Abstract

Boundary conditions play a crucial role in the solution of the transport of intensity equation (TIE). If not appropriately handled, they can create significant boundary artifacts across the reconstruction result. In a previous paper [Opt. Express 22, 9220 (2014)], we presented a new boundary-artifact-free TIE phase retrieval method with use of discrete cosine transform (DCT). Here we report its experimental investigations with applications to the micro-optics characterization. The experimental setup is based on a tunable lens based 4f system attached to a non-modified inverted bright-field microscope. We establish inhomogeneous Neumann boundary values by placing a rectangular aperture in the intermediate image plane of the microscope. Then the boundary values are applied to solve the TIE with our DCT-based TIE solver. Experimental results on microlenses highlight the importance of boundary conditions that often overlooked in simplified models, and confirm that our approach effectively avoid the boundary error even when objects are located at the image borders. It is further demonstrated that our technique is non-interferometric, accurate, fast, full-field, and flexible, rendering it a promising metrological tool for the micro-optics inspection.

© 2014 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
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  30. D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
    [Crossref] [PubMed]
  31. N. Streibl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. A 2(2), 121–127 (1985).
    [Crossref]
  32. E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206(3), 194–203 (2002).
    [Crossref] [PubMed]
  33. C. Campbell, “Wave-front sensing by use of a Green’s function solution to the intensity transport equation: comment,” J. Opt. Soc. Am. A 24(8), 2480–2481 (2007).
    [Crossref] [PubMed]
  34. S. C. Woods, H. I. Campbell, and A. H. Greenaway, “Wave-front sensing by use of a Green's function solution to the intensity transport equation: reply to comment,” J. Opt. Soc. Am. A 24(8), 2482–2484 (2007).
    [Crossref]
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    [Crossref]

2014 (2)

2013 (4)

2010 (3)

2008 (1)

2007 (2)

2006 (2)

2004 (2)

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref] [PubMed]

C. J. R. Sheppard, “Defocused transfer function for a partially coherent microscope and application to phase retrieval,” J. Opt. Soc. Am. A 21(5), 828–831 (2004).
[Crossref] [PubMed]

2003 (1)

2002 (2)

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206(3), 194–203 (2002).
[Crossref] [PubMed]

V. V. Volkov, Y. Zhu, and M. De Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33(5), 411–416 (2002).
[Crossref] [PubMed]

2001 (1)

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
[Crossref]

1999 (1)

1998 (2)

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[Crossref]

T. Gureyev and S. Wilkins, “On X-ray phase retrieval from polychromatic images,” Opt. Commun. 147(4-6), 229–232 (1998).
[Crossref]

1997 (1)

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
[Crossref]

1996 (1)

1995 (3)

T. Gureyev, A. Roberts, and K. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” JOSA A 12(9), 1932–1942 (1995).
[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(9), 1942–1946 (1995).
[Crossref]

I. W. Han, “New method for estimating wavefront from curvature signal by curve fitting,” Opt. Eng. 34(4), 1232–1237 (1995).
[Crossref]

1985 (1)

1983 (1)

1982 (1)

Allen, L. J.

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
[Crossref]

Altmeyer, S.

Aspert, N.

Asundi, A.

Barbastathis, G.

Barone-Nugent, E. D.

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206(3), 194–203 (2002).
[Crossref] [PubMed]

Barty, A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref] [PubMed]

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206(3), 194–203 (2002).
[Crossref] [PubMed]

Bevilacqua, F.

Bourquin, S.

Campbell, C.

Campbell, H. I.

Charrière, F.

Chen, Q.

Choo, C. O.

Colomb, T.

Coppola, G.

Cuche, E.

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(5), 411–416 (2002).
[Crossref] [PubMed]

De Nicola, S.

Depeursinge, C.

Emery, Y.

Falaggis, K.

Ferraro, P.

Fienup, J. R.

Finizio, A.

Frank, J.

Greenaway, A. H.

Grilli, S.

Gureyev, T.

T. Gureyev and S. Wilkins, “On X-ray phase retrieval from polychromatic images,” Opt. Commun. 147(4-6), 229–232 (1998).
[Crossref]

T. Gureyev, A. Roberts, and K. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” JOSA A 12(9), 1932–1942 (1995).
[Crossref]

Gureyev, T. E.

Han, I. W.

I. W. Han, “New method for estimating wavefront from curvature signal by curve fitting,” Opt. Eng. 34(4), 1232–1237 (1995).
[Crossref]

Kemper, B.

Kozacki, T.

Kühn, J.

Magro, C.

Marian, A.

Marquet, P.

Martinez-Carranza, J.

McMahon, P. J.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref] [PubMed]

Montfort, F.

Nugent, K.

