Do non gaussian-beams lose their shape in the far field? I know that gaussian beams have the same form in the near-field as in the far-field. But what about non gaussian beams? And what fundamental principle lies behind it?
 A: There's plenty of non-gaussian beams which get completely mangled in the transition from a focus to the far-field.
However, gaussian beams are far from alone in retaining their shape (modulo a scaling factor dictated by diffraction!) as they go from far-field into the focus and then out to the far-field again. The core examples of beam families that do this are the Hermite-Gauss solutions and the Laguerre-Gauss family, but there are plenty of other modes with that property.
A: The question is a bit unclear. What do you mean by nearfield? Nearfields appear in the context of light scattering from a particle, and are the terms that decay faster than 1/r^2, where r is the distance from the scattered.
I have the feeling that by nearfield and far field you just are referring to propagation of a light beam of a certain shape as a matter of distance. If so, then, the eigenvectors of a homogeneous medium are plane waves. So, any waveform that is not a plane wave is subject to change in its amplitude and shape while propagating. For example, a Gaussian beam will become wider and wider while propagating, although it will keep its Gaussian shape. this will apply to other waveform as well, though you can come up with beam shapes that change while propagating.
