Abstract

A new technique, microinterferometric optical phase tomography, is introduced for use in measuring small, asymmetric refractive-index differences in the profiles of optical fibers and fiber devices. The method combines microscopy-based fringe-field interferometry with parallel projection-based computed tomography to characterize fiber index profiles. The theory relating interference measurements to the projection set required for tomographic reconstruction is given, and discrete numerical simulations are presented for three test index profiles that establish the technique’s ability to characterize fiber with small, asymmetric index differences. An experimental measurement configuration and specific interferometry and tomography practices employed in the technique are discussed.

© 2005 Optical Society of America

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  1. L. Grüner-Nielsen, S. N. Knudsen, B. Edvold, T. Veng, D. Magnussen, C. C. Larsen, H. Damsgaard, “Dispersion compensating fibers,” Opt. Fiber Technol. 6, 164–180 (2000).
    [CrossRef]
  2. T. Erdogan, V. Mizrahi, “Characterization of UV-induced birefringence in photosensitive Ge-doped silica optical fibers,” J. Opt. Soc. Am. B 11, 2100–2105 (1994).
    [CrossRef]
  3. Y. Ishii, K. Shima, S. Okude, K. Nishide, A. Wada, “PDL suppression on long-period fiber gratings by azimuthally isotropic exposure,” IEICE Trans. Electron. E85-C, 934–939 (2002).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. K. Toga, N. Amano, K.-I. Noda, “Microscopic computer tomography measurement of nonaxisymmetrically distributed optical fiber refractive index,” J. Lightwave Technol. 6, 73–79 (1988).
    [CrossRef]
  15. T. Okoshi, M. Nishimura, “Measurement of axially non-symmetrical refractive-index distribution of a single-mode fiber by a multidirectional scattering-pattern method,” J. Lightwave Technol. 1, 9–14 (1983).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  23. B. V. Dorrio, J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33–R55 (1999).
    [CrossRef]
  24. D. Marcuse, H. Presby, “Index profile measurements of fibres and their evaluation,” Proc. IEEE 68, 666–688 (1980).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  30. J. M. Gauch, “Noise removal and contrast enhancement,” in The Colour Image Processing Handbook, S. J. Sangwine, R. E. N. Home, eds. (Chapman & Hall, New York, 1998), pp. 149–162.
    [CrossRef]
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2004 (1)

W. Górski, “The influence of diffraction in microinterferometry and microtomography of optical fibers,” Opt. Lasers Eng. 41, 563–583 (2004).
[CrossRef]

2003 (3)

2002 (4)

S. Vázquez-Montiel, J. J. Sánchez-Escobar, O. Fuentes, “Obtaining the phase of an interferogram by use of an evolution strategy. 1,” Appl. Opt. 41, 3448–3452 (2002).
[CrossRef]

K. Dossou, S. LaRochelle, M. Fontaine, “Numerical analysis of the contribution of the transverse asymmetry in the photo-induced index change profile to the birefringence of optical fiber,” J. Lightwave Technol. 20, 1463–1470 (2002).
[CrossRef]

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

Y. Ishii, K. Shima, S. Okude, K. Nishide, A. Wada, “PDL suppression on long-period fiber gratings by azimuthally isotropic exposure,” IEICE Trans. Electron. E85-C, 934–939 (2002).

2001 (1)

N. Barakat, H. A. El-Hennawi, E. A. El-Ghafar, H. El-Ghandoor, R. Hassan, F. El-Diasty, “Three-dimensional refractive index profile of a GRIN optical waveguide using multiple beam interference fringes,” Opt. Commun. 191, 39–47 (2001).
[CrossRef]

2000 (2)

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

L. Grüner-Nielsen, S. N. Knudsen, B. Edvold, T. Veng, D. Magnussen, C. C. Larsen, H. Damsgaard, “Dispersion compensating fibers,” Opt. Fiber Technol. 6, 164–180 (2000).
[CrossRef]

1999 (2)

N. H. Fontaine, M. Young, “Two-dimensional index profiling of fibers and waveguides,” Appl. Opt. 38, 6836–6844 (1999).
[CrossRef]

B. V. Dorrio, J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33–R55 (1999).
[CrossRef]

1997 (1)

S. T. Huntington, P. Mulvaney, A. Roberts, K. A. Nugent, M. Bazylenko, “Atomic force microscopy for the determination of refractive index profiles of optical fibers and waveguides: a quantitative study,” J. Appl. Phys. 82, 2730–2734 (1997).
[CrossRef]

