Abstract

A novel transmitted-light differential interference contrast (DIC) system is used for nondestructive measurement of the refractive-index profile (RIP) of an optical fiber. By means of this system the phase of a measured light beam can be modulated with an analyzer, and the phase distribution of a fiber is obtained by calculation of the various interference patterns. The measurement theory and structure and some typical applications of this system are demonstrated. The results of measuring RIPs in graded-index fiber are presented. Both the experimental results and theoretical analysis show that the system takes the advantage of high index resolution and of sufficient measurement accuracy for measuring the refractive index of the optical fiber. The system has strong ability to overcome environmental disturbance because of its common-path design. Moreover, one can use the system to measure the RIP along the fiber axis and acquire an image of the three-dimensional RIP of the fiber.

© 2004 Optical Society of America

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References

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  1. J. C. Palais, “Fiber optic communications system,” in Fiber Optic Communications, J. C. Palais, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1998), pp. 1–35.
  2. H. El-Ghandoor, E. A. El-Ghafar, R. Hassan, “Refractive index profiling of a GRIN optical fiber using a modulated speckled sheet of light,” Opt. Laser Technol. 31, 481–488 (1999).
    [CrossRef]
  3. N. Gisin, R. Passy, B. Perny, “Optical fiber characterization by simultaneous measurement of the transmitted and refracted near field,” J. Lightwave Technol. 11, 1875–1883 (1993).
    [CrossRef]
  4. R. Conde, C. Depeursinge, “Refractive index profile and geometry measurements in multicore fibres,” Pure Appl. Opt. 5, 269–274 (1996).
    [CrossRef]
  5. P. Oberson, B. Gisin, B. Huttner, N. Gisin, “Refracted near-field measurements of refractive index and geometry of silica-on-silicon integrated optical wave guides,” Appl. Opt. 37, 7268–7272 (1998).
    [CrossRef]
  6. K. W. Raine, J. G. N. Baines, D. E. Putland, “Refractive index profiling—state of the art,” J. Lightwave Technol. 7, 1162–1169 (1989).
    [CrossRef]
  7. M. Pluta, “Interference microscope of polymer fibers,” J. Microsc. 96, 309–332 (1972).
    [CrossRef]
  8. N. Barakat, “Interferometric studies on fibers. Theory of interferometric determination of indices of fibers. 1,” Text. Res. J. 41, 167–170 (1971).
    [CrossRef]
  9. A. A. Hamza, A. M. Ghander, M. A. Mabrouk, “Interferometric studies on polymer structure,” Radiat. Phys. Chem. 33, 231–235 (1989).
  10. A. A. Hamza, M. A. Mabrouk, W. A. Shams-Eldin, “Determination of GRIN optical fibre parameters from transverse interferograms considering the refraction of the incident ray by the fibre,” Opt. Commun. 200, 131–138 (2001).
    [CrossRef]
  11. H. El-Ghandoor, I. Nasser, M. A. Rahman, R. Hassan, “Theoretical model for the transverse interference pattern of GRIN optical fiber using a laser sheet of light,” Opt. Laser Technol. 32, 281–286 (2000).
    [CrossRef]
  12. L. Z. Cai, Y. R. Wang, X. M. Qu, “Real-time analysis of 3-D axisymmetric refractive index field by photorefractive holographic interferometry,” Opt. Laser Technol. 32, 171–175 (2000).
    [CrossRef]
  13. C. M. Vest, “Interferometry of strongly refracting axisymmetric phase objects,” Appl. Opt. 14, 1601–1606 (1975).
    [CrossRef] [PubMed]
  14. E. B. Van Munster, L. V. Vliet, J. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope,” J. Microsc. 188, 149–157 (1997).
    [CrossRef]
  15. W. Lang, “Nomarski differential interference contrast microscopy. II. Formation of the interference image,” Zeiss Inform. 17, 12–16 (1969).
  16. A. A. Hamza, “The interferometric methods applied to the studies of fibrous materials,” J. Microsc. 142, 35–47 (1986).
    [CrossRef]
  17. H. Gundlach, “Phase contrast and differential interference contrast instrumentation and application in cell, developmental, and marine biology,” Opt. Eng. 32, 3223–3228 (1993).
    [CrossRef]
  18. M. R. Roubin, J. S. Pharamond, “Application of laser scanning cytometry followed by epifluorescent and differential interference contrast microscopy for the detection and enumeration of Cryptosporidium and Giardia in raw and potable waters,” J. Appl. Microbiol. 93, 599–607 (2002).
    [CrossRef] [PubMed]
  19. D. L. Lessor, J. S. Hartman, R. L. Gordon, “Quantitative surface topography determination by Nomarski reflection microscopy. 1. Theory,” J. Opt. Soc. Am. 69, 357–366 (1979).
    [CrossRef]
  20. J. S. Hartman, R. L. Gordon, D. L. Lessor, “Quantitative surface topography determination by Nomarski reflection microscopy. 2. Microscope modification, calibration, and planar sample experiments,” Appl. Opt. 19, 2998–3009 (1980).
    [CrossRef] [PubMed]
  21. G. Franz, J. Kross, “Generation of two-dimensional surface profile from differential interference contrast (DIC)—images,” Optik 112, 363–367 (2001).
    [CrossRef]
  22. D. Malacara, M. Servin, Z. Malacara, “Interferogram analysis for optical testing,” in Phase Detection Algorithms, B. J. Thompson, ed. (Marcel Dekker, New York, 1998), pp. 169–245.
  23. M. Sochacka, F. L. Provost, “Implementation of phase-stepping interferometry to transmitted-light DIC microscopy for dielectric surface evaluation,” in Phase Contrast and Differential Interference Contrast Imaging Techniques and Applications, M. Pluta, M. Szyjer, eds., Proc. SPIE1846, 212–221 (1994).
    [CrossRef]
  24. C. M. Vest, “Interferometry of strongly refracting axisymmetric phase objects,” Appl. Opt. 14, 1601–1606 (1975).
    [CrossRef] [PubMed]
  25. R. L. Burden, J. D. Faries, “Iterative techniques in matrix algebra,” in Numerical Analysis, 7th ed., R. L. Burden, J. D. Faries, eds. (Wadsworth Group, Pacific Grove, Calif., 2001), pp. 417–462.

