  We calculate the pair correlation function and the order parameter
correlation function, which probe, respectively, the intra-pair and inter-pair
correlations of a Fermi gas with attractive inter-particle interaction, in
terms of a diagrammatic approach as a function of coupling throughout the
BCS-BEC crossover and of temperature, both in the superfluid and normal phase
across the critical temperature $T_{c}$. Several physical quantities are
obtained from this calculation, including the pair coherence and healing
lengths, the Tan's contact, the crossover temperature $T^{*}$ below which
inter-pair correlations begin to build up in the normal phase, and the
signature for the disappearance of the underlying Fermi surface which tends to
survive in spite of pairing correlations. A connection is also made with recent
experimental data on the temperature dependence of the normal coherence length
as extracted from the proximity effect measured in high-temperature (cuprate)
superconductors.
