Analytically guided design of an off-axis freeform telescope with a coaxial pupil for space-based gravitational-wave detection

To suppress tilt-to-length (TTL) coupling induced by the relative jitter between the optical bench and the telescope, a coaxial entrance–exit pupil configuration is required. Compared to existing non-coaxial schemes, this configuration enables direct beam exchange without auxiliary folding mirrors on the optical bench, thereby inherently eliminating a primary noise path induced by relative opto-mechanical motion. In an off-axis four-mirror telescope, achieving pupil co-alignment demands large tilts on tertiary (M3) and quaternary (M4) mirrors, which rapidly increase aberrations. We employ third-order vector aberration theory as an effective tool for screening initial configurations to establish the quantitative relationship between mirror tilt angle and wavefront root-mean-square (RMS), enabling the selection of a favorable initial configuration. Furthermore, circular-domain Zernike polynomials are analytically expanded under an elliptical pupil, providing initial coefficients for freeform optimization. Using these analytic initial values, a coaxial-pupil afocal telescope is designed, achieving a wavefront RMS≤0.004 λ at 1064 nm and an intrinsic TTL≤0.004pm/nrad, both within the science field of view.