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

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References

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  1. C. A. Mack, Fundamental Principles of Optical Lithography: The Science of Microfabrication (Wiley-Interscience, 2007).
    [CrossRef]
  2. V. Bakshi, EUV Lithography (SPIE Press, 2009).
  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 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]
  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]
  9. R. Shannon, R. Shack, J. Harvey, and R. Hooker, Robert Shannon and Roland Shack: legends in applied optics, Press Monograph (SPIE Press, 2005).
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    [CrossRef]
  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 λEUV/120 accuracy,” Opt. Lett. 28, 1534–1536 (2003).
    [CrossRef] [PubMed]
  12. P. Mercère, M. Idir, J. Floriot, X. Levecq, A. Erko, T. Krist, and A. Michette, Modern developments in X-Ray and neutron optics, (Springer BerlinHeidelberg, Berlin, Heidelberg, 2008).
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    [CrossRef]
  14. J. W. Goodman, Introduction to Fourier optics (Roberts and Company Publishers, 2005).
  15. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  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]
  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]
  20. With the autorization of Imagine Optic, patent no. Eur 1415133 - US 7,255,442 - Jap 4212472.
  21. D. Malacara, Optical Shop Testing (Wiley-Interscience, 2007).
    [CrossRef]
  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]

2011 (1)

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]

2010 (1)

2009 (1)

2007 (1)

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]

2004 (1)

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

2003 (1)

2002 (2)

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]

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

1996 (1)

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]

1994 (1)

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]

1993 (1)

1982 (1)

1980 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Andilla, J.

Bakshi, V.

V. Bakshi, EUV Lithography (SPIE Press, 2009).

Barrett, H. H.

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]

Barty, A.

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]

Bociort, F.

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]

Bradsher, L. S.

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]

Bucourt, S.

Carvalho, L. A.

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

Coleman, T. F.

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]

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]

Dainty, C.

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]

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]

Dillon, D. R.

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]

Douillet, D.

Dovillaire, G.

Erko, A.

P. Mercère, M. Idir, J. Floriot, X. Levecq, A. Erko, T. Krist, and A. Michette, Modern developments in X-Ray and neutron optics, (Springer BerlinHeidelberg, Berlin, Heidelberg, 2008).

Fienup, J. R

Fienup, J. R.

Floriot, J.

P. Mercère, M. Idir, J. Floriot, X. Levecq, A. Erko, T. Krist, and A. Michette, Modern developments in X-Ray and neutron optics, (Springer BerlinHeidelberg, Berlin, Heidelberg, 2008).

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Goldberg, K. A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier optics (Roberts and Company Publishers, 2005).

Harvey, J.

R. Shannon, R. Shack, J. Harvey, and R. Hooker, Robert Shannon and Roland Shack: legends in applied optics, Press Monograph (SPIE Press, 2005).

Hooker, R.

R. Shannon, R. Shack, J. Harvey, and R. Hooker, Robert Shannon and Roland Shack: legends in applied optics, Press Monograph (SPIE Press, 2005).

Idir, M.

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 λEUV/120 accuracy,” Opt. Lett. 28, 1534–1536 (2003).
[CrossRef] [PubMed]

P. Mercère, M. Idir, J. Floriot, X. Levecq, A. Erko, T. Krist, and A. Michette, Modern developments in X-Ray and neutron optics, (Springer BerlinHeidelberg, Berlin, Heidelberg, 2008).

Johnson, M. A.

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]

Krist, T.

P. Mercère, M. Idir, J. Floriot, X. Levecq, A. Erko, T. Krist, and A. Michette, Modern developments in X-Ray and neutron optics, (Springer BerlinHeidelberg, Berlin, Heidelberg, 2008).

Lara, D.

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]

Leroux, C.

Levecq, X.

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 λEUV/120 accuracy,” Opt. Lett. 28, 1534–1536 (2003).
[CrossRef] [PubMed]

P. Mercère, M. Idir, J. Floriot, X. Levecq, A. Erko, T. Krist, and A. Michette, Modern developments in X-Ray and neutron optics, (Springer BerlinHeidelberg, Berlin, Heidelberg, 2008).

Li, Y.

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]

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]

López-Quesada, C.

Mack, C. A.

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

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley-Interscience, 2007).
[CrossRef]

Martín-Badosa, E.

Mercère, P.

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 λEUV/120 accuracy,” Opt. Lett. 28, 1534–1536 (2003).
[CrossRef] [PubMed]

P. Mercère, M. Idir, J. Floriot, X. Levecq, A. Erko, T. Krist, and A. Michette, Modern developments in X-Ray and neutron optics, (Springer BerlinHeidelberg, Berlin, Heidelberg, 2008).

Michette, A.

P. Mercère, M. Idir, J. Floriot, X. Levecq, A. Erko, T. Krist, and A. Michette, Modern developments in X-Ray and neutron optics, (Springer BerlinHeidelberg, Berlin, Heidelberg, 2008).

Naulleau, P. P.

Neal, D. R.

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

Nguyen, N. Q.