1996 (1)

D. A. Viskoe, G. W. Donohoe, “Optimal computed tomography data acquisition techniques and filter selection for detection of small density variations,” IEEE Trans. Instrum. Meas. 45, 70–76 (1996).
[CrossRef]

1994 (4)

M. Sochacka, “Optical fiber profiling by phase-stepping transverse interferometry,” J. Lightwave Technol. 12, 19–23 (1994).
[CrossRef]

Q. Zhong, D. Inniss, “Characterization of the lightguiding structure of optical fibers by atomic force microscopy,” J. Light-wave Technol. 12, 1517–1523 (1994).
[CrossRef]

T. Erdogan, V. Mizrahi, “Characterization of UV-induced birefringence in photosensitive Ge-doped silica optical fibers,” J. Opt. Soc. Am. B 11, 2100–2105 (1994).
[CrossRef]

A. M. Vengsarkar, Q. Zhong, D. Inniss, W. A. Reed, P. J. Lemaire, S. G. Kosinski, “Birefringence reduction in side-written photoinduced fiber devices by a dual-exposure method,” Opt. Lett. 19, 1260–1262 (1994).
[CrossRef] [PubMed]

1988 (1)

K. Toga, N. Amano, K.-I. Noda, “Microscopic computer tomography measurement of nonaxisymmetrically distributed optical fiber refractive index,” J. Lightwave Technol. 6, 73–79 (1988).
[CrossRef]

1983 (1)

T. Okoshi, M. Nishimura, “Measurement of axially non-symmetrical refractive-index distribution of a single-mode fiber by a multidirectional scattering-pattern method,” J. Lightwave Technol. 1, 9–14 (1983).
[CrossRef]

1980 (1)

D. Marcuse, H. Presby, “Index profile measurements of fibres and their evaluation,” Proc. IEEE 68, 666–688 (1980).
[CrossRef]

1979 (2)

L. M. Boggs, H. M. Presby, D. Marcuse, “Rapid automatic index profiling of whole-fiber samples. 1,” Bell Syst. Tech. J. 58, 867–882 (1979).
[CrossRef]

D. Marcuse, H. M. Presby, “Focusing method for nondestructive measurement of optical fiber index profiles,” Appl. Opt. 18, 14–22 (1979).
[CrossRef] [PubMed]

1977 (1)

Y. Kokubun, K. Iga, “Precise measurement of the refractive index profile of optical fibers by a nondestructive interference method,” Trans. IECE Japan E60, 702–707 (1977).

Amano, N.

K. Toga, N. Amano, K.-I. Noda, “Microscopic computer tomography measurement of nonaxisymmetrically distributed optical fiber refractive index,” J. Lightwave Technol. 6, 73–79 (1988).
[CrossRef]

Anemogiannis, E.

Bachim, B. L.

Barakat, N.

N. Barakat, H. A. El-Hennawi, E. A. El-Ghafar, H. El-Ghandoor, R. Hassan, F. El-Diasty, “Three-dimensional refractive index profile of a GRIN optical waveguide using multiple beam interference fringes,” Opt. Commun. 191, 39–47 (2001).
[CrossRef]

Barrett, H. H.

H. H. Barrett, W. Swindell, Radiological Imaging: The Theory of Image Formation, Detection, and Processing (Academic, New York, 1981), Vol. 2.

Barty, A.

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

A. Barty, “Quantitative phase-amplitude microscopy,” Ph.D. dissertation (University of Melbourne, Parkville, Victoria, Australia, 2000).

Bazylenko, M.

S. T. Huntington, P. Mulvaney, A. Roberts, K. A. Nugent, M. Bazylenko, “Atomic force microscopy for the determination of refractive index profiles of optical fibers and waveguides: a quantitative study,” J. Appl. Phys. 82, 2730–2734 (1997).
[CrossRef]

Boggs, L. M.

L. M. Boggs, H. M. Presby, D. Marcuse, “Rapid automatic index profiling of whole-fiber samples. 1,” Bell Syst. Tech. J. 58, 867–882 (1979).
[CrossRef]

Choi, S.

Damsgaard, H.

L. Grüner-Nielsen, S. N. Knudsen, B. Edvold, T. Veng, D. Magnussen, C. C. Larsen, H. Damsgaard, “Dispersion compensating fibers,” Opt. Fiber Technol. 6, 164–180 (2000).
[CrossRef]

Donohoe, G. W.

