## Determination of wavefront structure for a Hartmann Wavefront Sensor using a phase-retrieval method |

Optics Express, Vol. 20, Issue 7, pp. 7822-7832 (2012)

http://dx.doi.org/10.1364/OE.20.007822

Acrobat PDF (2342 KB)

### Abstract

We apply a phase retrieval algorithm to the intensity pattern of a Hartmann wavefront sensor to measure with enhanced accuracy the phase structure of a Hartmann hole array. It is shown that the rms wavefront error achieved by phase reconstruction is one order of magnitude smaller than the one obtained from a typical centroid algorithm. Experimental results are consistent with a phase measurement performed independently using a Shack-Hartmann wavefront sensor.

© 2012 OSA

## 1. Introduction

4. G. E. Sommargren, D. W. Phillion, M. A. Johnson, N. Q. Nguyen, A. Barty, F. J. Snell, D. R. Dillon, and L. S. Bradsher, “100-picometer interferometry for EUVL,” Proc. SPIE **4688**, 316–328 (2002). [CrossRef]

5. A. Polo, F. Bociort, S. F. Pereira, and H. P. Urbach, “Wavefront measurement for EUV lithography system through Hartmann sensor,” Proc. SPIE **7971**, 79712R (2011). [CrossRef]

## 2. Hartmann wavefront sensor and its limits

*x*and

*y*directions results from where Φ is the wavefront function, Δ

*x*and Δ

_{i,j}*y*are the measured shifts along the

_{i,j}*x*and

*y*directions of the spot of hole (

*i, j*).

*L*is the distance between the sampling and detection planes. The displacement of the beamlets is usually calculated with a fast and efficient centroid algorithm [6

6. L. A. Carvalho, “A simple and effective algorithm for detection of arbitrary Hartmann-Shack patterns.” J. Biomed. Inf. **37**, 1–9 (2004). [CrossRef]

7. C. Leroux and C. Dainty, “Estimation of centroid positions with a matched-filter algorithm: relevance for aberrometry of the eye,” Opt. Express **18**, 1197–206 (2010). [CrossRef] [PubMed]

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

10. D. R. Neal, “Shack-Hartmann wavefront sensor precision and accuracy,” Proc. SPIE **4779**, 148–160 (2002). [CrossRef]

*λ*= 13.5 nm, it has already been demonstrated that Hartmann sensors can reach an accuracy of

*λ*/120 [11

11. P. Mercère, P. Zeitoun, M. Idir, S. L. Pape, D. Douillet, X. Levecq, G. Dovillaire, S. Bucourt, K. A. Goldberg, P. P. Naulleau, and S. Rekawa, “Hartmann wave-front measurement at 13.4 nm with *λ _{EUV}*/120 accuracy,” Opt. Lett.

**28**, 1534–1536 (2003). [CrossRef] [PubMed]

*x*and

*y*tilt component (Fig. 1(b)). Therefore, the information about the local curvature inside the sub-aperture of the Hartmann screen is lost. Secondly, Eq. (2) is only valid under the assumption of small deflection angles (Fig. 1(a)), i.e. sin(

*α*) ≃ tan(

_{x},α_{y}*α*) ≃ (

_{x},α_{y}*α*). This means that in the centroid algorithm the computation of the slope is linearized [13

_{x},α_{y}13. H. H. Barrett, C. Dainty, and D. Lara, “Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions,” J. Opt. Soc. Am. A. **24**, 391–414 (2007). [CrossRef]

## 3. Mathematical description of the phase retrieval algorithm for a Hartmann wavefront sensor

*A*(

*x,y*) be the amplitude of the complex field distribution after transmission by the Hartmann hole array and let Φ(

*x,y*) be its phase distribution. Hence the field in the aperture plane

*z*= 0 is given by: The field distribution at the detector plane

*U*(

*x,y,L*) is then given by: where

**denotes the two dimensional Fourier transform,

**[

*U*(

*x,y,*0)] is the angular spectrum of the plane waves and

*f*and

_{x}*f*are the spatial frequencies in the Fourier domain [14]. The quantity is the Rayleigh-Sommerfeld transfer function where

_{y}*L*represents the propagation distance and

*λ*is the wavelength. The phase retrieval procedure is to recover the phase distribution Φ from the measured intensity |