2002 (1)

M. R. Roubin, J. S. Pharamond, “Application of laser scanning cytometry followed by epifluorescent and differential interference contrast microscopy for the detection and enumeration of Cryptosporidium and Giardia in raw and potable waters,” J. Appl. Microbiol. 93, 599–607 (2002).
[CrossRef] [PubMed]

2001 (2)

G. Franz, J. Kross, “Generation of two-dimensional surface profile from differential interference contrast (DIC)—images,” Optik 112, 363–367 (2001).
[CrossRef]

A. A. Hamza, M. A. Mabrouk, W. A. Shams-Eldin, “Determination of GRIN optical fibre parameters from transverse interferograms considering the refraction of the incident ray by the fibre,” Opt. Commun. 200, 131–138 (2001).
[CrossRef]

2000 (2)

H. El-Ghandoor, I. Nasser, M. A. Rahman, R. Hassan, “Theoretical model for the transverse interference pattern of GRIN optical fiber using a laser sheet of light,” Opt. Laser Technol. 32, 281–286 (2000).
[CrossRef]

L. Z. Cai, Y. R. Wang, X. M. Qu, “Real-time analysis of 3-D axisymmetric refractive index field by photorefractive holographic interferometry,” Opt. Laser Technol. 32, 171–175 (2000).
[CrossRef]

1999 (1)

H. El-Ghandoor, E. A. El-Ghafar, R. Hassan, “Refractive index profiling of a GRIN optical fiber using a modulated speckled sheet of light,” Opt. Laser Technol. 31, 481–488 (1999).
[CrossRef]

1998 (1)

1997 (1)

E. B. Van Munster, L. V. Vliet, J. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope,” J. Microsc. 188, 149–157 (1997).
[CrossRef]

1996 (1)

R. Conde, C. Depeursinge, “Refractive index profile and geometry measurements in multicore fibres,” Pure Appl. Opt. 5, 269–274 (1996).
[CrossRef]

1993 (2)

N. Gisin, R. Passy, B. Perny, “Optical fiber characterization by simultaneous measurement of the transmitted and refracted near field,” J. Lightwave Technol. 11, 1875–1883 (1993).
[CrossRef]

H. Gundlach, “Phase contrast and differential interference contrast instrumentation and application in cell, developmental, and marine biology,” Opt. Eng. 32, 3223–3228 (1993).
[CrossRef]

1989 (2)

A. A. Hamza, A. M. Ghander, M. A. Mabrouk, “Interferometric studies on polymer structure,” Radiat. Phys. Chem. 33, 231–235 (1989).

