Abstract

We report theoretical and experimental results for imaging of electromagnetic phase edge effects in lithography photomasks. Our method starts from the transport of intensity equation (TIE), which solves for phase from through-focus intensity images. Traditional TIE algorithms make an implicit assumption that the underlying in-plane power flow is curl-free. Motivated by our current study, we describe a practical situation in which this assumption breaks down. Strong absorption gradients in mask features interact with phase edges to contribute a curl to the in-plane Poynting vector, causing severe artifacts in the phase recovered. We derive how curl effects are coupled into intensity measurements and propose an iterative algorithm that not only corrects the artifacts, but also recovers missing curl components.

© 2014 Optical Society of America

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  1. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [Crossref]
  2. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
    [Crossref]
  3. A. Barty, K. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
    [Crossref]
  4. E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
    [Crossref]
  5. C. Zuo, Q. Chen, Y. Yu, and A. Asundi, “Transport-of-intensity phase imaging using Savitzky–Golay differentiation filter-theory and applications,” Opt. Express 21, 5346–5362 (2013).
    [Crossref]
  6. Z. Jingshan, R. A. Claus, J. Dauwels, L. Tian, and L. Waller, “Transport of intensity phase imaging by intensity spectrum fitting of exponentially spaced defocus planes,” Opt. Express 22, 10661–10674 (2014).
    [Crossref]
  7. K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
    [Crossref]
  8. K. A. Nugent, “X-ray noninterferometric phase imaging: a unified picture,” J. Opt. Soc. Am. A 24, 536–547 (2007).
    [Crossref]
  9. M. Beleggia, M. Schofield, V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
    [Crossref]
  10. T. C. Petersen, V. J. Keast, and D. M. Paganin, “Quantitative TEM-based phase retrieval of MgO nano-cubes using the transport of intensity equation,” Ultramicroscopy 108, 805–815 (2008).
    [Crossref]
  11. K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. 54, 191–197 (2005).
    [Crossref]
  12. N. Loomis, L. Waller, and G. Barbastathis, “High-speed phase recovery using chromatic transport of intensity computation in graphics processing units,” in Biomedical Optics and 3-D Imaging (Optical Society of America, 2010), paper JMA7.
  13. M. Soto and E. Acosta, “Improved phase imaging from intensity measurements in multiple planes,” Appl. Opt. 46, 7978–7981 (2007).
    [Crossref]
  14. L. Waller, L. Tian, and G. Barbastathis, “Transport of intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18, 12552–12561 (2010).
    [Crossref]
  15. D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
    [Crossref]
  16. T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
    [Crossref]
  17. T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
    [Crossref]
  18. L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
    [Crossref]
  19. A. Lubk, G. Guzzinati, F. Börrnert, and J. Verbeeck, “Transport of intensity phase retrieval of arbitrary wave fields including vortices,” Phys. Rev. Lett. 111, 173902 (2013).
    [Crossref]
  20. V. Volkov and Y. Zhu, “Lorentz phase microscopy of magnetic materials,” Ultramicroscopy 98, 271–281 (2004).
    [Crossref]
  21. J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: validity of Teague’s method for solution of the transport-of-intensity equation,” Phys. Rev. A 84, 023808 (2011).
    [Crossref]
  22. J. A. Ferrari, G. A. Ayubi, J. L. Flores, and C. D. Perciante, “Transport of intensity equation: validity limits of the usually accepted solution,” Opt. Commun. 318, 133–136 (2014).
    [Crossref]
  23. A. Shanker, M. Sczyrba, B. Connolly, F. Kalk, A. Neureuther, and L. Waller, “Critical assessment of the transport of intensity equation as a phase recovery technique in optical lithography,” Proc. SPIE 9052, 90521D (2014).
    [Crossref]
  24. A. Shanker, L. Waller, and A. R. Neureuther, “Defocus based phase imaging for quantifying electromagnetic edge effects in photomasks,” Master’s thesis (University of California, 2014).
  25. C. Zuo, Q. Chen, L. Huang, and A. Asundi, “Phase discrepancy analysis and compensation for fast Fourier transform based solution of the transport of intensity equation,” Opt. Express 22, 17172–17186 (2014).
    [Crossref]
  26. K. Adam and A. Neureuther, “Simplified models for edge transitions in rigorous mask modeling,” Proc. SPIE 4346, 331–344 (2001).
    [Crossref]
  27. J. Tirapu-Azpiroz, P. Burchard, and E. Yablonovitch, “Boundary layer model to account for thick mask effects in photolithography,” Proc. SPIE 5040, 1611–1619 (2003).
    [Crossref]
  28. J. T. Azpiroz, A. E. Rosenbluth, K. Lai, C. Fonseca, and D. Yang, “Critical impact of mask electromagnetic effects on optical proximity corrections performance for 45 nm and beyond,” J. Vac. Sci. Technol. B 25, 164–168 (2007).
    [Crossref]
  29. T. Gureyev and K. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
  30. V. Volkov, Y. Zhu, and M. D. Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33, 411–416 (2002).
    [Crossref]
  31. A. Shanker, L. Tian, and L. Waller, “Defocus based quantitative phase imaging by coded illumination,” Proc. SPIE 8949, 89490R (2014).
  32. Y. Zhu, A. Pan, and G. Barbastathis, “Low-noise TIE phase imaging by structured illumination,” in Computational Optical Sensing and Imaging (Optical Society of America, 2014), paper CTh3C-5.
  33. A. M. Zysk, R. W. Schoonover, P. S. Carney, and M. A. Anastasio, “Transport of intensity and spectrum for partially coherent fields,” Opt. Lett. 35, 2239–2241 (2010).
    [Crossref]
  34. J. C. Petruccelli, L. Tian, and G. Barbastathis, “The Transport of intensity equation for optical path length recovery using partially coherent illumination,” Opt. Express 21, 14430–14441 (2013).
    [Crossref]
  35. Z. Jingshan, L. Tian, R. A. Claus, J. Dauwels, and L. Waller, “Partially coherent phase recovery by Kalman filtering,” in Frontiers in Optics 2013 Postdeadline (Optical Society of America, 2013), paper FW6A.9.
  36. M. H. Jenkins, J. M. Long, and T. K. Gaylord, “Multifilter phase imaging with partially coherent light,” Appl. Opt. 53, D29–D39 (2014).
    [Crossref]
  37. Z. Jingshan, L. Tian, J. Dauwels, and L. Waller, “Partially coherent phase microscopy with arbitrary illumination source shape,” in Computational Optical Sensing and Imaging (Optical Society of America, 2014), paper CTu1C5.
  38. M. Miller, “Mask edge effects in optical lithography and chip level modeling methods,” Ph.D. thesis (University of California, 2010).

