Rodica found a paper by Khazanov in which the phasefront distortion due to a TGG / DKDP crystal combination at high laser power was measured (see http://www.intechopen.com/books/advances-in-solid-state-lasers-development-and-applications/faraday-isolators-for-high-average-power-lasers). To see if this kind of distortion could be responsible for the measure beam shape distortion observed by Rodica and Cheryl in the above report, I ran an FFT simulation where the distortion measured by Khazanov was applied at the FI position, and the beam was them propagated through the focusing lens setup used by Rodica and Cheryl. The result showed that such a distortion was capable of producing the 'double-beam' like intensity pattern, shortly after the beam waist position. 

To understand the effect better, Giacomo created a semi-empirical functional form for the distortion that could roughly reproduce the phase distortion shape measured by Khazanov. We then used this functional form in FFT simulations, manually tweaking the parameters so as to match the intensity pattern measured by Rodica and Cheryl. This process was quite succesful; as shown in the attached plots we could reproduce Rodica and Cheryl's observations pretty well (to the eye).

The simulation study remains very much qualitative at this stage, but we hope that with a more complete set of beam profile measurements we can fit the phase distortion to the observed beam profiles, and pull out an estimate for the overall mode purity.