Based on the angular spectrum diffraction theory and the sampling theorem, the sampling conditions for calculation the Kirchhoff formula, the Rayleigh–Sommerfeld formula, the angular **spectrum diffraction** formula and the Fresnel diffraction formula in convolution form were studied. The results indicate that the diffraction calculation result is correct if the angular spectrum corresponding to the main energy of diffraction field can be fully transmitted by means of transfer functions when the samplings of initial wave fields satisfy the sampling condition. Compared with previous work, the sampling condition in this paper is less restricted. The diffraction calculation results are in agreement with experimental results.

## Research Highlights

► Based on the angular spectrum diffraction theory, the strict sampling conditions of diffraction calculations by using discrete convolution are discussed. ► It is testified by the theoretical analysis and experimental results that the correctness of diffraction calculation depends on the correct sampling of the initial wave. ► According to the Parseval theory and the relation between the discrete Fourier transform and Fourier transform, the correct sampling condition of the initial wave is proposed.

## Keywords

- Diffraction calculation;
- Fourier transform;
- Discrete Fourier transform