Two-dimensional superconducting fluctuations associated with charge density wave stripes in La$_{1.87}$Sr$_{0.13}$Cu$_{0.99}$Fe$_{0.01}$O$_4$

The presence of a small concentration of in-plane Fe dopants in La$_{1.87}$Sr$_{0.13}$Cu$_{0.99}$Fe$_{0.01}$O$_4$ is known to enhance stripe-like spin and charge density wave (SDW and CDW) order, and suppress the superconducting $T_c$. Here, we show that it also induces highly two-dimensional (2D) superconducting correlations that have been argued to be signatures of a new form of superconducting order, so-called pair-density-wave (PDW) order. In addition, using the resonant soft x-ray scattering, we find that the 2D superconducting fluctuation is strongly associated with the CDW stripe. In particular, the PDW signature first appears when the correlation length of the CDW stripe grows over eight times the lattice unit ($sim$ 8$a$). These results provide critical conditions for the formation of PDW order.