Bilinear Fourier integral operator and its boundedness

We consider the bilinear Fourier integral operator S(f, g)(x) = int Rd int Rd ei1(x,y)ei2(x,z)sigma(x,y ,z ) ˆ f(y)ˆg(z)dy dz, on modulation spaces. Our aim is to indicate this operator is well defined on S(Rd) and shall show the relationship between the bilinear operator and BFIO on modulation spaces.