  Electron-boson spectral density functions (EBSDFs) can be obtained from
measured spectra using various spectroscopic techniques, including optical
spectroscopy. EBSDFs, known as glue functions, have a magnetic origin. Here, we
investigated EBSDFs obtained from the measured optical spectra of hole-doped
cuprates with wide doping levels, from underdoped to overdoped cuprates. The
average frequency of an EBSDF provides the timescale for the spin fluctuations
to form Cooper pairs. This timescale is directly associated with retarded
interactions between electrons. Using this timescale and Fermi velocity, a
reasonable superconducting coherence length, which reflects the size of the
Cooper pair, can be extracted. The obtained coherence lengths were consistent
with those measured via other experimental techniques. Therefore, the formation
of Cooper pairs in cuprates can be explained by spin fluctuations, the
timescales of which appear in EBSDFs. Consequently, EBSDFs provide crucial
information on the timescale of the microscopic mechanism of Cooper pair
formation.
