Analytical Modeling of Far-Field Wavefront Error with Beam-Waist and Lateral-Shift Effects in Spaceborne Laser Interferometry

The coupling between far-field wavefront error (WFE) and laser pointing jitter is an important source of tilt-to-length (TTL) noise in spaceborne laser interferometric links. We extend the Nijboer--Zernike analytical model for far-field WFE of truncated Gaussian beams by incorporating two practical initial-condition parameters, the beam-waist-to-aperture ratio q and the normalized lateral spot-shift ratio sᵣ, to account for realistic beam truncation and alignment conditions. Based on this model, we analyze the influence of q on far-field WFE in addition to the conventional received-power trade-off, showing that decreasing q from 1 to 0.9 and from 0.9 to 0.8 reduces the mean far-field WFE by approximately 10% and 14%, respectively, in Monte Carlo simulations of random initial aberrations. We also derive the direct contribution of lateral spot shift and its coupling with transmitted WFE (constrained to λ/20). For the normalized lateral spot-shift ratio sᵣ, a 2~ entrance-pupil displacement in a Taiji-like telescope corresponds to sᵣ=0.001 and produces a phase-angle coupling coefficient of about 0.0892~pm/nrad, close to the typical far-field TTL requirement 0.1~pm/nrad, while the spot-shift--aberration coupling terms are much smaller and can be neglected in practical tolerance estimation. These results provide a theoretical basis for beam-parameter optimization and alignment tolerance design in future space-based gravitational-wave detection missions.