Abstract

We propose a novel tomographic measurement approach that enables a noise suppressed characterization of microstructures. The idea of this work is based on a finding that coherent noise in the input phase data generates an artificial circular structure whose magnitude is the highest at the centre of tomographic reconstruction. This method decreases the noise level by applying an unconventional tomographic measurement configuration with an object deliberately shifted with respect to the rotation axis. This enables a spatial separation between the reconstructed sample structure and the area of the largest refractive index perturbations. The input phase data defocusing that is a by-product of the introduced modification is numerically corrected with an automatic focus correction algorithm. The proposed method is validated with simulations and experimental measurements of an optical microtip.

© 2014 Optical Society of America

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  1. W. Górski, W. Osten, “Tomographic imaging of photonic crystal fibers,” Opt. Lett. 32(14), 1977–1979 (2007).
    [CrossRef] [PubMed]
  2. T. Kozacki, R. Krajewski, M. Kujawińska, “Reconstruction of refractive-index distribution in off-axis digital holography optical diffraction tomographic system,” Opt. Express 17(16), 13758–13767 (2009).
    [CrossRef] [PubMed]
  3. M. Kujawińska, V. Parat, M. Dudek, B. Siwicki, S. Wójcik, G. Baethge, B. Dahmani, “Interferometric and tomographic investigations of polymer microtips fabricated at the extremity of optical fibers,” Proc. SPIE8494 (2012).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  23. D. Verhoeven, “Limited-data computed tomography algorithms for the physical sciences,” Appl. Opt. 32(20), 3736–3754 (1993).
    [CrossRef] [PubMed]
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    [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  29. K.-H. Brenner, W. Singer, “Light propagation through microlenses: a new simulation method,” Appl. Opt. 32(26), 4984–4988 (1993).
    [CrossRef] [PubMed]
  30. R. Bachelot, C. Ecoffet, D. Deloeil, P. Royer, D. J. Lougnot, “Integration of Micrometer-Sized Polymer Elements at the End of Optical Fibers by Free-Radical Photopolymerization,” Appl. Opt. 40(32), 5860–5871 (2001).
    [CrossRef] [PubMed]

2013 (1)

J. Kostencka, T. Kozacki, K. Liżewski, “Autofocusing method for tilted image plane detection in digital holographic microscopy,” Opt. Commun. 297, 20–26 (2013).
[CrossRef]

2012 (2)

A. Yin, B. Chen, Y. Zhang, “Focusing evaluation method based on wavelet transform and adaptive genetic algorithm,” Opt. Eng. 51(2), 023201 (2012).
[CrossRef]

T. Kozacki, K. Falaggis, M. Kujawińska, “Computation of diffracted fields for the case of high numerical aperture using the angular spectrum method,” Appl. Opt. 51(29), 7080–7088 (2012).
[CrossRef] [PubMed]

2011 (1)

W. Górski, “Tomographic interferometry of optical phase microelements,” Opto-Electron. Rev. 9, 347–352 (2011).

2009 (4)

Y. Jeon, C. K. Hong, “Rotation error correction by numerical focus adjustment in tomographic phase microscopy,” Opt. Eng. 48(10), 105801 (2009).
[CrossRef]

T. Kozacki, R. Krajewski, M. Kujawińska, “Reconstruction of refractive-index distribution in off-axis digital holography optical diffraction tomographic system,” Opt. Express 17(16), 13758–13767 (2009).
[CrossRef] [PubMed]

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17(1), 266–277 (2009).
[CrossRef] [PubMed]

T. Kozacki, M. Józwik, R. Józwicki, “Determination of optical field generated by a microlens using digital holographic method,” Opto-Electron. Rev. 17(3), 58–63 (2009).
[CrossRef]

2008 (3)

2007 (3)

2006 (3)

2001 (1)

1997 (1)

1995 (2)

1993 (2)

1983 (1)

S. X. Pan, A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: Interpolation versus filtered-backpropagation,” Acoustics, Speech and Signal Processing 31(5), 1262–1275 (1983).
[CrossRef]

1982 (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4(4), 336–350 (1982).
[PubMed]

1974 (1)

R. Gordon, “A tutorial on ART (algebraic reconstruction technique),” IEEE Trans. Nucl. Sci. 21(3), 78–93 (1974).
[CrossRef]

Bachelot, R.

