## 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

The refractive index profile of an optical fiber is of fundamental interest since from it can be determined critical fiber properties such as mode field profiles, conditions for single-mode propagation and dispersion characteristics. Various methods have been developed to determine the refractive index profiles of optical fibers [1–7] and the industry standard is the refracted near-field (RNF) method [7]. Despite the fact that RNF is widely used, it is destructive and requires precise calibration. Recently, we have demonstrated that Quantitative Phase Microscopy (QPM) [8] can be used to determine the refractive index profiles of axially symmetric optical fibers [9]. This method is robust, straightforward, nondestructive and measurements can, in principle, be performed at the wavelength at which the fiber is designed to be used. Here we demonstrate a simplified variant on QPM that can be used to determine the refractive index profile of an axially symmetric fiber. Like QPM this method possesses a high spatial resolution and sensitivity to index changes that are comparable or superior to those obtainable using RNF.

Figure 1 shows schematic end-on and transverse diagrams of an axially symmetric fiber of radius *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, $\frac{\partial \varphi (x,y)}{\partial x}$ to which the inverse Abel transform is applied to obtain Δ*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

As with QPM, our method for determining the phase gradient is based on the transport of intensity equation [10] describing a paraxial optical field travelling in the *z*-direction:

where *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; ${\nabla}_{\perp}=\frac{\partial}{\partial x}\hat{\mathbf{x}}+\frac{\partial}{\partial y}\hat{\mathbf{y}}$. In the absence of phase singularities, Eq. (1) can be written in terms of Fourier transforms since derivative and the inverse Laplacian operators become multiplicative operators when acting on the Fourier representation of a function [10] and the phase gradient ∇_{⊥}
*ϕ*(*x*,*y*)can therefore be written:

where **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 ${k}_{r}^{2}$
=${k}_{x}^{2}$
+${k}_{y}^{2}$
. Once $\frac{\partial I(x,y)}{\partial z}$ has been estimated by obtaining defocused images, Eq. (2) forms the basis of an algorithm based on fast Fourier transforms to determine the phase gradients introduced into the wavefield by the specimen which, in the case of a cylindrically symmetric object, is sufficient to reconstruct the index profile of that specimen. Note that four Fourier transforms and inverse transforms are required to determine the phase gradient compared with the six to determine the phase using QPM. Furthermore, if the *y*-axis is taken to be along the optical axis of the fiber, then the transverse, *x*-component of the gradient, $\frac{\partial \varphi (x,y)}{\partial x}$, can be obtained with only two fast Fourier transforms.

Assuming the fiber is symmetric about its axis, once the transverse component of the phase gradient has been computed the inverse Abel transform:

can be used to recover the index profile.

Various methods have been demonstrated for computing the inverse Abel transform [11–17]. In this paper we use a variation on a Fourier based method [12]. The phase can be written as a Fourier cosine series:

where the *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:

Hence, by decomposing the transverse phase gradient $\frac{\partial \varphi (x,y)}{\partial x}$ as a Fourier sine series, it is possible to determine the coefficients *a*_{m}
(*y*) for *m* ≥1 and the method of [12] used to determine the refractive index profile. Note that *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

A short length of the fiber to be examined, stripped of its plastic coating, was placed on a microscope slide with two sections of fiber of the same diameter placed on either side of the fiber under study as spacers. A few drops of index matching fluid, prepared by mixing dimethyl sulphoxide (DMSO) (~87%) and distilled water, were placed on the fiber before a cover slip was placed across the fibers. The two fibers investigated here were GF1 a photosensitive fiber single-moded at 1300/1550 nm (Optical Fibre Technology Centre, Sydney) and Corning 62.5/125 multimode fiber.

Bright-field images were obtained of the fibers with a BX60 Olympus microscope using a 40×0.85NA (for the single-mode fiber) or 20×0.7NA UplanApo (for the multi-mode fiber) objective. The condenser aperture was set to 0.2 in order to increase the phase sensitivity by maximising the spatial coherence of the incident light. An in-focus image and images at ±2µm defocus were obtained with a 12-bit CCD camera equipped with a 1317×1035 pixel Kodak KAF-1400 CCD chip. The incident light was filtered through a bandpass filter with a central wavelength at 521 nm and a passband of 10nm. A quantitative phase gradient image of the optical fiber was calculated using an algorithm based on the transport of intensity equation described above. The inverse Abel transform procedure described above was then applied to the retrieved phase gradient images.

