Space-based gravitational-wave observatories (SGOs) promise to measure pico-meter variations in the gigameter separations of a triangular constellation. Telescopes play a crucial role in using transmitting and receiving laser beams measuring the constellation arms with heterodyne laser interferometry. The far-field phase noise induced by the coupling of the wavefront aberrations of optical telescopes with their pointing jitters is one of the major noise sources for the measurement. As phase noise suppression is a critical aspect for achieving the required comprehensive measuring stability, this paper theoretically analyzes the mechanism of the far-field phase noise, proposes an optimization strategy for the design of the optical telescopes in SGOs, and verifies it to pave the way for the comprehensive phase noise control in the design-to-manufacture process.

To analytically establish the relationship of the coupling coefficient with the aberrations in the form of polynomial expansions, the paper adopts the Fringe Zernike polynomials to represent the aberrations and further construct and describe the wavefront error. Then, the coupling coefficient defined as the modulus of the gradient of the far-field wavefront error is expressed as a polynomial function of the aberration coefficients and the tilt angles and is further simplified on the basis of the symmetry of the telescope. According to this relationship and the aberration characteristics of the telescope design residuals, the paper evaluates the effect of different aberrations on phase noise, revealing that defocus, primary astigmatism, and primary spherical aberration are the keys to controlling the coupling coefficient. Thus, an optimization strategy based on key aberration control is proposed. The performance of this method in far-field phase noise suppression is verified by examples of telescope design.

The wavefront quality of the telescopes before and after optimization (Table 1) by the above strategy is at the λ/20 (λ=1064 nm) level. Before optimization, the far-field wavefront changes significantly within the range of ±100 nrad. Accordingly, the coupling coefficient increases rapidly with the tilt angle to over 1 pm/nrad. After optimization, although the wavefront residuals are slightly worse (Table 2), the range of far-field wavefront error decreases markedly by more than 90% (Fig. 4). The corresponding coupling coefficient is smaller than 0.11 pm/nrad within the range of ±100 nrad. It is only 6% of that before optimization and much smaller than the required value (Fig. 5). These results indicate that the optimization strategy based on aberration control can effectively reduce the coupling coefficient of the far-field phase noise, even in the case of poor wavefront quality.

On the basis of theoretical analysis of the mechanism of the far-field phase noise, this paper determines the relationship of the coupling coefficient with the aberrations, develops an optimization strategy based on key aberration control, and verifies the strategy. The results reveal that suppressing the key aberrations deliberately instead of simply enhancing the demand for wavefront quality in the optimization process can reduce the sensitivity of the far-field phase to jitters more efficiently, improve the far-field phase stability of the telescope significantly, and balance the severe noise budget and the design freedom of the telescope to reserve sufficient margin for the rest optics.