Abstract

The topography of a phase plate is recovered from the phase reconstruction by solving the transport intensity equation (TIE). The TIE is solved using two different approaches: (a) the classical solution of solving the Poisson differential equation and (b) an algebraic approach with Zernike functions. In this paper we present and compare the topography reconstruction of a phase plate with these solution methods and justify why one solution is preferable over the other.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. M. Werdiger, S. Eliezer, S. Maman, Y. Horovitz, B. Arad, Z. Henis, and I. B. Goldberg, “Development of holographic methods for investigating a moving free surface, accelerated by laser-induced shock waves,” Laser Part. Beams 17, 653–660 (1999).
    [CrossRef]
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    [CrossRef] [PubMed]
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  28. T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).
    [CrossRef]
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2010 (1)

S. V. Pinhasi, R. Alimi, S. Eliezer, and L. Perelmutter, “Fast optical computerized topography,” Phys. Lett. A 374, 2798–2800 (2010).
[CrossRef]

2007 (1)

2006 (2)

Z. Pek’arek and R. Hrach, “A comparison of advanced Poisson equation solvers applied to the particle-in-cell plasma model,” in WDS’06 Proceedings (2006), pp. 187–192.

S. Vinikman-Pinhasi and E. N. Ribak, “Piezoelectric and piezooptic effects in porous silicon,” Appl. Phys. Lett. 88, 111905 (2006).
[CrossRef]

2005 (1)

2002 (2)

M. C. Lai, “A simple compact fourth-order Poisson solver on polar geometry,” J. Comput. Phys. 182, 337–345 (2002).
[CrossRef]

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

2000 (2)

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

A. Tonomura, “Direct observation of vortex motion in high-Tc superconductors by Lorentz microscopy,” Physica B 280, 227–228 (2000).
[CrossRef]

1999 (1)

M. Werdiger, S. Eliezer, S. Maman, Y. Horovitz, B. Arad, Z. Henis, and I. B. Goldberg, “Development of holographic methods for investigating a moving free surface, accelerated by laser-induced shock waves,” Laser Part. Beams 17, 653–660 (1999).
[CrossRef]

1998 (2)

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

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

1997 (2)

M. Werdiger, S. Eliezer, Z. Henis, B. Arad, Y. Horovitz, R. Shpitalnik, and S. Maman, “Off-axis holography of laser-induced shock wave targets,” Appl. Phys. Lett. 71, 211–212 (1997).
[CrossRef]

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

1996 (2)

T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).
[CrossRef]

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

1995 (1)

1993 (1)

1991 (1)

R. A. Sweet, W. L. Briggs, S. Oliveira, J. L. Porsche, and T. Turnbull, “FFTs and three-dimensional Poisson solvers for hypercubes,” Parallel Comput. 17, 121–131 (1991).
[CrossRef]

1990 (1)

1988 (2)

1984 (1)

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

1983 (1)

1981 (1)

L. a. Y. D. Hageman, Applied Iterative Methods (Academic, 1981).

1977 (1)

A. Brandt, “Multi-level adaptive solutions to boundary-value problems,” Math. Comput. 31, 333–390 (1977).
[CrossRef]

1970 (1)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1970).

1965 (1)

R. W. Hockney, “A fast direct solution of Poisson’s equation using Fourier analysis,” J. Assoc. Comput. Mach. 12, 95–113 (1965).
[CrossRef]

1934 (1)

v. F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica (Amsterdam) 1, 689–704 (1934).
[CrossRef]

Alimi, R.

S. V. Pinhasi, R. Alimi, S. Eliezer, and L. Perelmutter, “Fast optical computerized topography,” Phys. Lett. A 374, 2798–2800 (2010).
[CrossRef]

Arad, B.