Badizadegan, K.

Baethge, G.

M. Kujawińska, V. Parat, M. Dudek, B. Siwicki, S. Wójcik, G. Baethge, B. Dahmani, “Interferometric and tomographic investigations of polymer microtips fabricated at the extremity of optical fibers,” Proc. SPIE8494 (2012).

Brenner, K.-H.

Charrière, F.

Chen, B.

A. Yin, B. Chen, Y. Zhang, “Focusing evaluation method based on wavelet transform and adaptive genetic algorithm,” Opt. Eng. 51(2), 023201 (2012).
[CrossRef]

Choi, W.

Colomb, T.

Cuche, E.

Dahmani, B.

M. Kujawińska, V. Parat, M. Dudek, B. Siwicki, S. Wójcik, G. Baethge, B. Dahmani, “Interferometric and tomographic investigations of polymer microtips fabricated at the extremity of optical fibers,” Proc. SPIE8494 (2012).

Dasari, R. R.

Davis, C. S.

Deloeil, D.

Depeursinge, C.

Devaney, A. J.

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4(4), 336–350 (1982).
[PubMed]

Dirksen, D.

Dudek, M.

M. Kujawińska, V. Parat, M. Dudek, B. Siwicki, S. Wójcik, G. Baethge, B. Dahmani, “Interferometric and tomographic investigations of polymer microtips fabricated at the extremity of optical fibers,” Proc. SPIE8494 (2012).

Ecoffet, C.

Emery, Y.

Falaggis, K.

Fang-Yen, C.

Feld, M. S.

Gordon, R.

R. Gordon, “A tutorial on ART (algebraic reconstruction technique),” IEEE Trans. Nucl. Sci. 21(3), 78–93 (1974).
[CrossRef]

Górski, W.

W. Górski, “Tomographic interferometry of optical phase microelements,” Opto-Electron. Rev. 9, 347–352 (2011).

W. Górski, W. Osten, “Tomographic imaging of photonic crystal fibers,” Opt. Lett. 32(14), 1977–1979 (2007).
[CrossRef] [PubMed]

Heger, T. J.

Hong, C. K.

Y. Jeon, C. K. Hong, “Rotation error correction by numerical focus adjustment in tomographic phase microscopy,” Opt. Eng. 48(10), 105801 (2009).
[CrossRef]

Hu, Q.

Jeon, Y.

Y. Jeon, C. K. Hong, “Rotation error correction by numerical focus adjustment in tomographic phase microscopy,” Opt. Eng. 48(10), 105801 (2009).
[CrossRef]

Józwicki, R.

T. Kozacki, M. Józwik, R. Józwicki, “Determination of optical field generated by a microlens using digital holographic method,” Opto-Electron. Rev. 17(3), 58–63 (2009).
[CrossRef]

Józwik, M.

T. Kozacki, M. Józwik, R. Józwicki, “Determination of optical field generated by a microlens using digital holographic method,” Opto-Electron. Rev. 17(3), 58–63 (2009).
[CrossRef]

Kak, A. C.

S. X. Pan, A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: Interpolation versus filtered-backpropagation,” Acoustics, Speech and Signal Processing 31(5), 1262–1275 (1983).
[CrossRef]

Kemper, B.

Kniazewski, P.

T. Kozacki, M. Kujawińska, P. Kniażewski, “Investigation of limitations of optical diffraction tomography,” Opto-Electron. Rev. 15(2), 102–109 (2007).
[CrossRef]

Kostencka, J.

J. Kostencka, T. Kozacki, K. Liżewski, “Autofocusing method for tilted image plane detection in digital holographic microscopy,” Opt. Commun. 297, 20–26 (2013).
[CrossRef]

Kozacki, T.

