## Two-dimensional wave-front reconstruction from lateral shearing interferograms

Optics Express, Vol. 14, Issue 2, pp. 625-634 (2006)

http://dx.doi.org/10.1364/OPEX.14.000625

Acrobat PDF (579 KB)

### Abstract

An algorithm is proposed to reconstruct two-dimensional wave-front from phase differences measured by lateral shearing interferometer. Two one-dimensional phase profiles of object wave-front are computed using Fourier transform from phase differences, and then the two-dimensional wave-front distribution is retrieved by use of least-square fitting. The algorithm allows large shear amount and works fast based on fast Fourier transform. Investigations into reconstruction accuracy and reliability are carried out by numerical experiments, in which effects of different shear amounts and noises on reconstruction accuracy are evaluated. Optical measurement is made in a lateral shearing interferometer based on double-grating.

© 2006 Optical Society of America

## 1. Introduction

01. W. J. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc. **59**, 940–952 (1947). [CrossRef]

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

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

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

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

*et al*.[26

26. S. Okuda, T. Nomura, K. Kazuhide, H. Miyashiro, H. Hatsuzo, and K. Yoshikawa, “High precision analysis of lateral shearing interferogram using the integration method and polynomials,” Appl. Opt. , **39**, 5179–5186(2000). [CrossRef]

*N*along shearing direction is divisible by the shear amount s. Influence of shear amount on reconstruction accuracy is tested by numeric experiments, and performance on noise robustness is also evaluated. The 2-D surface profile of a microlens array is reconstructed by optical experiment, which is performed in a double-grating shearing interferometer. Numerical and optical test have confirmed the stability and accuracy of the proposed algorithm.

## 2. Principle

27. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. **26**, 2504–2505(1987). [CrossRef] [PubMed]

30. J. C. Wyant, “Double frequency grating lateral shear interferometer,” Appl. Opt. **12**, 2057–2060 (1973). [CrossRef] [PubMed]

*x*direction in the discrete form by

*D*(

_{x}*m,n*), and

*D*(

*m,n*) in the

*y*direction, they relate to the original phase φ(

*m,n*) by

*N*denotes the number of the sampling points along

*x*or

*y*direction, and

*s*is the shear amount. After performing 1-D Fourier transforms for both side of Eq. (1.a) in the

*x*direction and (1.b) in the

*y*direction, respectively, we get

*FT*represents a 1-D Fourier transform in the α direction, and

_{α}*v*is the corresponding spatial frequency and spans in (0,

_{α}*N*-1). If

*v*reaches a multiple of

_{α}*N*/

*s*, the denominator of the Eq. (2), also called shearing transfer function[22

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

*x*-dimensional, letting

*FT*(

_{x}*m,n*)=[

*FT*(

_{x}*m*+1,

*n*)+

*FT*(

_{x}*m*-1,

*n*)]/2. Even though such processing may result in possible reconstruction error, this minor error is tolerable providing that the shear

*s*is not so large that the number of these leaking points remains to be small, as discussed in detail in Section 3.

*x*-directional phase difference and in each column of data matrix of

*y*-directional phase difference. We recognize the distribution of inverse Fourier transform of the right side in Eq. (2) as the 1-D profile estimate of original phase, described by

*x*-directional profile

*f*(

_{x}*m,n*) and into each column of

*y*-directional profile

*f*(

_{y}*m,n*) to compensate for the deviation. As a result, the 1-D profile estimates must fit the original phase in the following way

*c*is the phase offset in the

_{n}*n*-th row of the

*x*-directional estimate and dm the offset in the

*m*-th column of the

*x*-directional estimate, respectively, and

*N*(

_{x}*m,n*) is the deviation of

*x*-directional phase estimate from the real phase and

*N*(

_{y}*m,n*) the

*y*-directional error. Then, the task of wave-front reconstruction reduces to find optimal phase offsets

*c*and

_{n}*d*for the 1-D phase estimates,

_{n}*f*(

_{x}*m,n*) and

*f*(

_{y}*m,n*). We adopt the squared sum of

*N*(

_{x}*m,n*) and

*N*(

_{y}*m,n*) as a residual error measure for wave-front reconstruction. Minimization of this residual error will produce the optimal phase. Following the LSQ procedure proposed by Tian and Yatagai [19