M. Werdiger, S. Eliezer, S. Maman, Y. Horovitz, B. Arad, Z. Henis, and I. B. Goldberg, “Development of holographic methods for investigating a moving free surface, accelerated by laser-induced shock waves,” Laser Part. Beams 17, 653–660 (1999).
[CrossRef]

M. Werdiger, S. Eliezer, Z. Henis, B. Arad, Y. Horovitz, R. Shpitalnik, and S. Maman, “Off-axis holography of laser-induced shock wave targets,” Appl. Phys. Lett. 71, 211–212 (1997).
[CrossRef]

Bajt, S.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

Barnea, Z.

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

Barty, A.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

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

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1970).

Brandt, A.

A. Brandt, “Multi-level adaptive solutions to boundary-value problems,” Math. Comput. 31, 333–390 (1977).
[CrossRef]

Briggs, W. L.

R. A. Sweet, W. L. Briggs, S. Oliveira, J. L. Porsche, and T. Turnbull, “FFTs and three-dimensional Poisson solvers for hypercubes,” Parallel Comput. 17, 121–131 (1991).
[CrossRef]

Cookson, D. F.

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

De Graef, M.

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

Dorrer, C.

Eliezer, S.

S. V. Pinhasi, R. Alimi, S. Eliezer, and L. Perelmutter, “Fast optical computerized topography,” Phys. Lett. A 374, 2798–2800 (2010).
[CrossRef]

M. Werdiger, S. Eliezer, S. Maman, Y. Horovitz, B. Arad, Z. Henis, and I. B. Goldberg, “Development of holographic methods for investigating a moving free surface, accelerated by laser-induced shock waves,” Laser Part. Beams 17, 653–660 (1999).
[CrossRef]

M. Werdiger, S. Eliezer, Z. Henis, B. Arad, Y. Horovitz, R. Shpitalnik, and S. Maman, “Off-axis holography of laser-induced shock wave targets,” Appl. Phys. Lett. 71, 211–212 (1997).
[CrossRef]

Goldberg, I. B.

M. Werdiger, S. Eliezer, S. Maman, Y. Horovitz, B. Arad, Z. Henis, and I. B. Goldberg, “Development of holographic methods for investigating a moving free surface, accelerated by laser-induced shock waves,” Laser Part. Beams 17, 653–660 (1999).
[CrossRef]

Gruppetta, S.

Gureyev, T. E.

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

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

T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).
[CrossRef]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A 12, 1932–1942 (1995).
[CrossRef]

Hageman, L. a. Y. D.

L. a. Y. D. Hageman, Applied Iterative Methods (Academic, 1981).

Henis, Z.

M. Werdiger, S. Eliezer, S. Maman, Y. Horovitz, B. Arad, Z. Henis, and I. B. Goldberg, “Development of holographic methods for investigating a moving free surface, accelerated by laser-induced shock waves,” Laser Part. Beams 17, 653–660 (1999).
[CrossRef]

M. Werdiger, S. Eliezer, Z. Henis, B. Arad, Y. Horovitz, R. Shpitalnik, and S. Maman, “Off-axis holography of laser-induced shock wave targets,” Appl. Phys. Lett. 71, 211–212 (1997).
[CrossRef]

Hockney, R. W.

R. W. Hockney, “A fast direct solution of Poisson’s equation using Fourier analysis,” J. Assoc. Comput. Mach. 12, 95–113 (1965).
[CrossRef]

Horovitz, Y.

M. Werdiger, S. Eliezer, S. Maman, Y. Horovitz, B. Arad, Z. Henis, and I. B. Goldberg, “Development of holographic methods for investigating a moving free surface, accelerated by laser-induced shock waves,” Laser Part. Beams 17, 653–660 (1999).
[CrossRef]

M. Werdiger, S. Eliezer, Z. Henis, B. Arad, Y. Horovitz, R. Shpitalnik, and S. Maman, “Off-axis holography of laser-induced shock wave targets,” Appl. Phys. Lett. 71, 211–212 (1997).
[CrossRef]

Hrach, R.

Z. Pek’arek and R. Hrach, “A comparison of advanced Poisson equation solvers applied to the particle-in-cell plasma model,” in WDS’06 Proceedings (2006), pp. 187–192.

Ichikawa, K.

Koechlin, L.

Lacombe, F.

Lai, M. C.

