Analysis of the inverse problem for determining nematic liquid crystal director profiles from optical measurements using singular value decomposition

We use the problem of determining nematic liquid crystal director profiles from optical measurements as an example to illustrate that what is often treated purely as a data fitting problem is really an inverse problem and that useful insights can be obtained by treating it in this way. Specifically we illustrate the analysis of the sufficiency of data and the sensitivity of a solution to measurement errors. We assume a stratified medium where the Berreman method can be used for the optical forward problem and we consider the inverse problem to be the determination of an anisotropic dielectric permittivity tensor from optical data. A numerical singular value decomposition (SVD) analysis reveals that although this inverse problem is severely ill-conditioned it is possible to determine depth-dependent information provided the medium is sufficiently birefringent and that, as one might expect, a larger range of incident angles gives greater information. Analytical solutions of the Berreman equations for general perturbations of an orthorhombic crystal confirm the uniqueness of solution for the linearized problem and give further insights into the severely ill-posed nature of the inverse problem.