Abstract

We present a simple method to determine the refractive indices of transparent specimens. The refractive index of an object under investigation is received by evaluating the optical path difference introduced by the object, while taking into account geometric parameters. The optical path difference that corresponds to the phase distribution is obtained by a noninterferometric, noniterative phase retrieval method based on Green’s functions. It will be shown that this technique is a highly accurate and quantitative method for refractive index determination.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Frank, S. Altmeyer, and G. Wernicke, “Non-interferometric, non-iterative phase retrieval by Green’s functions,” J. Opt. Soc. Am. A 27, 2244–2251 (2010).
    [CrossRef]
  2. M. A. El-Morsy, T. Yatagai, A. A. Hamza, M. A. Mabrouk, and T. Z. N. Sokkar, “Automatic refractive index profiling of fibers by phase analysis method using Fourier transform,” Opt. Lasers Eng. 38, 509–525 (2002).
    [CrossRef]
  3. A. A. Hamza, T. Z. N. Sokkar, M. A. El-Bakary, and A. M. Ali, “An interferometric method for studying the influence of temperature on the mean refractive indices and cross-sectional area of irregular fibres,” Polym. Test. 22, 83–91 (2003).
    [CrossRef]
  4. A. A. Hamza, M. A. Mabrouk, W. A. Ramadan, and H. H. Wahba, “Core-index determination of a thick fibre using lens-fibre interference (LFI) technique,” Opt. Lasers Eng. 42, 121–130 (2004).
    [CrossRef]
  5. I. Martincek, D. Kacik, I. Turek, and P. Peterka, “The determination of the refractive index profile in α-profile optical fibres by intermodal interference investigation,” Optik 115, 86–88 (2004).
    [CrossRef]
  6. Z. Liu, X. Dong, Q. Chen, C. Yin, Y. Xu, and Y. Zheng, “Nondestructive measurement of an optical fiber refractive-index profile by a transmitted-light differential interference contact microscope,” Appl. Opt. 43, 1485–1492 (2004).
    [CrossRef] [PubMed]
  7. T. Z. N. Sokkar, H. M. El Dessouky, M. A. Shams-Eldin, and M. A. El-Morsy, “Automatic fringe analysis of two-beam interference patterns for measurement of refractive index and birefringence profiles of fibres,” Opt. Lasers Eng. 45, 431–441 (2007).
    [CrossRef]
  8. H. H. Wahba and T. Kreis, “Characterization of graded index optical fiber by digital holographic interferometry,” Appl. Opt. 48, 1573–1582 (2009).
    [CrossRef] [PubMed]
  9. P. Stock, Manufacturing Department, FiberTech GmbH, Nalepastrasse 170–171, 12459 Berlin (personal communication, 2009).
  10. G. Fornaro, G. Franceschetti, and R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
    [CrossRef]
  11. H. Maitre and I. Lyuboshenko, “Robust algorithms for phase unwrapping in SAR interferometry,” Proc. SPIE 3217, 176–187 (1997).
    [CrossRef]
  12. I. Lyuboshenko and H. Maitre, “Phase unwrapping for interferometric synthetic aperture radar by use of Helmholtz equation eigenfunctions and the first Green’s identity,” J. Opt. Soc. Am. A 16, 378–395 (1999).
    [CrossRef]
  13. E. G. Abramochkin and V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in a Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
    [CrossRef]
  14. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  15. D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
    [CrossRef]
  16. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
    [CrossRef]
  17. E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
    [CrossRef] [PubMed]
  18. P. J. McMohan, E. D. Barone-Nugent, B. E. Allman, and K. A. Nugent, “Quantitative phase-amplitude microscopy II: differential interference contrast imaging for biological TEM,” J. Microsc. 206, 204–208 (2002).
    [CrossRef]
  19. C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. M. D. Delbridge, and K. A. Nugent, “Quantitative phase-amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62–69 (2004).
    [CrossRef] [PubMed]
  20. M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
    [CrossRef] [PubMed]
  21. C. Dorrer and J. D. Zuegel, “Optical testing using the transport-of-intensity equation,” Opt. Express 15, 7165–7175 (2007).
    [CrossRef] [PubMed]
  22. A. V. Martin, F.-R. Chen, W.-K. Hsieh, J.-J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy 106, 914–924(2006).
    [CrossRef] [PubMed]
  23. A. Koehler, “Ein neues Beleuchtungsverfahren fuer mikrophotographische Zwecke,” Z. Wiss. Mikrosk. Mikrosk. Tech. 10, 443–440 (1893).
  24. D. Paganin, A. Barty, P. J. McMohan, and K. A. Nugent, “Quantitative phase-amplitude microscopy III: the effects of noise,” J. Microsc. 214, 51–61 (2004).
    [CrossRef] [PubMed]
  25. S. Altmeyer and J. Frank, “Optics and information technology,” in Technology Guide Principles—Applications—Trends, H.-J.Bullinger, ed. (Springer, 2009), pp. 98–103.
    [CrossRef]
  26. FiberTech GmbH, “Quartz/quartz fibers—all silica (AS), specifications,” http://www.leoni-fiber-optics.com/Quarz-Quarz-Fasern-All-Silica-AS.11918.0.html?&L=1.
  27. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).
  28. A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
    [CrossRef]
  29. A. Roberts, E. Ampem-Lassen, A. Barty, K. A. Nugent, G. W. Baxter, N. M. Dragomir, and S. T. Huntington, “Refractive-index profiling of optical fibers with axial symmetry by use of quantitative phase microscopy,” Opt. Lett. 27, 2061–2063(2002).
    [CrossRef]
  30. E. Ampem-Lassen, S. T. Huntington, N. M. Dragomir, K. A. Nugent, and A. Roberts, “Refractive index profiling of axially symmetric optical fibers: a new technique,” Opt. Express 13, 3277–3282 (2005).
    [CrossRef] [PubMed]

