The panoramic annular lens (PAL) system can capture plentiful scene information in real time. The locally described freeform surface can provide more degrees of freedom for the optical system design than the globally described one. In this paper, we propose a locally described annular Gaussian radial basis function freeform surface that enables a high-performance yet compact PAL system. The designed PAL system achieves an ultra-wide field of view (FOV) of (30∘∼125∘)×360∘. The F-theta distortion of the system is less than 3%, and the modulation transfer function across the whole FOV is greater than 0.5 at 100 lp/mm. The proposed system can be implemented for various applications such as video conferencing or robotic navigation.

The numerical method based on the fast Fourier transform (FFT) is generally applied to calculate the Fresnel diffraction field, which would suffer from sampling constraints. To break this limit, in this Letter, the semi-analytic Fresnel diffraction calculation method is proposed based on polynomial decomposition. The diffraction field is computed by using properly analytic Fresnel diffraction basis functions (FDBFs) according to the application requirements. Analytic FDBF is calculated based on Legendre or Chebyshev polynomials by using the object-domain frequency division multiplexing method. The proposed method offers arbitrary sampling, high-flexibility, and high-accuracy diffraction calculation in the full Fresnel region. The computational efficiency and accuracy of the proposed method are compared with FFT-based methods. It has potential application in light field analysis, wavefront sensing, and image processing.