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

Putland, Gavin Richard


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    <subfield code="a">wave optics, diffraction, obliquity factor, inclination factor, spatiotemporal dipoles, Huygens' principle, Huygens-Fresnel, Huygens-Kirchhoff, Fresnel-Kirchhoff, Helmholtz-Kirchhoff, saltus problem, Maggi-Rubinowicz, Poisson's spot, Arago's spot</subfield>
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    <subfield code="a">Exact derivation of Kirchhoff's integral theorem and diffraction formula using high-school math</subfield>
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    <subfield code="a">&lt;p&gt;A theory of diffraction is developed from three premises: causality (including the assumption that the wave function had a beginning), superposition, and the form of the wave function due to a monopole source. It is shown that the wave function in a region &lt;em&gt;R&lt;/em&gt;, due to sources outside &lt;em&gt;R&lt;/em&gt; (that is, in &lt;em&gt;R&amp;#39;&lt;/em&gt;), is identical to that due to a distribution of spatiotemporal dipole (STD) sources on the surface &lt;em&gt;S&lt;/em&gt; separating &lt;em&gt;R&amp;#39;&lt;/em&gt; and &lt;em&gt;R&lt;/em&gt;, facing &lt;em&gt;R&lt;/em&gt;, with a strength density given by the original wave function on &lt;em&gt;S&lt;/em&gt; (divided by 4&lt;em&gt;&amp;pi;&lt;/em&gt;, under the adopted definition of strength). The Kirchhoff integral theorem and consequent diffraction formulae are obtained by superposing the elemental wave functions due to the sources on &lt;em&gt;S&lt;/em&gt;. Whereas D.A.B. Miller (1991) justified the STDs by comparison with Kirchhoff&amp;#39;s integral, I derive the integral (with a near-primary-source correction to Miller&amp;#39;s form) solely from the STDs. Diffraction by an aperture in an opaque screen is modeled by assuming that we retain only the sources on that part of &lt;em&gt;S&lt;/em&gt; which spans the aperture. This assumption avoids the notorious inconsistency in Kirchhoff&amp;#39;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 &lt;em&gt;R&lt;/em&gt;. Finally, the diffraction integral for a monopole primary source is transformed to a geometrical-optics term plus an integral along the edge of the aperture, and applied to the Poisson-Arago spot. The transformation (due to Maggi and Rubinowicz) is first derived from a far-field/short-wavelength assumption, but later re-derived exactly for general time-dependence, and shown to be a valid approximation for a weakly directional primary source.&lt;/p&gt;</subfield>
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