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]

Pape, S. L.

Pereira, S. F.

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]

Phillion, D. W.

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]

Polo, A.

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]

Rekawa, S.

Saathof, R.

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

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Shack, R.

R. Shannon, R. Shack, J. Harvey, and R. Hooker, Robert Shannon and Roland Shack: legends in applied optics, Press Monograph (SPIE Press, 2005).

Shannon, R.

R. Shannon, R. Shack, J. Harvey, and R. Hooker, Robert Shannon and Roland Shack: legends in applied optics, Press Monograph (SPIE Press, 2005).

Snell, F. J.

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]

Sommargren, G. E.

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]

Southwell, W. H

Urbach, H. P.

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]

Zeitoun, P.

Appl. Opt. (3)

J. Biomed. Inf. (1)

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A. (1)

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]

Math. Program. (1)

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]

Opt. Express (1)

Opt. Lett. (1)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Proc. SPIE (3)

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

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]

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]

SIAM J. Optim. (1)

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]

Other (8)

P. Mercère, M. Idir, J. Floriot, X. Levecq, A. Erko, T. Krist, and A. Michette, Modern developments in X-Ray and neutron optics, (Springer BerlinHeidelberg, Berlin, Heidelberg, 2008).

J. W. Goodman, Introduction to Fourier optics (Roberts and Company Publishers, 2005).

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).

R. Shannon, R. Shack, J. Harvey, and R. Hooker, Robert Shannon and Roland Shack: legends in applied optics, Press Monograph (SPIE Press, 2005).

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]

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Figures (11)

Fig. 1
Fig. 1

(a) graphical representation of a HWS. (b) sampled wavefront approximated by a flat wavefront in the wavefront measurement with a HWS

Fig. 2
Fig. 2

(a) Detail of the intensity distribution on the detector. (b) Normalized statistical representation of the noise distribution

Fig. 3
Fig. 3

Wavefront used for simulation. (a) Defocus distribution (RMS = 0.0035λ, PV = 0.0135λ), (b) Zernike coefficient at the defocus simulation. (c) Astigmatism distribution (RMS = 0.0017λ, P–V = 0.0084λ), (d) Zernike coefficient at the astigmatism simulation.

Fig. 4
Fig. 4

Reconstructed phase distribution based on the slope measurement method for (a) defocus (RMS = 0.0043λ, P – V = 0.0220λ) and (c) astigmatism (RMS = 0.0130λ, P – V = 0.0072λ) respectively. The obtained wavefront reconstruction error of (b) defocus (RMS error = 0.0024λ) and (d) astigmatism (RMS error = 0.0023λ).

Fig. 5
Fig. 5

Reconstructed phase distribution based on the phase retrieval procedure for (a) defocus (RMS = 0.0037λ, P – V = 0.0135λ) and (c) astigmatism (RMS = 0.0017λ, P – V = 0.0083λ). The obtained reconstruction wavefront error of (b) defocus (RMS error = 5.46 × 10−4λ) and (d) astigmatism (RMS error = 2.24 × 10−4λ).

Fig. 6
Fig. 6

Zernike spectrum of the input aberration compared with the reconstructed wavefront for both the slope measurement and phase retrieval method. (a) Defocus case, (b) astigmatism case.

Fig. 7
Fig. 7

Optical bench working at λ = 638 nm with the SLM as a wavefront generator and the Hartmann grid as a wavefront measuring device

Fig. 8
Fig. 8

(a) Wavefront distribution measured with a Shack-Hartmann wavefront sensor revealing the systematic error in the SLM. (b) Wavefront distribution after correcting for the non-flatness of the SLM surface. (c) Phase and intensity shift of the Holoeye LC-R 2500 measured in situ with a Shack-Hartmann wavefront sensor

Fig. 9
Fig. 9

Shack-Hartmann wavefront measurement. (a) Tilt case, (b) defocus case

Fig. 10
Fig. 10

Phase retrieval results for tilt aberration: (a) intensity pattern on which the phase retrieval is applied, (b) retrieved phase distribution, (c) convergence of the optimization algorithm

Fig. 11
Fig. 11

Phase retrieval results for defocus aberration: (a) intensity pattern on which the phase retrieval is applied, (b) retrieved phase distribution, (c) convergence of the optimization algorithm

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

CD = k 1 λ NA ,
Φ x ( x i , y j ) Δ x i , j L Φ y ( x i , y j ) Δ y i , j L ,
U ( x , y , 0 ) = A ( x , y ) exp [ i k Φ ] .
U ( x , y , L ) = 1 [ [ [ U ( x , y , 0 ) ] exp [ i 2 π λ L 1 ( λ f x ) 2 ( λ f y ) 2 ] ]
H ( f x , f y ) = exp [ i 2 π λ L 1 ( λ f x ) 2 ( λ f y ) 2 ) ]
Φ ( x , y ) = k = 1 K α k Z k ( x , y ) ,
E = i , j [ | F ( x i , y j ) | | U ( x i , y j , L ) | ] 2
F n = d 2 / λ L ,

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