J. Kostencka, T. Kozacki, K. Liżewski, “Autofocusing method for tilted image plane detection in digital holographic microscopy,” Opt. Commun. 297, 20–26 (2013).
[CrossRef]

T. Kozacki, K. Falaggis, M. Kujawińska, “Computation of diffracted fields for the case of high numerical aperture using the angular spectrum method,” Appl. Opt. 51(29), 7080–7088 (2012).
[CrossRef] [PubMed]

T. Kozacki, M. Józwik, R. Józwicki, “Determination of optical field generated by a microlens using digital holographic method,” Opto-Electron. Rev. 17(3), 58–63 (2009).
[CrossRef]

T. Kozacki, R. Krajewski, M. Kujawińska, “Reconstruction of refractive-index distribution in off-axis digital holography optical diffraction tomographic system,” Opt. Express 17(16), 13758–13767 (2009).
[CrossRef] [PubMed]

T. Kozacki, M. Kujawińska, P. Kniażewski, “Investigation of limitations of optical diffraction tomography,” Opto-Electron. Rev. 15(2), 102–109 (2007).
[CrossRef]

Krajewski, R.

Kuehn, J.

Kühn, J.

Kujawinska, M.

T. Kozacki, K. Falaggis, M. Kujawińska, “Computation of diffracted fields for the case of high numerical aperture using the angular spectrum method,” Appl. Opt. 51(29), 7080–7088 (2012).
[CrossRef] [PubMed]

T. Kozacki, R. Krajewski, M. Kujawińska, “Reconstruction of refractive-index distribution in off-axis digital holography optical diffraction tomographic system,” Opt. Express 17(16), 13758–13767 (2009).
[CrossRef] [PubMed]

T. Kozacki, M. Kujawińska, P. Kniażewski, “Investigation of limitations of optical diffraction tomography,” Opto-Electron. Rev. 15(2), 102–109 (2007).
[CrossRef]

M. Kujawińska, V. Parat, M. Dudek, B. Siwicki, S. Wójcik, G. Baethge, B. Dahmani, “Interferometric and tomographic investigations of polymer microtips fabricated at the extremity of optical fibers,” Proc. SPIE8494 (2012).

Langehanenberg, P.

Li, W.

Lizewski, K.

J. Kostencka, T. Kozacki, K. Liżewski, “Autofocusing method for tilted image plane detection in digital holographic microscopy,” Opt. Commun. 297, 20–26 (2013).
[CrossRef]

Loomis, N. C.

Lougnot, D. J.

Marian, A.

Marquet, P.

Mitchell, E. A. D.

Montfort, F.

Osten, W.

Pan, S. X.

S. X. Pan, A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: Interpolation versus filtered-backpropagation,” Acoustics, Speech and Signal Processing 31(5), 1262–1275 (1983).
[CrossRef]

Parat, V.

M. Kujawińska, V. Parat, M. Dudek, B. Siwicki, S. Wójcik, G. Baethge, B. Dahmani, “Interferometric and tomographic investigations of polymer microtips fabricated at the extremity of optical fibers,” Proc. SPIE8494 (2012).

Pavillon, N.

Rappaz, B.

Royer, P.

Singer, W.

Siwicki, B.

M. Kujawińska, V. Parat, M. Dudek, B. Siwicki, S. Wójcik, G. Baethge, B. Dahmani, “Interferometric and tomographic investigations of polymer microtips fabricated at the extremity of optical fibers,” Proc. SPIE8494 (2012).

Stamnes, J. J.

Sung, Y.

Verhoeven, D.

von Bally, G.

Wedberg, T. C.

Weible, K.

Wójcik, S.

M. Kujawińska, V. Parat, M. Dudek, B. Siwicki, S. Wójcik, G. Baethge, B. Dahmani, “Interferometric and tomographic investigations of polymer microtips fabricated at the extremity of optical fibers,” Proc. SPIE8494 (2012).

Yamaguchi, I.

Yin, A.

A. Yin, B. Chen, Y. Zhang, “Focusing evaluation method based on wavelet transform and adaptive genetic algorithm,” Opt. Eng. 51(2), 023201 (2012).
[CrossRef]

Zhang, T.

Zhang, Y.

