Consistent derivation of Kirchhoff's integral theorem and diffraction formula and the Maggi-Rubinowicz transformation using high-school math
Creators: name: Putland, Gavin Richard
Publisher: name: Zenodo
Year: 2022
Subjects: wave optics, diffraction, saltus conditions, saltus problem, Huygens' principle, Helmholtz formula, Helmholtz integral, Helmholtz-Kirchhoff, obliquity factor, inclination factor, Huygens-Kirchhoff, Fresnel-Kirchhoff, Huygens-Fresnel, Maggi-Rubinowicz, generalized spatiotemporal dipole, Poisson's spot, Arago's spot

description: A theory of diffraction is constructed on three principles: causality, superposition, and the assumption that the wave function had a beginning. Given a region <em>R</em> containing no sources and separated from the remaining region <em>R'</em> by a surface <em>S</em>, these principles suffice to show that if the ("primary") sources in <em>R'</em> are <em>replaced</em> by a distribution of ("secondary") sources on <em>S</em>, such that the step-changes in the wave function and its normal derivative, in crossing from <em>R'</em> to <em>R</em>, are respectively equal to the original wave function and its normal derivative on <em>S</em>, then we get the original wave function in <em>R</em> and a null wave function in <em>R'</em> (no backward secondary waves). By further assuming the form of the wave function due to a monopole source, we obtain the distribution of monopole and dipole sources over <em>S</em> that causes the desired step-changes ("saltus conditions"). Adding the contributions from these secondary sources yields an integral expressing the wave function in <em>R</em> in terms of the boundary conditions at the <em>R</em> side of <em>S</em>. Combining this formula with the null wave function in <em>R'</em>, we have the Helmholtz integral theorem for general time-dependence (whereas the traditional derivation is for sinusoidal time-dependence). The Kirchhoff integral theorem follows by elementary rules of differentiation. Diffraction by an aperture in a baffle is modeled by letting <em>S</em> comprise two segments, namely <em>S<sub>a</sub></em> spanning the aperture and <em>S<sub>b</sub></em> on the <em>R</em> side of the baffle, and <em>assuming</em> that the baffle simply eliminates the secondary sources on <em>S<sub>b</sub></em> (without changing those on <em>S<sub>a</sub></em>), with the result that the range of integration is limited to <em>S<sub>a</sub></em> while the integrand is unchanged. This "secondary-source selection" assumption circumvents the need to assume boundary conditions on both the wave function and its normal derivative, and thereby avoids the notorious inconsistency in Kirchhoff's boundary conditions, but yields the same result. The "secondary-source selection" assumption has a saltus interpretation and is easily shown to be equivalent to Kottler's saltus formulation—involving saltus conditions at <em>S<sub>b</sub></em>—which also has a more intuitive interpretation based on secondary sources. The case of a <em>single monopole primary source</em> leads to two simplifications. First, the spatial derivatives in the Kirchhoff integrand are expressed in terms of angles, yielding the Kirchhoff diffraction formula and its far-field obliquity factor, which reduces to the familiar (1 + cos <em>χ</em>)/2 if <em>S</em> is a primary wavefront. Second, the Helmholtz form of the integral over an aperture is expressed as a geometrical-optics term plus an integral over the conical boundary of the geometric shadow, and the latter term is converted to an integral along the edge of the aperture (Maggi-Rubinowicz transformation, for general time-dependence) by a simplified method based on a short-wavelength approximation; an exact conversion modeled on Rubinowicz's, yielding the same result, is given in an appendix. A single monopole primary source also allows the monopole and dipole secondary sources on a surface element <em>dS</em> to be replaced by a single "generalized spatiotemporal dipole" (GSTD), in which the inverted monopole is delayed and attenuated relative to the uninverted one. If <em>S</em> is a primary wavefront in the far field of the primary source, the GSTD reduces to the "spatiotemporal dipole" proposed by D.A.B. Miller in 1991. The above results, having been derived for general time-dependence, are then restated for sinusoidal time-dependence, showing that previously mentioned "far-field" and "short-wavelength" approximations are synonymous. A <em>consistent</em> introduction to complex numbers and the "cis" representation is included. However, the sinusoidal (monochromatic) diffraction integrals are not cited in the subsequent analysis of Poisson's spot and the field on the axis of a circular aperture; these cases can be described in terms of the unobstructed wave function and a delayed version thereof.