1. Introduction

The development of an increasingly wide range of optical fibers with complex index profiles means that the ability to accurately determine the physical properties of the fiber is essential. Quantitative Phase Microscopy (QPM) [1] has been used to obtain transverse phase images of optical fibers and from these the refractive index profiles of axially symmetric optical fibers have been obtained by applying the inverse Abel transform [2,3]. This has been extended to obtaining three dimensional index profiles using tomography where a series of phase images of a specimen is obtained as it is rotated about its axis [4,5]. This method, however, does require a suitable rotation mount and considerable post-processing of data. It is also unsuitable for large diameter and highly asymmetric fibers [6].

Knowledge of the refractive index profile is invaluable since from it fiber properties, such as the mode profile [7], dispersion characteristics, and cutoff wavelength can be calculated. There currently exist a range of methods for determining the refractive index profile of an optical fiber [8, 9]. The most popular of these and the technique that forms the basis of most commercial index profiling instruments is the Refracted Near-Field (RNF) method [10]. Commercial instruments, however, are relatively expensive and the calibration of any instrument based on the RNF technique is also critical if accurate values of the refractive index change δn are required. A range of other methods have also been considered. The method that is closest in spirit to the work described here is axial interferometry [11], although the accuracy of the interferometric method is limited by background phase variations due to multiple coherent reflections in various components of the microscope. For the interferometric method, therefore, a phase compensator is needed to directly measure the optical path to remove the uncertainties that can arise in interpreting the intensity distribution of the interferogram. This method makes measurements of a complete profile very difficult.

In this paper we demonstrate the use of a non-interferometric phase imaging technique [1] to retrieve the refractive index profile from a phase map of a thin transverse slice of a fiber that has been lapped and polished to a few micrometers thickness. Unlike the interferometric measurement that is susceptible to environmental perturbations such as temperature variations and poor stability, the non-interferometric technique has more relaxed experimental conditions and can be performed using a simple brightfield microscope. Furthermore, it is not necessary to ‘unwrap’ the phase of images obtained using QPM when the phase range exceeds 2π and, since QPM can be used with partially coherent light, superior spatial resolution can be obtained.

2. Axial phase retrieval and index profiling

Quantitative Phase Microscopy (QPM) is a microscopic, non-interferometric technique for obtaining quantitative information about phase shifts introduced into a wavefield by a transparent or partially transparent specimen. It involves the acquisition of an in-focus image of the specimen and positively and negatively defocused images which are then analysed using an algorithm based on the solution of the transport of intensity equation [12],

where k is the wave number of the light, I(x, y, z) is the intensity of the optical wavefield at transverse position (x,y) in axial plane z, ∂_z I(x, y, z) is the gradient in the direction of propagation z, and ∇_˔ ϕ(x, y, z) the transverse gradient of the phase. An algorithm based on the transport of intensity equation computes the phase shift ϕ(x, y, z) in the plane of interest z to within an arbitrary constant. QPM can be used with partially coherent light in a conventional bright field microscope. In contrast to other phase imaging techniques it provides simultaneous, but separate, quantitative information about both the phase change and the attenuation of the wavefield by a specimen. The method has been shown to be robust, repeatable and accurate to within approximately ±5 %. The algorithm as used here is written in the IDL programming language.

In the research presented here, the phase shift, ϕ(x, y, z)introduced by a thin slice of a fiber is obtained using QPM. The refractive index difference δn(x,y) across the slice can then be calculated from the phase images assuming the slices are of uniform thickness and phase shifts are introduced only by refractive index variations within the samples. Hence, once the phase has been calculated, it is straightforward to compute the refractive index change using the relationship:

where λ is the wavelength at which the phase is measured and t is the thickness of the slice.

3. Experiment

The polymer coating on the fibers to be studied was removed and the fiber cut into lengths of approximately 1–2 cm. A close-packed bundle of fibers was then bonded using Gatan G1 (601.07260) epoxy with a mixture of resin and hardener in the ratio of 10:1. The bundle of fibers was then placed in a cylindrical metal frame made from stainless steel or bronze. These frames have inner and outer diameters of 1 and 3 mm respectively. It is critical that fibers are oriented correctly to ensure that the polished endface of the fibers are orthogonal to the fiber axis. If this requirement is not closely met, significant artefacts will appear in the resulting phase images. The fiber assembly is then placed into an oven at a cure temperature of 100°C for approximately four hours and then left at room temperature overnight. The fiber assembly was then mounted onto a specimen stub and one end was polished using a model 590 tripod polisher (South Bay Technology) or a model 656/3 precision dimple grinder (Gatan) with 30, 15, 9, 3, and finally 1 μm diamond lapping film until a scratch-free surface was achieved. The surface was then finished with an alumina polishing suspension (Gatan) for a few minutes. The second face was then polished using the same procedure until the required sample thickness was achieved and finished in the same way.

4. Results and discussion

Brightfield images were taken of the resulting slices of the fiber bundle using an Olympus BX-60 microscope with either a 20× 0.7 NA or a 40× 0.85 NA UplanApo microscope objective with a 12-bit Photometrics liquid-cooled CH250 camera containing a Kodak KAF1400 CCD chip. These images were taken with the numerical aperture of the condenser set to 0.2 to optimize the phase resolution. The light was filtered using a bandpass filter with a central wavelength of 521 nm and a passband of 10 nm. A through-focal series of three brightfield images was obtained of each specimen extending from positive to negative defocus. The image obtained with the focal plane corresponding to the centre of the slice along with two defocused images (one positive and the other negative) were used to obtain quantitative phase images using the algorithm based on the transport of intensity equation [10]. The optical thickness of the sample was measured under the microscope system at a wavelength of 521 ± 5 nm. The vertical stage position of the microscope was manually moved until the lower surface of the microscope slide on which the fiber slice is, was in focus. The reading on the stage position was noted. The vertical stage was moved again until the upper surface of the fibre slice was in focus. The results obtained were used to determine the thickness of the slice. Results for two fibers are shown here: (i) a (symmetric) multimode fiber with a core diameter of 62.5μm and (ii) a high-birefringence ‘bow-tie’ fiber (OFTC, Sydney) which is not symmetric about its axis.