2014 (6)

Z. Jingshan, R. A. Claus, J. Dauwels, L. Tian, and L. Waller, “Transport of intensity phase imaging by intensity spectrum fitting of exponentially spaced defocus planes,” Opt. Express 22, 10661–10674 (2014).
[Crossref]

J. A. Ferrari, G. A. Ayubi, J. L. Flores, and C. D. Perciante, “Transport of intensity equation: validity limits of the usually accepted solution,” Opt. Commun. 318, 133–136 (2014).
[Crossref]

A. Shanker, M. Sczyrba, B. Connolly, F. Kalk, A. Neureuther, and L. Waller, “Critical assessment of the transport of intensity equation as a phase recovery technique in optical lithography,” Proc. SPIE 9052, 90521D (2014).
[Crossref]

C. Zuo, Q. Chen, L. Huang, and A. Asundi, “Phase discrepancy analysis and compensation for fast Fourier transform based solution of the transport of intensity equation,” Opt. Express 22, 17172–17186 (2014).
[Crossref]

A. Shanker, L. Tian, and L. Waller, “Defocus based quantitative phase imaging by coded illumination,” Proc. SPIE 8949, 89490R (2014).

M. H. Jenkins, J. M. Long, and T. K. Gaylord, “Multifilter phase imaging with partially coherent light,” Appl. Opt. 53, D29–D39 (2014).
[Crossref]

2013 (3)

2011 (1)

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: validity of Teague’s method for solution of the transport-of-intensity equation,” Phys. Rev. A 84, 023808 (2011).
[Crossref]

2010 (2)

2008 (1)

T. C. Petersen, V. J. Keast, and D. M. Paganin, “Quantitative TEM-based phase retrieval of MgO nano-cubes using the transport of intensity equation,” Ultramicroscopy 108, 805–815 (2008).
[Crossref]

2007 (3)

M. Soto and E. Acosta, “Improved phase imaging from intensity measurements in multiple planes,” Appl. Opt. 46, 7978–7981 (2007).
[Crossref]

K. A. Nugent, “X-ray noninterferometric phase imaging: a unified picture,” J. Opt. Soc. Am. A 24, 536–547 (2007).
[Crossref]

