Coherent diffraction surface imaging in reflection geometry

Shashidhara Marathe; S. S. Kim; S. N. Kim; Chan Kim; H. C. Kang; P. V. Nickles; D. Y. Noh

doi:10.1364/OE.18.007253

Coherent diffraction surface imaging in reflection geometry

Shashidhara Marathe, S. S. Kim, S. N. Kim, Chan Kim, H. C. Kang, P. V. Nickles, and D. Y. Noh

Optics Express
Vol. 18,
Issue 7,
pp. 7253-7262
(2010)
•https://doi.org/10.1364/OE.18.007253

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Shashidhara Marathe, S. S. Kim, S. N. Kim, Chan Kim, H. C. Kang, P. V. Nickles, and D. Y. Noh, "Coherent diffraction surface imaging in reflection geometry," Opt. Express 18, 7253-7262 (2010)

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Related Topics
Table of Contents Category
- Imaging Systems
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Phase shift (050.5080)
- Phase retrieval (100.5070)
- Coherence imaging (110.1650)

About this Article
History
- Original Manuscript: January 5, 2010
- Revised Manuscript: February 26, 2010
- Manuscript Accepted: March 17, 2010
- Published: March 24, 2010

Accessible

Open Access

Abstract

We present a reflection based coherent diffraction imaging method which can be used to reconstruct a non periodic surface image from a diffraction amplitude measured in reflection geometry. Using a He-Ne laser, we demonstrated that a surface image can be reconstructed solely from the reflected intensity from a surface without relying on any prior knowledge of the sample object or the object support. The reconstructed phase image of the exit wave is particularly interesting since it can be used to obtain quantitative information of the surface depth profile or the phase change during the reflection process. We believe that this work will broaden the application areas of coherent diffraction imaging techniques using light sources with limited penetration depth.

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References

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Author
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Publication

D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5(6), 843 (1952).
[Crossref]
R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik (Stuttg.) 61, 247–262 (1982).
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[Crossref]
J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67(17), 174104 (2003).
[Crossref]
R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–246 (1972).
J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3(1), 27–29 (1978).
[Crossref] [PubMed]
J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982).
[Crossref] [PubMed]
J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3(11), 1897–1907 (1986).
[Crossref]
J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4(1), 118–123 (1987).
[Crossref]
T. R. Crimmins, J. R. Fienup, and B. J. Thelen, “Improved bounds on object support from autocorrelation support and application to phase retrieval,” J. Opt. Soc. Am. A 7(1), 3–13 (1990).
[Crossref]
J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999).
[Crossref]
I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87(19), 195505 (2001).
[Crossref] [PubMed]
J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89(8), 088303 (2002).
[Crossref] [PubMed]
Y. Nishino, J. Miao, and T. Ishikawa, “Image reconstruction of nanostructured nonperiodic objects only from oversampled hard x-ray diffraction intensities,” Phys. Rev. B 68(22), 220101 (2003).
[Crossref]
S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68(14), 140101 (2003).
[Crossref]
H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düsterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. van der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-X-ray free- electron laser,” Nat. Phys. 2(12), 839–843 (2006).
[Crossref]
J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[Crossref] [PubMed]
P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]
A. Taguchi, T. Miyoshi, Y. Takaya, S. Takahashi, and K. Saito, “3D Micro-Profile Measurement using Optical Inverse Scattering Phase Method,” Annals CIRP 49(1), 423–426 (2000).
[Crossref]
A. Taguchi, T. Miyoshi, Y. Takaya, and S. Takahashi, “Optical 3D profilometer in- process measurement of microsurface based on phase retrieval technique,” Precis. Eng. 28(2), 152–163 (2004).
[Crossref]
J. R. Fienup, “Lensless coherent imaging by phase retrieval with an illumination pattern constraint,” Opt. Express 14(2), 498–508 (2006).
[Crossref] [PubMed]
M. Born, and E. Wolf, Principles of Optics (Cambridge University press, 1999),7th Edition.
J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill Companies, Inc., 1996), 2nd ed.
D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102(43), 15343–15346 (2005).
[Crossref] [PubMed]

2008 (1)

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

2007 (1)

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[Crossref] [PubMed]

2006 (2)

J. R. Fienup, “Lensless coherent imaging by phase retrieval with an illumination pattern constraint,” Opt. Express 14(2), 498–508 (2006).
[Crossref] [PubMed]

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plönjes, M. Kuhlmann, R. Treusch, S. Düsterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Möller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. van der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szöke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-X-ray free- electron laser,” Nat. Phys. 2(12), 839–843 (2006).
[Crossref]

2005 (1)

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102(43), 15343–15346 (2005).
[Crossref] [PubMed]

2004 (1)

A. Taguchi, T. Miyoshi, Y. Takaya, and S. Takahashi, “Optical 3D profilometer in- process measurement of microsurface based on phase retrieval technique,” Precis. Eng. 28(2), 152–163 (2004).
[Crossref]

2003 (3)

