The sparse aperture optical system employs multiple discrete sub-apertures to replace the full aperture and achieves the resolution equivalent to that of the full aperture optical system while reducing the volume, quality, and costs. The sub-aperture's wavefront aberrations of the sparse aperture optical system exert impacts on the imaging performance of the whole system. In most studies, the system's field of view is not taken into account during the analysis of the imaging performance and sub-apertures' wavefronts of the sparse aperture optical system. Starting from the generalized pupil function, this paper develops the sparse aperture imaging model considering the system's field of view, thereby providing a theoretical basis for predicting the imaging performance and image restoration of the sparse aperture optical system under different fields of view.

The generalized pupil function of the sparse aperture optical system considering the field of view is derived on the theoretical basis of double Zernike polynomials (DZPs). The modulation transfer function (MTF) of the system is obtained by the Fourier transform. The Golay3 sparse aperture imaging system designed by the ZEMAX optical software is taken as an example. According to the design results, the coefficients of double Zernike polynomials are fitted. The theoretical calculation results and optical design results are compared to verify the sparse aperture imaging theory considering the field of view. The Wiener filter is constructed according to the optical transfer function (OTF) for image restoration to improve the imaging quality of the system under different fields of view.

According to the theoretical model, the results show that when the field of view is 0°, the imaging of the sparse aperture optical system approaches the diffraction limit as shown in Fig. 3(a). Figs. 3(b)-(e) indicate that under the same field of view, the main lobe and side lobe of MTFs decrease rapidly, and the main lobe shows different divergent directions corresponding to the directions of the incident light. As the field of view rises, the main lobe of MTFs further narrows, and the imaging performance of the optical system decreases significantly. MTFs calculated by DZPs are similar to those obtained by ZEMAX software.

The contrasts of each line pair in the image simulated by the sparse aperture optical system are calculated under different fields of view. The images are processed by the Wiener filter, and the contrast curves are drawn, as shown in Figs. 9 (a)-(d). The figures demonstrate that the image contrasts of each field of view in horizontal and vertical directions can be greatly improved by the Wiener filter. Under the same field of view and different directions, the restored image has different contrasts in the horizontal and vertical directions. As shown in Fig. 9(b), when the field of view is (0.05°, 0°), the contrast ranges in the horizontal and vertical directions are 0.84-0.99 and 0.62-0.99, respectively. In Fig. 9(c), when the field of view is (0°, 0.05°), the contrast ranges in the horizontal and vertical directions are 0.44-0.84 and 0.89-0.99, respectively. As the field of view further increases, the contrasts of the image processed by the Wiener filter gradually decrease. In Fig. 9(d), when the field of view is (0.1°, 0°), the contrast range in the vertical direction of the image before and after restoration is 0.13-0.26 and 0.30-0.43, respectively.

The sub-aperture's wavefront of the sparse aperture optical system under a non-zero field of view is represented by the DZP. When the generalized pupil function is constructed, the MTFs of the system under different fields of view are calculated by the Fourier transform, and the optical design of the system is carried out by ZEMAX software. Upon the fitting of the DZPs, the calculated MTFs of the system are proven to be consistent with those of the ZEMAX software, which verifies the method of utilizing DZPs to describe the wavefront of the sparse aperture imaging system under different fields of view. The Wiener filter related to the field of view is constructed on the basis of the OTF of the optical system. The image restoration using the Wiener filter effectively improves the imaging quality of the sparse aperture optical system under different fields of view.