M. C. Lai, “A simple compact fourth-order Poisson solver on polar geometry,” J. Comput. Phys. 182, 337–345 (2002).
[CrossRef]

Lohmann, A. W.

Maman, S.

M. Werdiger, S. Eliezer, S. Maman, Y. Horovitz, B. Arad, Z. Henis, and I. B. Goldberg, “Development of holographic methods for investigating a moving free surface, accelerated by laser-induced shock waves,” Laser Part. Beams 17, 653–660 (1999).
[CrossRef]

M. Werdiger, S. Eliezer, Z. Henis, B. Arad, Y. Horovitz, R. Shpitalnik, and S. Maman, “Off-axis holography of laser-induced shock wave targets,” Appl. Phys. Lett. 71, 211–212 (1997).
[CrossRef]

McCartney, M.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

Millane, R. P.

Nugent, K. A.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

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

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

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

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

T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).
[CrossRef]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A 12, 1932–1942 (1995).
[CrossRef]

Oliveira, S.

R. A. Sweet, W. L. Briggs, S. Oliveira, J. L. Porsche, and T. Turnbull, “FFTs and three-dimensional Poisson solvers for hypercubes,” Parallel Comput. 17, 121–131 (1991).
[CrossRef]

Paganin, D.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

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

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

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

Pek’arek, Z.

Z. Pek’arek and R. Hrach, “A comparison of advanced Poisson equation solvers applied to the particle-in-cell plasma model,” in WDS’06 Proceedings (2006), pp. 187–192.

Perelmutter, L.

S. V. Pinhasi, R. Alimi, S. Eliezer, and L. Perelmutter, “Fast optical computerized topography,” Phys. Lett. A 374, 2798–2800 (2010).
[CrossRef]

Pinhasi, S. V.

S. V. Pinhasi, R. Alimi, S. Eliezer, and L. Perelmutter, “Fast optical computerized topography,” Phys. Lett. A 374, 2798–2800 (2010).
[CrossRef]

Porsche, J. L.

R. A. Sweet, W. L. Briggs, S. Oliveira, J. L. Porsche, and T. Turnbull, “FFTs and three-dimensional Poisson solvers for hypercubes,” Parallel Comput. 17, 121–131 (1991).
[CrossRef]

Puget, P.

Reed Teague, M.

Ribak, E. N.

S. Vinikman-Pinhasi and E. N. Ribak, “Piezoelectric and piezooptic effects in porous silicon,” Appl. Phys. Lett. 88, 111905 (2006).
[CrossRef]

Roberts, A.

Roddier, C.

Roddier, F.

Shpitalnik, R.

M. Werdiger, S. Eliezer, Z. Henis, B. Arad, Y. Horovitz, R. Shpitalnik, and S. Maman, “Off-axis holography of laser-induced shock wave targets,” Appl. Phys. Lett. 71, 211–212 (1997).
[CrossRef]

Streibl, N.

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

Sweet, R. A.

R. A. Sweet, W. L. Briggs, S. Oliveira, J. L. Porsche, and T. Turnbull, “FFTs and three-dimensional Poisson solvers for hypercubes,” Parallel Comput. 17, 121–131 (1991).
[CrossRef]

Takeda, M.

Tonomura, A.

A. Tonomura, “Direct observation of vortex motion in high-Tc superconductors by Lorentz microscopy,” Physica B 280, 227–228 (2000).
[CrossRef]

Turnbull, T.

R. A. Sweet, W. L. Briggs, S. Oliveira, J. L. Porsche, and T. Turnbull, “FFTs and three-dimensional Poisson solvers for hypercubes,” Parallel Comput. 17, 121–131 (1991).
[CrossRef]

Vinikman-Pinhasi, S.

S. Vinikman-Pinhasi and E. N. Ribak, “Piezoelectric and piezooptic effects in porous silicon,” Appl. Phys. Lett. 88, 111905 (2006).
[CrossRef]

Volkov, V. V.

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

Wall, M.

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

Werdiger, M.

