Optimum telescope focal ratios for microlens-to-fiber coupled integral field spectrographs

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We describe the optimum telescope focal ratio for a two-element, three surface, telecentric image-transfer microlens-to-fiber coupled integral field unit within the constraints imposed by micro-optics fabrication and optical aberrations. We create a generalized analytical description of the micro-optics optical parameters from first principals. We find that the optical performance, including all aberrations, of a design constrained by an analytic model considering only spherical aberration and diffraction matches within +/-4% of a design optimized by ray-tracing software such as Zemax. The analytical model does not require any compromise on the available clear aperture; about 90% mechanical aperture of a hexagonal microlens is available for light collection. The optimum telescope f-ratio for a 200{\mu}m core fiber fed at f/3.5 is between f/7 and f/12. We find the optimum telescope focal ratio changes as a function of fiber core diameter and fiber input beam speed. A telescope focal ratio of f/8 would support the largest range of fiber diameters (100 to 500{\mu}m) and fiber injection speeds (between f/3 and f/5). The optimization of telescope and lenslet-coupled fibers is relevant for the design of high-efficiency dedicated survey telescopes, and for retro-fitting existing facilities via introducing focal macro-optics to match the instrument input requirements.