Abstract

In the paper the optical diffraction tomographic system for reconstruction of the internal refractive index distribution in optical fiber utilizing grating Mach-Zehnder interferometer configuration is explored. The setup applies afocal imaging. Conventional grating application gives, however, highly aberrated object beam producing incorrect refractive-index reconstructions. The grating inherent aberrations are characterized, its influence on both image projections and refractive index reconstructions is presented. To remove aberrations and enable tomographic reconstruction a novel digital holographic algorithm, correcting optical system imaging, is developed. The algorithm uses plane wave spectrum decomposition of optical field for solving diffraction problem between parallel and tilted planes and enabling correction of imaging system aberrations. The algorithm concept was successfully proved in simulations and the experiment.

© 2009 Optical Society of America

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References

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2008 (1)

T. Kozacki, "Numerical errors of diffraction computing using plane wave spectrum decomposition," Opt. Commun. 281, 4219-4223 (2008).
[CrossRef]

2007 (1)

T. Kozacki, M. Kujawinska and P. Kniazewski, "Investigating the limitation of optical scalar field tomography," Opto-Electron.Rev. 15, 102-109 (2007).
[CrossRef]

2006 (1)

2003 (2)

2002 (2)

L. Yu, Y. An, and L. Cai, "Numerical reconstruction of digital holograms with variable viewing angles," Opt. Express 10, 1250-1257 (2002).
[PubMed]

W. Górski and M. Kujawińska, "Three-dimensional reconstruction of refractive index inhomogeneities in optical phase elements," Opt. Lasers Eng. 38,373-385 (2002).
[CrossRef]

2001 (1)

L. Sałbut and M. Kujawińska, "The optical measurement station for complex testing of microelements," Opt. Laser Eng.  36, 225-240 (2001) .
[CrossRef]

2000 (1)

1999 (2)

S. K. Mangal, and K. Ramesh, "Determination of characteristic parameters in integrated photoelasticity by phase-shifting technique," Opt. Laser Eng. 31, 263-278 (1999).
[CrossRef]

T.M. Lehmann, C. Gonner, and K. Spitzer, "Survey: interpolation methods in medical image processing," IEEE Trans. Med. Imaging 18, 1049-1075 (1999).
[CrossRef]

1998 (1)

1997 (1)

1995 (1)

1993 (1)

1992 (1)

1990 (1)

R. Czarnek, "High sensitivity moiré interferometry with compact achromatic interferometer," Opt. Laser Eng. 13, 93-101 (1990).

1986 (1)

1982 (1)

A.J. Devaney, "A filtered backpropagation algorithm for diffraction tomography," Ultrasonic imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

1966 (1)

An, Y.

Benkouider, A.

Brenner, K.-H.

Buchele, D. R.

Cai, L.

Coëtmellec, S.

Czarnek, R.

R. Czarnek, "High sensitivity moiré interferometry with compact achromatic interferometer," Opt. Laser Eng. 13, 93-101 (1990).

Dasch, C. J.

De Nicola, S.

Delen, N.

Devaney, A.J.

A.J. Devaney, "A filtered backpropagation algorithm for diffraction tomography," Ultrasonic imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

Ferraro, P.

Finizio, A.

Gonner, C.

T.M. Lehmann, C. Gonner, and K. Spitzer, "Survey: interpolation methods in medical image processing," IEEE Trans. Med. Imaging 18, 1049-1075 (1999).
[CrossRef]

Górski, W.

W. Górski and M. Kujawińska, "Three-dimensional reconstruction of refractive index inhomogeneities in optical phase elements," Opt. Lasers Eng. 38,373-385 (2002).
[CrossRef]

Hooker, B.

Howes, W. L.

Kniazewski, P.

T. Kozacki, M. Kujawinska and P. Kniazewski, "Investigating the limitation of optical scalar field tomography," Opto-Electron.Rev. 15, 102-109 (2007).
[CrossRef]

Kozacki, T.

