  We analyze the ultra high resolution laser angle resolved photo-emission
spectroscopy (ARPES) intensity from the slightly underdoped
Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ in the superconductive (SC) state. The
momentum distribution curves (MDC) were fitted at each energy $\w$ employing
the SC Green's function along several cuts perpendicular to the Fermi surface
with the tilt angle $\theta$ with respect to the nodal cut. The clear
observation of particle-hole mixing was utilized such that the complex
self-energy as a function of $\omega$ is directly obtained from the fitting.
The obtained angle resolved self-energy is then used to deduce the Eliashberg
function $\alpha^2 F^{(+)}(\th,\w)$ in the diagonal channel by inverting the
d-wave Eliashberg equation using the maximum entropy method. Besides a broad
featureless spectrum up to the cutoff energy $\omega_c$, the deduced $\alpha^2
F$ exhibits two peaks around 0.05 eV and 0.015 eV. The former and the broad
feature are already present in the normal state, while the latter emerges only
below $T_c$. Both peaks become enhanced as $T$ is lowered or the angle $\th$
moves away from the nodal direction. The implication of these findings are
discussed.