T. Gureyev, A. Roberts, and K. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” JOSA A 12(9), 1932–1942 (1995).
[Crossref]

Nugent, K. A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref] [PubMed]

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206(3), 194–203 (2002).
[Crossref] [PubMed]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[Crossref]

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
[Crossref]

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(8), 1670–1682 (1996).
[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(9), 1942–1946 (1995).
[Crossref]

Oxley, M. P.

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
[Crossref]

Paganin, D.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref] [PubMed]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[Crossref]

Pierattini, G.

Qu, W.

Reed Teague, M.

Roberts, A.

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(9), 1942–1946 (1995).
[Crossref]

T. Gureyev, A. Roberts, and K. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” JOSA A 12(9), 1932–1942 (1995).
[Crossref]

Sheppard, C. J. R.

Streibl, N.

Tian, L.

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(5), 411–416 (2002).
[Crossref] [PubMed]

von Bally, G.

Waller, L.

Weible, K.

Weijuan, Q.

Wernicke, G.

Wilkins, S.

T. Gureyev and S. Wilkins, “On X-ray phase retrieval from polychromatic images,” Opt. Commun. 147(4-6), 229–232 (1998).
[Crossref]

Woods, S. C.

Yingjie, Y.

Yu, Y.

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(5), 411–416 (2002).
[Crossref] [PubMed]

Zuo, C.

Appl. Opt. (5)

J. Microsc. (2)

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206(3), 194–203 (2002).
[Crossref] [PubMed]

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

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(8), 1670–1682 (1996).
[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(9), 1942–1946 (1995).
[Crossref]

J. Frank, S. Altmeyer, and G. Wernicke, “Non-interferometric, non-iterative phase retrieval by Green’s functions,” J. Opt. Soc. Am. A 27(10), 2244–2251 (2010).
[Crossref] [PubMed]

N. Streibl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. A 2(2), 121–127 (1985).
[Crossref]

C. Campbell, “Wave-front sensing by use of a Green’s function solution to the intensity transport equation: comment,” J. Opt. Soc. Am. A 24(8), 2480–2481 (2007).
[Crossref] [PubMed]

S. C. Woods, H. I. Campbell, and A. H. Greenaway, “Wave-front sensing by use of a Green's function solution to the intensity transport equation: reply to comment,” J. Opt. Soc. Am. A 24(8), 2482–2484 (2007).
[Crossref]

T. Colomb, F. Montfort, J. Kühn, N. Aspert, E. Cuche, A. Marian, F. Charrière, S. Bourquin, P. Marquet, and C. Depeursinge, “Numerical parametric lens for shifting, magnification, and complete aberration compensation in digital holographic microscopy,” J. Opt. Soc. Am. A 23(12), 3177–3190 (2006).
[Crossref] [PubMed]

C. J. R. Sheppard, “Defocused transfer function for a partially coherent microscope and application to phase retrieval,” J. Opt. Soc. Am. A 21(5), 828–831 (2004).
[Crossref] [PubMed]

JOSA A (1)

T. Gureyev, A. Roberts, and K. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” JOSA A 12(9), 1932–1942 (1995).
[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(5), 411–416 (2002).
[Crossref] [PubMed]

Opt. Commun. (3)

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
[Crossref]

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
[Crossref]

T. Gureyev and S. Wilkins, “On X-ray phase retrieval from polychromatic images,” Opt. Commun. 147(4-6), 229–232 (1998).
[Crossref]

Opt. Eng. (1)

I. W. Han, “New method for estimating wavefront from curvature signal by curve fitting,” Opt. Eng. 34(4), 1232–1237 (1995).
[Crossref]

Opt. Express (4)

Opt. Lett. (4)

Phys. Rev. Lett. (1)

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[Crossref]

Other (4)

D. A. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, 2001), Vol. 224.

J. Martinez-Carranza, K. Falaggis, T. Kozacki, and M. Kujawinska, “Effect of imposed boundary conditions on the accuracy of transport of intensity equation based solvers,” Proc. SPIE, 87890N (2013).

C. Zuo, Q. Chen, and A. Asundi, “Comparison of Digital Holography and Transport of Intensity for Quantitative Phase Contrast Imaging,” in Fringe 2013, W. Osten, ed. (Springer Berlin Heidelberg, 2014), pp. 137–142.

F. Morse, Methods of Theoretical Physics (2-Volume set),” (McGraw-Hill, 1981).

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Figures (9)

Fig. 1
Fig. 1

Phase retrieval for a complex object covering the image boundary. (a) Phase distribution. (b) Intensity distribution. (c) Axial intensity derivative. (d) Phase retrieved by the FFT-based method (periodic boundary conditions). (e) Phase retrieved by the even symmetrization method (Zero Dirichlet boundary conditions). (f) Phase retrieved by the odd symmetrization method (Zero Neumann boundary conditions).

