Equivalence between two consistent formulations of Kirchhoff’s diffraction theory

Kirchhoff’s theory of diffraction, as usually formulated, leads to an internal mathematical inconsistency because the solution does not reduce to the assumed boundary conditions as the observation point approaches the plane of the diffracting aperture. However, it is in excellent agreement with experiment. We show that two different, consistent formulations of Kirchhoff’s theory [ F. Kottler , Ann. Phys.70, 405 ( 1923); E. W. Marchand E. Wolf , J. Opt. Soc. Am.56, 1712 ( 1966)] are equivalent.