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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 assumption that the wave function had a beginning in time, and the form of the wave function due to a monopole source.  It is shown that the wave function in a region R, due to sources outside R (that is, in R'), is identical to that due to a distribution of spatiotemporal dipole (STD) sources on the surface S separating R' and R, facing R, with a strength density given by the original wave function on S (divided by 4π, under the adopted definition of strength).  The Kirchhoff integral theorem and consequent diffraction formulae are then obtained by superposing the elemental wave functions due to the sources on S.  Whereas D.A.B. Miller (1991) justified the STDs by comparison with Kirchhoff’s integral, I derive 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 is modeled on the assumption that we retain only the sources on that part of S which spans the aperture.  This assumption avoids the notorious inconsistency in Kirchhoff’s boundary conditions (1882), but gives the same diffraction integral.  It also yields the consistent saltus conditions of Kottler (1923), and a more direct saltus description of the aperture giving the same solution in R.  Finally, the diffraction integral for a monopole primary source, in the far field of the aperture, is transformed to the Maggi-Rubinovicz edge integral and applied to the Poisson-Arago spot.

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