We consider the focusing nonlinear Schrödinger equation on the quarter plane. The initial data are vanishing at infinity while the boundary data are time- periodic, of the form. The goal of this paper is to study the asymptotic behavior of the solution of this initial-boundary-value problem. The main tool is the asymptotic analysis of an associated matrix Riemann-Hilbert problem. We show that for the solution of the IBV problem has different asymptotic behaviors in different regions. In the region, where, the solution takes the form of the Zakharov-Manakov vanishing asymptotics. In a region of type, where N is any integer, the solution is asymptotic to a train of asymptotic solitons. In the region, the solution takes the form of a modulated elliptic wave. In the region, the solution takes the form of a plane wave.