K. W. Raine, J. G. N. Baines, D. E. Putland, “Refractive index profiling—state of the art,” J. Lightwave Technol. 7, 1162–1169 (1989).
[CrossRef]

1986 (1)

A. A. Hamza, “The interferometric methods applied to the studies of fibrous materials,” J. Microsc. 142, 35–47 (1986).
[CrossRef]

1980 (1)

1979 (1)

1975 (2)

1972 (1)

M. Pluta, “Interference microscope of polymer fibers,” J. Microsc. 96, 309–332 (1972).
[CrossRef]

1971 (1)

N. Barakat, “Interferometric studies on fibers. Theory of interferometric determination of indices of fibers. 1,” Text. Res. J. 41, 167–170 (1971).
[CrossRef]

1969 (1)

W. Lang, “Nomarski differential interference contrast microscopy. II. Formation of the interference image,” Zeiss Inform. 17, 12–16 (1969).

Aten, J.

E. B. Van Munster, L. V. Vliet, J. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope,” J. Microsc. 188, 149–157 (1997).
[CrossRef]

Baines, J. G. N.

K. W. Raine, J. G. N. Baines, D. E. Putland, “Refractive index profiling—state of the art,” J. Lightwave Technol. 7, 1162–1169 (1989).
[CrossRef]

Barakat, N.

N. Barakat, “Interferometric studies on fibers. Theory of interferometric determination of indices of fibers. 1,” Text. Res. J. 41, 167–170 (1971).
[CrossRef]

Burden, R. L.

R. L. Burden, J. D. Faries, “Iterative techniques in matrix algebra,” in Numerical Analysis, 7th ed., R. L. Burden, J. D. Faries, eds. (Wadsworth Group, Pacific Grove, Calif., 2001), pp. 417–462.

Cai, L. Z.

L. Z. Cai, Y. R. Wang, X. M. Qu, “Real-time analysis of 3-D axisymmetric refractive index field by photorefractive holographic interferometry,” Opt. Laser Technol. 32, 171–175 (2000).
[CrossRef]

Conde, R.

R. Conde, C. Depeursinge, “Refractive index profile and geometry measurements in multicore fibres,” Pure Appl. Opt. 5, 269–274 (1996).
[CrossRef]

Depeursinge, C.

R. Conde, C. Depeursinge, “Refractive index profile and geometry measurements in multicore fibres,” Pure Appl. Opt. 5, 269–274 (1996).
[CrossRef]

El-Ghafar, E. A.

H. El-Ghandoor, E. A. El-Ghafar, R. Hassan, “Refractive index profiling of a GRIN optical fiber using a modulated speckled sheet of light,” Opt. Laser Technol. 31, 481–488 (1999).
[CrossRef]

El-Ghandoor, H.

H. El-Ghandoor, I. Nasser, M. A. Rahman, R. Hassan, “Theoretical model for the transverse interference pattern of GRIN optical fiber using a laser sheet of light,” Opt. Laser Technol. 32, 281–286 (2000).
[CrossRef]

H. El-Ghandoor, E. A. El-Ghafar, R. Hassan, “Refractive index profiling of a GRIN optical fiber using a modulated speckled sheet of light,” Opt. Laser Technol. 31, 481–488 (1999).
[CrossRef]

Faries, J. D.

R. L. Burden, J. D. Faries, “Iterative techniques in matrix algebra,” in Numerical Analysis, 7th ed., R. L. Burden, J. D. Faries, eds. (Wadsworth Group, Pacific Grove, Calif., 2001), pp. 417–462.

Franz, G.