A. Yin, B. Chen, Y. Zhang, “Focusing evaluation method based on wavelet transform and adaptive genetic algorithm,” Opt. Eng. 51(2), 023201 (2012).
[CrossRef]

Acoustics, Speech and Signal Processing (1)

S. X. Pan, A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: Interpolation versus filtered-backpropagation,” Acoustics, Speech and Signal Processing 31(5), 1262–1275 (1983).
[CrossRef]

Appl. Opt. (8)

D. Verhoeven, “Limited-data computed tomography algorithms for the physical sciences,” Appl. Opt. 32(20), 3736–3754 (1993).
[CrossRef] [PubMed]

P. Langehanenberg, B. Kemper, D. Dirksen, G. von Bally, “Autofocusing in digital holographic phase contrast microscopy on pure phase objects for live cell imaging,” Appl. Opt. 47(19), D176–D182 (2008).
[CrossRef] [PubMed]

T. Kozacki, K. Falaggis, M. Kujawińska, “Computation of diffracted fields for the case of high numerical aperture using the angular spectrum method,” Appl. Opt. 51(29), 7080–7088 (2012).
[CrossRef] [PubMed]

K.-H. Brenner, W. Singer, “Light propagation through microlenses: a new simulation method,” Appl. Opt. 32(26), 4984–4988 (1993).
[CrossRef] [PubMed]

R. Bachelot, C. Ecoffet, D. Deloeil, P. Royer, D. J. Lougnot, “Integration of Micrometer-Sized Polymer Elements at the End of Optical Fibers by Free-Radical Photopolymerization,” Appl. Opt. 40(32), 5860–5871 (2001).
[CrossRef] [PubMed]

T. C. Wedberg, J. J. Stamnes, W. Singer, “Comparison of the filtered backpropagation and the filtered backprojection algorithms for quantitative tomography,” Appl. Opt. 34(28), 6575–6581 (1995).
[CrossRef] [PubMed]

F. Charrière, J. Kühn, T. Colomb, F. Montfort, E. Cuche, Y. Emery, K. Weible, P. Marquet, C. Depeursinge, “Characterization of microlenses by digital holographic microscopy,” Appl. Opt. 45(5), 829–835 (2006).
[CrossRef] [PubMed]

B. Kemper, G. von Bally, “Digital holographic microscopy for live cell applications and technical inspection,” Appl. Opt. 47(4), A52–A61 (2008).
[CrossRef] [PubMed]

IEEE Trans. Nucl. Sci. (1)

R. Gordon, “A tutorial on ART (algebraic reconstruction technique),” IEEE Trans. Nucl. Sci. 21(3), 78–93 (1974).
[CrossRef]

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

Opt. Commun. (1)

J. Kostencka, T. Kozacki, K. Liżewski, “Autofocusing method for tilted image plane detection in digital holographic microscopy,” Opt. Commun. 297, 20–26 (2013).
[CrossRef]

Opt. Eng. (2)

A. Yin, B. Chen, Y. Zhang, “Focusing evaluation method based on wavelet transform and adaptive genetic algorithm,” Opt. Eng. 51(2), 023201 (2012).
[CrossRef]

Y. Jeon, C. K. Hong, “Rotation error correction by numerical focus adjustment in tomographic phase microscopy,” Opt. Eng. 48(10), 105801 (2009).
[CrossRef]

Opt. Express (3)

Opt. Lett. (4)

Opto-Electron. Rev. (3)

T. Kozacki, M. Józwik, R. Józwicki, “Determination of optical field generated by a microlens using digital holographic method,” Opto-Electron. Rev. 17(3), 58–63 (2009).
[CrossRef]

W. Górski, “Tomographic interferometry of optical phase microelements,” Opto-Electron. Rev. 9, 347–352 (2011).

T. Kozacki, M. Kujawińska, P. Kniażewski, “Investigation of limitations of optical diffraction tomography,” Opto-Electron. Rev. 15(2), 102–109 (2007).
[CrossRef]

Pure Appl. Opt. (1)

T. C. Wedberg, J. J. Stamnes, “Comparison of phase-retrieval methods for optical diffraction tomography,” Pure Appl. Opt. 4(1), 39–54 (1995).
[CrossRef]

Ultrason. Imaging (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4(4), 336–350 (1982).
[PubMed]

Other (4)

T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH Verlag GmbH & Co. KGaA, 2005).

M. Kujawińska, V. Parat, M. Dudek, B. Siwicki, S. Wójcik, G. Baethge, B. Dahmani, “Interferometric and tomographic investigations of polymer microtips fabricated at the extremity of optical fibers,” Proc. SPIE8494 (2012).

