Extinction Correction Significantly Influences the Estimate of
  Ly$\alpha$ Escape Fraction
  The Ly$\alpha$ escape fraction is a key measure to constrain the neutral
state of the intergalactic medium and then to understand how the universe was
fully reionized. We combine deep narrowband imaging data from the custom-made
filter NB393 and the $H_{2}S$1 filter centered at 2.14 $\mu$m to examine the
Ly$\alpha$ emitters and H$\alpha$ emitters at the same redshift $z=2.24$. The
combination of these two populations allows us to determine the Ly$\alpha$
escape fraction at $z=2.24$. Over an area of 383 arcmin$^{2}$ in the Extended
Chandra Deep Field South (ECDFS), 124 Ly$\alpha$ emitters are detected down to
NB393 = 26.4 mag at the 5$\sigma$ level, and 56 H$\alpha$ emitters come from
An14. Of these, four have both Ly$\alpha$ and H$\alpha$ emissions (LAHAEs). We
measure the individual/volumetric Ly$\alpha$ escape fraction by comparing the
observed Ly$\alpha$ luminosity/luminosity density to the extinction-corrected
H$\alpha$ luminosity/luminosity density. We revisit the extinction correction
for H$\alpha$ emitters using the Galactic extinction law with the color excess
for nebular emission. We also adopt the Calzetti extinction law together with
an identical color excess for stellar and nebular regions to explore how the
uncertainties in extinction correction affect our results. In both cases, an
anti-correlation between the Ly$\alpha$ escape fraction and dust attenuation is
found among the LAHAEs, suggesting that dust absorption is responsible for the
suppression of the escaping Ly$\alpha$ photons. However, the estimated
Ly$\alpha$ escape fraction of individual LAHAEs varies up to ~3 percentage
points between the two methods of extinction correction. We find the global
Ly$\alpha$ escape fraction at $z=2.24$ to be ($3.7\pm1.4$)% in the ECDFS. The
variation in the color excess of the extinction causes a discrepancy of ~1
percentage point in the global Ly$\alpha$ escape fraction.