*U*(

*x,y,L*)|

^{2}and the intensity |

*U*(

*x,y,*0)|

^{2}in the aperture plane which is assumed to be uniform and known [1

1. C. A. Mack, *Fundamental Principles of Optical Lithography: The Science of Microfabrication* (Wiley-Interscience, 2007). [CrossRef]

16. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–69 (1982). [CrossRef] [PubMed]

17. J. R Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. **32**, 1737–1746 (1993). [CrossRef] [PubMed]

*Z*(

_{k}*x, y*) is the

*k*polynomial. The coefficients

^{th}*α*are the the optimization variables of the problem. The chosen object function for minimization is defined by the sum of the squared differences between calculated and measured pixel intensities in the CCD:

_{k}*F*(

*x*) is the square root of the measured intensity on the pixel (

_{i}, y_{j}*i, j*), (i.e., the amplitude in the pixel (

*i, j*)). The sum is over all discrete values in the 2D distribution at the detector (i.e., over the number of CCD pixels). A gradient search method is usually chosen to perform the minimization. In this paper we use, alternatively, a trust-region reflective algorithm [18

18. T. F. Coleman and Y. Li, “An interior trust region approach for nonlinear minimization subject to bounds,” SIAM J. Optim. **6**, 418–445 (1996). [CrossRef]

19. T. F. Coleman and Y. Li, “On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds,” Math. Program. **67**, 189–224 (1994). [CrossRef]

## 4. Simulation results

*d*and distance between the array and detector

*L*) has been scaled to visible light of wavelength

*λ*= 638

*nm*, such that the Fresnel Number of the system is the same as for the EUV radiation. In particular we model a hole array consisting of 29 × 29 square holes over a 25 × 25 mm area. Each hole is rotated by 25° compared to its neighbours to minimize the overlap of the diffraction orders with the adjacent ones in the detector plane [20]. The size of each hole is 200

*μ*m (which is equivalent to 80

*μ*m at the EUV wavelength) and the spacing between two adjacent holes is 562.5

*μ*m. The distance between the array and the detector is chosen to be 50 mm (400 mm at EUV wavelength) to obtain a well defined spot size on the detector. The detector is assumed to have 1024 × 1024 pixels with a pixel size of 16.5

*μm*. The spot size due to one hole on the detector is about 150

*μm*and covers about 15×15 pixels on the CCD camera as shown in Fig. 2(a).

*μ*= 0.0076 and a standard deviation

*σ*= 4 × 10

^{−4}.

*Zernike coefficient (the Malacara system [21*

^{th}21. D. Malacara, *Optical Shop Testing* (Wiley-Interscience, 2007). [CrossRef]

*coefficient and is called the “astigmatism distribution” (Fig. 3((c)–3(d)). The amplitude of the complex field is constant.*

^{th}*λ*in the first case and 0.0023

*λ*in the second case. These values have the same order of magnitude as those reported in Ref. [11

11. P. Mercère, P. Zeitoun, M. Idir, S. L. Pape, D. Douillet, X. Levecq, G. Dovillaire, S. Bucourt, K. A. Goldberg, P. P. Naulleau, and S. Rekawa, “Hartmann wave-front measurement at 13.4 nm with *λ _{EUV}*/120 accuracy,” Opt. Lett.

**28**, 1534–1536 (2003). [CrossRef] [PubMed]

^{®}using the included

*lsqnonlin*routine [19

19. T. F. Coleman and Y. Li, “On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds,” Math. Program. **67**, 189–224 (1994). [CrossRef]

18. T. F. Coleman and Y. Li, “An interior trust region approach for nonlinear minimization subject to bounds,” SIAM J. Optim. **6**, 418–445 (1996). [CrossRef]

^{−4}

*λ*and 2.24×10

^{−4}

*λ*, respectively, correspond to an improvement of almost one order of magnitude in the rms wavefront error reconstruction.

*λ*/100). In contrast, with the phase retrieval procedure we can get more information from the intensity pattern and the phase distribution of the field under test is correctly recovered.