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Fig. 2
Fig. 2

Phase retrieval for an isolated object located in the central position. (a) Phase distribution. (b) Intensity distribution. (c) Axial intensity derivative. (d) Phase retrieved by the FFT-based method (periodic boundary conditions). (e) Phase retrieved by the even symmetrization method (Zero Dirichlet boundary conditions). (f) Phase retrieved by the odd symmetrization method (Zero Neumann boundary conditions).

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Fig. 3
Fig. 3

Phase retrieval for a complex object covering the image boundary with the DCT-based method. (a) Phase distribution. (b) Intensity distribution (with a square aperture). (c) Axial intensity derivative. (d) Enlarged region corresponding to the lower right quarter of Fig. 1(c). (e) Enlarged region corresponding to the lower right quarter of Fig. 3(c), boundary signals can be clearly observed along the aperture edge (red shaded areas). (f) Phase retrieved by the DCT-based method (inhomogeneous Neumann boundary conditions). The red box in (f) outlines the aperture edge.

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Fig. 4
Fig. 4

Experimental configuration. A 4f system with OL/ETL located at the Fourier plane is attached to Olympus IX71 bright field microscope. The Fourier lens L1 relays the back focal plane of the objective onto the OL/ETL. Fourier lens L2 reconstructs the final image at the CCD plane, which is conjugated with the intermediate image plane. A rectangular aperture is placed in the intermediate image plane (conjugated object plane) of the microscope to fit the camera sensor area. The focal lengths of the lens L1 and L2 are both 150mm. The tunable focal length range of the ETL is from + 50 to + 200 mm. The change of the focus positions can be realized by adjusting the focal length of the ETL.

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Fig. 5
Fig. 5

Characterization of a plano-convex quartz microlens array (pitch 250 μm) (a) In-focus intensity. (b) Defocused intensity distribution ( Δz=−550 μm). (c) Axial intensity derivative, the inset shows the enlarged boxed region. (d) Retrieved phase. (e) Rendered surface plot. (f) Confocal microscopic result. (g) 3-D topography by confocal microscopy. (f) Comparison of the line profiles of single lens.

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Fig. 6
Fig. 6

Comparison of different TIE solutions. The first row shows the recovered phase by (a) our algorithm on the correct domain Ω ¯ (with boundary signals), (b) FFT-based algorithm on the correct domain Ω ¯ , (c) our algorithm on the reduced domain Ω (without boundary signals), (d) FFT-based algorithm on the reduced domain Ω . The second row shows the corresponding digitally rewrapped phases of a small portion of from the top-left corner of each result (e)-(f).

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Fig. 7
Fig. 7

Aberration and roughness analysis using the Zernike expansion. (a) Raw topography of an individual microlens measured by the TIE. (b) The first 21 Zernike functions used as the basis for the decomposition. (c) Calculated Zernike coefficients. (d) Reconstructed microlens topology using the 21 Zernike coefficients shown in (c). (e) Fitting residual.

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Fig. 8
Fig. 8

Measurement results with the misaligned system. (a) and (b) show the axial intensity derivative and the recovered phase when the ETL was slightly shifted horizontally. (c) and (d) show the axial intensity derivative and the recovered phase when the ETL was slightly translated longitudinally.

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Fig. 9
Fig. 9

Measurement results of different types of micro-optics components. First row: a 100μm pitch plano-convex quartz microlens array. Second row: a 100μm diameter cylindrical lens. Third row: a Fresnel lens. The retrieved phase map and the 3-D height distribution are respectively given in the first and second columns.

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Equations (11)

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

−k ∂I( r ) ∂z =∇⋅[ I( r )∇ϕ( r ) ],
ϕ| ∂Ω =g.
I ∂ϕ ∂n | ∂Ω =g.
ϕ| ∂Ω =0.
I ∂ϕ ∂n | ∂Ω =0.
I= A Ω I 0 ={ I 0                                     r∈ Ω ¯ 0                                     others   ,
−k ∂I ∂z = A Ω ( I 0 ∇ 2 ϕ+∇ I 0 ⋅∇ϕ )−I ∂ϕ ∂n δ ∂Ω .
∬ Ω ¯ −k ∂I( r ) ∂z dr= ∬ Ω −k ∂I( r ) ∂z dr− ∮ ∂Ω I( r ) ∂ϕ( r ) ∂n ds=0,
ϕ( r )=−k ∇ DCT −2 ∇ DCT [ I −1 ( r ) ∇ DCT ∇ DCT −2 ∂I( r ) ∂z ],
h(x,y)= OPD(x,y) Δn = λ n o − n m ϕ(x,y) 2π ,
ROC=h/2 + D 2 / 8h ,

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