Vector Focus Wave Modes with Elliptic Cross-Section

Nondiffracting pulsed beams are well studied nowadays and can be as short as a few femtoseconds. The nondiffracting pulsed beams not only resist diffraction but also propagate without changes due to the dispersion of a linear dispersive medium. A promising member of non-diffracting beam family is the so-called Mathieu beam which is a solution of Helmholtz wave equation in elliptical coordinate system. Zeroth order even Mathieu beams have unique asymmetric cross-section which makes these beams suitable for precise material processing. In this work we derive vectorial Mathieu beams using classical techniques and superpose them to create femtosecond pulsed beams. For these pulsed beams diffraction spreading and dispersive broadening is compensated by a given angular dispersion. Various intensity distributions, durations, angular dispersions and polarization states of different vector Mathieu focus wave modes are presented and discussed in detail.