Working paper Open Access

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

Putland, Gavin Richard

DCAT Export

<?xml version='1.0' encoding='utf-8'?>
<rdf:RDF xmlns:rdf="" xmlns:adms="" xmlns:dc="" xmlns:dct="" xmlns:dctype="" xmlns:dcat="" xmlns:duv="" xmlns:foaf="" xmlns:frapo="" xmlns:geo="" xmlns:gsp="" xmlns:locn="" xmlns:org="" xmlns:owl="" xmlns:prov="" xmlns:rdfs="" xmlns:schema="" xmlns:skos="" xmlns:vcard="" xmlns:wdrs="">
  <rdf:Description rdf:about="">
    <rdf:type rdf:resource=""/>
    <dct:type rdf:resource=""/>
    <dct:identifier rdf:datatype=""></dct:identifier>
    <foaf:page rdf:resource=""/>
      <rdf:Description rdf:about="">
        <rdf:type rdf:resource=""/>
        <dct:identifier rdf:datatype="">0000-0003-4757-6341</dct:identifier>
        <foaf:name>Putland, Gavin Richard</foaf:name>
        <foaf:givenName>Gavin Richard</foaf:givenName>
    <dct:title>Exact derivation of Kirchhoff's integral theorem and diffraction formula using high-school math</dct:title>
    <dct:issued rdf:datatype="">2020</dct:issued>
    <dcat:keyword>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</dcat:keyword>
    <dct:issued rdf:datatype="">2020-04-09</dct:issued>
    <dct:language rdf:resource=""/>
    <owl:sameAs rdf:resource=""/>
        <skos:notation rdf:datatype=""></skos:notation>
    <dct:isVersionOf rdf:resource=""/>
    <dct:description>&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;</dct:description>
    <dct:accessRights rdf:resource=""/>
      <dct:RightsStatement rdf:about="info:eu-repo/semantics/openAccess">
        <rdfs:label>Open Access</rdfs:label>
        <dct:license rdf:resource=""/>
        <dcat:accessURL rdf:resource=""/>
All versions This version
Views 1,482873
Downloads 1,527498
Data volume 683.0 MB263.8 MB
Unique views 1,330816
Unique downloads 1,391453


Cite as