MOFR-EEM: An Encircled-Energy-Based Evaluation of Microlens Array Output Focal Ratio for Fiber Integral Field Units

Microlens-array (MLA)-based integral field units (IFUs) are widely employed in modern astronomical instrumentation. While focal-ratio degradation (FRD) in the fiber affects the final fiber-output beam, the focal ratio generated by the microlens array immediately before fiber injection constitutes a critical parameter for focal-ratio matching and system design. Under the parameter configuration of Fiber Arrayed Solar Optical Telescope (FASOT)--1B, this study focuses only on the effects of MLA parameters on output focal ratio. An encircled-energy-based evaluation method, referred to as Microlens-array Output Focal Ratio based on the Encircled Energy Method (MOFR-EEM), was developed to quantitatively investigate the dependence of MLA output focal ratio on exit-pupil diameter, microlens diameter, and input focal ratio. The results show that the output focal ratio increases with exit-pupil diameter, decreases with microlens diameter, and scales with the input focal ratio, the last dependence being obtained from a sequence of re-optimized MLA configurations with the exit-pupil diameter held fixed. In addition, under the parameter configuration of FASOT-1B, the output focal ratio was found to be nearly independent of field position within the MLA. By introducing a shape-dependent correction factor Α shape ( EE ), a quantitative relationship between the focal ratio corresponding to a given encircled energy and the focal ratio predicted by geometric optics is established. The proposed MOFR-EEM provides a practical methodology for characterizing the output focal ratio of MLA-coupled fiber systems and enables quantitative assessment of the influence of key MLA parameters within the investigated FASOT-1B parameter space.