G. Franz, J. Kross, “Generation of two-dimensional surface profile from differential interference contrast (DIC)—images,” Optik 112, 363–367 (2001).
[CrossRef]

Ghander, A. M.

A. A. Hamza, A. M. Ghander, M. A. Mabrouk, “Interferometric studies on polymer structure,” Radiat. Phys. Chem. 33, 231–235 (1989).

Gisin, B.

Gisin, N.

P. Oberson, B. Gisin, B. Huttner, N. Gisin, “Refracted near-field measurements of refractive index and geometry of silica-on-silicon integrated optical wave guides,” Appl. Opt. 37, 7268–7272 (1998).
[CrossRef]

N. Gisin, R. Passy, B. Perny, “Optical fiber characterization by simultaneous measurement of the transmitted and refracted near field,” J. Lightwave Technol. 11, 1875–1883 (1993).
[CrossRef]

Gordon, R. L.

Gundlach, H.

H. Gundlach, “Phase contrast and differential interference contrast instrumentation and application in cell, developmental, and marine biology,” Opt. Eng. 32, 3223–3228 (1993).
[CrossRef]

Hamza, A. A.

A. A. Hamza, M. A. Mabrouk, W. A. Shams-Eldin, “Determination of GRIN optical fibre parameters from transverse interferograms considering the refraction of the incident ray by the fibre,” Opt. Commun. 200, 131–138 (2001).
[CrossRef]

A. A. Hamza, A. M. Ghander, M. A. Mabrouk, “Interferometric studies on polymer structure,” Radiat. Phys. Chem. 33, 231–235 (1989).

A. A. Hamza, “The interferometric methods applied to the studies of fibrous materials,” J. Microsc. 142, 35–47 (1986).
[CrossRef]

Hartman, J. S.

Hassan, R.

H. El-Ghandoor, I. Nasser, M. A. Rahman, R. Hassan, “Theoretical model for the transverse interference pattern of GRIN optical fiber using a laser sheet of light,” Opt. Laser Technol. 32, 281–286 (2000).
[CrossRef]

H. El-Ghandoor, E. A. El-Ghafar, R. Hassan, “Refractive index profiling of a GRIN optical fiber using a modulated speckled sheet of light,” Opt. Laser Technol. 31, 481–488 (1999).
[CrossRef]

Huttner, B.

Kross, J.

G. Franz, J. Kross, “Generation of two-dimensional surface profile from differential interference contrast (DIC)—images,” Optik 112, 363–367 (2001).
[CrossRef]

Lang, W.

W. Lang, “Nomarski differential interference contrast microscopy. II. Formation of the interference image,” Zeiss Inform. 17, 12–16 (1969).

Lessor, D. L.

Mabrouk, M. A.

A. A. Hamza, M. A. Mabrouk, W. A. Shams-Eldin, “Determination of GRIN optical fibre parameters from transverse interferograms considering the refraction of the incident ray by the fibre,” Opt. Commun. 200, 131–138 (2001).
[CrossRef]

A. A. Hamza, A. M. Ghander, M. A. Mabrouk, “Interferometric studies on polymer structure,” Radiat. Phys. Chem. 33, 231–235 (1989).

Malacara, D.

D. Malacara, M. Servin, Z. Malacara, “Interferogram analysis for optical testing,” in Phase Detection Algorithms, B. J. Thompson, ed. (Marcel Dekker, New York, 1998), pp. 169–245.

Malacara, Z.

D. Malacara, M. Servin, Z. Malacara, “Interferogram analysis for optical testing,” in Phase Detection Algorithms, B. J. Thompson, ed. (Marcel Dekker, New York, 1998), pp. 169–245.

Nasser, I.

H. El-Ghandoor, I. Nasser, M. A. Rahman, R. Hassan, “Theoretical model for the transverse interference pattern of GRIN optical fiber using a laser sheet of light,” Opt. Laser Technol. 32, 281–286 (2000).
[CrossRef]

Oberson, P.

Palais, J. C.

J. C. Palais, “Fiber optic communications system,” in Fiber Optic Communications, J. C. Palais, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1998), pp. 1–35.

Passy, R.