D. A. Viskoe, G. W. Donohoe, “Optimal computed tomography data acquisition techniques and filter selection for detection of small density variations,” IEEE Trans. Instrum. Meas. 45, 70–76 (1996).
[CrossRef]

Dorrio, B. V.

B. V. Dorrio, J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33–R55 (1999).
[CrossRef]

Dossou, K.

Edvold, B.

L. Grüner-Nielsen, S. N. Knudsen, B. Edvold, T. Veng, D. Magnussen, C. C. Larsen, H. Damsgaard, “Dispersion compensating fibers,” Opt. Fiber Technol. 6, 164–180 (2000).
[CrossRef]

El-Diasty, F.

N. Barakat, H. A. El-Hennawi, E. A. El-Ghafar, H. El-Ghandoor, R. Hassan, F. El-Diasty, “Three-dimensional refractive index profile of a GRIN optical waveguide using multiple beam interference fringes,” Opt. Commun. 191, 39–47 (2001).
[CrossRef]

El-Ghafar, E. A.

N. Barakat, H. A. El-Hennawi, E. A. El-Ghafar, H. El-Ghandoor, R. Hassan, F. El-Diasty, “Three-dimensional refractive index profile of a GRIN optical waveguide using multiple beam interference fringes,” Opt. Commun. 191, 39–47 (2001).
[CrossRef]

El-Ghandoor, H.

N. Barakat, H. A. El-Hennawi, E. A. El-Ghafar, H. El-Ghandoor, R. Hassan, F. El-Diasty, “Three-dimensional refractive index profile of a GRIN optical waveguide using multiple beam interference fringes,” Opt. Commun. 191, 39–47 (2001).
[CrossRef]

El-Hennawi, H. A.

N. Barakat, H. A. El-Hennawi, E. A. El-Ghafar, H. El-Ghandoor, R. Hassan, F. El-Diasty, “Three-dimensional refractive index profile of a GRIN optical waveguide using multiple beam interference fringes,” Opt. Commun. 191, 39–47 (2001).
[CrossRef]

Erdogan, T.

Fernández, J. L.

B. V. Dorrio, J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33–R55 (1999).
[CrossRef]

Fontaine, M.

Fontaine, N. H.

Fuentes, O.

Gauch, J. M.

J. M. Gauch, “Noise removal and contrast enhancement,” in The Colour Image Processing Handbook, S. J. Sangwine, R. E. N. Home, eds. (Chapman & Hall, New York, 1998), pp. 149–162.
[CrossRef]

Gaylord, T. K.

Glytsis, E. N.

Górski, W.

W. Górski, “The influence of diffraction in microinterferometry and microtomography of optical fibers,” Opt. Lasers Eng. 41, 563–583 (2004).
[CrossRef]

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

Grüner-Nielsen, L.

L. Grüner-Nielsen, S. N. Knudsen, B. Edvold, T. Veng, D. Magnussen, C. C. Larsen, H. Damsgaard, “Dispersion compensating fibers,” Opt. Fiber Technol. 6, 164–180 (2000).
[CrossRef]

Hassan, R.

N. Barakat, H. A. El-Hennawi, E. A. El-Ghafar, H. El-Ghandoor, R. Hassan, F. El-Diasty, “Three-dimensional refractive index profile of a GRIN optical waveguide using multiple beam interference fringes,” Opt. Commun. 191, 39–47 (2001).
[CrossRef]

Hsieh, J.

J. Hsieh, Computed Tomography: Principles, Design, Artifacts, and Recent Advances (SPIE Press, Bellingham, Wash., 2003).

Huntington, S. T.

S. T. Huntington, P. Mulvaney, A. Roberts, K. A. Nugent, M. Bazylenko, “Atomic force microscopy for the determination of refractive index profiles of optical fibers and waveguides: a quantitative study,” J. Appl. Phys. 82, 2730–2734 (1997).
[CrossRef]

Iga, K.

Y. Kokubun, K. Iga, “Precise measurement of the refractive index profile of optical fibers by a nondestructive interference method,” Trans. IECE Japan E60, 702–707 (1977).

Inniss, D.

Q. Zhong, D. Inniss, “Characterization of the lightguiding structure of optical fibers by atomic force microscopy,” J. Light-wave Technol. 12, 1517–1523 (1994).
[CrossRef]

A. M. Vengsarkar, Q. Zhong, D. Inniss, W. A. Reed, P. J. Lemaire, S. G. Kosinski, “Birefringence reduction in side-written photoinduced fiber devices by a dual-exposure method,” Opt. Lett. 19, 1260–1262 (1994).
[CrossRef] [PubMed]

Ishii, Y.