J. T. Azpiroz, A. E. Rosenbluth, K. Lai, C. Fonseca, and D. Yang, “Critical impact of mask electromagnetic effects on optical proximity corrections performance for 45 nm and beyond,” J. Vac. Sci. Technol. B 25, 164–168 (2007).
[Crossref]

2006 (1)

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
[Crossref]

2005 (1)

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. 54, 191–197 (2005).
[Crossref]

2004 (2)

M. Beleggia, M. Schofield, V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref]

V. Volkov and Y. Zhu, “Lorentz phase microscopy of magnetic materials,” Ultramicroscopy 98, 271–281 (2004).
[Crossref]

2003 (1)

J. Tirapu-Azpiroz, P. Burchard, and E. Yablonovitch, “Boundary layer model to account for thick mask effects in photolithography,” Proc. SPIE 5040, 1611–1619 (2003).
[Crossref]

2002 (2)

V. Volkov, Y. Zhu, and M. D. Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33, 411–416 (2002).
[Crossref]

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[Crossref]

2001 (2)

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[Crossref]

K. Adam and A. Neureuther, “Simplified models for edge transitions in rigorous mask modeling,” Proc. SPIE 4346, 331–344 (2001).
[Crossref]

1998 (2)

A. Barty, K. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[Crossref]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[Crossref]

1997 (1)

T. Gureyev and K. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).

1996 (1)

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[Crossref]

1995 (1)

1984 (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[Crossref]

1983 (1)

Acosta, E.

Adam, K.

K. Adam and A. Neureuther, “Simplified models for edge transitions in rigorous mask modeling,” Proc. SPIE 4346, 331–344 (2001).
[Crossref]

Allen, L. J.

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[Crossref]

Allman, B.

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. 54, 191–197 (2005).
[Crossref]

Anastasio, M. A.

Asundi, A.

Ayubi, G. A.

J. A. Ferrari, G. A. Ayubi, J. L. Flores, and C. D. Perciante, “Transport of intensity equation: validity limits of the usually accepted solution,” Opt. Commun. 318, 133–136 (2014).
[Crossref]

Azpiroz, J. T.

J. T. Azpiroz, A. E. Rosenbluth, K. Lai, C. Fonseca, and D. Yang, “Critical impact of mask electromagnetic effects on optical proximity corrections performance for 45 nm and beyond,” J. Vac. Sci. Technol. B 25, 164–168 (2007).
[Crossref]

Barbastathis, G.

J. C. Petruccelli, L. Tian, and G. Barbastathis, “The Transport of intensity equation for optical path length recovery using partially coherent illumination,” Opt. Express 21, 14430–14441 (2013).
[Crossref]

L. Waller, L. Tian, and G. Barbastathis, “Transport of intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18, 12552–12561 (2010).
[Crossref]

Y. Zhu, A. Pan, and G. Barbastathis, “Low-noise TIE phase imaging by structured illumination,” in Computational Optical Sensing and Imaging (Optical Society of America, 2014), paper CTh3C-5.

N. Loomis, L. Waller, and G. Barbastathis, “High-speed phase recovery using chromatic transport of intensity computation in graphics processing units,” in Biomedical Optics and 3-D Imaging (Optical Society of America, 2010), paper JMA7.

Barnea, Z.

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[Crossref]

Barone-Nugent, E. D.

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[Crossref]

Barty, A.

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[Crossref]

A. Barty, K. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[Crossref]

Beleggia, M.

M. Beleggia, M. Schofield, V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref]

Börrnert, F.

A. Lubk, G. Guzzinati, F. Börrnert, and J. Verbeeck, “Transport of intensity phase retrieval of arbitrary wave fields including vortices,” Phys. Rev. Lett. 111, 173902 (2013).
[Crossref]

Burchard, P.

J. Tirapu-Azpiroz, P. Burchard, and E. Yablonovitch, “Boundary layer model to account for thick mask effects in photolithography,” Proc. SPIE 5040, 1611–1619 (2003).
[Crossref]

Carney, P. S.

Chen, Q.

Claus, R. A.

Z. Jingshan, R. A. Claus, J. Dauwels, L. Tian, and L. Waller, “Transport of intensity phase imaging by intensity spectrum fitting of exponentially spaced defocus planes,” Opt. Express 22, 10661–10674 (2014).
[Crossref]

Z. Jingshan, L. Tian, R. A. Claus, J. Dauwels, and L. Waller, “Partially coherent phase recovery by Kalman filtering,” in Frontiers in Optics 2013 Postdeadline (Optical Society of America, 2013), paper FW6A.9.

