Using deep narrow-band $H_2S1$ and $K_{s}$-band imaging data obtained with CFHT/WIRCam, we identify a sample of 56 H$alpha$ emission-line galaxies (ELGs) at $z=2.24$ with the 5$sigma$ depths of $H_2S1=22.8$ and $K_{s}=24.8$ (AB) over 383 arcmin$^{2}$ area in the ECDFS. A detailed analysis is carried out with existing multi-wavelength data in this field. Three of the 56 H$alpha$ ELGs are detected in Chandra 4 Ms X-ray observation and two of them are classified as AGNs. The rest-frame UV and optical morphologies revealed by HST/ACS and WFC3 deep images show that nearly half of the H$alpha$ ELGs are either merging systems or with a close companion, indicating that the merging/interacting processes play a key role in regulating star formation at cosmic epoch z=2-3; About 14% are too faint to be resolved in the rest-frame UV morphology due to high dust extinction. We estimate dust extinction from SEDs. We find that dust extinction is generally correlated with H$alpha$ luminosity and stellar mass (SM). Our results suggest that H$alpha$ ELGs are representative of star-forming galaxies (SFGs). Applying extinction correction for individual objects, we examine the intrinsic H$alpha$ luminosity function (LF) at $z=2.24$, obtaining a best-fit Schechter function characterized by a faint-end slope of $alpha=-1.3$. This is shallower than the typical slope of $alpha sim -1.6$ in previous works based on constant extinction correction. We demonstrate that this difference is mainly due to the different extinction corrections. The proper extinction correction is thus key to recovering the intrinsic LF as the extinction globally increases with H$alpha$ luminosity. Moreover, we find that our H$alpha$ LF mirrors the SM function of SFGs at the same cosmic epoch. This finding indeed reflects the tight correlation between SFR and SM for the SFGs, i.e., the so-called main sequence.
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