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, (Society for Industrial and Applied Mathematics, New York, 2001), Chap. 3.

S. Vertu, M. Ochiai, M. Shuzo, I. Yamada, J. Delaunay, O. Haeberlé, and Y. Okamoto, “Optical projection microtomography of transparent objects,” SPIE Proc. 6627 (2007).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

The scheme of Mach-Zehnder holographic microscope setup: SF–spatial filter; C–collimator lens; M1, M2–mirrors; PZT-M–mirror mounted on PZT; BS-beam splitter; BC–beam combiner; MO1, MO2–microscope objectives; IL–imaging lens; RH–rotational holder; IC–cuvette with immersion liquid; MS–measured sample; OB–object beam; RB–reference beam.

Fig. 2
Fig. 2

Illustration of the FBPJ tomographic reconstruction process.

Fig. 3
Fig. 3

The tomographic representation of a single point of the sample: (a) a sine wave in a sinogram; (b) a point in the reconstruction.

Fig. 4
Fig. 4

The tomographic representation of a single speckle: (a) line in a sinogram; (b) arc in the reconstruction.

Fig. 5
Fig. 5

Tomographic reconstruction of a speckle pattern: (a) sinogram with a set of horizontal lines of the same magnitude representing evenly distributed phase noise; (b) section A-A through the central column of the sinogram; (c) reconstructed refractive index distribution Δn; (d) section B-B through Δn.

Fig. 6
Fig. 6

Tomographic evaluation of typical background images captured in the DH microscope system: (a) sinogram; (b) reconstructed distribution of refractive index variations; (c) section through the middle row of ∆n.

Fig. 7
Fig. 7

Schematic diagram of the sample trajectory during the off-axis rotation.

Fig. 8
Fig. 8

The FBPJ reconstructions of cylinders with refractive index variation Δn = 0.01 and diameters Φ = 5λ (a,c) and Φ = 20λ (b,d) obtained for on-axis (upper row) and off-axis (lower row) measurement configuration.

Fig. 9
Fig. 9

Reconstruction analysis for the case of a cylindrical objects (∆n = 0.01) (a) error E 1 of noisy defocus corrected reconstructions calculated with respect to the noiseless results; (b) reduction of an error E1 with respect to the on-axis measurement; (c) error E2 of noiseless off-axis reconstructions obtained without defocus correction calculated with respect to the on-axis case; (d) error E 3 of the noiseless defocus corrected reconstructions calculated with respect to the ideal refractive index distribution.

Fig. 10
Fig. 10

The error E 1 of noisy reconstructions obtained for the cases of (a) on-axis and (b,c) off-axis measurement with radial run-out of R = 5μm (b) and R = 20μm (c).

Fig. 11
Fig. 11

Hologram of an optical microtip.

Fig. 12
Fig. 12

Defocus correction for the exemplary sample view α = 60: (a) amplitude and (b) unwrapped phase reconstructed from the hologram in the measurement plane; (c) amplitude and (d) unwrapped phase after propagation to the in-focus position; (e) variance of the amplitude distribution in the function of an axial position.

Fig. 13
Fig. 13

The tomographic reconstruction of refractive index distribution in the microtip: (a) 3D representation and three central cross-sections of the refractive index reconstructions obtained with (b)-(d) and without (e)-(g) numerical defocus correction for the off-axis configuration; (h)-(j) reconstruction obtained for the on-axis measurement case.

Equations (7)

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

Δ n α ( x , y , z ) = n α ( x , y , z ) n 0 ,
φ α ( x , y ) = 2 π λ Δ n α ( x , y , z ) d z .
S f ( x , α ) = + | f | ( + S ( x , α ) exp ( i 2 π f x ) d x ) exp ( 2 π f x ) d f = F T 1 { | f | F T { S ( x , α ) } } ,
Φ ( x , y 0 , z ) = 0 π S f ( x cos α + z sin α , α ) d α .
Δ n ( x , y , z ) = λ 2 π Δ x Φ ( x , y , z ) ,
φ α ( x , y 0 ) d x = 2 π λ n α ( x , y 0 , z ) d x d z
E ( Δ n | Δ n r e f ) = ( Δ n Δ n r e f ) 2 ( Δ n r e f ) 2

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