A differential formulation of diffraction theory for the undergraduate optics course

Abstract

The integral formulation of the Huygens–Fresnel principle embodied by the Fresnel–Kirchhoff and Rayleigh–Sommerfeld diffraction formulas constitutes standard treatment of scalar diffraction theory. It is not generally appreciated that the same results can be obtained by using standard methods to solve the relevant partial differential equations: the exact Rayleigh–Sommerfeld integral is equivalent to the scalar Helmholtz equation, and the Rayleigh–Sommerfeld integral in the Fresnel approximation is equivalent to the paraxial wave equation. In view of students’ familiarity with the latter partial differential equation, some pedagogical advantages may be realized if diffraction theory is also formulated in terms of partial differential equations as a supplement to the usual integral formulation.