2010 (1)

2009 (1)

2007 (2)

C. Dorrer and J. D. Zuegel, “Optical testing using the transport-of-intensity equation,” Opt. Express 15, 7165–7175 (2007).
[CrossRef] [PubMed]

T. Z. N. Sokkar, H. M. El Dessouky, M. A. Shams-Eldin, and M. A. El-Morsy, “Automatic fringe analysis of two-beam interference patterns for measurement of refractive index and birefringence profiles of fibres,” Opt. Lasers Eng. 45, 431–441 (2007).
[CrossRef]

2006 (1)

A. V. Martin, F.-R. Chen, W.-K. Hsieh, J.-J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy 106, 914–924(2006).
[CrossRef] [PubMed]

2005 (1)

2004 (6)

Z. Liu, X. Dong, Q. Chen, C. Yin, Y. Xu, and Y. Zheng, “Nondestructive measurement of an optical fiber refractive-index profile by a transmitted-light differential interference contact microscope,” Appl. Opt. 43, 1485–1492 (2004).
[CrossRef] [PubMed]

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

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. M. D. Delbridge, and K. A. Nugent, “Quantitative phase-amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62–69 (2004).
[CrossRef] [PubMed]

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[CrossRef] [PubMed]

A. A. Hamza, M. A. Mabrouk, W. A. Ramadan, and H. H. Wahba, “Core-index determination of a thick fibre using lens-fibre interference (LFI) technique,” Opt. Lasers Eng. 42, 121–130 (2004).
[CrossRef]

I. Martincek, D. Kacik, I. Turek, and P. Peterka, “The determination of the refractive index profile in α-profile optical fibres by intermodal interference investigation,” Optik 115, 86–88 (2004).
[CrossRef]

2003 (1)

A. A. Hamza, T. Z. N. Sokkar, M. A. El-Bakary, and A. M. Ali, “An interferometric method for studying the influence of temperature on the mean refractive indices and cross-sectional area of irregular fibres,” Polym. Test. 22, 83–91 (2003).
[CrossRef]

2002 (4)

M. A. El-Morsy, T. Yatagai, A. A. Hamza, M. A. Mabrouk, and T. Z. N. Sokkar, “Automatic refractive index profiling of fibers by phase analysis method using Fourier transform,” Opt. Lasers Eng. 38, 509–525 (2002).
[CrossRef]

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

P. J. McMohan, E. D. Barone-Nugent, B. E. Allman, and K. A. Nugent, “Quantitative phase-amplitude microscopy II: differential interference contrast imaging for biological TEM,” J. Microsc. 206, 204–208 (2002).
[CrossRef]