M. Werdiger, S. Eliezer, S. Maman, Y. Horovitz, B. Arad, Z. Henis, and I. B. Goldberg, “Development of holographic methods for investigating a moving free surface, accelerated by laser-induced shock waves,” Laser Part. Beams 17, 653–660 (1999).
[CrossRef]

M. Werdiger, S. Eliezer, Z. Henis, B. Arad, Y. Horovitz, R. Shpitalnik, and S. Maman, “Off-axis holography of laser-induced shock wave targets,” Appl. Phys. Lett. 71, 211–212 (1997).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1970).

Zernike, v. F.

v. F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica (Amsterdam) 1, 689–704 (1934).
[CrossRef]

Zhu, Y.

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

Zuegel, J. D.

Appl. Opt. (2)

Appl. Phys. Lett. (2)

M. Werdiger, S. Eliezer, Z. Henis, B. Arad, Y. Horovitz, R. Shpitalnik, and S. Maman, “Off-axis holography of laser-induced shock wave targets,” Appl. Phys. Lett. 71, 211–212 (1997).
[CrossRef]

S. Vinikman-Pinhasi and E. N. Ribak, “Piezoelectric and piezooptic effects in porous silicon,” Appl. Phys. Lett. 88, 111905 (2006).
[CrossRef]

J. Assoc. Comput. Mach. (1)

R. W. Hockney, “A fast direct solution of Poisson’s equation using Fourier analysis,” J. Assoc. Comput. Mach. 12, 95–113 (1965).
[CrossRef]

J. Comput. Phys. (1)

M. C. Lai, “A simple compact fourth-order Poisson solver on polar geometry,” J. Comput. Phys. 182, 337–345 (2002).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Laser Part. Beams (1)

M. Werdiger, S. Eliezer, S. Maman, Y. Horovitz, B. Arad, Z. Henis, and I. B. Goldberg, “Development of holographic methods for investigating a moving free surface, accelerated by laser-induced shock waves,” Laser Part. Beams 17, 653–660 (1999).
[CrossRef]

Math. Comput. (1)

A. Brandt, “Multi-level adaptive solutions to boundary-value problems,” Math. Comput. 31, 333–390 (1977).
[CrossRef]

Micron (1)

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

Opt. Commun. (2)

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

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

Opt. Express (1)

Opt. Lett. (2)

Parallel Comput. (1)

R. A. Sweet, W. L. Briggs, S. Oliveira, J. L. Porsche, and T. Turnbull, “FFTs and three-dimensional Poisson solvers for hypercubes,” Parallel Comput. 17, 121–131 (1991).
[CrossRef]

Phys. Lett. A (1)

S. V. Pinhasi, R. Alimi, S. Eliezer, and L. Perelmutter, “Fast optical computerized topography,” Phys. Lett. A 374, 2798–2800 (2010).
[CrossRef]

Phys. Rev. Lett. (2)

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

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

Physica (Amsterdam) (1)

v. F. Zernike, “Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode,” Physica (Amsterdam) 1, 689–704 (1934).
[CrossRef]

Physica B (1)

A. Tonomura, “Direct observation of vortex motion in high-Tc superconductors by Lorentz microscopy,” Physica B 280, 227–228 (2000).
[CrossRef]

Ultramicroscopy (1)

S. Bajt, A. Barty, K. A. Nugent, M. McCartney, M. Wall, and D. Paganin, “Quantitative phase-sensitive imaging in a transmission electron microscope,” Ultramicroscopy 83, 67–73 (2000).
[CrossRef] [PubMed]

Other (3)

Z. Pek’arek and R. Hrach, “A comparison of advanced Poisson equation solvers applied to the particle-in-cell plasma model,” in WDS’06 Proceedings (2006), pp. 187–192.

L. a. Y. D. Hageman, Applied Iterative Methods (Academic, 1981).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1970).

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Figures (7)

Fig. 1
Fig. 1

Anti-symmetrization procedure that duplicates the experimental image in order to obtain the required NBCs.

