Collimated light directly into a multi-mode fiber This is my question - Will a collimated, incoherent light, consists of a limited angle, that "hits" a multi-mode fiber facet exit the fiber as collimated? (just to be clear, no lens is involved. I assume the light is emerging as collimated at various angles).  
Thanks all in advance,
Dror
 A: In general, the answer is no, but with careful engineering you can achieve reasonable collimation at the output.
Depending on the exact fiber, the field is confined to a region of diameter $d$, where $d$ is typically less than $100\mu$. So even if the field arrives at the output with no phasefront curvature, the divergence angle is of the order of $\lambda/d$, i.e. of the order of 10 milliradians. You can only reduce this angle by broadening the width of the field.
Now we strike a big problem. The field is transported through the fiber by a system of modes, which all have different propagation speeds along the fiber. So a collimated input will in general be output with a hopelessly scrambled phasefront, particularly if the fiber is of the "bendy" kind typically used for communications. You'll simply see a mess in the farfield.
However, if you engineer the fiber refractive index profile e.g. varying like $n(r) \approx n_0\,\operatorname{sech}(g\,r)$ where $r$ is the distance from the optical axis, and can keep the fiber straight and strain free, then a collimated field will be periodically reassembled with distance along the lens; the period length is $2\pi/g$. So you need to engineer the length and profile correctly, and take steps to make sure that the rod is not strained. You now have a GRIN lens, rather than a fiber!