A. Roberts, E. Ampem-Lassen, A. Barty, K. A. Nugent, G. W. Baxter, N. M. Dragomir, and S. T. Huntington, “Refractive-index profiling of optical fibers with axial symmetry by use of quantitative phase microscopy,” Opt. Lett. 27, 2061–2063(2002).
[CrossRef]

1999 (1)

1998 (2)

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[CrossRef]

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

1997 (1)

H. Maitre and I. Lyuboshenko, “Robust algorithms for phase unwrapping in SAR interferometry,” Proc. SPIE 3217, 176–187 (1997).
[CrossRef]

1996 (1)

G. Fornaro, G. Franceschetti, and R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

1989 (1)

E. G. Abramochkin and V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in a Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
[CrossRef]

1984 (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

1983 (1)

1893 (1)

A. Koehler, “Ein neues Beleuchtungsverfahren fuer mikrophotographische Zwecke,” Z. Wiss. Mikrosk. Mikrosk. Tech. 10, 443–440 (1893).

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in a Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
[CrossRef]

Ali, A. M.

A. A. Hamza, T. Z. N. Sokkar, M. A. El-Bakary, and A. M. Ali, “An interferometric method for studying the influence of temperature on the mean refractive indices and cross-sectional area of irregular fibres,” Polym. Test. 22, 83–91 (2003).
[CrossRef]

Allen, L. J.

A. V. Martin, F.-R. Chen, W.-K. Hsieh, J.-J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy 106, 914–924(2006).
[CrossRef] [PubMed]

Allman, B. E.

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. M. D. Delbridge, and K. A. Nugent, “Quantitative phase-amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62–69 (2004).
[CrossRef] [PubMed]

P. J. McMohan, E. D. Barone-Nugent, B. E. Allman, and K. A. Nugent, “Quantitative phase-amplitude microscopy II: differential interference contrast imaging for biological TEM,” J. Microsc. 206, 204–208 (2002).
[CrossRef]

Altmeyer, S.

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

S. Altmeyer and J. Frank, “Optics and information technology,” in Technology Guide Principles—Applications—Trends, H.-J.Bullinger, ed. (Springer, 2009), pp. 98–103.
[CrossRef]

Ampem-Lassen, E.

Barone-Nugent, E. D.

P. J. McMohan, E. D. Barone-Nugent, B. E. Allman, and K. A. Nugent, “Quantitative phase-amplitude microscopy II: differential interference contrast imaging for biological TEM,” J. Microsc. 206, 204–208 (2002).
[CrossRef]

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

Barty, A.

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

A. Roberts, E. Ampem-Lassen, A. Barty, K. A. Nugent, G. W. Baxter, N. M. Dragomir, and S. T. Huntington, “Refractive-index profiling of optical fibers with axial symmetry by use of quantitative phase microscopy,” Opt. Lett. 27, 2061–2063(2002).
[CrossRef]

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

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[CrossRef]

Baxter, G. W.

Beleggia, M.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[CrossRef] [PubMed]

Bellair, C. J.

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. M. D. Delbridge, and K. A. Nugent, “Quantitative phase-amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62–69 (2004).
[CrossRef] [PubMed]

Chen, F.-R.

A. V. Martin, F.-R. Chen, W.-K. Hsieh, J.-J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy 106, 914–924(2006).
[CrossRef] [PubMed]

Chen, Q.

Curl, C. L.

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. M. D. Delbridge, and K. A. Nugent, “Quantitative phase-amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62–69 (2004).
[CrossRef] [PubMed]

Delbridge, L. M. D.

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. M. D. Delbridge, and K. A. Nugent, “Quantitative phase-amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62–69 (2004).
[CrossRef] [PubMed]

Dong, X.

Dorrer, C.

Dragomir, N. M.

El Dessouky, H. M.

T. Z. N. Sokkar, H. M. El Dessouky, M. A. Shams-Eldin, and M. A. El-Morsy, “Automatic fringe analysis of two-beam interference patterns for measurement of refractive index and birefringence profiles of fibres,” Opt. Lasers Eng. 45, 431–441 (2007).
[CrossRef]

El-Bakary, M. A.

