  In the past decade superconducting nanowire single photon detectors (SNSPDs)
have gradually become an indispensable part of any demanding quantum optics
experiment. Until now, most SNSPDs are coupled to single-mode fibers. SNSPDs
coupled to multimode fibers have shown promising efficiencies but are yet to
achieve high time resolution. For a number of applications ranging from quantum
nano-photonics to bio-optics, high efficiency and high time-resolution are
desired at the same time. In this paper, we demonstrate the role of
polarization on the efficiency of multi-mode fiber coupled detectors, and show
how it can be addressed. We fabricated high performance 20, 25 and 50{\mu}m
diameter detectors targeted for visible, near infrared, and telecom
wavelengths. A custom-built setup was used to simulate realistic experiments
with randomized modes in the fiber. We simultaneously achieved system
efficiency >80% and time resolution <20 ps and made large detectors that offer
outstanding performances.
