Optical diffraction tomography of second-order nonlinear structures in weak scattering media: theoretical analysis and experimental consideration

Computed tomography (CT) allows for high lateral and axial resolution imaging of the endogenous structure of matter thanks to its large spatial frequency support and has been realized in X-ray and linear optical domain known as optical diffraction tomography (ODT). Here, we present the theoretical basis and experimental considerations for ODT of second-order nonlinear structures in weak scattering media. We have derived the relation between second harmonic wave and the anisotropic nonlinear tensor in spatial frequency domain under first-order Born approximation. Our results show that, under a plane wave illumination, the two dimensional (2D) spatial spectra of generated second harmonic complex field relates to the inverse lattice of nonlinear structure on Ewald sphere shells. The centers of the Ewald spheres are determined by 2 times wavevector of the incident fundamental wave and the radii are determined by the modulus of the second harmonic wavevector. More importantly, it shows that the 2D spatial spectra is a superposition of the Ewald spheres of different components of the anisotropic nonlinear tensor. We propose to solve the inverse problem by controlling the polarizations of the fundamental and second harmonic signal. We tested the feasibility of the proposed method using a numerical phantom and make some discussions on practical implementations, including angular scanning schemes, polarization detection and illumination profile for optimizing reconstruction region. Possessing high resolution, wide-field imaging and polarization-sensitive property, we believe that the proposed scheme would have important applications in nonlinear microscopy.