 

Fig. 1. (a) Brightfield intensity image of a slice of a bundle 62.5/125 multimode fiber sample. (b) Brightfield intensity image of a single slice (detail) from (a). Image size of (a) is 836μm × 715μm. Image size of (b) is 140μm × 140μm. Thickness of slice is 36 ± 1 μm..

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Results for the multimode fibre are shown in Fig. 2. The phase shift across the core of this fiber was calculated using images defocused by ± 2 μm and the thickness of this specimen was estimated to be 36 ± 1 μm. This defocus distance was selected as a compromise between the small defocus required to optimise spatial resolution and the fact that larger defocus distances reduce the appearance of artefacts associated with image noise. The measured phase shift (and hence the refractive index profile) of the fiber is very nearly parabolic (dashed line) as expected (Fig. 2(a), 2(b)). Note the large phase excursion shown in Fig. 2(b) which required no phase unwrapping despite the fact that the phase range exceeded 2π. The refractive index profile determined using this technique (solid line) is compared with the results of the S14 commercial profiler (dash dot line) in Fig. 2(c). There is excellent qualitative and quantitative agreement between the refractive index profile obtained of the fiber slice and the S14 refractive index profile. The small discrepancy observable in δn may be due to uncertainties in the measurement of the thickness of the fiber slice, small variations in thickness and the inherent uncertainty in the retrieved phase (estimated to be within 10%).

There are some features (ripples) that are detectable on the recovered refractive index profile from 5μm – 20μm. These ripples are also detectable observed in an Atomic Force Microscope (AFM) profile of the same fiber etched in 48% hydrofluoric acid at room temperature for 20 seconds. These correspond to layers deposited during the fabrication process (see Fig. 2d) [13,14]. Note that due to the complex nonlinear relationship between etch rate as measured by the AFM and the refractive index profile, that is exacerbated in this case by the presence of multiple dopants, it is not appropriate to directly compare the AFM surface topography profile with the index profile. This measurement does, however, provide confirmation of the measurement of the fiber’s geometric properties.

In order to demonstrate the flexibility of this technique, results for a high-birefringence (HiBi) ‘bow-tie’ optical fiber are shown in Fig. 3. Phase images were obtained with images defocused at ± 2 μm from the plane of best focus. The thickness of the specimen was estimated to be 26 ± 1 μm. Figure 3(a) shows the magnified central bow-tie region of the fiber, while profiles along the two cross-sectional principal axes of the refractive index profile of the fiber are shown in Fig. 3(b), the refractive index of the fiber has been computed from the phase profile using the equation described above. Note that the subtle variation apparent in the recovered index in the cladding region in Fig. 3(a) appears in the phase images as a consequence of noise in the original intensity images.

The refractive index profile is compared with the profile obtained using an AFM of the cleaved end of this fiber after it has been etched in 48% hydrofluoric acid for 30 seconds at room temperature (Fig. 4(a) and 4(b)). This technique is known to elucidate the dopant profiles of fibers with nanometric spatial resolution [14] but is not necessarily proportional to the refractive index. The core and large depressed index region are clearly visible in both Fig. 4(a) and 4(b), although the data obtained using the AFM, despite possessing a clearly superior spatial resolution does not provide the quantitative phase and refractive index information obtainable from QPM.

 

Fig. 2. Images of a multimode fiber: (a) is the phase image retrieved from brightfield intensity images defocused at ± 2 μm at λ=521 ± 5 nm of Fig.1 (a). Image size of (a) is 100 μm × 100 μm. (b) is the lateral profile across 2(a). The solid line is the phase profile retrieved and a parabolic fit with the assumption that the fibre has a core diameter of 62.5 μm (dashed line). Panel (c) shows the refractive index profile retrieved from the phase image (solid line) compared with a parabolic fit of diameter 62.5 μm (dashed line) and S14 commercial refractive index profile (dash dot line). Panel (d) shows the AFM profile which shows the etch depth and the structural features of the same fiber.

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It is clear from Fig. 2 and 3 that the phase images obtained in this way provide direct quantitative information about the two-dimensional refractive index profile of the fiber. The method can be performed with no prior calibration on fibers with arbitrary index profiles. Although the sample preparation can be laborious, economies of scale can be achieved by preparing a bundle of several different fibers at the same time

 

Fig. 3. (a) Refractive index of ‘Bowtie’ fiber obtained with a 40× 0.85 UplanApo objective at λ= 521 ± 5 nm. (b) shows profiles through the refractive index image obtained in (a) for a thickness of 26 ± 1 μm. Profiles are taken along the long (dashed line) and short (solid line) axes of the core. Size of (a) is 60.7 μm × 60.7 μm.

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Fig. 4. (a) AFM image of bow-tie fiber etched in 48% HF solution at room temperature for 30 seconds. The size of the image is 55 μm × 55 μm. 4(b) is the profile across both the short (solid line) and the long axes (dashed line) of (a)..

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5. Conclusion

In this paper quantitative phase images and refractive index profiles of thin polished transverse sections of optical fiber have been presented. It can be seen that the images are proportional to the expected refractive index profile. This technique provides an extremely useful method for obtaining qualitative and quantitative information about the refractive index profiles of symmetric and non-symmetric optical fibers. This provides an important adjunct to existing methods for retrieving the refractive index profile.