## 5. Experiments

*λ*= 638 nm coupled by a single mode fiber. The beam coming from the fiber is first collimated by a collimator lens and then clipped by an iris of 10 mm diameter. The iris is then conjugated with a Spatial Light Modulator (SLM) (Holoeye LC-R 2500), that is used to generate a certain phase distribution in the beam. The SLM consists of a LCOS display with a resolution of 1024×768 pixels over an image array dimension of 19.5 × 14.6 mm

^{2}and supports a digital video interface (DVI) signal. The phase shift induced by the SLM was characterized in situ on the same optical bench using a Shack-Hartmann wavefront sensor (Thorlabs WFS150-5C, 1.3 Mpixel,

*λ*/50 rms accuracy). During the characterization, a strong astigmatism component was measured by the SH sensor at the SLM plane due to the non-perfect flatness of the surface (Fig. 8(a)). This measured phase distortion was then used to correct for this systematic error of the SLM [22

22. C. López-Quesada, J. Andilla, and E. Martín-Badosa, “Correction of aberration in holographic optical tweezers using a Shack-Hartmann sensor,” Appl. Opt. **48**, 1084–1090 (2009). [CrossRef]

*μ*m thick layer of Chromium. For the manufacturing of the chromium mask an Electron Beam Pattern Generator (EBPG) was used. The diffraction pattern at a distance of 50 mm is then collected by a CCD camera (1280 × 960 pixel) through a camera lens (L3). Between the beam expander and the Hartmann grid a beamsplitter is placed in order to direct half of the beam to a Shack-Hartmann wavefront sensor (S-H WFS) through a 4x beam reducer (the clear aperture of this latter device is 4.8 mm) to monitor in real time the wavefront distribution. The measurements taken with the latter device are used as a reference for a comparison with the phase measured by the phase retrieval procedure.

*x*-direction and defocus. The phase is measured with the Shack-Hartmann wavefront sensor and the results are shown in the Fig. 9.

*λ*and 0.258

*λ*respectively. The phase retrieval results are presented in Fig. 10 and 11. Also, a image of the intensity pattern to which the phase retrieval algorithm is applied is shown, as well as the convergence of the optimization.

*λ*and 0.285

*λ*, in a reasonable agreement with the measurement using a Shack-Hartmann sensor.

## 6. Conclusion

## Acknowledgments

## References and links

1. | C. A. Mack, |

2. | V. Bakshi, |

3. | R. Saathof, Precision and Microsystem Engineering Dept., Delft University of Technology (private communication). |

4. | G. E. Sommargren, D. W. Phillion, M. A. Johnson, N. Q. Nguyen, A. Barty, F. J. Snell, D. R. Dillon, and L. S. Bradsher, “100-picometer interferometry for EUVL,” Proc. SPIE |

5. | A. Polo, F. Bociort, S. F. Pereira, and H. P. Urbach, “Wavefront measurement for EUV lithography system through Hartmann sensor,” Proc. SPIE |

6. | L. A. Carvalho, “A simple and effective algorithm for detection of arbitrary Hartmann-Shack patterns.” J. Biomed. Inf. |

7. | C. Leroux and C. Dainty, “Estimation of centroid positions with a matched-filter algorithm: relevance for aberrometry of the eye,” Opt. Express |

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

9. | R. Shannon, R. Shack, J. Harvey, and R. Hooker, |

10. | D. R. Neal, “Shack-Hartmann wavefront sensor precision and accuracy,” Proc. SPIE |

11. | P. Mercère, P. Zeitoun, M. Idir, S. L. Pape, D. Douillet, X. Levecq, G. Dovillaire, S. Bucourt, K. A. Goldberg, P. P. Naulleau, and S. Rekawa, “Hartmann wave-front measurement at 13.4 nm with 28, 1534–1536 (2003). [CrossRef] [PubMed] |

12. | P. Mercère, M. Idir, J. Floriot, X. Levecq, A. Erko, T. Krist, and A. Michette, |

13. | H. H. Barrett, C. Dainty, and D. Lara, “Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions,” J. Opt. Soc. Am. A. |

14. | J. W. Goodman, |

15. | R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik |

16. | J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. |

17. | J. R Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. |

18. | T. F. Coleman and Y. Li, “An interior trust region approach for nonlinear minimization subject to bounds,” SIAM J. Optim. |

19. | T. F. Coleman and Y. Li, “On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds,” Math. Program. |

20. | With the autorization of Imagine Optic, patent no. Eur 1415133 - US 7,255,442 - Jap 4212472. |