A. A. Hamza, T. Z. N. Sokkar, M. A. El-Bakary, and A. M. Ali, “An interferometric method for studying the influence of temperature on the mean refractive indices and cross-sectional area of irregular fibres,” Polym. Test. 22, 83–91 (2003).
[CrossRef]

El-Morsy, M. A.

T. Z. N. Sokkar, H. M. El Dessouky, M. A. Shams-Eldin, and M. A. El-Morsy, “Automatic fringe analysis of two-beam interference patterns for measurement of refractive index and birefringence profiles of fibres,” Opt. Lasers Eng. 45, 431–441 (2007).
[CrossRef]

M. A. El-Morsy, T. Yatagai, A. A. Hamza, M. A. Mabrouk, and T. Z. N. Sokkar, “Automatic refractive index profiling of fibers by phase analysis method using Fourier transform,” Opt. Lasers Eng. 38, 509–525 (2002).
[CrossRef]

Findlay, S. D.

A. V. Martin, F.-R. Chen, W.-K. Hsieh, J.-J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy 106, 914–924(2006).
[CrossRef] [PubMed]

Fornaro, G.

G. Fornaro, G. Franceschetti, and R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

Franceschetti, G.

G. Fornaro, G. Franceschetti, and R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

Frank, J.

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

S. Altmeyer and J. Frank, “Optics and information technology,” in Technology Guide Principles—Applications—Trends, H.-J.Bullinger, ed. (Springer, 2009), pp. 98–103.
[CrossRef]

Hamza, A. A.

A. A. Hamza, M. A. Mabrouk, W. A. Ramadan, and H. H. Wahba, “Core-index determination of a thick fibre using lens-fibre interference (LFI) technique,” Opt. Lasers Eng. 42, 121–130 (2004).
[CrossRef]

A. A. Hamza, T. Z. N. Sokkar, M. A. El-Bakary, and A. M. Ali, “An interferometric method for studying the influence of temperature on the mean refractive indices and cross-sectional area of irregular fibres,” Polym. Test. 22, 83–91 (2003).
[CrossRef]

M. A. El-Morsy, T. Yatagai, A. A. Hamza, M. A. Mabrouk, and T. Z. N. Sokkar, “Automatic refractive index profiling of fibers by phase analysis method using Fourier transform,” Opt. Lasers Eng. 38, 509–525 (2002).
[CrossRef]

Harris, P. J.

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. M. D. Delbridge, and K. A. Nugent, “Quantitative phase-amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62–69 (2004).
[CrossRef] [PubMed]

Hsieh, W.-K.

A. V. Martin, F.-R. Chen, W.-K. Hsieh, J.-J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy 106, 914–924(2006).
[CrossRef] [PubMed]

Huntington, S. T.

Kacik, D.

I. Martincek, D. Kacik, I. Turek, and P. Peterka, “The determination of the refractive index profile in α-profile optical fibres by intermodal interference investigation,” Optik 115, 86–88 (2004).
[CrossRef]

Kai, J.-J.

A. V. Martin, F.-R. Chen, W.-K. Hsieh, J.-J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy 106, 914–924(2006).
[CrossRef] [PubMed]

Koehler, A.

A. Koehler, “Ein neues Beleuchtungsverfahren fuer mikrophotographische Zwecke,” Z. Wiss. Mikrosk. Mikrosk. Tech. 10, 443–440 (1893).

Kreis, T.

Lanari, R.

G. Fornaro, G. Franceschetti, and R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

Liu, Z.

Lyuboshenko, I.

Mabrouk, M. A.

A. A. Hamza, M. A. Mabrouk, W. A. Ramadan, and H. H. Wahba, “Core-index determination of a thick fibre using lens-fibre interference (LFI) technique,” Opt. Lasers Eng. 42, 121–130 (2004).
[CrossRef]

M. A. El-Morsy, T. Yatagai, A. A. Hamza, M. A. Mabrouk, and T. Z. N. Sokkar, “Automatic refractive index profiling of fibers by phase analysis method using Fourier transform,” Opt. Lasers Eng. 38, 509–525 (2002).
[CrossRef]

Maitre, H.

Martin, A. V.

A. V. Martin, F.-R. Chen, W.-K. Hsieh, J.-J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy 106, 914–924(2006).
[CrossRef] [PubMed]

Martincek, I.

