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Freeform surfaces are widely used in off-axis optical systems with large aperture, large field of view, and long focal length. The polarization effect caused by the non-rotationally symmetrical shape has an impact on the system's polarization imaging quality and measurement accuracy. Based on Jones’ notation, this paper proposes a polarization aberration analysis method for fringed Zernike polynomial freeform optical systems. It can solve the polarization effect problem introduced by the off-axis freeform surface optical system. This paper constructs a full-field polarisation aberration analysis model of non-rotationally symmetric freeform reflective optical systems. The role of the freeform surface on the polarization aberration distribution characteristics of non-rotationally symmetric optical systems is demonstrated. The multi-degree-of-freedom characteristic of the freeform surface makes the curvature of each point in the entrance pupil different, and the incident angle also changes slightly. The light propagation vector k is added on the basis of the two-dimensional ray tracing algorithm. By tracing the full-field polarized light of the off-axis optical system in the field of view, the Jones pupil diagram is obtained. The phase aberration, diattenuation and retardance are separated by Pauli decomposition and SVD decomposition. The theoretical relationship between the phase aberration and the freeform surface represented by the fringe Zernike polynomial is deduced. The distribution law of the diattenuation and retardance in the full aperture and full field of view is analyzed. The off-axis freeform surface optical system is designed. The analysis results show that the phase aberration of the off-axis freeform surface optical system is directly related to the surface shape of the freeform surface. The changes of the freeform surface to the diattenuation and retardance are both 56% of the diattenuation and retardance of the sistem. For deep-space telescopes and lithography systems, it is of great significance for improving system accuracy to master this change.
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