Y. Ishii, K. Shima, S. Okude, K. Nishide, A. Wada, “PDL suppression on long-period fiber gratings by azimuthally isotropic exposure,” IEICE Trans. Electron. E85-C, 934–939 (2002).

Kim, D. Y.

Knudsen, S. N.

L. Grüner-Nielsen, S. N. Knudsen, B. Edvold, T. Veng, D. Magnussen, C. C. Larsen, H. Damsgaard, “Dispersion compensating fibers,” Opt. Fiber Technol. 6, 164–180 (2000).
[CrossRef]

Kokubun, Y.

Y. Kokubun, K. Iga, “Precise measurement of the refractive index profile of optical fibers by a nondestructive interference method,” Trans. IECE Japan E60, 702–707 (1977).

Kosinski, S. G.

Kujawinska, M.

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

LaRochelle, S.

Larsen, C. C.

L. Grüner-Nielsen, S. N. Knudsen, B. Edvold, T. Veng, D. Magnussen, C. C. Larsen, H. Damsgaard, “Dispersion compensating fibers,” Opt. Fiber Technol. 6, 164–180 (2000).
[CrossRef]

Lemaire, P. J.

Magnussen, D.

L. Grüner-Nielsen, S. N. Knudsen, B. Edvold, T. Veng, D. Magnussen, C. C. Larsen, H. Damsgaard, “Dispersion compensating fibers,” Opt. Fiber Technol. 6, 164–180 (2000).
[CrossRef]

Marcuse, D.

D. Marcuse, H. Presby, “Index profile measurements of fibres and their evaluation,” Proc. IEEE 68, 666–688 (1980).
[CrossRef]

D. Marcuse, H. M. Presby, “Focusing method for nondestructive measurement of optical fiber index profiles,” Appl. Opt. 18, 14–22 (1979).
[CrossRef] [PubMed]

L. M. Boggs, H. M. Presby, D. Marcuse, “Rapid automatic index profiling of whole-fiber samples. 1,” Bell Syst. Tech. J. 58, 867–882 (1979).
[CrossRef]

Mizrahi, V.

Mulvaney, P.

S. T. Huntington, P. Mulvaney, A. Roberts, K. A. Nugent, M. Bazylenko, “Atomic force microscopy for the determination of refractive index profiles of optical fibers and waveguides: a quantitative study,” J. Appl. Phys. 82, 2730–2734 (1997).
[CrossRef]

Nishide, K.

Y. Ishii, K. Shima, S. Okude, K. Nishide, A. Wada, “PDL suppression on long-period fiber gratings by azimuthally isotropic exposure,” IEICE Trans. Electron. E85-C, 934–939 (2002).

Nishimura, M.

T. Okoshi, M. Nishimura, “Measurement of axially non-symmetrical refractive-index distribution of a single-mode fiber by a multidirectional scattering-pattern method,” J. Lightwave Technol. 1, 9–14 (1983).
[CrossRef]

Noda, K.-I.

K. Toga, N. Amano, K.-I. Noda, “Microscopic computer tomography measurement of nonaxisymmetrically distributed optical fiber refractive index,” J. Lightwave Technol. 6, 73–79 (1988).
[CrossRef]

Nugent, K. A.

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

S. T. Huntington, P. Mulvaney, A. Roberts, K. A. Nugent, M. Bazylenko, “Atomic force microscopy for the determination of refractive index profiles of optical fibers and waveguides: a quantitative study,” J. Appl. Phys. 82, 2730–2734 (1997).
[CrossRef]

Oh, K.

Okoshi, T.

T. Okoshi, M. Nishimura, “Measurement of axially non-symmetrical refractive-index distribution of a single-mode fiber by a multidirectional scattering-pattern method,” J. Lightwave Technol. 1, 9–14 (1983).
[CrossRef]

Okude, S.

Y. Ishii, K. Shima, S. Okude, K. Nishide, A. Wada, “PDL suppression on long-period fiber gratings by azimuthally isotropic exposure,” IEICE Trans. Electron. E85-C, 934–939 (2002).

Paek, U. C.

Paganin, D.

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

Park, Y.

Pluta, M.