Y. Nishino, J. Miao, and T. Ishikawa, “Image reconstruction of nanostructured nonperiodic objects only from oversampled hard x-ray diffraction intensities,” Phys. Rev. B 68(22), 220101 (2003).
[Crossref]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68(14), 140101 (2003).
[Crossref]

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67(17), 174104 (2003).
[Crossref]

2002 (1)

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89(8), 088303 (2002).
[Crossref] [PubMed]

2001 (1)

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87(19), 195505 (2001).
[Crossref] [PubMed]

2000 (1)

A. Taguchi, T. Miyoshi, Y. Takaya, S. Takahashi, and K. Saito, “3D Micro-Profile Measurement using Optical Inverse Scattering Phase Method,” Annals CIRP 49(1), 423–426 (2000).
[Crossref]

1999 (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999).
[Crossref]

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik (Stuttg.) 61, 247–262 (1982).

1978 (1)

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3(1), 27–29 (1978).
[Crossref] [PubMed]

1972 (1)

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

1952 (1)

D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5(6), 843 (1952).
[Crossref]

Anderson, E. H.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89(8), 088303 (2002).
[Crossref] [PubMed]

Bajt, S.

Barty, A.

Bates, R. H. T.

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik (Stuttg.) 61, 247–262 (1982).

Beetz, T.

Benner, W. H.

Bergh, M.

Bogan, M. J.

Bostedt, C.

Boutet, S.

Bunk, O.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

Burmeister, F.

Caleman, C.

Chapman, H. N.

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15(6), 1662–1669 (1998).
[Crossref]

Charalambous, P.

Crimmins, T. R.

Cullis, A. G.

David, C.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

Dierolf, M.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

Dobson, B. R.

Düsterer, S.

Elser, V.

Fienup, J. R.

J. R. Fienup, “Lensless coherent imaging by phase retrieval with an illumination pattern constraint,” Opt. Express 14(2), 498–508 (2006).
[Crossref] [PubMed]

J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4(1), 118–123 (1987).
[Crossref]

J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3(11), 1897–1907 (1986).
[Crossref]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982).
[Crossref] [PubMed]

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3(1), 27–29 (1978).
[Crossref] [PubMed]

Frank, M.

Gerchberg, R. W.

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

Hajdu, J.

Hau-Riege, S. P.

He, H.

Hodgson, K. O.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89(8), 088303 (2002).
[Crossref] [PubMed]

Hoener, M.

Howells, M.

Howells, M. R.

Huldt, G.

Hurst, A. C.

Ishikawa, T.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89(8), 088303 (2002).
[Crossref] [PubMed]

Jacobsen, C.

Jefimovs, K.

Johnson, B.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89(8), 088303 (2002).
[Crossref] [PubMed]

Johnson, I.

Kirz, J.

Kuhlmann, M.

Lai, B.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89(8), 088303 (2002).
[Crossref] [PubMed]

Lee, R. W.

Lima, E.

London, R. A.

Maia, F. R. N. C.

Marchesini, S.

Menzel, A.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

Miao, H.

Miao, J.

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89(8), 088303 (2002).
[Crossref] [PubMed]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15(6), 1662–1669 (1998).
[Crossref]

Miyoshi, T.

A. Taguchi, T. Miyoshi, Y. Takaya, S. Takahashi, and K. Saito, “3D Micro-Profile Measurement using Optical Inverse Scattering Phase Method,” Annals CIRP 49(1), 423–426 (2000).
[Crossref]

Möller, T.

Neiman, A. M.

Nishino, Y.

Noy, A.

Pfeifer, M. A.

Pfeiffer, F.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

Pitney, J. A.

Plönjes, E.

Robinson, I. K.

Rodenburg, J. M.

Saito, K.

A. Taguchi, T. Miyoshi, Y. Takaya, S. Takahashi, and K. Saito, “3D Micro-Profile Measurement using Optical Inverse Scattering Phase Method,” Annals CIRP 49(1), 423–426 (2000).
[Crossref]

Saxton, W. O.

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

Sayre, D.

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15(6), 1662–1669 (1998).
[Crossref]

D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5(6), 843 (1952).
[Crossref]

Schneider, J. R.

Seibert, M. M.

Shapiro, D.

Shapiro, D. A.

Spence, J. C. H.

Spiller, E.

Szöke, A.

Taguchi, A.

A. Taguchi, T. Miyoshi, Y. Takaya, S. Takahashi, and K. Saito, “3D Micro-Profile Measurement using Optical Inverse Scattering Phase Method,” Annals CIRP 49(1), 423–426 (2000).
[Crossref]

Takahashi, S.

A. Taguchi, T. Miyoshi, Y. Takaya, S. Takahashi, and K. Saito, “3D Micro-Profile Measurement using Optical Inverse Scattering Phase Method,” Annals CIRP 49(1), 423–426 (2000).
[Crossref]

Takaya, Y.

A. Taguchi, T. Miyoshi, Y. Takaya, S. Takahashi, and K. Saito, “3D Micro-Profile Measurement using Optical Inverse Scattering Phase Method,” Annals CIRP 49(1), 423–426 (2000).
[Crossref]

Thelen, B. J.

