Equivalent of the point spread function for partially coherent imaging

Angularly diverse or partially coherent illumination is widely used for optical, x-ray, and electron microscopy. A long-standing challenge in developing new partially coherent approaches is that the nonlinear image formation model does not allow physical intuition into how the imaging and illumination pupils impact contrast and resolution. We report a phase-space model, the phase-space imaging kernel, for partially coherent systems that describes image formation in terms of a convolution and is analogous to the point spread function model for coherent imaging. We simulate phase-space imaging kernels for brightfield and differential interference contrast (DIC) microscopes to explain a seemingly paradoxical experimental result that the DIC image of a point depends on the coherence of the illumination. We discuss interpretation of the spatial and spatial-frequency marginals of the kernel. We expect this intuitive model and simulations to facilitate design of novel computational schemes for phase imaging and optical lithography.