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

*c*and

_{n}*d*, expressed by

_{m}*φ(m, n)*distributes in the (ROI):

*m,n*∈[0,

*N*-1], the resulting phase difference

*D*(

_{x}*m, n*) can only distribute in a smaller region:

*m*∈ [

*s, N*-1] and

*n*∈ [0,

*N*-1]; i.e., the dimension of the phase difference matrix is

*N*× (

*N*-

*s*) other than the expected

*N*×

*N*. Therefore,

*D*(

_{x}*m, n*) is lack of

*N*×

*s*data when we calculate the 1-D wave-front estimate according to Eq.(3), and so is

*D*(

_{y}*m, n*). To solve this problem, we extend the original phase

*φ(m, n)*periodically in

*x*direction and

*y*direction, respectively, i.e., let

*φ*(

*m*+

*kN*,

*n*+

*lN*)=

*φ*(

*m, n*), where

*k*and

*1*=1, 2, …. If the dimension of the original phase distribution

*N*is a multiple of the shear

*s*, the following relationship is obtained:

*N*must be chosen so that

*N*/

*s*is an integer.

*et al*[21

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

## 3. Computer simulation

*φ*(

*x,y*) for reconstruction simulation:

*s*is chosen to be 16 in the simulation and the phase differences in two orthogonal directions are plotted in Figs.1(b) and (c), respectively. Figure 1(d) shows the reconstructed phase. The deviation of the reconstructed from the original phase is computed, using the root mean square (RMS) and peak-to-valley (PV) value of deviation as the error measure. In the numeric result, the unit of error is converted to wavelength λ, taking into account that the 2π radian is equivalent to an optical length of λ. In this numeric test, the RMS is 3.00×10

^{-7}λ and the PV value is 3.27×10

^{-4}λ.

*s*in simulations. The numeric results of the reconstruction are presented in Table 1. From Table 1 we know that, as

*s*increases, both RMS and PV increase, because the amount of missing data of the phase difference increases accordingly. This test also shows that if the shear amount is not over 64 (i.e., 1/4 of the ROI dimension in this case), the RMS is less than λ/1000 and the PV is not up to the magnitude order of λ/10. Of course, the shear 64 is not the limit for the tested phase; the largest shear amount of the tested phase can be

*N*/2 because the sampling dimension should be a multiple of the shear amount. Other phase functions, including continuous and discontinuous phase distributions, are also tested and give similar results. Therefore, the simulation results confirm that the proposed algorithm can yield satisfactory reconstruction precision even for a fairly large shear.

^{3}and the PV is below λ/10, which is acceptable in interferometry practice.

## 4. Optical experiment

30. J. C. Wyant, “Double frequency grating lateral shear interferometer,” Appl. Opt. **12**, 2057–2060 (1973). [CrossRef] [PubMed]

27. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. **26**, 2504–2505(1987). [CrossRef] [PubMed]

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

*d*between the top and bottom of the binary relief of the element relates to the measured phase

*φ*through

*d*=λ

*φ*/(2π∆

*n*), where the wavelength of the light source is λ=632.8 nm and ∆

*n*is the difference between the refractive index of the glass and that of the air. The refractive index of the glass is 1.457 at the 632.8 nm wavelength. As a result, the relief height of the element is measured to be around 690 nm. For comparison, the step height is measured with the Dektek 3 profilometer from Veeco Instruments Inc.. The measured value from the profiler is 630 nm, which yields a difference of about 10% between the two measurements. This may result from measuring process; the profilometer scans across only a line in a local area that is different from one tested by the interferometer.

## 5. Conclusion

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

## Acknowledgments

## Reference and links

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

02. | V. Ronchi, “Forty years of history of a grating interferometers,” Appl. Opt. |

03. | M. V. R. K. Murty, “The use of a single plane parallel plate as a lateral shearing interferometer with a visible gas laser source,” Appl. Opt. |

04. | M. P. Rimmer, “Method for evaluating lateral shearing interferograms,” App. Opt. |

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

06. | R. R. Gruenzel, “Application of lateral shearing interferometry to stochastic inputs,” J. Opt. Soc. Am. , |

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

08. | T. Yatagai and T. Kanou, “Aspherical surface testing with shearing interferometer using fringe scanning detection method,” Opt. Eng. |

