Theoretical analysis about 2π ambiguity of the piston error in sparse optical synthetic aperture imaging system

In recent years, sparse-optical-synthetic-aperture telescope has emerged as an important developing trend for large-scale telescope, where the correction of piston error among different sub-apertures plays an important role for achieving high resolution. The 2π ambiguity effect is the most important aspect of piston error's influences upon system's far-field intensity, which was theoretically analyzed at first. Then the bi-sub-aperture system was taken as an example to test the correctness of the theoretical model through distributions of the system's far-field intensity. In addition, a theoretical proof was made to get some important properties of piston errors within such kind of system. From the results, it is shown that the 2π ambiguity comes along with any piston error in the system, and this effect stands independent for different piston errors between different pair of sub-apertures. The derivation also provides effective theoretical model for related concerns within such kind of system.