Studying fundamental limit of optical fiber links to $10^{-21}$ level

We present an hybrid fiber link combining effective optical frequency transfer and evaluation of performances with a self-synchronized two-way comparison. It enables us to detect the round-trip fiber noise and each of the forward and backward one-way fiber noises simultaneously. The various signals acquired with this setup allow us to study quantitatively several properties of optical fiber links. We check the reciprocity of the accumulated noise forth and back over a bi-directional fiber to the level of $3.1(\pm 3.9)\times 10^{-20}$ based on a 160000s continuous data. We also analyze the noise correlation between two adjacent fibers and show the first experimental evidence of interferometric noise at very low Fourier frequency. We estimate redundantly and consistently the stability and accuracy of the transferred optical frequency over 43~km at $4\times 10^{-21}$ level after 16 days of integration and demonstrate that frequency comparison with instability as low as $8\times 10^{-18}$ would be achievable with uni-directional fibers in urban area.