09. | T. Nomura, K. Kamiya, S. Okuda, H. Miyashiro, K. Yoshikawa, and H. Tashiro, “Shape measurements of mirror surfaces with a lateral-shearing interferometer during machine running,” Precis. Eng. , |

10. | L. Erdman and R. Kowarschik, “Testing of refractive silicon microlenses by use of a lateral shearing interferometer in transmission,” Appl. Opt. |

11. | F. Quercioli, B. Tiribilli, and M. Vassalli, “Wavefront-division lateral shearing autocorrelator for ultrafast laser microscopy,” Opt. Express |

12. | D. Mehta, S. Dubey, M. Hossain, and C. Shakher, “Simple multifrequency and phase-shifting fringe-projection system based on two-wavelength lateral shearing interferometry for three-dimensional profilometry,” Appl. Opt. |

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

14. | D. L. Fried, “Least-squares fitting a wavefront distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. |

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

16. | W. H. Southwell, “Wavefront estimation from wavefront slope measurements,” J. Opt. Soc. Am. |

17. | K. R. Freischlad and C. L. Koliopoulos, “Modal estimation of wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A |

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

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

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

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

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

23. | C. Elster and I. Weingartner, “Solution to the shearing problem,” Appl. Opt. |

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

25. | A. Dubra, C. Paterson, and C. Dainty, “Wavefront reconstruction from shear phase maps using the discrete Fourier transform,” Appl. Opt. |

26. | S. Okuda, T. Nomura, K. Kazuhide, H. Miyashiro, H. Hatsuzo, and K. Yoshikawa, “High precision analysis of lateral shearing interferogram using the integration method and polynomials,” Appl. Opt. , |

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

28. | 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. |

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

30. | J. C. Wyant, “Double frequency grating lateral shear interferometer,” Appl. Opt. |

31. | K. Patorski, “Grating shearing interferometer with variable shear and fringe orientation,” Appl. Opt. |

32. | 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. |

33. | H. Schreiber and J. Schwider. “Lateral shearing interferometer based on two Ronchi phase gratings in series,” Appl. Opt. |

**OCIS Codes**

(070.2590) Fourier optics and signal processing : ABCD transforms

(120.3180) Instrumentation, measurement, and metrology : Interferometry

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

**ToC Category:**

Instrumentation, Measurement, and Metrology

**Citation**

Peiying Liang, Jianping Ding, Zhou Jin, Cheng-Shan Guo, and Hui-tian Wang, "Two-dimensional wave-front reconstruction from lateral shearing interferograms," Opt. Express **14**, 625-634 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-2-625

