, “Analog optical information processing,” in Introduction to Fourier optics, 2nd ed., (McGraw-Hill, 1996), Chap. 8.

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.