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# Exact derivation of Kirchhoff's integral theorem and diffraction formula using high-school math

Gavin R. Putland

A comprehensive theory of diffraction is developed from three elementary premises: superposition, the form of the wave function due to a monopole source, and the assumption that the wave function had a beginning in time. It is shown that the wave function in a region R, due to sources outside R (i.e., in R'), is identical to that due to a distribution of sources on the surface S separating R' and R, such that the step-change (saltus) in the wave function, in crossing S from R' to R, is equal to the original wave function on S. The necessary sources are shown to be spatiotemporal dipoles (STDs), as discovered by D.A.B. Miller (1991). The Kirchhoff integral theorem and consequent diffraction formulae are then obtained by superposing the elemental wave functions due to the sources on S. Whereas Miller justified the STDs by comparison with Kirchhoff's integral, this paper derives the integral (including a near-primary-source correction to Miller's form thereof) solely from the STDs. The case of diffraction by an aperture in an opaque screen can be handled on the assumption that we retain only those sources on the part of S that spans the aperture. This assumption avoids the notorious inconsistency in Kirchhoff's boundary conditions (1882), but gives the same diffraction integral, and yields the consistent saltus condition of Kottler (1923).

Update (27 February 2020): Version 1 of this paper has a logical gap (a "saltus condition" of an undesirable sort). Version 2 is in preparation.
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