T. Kozacki, "Numerical errors of diffraction computing using plane wave spectrum decomposition," Opt. Commun. 281, 4219-4223 (2008).
[CrossRef]

T. Kozacki, M. Kujawinska and P. Kniazewski, "Investigating the limitation of optical scalar field tomography," Opto-Electron.Rev. 15, 102-109 (2007).
[CrossRef]

Kujawinska, M.

T. Kozacki, M. Kujawinska and P. Kniazewski, "Investigating the limitation of optical scalar field tomography," Opto-Electron.Rev. 15, 102-109 (2007).
[CrossRef]

W. Górski and M. Kujawińska, "Three-dimensional reconstruction of refractive index inhomogeneities in optical phase elements," Opt. Lasers Eng. 38,373-385 (2002).
[CrossRef]

Lebrun, D.

Lehmann, T.M.

T.M. Lehmann, C. Gonner, and K. Spitzer, "Survey: interpolation methods in medical image processing," IEEE Trans. Med. Imaging 18, 1049-1075 (1999).
[CrossRef]

Leith, E.N.

Malek, M.

Mangal, S. K.

S. K. Mangal, and K. Ramesh, "Determination of characteristic parameters in integrated photoelasticity by phase-shifting technique," Opt. Laser Eng. 31, 263-278 (1999).
[CrossRef]

Matsushima, K.

Pierattini, G.

Ramesh, K.

S. K. Mangal, and K. Ramesh, "Determination of characteristic parameters in integrated photoelasticity by phase-shifting technique," Opt. Laser Eng. 31, 263-278 (1999).
[CrossRef]

Salbut, L.

L. Sałbut and M. Kujawińska, "The optical measurement station for complex testing of microelements," Opt. Laser Eng.  36, 225-240 (2001) .
[CrossRef]

Schimmel, H.

Shen, F.

Shentu, L.

Singer, W.

Spitzer, K.

T.M. Lehmann, C. Gonner, and K. Spitzer, "Survey: interpolation methods in medical image processing," IEEE Trans. Med. Imaging 18, 1049-1075 (1999).
[CrossRef]

Stamnes, J.J.

Wang, A.

Wedberg, T.C.

Wyrowski, F.

Yamaguchi, I.

Yu, L.

Zhang, T.

Appl. Opt. (6)

IEEE Trans. Med. Imaging (1)

T.M. Lehmann, C. Gonner, and K. Spitzer, "Survey: interpolation methods in medical image processing," IEEE Trans. Med. Imaging 18, 1049-1075 (1999).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

T. Kozacki, "Numerical errors of diffraction computing using plane wave spectrum decomposition," Opt. Commun. 281, 4219-4223 (2008).
[CrossRef]

Opt. Express (2)

Opt. Laser Eng. (3)

L. Sałbut and M. Kujawińska, "The optical measurement station for complex testing of microelements," Opt. Laser Eng.  36, 225-240 (2001) .
[CrossRef]

S. K. Mangal, and K. Ramesh, "Determination of characteristic parameters in integrated photoelasticity by phase-shifting technique," Opt. Laser Eng. 31, 263-278 (1999).
[CrossRef]

R. Czarnek, "High sensitivity moiré interferometry with compact achromatic interferometer," Opt. Laser Eng. 13, 93-101 (1990).

Opt. Lasers Eng. (1)

W. Górski and M. Kujawińska, "Three-dimensional reconstruction of refractive index inhomogeneities in optical phase elements," Opt. Lasers Eng. 38,373-385 (2002).
[CrossRef]

Opt. Lett. (1)

Rev. (1)

T. Kozacki, M. Kujawinska and P. Kniazewski, "Investigating the limitation of optical scalar field tomography," Opto-Electron.Rev. 15, 102-109 (2007).
[CrossRef]

Ultrasonic imaging (1)

A.J. Devaney, "A filtered backpropagation algorithm for diffraction tomography," Ultrasonic imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

Other (5)

J. Hseigh, Computed Tomograph, (SPIE Press, Washington, 2003).

L. P. Yaroslavskii and N. Merzlyakov, Methods of digital holography (Consultants Bureau, New York, 1980).

R. Krajewski, M. Kujawińska, B. Volckaerts, and H. Thienpont "Low-cost microinterferomtric tomograpy system for 3D refraction index distribution Measurements in optical fiber splices, Proc. SPIE 5855, 17th OFS conference Brugge, 347-351 (2005).
[CrossRef]

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol 1986).