Connolly, B.

A. Shanker, M. Sczyrba, B. Connolly, F. Kalk, A. Neureuther, and L. Waller, “Critical assessment of the transport of intensity equation as a phase recovery technique in optical lithography,” Proc. SPIE 9052, 90521D (2014).
[Crossref]

Cookson, D.

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[Crossref]

Dauwels, J.

Z. Jingshan, R. A. Claus, J. Dauwels, L. Tian, and L. Waller, “Transport of intensity phase imaging by intensity spectrum fitting of exponentially spaced defocus planes,” Opt. Express 22, 10661–10674 (2014).
[Crossref]

Z. Jingshan, L. Tian, R. A. Claus, J. Dauwels, and L. Waller, “Partially coherent phase recovery by Kalman filtering,” in Frontiers in Optics 2013 Postdeadline (Optical Society of America, 2013), paper FW6A.9.

Z. Jingshan, L. Tian, J. Dauwels, and L. Waller, “Partially coherent phase microscopy with arbitrary illumination source shape,” in Computational Optical Sensing and Imaging (Optical Society of America, 2014), paper CTu1C5.

Faulkner, H. M. L.

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[Crossref]

Ferrari, J. A.

J. A. Ferrari, G. A. Ayubi, J. L. Flores, and C. D. Perciante, “Transport of intensity equation: validity limits of the usually accepted solution,” Opt. Commun. 318, 133–136 (2014).
[Crossref]

Flores, J. L.

J. A. Ferrari, G. A. Ayubi, J. L. Flores, and C. D. Perciante, “Transport of intensity equation: validity limits of the usually accepted solution,” Opt. Commun. 318, 133–136 (2014).
[Crossref]

Fonseca, C.

J. T. Azpiroz, A. E. Rosenbluth, K. Lai, C. Fonseca, and D. Yang, “Critical impact of mask electromagnetic effects on optical proximity corrections performance for 45 nm and beyond,” J. Vac. Sci. Technol. B 25, 164–168 (2007).
[Crossref]

Gaylord, T. K.

Graef, M. D.

V. Volkov, Y. Zhu, and M. D. Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33, 411–416 (2002).
[Crossref]

Gureyev, T.

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
[Crossref]

T. Gureyev and K. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[Crossref]

Gureyev, T. E.

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: validity of Teague’s method for solution of the transport-of-intensity equation,” Phys. Rev. A 84, 023808 (2011).
[Crossref]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
[Crossref]

Guzzinati, G.

A. Lubk, G. Guzzinati, F. Börrnert, and J. Verbeeck, “Transport of intensity phase retrieval of arbitrary wave fields including vortices,” Phys. Rev. Lett. 111, 173902 (2013).
[Crossref]

Huang, L.

Ishizuka, K.

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. 54, 191–197 (2005).
[Crossref]

Jenkins, M. H.

Jingshan, Z.

Z. Jingshan, R. A. Claus, J. Dauwels, L. Tian, and L. Waller, “Transport of intensity phase imaging by intensity spectrum fitting of exponentially spaced defocus planes,” Opt. Express 22, 10661–10674 (2014).
[Crossref]

Z. Jingshan, L. Tian, J. Dauwels, and L. Waller, “Partially coherent phase microscopy with arbitrary illumination source shape,” in Computational Optical Sensing and Imaging (Optical Society of America, 2014), paper CTu1C5.

Z. Jingshan, L. Tian, R. A. Claus, J. Dauwels, and L. Waller, “Partially coherent phase recovery by Kalman filtering,” in Frontiers in Optics 2013 Postdeadline (Optical Society of America, 2013), paper FW6A.9.

Kalk, F.

A. Shanker, M. Sczyrba, B. Connolly, F. Kalk, A. Neureuther, and L. Waller, “Critical assessment of the transport of intensity equation as a phase recovery technique in optical lithography,” Proc. SPIE 9052, 90521D (2014).
[Crossref]

Keast, V. J.

T. C. Petersen, V. J. Keast, and D. M. Paganin, “Quantitative TEM-based phase retrieval of MgO nano-cubes using the transport of intensity equation,” Ultramicroscopy 108, 805–815 (2008).
[Crossref]

Lai, K.

J. T. Azpiroz, A. E. Rosenbluth, K. Lai, C. Fonseca, and D. Yang, “Critical impact of mask electromagnetic effects on optical proximity corrections performance for 45 nm and beyond,” J. Vac. Sci. Technol. B 25, 164–168 (2007).
[Crossref]

Long, J. M.

