  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.
