## Two-dimensional wavefront reconstruction from lateral multi-shear interferograms |

Optics Express, Vol. 20, Issue 14, pp. 15723-15733 (2012)

http://dx.doi.org/10.1364/OE.20.015723

Acrobat PDF (2182 KB)

### 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

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

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]

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]

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]

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]

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]

## 2. Principle

*φ*(

*m*,

*n*) can be expanded in a set of basis functions

*z*(

_{pq}*m*,

*n*) as follows: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

*α*are determined from optical interferometry. If we choose the following complex exponential as the basis function:then Eq. (1) represents a two-dimensional discrete Fourier transform; thus, the coefficients

_{pq}*α*are the Fourier expansion coefficients of the desired wavefront.

_{pq}*x*and

*y*directions are denoted by

*FT*represents the 2-D Fourier transform and

_{pq}*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]

*K*(i.e.,

*K*groups of phase differences), we reconstitute the error function as follows:

*s*represents the

^{j}*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

*α*.

_{pq}## 3. Error propagation

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]

## 4. Computer simulation

*φ*(

*x*,

*y*) for a reconstruction simulation:

*s*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

^{j}*s*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,

^{j}*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}%.

*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.

^{−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.

*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 , 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.

*φ*(

*x*,

*y*) and the letter “B” and of the noise in the phase differences are denoted by

*C*, is computed according to Eq. (17), and the data in the third row represent the expected RMS of the reconstructed phase. The fourth and fifth rows present the actual RMS of the two phases. The numerical simulation is in agreement with the theoretical analysis in the sense of the noise propagation, and justifies the conclusion that the reconstruction error decreases significantly as the number of shear steps increases

## 5. Optical experiment

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]

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]

## 6. Conclusion

## Acknowledgments

## References and links

1. | W. J. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc. |

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

3. | R. S. Kasana and K.-J. Rosenbruch, “Determination of the refractive index of a lens using the Murty shearing interferometer,” Appl. Opt. |

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. |

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 |

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. |

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 |

9. | A. Dubra, C. Paterson, and C. Dainty, “Wave-front reconstruction from shear phase maps by use of the discrete Fourier transform,” Appl. Opt. |

10. | X. Tian, M. Itoh, and T. Yatagai, “Simple algorithm for large-grid phase reconstruction of lateral-shearing interferometry,” Appl. Opt. |

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

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

13. | A. Talmi and E. N. Ribak, “Wavefront reconstruction from its gradients,” J. Opt. Soc. Am. A |

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. |

15. | M. Servin, D. Malacara, and J. L. Marroquin, “Wave-front recovery from two orthogonal sheared interferograms,” Appl. Opt. |

16. | C. Elster and I. Weingärtner, “Exact wave-front reconstruction from two lateral shearing interferograms,” J. Opt. Soc. Am. A |

17. | C. Elster, “Exact two-dimensional wave-front reconstruction from lateral shearing interferograms with large shears,” Appl. Opt. |

18. | C. Falldorf, Y. Heimbach, C. von Kopylow, and W. Jüptner, “Efficient reconstruction of spatially limited phase distributions from their sheared representation,” Appl. Opt. |

19. | R. Legarda-Sáenz, M. Rivera, R. Rodríguez-Vera, and G. Trujillo-Schiaffino, “Robust wave-front estimation from multiple directional derivatives,” Opt. Lett. |

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. |

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 |

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 |

23. | P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. |

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. |

25. | T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A |

26. | G. Fornaro, G. Franceschetti, R. Lanari, and E. Sansosti, “Robust phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. A |

27. | S. W. Bahk, “Highly accurate wavefront reconstruction algorithms over broad spatial-frequency bandwidth,” Opt. Express |

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

**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

- W. J. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc.59(6), 940–950-2 (1947). [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]
- 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]
- 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]
- M. V. Mantravadi, “Lateral shearing interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992), 123–172.
- 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]
- 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]
- 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]
- 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]
- 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]
- 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. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am.70(8), 998–1006 (1980). [CrossRef]
- A. Talmi and E. N. Ribak, “Wavefront reconstruction from its gradients,” J. Opt. Soc. Am. A23(2), 288–297 (2006). [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]
- 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]
- C. Elster and I. Weingärtner, “Exact wave-front reconstruction from two lateral shearing interferograms,” J. Opt. Soc. Am. A16(9), 2281–2285 (1999). [CrossRef]
- C. Elster, “Exact two-dimensional wave-front reconstruction from lateral shearing interferograms with large shears,” Appl. Opt.39(29), 5353–5359 (2000). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A14(10), 2692–2701 (1997). [CrossRef]
- G. Fornaro, G. Franceschetti, R. Lanari, and E. Sansosti, “Robust phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. A13(12), 2355–2366 (1996). [CrossRef]
- S. W. Bahk, “Highly accurate wavefront reconstruction algorithms over broad spatial-frequency bandwidth,” Opt. Express19(20), 18997–19014 (2011). [CrossRef] [PubMed]
- R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am.67(3), 375–378 (1977). [CrossRef]

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