Scattering of Ly{\alpha} Photons through the Reionizing Intergalactic
  Medium: I. Spectral Energy Distribution
  During reionization, a fraction of galactic Ly$\alpha$ emission is scattered
in the intergalactic medium (IGM) and appears as a diffuse light extending
megaparsecs from the source. We investigate how to probe the properties of the
early galaxies and their surrounding IGM using this scattered light. We create
a Monte Carlo algorithm to track individual photons and reproduce several test
cases from previous literature. Then, we run our code on the simulated IGM of
the CoDaII simulation. We find that the scattered light can leave an observable
imprint on the emergent spectrum if collected over $\sim{\rm arcmin}^2$.
Scattering can redden the emission by increasing the path lengths of photons,
but it can also make the photons bluer by upscattering them according to the
peculiar motion of the scatterer. The photons emitted on the far blue side of
the resonance appear more extended in both frequency and space compared to
those emitted near the resonance. This provides a discriminating feature for
the blueward emission, which cannot be constrained from the unscattered light
directly coming from the source. The ionization state of the IGM also affects
the scattered light spectrum. When the source is in a small HII region, the
emission goes through more scatterings in the surrounding HI region regardless
of the initial frequency and ends up more redshifted and spatially extended.
This can result in a weakening of the scattered light toward high-$z$ during
reionization. Our results provide a framework for interpreting the scattered
light to be measured by high-$z$ integral-field-unit surveys.
