A sparse aperture refers to a configuration where multiple sub-mirrors are arranged in a non-redundant pattern, utilizing interference techniques. This results in a system with a smaller light-receiving area than a single large aperture, while still capturing comparable information. The polarization-induced aberrations in each optical element of a sparse aperture optical system significantly influence the overall imaging performance. However, limited research has been conducted on the polarization characteristics of such systems. We systematically examine the polarization aberrations of sparse aperture imaging systems using the polarization ray tracing theory.

In this study, we use the Golay3 sparse aperture imaging system, designed with ZEMAX optical software, as a case study. Based on polarization ray tracing theory, we calculate the polarization aberrations of the system, including diattenuation and phase retardation. The system’s Jones pupil is derived, and through Fourier transformation, we calculate the system’s amplitude response matrix (ARM) and modulation transfer matrix (MTM).

Our theoretical model reveals that, at a zero field of view (FOV), the diattenuation and phase retardation of the sparse aperture optical system exhibit rotational symmetry. The maximum values of the system’s diattenuation and phase retardation are 2.313×10^-3 and 2.315×10^-2 rad, respectively, as shown in Fig. 6 and Fig. 7 indicate that across the full FOV, the Peak-to-Valley (PV) values of diattenuation and phase retardation share a consistent distribution characteristic, exhibiting symmetrical distribution along the Y-field. By performing a Fourier transformation on the Jones pupil, we obtain the system’s ARM, as shown in Fig. 10. The diagonal matrices M_ARM,XX and M_ARM,YY of ARM are close to the amplitude response function of a diffraction-limited system. Figs. 10(b) and (c) illustrate that the non-diagonal matrices M_ARM,XY and M_ARM,YX have equal magnitudes, exhibiting a symmetrical structure not centered at the origin but symmetric around a horizontal line with four peak points. We calculate the MTM of the sparse aperture optical system under different filling factors. Fig. 12(a) demonstrates that the matrices in the horizontal direction of MTM are symmetrically distributed about the diagonal. From the MTF of the main diagonal, it is evident that a higher filling factor results in higher MTF values at the same spatial frequency, with consistent trends in the MTF curves. The maximum MTF positions of the non-diagonal MTFs are not at zero frequency. M₀₁, M₀₂, M₂₃, and M₀₃ show that as the filling factor decreases, the peak value of the MTF curve lowers, and the cutoff frequency shifts to lower frequencies. Fig. 12(b) indicates that the MTF values on the diagonal decrease as the filling factor decreases. On the non-diagonal lines of MTM, the maximum MTF values are at non-zero frequencies, exhibiting the same distribution trend across different filling factors, with the maximum values increasing as the filling factor decreases. The MTF curves show significant variations at mid-to-high frequencies.

Our study uses polarization ray tracing methods to calculate the polarization aberrations of the Golay3 sparse aperture optical system. The results indicate that under a 0° FOV condition, the system’s diattenuation and phase retardation are primarily caused by the sub-mirrors. Polarization aberrations are closely linked to the structural characteristics of sparse aperture optical systems. All mirrors in the Golay3 system are rotationally symmetric around the optical axis, leading to rotational symmetry in diattenuation and phase retardation. Across the full FOV, the PV values are symmetrically distributed along the Y-field, increasing with the Y-field and decreasing towards -0.1° along the X-field. Polarization crosstalk in the Jones pupil and ARM exhibits horizontal symmetrical distribution, with four peak points in the latter. As the filling factor increases, the non-diagonal matrices of the MTM decrease in the horizontal direction’s MTF, while the vertical direction’s MTF increases. Polarization aberrations of sparse apertures are closely related to the arrangement of sub-aperture arrays, and their presence can reduce system imaging quality.