On one dimensional inverse problems arising from polarimetric measurements of nematic liquid crystals

We revisit the problem of determining dielectric parameters in layered nematic liquid crystals from polarimetric measurements originally introduced by Lionheart & Newton. After a detailed analysis of the model, of the scales involved, and of natural obstacles to the reconstruction of more than one dielectric parameters, we produce two simple one-dimensional inverse problems which can be studied without any expertise in liquid crystals. We then confirm that very little can be recovered about the internal configuration of smooth dielectric parameters from these measurements, and give a uniqueness result for one of the two problem, when the unknown parameter satisfies a monotonicity property. In that case, the available data can be expressed in terms of Laplace and Hankel transforms.