## Refractive index profiling of axially symmetric optical fibers: a new technique

Optics Express, Vol. 13, Issue 9, pp. 3277-3282 (2005)

http://dx.doi.org/10.1364/OPEX.13.003277

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### Abstract

We present a new technique for determining the refractive index profiles of axially symmetric optical fibers based on imaging phase gradients introduced into a transmitted optical field by a fiber sample. An image of the phase gradients within the field is obtained using a new non-interferometric technique based on bright field microscopy. This provides sufficient information to reconstruct the refractive index profile using the inverse Abel transform. The technique is robust, rapid and possesses high spatial resolution and we demonstrate its application to the reconstruction of the refractive index profiles of a single-mode and a multimode optical fiber.

© 2005 Optical Society of America

## 1. Introduction

1. Y. Park, N. H. Seong, Y. Youk, and D.Y Kim, “Simple scanning fibre-optic confocal microscopy for the refractive index profile measurement of an optical fibre,” Meas. Sci Technol. **13**, 695–699 (2002). [CrossRef]

7. K. I. White, “Practical application of refracted near-field technique for the measurement of optical fibre refractive index profile,” Opt. and Quant. Electron. **11**, 185–196 (1979). [CrossRef]

7. K. I. White, “Practical application of refracted near-field technique for the measurement of optical fibre refractive index profile,” Opt. and Quant. Electron. **11**, 185–196 (1979). [CrossRef]

8. A. Barty, K.A Nugent, A Roberts, and D. Paganin, “Quantitative Phase Microscopy,” Opt. Lett. **23**, 817–819 (1998) [CrossRef]

9. 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]

*R*. We take the y-coordinate to be along the axis of the fiber and the

*x*to be the transverse coordinate. Since the technique involves obtaining transverse images of the fiber, the

*z*-direction is taken to be along the optical axis of the imaging system, i.e. perpendicular to the axis of the fiber. We wish to determine the refractive index relative to the cladding index, Δ

*n*(

*r*,

*y*), where

*r*is the distance from the axis of the fiber. Rather than compute the phase shift, φ(

*x*,

*y*), introduced into the optical field by the fiber, as is the case with QPM, we show that it is sufficient to obtain only the

*transverse gradient*of the phase,

*n*(

*r*,

*y*). This quantity can be easily determined by taking in-focus and defocused bright-field images with a microscope and applying an algorithm based on two Fourier transforms. This leads to a significant computational saving compared to QPM.

## 2. Phase gradient imaging

10. M. R. Teague, “Irradiance movements: their propagation and use for unique retrieval of phase,” J. Opt. Soc. Am. **72**, 1199–1209 (1982). [CrossRef]

*z*-direction:

*k*=2

*π*/

*λ*is the wavenumber of the optical field,

*λ*is the wavelength,

*I*(

*x*,

*y*,

*z*) the intensity of the optical field measured in a transverse plane

*z*and Δ

_{⊥}is the transverse gradient operator;

10. M. R. Teague, “Irradiance movements: their propagation and use for unique retrieval of phase,” J. Opt. Soc. Am. **72**, 1199–1209 (1982). [CrossRef]

_{⊥}

*ϕ*(

*x*,

*y*)can therefore be written:

**F**and

**F**

^{-1}denote Fourier transformation and inverse transformation respectively,

*k*

_{x}and

*k*

_{y}are the variables conjugate to

*x*and

*y*, and

*y*-axis is taken to be along the optical axis of the fiber, then the transverse,

*x*-component of the gradient,

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

17. L. S. Fan and W. Squire, “Inversion of Abel’s integral equation by a direct method,” Comp. Phys. Commun. **10**, 98–103 (1975). [CrossRef]

12. M. Kalal and K. A. Nugent, “Abel inversion using fast Fourier transforms,” Appl. Opt. **27**, 1956–1959 (1988). [CrossRef] [PubMed]

*a*

_{m}(y) are Fourier coefficients, and the inverse Abel transform is applied to each Fourier component. Using this same notation, the transverse phase gradient is given by:

*a*

_{m}(

*y*) for

*m*≥1 and the method of [12

12. M. Kalal and K. A. Nugent, “Abel inversion using fast Fourier transforms,” Appl. Opt. **27**, 1956–1959 (1988). [CrossRef] [PubMed]

*a*

_{0}(

*y*) cannot be determined and corresponds to an irrelevant constant phase offset. Hence, once the transverse phase gradient has been determined using an algorithm derived from Eq. (2), it is possible to reconstruct the index profile at any point along the fiber using the technique of reference 11 or other inverse Abel transform algorithm.

## 3. Experiment

## 4. Results and discussions

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

^{-4}. This value is comparable with that quoted in specifications provided by manufacturers of index profiling instruments based on the RNF method.

## 5. Conclusion

## Acknowledgments

## References and links

1. | Y. Park, N. H. Seong, Y. Youk, and D.Y Kim, “Simple scanning fibre-optic confocal microscopy for the refractive index profile measurement of an optical fibre,” Meas. Sci Technol. |

2. | T. Okoshi and K. Hotate, “Refractive index profile of an optical fibre: its measurements by scattering pattern method,” Appl. Opt. |

3. | D. Marcuse, “Refractive index determination by the focusing method,” Appl. Opt. |

4. | D. J. Butler, K. A. Nugent, and A. Roberts, “Characterisation of optical fibres using near-field scanning optical microscopy,” J. Appl. Phys. |

5. | E. Brinkmeyer, “Refractive index profile determination from the diffraction pattern,” App. Opt. |

6. | M. Tateda, “Single mode fiber refractive index profile measurement by reflection method,” Appl. Opt. |