N. Gisin, R. Passy, B. Perny, “Optical fiber characterization by simultaneous measurement of the transmitted and refracted near field,” J. Lightwave Technol. 11, 1875–1883 (1993).
[CrossRef]

Perny, B.

N. Gisin, R. Passy, B. Perny, “Optical fiber characterization by simultaneous measurement of the transmitted and refracted near field,” J. Lightwave Technol. 11, 1875–1883 (1993).
[CrossRef]

Pharamond, J. S.

M. R. Roubin, J. S. Pharamond, “Application of laser scanning cytometry followed by epifluorescent and differential interference contrast microscopy for the detection and enumeration of Cryptosporidium and Giardia in raw and potable waters,” J. Appl. Microbiol. 93, 599–607 (2002).
[CrossRef] [PubMed]

Pluta, M.

M. Pluta, “Interference microscope of polymer fibers,” J. Microsc. 96, 309–332 (1972).
[CrossRef]

Provost, F. L.

M. Sochacka, F. L. Provost, “Implementation of phase-stepping interferometry to transmitted-light DIC microscopy for dielectric surface evaluation,” in Phase Contrast and Differential Interference Contrast Imaging Techniques and Applications, M. Pluta, M. Szyjer, eds., Proc. SPIE1846, 212–221 (1994).
[CrossRef]

Putland, D. E.

K. W. Raine, J. G. N. Baines, D. E. Putland, “Refractive index profiling—state of the art,” J. Lightwave Technol. 7, 1162–1169 (1989).
[CrossRef]

Qu, X. M.

L. Z. Cai, Y. R. Wang, X. M. Qu, “Real-time analysis of 3-D axisymmetric refractive index field by photorefractive holographic interferometry,” Opt. Laser Technol. 32, 171–175 (2000).
[CrossRef]

Rahman, M. A.

H. El-Ghandoor, I. Nasser, M. A. Rahman, R. Hassan, “Theoretical model for the transverse interference pattern of GRIN optical fiber using a laser sheet of light,” Opt. Laser Technol. 32, 281–286 (2000).
[CrossRef]

Raine, K. W.

K. W. Raine, J. G. N. Baines, D. E. Putland, “Refractive index profiling—state of the art,” J. Lightwave Technol. 7, 1162–1169 (1989).
[CrossRef]

Roubin, M. R.

M. R. Roubin, J. S. Pharamond, “Application of laser scanning cytometry followed by epifluorescent and differential interference contrast microscopy for the detection and enumeration of Cryptosporidium and Giardia in raw and potable waters,” J. Appl. Microbiol. 93, 599–607 (2002).
[CrossRef] [PubMed]

Servin, M.

D. Malacara, M. Servin, Z. Malacara, “Interferogram analysis for optical testing,” in Phase Detection Algorithms, B. J. Thompson, ed. (Marcel Dekker, New York, 1998), pp. 169–245.

Shams-Eldin, W. A.

A. A. Hamza, M. A. Mabrouk, W. A. Shams-Eldin, “Determination of GRIN optical fibre parameters from transverse interferograms considering the refraction of the incident ray by the fibre,” Opt. Commun. 200, 131–138 (2001).
[CrossRef]

Sochacka, M.

M. Sochacka, F. L. Provost, “Implementation of phase-stepping interferometry to transmitted-light DIC microscopy for dielectric surface evaluation,” in Phase Contrast and Differential Interference Contrast Imaging Techniques and Applications, M. Pluta, M. Szyjer, eds., Proc. SPIE1846, 212–221 (1994).
[CrossRef]

Van Munster, E. B.

E. B. Van Munster, L. V. Vliet, J. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope,” J. Microsc. 188, 149–157 (1997).
[CrossRef]

Vest, C. M.

Vliet, L. V.

E. B. Van Munster, L. V. Vliet, J. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope,” J. Microsc. 188, 149–157 (1997).
[CrossRef]

Wang, Y. R.

