## Use of numerical orthogonal transformation for the Zernike analysis of lateral shearing interferograms |

Optics Express, Vol. 20, Issue 2, pp. 1530-1544 (2012)

http://dx.doi.org/10.1364/OE.20.001530

Acrobat PDF (1219 KB)

### Abstract

A numerical orthogonal transformation method for reconstructing a wavefront by use of Zernike polynomials in lateral shearing interferometry is proposed. The difference fronts data in two perpendicular directions are fitted to numerical orthonormal polynomials instead of Zernike polynomials, and then the orthonormal coefficients are used to evaluate the Zernike coefficients of the original wavefront by use of a numerical shear matrix. Due to the fact that the dimensions of the shear matrix are finite, the high-order terms of the original wavefront above a certain order have to be neglected. One of advantages of the proposed method is that the impact of the neglected high-order terms on the outcomes of the lower-order terms can be decreased, which leads to a more accurate reconstruction result. Another advantage is that the proposed method can be applied to reconstruct a wavefront on an aperture of arbitrary shape from its difference fronts. Theoretical analysis and numerical simulations shows that the proposed method is correct and its reconstruction error is obviously smaller than that of Rimmer-Wyant method.

© 2012 OSA

## 1. Introduction

## 2. Numerical orthogonal transformation

### 2.1. Calculation of the numerical orthonormal polynomials

24. G. M. Dai and V. N. Mahajan, “Nonrecursive determination of orthonormal polynomials with matrix formulation,” Opt. Lett. **32**(1), 74–76 (2007). [CrossRef] [PubMed]

### 2.2 Derivation of the corresponding shear matrix

### 2.3 Wavefront reconstruction

16. 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]

## 3. Impact of remaining high-order terms

*J*stands for infinity. Therefore, any practical wavefront can be represented completely by Eq. (1). Now, returning to Eq. (10), under the assumption that

*J*stands for infinity, the column number of the matrix

### 3.1 Least-square fitting of difference fronts

### 3.2 Splitting of the shear matrixes

### 3.3 Reconstruction with finite-dimensional shear matrixes

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

18. G.- Dai, “Modal wavefront reconstruction with Zernike polynomials and Karhunen-Loève functions,” J. Opt. Soc. Am. A **13**(6), 1218–1225 (1996). [CrossRef]

### 3.4 Same analysis of numerical orthogonal transformation method

## 4. Numerical simulation

### 4.1 Simulation condition

### 4.2 Reconstruction without remaining high-order terms

### 4.3 Reconstruction under the impact of remaining high-order terms

*K*=15”column of Table 1. The notations of

### 4.4 Comparison of RMS and PV value

### 4.5 Evaluation of the cross-coupling matrix

### 4.6. Comparison of the computation time

*x*direction can be obtained directly by

### 4.7. Simulation with a general wavefront

28. Y. Zhu, S. Odate, A. Sugaya, K. Otaki, K. Sugisaki, C. Koike, T. Koike, and K. Uchikawa, “Method for designing phase-calculation algorithms for two-dimensional grating phase-shifting interferometry,” Appl. Opt. **50**(18), 2815–2822 (2011). [CrossRef] [PubMed]

## 5. Conclusion

## Acknowledgments

## References and links

1. | D. Malacara, |

2. | A. Dubra, C. Paterson, and C. Dainty, “Study of the tear topography dynamics using a lateral shearing interferometer,” Opt. Express |

3. | Y. Zhu, K. Sugisaki, M. Okada, K. Otaki, Z. Liu, J. Kawakami, M. Ishii, J. Saito, K. Murakami, M. Hasegawa, C. Ouchi, S. Kato, T. Hasegawa, A. Suzuki, H. Yokota, and M. Niibe, “Wavefront measurement interferometry at the operational wavelength of extreme-ultraviolet lithography,” Appl. Opt. |

4. | M. P. Rimmer, “Method for evaluating lateral shearing interferograms,” Appl. Opt. |

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

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

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

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

9. | X. Liu, Y. Gao, and M. Chang, “A partial differential equation algorithm for wavefront reconstruction in lateral shearing interferometry,” J. Opt. A, Pure Appl. Opt. |

10. | 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. |

11. | P. Liang, J. Ding, Z. Jin, C. S. Guo, and H. T. Wang, “Two-dimensional wave-front reconstruction from lateral shearing interferograms,” Opt. Express |

12. | J. Herrmann, “Cross coupling and aliasing in modal wavefront estimation,” J. Opt. Soc. Am. |

13. | 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 |

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

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

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

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

18. | G.- Dai, “Modal wavefront reconstruction with Zernike polynomials and Karhunen-Loève functions,” J. Opt. Soc. Am. A |

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

20. | J. C. Wyant and K. Creath, |

21. | V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. |

22. | R. Upton and B. Ellerbroek, “Gram-Schmidt orthogonalization of the Zernike polynomials on apertures of arbitrary shape,” Opt. Lett. |

23. | V. N. Mahajan and G. M. Dai, “Orthonormal polynomials for hexagonal pupils,” Opt. Lett. |

24. | G. M. Dai and V. N. Mahajan, “Nonrecursive determination of orthonormal polynomials with matrix formulation,” Opt. Lett. |