M. Born and E. Wolf, Principles of Optics 7th (expanded) edition, (Cambridge University Press, 1999).

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Figures (10)

Fig. 1.
Fig. 1.

Two grating optical diffraction tomographic system; (a) free space configuration SF - spatial filter, C - collimator lens, G1 and G2 - phase gratings with maximum diffraction efficiency of ±1 orders (period 1 µm), M - mirror, MO - microscope objective, IL - imaging lens, PS - phase shifter, refractive index of immersion liquid 1.459 for 0.633 µm, (b) compact integrated GODT system design.

Fig. 2.
Fig. 2.

The intensity image of an object beam generated by a multimode fiber.

Fig. 3.
Fig. 3.

Object beam imaging in the GODT system.

Fig. 4.
Fig. 4.

Object Fourier spectrum modification in off-axis tomography system for Λ=1 µm and λ=633 nm.

Fig. 5.
Fig. 5.

Comparison of the phase projection image given by the grating and the object phase projection: Arg[u d ] - phase distribution of optical field at distorted image plane, φ p - object phase projection.

Fig. 6.
Fig. 6.

Comparison of the refractive index distribution (cross-sections): n drec - the reconstruction from phase given at distorted image plane with Abel inversion [20], n o - the object distribution.

Fig. 7.
Fig. 7.

Scheme of the digital holography algorithm for optical reconstruction of a projection image.

Fig. 8.
Fig. 8.

The spatial shift introduced by the free space propagation algorithm of an optical field between planes Xh’ and XO’.

Fig. 9.
Fig. 9.

Comparison of the phase projections obtained for the multimode fiber as an object, Arg[u] - the received image projection computed with the algorithm (Eq. (11)) for optical multimode fiber, φ p - the object phase projection.

Fig. 10.
Fig. 10.

The experimental data obtained for a multimode fiber: (a) fringe pattern at the hologram plane, (b) phase image at the hologram plane, (c) phase projection image at the object plane, (d) tomographic reconstruction of refractive index.

Equations (11)

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fxo=fxi 1λ2Λ2 λΛ1λ2fxi2+Λ1,
UX(ft)=u(x) exp {i2πft·x}d x ,
UXo(ft)=UXh (fxΛ1,fy) exp {ikz1(1λ2(fxΛ1)2+λ2fy2)12} exp {ikz2(1λ2ft·ft)12} .
Uxo (fts)=Uxh(fts)exp{ikz1(1λ2fts·fts)12}exp{ikz2(1λ2(fxs+Λ1)2+λ2fy2)12} .
UXO (fts)=UXh (fts) H (fts) ,
Xlf=(ARG[H(fxs)])2πdfxs= λz1fxs(1λ2fxs2)12+λz2Λ(1Λ2λ2)12+λ(fxs+Λ1)z2(1λ2(fxs+Λ1)2)12 .
Nx > 2λΔ(z1fm(1λ2fm2)12+z2Λ(1Λ2λ2)12+(fm+Λ1)z2(1λ2(fm+Λ1)2)12).
T=[cosα0sinα010sinα0cosα]
UX(ft)=UX(fxcosα(λ2ft·ft)12sinα,fy) .
UX(ft)=UX ((fx+Λ1)cosα(λ2(fx+Λ1)2fy2)12 sin α , fy ) .
u (x)= UX (ft) J(ft)exp{2πix.ft}df.

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