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Optical retrieval of the structure of transparent objects at the nano-scale requires adapted methods capable of probing their interaction with suitable light. Scattering events, for instance, depend on the content and the arrangement of the medium encountered by the light along its propagation path. Physical models of scattering were used in the past to understand image formation and phase retrieval in transparent objects. Here, we considered one of them, which is based on the acquisition of defocused images obtained with partially coherent illumination, and explored its phase retrieval capability over a wide range of illumination angles. We used a basic transmission bright-field microscope with a bandpass filter and an adjustable illumination aperture. The bandpass filter was used to select a specific wavelength range to avoid light dispersion in the sample and maximize illumination capabilities. The adjustable illumination aperture was used to probe and assess the calibration over a wide range of illumination angles, which give access to different parts of the spatial frequency spectrum of the sample. We subsequently employed a computational algorithm to retrieve the local 3-dimensional phase-shift induced on the light field by the scattering through the sample. We imaged several types of samples to explore the calibration and the results in different experimental configurations. We employed: (1) commercial dielectric nanospheres to assess the phase calibration when measured along the optical axis, (2) custom-made nanosteps micropatterned in a glass substrate to assess the phase calibration when measured along the transversal axis. We first verified the model prediction in the spatial frequency domain for the scattering induced by our samples, and subsequently obtained a consistent and linear phase-calibration for illumination numerical apertures ranging from 0.1 to 0.5. This enabled us to calibrate a very simple optical system in a wide range of illumination angles to obtain quantitative 3D metrology of transparent samples at the nanoscale.
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