Optical coherence tomography (OCT) is a non-invasive imaging technique based on the use of light sources exhibiting a low degree of coherence. Low coherence interferometric microscopes have been successful in producing internal images of thin pieces of biological tissue; typically samples of the order of 1 milimeter in depth have been imaged, with a resolution of the order of 10 to 20 microns in some portions of the sample. In this thesis, I deal with the imaging problem of determining the internal structure of a body from backscattered laser light and low-coherence interferometry. In detail, I formulate and solve an inverse problem which, using the interference fringes that result as the back-scattering of low-coherence light is made to interfere with a reference beam, produces maps detailing the values of the refractive index within the imaged sample. Unlike previous approaches to this imaging problem, the solver I introduce does not require processing at data collection time, and it can therefore produce solutions for inverse problems of multi-layered structures containing thousands of layers from back-scattering interference fringes only. We expect that the approach presented in this work, which accounts fully for the statistical nature of the coherence phenomenon, should prove of interest in the fields of medicine, biology and materials science.
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