Beyond piston: three-dimensional dispersed fringe sensing

Dispersed fringe sensing (DFS) is an efficient co-phasing method for segmented mirrors, though conventionally limited to one-dimensional piston measurement. We propose a two-stage orthogonal decoupling method based on the multi-trace Hilbert transform. By combining frequency-domain carrier isolation with spatially robust fitting, this approach enables full-dimensional pose error retrieval using a standard DFS. Simulations show that under a signal-to-noise ratio (SNR) of 5 dB, the piston retrieval error is below 0.6 nm, and the tip-tilt error is within 0.5 μrad. Experimental validations yield an average piston retrieval error of 50.67 nm, with absolute tip and tilt errors below 0.63 μrad and 0.70 μrad, respectively. This extends traditional DFS into a three-dimensional coarse co-phasing solution for large-aperture telescopes.