7. | K. I. White, “Practical application of refracted near-field technique for the measurement of optical fibre refractive index profile,” Opt. and Quant. Electron. |

8. | A. Barty, K.A Nugent, A Roberts, and D. Paganin, “Quantitative Phase Microscopy,” Opt. Lett. |

9. | 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. |

10. | M. R. Teague, “Irradiance movements: their propagation and use for unique retrieval of phase,” J. Opt. Soc. Am. |

11. | D. Paganin and K. A. Nugent, “Non-interferometric phase imaging with partially coherent light,” Phys. Rev. Lett. |

12. | M. Kalal and K. A. Nugent, “Abel inversion using fast Fourier transforms,” Appl. Opt. |

13. | C. J. Cramer and R.C. Birkebak, “Application of the Abel integral equation to spectrographic data,” Appl. Opt. |

14. | G. N. Minerbo and M. E. Levy, “Inversion of Abel integral equation by means of orthogonal polynomials,” SIAM J. Numer. Anal. , |

15. | H. Brunner, “The numerical solution of a class of Abel integral equations by piecewise polynomials,” J. Comp. Phys. , |

16. | C. Fleurier and J. Chapelle, “Inversion of Abel’s integral equation-application to plasma spectroscopy,” Comp. Phys. Commun. |

17. | L. S. Fan and W. Squire, “Inversion of Abel’s integral equation by a direct method,” Comp. Phys. Commun. |

18. | G. Makosch and B. Solf
, “Surface profiling by electro-optical phase measurement”, in |

19. | E.D. Barone-Nugent, A. Barty, and K.A. Nugent, “Quantitative phase-amplitude micrscopy I: optical microscopy,” J. Microsc. |

**OCIS Codes**

(060.2270) Fiber optics and optical communications : Fiber characterization

(060.2300) Fiber optics and optical communications : Fiber measurements

(110.0180) Imaging systems : Microscopy

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 15, 2005

Revised Manuscript: April 14, 2005

Published: May 2, 2005

**Citation**

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)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-9-3277

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### References

- Y. Park, N. H. Seong, Y. Youk and D.Y Kim, �??Simple scanning fibre-optic confocal microscopy for the refractive index profile measurement of an optical fibre,�?? Meas. Sci Technol. 13, 695-699 (2002). [CrossRef]
- T. Okoshi, and K. Hotate, �??Refractive index profile of an optical fibre: its measurements by scattering pattern method,�?? Appl. Opt. 15, 2756-2764 (1976). [CrossRef] [PubMed]
- D. Marcuse, �??Refractive index determination by the focusing method,�?? Appl. Opt. 18, 9-13 (1979). [CrossRef] [PubMed]
- D. J. Butler, K. A. Nugent and A. Roberts, �??Characterisation of optical fibres using near-field scanning optical microscopy,�?? J. Appl. Phys. 75, 2753-2756 (1994). [CrossRef]
- E. Brinkmeyer, �??Refractive index profile determination from the diffraction pattern,�?? App. Opt. 16, 2802- 2803 (1977). [CrossRef]
- M. Tateda, �??Single mode fiber refractive index profile measurement by reflection method,�?? Appl. Opt. 17, 475-478 (1978). [CrossRef] [PubMed]
- K. I. White, �??Practical application of refracted near-field technique for the measurement of optical fibre refractive index profile,�?? Opt. and Quant. Electron. 11, 185-196 (1979). [CrossRef]
- A.Barty, K.A Nugent, A Roberts and D.Paganin, �??Quantitative Phase Microscopy,�?? Opt. Lett. 23, 817-819 (1998) [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]
- M. R. Teague, �??Irradiance movements: their propagation and use for unique retrieval of phase,�?? J. Opt. Soc. Am. 72, 1199-1209 (1982). [CrossRef]
- D. Paganin and K. A. Nugent, �??Non-interferometric phase imaging with partially coherent light,�?? Phys. Rev. Lett. 80, 2586-2589 (1998). [CrossRef]
- M. Kalal and K. A. Nugent, �??Abel inversion using fast Fourier transforms,�?? Appl. Opt. 27, 1956-1959 (1988). [CrossRef] [PubMed]
- C. J. Cramer and R.C. Birkebak, �??Application of the Abel integral equation to spectrographic data,�?? Appl. Opt. 5, 1057-1064 (1966). [CrossRef]
- G. N. Minerbo and M. E. Levy, �??Inversion of Abel integral equation by means of orthogonal polynomials,�?? SIAM J. Numer. Anal.,6, 598-616 (1969). [CrossRef]
- H. Brunner, �??The numerical solution of a class of Abel integral equations by piecewise polynomials,�?? J. Comp. Phys., 12, 412-416 (1973) [CrossRef]
- C. Fleurier and J. Chapelle, �??Inversion of Abel�??s integral equation-application to plasma spectroscopy,�?? Comp. Phys. Commun. 7, 200-206 (1974). [CrossRef]
- L. S. Fan and W. Squire, �??Inversion of Abel�??s integral equation by a direct method,�?? Comp. Phys. Commun. 10, 98-103 (1975). [CrossRef]
- G. Makosch and B. Solf, �??Surface profiling by electro-optical phase measurement�??, in High Resolution Soft X-Ray Optics, E. Spiller, ed., Proc. SPIE 316, 43-53 (1981)
- E.D. Barone-Nugent, A. Barty and K.A. Nugent, �??Quantitative phase-amplitude micrscopy I: optical microscopy,�?? J. Microsc. 206, 194-203 (2002). [CrossRef] [PubMed]

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