Loomis, N.

N. Loomis, L. Waller, and G. Barbastathis, “High-speed phase recovery using chromatic transport of intensity computation in graphics processing units,” in Biomedical Optics and 3-D Imaging (Optical Society of America, 2010), paper JMA7.

Lubk, A.

A. Lubk, G. Guzzinati, F. Börrnert, and J. Verbeeck, “Transport of intensity phase retrieval of arbitrary wave fields including vortices,” Phys. Rev. Lett. 111, 173902 (2013).
[Crossref]

Miller, M.

M. Miller, “Mask edge effects in optical lithography and chip level modeling methods,” Ph.D. thesis (University of California, 2010).

Nesterets, Y.

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
[Crossref]

Neureuther, A.

A. Shanker, M. Sczyrba, B. Connolly, F. Kalk, A. Neureuther, and L. Waller, “Critical assessment of the transport of intensity equation as a phase recovery technique in optical lithography,” Proc. SPIE 9052, 90521D (2014).
[Crossref]

K. Adam and A. Neureuther, “Simplified models for edge transitions in rigorous mask modeling,” Proc. SPIE 4346, 331–344 (2001).
[Crossref]

Neureuther, A. R.

A. Shanker, L. Waller, and A. R. Neureuther, “Defocus based phase imaging for quantifying electromagnetic edge effects in photomasks,” Master’s thesis (University of California, 2014).

Nugent, K.

A. Barty, K. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[Crossref]

T. Gureyev and K. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[Crossref]

Nugent, K. A.

K. A. Nugent, “X-ray noninterferometric phase imaging: a unified picture,” J. Opt. Soc. Am. A 24, 536–547 (2007).
[Crossref]

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[Crossref]

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[Crossref]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[Crossref]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
[Crossref]

Oxley, M. P.

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[Crossref]

Paganin, D.

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
[Crossref]

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[Crossref]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[Crossref]

A. Barty, K. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[Crossref]

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[Crossref]

Paganin, D. M.

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: validity of Teague’s method for solution of the transport-of-intensity equation,” Phys. Rev. A 84, 023808 (2011).
[Crossref]

T. C. Petersen, V. J. Keast, and D. M. Paganin, “Quantitative TEM-based phase retrieval of MgO nano-cubes using the transport of intensity equation,” Ultramicroscopy 108, 805–815 (2008).
[Crossref]

Pan, A.

Y. Zhu, A. Pan, and G. Barbastathis, “Low-noise TIE phase imaging by structured illumination,” in Computational Optical Sensing and Imaging (Optical Society of America, 2014), paper CTh3C-5.

Pavlov, K. M.

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: validity of Teague’s method for solution of the transport-of-intensity equation,” Phys. Rev. A 84, 023808 (2011).
[Crossref]

Perciante, C. D.

J. A. Ferrari, G. A. Ayubi, J. L. Flores, and C. D. Perciante, “Transport of intensity equation: validity limits of the usually accepted solution,” Opt. Commun. 318, 133–136 (2014).
[Crossref]

Petersen, T. C.

T. C. Petersen, V. J. Keast, and D. M. Paganin, “Quantitative TEM-based phase retrieval of MgO nano-cubes using the transport of intensity equation,” Ultramicroscopy 108, 805–815 (2008).
[Crossref]

Petruccelli, J. C.

Pogany, A.

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
[Crossref]

Roberts, A.

Rosenbluth, A. E.

J. T. Azpiroz, A. E. Rosenbluth, K. Lai, C. Fonseca, and D. Yang, “Critical impact of mask electromagnetic effects on optical proximity corrections performance for 45 nm and beyond,” J. Vac. Sci. Technol. B 25, 164–168 (2007).
[Crossref]

Schmalz, J. A.

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: validity of Teague’s method for solution of the transport-of-intensity equation,” Phys. Rev. A 84, 023808 (2011).
[Crossref]

Schofield, M.

M. Beleggia, M. Schofield, V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref]

Schoonover, R. W.

Sczyrba, M.

A. Shanker, M. Sczyrba, B. Connolly, F. Kalk, A. Neureuther, and L. Waller, “Critical assessment of the transport of intensity equation as a phase recovery technique in optical lithography,” Proc. SPIE 9052, 90521D (2014).
[Crossref]

Shanker, A.

