Focal shift in vector Mathieu-Gauss beams

We study the three dimensional distributioin of vector Mathieu-Gauss (vMG) beams near the focus of a non-aperturing thin lens. These beams constitute a family of solutions to the vectorial wave equation in the paraxial approximation. We adopt an energy criterion for defining the width of the beam as well as the focal plane. It is observed in our analysis that the focal plane does not coincide with the geometric focus, but it is slightly shifted towards the lens. The dependence of the focal shift on the different parameters of the vMG beams is discussed for beams with different polarizations.