  The Zwicky Transient Facility (ZTF) performs a systematic neutrino follow-up
program, searching for optical counterparts to high-energy neutrinos with
dedicated Target-of-Opportunity (ToO) observations. Since first light in March
2018, ZTF has taken prompt observations for 24 high-quality neutrino alerts
from the IceCube Neutrino Observatory, with a median latency of 12.2 hours from
initial neutrino detection. From two of these campaigns, we have already
reported tidal disruption event (TDE) AT 2019dsg and likely TDE AT 2019fdr as
probable counterparts, suggesting that TDEs contribute >7.8% of the
astrophysical neutrino flux. We here present the full results of our program
through to December 2021. No additional candidate neutrino sources were
identified by our program, allowing us to place the first constraints on the
underlying optical luminosity function of astrophysical neutrino sources.
Transients with optical absolutes magnitudes brighter that $-21$ can contribute
no more than 87% of the total, while transients brighter than $-22$ can
contribute no more than 58% of the total, neglecting the effect of extinction
and assuming they follow the star formation rate. These are the first
observational constraints on the neutrino emission of bright populations such
as superluminous supernovae. None of the neutrinos were coincident with bright
optical AGN flares comparable to that observed for TXS 0506+056/IC170922A, with
such optical blazar flares producing no more than 26% of the total neutrino
flux. We highlight the outlook for electromagnetic neutrino follow-up programs,
including the expected potential for the Rubin Observatory.
