Tilt-to-length (TTL) noise presents a dominant limitation for space-based gravitational wave (GW) detection. The coupling between pointing jitter and wavefront errors induces phase noise in far-field detection, which emerges as a principal component of the TTL noise budget. We derive an analytical model for this TTL noise using the Zernike polynomial decomposition. Our framework expresses the TTL coefficients via matrix multiplication, providing analytical insight into the interaction mechanism between pointing jitter and wavefront errors. We numerically validate the theoretical expression and perform ${10}^4$ Monte Carlo simulations to quantify the TTL noise at both the maximum intensity point (Imax) and the phase stationary point. Moreover, we demonstrated that optimizing the beam-waist radius to 30 mm efficiently suppresses the TTL noise to $1\mathrm{pm}{\mathrm{Hz}}^{-1/2}$, without compromising the power budget of the geocentric GW detections.

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• Gravitational wave detection