  Imaging 3D nano-structures at very high resolution is crucial in a variety of
scientific fields. However, due to fundamental limitations of light propagation
we can only measure the object indirectly via 2D intensity measurements of the
3D specimen through highly nonlinear projection mappings where a variety of
information (including phase) is lost. Reconstruction therefore involves
inverting highly non-linear and seemingly non-invertible mappings. In this
paper, we introduce a novel technique where the 3D object is directly
reconstructed from an accurate non-linear propagation model. Furthermore, we
characterize the ambiguities of this model and leverage a priori knowledge to
mitigate their effect and also significantly reduce the required number of
measurements and hence the acquisition time. We demonstrate the performance of
our algorithm via numerical experiments aimed at nano-scale reconstruction of
3D integrated circuits. Moreover, we provide rigorous theoretical guarantees
for convergence to stationarity.
