  Optical diffraction tomography (ODT) has emerged as an important label-free
tool in biomedicine to measure the three-dimensional (3D) structure of a
biological sample. In this paper, we describe ODT using second-harmonic
generation (SHG) which is a coherent nonlinear optical process with a strict
symmetry selectivity and has several advantages over traditional fluorescence
methods. We report the tomographic retrieval of the 3D second-order nonlinear
optical susceptibility using two-dimensional holographic measurements of the
SHG fields at different illumination angles and polarization states. The method
is a generalization of the conventional linear ODT to the nonlinear scenario.
We demonstrate the method with a numerically simulated nanoparticle
distribution and an experiment with muscle tissue fibers. Our results show that
SHG ODT does not only provide an effective contrast mechanism for label-free
imaging but also due to the symmetry requirement enables the visualization of
properties that are not otherwise accessible.
