  In this work we study the initial boundary value problem associated with the
coupled Schr\"odinger equations {with quadratic nonlinearities, that appears in
nonlinear optics}, on the half-line. We obtain local well-posedness for data
{in Sobolev spaces} with low regularity, by using a forcing problem on the full
line with a presence of a forcing term in order to apply the Fourier
restriction method of Bourgain. The crucial point in this work is the new
bilinear estimates on the classical Bourgain spaces $X^{s,b}$ with $b<\frac12$,
jointly with bilinear estimates in adapted Bourgain spaces that will used to
treat the traces of nonlinear part of the solution. Here the understanding of
the dispersion relation is the key point in these estimates, where the set of
regularity depends strongly of the constant $a$ measures the
scaling-diffraction magnitude indices.
