  This paper is concerned with the Cauchy problem of the quadratic nonlinear
Schr\"{o}dinger equation in $\mathbb{R} \times \mathbb{R}^2$ with the
nonlinearity $\eta |u|^2$ where $\eta \in \mathbb{C} \setminus \{0\}$ and low
regularity initial data. If $s < -1/4$, the ill-posedness result in the Sobolev
space $H^{s}(\mathbb{R}^2)$ is known. We will prove the well-posedness in
$H^s(\mathbb{R}^2)$ for $-1/2 < s < -1/4$ by assuming some angular regularity
on initial data. The key tools are the modified Fourier restriction norm and
the convolution estimate on thickened hypersurfaces.