L. Z. Cai, Y. R. Wang, X. M. Qu, “Real-time analysis of 3-D axisymmetric refractive index field by photorefractive holographic interferometry,” Opt. Laser Technol. 32, 171–175 (2000).
[CrossRef]

Appl. Opt. (4)

J. Appl. Microbiol. (1)

M. R. Roubin, J. S. Pharamond, “Application of laser scanning cytometry followed by epifluorescent and differential interference contrast microscopy for the detection and enumeration of Cryptosporidium and Giardia in raw and potable waters,” J. Appl. Microbiol. 93, 599–607 (2002).
[CrossRef] [PubMed]

J. Lightwave Technol. (2)

K. W. Raine, J. G. N. Baines, D. E. Putland, “Refractive index profiling—state of the art,” J. Lightwave Technol. 7, 1162–1169 (1989).
[CrossRef]

N. Gisin, R. Passy, B. Perny, “Optical fiber characterization by simultaneous measurement of the transmitted and refracted near field,” J. Lightwave Technol. 11, 1875–1883 (1993).
[CrossRef]

J. Microsc. (3)

M. Pluta, “Interference microscope of polymer fibers,” J. Microsc. 96, 309–332 (1972).
[CrossRef]

A. A. Hamza, “The interferometric methods applied to the studies of fibrous materials,” J. Microsc. 142, 35–47 (1986).
[CrossRef]

E. B. Van Munster, L. V. Vliet, J. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope,” J. Microsc. 188, 149–157 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

A. A. Hamza, M. A. Mabrouk, W. A. Shams-Eldin, “Determination of GRIN optical fibre parameters from transverse interferograms considering the refraction of the incident ray by the fibre,” Opt. Commun. 200, 131–138 (2001).
[CrossRef]

Opt. Eng. (1)

H. Gundlach, “Phase contrast and differential interference contrast instrumentation and application in cell, developmental, and marine biology,” Opt. Eng. 32, 3223–3228 (1993).
[CrossRef]

Opt. Laser Technol. (3)

H. El-Ghandoor, I. Nasser, M. A. Rahman, R. Hassan, “Theoretical model for the transverse interference pattern of GRIN optical fiber using a laser sheet of light,” Opt. Laser Technol. 32, 281–286 (2000).
[CrossRef]

L. Z. Cai, Y. R. Wang, X. M. Qu, “Real-time analysis of 3-D axisymmetric refractive index field by photorefractive holographic interferometry,” Opt. Laser Technol. 32, 171–175 (2000).
[CrossRef]

H. El-Ghandoor, E. A. El-Ghafar, R. Hassan, “Refractive index profiling of a GRIN optical fiber using a modulated speckled sheet of light,” Opt. Laser Technol. 31, 481–488 (1999).
[CrossRef]

Optik (1)

G. Franz, J. Kross, “Generation of two-dimensional surface profile from differential interference contrast (DIC)—images,” Optik 112, 363–367 (2001).
[CrossRef]

Pure Appl. Opt. (1)

R. Conde, C. Depeursinge, “Refractive index profile and geometry measurements in multicore fibres,” Pure Appl. Opt. 5, 269–274 (1996).
[CrossRef]

Radiat. Phys. Chem. (1)

A. A. Hamza, A. M. Ghander, M. A. Mabrouk, “Interferometric studies on polymer structure,” Radiat. Phys. Chem. 33, 231–235 (1989).

Text. Res. J. (1)

N. Barakat, “Interferometric studies on fibers. Theory of interferometric determination of indices of fibers. 1,” Text. Res. J. 41, 167–170 (1971).
[CrossRef]

Zeiss Inform. (1)

W. Lang, “Nomarski differential interference contrast microscopy. II. Formation of the interference image,” Zeiss Inform. 17, 12–16 (1969).

Other (4)

J. C. Palais, “Fiber optic communications system,” in Fiber Optic Communications, J. C. Palais, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1998), pp. 1–35.

R. L. Burden, J. D. Faries, “Iterative techniques in matrix algebra,” in Numerical Analysis, 7th ed., R. L. Burden, J. D. Faries, eds. (Wadsworth Group, Pacific Grove, Calif., 2001), pp. 417–462.

D. Malacara, M. Servin, Z. Malacara, “Interferogram analysis for optical testing,” in Phase Detection Algorithms, B. J. Thompson, ed. (Marcel Dekker, New York, 1998), pp. 169–245.

