Optical imaging, as a cornerstone of modern optical research, bears significant application value in both scientific exploration and engineering technologies. Fresnel diffraction theory reveals that when imaging a point source through an optical system, the interplay between light’s wave nature and finite lens aperture induces diffraction effects, generating an Airy disk whose spatial extent defines the Rayleigh diffraction limit. While current undergraduate curricula predominantly utilize Fresnel diffraction integrals for imaging resolution analysis, this study adopts a Fourier optics framework to rigorously derive the angular spectrum propagation of point-source radiation, thereby fundamentally unveiling the diffraction-limited nature of conventional lens systems. The first zero point of the intensity distribution in the circular aperture diffraction pattern is obtained through numerical solution based on the Fourier angular spectrum method, and the result is compared with that calculated using the Rayleigh criterion. By further integrating cutting-edge developments in negative-refraction optics, we systematically elucidate the physical principles enabling negative-index metamaterials to overcome classical diffraction constraints. This work not only establishes a novel pedagogical paradigm for optical imaging theory but also provides a critical analytical framework for super-resolution imaging research, serving as a conceptual bridge between undergraduate optics education and advanced graduate studies in photonic innovation.

Optics is an important branch of physics and a required foundational course for students majoring in physics, optical engineering, and optoelectronics, among others. Snell's law and Fresnel's formulas are crucial topics in optical teaching, but recent research has revealed anomalous reflection and refraction phenomena at non-uniform interfaces, rendering traditional Snell's law obsolete. While researchers have derived the generalized Snell's law using Fermat's principle to calculate reflection and refraction angles at non-uniform interfaces, the generalized Fresnel formula for calculating the reflection coefficients has remained unexplored. In this paper, we derive the generalized Snell's law and Fresnel's formulas using Maxwell's equations and boundary conditions, providing a comprehensive framework for describing the relationships between reflection and refraction angles at both uniform and non-uniform interfaces, as well as calculating the efficiency of reflection and refraction. The theoretical formulas derived align with simulation results, confirming the correctness of the generalized Snell's law and Fresnel's formulas. This work presents a valuable attempt at innovating the teaching content of optics in universities.