  Photonic computing has the potential of harnessing the full degrees of
freedom (DOFs) of the light field, including wavelength, spatial mode, spatial
location, phase quadrature, and polarization, to achieve higher level of
computation parallelization and scalability than digital electronic processors.
While multiplexing using wavelength and other DOFs can be readily integrated on
silicon photonics platforms with compact footprints, conventional mode-division
multiplexed (MDM) photonic designs occupy areas exceeding tens to hundreds of
microns for a few spatial modes, significantly limiting their scalability. Here
we utilize inverse design to demonstrate an ultracompact photonic computing
core that calculates vector dot-products based on MDM coherent mixing within a
nominal footprint of 5 um x 3 um. Our dot-product core integrates the
functionalities of 2 mode multiplexers and 1 multi-mode coherent mixers, all
within the footprint, and could be applied to various computation and computer
vision tasks, with high computing throughput density. We experimentally
demonstrate computing examples on the fabricated core, including complex number
multiplication and motion estimation using optical flow.
