  We present a deep $K^{\prime}$-band (2.12$\mu$m) imaging of 1\arcmin\
$\times$ 1\arcmin Subaru Super Deep Field (SSDF) taken with the Subaru adaptive
optics (AO) system. Total integration time of 26.8 hours results in the
limiting magnitude of $K^{\prime} \sim 24.7$ (5$\sigma$, 0\farcs2 aperture) for
point sources and $K^{\prime} \sim 23.5$ (5$\sigma$, 0\farcs6 aperture) for
galaxies, which is the deepest limit ever achieved in the $K^{\prime}$ band.
The average stellar FWHM of the co-added image is 0\farcs18. Based on the
photometric measurements of detected galaxies, we obtained the differential
galaxy number counts, for the first time, down to $K^{\prime} \sim 25$, which
is more than 0.5 mag deeper than the previous data. We found that the number
count slope $d\log N/dm$ is about 0.15 at $22 < K^{\prime} < 25$, which is
flatter than the previous data. Therefore, detected galaxies in the SSDF have
only negligible contribution to the near-infrared extragalactic background
light (EBL), and the discrepancy claimed so far between the diffuse EBL
measurements and the estimated EBL from galaxy count integration has become
more serious . The size distribution of detected galaxies was obtained down to
the area size of less than 0.1 arcsec$^2$, which is less than a half of the
previous data in the $K^{\prime}$ band. We compared the observed size-magnitude
relation with a simple pure luminosity evolution model allowing for intrinsic
size evolution, and found that a model with no size evolution gives the best
fit to the data. It implies that the surface brightness of galaxies at high
redshift is not much different from that expected from the size-luminosity
relation of present-day galaxies.

Astrophysics