Sort: Journal | Reset

### References

- W. J. Bates, "A wavefront shearing interferometer," Proc. Phys. Soc. 59, 940-952 (1947). [CrossRef]
- V. Ronchi, "Forty years of history of a grating interferometers," Appl. Opt. 3, 427-450 (1964). [CrossRef]
- M. V. R. K. Murty, "The use of a single plane parallel plate as a lateral shearing interferometer with a visible gas laser source," Appl. Opt. 3, 531-534 (1964). [CrossRef]
- M. P. Rimmer, "Method for evaluating lateral shearing interferograms," App. Opt. 13, 623-629 (1974). [CrossRef]
- M. P. Rimmer and J. C. Wyant, "Evaluation of large aberration using a lateral-shear interferometer having variable shear," App. Opt. 14, 142-150 (1975).
- R. R. Gruenzel, "Application of lateral shearing interferometry to stochastic inputs," J. Opt. Soc. Am. 66, 1341-1347 (1976). [CrossRef]
- R. S. Kasana and K. J. Rosenbruch, "Determination of the refractive index of a lens using the Murty shearing interferometer," Appl. Opt. 22, 3526-3531 (1983). [CrossRef] [PubMed]
- T. Yatagai and T. Kanou, "Aspherical surface testing with shearing interferometer using fringe scanning detection method," Opt. Eng. 23, 357-360 (1984).
- T. Nomura, K. Kamiya, S. Okuda, H. Miyashiro, K. Yoshikawa, and H. Tashiro, "Shape measurements of mirror surfaces with a lateral-shearing interferometer during machine running," Precis. Eng. 22, 185-189 (1998). [CrossRef]
- L. Erdman and R. Kowarschik, "Testing of refractive silicon microlenses by use of a lateral shearing interferometer in transmission," Appl. Opt. 37, 676-682 (1998). [CrossRef]
- F. Quercioli, B. Tiribilli, and M. Vassalli, "Wavefront-division lateral shearing autocorrelator for ultrafast laser microscopy," Opt. Express 12, 4303-4312 (2004). [CrossRef] [PubMed]
- D. Mehta, S. Dubey, M. Hossain, and C. Shakher, "Simple multifrequency and phase-shifting fringe-projection system based on two-wavelength lateral shearing interferometry for three-dimensional profilometry," Appl. Opt. 44, 7515-7521 (2005). [CrossRef] [PubMed]
- R. H. Hudgin, "Wave-front reconstruction for compensated imaging," J. Opt. Soc. Am. 67, 375-378 (1977). [CrossRef]
- D. L. Fried, "Least-squares fitting a wavefront distortion estimate to an array of phase-difference measurements," J. Opt. Soc. Am. 67, 370-375 (1977). [CrossRef]
- B. R. Hunt, "Matrix formulation of the reconstruction of phase values from phase differences," J. Opt. Soc. Am. 69, 393-399 (1979). [CrossRef]
- W. H. Southwell, "Wavefront estimation from wavefront slope measurements," J. Opt. Soc. Am. 70, 998-1006 (1980). [CrossRef]
- K. R. Freischlad and C. L. Koliopoulos, "Modal estimation of wave front from difference measurements using the discrete Fourier transform," J. Opt. Soc. Am. A 3, 1852-1861 (1986). [CrossRef]
- G. Harbers, P. J. Kunst and G. W. R. Leibbrandt, "Analysis of lateral shearing interferograms by use of Zernike polynomials," Appl. Opt. 35, 6162-6172 (1996). [CrossRef] [PubMed]
- X. Tian and T. Yatagai, "Simple algorithm for large-grid phase reconstruction of lateral-shearing interferometry," Appl. Opt. 34, 7213-7220 (1995). [CrossRef] [PubMed]
- H. von Brug, "Zernike polynomials as basis for wave-front fitting in lateral shearing interferometry," Appl. Opt. 36, 2788-2790 (1997). [CrossRef]
- M. Servin, D. Malacara, and J. L. Marroquin, "Wave-front recovery from two orthogonal sheared interferograms," Appl. Opt. 35, 4343-4348 (1996). [CrossRef] [PubMed]
- C. Elster and I. Weingartner, "Exact wave-front reconstruction from two lateral shearing interferograms," J. Opt. Soc. Am. A. 16, 2281-2285 (1999). [CrossRef]
- C. Elster and I. Weingartner, "Solution to the shearing problem," Appl. Opt. 38, 5024-5031 ( 1999). [CrossRef]
- C. Elster, "Exact two-dimensional wave-front reconstruction from lateral shearing interferograms with large shears," Appl. Opt. 39, 5353-5359 (2000). [CrossRef]
- A. Dubra, C. Paterson, and C. Dainty, "Wavefront reconstruction from shear phase maps using the discrete Fourier transform," Appl. Opt. 43, 1108-1113 (2004). [CrossRef] [PubMed]
- S. Okuda, T. Nomura, K. Kazuhide, H. Miyashiro, H. Hatsuzo, and K. Yoshikawa, "High precision analysis of lateral shearing interferogram using the integration method and polynomials," Appl. Opt. 39, 5179-5186 (2000). [CrossRef]
- P. Hariharan, B. F. Oreb, and T. Eiju, "Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm," Appl. Opt. 26, 2504-2505 (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, 156-160 (1982). [CrossRef]
- T. J. Flynn, "Two-dimensional phase unwrapping with minimum weighted discontinuity," J. Opt. Soc. Am. A. 14, 2692-2701 (1997). [CrossRef]
- J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057-2060 (1973). [CrossRef] [PubMed]
- K. Patorski, "Grating shearing interferometer with variable shear and fringe orientation," Appl. Opt. 25, 4192-4198 (1986) [CrossRef] [PubMed]
- 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, 6151-6161 (1996) [CrossRef] [PubMed]
- H. Schreiber and J. Schwider, "Lateral shearing interferometer based on two Ronchi phase gratings in series," Appl. Opt. 36, 5321-5324 (1997). [CrossRef] [PubMed]

## 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.