A. Shanker, M. Sczyrba, B. Connolly, F. Kalk, A. Neureuther, and L. Waller, “Critical assessment of the transport of intensity equation as a phase recovery technique in optical lithography,” Proc. SPIE 9052, 90521D (2014).
[Crossref]

A. Shanker, L. Tian, and L. Waller, “Defocus based quantitative phase imaging by coded illumination,” Proc. SPIE 8949, 89490R (2014).

A. Shanker, L. Waller, and A. R. Neureuther, “Defocus based phase imaging for quantifying electromagnetic edge effects in photomasks,” Master’s thesis (University of California, 2014).

Soto, M.

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[Crossref]

Teague, M. R.

Tian, L.

Z. Jingshan, R. A. Claus, J. Dauwels, L. Tian, and L. Waller, “Transport of intensity phase imaging by intensity spectrum fitting of exponentially spaced defocus planes,” Opt. Express 22, 10661–10674 (2014).
[Crossref]

A. Shanker, L. Tian, and L. Waller, “Defocus based quantitative phase imaging by coded illumination,” Proc. SPIE 8949, 89490R (2014).

J. C. Petruccelli, L. Tian, and G. Barbastathis, “The Transport of intensity equation for optical path length recovery using partially coherent illumination,” Opt. Express 21, 14430–14441 (2013).
[Crossref]

L. Waller, L. Tian, and G. Barbastathis, “Transport of intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18, 12552–12561 (2010).
[Crossref]

Z. Jingshan, L. Tian, J. Dauwels, and L. Waller, “Partially coherent phase microscopy with arbitrary illumination source shape,” in Computational Optical Sensing and Imaging (Optical Society of America, 2014), paper CTu1C5.

Z. Jingshan, L. Tian, R. A. Claus, J. Dauwels, and L. Waller, “Partially coherent phase recovery by Kalman filtering,” in Frontiers in Optics 2013 Postdeadline (Optical Society of America, 2013), paper FW6A.9.

Tirapu-Azpiroz, J.

J. Tirapu-Azpiroz, P. Burchard, and E. Yablonovitch, “Boundary layer model to account for thick mask effects in photolithography,” Proc. SPIE 5040, 1611–1619 (2003).
[Crossref]

Verbeeck, J.

A. Lubk, G. Guzzinati, F. Börrnert, and J. Verbeeck, “Transport of intensity phase retrieval of arbitrary wave fields including vortices,” Phys. Rev. Lett. 111, 173902 (2013).
[Crossref]

Volkov, V.

V. Volkov and Y. Zhu, “Lorentz phase microscopy of magnetic materials,” Ultramicroscopy 98, 271–281 (2004).
[Crossref]

M. Beleggia, M. Schofield, V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref]

V. Volkov, Y. Zhu, and M. D. Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33, 411–416 (2002).
[Crossref]

Waller, L.

A. Shanker, L. Tian, and L. Waller, “Defocus based quantitative phase imaging by coded illumination,” Proc. SPIE 8949, 89490R (2014).

A. Shanker, M. Sczyrba, B. Connolly, F. Kalk, A. Neureuther, and L. Waller, “Critical assessment of the transport of intensity equation as a phase recovery technique in optical lithography,” Proc. SPIE 9052, 90521D (2014).
[Crossref]

Z. Jingshan, R. A. Claus, J. Dauwels, L. Tian, and L. Waller, “Transport of intensity phase imaging by intensity spectrum fitting of exponentially spaced defocus planes,” Opt. Express 22, 10661–10674 (2014).
[Crossref]

L. Waller, L. Tian, and G. Barbastathis, “Transport of intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18, 12552–12561 (2010).
[Crossref]

Z. Jingshan, L. Tian, R. A. Claus, J. Dauwels, and L. Waller, “Partially coherent phase recovery by Kalman filtering,” in Frontiers in Optics 2013 Postdeadline (Optical Society of America, 2013), paper FW6A.9.

Z. Jingshan, L. Tian, J. Dauwels, and L. Waller, “Partially coherent phase microscopy with arbitrary illumination source shape,” in Computational Optical Sensing and Imaging (Optical Society of America, 2014), paper CTu1C5.

A. Shanker, L. Waller, and A. R. Neureuther, “Defocus based phase imaging for quantifying electromagnetic edge effects in photomasks,” Master’s thesis (University of California, 2014).

N. Loomis, L. Waller, and G. Barbastathis, “High-speed phase recovery using chromatic transport of intensity computation in graphics processing units,” in Biomedical Optics and 3-D Imaging (Optical Society of America, 2010), paper JMA7.

