  This work introduces a novel reinterpretation of structured illumination (SI)
microscopy for coherent imaging that allows three-dimensional imaging of
complex refractive index (RI). To do so, we show that coherent SI is
mathematically equivalent to a superposition of angled illuminations. It
follows that raw acquisitions for standard SI-enhanced quantitative-phase
images can be processed into complex electric-field maps describing sample
diffraction under angled illuminations. Standard diffraction tomography (DT)
computation can then be used to reconstruct the sample 3D RI distribution at
sub-diffraction resolutions. We demonstrate this concept by using a
SI-quantitative-phase imaging system to computationally reconstruct 3D RI
distributions of human breast (MCF-7) and colorectal (HT-29) adenocarcinoma
cells. Our experimental setup uses a spatial light modulator to generate
structured patterns at the sample and collects angle-dependent sample
diffraction using a common-path, off-axis interference configuration with no
moving components. Furthermore, this technique holds promise for easy pairing
with SI fluorescence microscopy, and important future extensions may include
multimodal, sub-diffraction resolution, 3D RI and fluorescent visualizations.
