What is the simplest way to get the light of 50 visible LEDs into a multimode glass fiber? LEDs are cheap. How can I get all their light into a fiber? Even if it is only 1% from each LED, it would be useful.
Any proposal is welcome. I read about fiber combiners, but they are not helpful. I read about polarizing combiners, but they do not help either.
Is there a way?
 A: It is important to remember that light respects some conservation laws: the radiance can be decreased or at least conserved, but not increased by passive elements (such as lenses and mirrors).
In your case, you want to use a set of LEDs. They have an emitting area, with a given surface, emitting on a wide angle. Typically, the emitting surface is bigger than the section of the fiber, and the emission angle is wider than the acceptance angle of the fiber. Any optical system will cut both the section and the angle! The highest power density (power per surface) that you can obtain, with whatever passive optical system you use, equals the power density of the emitting element. Roughly, what we can do is to couple a part of the emitting area, with a large collection angle, to the section of the fiber optics, possibly matching the acceptance angle.
Of course, we can design an optical system collecting light from two LEDs. However, the radiance of two identical LEDs is the same of the radiance of a single LED (see Wikipedia for the definition of "radiance"). So, roughly speaking, if you couple one LED to the fiber, capturing - say - 30% of light, coupling two LEDs will couple roughly 15% of light from each.
In the case of identical LEDs, I do no see any solution. We cannot circumvent the conservation of radiance. However, in other cases, there can be clever solutions. For example, we can notice that the "spectral" radiance is defined on the spectral bandwidth. Thus we can combine several LEDs, with different wavelengths, by means of beam splitters that are selective for the wave length. Or exploit the radiance per polarization direction.
Finally, if you want a powerful fiberoptics source, I suggest you a superradiant laser, or a laser under threshold. See e.g. Superlum sources.
