Geometrization of the Huygens–Fresnel principle: Applications to Fraunhofer diffraction

Abstract

The classical wave theory can trace its historical origins to the seminal works of Christian Huygens, Thomas Young, and Augustin Fresnel. To explain some of light’s observed properties, such as rectilinear propagation, reflection, and refraction, Huygens proposed a simple geometrical construction of secondary spherical wavelets with centers of disturbance located on a primary wavefront. More than a century later, Young formulated the law of interference to both predict the formation of fringes in his now famous double slit experiment and also to estimate the wavelengths associated with different colors. A decade after that, Fresnel combined Huygens’ construction with Young’s interference law to qualitatively and quantitatively describe diffraction, which is the bending of light upon encountering an obstacle or an aperture. This grand synthesis, called the Huygens–Fresnel principle, acts as a powerful pictorial aid and conceptual tool that can describe a wide variety of complicated optical phenomena. However, the applications of the principle and its later developments, such as the Kirchhoff–Fresnel integral, are strewn with several simplifying assumptions and approximations that are aimed at minimizing the mathematical challenges involved. Consequently, two distinct formalisms are necessary to account for diffraction effects when the source of light or observation screen is placed nearby and far away from the aperture or obstacle. Recently, a hyperbola framework for analyzing wave interference at a multi-slit barrier was shown to successfully circumvent all conventionally imposed ad hoc conditions. The method commences directly from the Huygens–Fresnel principle and the ensuing predictions pertaining to the distribution of fringe characteristics, namely, positions, widths, and intensities on a detection screen can, therefore, justifiably claim accuracy in both the near field (Fresnel regime) and the far field (Fraunhofer regime). In this paper, the analysis that was previously carried out for the special case of slits of negligible widths is further extended to encompass slits of finite widths as well.