21. | D. Malacara, |

22. | C. López-Quesada, J. Andilla, and E. Martín-Badosa, “Correction of aberration in holographic optical tweezers using a Shack-Hartmann sensor,” Appl. Opt. |

**OCIS Codes**

(100.5070) Image processing : Phase retrieval

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

(220.1080) Optical design and fabrication : Active or adaptive optics

**ToC Category:**

Adaptive Optics

**History**

Original Manuscript: December 9, 2011

Revised Manuscript: March 3, 2012

Manuscript Accepted: March 15, 2012

Published: March 21, 2012

**Citation**

A. Polo, V. Kutchoukov, F. Bociort, S.F. Pereira, and H.P. Urbach, "Determination of wavefront structure for a Hartmann Wavefront Sensor using a phase-retrieval method," Opt. Express **20**, 7822-7832 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-7822

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### References

- C. A. Mack, Fundamental Principles of Optical Lithography: The Science of Microfabrication (Wiley-Interscience, 2007). [CrossRef]
- V. Bakshi, EUV Lithography (SPIE Press, 2009).
- R. Saathof, Precision and Microsystem Engineering Dept., Delft University of Technology (private communication).
- G. E. Sommargren, D. W. Phillion, M. A. Johnson, N. Q. Nguyen, A. Barty, F. J. Snell, D. R. Dillon, L. S. Bradsher, “100-picometer interferometry for EUVL,” Proc. SPIE 4688, 316–328 (2002). [CrossRef]
- A. Polo, F. Bociort, S. F. Pereira, H. P. Urbach, “Wavefront measurement for EUV lithography system through Hartmann sensor,” Proc. SPIE 7971, 79712R (2011). [CrossRef]
- L. A. Carvalho, “A simple and effective algorithm for detection of arbitrary Hartmann-Shack patterns.” J. Biomed. Inf. 37, 1–9 (2004). [CrossRef]
- C. Leroux, C. Dainty, “Estimation of centroid positions with a matched-filter algorithm: relevance for aberrometry of the eye,” Opt. Express 18, 1197–206 (2010). [CrossRef] [PubMed]
- W. H Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980). [CrossRef]
- R. Shannon, R. Shack, J. Harvey, R. Hooker, Robert Shannon and Roland Shack: legends in applied optics, Press Monograph (SPIE Press, 2005).
- D. R. Neal, “Shack-Hartmann wavefront sensor precision and accuracy,” Proc. SPIE 4779, 148–160 (2002). [CrossRef]
- P. Mercère, P. Zeitoun, M. Idir, S. L. Pape, D. Douillet, X. Levecq, G. Dovillaire, S. Bucourt, K. A. Goldberg, P. P. Naulleau, S. Rekawa, “Hartmann wave-front measurement at 13.4 nm with λEUV/120 accuracy,” Opt. Lett. 28, 1534–1536 (2003). [CrossRef] [PubMed]
- P. Mercère, M. Idir, J. Floriot, X. Levecq, A. Erko, T. Krist, A. Michette, Modern developments in X-Ray and neutron optics, (Springer BerlinHeidelberg, Berlin, Heidelberg, 2008).
- H. H. Barrett, C. Dainty, D. Lara, “Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions,” J. Opt. Soc. Am. A. 24, 391–414 (2007). [CrossRef]
- J. W. Goodman, Introduction to Fourier optics (Roberts and Company Publishers, 2005).
- R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
- J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–69 (1982). [CrossRef] [PubMed]
- J. R Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993). [CrossRef] [PubMed]
- T. F. Coleman, Y. Li, “An interior trust region approach for nonlinear minimization subject to bounds,” SIAM J. Optim. 6, 418–445 (1996). [CrossRef]
- T. F. Coleman, Y. Li, “On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds,” Math. Program. 67, 189–224 (1994). [CrossRef]
- With the autorization of Imagine Optic, patent no. Eur 1415133 - US 7,255,442 - Jap 4212472.
- D. Malacara, Optical Shop Testing (Wiley-Interscience, 2007). [CrossRef]
- C. López-Quesada, J. Andilla, E. Martín-Badosa, “Correction of aberration in holographic optical tweezers using a Shack-Hartmann sensor,” Appl. Opt. 48, 1084–1090 (2009). [CrossRef]

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