Propagation of generalized Mathieu-Gauss beams through paraxial misaligned optical system

Recently, the topic of nondiffracting laser beams has been studied extensively. Mathieu beams are nondiffracting beams because they are solutions of Helmholtz equation in elliptic coordinates. Mathieu beams have been applied in photonic lattices, transfer of angular moments, localized X waves, and so on. Mathieu beams have an infinite extent and energy, they cannot be realized in practice. The apodization of Mathieu beams by a Gaussian transmittance lead to the Mathieu-Gauss beams (MGBs). These beams can be generated experimentally and they are a very good approximation of nondiffracting Mathieu beams. MGBs are introduced first by Vega et al. as a member of Helmholtz-Gauss beams family. Although the propagation characteristics through paraxial aligned optical systems of MGBs are well understood, their propagation through misaligned optical systems has not been investigated until now. In practice, the small perturbations of the optical systems, the inevitable errors of the adjustment of beams path, the tolerance in designing optical elements and the thermal deformation or displacement of optical elements etc, lead to the misalignment of optical systems. Thus it is necessary to study the propagation of the considered laser beams through this kind of systems. In this work, the generalized diffraction integral is used to study the propagation of MGBs through a slightly misaligned optical system. So, an analytical propagation equation is derived by using this integral in the case of apertured misaligned optical system. Using this approach, the propagation of ordinary and modified zeroth order MGB through a misaligned thin lens is presented as an example.