Closed-loop cophasing method for segmented mirrors based on slope signals derived from dispersed interferograms

To achieve diffraction-limited performance, the segmented mirror telescope must maintain cophasing accuracy within a small fraction of the operating wavelength. Nevertheless, contemporary cophasing detection methodologies are fundamentally bounded by inherent trade-offs between achievable accuracy, dynamic range limitations, calibration complexity, computational overhead, and temporal resolution requirements. This Letter presents a closed-loop cophasing correction method based on slope analysis of dispersive interferometric images for Fizeau interferometric telescopes. By extracting the intensity ratios of the second and third peaks in dispersive interferograms, the method constructs a ratio vector distribution along the dispersion direction and derives its slope parameter through fitting. Theoretical analysis validates a proportional relationship between the slope parameter and the cophasing error, enabling its use as a feedback signal for iterative correction. A normalized-ratio-based error objective function is defined as the convergence criterion to optimize the correction process. Notably, the proposed method eliminates the inherent slope offset present in traditional approaches, achieving both a null error function and zero slope at perfect cophasing alignment. The experimental results demonstrate that the method successfully reduces residual cophasing error from 30 μm to 5 nm within several iterations, requiring neither complex computations nor system pre-calibration. This is the first, to the best of our knowledge, demonstration of the proposed method for closed-loop cophasing using the DFS signal.