To achieve extremely low wavefront error and high throughput with vignetting-free reflective relay optics, a new design approach, “co-axis double three-mirror anastigmat,” is proposed. This approach enables the elimination of rotationally asymmetric aberrations that becomes a problem for vignetting-free off-axis reflective optics by utilizing two components of on-axis telescopes and placing the central axis of the two components in parallel. The validity of the proposed approach has been confirmed by a next-generation astronomical instrument.