Propagation properties of vector Mathieu–Gauss beams

The vector Helmholtz–Gauss (vHzG) beam is known as a general family of localized vector beam solutions of the Maxwell equations in the paraxial limit and the vector Mathieu–Gauss beam constitutes its version in elliptic cylindrical coordinates system. In this work, starting from the expansion of the scalar Mathieu–Gauss beam in term of Bessel–Gauss beams, we give a general expression of vector Mathieu–Gauss beams in cylindrical coordinates. Within the frame work of the Collins diffraction integral formula we derive the analytical expressions of transverse vector Mathieu–Gauss beams through an axisymmetric ABCD optical system. Some numerical calculations are performed to illustrate the propagation of the vector Mathieu–Gauss beam in free space and through a simple lens system. The results are analyzed and discussed.