## 4. Results and discussions

Figure 2(a) shows the transverse phase gradient image of single-mode fiber and the reconstructed refractive index is shown in Fig. 2(b). Figures 3(a) and 3(b) show the transverse phase gradients and the recovered index map of the Corning 62.5/125 multimode fiber.

A comparison of the refractive index profile of the single-mode fiber obtained using the technique described here (solid curve) and that obtained using a commercial profiling instrument (Photon Kinetics, S14) based on the RFN technique (dashed curve) is shown in Fig. 4(a). It can be seen that there is very good agreement between the two profiles, but the superior spatial resolution obtainable using the quantitative phase gradient imaging approach is apparent. In particular, the fabrication artifact seen near the peak of the index profile near a radius of 3µm has not been resolved in the profile obtained using the commercial profiler. The deposition layers in the depressed inner cladding are also more clearly resolved in the phase gradient technique than that in the commercial profiling technique. The most likely source of the discrepancy between the index determined using the phase gradient technique and the RNF method is due to the inherent complexities associated with the RNF method.

The refractive index profile obtained for Corning 62.5/125 multimode fiber is shown as a solid line in Fig. 4(b). This is compared with the expected refractive index profile with the assumption that the fiber is parabolic (dashed curve), using approximate index changes information provided by the manufacturer. It can be seen that the refractive index obtained is in favorable agreement with the manufacturer’ specifications.

Note that the results obtained here with the phase gradient method are identical to those obtained with the QPM technique with the same set of bright-field images and the same inverse Abel technique, hence, for the purposes of index profiling axially symmetric fibers there are no benefits to obtaining complete phase information.

It should also be noted that, as is the case with QPM, the ultimate spatial resolution and phase (and, hence, refractive index) sensitivity obtainable with the phase gradient technique are influenced by the choice of defocus distance. As the defocus distance increases, the sensitivity to index variations also increases, but this is at the expense of poorer spatial resolution. For small defocus distances, it has been shown that the spatial resolution is diffraction limited [19] as is the case with the RNF method. The ripples in index in the inner cladding of the fiber shown in Fig. 4(a) have an amplitude of approximately 0.0001 and, hence, with a defocus of ± 2µm the index sensitivity is of the order of better than 10^{-4}. This value is comparable with that quoted in specifications provided by manufacturers of index profiling instruments based on the RNF method.

## 5. Conclusion

We have successfully demonstrated that a non-interferometric technique for the quantitative determination of the transverse phase gradients introduced into an optical field by an optical fiber can be used to determine the refractive index profile of axially symmetric optical fibers. The method is relatively non-destructive, robust, sensitive and accurate with a spatial resolution rivalling that of commercial profiling instruments. Note that this technique requires no calibration and presents a straightforward method for determining the index profile of a fiber at different points along its axis.

## Acknowledgments

The authors wish to acknowledge OFTC especially John Canning and Tom Ryan for providing GFI fiber and its index profile using the S14 profiler. The financial support of the Australian Research Council is acknowledged. Eric Ampem-Lassen acknowledges the receipt of an Australian International Postgraduate Research Scholarship and a Melbourne International Research Scholarship.

## 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. **13**, 695–699 (2002). [CrossRef]

**2. **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]

**3. **D. Marcuse, “Refractive index determination by the focusing method,” Appl. Opt. **18**, 9–13 (1979). [CrossRef] [PubMed]

**4. **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]

**5. **E. Brinkmeyer, “Refractive index profile determination from the diffraction pattern,” App. Opt. **16**, 2802–2803 (1977). [CrossRef]

**6. **M. Tateda, “Single mode fiber refractive index profile measurement by reflection method,” Appl. Opt. **17**, 475–478 (1978). [CrossRef] [PubMed]

**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]

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

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

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

**13. **C. J. Cramer and R.C. Birkebak, “Application of the Abel integral equation to spectrographic data,” Appl. Opt. **5**, 1057–1064 (1966). [CrossRef]

**14. **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]

**15. **H. Brunner, “The numerical solution of a class of Abel integral equations by piecewise polynomials,” J. Comp. Phys. , **12**, 412–416 (1973) [CrossRef]

**16. **C. Fleurier and J. Chapelle, “Inversion of Abel’s integral equation-application to plasma spectroscopy,” Comp. Phys. Commun. **7**, 200–206 (1974). [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]

**18. **G. Makosch and B. Solf
, “Surface profiling by electro-optical phase measurement”, in *High Resolution Soft X-Ray Optics*,
E. Spiller, ed., Proc. SPIE316, 43–53 (1981)

**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]