M. Sochacka, F. L. Provost, “Implementation of phase-stepping interferometry to transmitted-light DIC microscopy for dielectric surface evaluation,” in Phase Contrast and Differential Interference Contrast Imaging Techniques and Applications, M. Pluta, M. Szyjer, eds., Proc. SPIE1846, 212–221 (1994).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Schematic of the DIC setup for measuring the RIP of a fiber.

Fig. 2
Fig. 2

Schematic diagrams of a fiber when a beam passes through (a) one layer, (b) the second layer only, (c) the third layer only, and (d) the kth layer.

Fig. 3
Fig. 3

Five interference patterns: (a) φ = -π/2, (b) φ = -π/4, (c) φ = 0, (d) φ = π/4, (e) φ = π/2.

Fig. 4
Fig. 4

Phase profile in a graded-index fiber.

Fig. 5
Fig. 5

Optical length profile in a graded-index fiber.

Fig. 6
Fig. 6

Comparison of the result obtained by our method and from the NR-9200 fiber analyzer (RNF method).

Fig. 7
Fig. 7

Three-dimensional RIP of the fiber.

Fig. 8
Fig. 8

Condition number of the M matrix when the number of layers N changes.

Fig. 9
Fig. 9

Two sets of RIPs when N = 25 and α = 2.5.

Fig. 10
Fig. 10

Two sets of RIPs when N = 25 and α = 2.0.

Fig. 11
Fig. 11

Error distribution of Figs. 9 and 10.

Fig. 12
Fig. 12

Maximum and average simulation errors.

Equations (24)

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

s=α×f,
I1x, y=a+b sinφx, y-π/2,I2x, y=a+b sinφx, y-π/4,I3x, y=a+b sinφx, y,I4x, y=a+b sinφx, y+π/4,I5x, y=a+b sinφx, y+π/2,
φx, y=arctan22I3-I5-I12I4-I2.
dHdx=λ2πs φx, y,Hx, y= λ2πs φx, ydx,
Δd=r/N.
hSS1=hSA+hAA+hAS=2n0r-r2-d12+2n1r2-d12,
d1=r-Δd/2
hSS2=2hSA+2hAB+hBB=2n0r-r2-d22+2n1r2-d22-r-Δd2-d22+2n2r-Δd2-d22.
d2=r-3/2Δd.
hSS3=2hSA+2hAB+2hBC+hCC=2n0r-r2-d32+2n1r2-d32-r-Δd2-d32+2n2r-Δd2-d32-r-2Δd2-d32+2n3r-2Δd2-d32,
d3=r-5/2Δd.
hSSk=2hSA+2hAB+2hBB++hcenter=2n0r-r2-dk2+2 i=1k-1 ni×r-i-1Δd2-dk2-r-iΔd2-dk2+2nkr-k-1Δd2-dk2,
dk=r-2k-1/2Δd.
hSSk=2n0r-r2-dk2+2 dRnrrdrr2-d2.
nr=-1πdRdhSSk/dxdxx2-r2.
Mñ+c˜=h˜,
ñ=n1n2n3nN, c˜=n0r-r2-d12n0r-r2-d22n0r-r2-d32n0r-r2-dN2, h˜=hSS1hSS2hSS3hSSN,
M=2r2-d1202r2-d22-r-Δd2-d222r-Δd2-d222r2-d32-r-Δd2-d322r-Δd2-d32-r-2Δd2-d322r2-dk2-r-Δd2-dk22r-Δd2-dk2-r-2Δd2-dk22r2-dN2-r-Δd2-dN22r-Δd2-dN2-r-2Δd2-dN2002r - 2Δd2 - d322r - 2Δd2 - dk2 - r - 3Δd2 - dk22r - 2Δd2 - dN2 - r - 3Δd2 - dN200 000 000 0 02r - k - 1Δd2 - dk2 0 0 2r - (N - 1)Δd2 - dN2,
MN×NMN×NñN×1+c˜N×1c˜N×1=h˜N×1h˜N×1.
M · ñ=h˜N×1+h˜N×1(-c˜N×1-c˜N×1/2.
dk=r1-k-0.5N.
M=1/rM.
cond2M=M2M-12, M2=eigMTM1/2,
nd=ncore-ncore-ncld/rα,

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