We describe the design of the fiber-optic coupling and light transfer system of the WISDOM (WIYN Spectrograph for DOppler Monitoring) instrument. As a next-generation Precision Radial Velocity (PRV) spectrometer, WISDOM incorporates lessons learned from HARPS about thermal, pressure, and gravity control, but also takes new measures to stabilize the spectrograph illumination, a subject that has been overlooked until recently. While fiber optic links provide more even illumination than a conventional slit, careful engineering of the interface is required to realize their full potential. Conventional round fiber core geometries have been used successfully in conjunction with optical double scramblers, but such systems still retain a memory of the input illumination that is visible in systems seeking sub-m/s PRV precision. Noncircular fibers, along with advanced optical scramblers, and careful optimization of the spectrograph optical system itself are therefore necessary to study Earth-sized planets. For WISDOM, we have developed such a state-of-the-art fiber link concept. Its design is driven primarily by PRV requirements, but it also manages to preserve high overall throughput. Light from the telescope is coupled into a set of six, 32 μm diameter octagonal core fibers, as high resolution is achieved via pupil slicing. The low-OH, step index, fused silica, FBPI-type fibers are custom designed for their numerical aperture that matches the convergence of the feeding beam and thus minimizes focal ratio degradation at the output. Given the demanding environment at the telescope the fiber end tips are mounted in a custom fused silica holder, providing a perfect thermal match. We used a novel process, chemically assisted photo etching, to manufacture this glass fiber holder. A single ball-lens scrambler is inserted into the 25m long fibers. Employing an anti-reflection (AR) coated, high index, cubic-zirconia ball lens the alignment of the scrambler components are straightforward, as the fiber end tips (also AR coated) by design touch the ball lens and thus eliminate spacing tolerances. A clever and simple opto-mechanical design and assembly process assures micron-level self-alignment, yielding a ~87% throughput and a scrambling gain of >20,000. To mitigate modal noise the individual fibers then subsequently combined into a pair of rectangular fibers, providing a much larger modal area thanks to the 34x106 micron diameter. To minimize slit height, and thus better utilize detector area, the octagonal cores are brought very close together in this transition. The two outer fibers are side polished at one side, into a D-shaped cladding, while the central fiber has a dual side polish. These tapered, side-flattening operations are executed with precise alignment to the octagonal core. Thus the cores of the 3 fibers are brought together and aligned within few microns of each other before spliced onto the rectangular fiber. Overall throughput kept high and FRD at bay by careful management of fiber mounting, vacuum feed-through, application of efficient AR coatings, and implementation of thermal breaks that allow for independent expansion of the fibers and the protective tubing.