M. Pluta, Measuring Techniques, Vol. 3, Advanced Light Microscopy (Elsevier, New York, 1993).

M. Pluta, “Profile refractometry of optical fibers using double-refracting microinterferometry,” in radient-Index Optics in Science and Engineering, M. Pluta, M. Szyjer, eds., Proc. SPIE2943, 113–127 (1996).
[CrossRef]

Presby, H.

D. Marcuse, H. Presby, “Index profile measurements of fibres and their evaluation,” Proc. IEEE 68, 666–688 (1980).
[CrossRef]

Presby, H. M.

D. Marcuse, H. M. Presby, “Focusing method for nondestructive measurement of optical fiber index profiles,” Appl. Opt. 18, 14–22 (1979).
[CrossRef] [PubMed]

L. M. Boggs, H. M. Presby, D. Marcuse, “Rapid automatic index profiling of whole-fiber samples. 1,” Bell Syst. Tech. J. 58, 867–882 (1979).
[CrossRef]

Reed, W. A.

Roberts, A.

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

S. T. Huntington, P. Mulvaney, A. Roberts, K. A. Nugent, M. Bazylenko, “Atomic force microscopy for the determination of refractive index profiles of optical fibers and waveguides: a quantitative study,” J. Appl. Phys. 82, 2730–2734 (1997).
[CrossRef]

Sánchez-Escobar, J. J.

Schwider, J.

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1990), Vol. 28, pp. 271–359.
[CrossRef]

Shima, K.

Y. Ishii, K. Shima, S. Okude, K. Nishide, A. Wada, “PDL suppression on long-period fiber gratings by azimuthally isotropic exposure,” IEICE Trans. Electron. E85-C, 934–939 (2002).

Sochacka, M.

M. Sochacka, “Optical fiber profiling by phase-stepping transverse interferometry,” J. Lightwave Technol. 12, 19–23 (1994).
[CrossRef]

Swindell, W.

H. H. Barrett, W. Swindell, Radiological Imaging: The Theory of Image Formation, Detection, and Processing (Academic, New York, 1981), Vol. 2.

Toga, K.

K. Toga, N. Amano, K.-I. Noda, “Microscopic computer tomography measurement of nonaxisymmetrically distributed optical fiber refractive index,” J. Lightwave Technol. 6, 73–79 (1988).
[CrossRef]

Vázquez-Montiel, S.

Veng, T.

L. Grüner-Nielsen, S. N. Knudsen, B. Edvold, T. Veng, D. Magnussen, C. C. Larsen, H. Damsgaard, “Dispersion compensating fibers,” Opt. Fiber Technol. 6, 164–180 (2000).
[CrossRef]

Vengsarkar, A. M.

Viskoe, D. A.

D. A. Viskoe, G. W. Donohoe, “Optimal computed tomography data acquisition techniques and filter selection for detection of small density variations,” IEEE Trans. Instrum. Meas. 45, 70–76 (1996).
[CrossRef]

Wada, A.

Y. Ishii, K. Shima, S. Okude, K. Nishide, A. Wada, “PDL suppression on long-period fiber gratings by azimuthally isotropic exposure,” IEICE Trans. Electron. E85-C, 934–939 (2002).

Young, M.

Zhong, Q.

A. M. Vengsarkar, Q. Zhong, D. Inniss, W. A. Reed, P. J. Lemaire, S. G. Kosinski, “Birefringence reduction in side-written photoinduced fiber devices by a dual-exposure method,” Opt. Lett. 19, 1260–1262 (1994).
[CrossRef] [PubMed]

Q. Zhong, D. Inniss, “Characterization of the lightguiding structure of optical fibers by atomic force microscopy,” J. Light-wave Technol. 12, 1517–1523 (1994).
[CrossRef]

Appl. Opt. (5)

Bell Syst. Tech. J. (1)

L. M. Boggs, H. M. Presby, D. Marcuse, “Rapid automatic index profiling of whole-fiber samples. 1,” Bell Syst. Tech. J. 58, 867–882 (1979).
[CrossRef]

IEEE Trans. Instrum. Meas. (1)

D. A. Viskoe, G. W. Donohoe, “Optimal computed tomography data acquisition techniques and filter selection for detection of small density variations,” IEEE Trans. Instrum. Meas. 45, 70–76 (1996).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

(a) Illustration of refractive-index projections [optical path length (OPL)] of a twin-core optical fiber taken 90° apart. (b) Relationship between the fixed coordinate system (x, y) of the optical fiber and the rotated coordinate system (d, L) of the projection, p(d, θ), at angle θ. The projections go to zero outside the spatial limits of the fiber cross sections.

