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Optical fibers transport light within them using internal reflection, but the light only enters from one end of the fiber. Is it possible to somehow have light entering from the sides of the fiber (the cylindrical face) and become encapsulated inside the fiber and then transported as usual? Sort of like a greenhouse; but just that light should enter from the sides, not the end, and then not escape it, but instead be transported. What kind of materials/methods can be used to build such a fiber?

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Non-absorbing waveguides, such as optical fibers, are reciprocal optical elements, which means that as easily as light can get in, it can also get out. (This is also true for greenhouses, which work by converting the light to heat and preventing the heat from getting out by eliminating convection with the outside air) So there is no fiber that behaves exactly how I think you are talking about. The basic problem is that you need to couple to the propagating mode of the fiber, and you can't do that simply by shining light at the side (if you could, the fiber would be extraordinary lossy). But there are some special engineering steps you can take to couple light into the propagating mode via the side of a fiber at a particular location. For example, you could use grating couplers, or you could remove the fiber cladding and (inefficiently) couple in light by directing it along the length of the fiber in that region.

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