Thibault, P.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

Timneanu, N.

Treusch, R.

Tschentscher, T.

van der Spoel, D.

Vartanyants, I. A.

Wackerman, C. C.

J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3(11), 1897–1907 (1986).
[Crossref]

Weierstall, U.

Williams, G. J.

Woods, B. W.

Acta Crystallogr. (1)

D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5(6), 843 (1952).
[Crossref]

Annals CIRP (1)

A. Taguchi, T. Miyoshi, Y. Takaya, S. Takahashi, and K. Saito, “3D Micro-Profile Measurement using Optical Inverse Scattering Phase Method,” Annals CIRP 49(1), 423–426 (2000).
[Crossref]

Appl. Opt. (1)

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (4)

J. R. Fienup and C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3(11), 1897–1907 (1986).
[Crossref]

J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4(1), 118–123 (1987).
[Crossref]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15(6), 1662–1669 (1998).
[Crossref]

Nat. Phys. (1)

Nature (1)

Opt. Express (1)

J. R. Fienup, “Lensless coherent imaging by phase retrieval with an illumination pattern constraint,” Opt. Express 14(2), 498–508 (2006).
[Crossref] [PubMed]

Opt. Lett. (1)

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3(1), 27–29 (1978).
[Crossref] [PubMed]

Optik (Jena) (1)

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

Optik (Stuttg.) (1)

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I: Underlying theory,” Optik (Stuttg.) 61, 247–262 (1982).

Phys. Rev. B (3)

Phys. Rev. Lett. (3)

J. Miao, T. Ishikawa, B. Johnson, E. H. Anderson, B. Lai, and K. O. Hodgson, “High resolution 3D x-ray diffraction microscopy,” Phys. Rev. Lett. 89(8), 088303 (2002).
[Crossref] [PubMed]

Precis. Eng. (1)

Proc. Natl. Acad. Sci. U.S.A. (1)

Science (1)

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

Other (2)

M. Born, and E. Wolf, Principles of Optics (Cambridge University press, 1999),7th Edition.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill Companies, Inc., 1996), 2nd ed.

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Fig. 1 Schematic of the reflection geometry used in this experiment.

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Fig. 2 Schematic diagram of the experimental setup. A He-Ne laser beam passing through a 500µm aperture illuminates the sample defined by another 500µm aperture. The reflected light is collected by a biconvex lens that focuses the beam on to a CCD. A microscope image of the first test sample is also illustrated.

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Fig. 3 Reflection pattern obtained at 15°, 30°, 45° incident angles respectively. Each of the diffraction patterns were generated by combining 30, 70, 200, 400, 800, 1500 msec, exposures. As the incident angle is increased, the diffraction pattern is elongated in the Q_x direction.

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Fig. 4 Reconstructed amplitudes (a,b,c) and phases (d,e,f) at 15°, 30°, 45° incident angles. The images are obtained using shrinkwrap+HIO algorithm. Each image was generated by adding 11 different reconstructed amplitude/phase reconstructions starting from a random initial phase value and multiplied by the support constraint. Images are multiplied with the oblique factor of 1/cosθ _i on to x scale.

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Fig. 5 (a) 3D rendering of the reconstructed phase map for the data obtained at an incident angle of 15°. In this 3D plotting, the phase value outside the central 70 pixels (in radius) was set to zero to illustrate the variation in phase value clearly. (b) Non-linear least squares fit for the line profile across center of phase map for the 15° incidence. The line profiles were fit to a series of step functions whose edges are smoothened by an error function. (c) Ratio of the phase error to the phase difference at the top and bottom layers.

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Fig. 6 Optical microscope image of a Cr letter patterned on an AR coated substrate using UV lithography.

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Fig. 7 Measured diffraction patterns (a,b,c) and the reconstructed amplitudes (d,e,f) and phases (g,h,i) from the sample ‘2’on an anti-reflection coated substrate at 15°, 30°, and 45° incident angles. As the angle of incidence is increased, the diffraction pattern is elongated in the Q_x direction. Plotted reconstructed image size is 301x301 pixels. Image is multiplied by the support. We also multiplied the reconstructed image by the oblique factor 1/cosθ _i in the x direction.

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Equations (3)

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ψ_{r} (x, y) = ψ_{0} \exp [i k (x \sin θ_{i} - z (x, y) \cos θ_{i})] r (x, y)

ψ (X, Y) = - \frac{i k e^{i k (d + \frac{X^{2} + Y^{2}}{2 d})}}{2 π} \int ψ_{r} (x, y) e^{- i k x \sin θ_{i}} e^{- i k z (\cos θ_{i} \frac{X \sin θ_{i}}{d})} e^{- i k \frac{Y y + X x \cos θ_{i}}{d}} d x d y

ψ (X, Y) = - \frac{i k e^{i k (d + \frac{X^{2} + Y^{2}}{2 d})}}{2 π} ψ_{0} \int e^{- 2 i k z (x, y) \cos θ_{i}} r (x, y) e^{- i k \frac{Y y + X x \cos θ_{i}}{d}} d x d y .

Abstract

References

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