The displacement theorem and frequency shifting theorem are not only important concepts in information optics, but they can also provide specific control over the spectrum of a hologram and the spatial position of the reconstructed image. In this paper, the differing relationships of the displacement and frequency shifting characteristics in space and frequency domains of the computer-generated holography (CGH) are discussed in detail based on Fresnel diffraction. Their intrinsic connection and effects on the holographic light field are explained and analyzed. Furthermore, by combining these two operations, it is possible to arbitrarily manipulate the hologram spectrum without affecting the position of reconstructed images, which facilitates functions such as diffraction order filtering of holograms and frequency division multiplexing. It can be demonstrated that the fundamental concepts have practical applications in addressing practical advanced research problems. The above discussion also deepens the understanding of the tangible concepts to the displacement theorem and frequency shifting theorem in Fourier transforms during diffraction propagation.

The aperture stop is an important concept in applied optics and physical optics, determining the performance of optical systems such as resolution and imaging brightness. This paper discusses the application of the aperture stop in both microscopic and long-distance systems within the forefront of current computational imaging, specifically in Fourier ptychographic high-resolution imaging systems. Firstly, based on the thin lens imaging model, the coherent transfer functions corresponding to the pupil function in the object and image space are provided. Secondly, the actual settings of the aperture stop in two types of Fourier ptychographic imaging systems are analyzed. Finally, the equivalence of high-resolution image reconstruction from both object and image space data is verified through numerical simulation and optical experiments. The above discussions deepen the understanding of the concept of the aperture stop and its application in practical imaging systems.