Wilkins, S.

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
[Crossref]

Yablonovitch, E.

J. Tirapu-Azpiroz, P. Burchard, and E. Yablonovitch, “Boundary layer model to account for thick mask effects in photolithography,” Proc. SPIE 5040, 1611–1619 (2003).
[Crossref]

Yang, D.

J. T. Azpiroz, A. E. Rosenbluth, K. Lai, C. Fonseca, and D. Yang, “Critical impact of mask electromagnetic effects on optical proximity corrections performance for 45 nm and beyond,” J. Vac. Sci. Technol. B 25, 164–168 (2007).
[Crossref]

Yu, Y.

Zhu, Y.

V. Volkov and Y. Zhu, “Lorentz phase microscopy of magnetic materials,” Ultramicroscopy 98, 271–281 (2004).
[Crossref]

M. Beleggia, M. Schofield, V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref]

V. Volkov, Y. Zhu, and M. D. Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33, 411–416 (2002).
[Crossref]

Y. Zhu, A. Pan, and G. Barbastathis, “Low-noise TIE phase imaging by structured illumination,” in Computational Optical Sensing and Imaging (Optical Society of America, 2014), paper CTh3C-5.

Zuo, C.

Zysk, A. M.

Appl. Opt. (2)

J. Electron Microsc. (1)

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. 54, 191–197 (2005).
[Crossref]

J. Microsc. (1)

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Vac. Sci. Technol. B (1)

J. T. Azpiroz, A. E. Rosenbluth, K. Lai, C. Fonseca, and D. Yang, “Critical impact of mask electromagnetic effects on optical proximity corrections performance for 45 nm and beyond,” J. Vac. Sci. Technol. B 25, 164–168 (2007).
[Crossref]

Micron (1)

V. Volkov, Y. Zhu, and M. D. Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33, 411–416 (2002).
[Crossref]

Opt. Commun. (4)

T. Gureyev and K. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).

T. Gureyev, Y. Nesterets, D. Paganin, A. Pogany, and S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569–580 (2006).
[Crossref]

J. A. Ferrari, G. A. Ayubi, J. L. Flores, and C. D. Perciante, “Transport of intensity equation: validity limits of the usually accepted solution,” Opt. Commun. 318, 133–136 (2014).
[Crossref]

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[Crossref]

Opt. Express (5)

Opt. Lett. (2)

Phys. Rev. A (1)

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter-wave fields: validity of Teague’s method for solution of the transport-of-intensity equation,” Phys. Rev. A 84, 023808 (2011).
[Crossref]

Phys. Rev. E (1)

L. J. Allen, H. M. L. Faulkner, K. A. Nugent, M. P. Oxley, and D. Paganin, “Phase retrieval from images in the presence of first-order vortices,” Phys. Rev. E 63, 037602 (2001).
[Crossref]

Phys. Rev. Lett. (3)

A. Lubk, G. Guzzinati, F. Börrnert, and J. Verbeeck, “Transport of intensity phase retrieval of arbitrary wave fields including vortices,” Phys. Rev. Lett. 111, 173902 (2013).
[Crossref]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[Crossref]

K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[Crossref]

Proc. SPIE (4)

A. Shanker, M. Sczyrba, B. Connolly, F. Kalk, A. Neureuther, and L. Waller, “Critical assessment of the transport of intensity equation as a phase recovery technique in optical lithography,” Proc. SPIE 9052, 90521D (2014).
[Crossref]

K. Adam and A. Neureuther, “Simplified models for edge transitions in rigorous mask modeling,” Proc. SPIE 4346, 331–344 (2001).
[Crossref]

J. Tirapu-Azpiroz, P. Burchard, and E. Yablonovitch, “Boundary layer model to account for thick mask effects in photolithography,” Proc. SPIE 5040, 1611–1619 (2003).
[Crossref]

A. Shanker, L. Tian, and L. Waller, “Defocus based quantitative phase imaging by coded illumination,” Proc. SPIE 8949, 89490R (2014).

Ultramicroscopy (3)

M. Beleggia, M. Schofield, V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[Crossref]

T. C. Petersen, V. J. Keast, and D. M. Paganin, “Quantitative TEM-based phase retrieval of MgO nano-cubes using the transport of intensity equation,” Ultramicroscopy 108, 805–815 (2008).
[Crossref]

V. Volkov and Y. Zhu, “Lorentz phase microscopy of magnetic materials,” Ultramicroscopy 98, 271–281 (2004).
[Crossref]

Other (6)

N. Loomis, L. Waller, and G. Barbastathis, “High-speed phase recovery using chromatic transport of intensity computation in graphics processing units,” in Biomedical Optics and 3-D Imaging (Optical Society of America, 2010), paper JMA7.

