Trefftz discontinuous Galerkin (TDG) methods have recently shown potential [6, 27] for numerical approximation of transport equations with exponential modes. This paper focus on a proof of convergence in two-space dimension for the TDG method through the study of the approximation properties of the exponential solutions constructed in [6]. We show that these vectorial exponential functions can achieve high order convergence with a significant gain in term of the number of basis functions compare to more standard discontinuous Galerkin schemes. The fundamental part of the proof is based on discrete Fourier techniques conveniently adapted to the matrices of the problem.