I. Martincek, D. Kacik, I. Turek, and P. Peterka, “The determination of the refractive index profile in α-profile optical fibres by intermodal interference investigation,” Optik 115, 86–88 (2004).
[CrossRef]

McMohan, P. J.

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

P. J. McMohan, E. D. Barone-Nugent, B. E. Allman, and K. A. Nugent, “Quantitative phase-amplitude microscopy II: differential interference contrast imaging for biological TEM,” J. Microsc. 206, 204–208 (2002).
[CrossRef]

Nugent, K. A.

E. Ampem-Lassen, S. T. Huntington, N. M. Dragomir, K. A. Nugent, and A. Roberts, “Refractive index profiling of axially symmetric optical fibers: a new technique,” Opt. Express 13, 3277–3282 (2005).
[CrossRef] [PubMed]

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. M. D. Delbridge, and K. A. Nugent, “Quantitative phase-amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62–69 (2004).
[CrossRef] [PubMed]

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

P. J. McMohan, E. D. Barone-Nugent, B. E. Allman, and K. A. Nugent, “Quantitative phase-amplitude microscopy II: differential interference contrast imaging for biological TEM,” J. Microsc. 206, 204–208 (2002).
[CrossRef]

A. Roberts, E. Ampem-Lassen, A. Barty, K. A. Nugent, G. W. Baxter, N. M. Dragomir, and S. T. Huntington, “Refractive-index profiling of optical fibers with axial symmetry by use of quantitative phase microscopy,” Opt. Lett. 27, 2061–2063(2002).
[CrossRef]

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

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

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[CrossRef]

Paganin, D.

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

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[CrossRef]

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

Peterka, P.

I. Martincek, D. Kacik, I. Turek, and P. Peterka, “The determination of the refractive index profile in α-profile optical fibres by intermodal interference investigation,” Optik 115, 86–88 (2004).
[CrossRef]

Ramadan, W. A.

A. A. Hamza, M. A. Mabrouk, W. A. Ramadan, and H. H. Wahba, “Core-index determination of a thick fibre using lens-fibre interference (LFI) technique,” Opt. Lasers Eng. 42, 121–130 (2004).
[CrossRef]

Roberts, A.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).

Schofield, M. A.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[CrossRef] [PubMed]

Shams-Eldin, M. A.

T. Z. N. Sokkar, H. M. El Dessouky, M. A. Shams-Eldin, and M. A. El-Morsy, “Automatic fringe analysis of two-beam interference patterns for measurement of refractive index and birefringence profiles of fibres,” Opt. Lasers Eng. 45, 431–441 (2007).
[CrossRef]

Sokkar, T. Z. N.

T. Z. N. Sokkar, H. M. El Dessouky, M. A. Shams-Eldin, and M. A. El-Morsy, “Automatic fringe analysis of two-beam interference patterns for measurement of refractive index and birefringence profiles of fibres,” Opt. Lasers Eng. 45, 431–441 (2007).
[CrossRef]

A. A. Hamza, T. Z. N. Sokkar, M. A. El-Bakary, and A. M. Ali, “An interferometric method for studying the influence of temperature on the mean refractive indices and cross-sectional area of irregular fibres,” Polym. Test. 22, 83–91 (2003).
[CrossRef]

M. A. El-Morsy, T. Yatagai, A. A. Hamza, M. A. Mabrouk, and T. Z. N. Sokkar, “Automatic refractive index profiling of fibers by phase analysis method using Fourier transform,” Opt. Lasers Eng. 38, 509–525 (2002).
[CrossRef]

Stock, P.

P. Stock, Manufacturing Department, FiberTech GmbH, Nalepastrasse 170–171, 12459 Berlin (personal communication, 2009).

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

Teague, M. R.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).

Turek, I.

I. Martincek, D. Kacik, I. Turek, and P. Peterka, “The determination of the refractive index profile in α-profile optical fibres by intermodal interference investigation,” Optik 115, 86–88 (2004).
[CrossRef]

Volkov, V. V.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[CrossRef] [PubMed]

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in a Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
[CrossRef]

Wahba, H. H.

