  Many important microscopy samples, such as liquid crystals, biological
tissue, or starches, are birefringent in nature. They scatter light differently
depending on the light polarization and molecular orientations. The complete
characterization of a birefringent sample is a challenging task because its 3 x
3 dielectric tensor must be reconstructed at every three-dimensional position.
Moreover, obtaining a birefringent tomogram is more arduous for thick samples,
where multiple light scattering should also be considered. In this study, we
developed a new dielectric tensor tomography algorithm that enables full
characterization of highly scattering birefringent samples by solving the
vectoral inverse scattering problem considering multiple light scattering. We
proposed a discrete image-processing theory to compute the error
backpropagation of vectorially diffracting light. Finally, our theory was
experimentally demonstrated using both synthetic and biologically birefringent
samples.
