Can a fibre preserve the shape of the light being transported through it? Can some fibres retain the shape of the light being transported through them?
If I emit radiation towards a sample (in the shape of an "S") and this is reflected back into a fibre, will the same image come out of the other side of the fibre? Will I obtain an "S" shaped beam on the other side?
 A: A conventional fiber cannot do this because the modes inside the fiber determine the distribution of energy of transported light. Thus the output distribution is entirely independent of the input distribution (single mode fiber) or scrambled (multimode fiber).
For imaging, so called "coherent fiber bundles" are used. These are an array of tightly packed multimode fibers. Each fiber acts as a pixel and relays the intensity arriving on it to the backside. If you look at the back face with a camera you will see a honeycomb pattern (the thin gaps between cores) superimposed over whatever the fiber face sees:

Note that the number of fibers is typically limited (1000-10,000s), so do not expect an HD image.
See: https://opg.optica.org/ol/abstract.cfm?uri=ol-36-16-3212
A: No, consider that since fiber optics are flexible and operate via total internal reflection that the number of reflections undergone by an extreme part of the image is variable with the variable cable geometry, therefore a single fiber optic cable doesn't have a definite focal length, and cannot maintain shape.
A: Yes it can, an it is used in endoscopes, look it up in wiki https://en.wikipedia.org/wiki/Endoscope