H. H. Wahba and T. Kreis, “Characterization of graded index optical fiber by digital holographic interferometry,” Appl. Opt. 48, 1573–1582 (2009).
[CrossRef] [PubMed]

A. A. Hamza, M. A. Mabrouk, W. A. Ramadan, and H. H. Wahba, “Core-index determination of a thick fibre using lens-fibre interference (LFI) technique,” Opt. Lasers Eng. 42, 121–130 (2004).
[CrossRef]

Wernicke, G.

Xu, Y.

Yatagai, T.

M. A. El-Morsy, T. Yatagai, A. A. Hamza, M. A. Mabrouk, and T. Z. N. Sokkar, “Automatic refractive index profiling of fibers by phase analysis method using Fourier transform,” Opt. Lasers Eng. 38, 509–525 (2002).
[CrossRef]

Yin, C.

Zheng, Y.

Zhu, Y.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[CrossRef] [PubMed]

Zuegel, J. D.

Appl. Opt. (2)

IEEE Trans. Geosci. Remote Sens. (1)

G. Fornaro, G. Franceschetti, and R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

J. Microsc. (4)

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

P. J. McMohan, E. D. Barone-Nugent, B. E. Allman, and K. A. Nugent, “Quantitative phase-amplitude microscopy II: differential interference contrast imaging for biological TEM,” J. Microsc. 206, 204–208 (2002).
[CrossRef]

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. M. D. Delbridge, and K. A. Nugent, “Quantitative phase-amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62–69 (2004).
[CrossRef] [PubMed]

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

J. Opt. Soc. Am. (1)

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

Opt. Commun. (2)

E. G. Abramochkin and V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in a Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
[CrossRef]

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

Opt. Express (2)

Opt. Lasers Eng. (3)

M. A. El-Morsy, T. Yatagai, A. A. Hamza, M. A. Mabrouk, and T. Z. N. Sokkar, “Automatic refractive index profiling of fibers by phase analysis method using Fourier transform,” Opt. Lasers Eng. 38, 509–525 (2002).
[CrossRef]

A. A. Hamza, M. A. Mabrouk, W. A. Ramadan, and H. H. Wahba, “Core-index determination of a thick fibre using lens-fibre interference (LFI) technique,” Opt. Lasers Eng. 42, 121–130 (2004).
[CrossRef]

T. Z. N. Sokkar, H. M. El Dessouky, M. A. Shams-Eldin, and M. A. El-Morsy, “Automatic fringe analysis of two-beam interference patterns for measurement of refractive index and birefringence profiles of fibres,” Opt. Lasers Eng. 45, 431–441 (2007).
[CrossRef]

Opt. Lett. (2)

Optik (1)

I. Martincek, D. Kacik, I. Turek, and P. Peterka, “The determination of the refractive index profile in α-profile optical fibres by intermodal interference investigation,” Optik 115, 86–88 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

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

Polym. Test. (1)

A. A. Hamza, T. Z. N. Sokkar, M. A. El-Bakary, and A. M. Ali, “An interferometric method for studying the influence of temperature on the mean refractive indices and cross-sectional area of irregular fibres,” Polym. Test. 22, 83–91 (2003).
[CrossRef]

Proc. SPIE (1)

H. Maitre and I. Lyuboshenko, “Robust algorithms for phase unwrapping in SAR interferometry,” Proc. SPIE 3217, 176–187 (1997).
[CrossRef]

Ultramicroscopy (2)

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[CrossRef] [PubMed]

A. V. Martin, F.-R. Chen, W.-K. Hsieh, J.-J. Kai, S. D. Findlay, and L. J. Allen, “Spatial incoherence in phase retrieval based on focus variation,” Ultramicroscopy 106, 914–924(2006).
[CrossRef] [PubMed]

Z. Wiss. Mikrosk. Mikrosk. Tech. (1)

A. Koehler, “Ein neues Beleuchtungsverfahren fuer mikrophotographische Zwecke,” Z. Wiss. Mikrosk. Mikrosk. Tech. 10, 443–440 (1893).

Other (4)

S. Altmeyer and J. Frank, “Optics and information technology,” in Technology Guide Principles—Applications—Trends, H.-J.Bullinger, ed. (Springer, 2009), pp. 98–103.
[CrossRef]

FiberTech GmbH, “Quartz/quartz fibers—all silica (AS), specifications,” http://www.leoni-fiber-optics.com/Quarz-Quarz-Fasern-All-Silica-AS.11918.0.html?&L=1.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).

