Visualizing the Wave to Geometrical Transition in Optics:A Pedagogical Framework

Abstract

This paper presents an accessible pedagogical framework and a low-cost experiment to illustrate the continuous transition from wave optics to geometric optics using single-slit diffraction. Key contributions of the paper:1. A conceptual bridge connecting Richard Feynman’s path-integral formulation (stationary phase) directly to Fermat’s Principle of Least Time.2. A proposed and executed low-cost single-slit diffraction experiment that quantitatively tracks the ratio of slit width to wavelength (a/λ) to show exactly where the geometric approximation becomes valid.3. An error analysis and limitation note regarding the Fraunhofer (far-field) diffraction condition, demonstrating scientific rigor and the learning process of experimental physics.4. A dual derivation of Snell’s Law (Fermat’s Principle and electromagnetic boundary conditions) to illustrate how the geometric approach serves as a powerful, simplified shortcut to complex wave equations. Keywords: Wave optics, geometric optics, single-slit diffraction, Fermat's principle, Feynman path integral, pedagogical physics, independent high school research.