Fig. 2
Fig. 2

Experimental setup: Two CCD cameras record illumination I 1 and I 2 reflected by the target at two imaging planes separated by distance Δ z . A 532 nm pulsed Nd-YAG, 7 ns pulse, was used as the illumination source.

Fig. 3
Fig. 3

Typical cross-section surface profile of the target. The experimental reference denoted by (0) is the actual surface determined by a Hartmann–Shack sensor. The dotted curve and the cross markers represent the cross-section of the surface reconstruction using the Poisson solver for uniform (1) and non-uniform (2) reflectivities, respectively. The square markers represent the cross-section of the surface reconstruction using polynomial expansion (3).

Fig. 4
Fig. 4

Reconstructed three-dimensional topography using the Poisson solver for uniform reflectivity (first method). (a) The direct output of the Poisson solver. (b) Surface topography after removing the optical aberrations. (c) and (d) are the surface topography after removing the first 15 and 21 Zernike components, respectively.

Fig. 5
Fig. 5

Reconstructed three-dimensional topography using the Poisson solver twice (second method). (a) The direct output of the Poisson solver. It contains both the object phase and the optical aberrations. (b) Surface topography after removing the first four Zernike components. (c) and (d) are the surface topography after removing the first 15 and 21 Zernike components, respectively.

Fig. 6
Fig. 6

Flat glass surface three-dimensional topography using polynomial expansion. (a) The calculated topography using all the 26 first Zernike polynomials. (b) When the first four Zernike polynomials were subtracted the flat surface was reconstructed within an accuracy of half-wavelength.

Fig. 7
Fig. 7

Three-dimensional topography calculation of the target using the polynomial expansion method (third method). (a) The calculated topography using the 26 first Zernike polynomials. (b) The topography reconstruction after deducting the optical aberrations.

Equations (24)

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( x y 2 + 2 i k z ) u ( x , y ; z ) = 0 ,
k I ( x , y ; z ) z = x y [ I ( x , y ; z ) x y ϕ ( x , y ; z ) ] .
k I ( x , y ; z ) z k I 2 ( x , y ; Δ z ) I 1 ( x , y ; 0 ) Δ z .
k I 0 I 2 ( x , y ; Δ z ) I 1 ( x , y ; 0 ) Δ z = x y 2 ϕ ( x , y ; z ) .
( x y ϕ ) n ̂ = 0.
S d S x y f = l ( f n ̂ ) d l .
k I z d s = x y ( I x y ) d s = ( I x y ) n ̂ d l = 0.
k I z d s = 0.
ψ = I ϕ .
2 ψ = k I z .
( ψ I ) = 2 ϕ .
I ( r , θ ) > 0 ,     r < R ,
I ( r , θ ) = 0 ,     r R ,
ϕ ( r , θ ) = i = 0 φ i Z i ( r / R , θ ) .
R 2 0 2 π 0 R r , θ [ I ( r , θ ; z ) r , θ ϕ ( r , θ ) ] Z j ( r / R , θ ) r d r d θ = R 2 0 2 π 0 R k I ( r , θ ; z ) z Z j ( r / R , θ ) r d r d θ F j ( R , z ) ,
i = 0 φ i R 2 0 2 π 0 R I ( r , θ ; z ) r , θ Z i ( r / R , θ ) r , θ Z j ( r / R , θ ) r d r d θ = F j ( R , z ) ,     i , j = 1 , 2 , 3 , .
M i j 0 2 π 0 R I ( r , θ ; z ) r , θ Z i ( r / R , θ ) r , θ Z j ( r / R , θ ) r d r d θ .
i = 1 M i j φ i = R 2 F j .
z = λ 2 π ϕ .
z ( r , θ ) = 5.00 × 10 3 r 2 + 9.00 × 10 4 r 4 3.40 × 10 5 r 6 ,     r < 8   mm ,
z ( r , θ ) = 0 ,     r > 8   mm .
ϕ = ϕ obj + ϕ abb .
ϕ abb = n = 1 4 φ i z i ( r , θ ) ,
ϕ obj = n = 5 26 φ i z i ( r , θ ) ,

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