Quantitative phase tomography

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Abstract

We describe the application of a new technique for the simultaneous determination of three-dimensional absorption and refractive index distributions using a combination of quantitative phase-amplitude microscopy and tomographic reconstruction techniques. We briefly review the phase-amplitude microscopy technique and present experimental results in which we have successfully reconstructed the refractive index profile of two different optical fibres.

Introduction

Optical microscopy is a technique with a long history of ongoing development. As microscopy-based research becomes more sophisticated, so the demands on microscopic techniques increase. In particular, the ability of optical microscopy to obtain three-dimensional structural information has become of increased interest [1].

The confocal microscope using fluorescence provides the most reliable three-dimensional information. The properties of confocal microscopy are such that true 3D images are possible, however the problem of obtaining three-dimensional transmission images remains [2]. Moreover, most samples of biological interest have considerable phase structure but limited amplitude information. There is thus a strong motivation for developing three-dimensional phase imaging techniques.

Non-tomographic three-dimensional optical imaging is plagued by the so-called `missing cones' problem [1]. It can be shown that these techniques do not sample the spatial frequencies contained in two cone-shaped regions of frequency space. This is a fundamental limitation of the imaging process and may only be overcome using illumination over a larger range of angles, although non-linear data extrapolation techniques have been used to estimate the missing spatial frequencies. However, it is apparent that true three-dimensional transmission imaging requires a tomographic approach.

Tomographic techniques are well established in the context of X-ray imaging as a means of determining three-dimensional absorption profiles [3], and more recently X-ray phase contrast tomography has been demonstrated by reconstructing the three-dimensional Laplacian of a weakly absorbing sample 4, 5. The 3D Laplacian can be readily interpreted as a phase contrast image in the case of negligible attenuation, however interpretation of the data is more difficult when the sample has a non-negligible absorption profile. Whilst well established in the context of X-ray imaging, tomographic techniques have attracted relatively little attention in the context of optical microscopy [6].

In the context of three-dimensional optical phase microscopy diffraction tomography has been investigated as a means of recovering information about the complex refractive index distribution in an object 7, 8, but existing algorithms generally rely on an iterative approach to determine the phase of the projection data and so suffer from instabilities under certain experimental configurations [7]. Interferometric techniques can be used for phase measurement in an optical microscope, although they are sensitive to alignment, require highly coherent illumination, and are difficult to implement in the transmission geometry required for three-dimensional tomographic reconstruction. Furthermore, interferometric techniques are plagued by problems of phase unwrapping when the phase excursion exceeds 2π. Transmission optical microscopes are common and readily available, thus there is considerable advantage in developing means to measure the phase distribution directly from the transmitted intensity distribution using the partially coherent illumination available on conventional transmission microscopes.

In this paper we demonstrate a method for reconstructing three-dimensional phase data using a deterministic phase retrieval algorithm combined with tomographic reconstruction techniques. The technique we present here is capable of separating the phase and intensity information into separate images. Significantly, this technique does not require the insertion of additional, specialised optics into the microscope and, moreover, recovers the phase directly, obviating the need for the use of phase unwrapping algorithms. Although we provide an experimental demonstration in the optical regime, we point out that the approach is equally applicable to X-ray phase tomography.

Section snippets

Theoretical description

In an earlier paper, we described a technique for determining the phase of a partially coherent wave field in the presence of significant sample absorption from non-interferometric measurements of the field intensity [9]. This approach relies on the observation that, although the two dimensional intensity of a wave-field provides no information about the phase over the observation surface, the three-dimensional intensity distribution uniquely specifies the phase provided there are no

Experimental demonstration

In the work described here, we place a well-characterised sample under the microscope and determine the quantitative phase data by obtaining images in-focus as well as images slightly positively and negatively defocused. The object is then rotated a known amount and the process repeated.

The experimental arrangement consists of a classic tomographic imaging configuration in which the sample is rotated within the imaging system. In this case the imaging system is an optical microscope in which

Conclusions

In this paper we have applied the new technique of quantitative optical phase microscopy to the tomographic reconstruction of three-dimensional refractive index distributions, and shown that the results obtained are in quantitative agreement with the known refractive index distribution. The phase algorithm used in this work is capable of separating the amplitude and phase components of the sample structure, thereby opening up the possibility of doing full phase-amplitude tomography. The

Acknowledgements

The authors acknowledge the support of the Australian Research Council. A.B. wishes to thank Shane Huntington for numerous helpful discussions regarding fibre characterisation, and also acknowledges the support of an Australian Postgraduate Award. We also thank John Arkwright of Siemens Australia for provision of the twin-core fibre images in this paper.

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