Y. Zhu, A. Pan, and G. Barbastathis, “Low-noise TIE phase imaging by structured illumination,” in Computational Optical Sensing and Imaging (Optical Society of America, 2014), paper CTh3C-5.

Z. Jingshan, L. Tian, R. A. Claus, J. Dauwels, and L. Waller, “Partially coherent phase recovery by Kalman filtering,” in Frontiers in Optics 2013 Postdeadline (Optical Society of America, 2013), paper FW6A.9.

Z. Jingshan, L. Tian, J. Dauwels, and L. Waller, “Partially coherent phase microscopy with arbitrary illumination source shape,” in Computational Optical Sensing and Imaging (Optical Society of America, 2014), paper CTu1C5.

M. Miller, “Mask edge effects in optical lithography and chip level modeling methods,” Ph.D. thesis (University of California, 2010).

A. Shanker, L. Waller, and A. R. Neureuther, “Defocus based phase imaging for quantifying electromagnetic edge effects in photomasks,” Master’s thesis (University of California, 2014).

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Figures (5)

Fig. 1.
Fig. 1.

Photolithography masks incur polarization-dependent electromagnetic edge effects. (a) Because the mask is relatively thick, the electric field accumulates an unwanted phase delay at the feature edges, due to diffraction. (b) An ideal mask has only absorption variations. (c) Phase edge effects can be modeled by phase strips at the feature edges, depending on polarization [2628].

Fig. 2.
Fig. 2.

(Left) Simulation of a 240 nm square absorbing feature on a photomask with phase edges added along the vertical sides, causing a nonzero curl in the Poynting vector near the feature corners, where the phase gradient is tangential to intensity contours. (Right) When through-focus images are simulated for this complex field and are used as input to the traditional TIE solver, the phase recovered suffers serious errors due to the curl effects.

Fig. 3.
Fig. 3.

Simulation showing the first iteration of our algorithm. Teague’s solver recovers the initial phase estimate; then we plug that into the TIE to find the estimated intensity derivative. The residual between the measured and estimated intensity derivatives is plugged into Teague’s solver a second time in order to estimate the phase residual, which is subtracted from the recovered phase for an improved estimate.

Fig. 4.
Fig. 4.

Estimating curl components with the iterative TIE. The top row shows the true curl for the simulated mask, and the corresponding curl recovered by our iterative algorithm. The bottom plot shows that the RMS error in our estimate of the curl diminishes progressively as the algorithm iterates. Errors in the curl component estimation for the first three iterations of the algorithm are shown as insets.

Fig. 5.
Fig. 5.

Experimental results for our iterative TIE method, as compared with those for the traditional Teague’s solver, with a 240 nm square feature on an OMOG mask. The top row shows the phase recovered by Teague’s solver, with nonphysical saddle artifacts due to the Poynting vector curl at the feature corners. The bottom row shows the phase recovered by the iterative solver, where artifacts have been corrected, clearly showing the presence of phase edges that match well the theoretical predictions. On the left is the result for illumination polarized in the horizontal direction, whereas on the right is that for the vertical direction, showing strong polarization dependence, as expected.

Equations (14)

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

dIdz=λ2π∇⃗·I∇⃗ϕ,
dIdz=λ2π2ψ.
∇⃗·(∇⃗ψ/I)=2ϕ,
∇⃗×(I∇⃗ϕ)=∇⃗I×∇⃗ϕ=0.
I∇⃗ϕ=∇⃗ψ+∇⃗×A⃗1,
∇⃗·∇⃗ψI+∇⃗·∇⃗×A1I=2ϕ,
2ϕTIE+2ϕres=2ϕ,
dIdz|est=λ2π∇⃗·I∇⃗ϕTIE.
dIdz|res=dIdzdIdz|est,
dIdz|res=λ2π∇⃗·I∇⃗ϕr1,
∇⃗ψ/I=∇⃗ϕTIE+∇⃗×A⃗2,
dIdz|res=λ2π{2ψ1∇⃗·I∇⃗ϕTIE},
dIdz|res=λ2π∇⃗·I∇⃗×A⃗2.
××A⃗2=×(×A⃗1I),

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