  Light field microscopy (LFM) uses a microlens array (MLA) near the sensor
plane of a microscope to achieve single-shot 3D imaging of a sample without any
moving parts. Unfortunately, the 3D capability of LFM comes with a significant
loss of lateral resolution at the focal plane. Placing the MLA near the pupil
plane of the microscope, instead of the image plane, can mitigate the artifacts
and provide an efficient forward model, at the expense of field-of-view (FOV).
Here, we demonstrate improved resolution across a large volume with Fourier
DiffuserScope, which uses a diffuser in the pupil plane to encode 3D
information, then computationally reconstructs the volume by solving a
sparsity-constrained inverse problem. Our diffuser consists of randomly placed
microlenses with varying focal lengths; the random positions provide a larger
FOV compared to a conventional MLA, and the diverse focal lengths improve the
axial depth range. To predict system performance based on diffuser parameters,
we for the first time establish a theoretical framework and design guidelines,
which are verified by numerical simulations, then build an experimental system
that achieves $< 3$ um lateral and $4$ um axial resolution over a $1000 \times
1000 \times 280$ um$^3$ volume. Our diffuser design outperforms the MLA used in
LFM, providing more uniform resolution over a larger volume, both laterally and
axially.