Fig. 2
Fig. 2

(a) Diagram of a typical ray passing through the optical fiber sample. The quantities d and L are the rotated coordinate system axes, n(d, L) is the two-dimensional transverse refractive-index profile of the sample, noil is the index of the matching oil, δref is the phase of a ray traveling through the oil in the reference arm, δsamp is the accumulated phase of a ray traveling through the sample, dr is the distance from the fiber core to the sample ray, Lf is the length of the sample through which the ray passes, and Lr is an arbitrary reference length. (b) Interference image of optical fiber. D is the fringe separation distance and Qd is the relative fringe shift.

Fig. 3
Fig. 3

Experimental configuration for measuring interference images of an optical fiber test object at various projection angles. An optical fiber sample, secured in the holder, can be rotated about its axis to enable interference images to be recorded at any angle. The measurement system is automated easily by incorporation of a motion controller and a frame grabber.

Fig. 4
Fig. 4

Gray-scale plot of a transverse optical fiber refractive-index profile relative to the matching oil’s index. Simulated profiles like this one are used for generating interference images and testing the fringe analysis reconstruction programs. This particular simulated profile is circularly symmetric and possesses outer cladding, inner cladding, and core regions.

Fig. 5
Fig. 5

Example interference image generated by use of Eq. (11) from the test profile shown in Fig. 3. As the profile is symmetric, all the projections are identical (except for additive noise).

Fig. 6
Fig. 6

Gray-scale plot of the reconstructed index profile of the circularly symmetric optical fiber.

Fig. 7
Fig. 7

Symmetric optical fiber simulation results. (a) Comparison of test and reconstructed profiles taken along the length at the center of the width. (b) Absolute index difference between test and reconstructed profiles shown in (a). The noise in the interior cladding regions is lower than that near the edges and in the core because of the modified filter used in reconstruction.

Fig. 8
Fig. 8

(a) Gray-scale plot of the generated transverse refractive-index profile of a twin-core optical fiber relative to the matching oil’s index. The profile is not circularly symmetric because of the two offset (from center) cores. (b) Reconstructed index profile.

Fig. 9
Fig. 9

Twin-core optical fiber simulation results. (a) Comparison of test and reconstructed profiles taken along the length at the center of the width. (b) Absolute index difference between test and reconstructed profiles shown in (a). Noise levels are roughly similar in the cladding and cores and near the edges because only the basic ramp-type filter was used.

Fig. 10
Fig. 10

(a) Gray-scale plot of the generated transverse refractive-index profile of a single-mode optical fiber relative to the matching oil’s index. The exponential variation originates from one side and was calculated from an equation of Dossou et al.5, but applied over the entire cross section. (b) Reconstructed index profile. A shorter relative index range is used to highlight index variations in the cladding region (core features are not shown).

Fig. 11
Fig. 11

Asymmetric (exponential) profile optical fiber simulation results. (a) Comparison of test and reconstructed profiles taken along the length at the center of the width. The exponential variation over the length is evident in the reconstructed profile. (b) Absolute index difference between test and reconstructed profiles shown in (a).

Equations (11)

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p ( d , θ ) = - n ( d , L ) d L ,
n ( x , y ) = 0 2 π d θ 0 P ( ω , θ ) ω exp ( i 2 π ω d ) d ω ,
δ samp = k 0 n oil [ L r - L f ] + k 0 L f n ( d , L ) d L ,
δ ref = k 0 n oil L r = k 0 n oil [ L r - L f ] + k 0 n oil L f .
Δ δ = δ samp - δ ref = k 0 n oil [ L r - L f ] + k 0 L f n ( d , L ) d L - k 0 n oil [ L r - L f ] - k 0 n oil L f ,
Δ δ = k 0 L f n ( d , L ) d L - k 0 n oil L f .
k 0 n oil L f = k 0 L f n oil d L .
Δ δ = k 0 L f n ( d , L ) d L - k 0 L f n oil d L = k 0 L f [ n ( d , L ) - n oil ] d L .
Δ δ = 2 π Q d D
P r ( d , θ ) = L f [ n ( d , L ) - n oil ] d L = 2 π Q d k 0 D = Q d D λ 0 ,
I ( p , q ) = { A + B cos [ k 0 W ( p , q ) ] } + N ( p , q ) ,

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