25. | V. N. Mahajan and G. M. Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A |

26. | G. M. Dai and V. N. Mahajan, “Orthonormal polynomials in wavefront analysis: error analysis,” Appl. Opt. |

27. | M. Hasegawa, C. Ouchi, T. Hasegawa, S. Kato, A. Ohkubo, A. Suzuki, K. Sugisaki, M. Okada, K. Otaki, K. Murakami, J. Saito, M. Niibe, and M. Takeda, “Recent progress of EUV wavefront metrology in EUVA,” Proc. SPIE |

28. | Y. Zhu, S. Odate, A. Sugaya, K. Otaki, K. Sugisaki, C. Koike, T. Koike, and K. Uchikawa, “Method for designing phase-calculation algorithms for two-dimensional grating phase-shifting interferometry,” Appl. Opt. |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.3940) Instrumentation, measurement, and metrology : Metrology

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: November 11, 2011

Revised Manuscript: December 15, 2011

Manuscript Accepted: December 26, 2011

Published: January 10, 2012

**Citation**

Fengzhao Dai, Feng Tang, Xiangzhao Wang, Peng Feng, and Osami Sasaki, "Use of numerical orthogonal transformation for the Zernike analysis of lateral shearing interferograms," Opt. Express **20**, 1530-1544 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-1530

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### References

- D. Malacara, Optical Shop Testing, 3rd ed, (CRC Press, Taylor& Francis, 2007).
- A. Dubra, C. Paterson, and C. Dainty, “Study of the tear topography dynamics using a lateral shearing interferometer,” Opt. Express12(25), 6278–6288 (2004). [CrossRef] [PubMed]
- Y. Zhu, K. Sugisaki, M. Okada, K. Otaki, Z. Liu, J. Kawakami, M. Ishii, J. Saito, K. Murakami, M. Hasegawa, C. Ouchi, S. Kato, T. Hasegawa, A. Suzuki, H. Yokota, and M. Niibe, “Wavefront measurement interferometry at the operational wavelength of extreme-ultraviolet lithography,” Appl. Opt.46(27), 6783–6792 (2007). [CrossRef] [PubMed]
- M. P. Rimmer, “Method for evaluating lateral shearing interferograms,” Appl. Opt.13(3), 623–629 (1974). [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), 370–375 (1977). [CrossRef]
- R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am.67(3), 375–378 (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 with minimum norm,” J. Opt. Soc. Am.70(1), 28–35 (1980). [CrossRef]
- X. Liu, Y. Gao, and M. Chang, “A partial differential equation algorithm for wavefront reconstruction in lateral shearing interferometry,” J. Opt. A, Pure Appl. Opt.11(4), 045702 (2009). [CrossRef]
- 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]
- 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]
- J. Herrmann, “Cross coupling and aliasing in modal wavefront 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]
- 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]
- W. Shen, M. W. Chang, and D. S. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng.36(3), 905–913 (1997). [CrossRef]
- 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]
- 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]
- G.- Dai, “Modal wavefront reconstruction with Zernike polynomials and Karhunen-Loève functions,” J. Opt. Soc. Am. A13(6), 1218–1225 (1996). [CrossRef]
- W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am.70(8), 998–1006 (1980). [CrossRef]
- J. C. Wyant and K. Creath, Basic Wavefront Aberration Theory for Optical Metrology, Vol. XI of Applied Optics and Optical Engineering Series (Academic, 1992), 28.
- V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am.71(1), 75–85 (1981). [CrossRef]
- R. Upton and B. Ellerbroek, “Gram-Schmidt orthogonalization of the Zernike polynomials on apertures of arbitrary shape,” Opt. Lett.29(24), 2840–2842 (2004). [CrossRef] [PubMed]
- V. N. Mahajan and G. M. Dai, “Orthonormal polynomials for hexagonal pupils,” Opt. Lett.31(16), 2462–2464 (2006). [CrossRef] [PubMed]
- G. M. Dai and V. N. Mahajan, “Nonrecursive determination of orthonormal polynomials with matrix formulation,” Opt. Lett.32(1), 74–76 (2007). [CrossRef] [PubMed]
- V. N. Mahajan and G. M. Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A24(9), 2994–3016 (2007). [CrossRef] [PubMed]
- G. M. Dai and V. N. Mahajan, “Orthonormal polynomials in wavefront analysis: error analysis,” Appl. Opt.47(19), 3433–3445 (2008). [CrossRef] [PubMed]
- M. Hasegawa, C. Ouchi, T. Hasegawa, S. Kato, A. Ohkubo, A. Suzuki, K. Sugisaki, M. Okada, K. Otaki, K. Murakami, J. Saito, M. Niibe, and M. Takeda, “Recent progress of EUV wavefront metrology in EUVA,” Proc. SPIE5533, 27–36 (2004). [CrossRef]
- Y. Zhu, S. Odate, A. Sugaya, K. Otaki, K. Sugisaki, C. Koike, T. Koike, and K. Uchikawa, “Method for designing phase-calculation algorithms for two-dimensional grating phase-shifting interferometry,” Appl. Opt.50(18), 2815–2822 (2011). [CrossRef] [PubMed]

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