P. Stock, Manufacturing Department, FiberTech GmbH, Nalepastrasse 170–171, 12459 Berlin (personal communication, 2009).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Schematic diagram of the optical setup. The system is composed of a microscope objective and a tube lens that images a magnified copy of the specimen onto a sensor. The motorized sample stage allows precise defocusing of the sample by moving the stage out of the focal plane. The three images ( I + , I 0 , I ) of the object, out of slightly different planes, are captured one after the other by a CCD sensor.

Fig. 2
Fig. 2

Expected phase disturbance of a plane wave after passing through a fiber sample: The step-index fiber is placed on a microscope slide, together with an index matching fluid and covered by a coverslip while n cladding < n match < n core . Disturbance of the initial phase distribution occurs due to the fact that the fiber induces an optical path difference given by the fiber’s geometry and refractive index.

Fig. 3
Fig. 3

Input data for the phase retrieval algorithm. Captured intensity distributions of the microsphere (top row) and step-index fiber (bottom row). (a), (c), (d), and (f) are defocused images ( I + , I ) of the objects while (b) and (e) are in focus ( I 0 ). Artifacts in the images, such as an interference pattern, arise from dust and dirt particles and internal reflections in the optical system due to the coherent illumination (sodium gas discharge lamp, He–Ne laser).

Fig. 4
Fig. 4

Retrieved phase distribution for the borosilicate microsphere with a diameter of d sphere 15.2 μm considering Dirichlet boundary conditions. (a) The two-dimensional phase distribution. The phase was obtained by capturing one focused and two defocused images ( ± 9 μm ) of the object [see Figs. 3a, 3b, 3c] and using them as input data in the presented phase retrieval algorithm. The colors black and white indicate a maximum phase shift of Δ Φ 0.5 rad . (b) A cross section through (a) (dashed curve), together with the corresponding fit function (solid curve), is shown.

Fig. 5
Fig. 5

Retrieved phase distribution for the optical fiber AS 100/140 IRAN from the company FiberTech GmbH considering Neumann boundary conditions. (a) The two- dimensional phase distribution. The dimensions of the image are 512 × 512 pixels and 262 μm × 262 μm . The phase was obtained by capturing one focused and two defocused images ( ± 10 μm ) of the object [see Figs. 3d, 3e, 3f] and using them as input data. The colors black and white indicate a maximum phase shift of Δ Φ 14.6 rad . (b) A cross section through (a) (dashed curve), together with the corresponding fit function (solid curve), is shown.

Equations (7)

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

Φ ( r ) = R [ Φ ( r ) · G ( r , r ) ] d r ,
Φ ( r ) = k I z ( r ) R [ I z ( r ) z · G ( r , r ) ] d r .
G N x ( x , y , x , y ) = 2 a m = 1 cos ( m π a x ) sin ( m π a x ) × { cosh ( π m a [ b y ] ) cosh ( π m a y ) sinh ( π b a m ) , for     y > y cosh ( π m a [ b y ] ) cosh ( π m a y ) sinh ( π b a m ) , for     y < y ,
G N y ( x , y , x , y ) = 2 b n = 1 cos ( n π b y ) sin ( n π b y ) × { cosh ( π n b [ a x ] ) cosh ( π n b x ) sinh ( π a b n ) , for     x > x cosh ( π n b [ a x ] ) cosh ( π n b x ) sinh ( π a b n ) , for     x < x ,
G D x ( x , y , x , y ) = 2 a m = 1 sin ( m π a x ) cos ( m π a x ) × { sinh ( π m a [ b y ] ) sinh ( π m a y ) sinh ( π b a m ) , for     y > y sinh ( π m a [ b y ] ) sinh ( π m a y ) sinh ( π b a m ) , for     y < y ,
G D y ( x , y , x , y ) = 2 b n = 1 sin ( n π b y ) cos ( n π b y ) × { sinh ( π n b [ a x ] ) sinh ( π n b x ) sinh ( π a b n ) , for     x > x sinh ( π n b [ a x ] ) sinh ( π n b x ) sinh ( π a b n ) , for     x < x ,
I 0 z I + I Δ z ,

Metrics