The adopted scalar aperture-diffraction model

Assume the Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) as carried by Outgoing source Helmholtz field: fundamental solution, Fraunhofer and Fresnel error terms and Exact Helmholtz boundary representation with actual traces (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)); the smooth modal full-Choice branch of the latter is not used. Adopt as a model assumption of the framework the following scalar aperture-diffraction model.

(a) [Data.] Fix an angular frequency $\omega >0$ and wavenumber $k:=\omega \mathrm{/}c$, with $c$ the vacuum speed of Framework-local SI units, the vacuum constants and the exact/measured status of their values. Let $S\subset \{y\in {\mathbb{R}}^3:y_3=0\}\cap \{y:\mathrm{\mid }y\mathrm{\mid }\le a\}$ be a nonempty compact subset of a declared aperture plane $y_3=0$ with aperture radius $a>0$ (the screening plane and spatial region are read in the laboratory chart of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields), and let $\alpha$ be a finite complex measure on ${\mathbb{R}}^3$ supported in $S$ (A complex measure is a finite-valued countably additive set function, The total variation |nu|(E) from countable measurable partitions) — the aperture distribution — with $\alpha$ read in volts so that the model field below carries the unit $\mathrm{V}{\mathrm{m}}^{-1}$ of an electric-field component and its pairings are those of Integration against a signed or complex measure, and the class L^1(nu) = L^1(|nu|).

(b) [Model field.] The model assigns to every observation point $x$ with $x_3>0$ the complex scalar amplitude

$u(x):=\int G_k(x-y)d\alpha (y),$

where $G_k(z)=e^{ik\mathrm{\mid }z\mathrm{\mid }}\mathrm{/}(4\pi \mathrm{\mid }z\mathrm{\mid })$ for $z\mathrm{\ne }0$ is the outgoing Helmholtz kernel of Outgoing source Helmholtz field: fundamental solution, Fraunhofer and Fresnel error terms, together with the real scalar field $U(x,t):=\mathrm{Re}(u(x)e^{-i\omega t})$ and the model period-averaged squared field $(1\mathrm{/}2)\mathrm{\mid }u(x){\mathrm{\mid }}^2$. All model predictions are declared comparisons of this squared field between observation points; no absolute radiometric power or intensity calibration is adopted.

(c) [Adopted regime and status.] The model is the declared polarization-insensitive scalar approximation of aperture diffraction. It ignores the vector nature and polarization of the electromagnetic field, takes the aperture distribution rather than the exact screen traces as its datum, assigns no back-reaction of the diffracted field on the illumination and no screen interaction beyond the aperture, and is not derived from either branch of The microscopic vacuum Maxwell equations: smooth and weak branches or from the material-closure branches of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch; it is a declared optical approximation, not an exact electromagnetic model. It does not assert that $u$ or $U$ satisfies the Maxwell equations exactly, and it does not identify $\alpha$ with the unknown exact Maxwell boundary traces: assigning both $u$ and ${\mathrm{\partial }}_{n}u$ on the aperture independently is not a well-posed exact boundary problem, as recorded in Exact Helmholtz boundary representation with actual traces, and the error terms of Outgoing source Helmholtz field: fundamental solution, Fraunhofer and Fresnel error terms are the controlled content of the model. No measured intensity, apparatus, sampling model, uncertainty budget, empirical comparison or quantitative radiometric calibration is adopted.

Well-definedness and non-derivation notes

For fixed $x$ with $x_3>0$ and $y$ ranging over the compact support $S$ of $\alpha$, one has $\mathrm{\mid }x-y\mathrm{\mid }\ge x_3>0$, so $y\mapsto G_k(x-y)$ is continuous on $S$, bounded there by $1\mathrm{/}(4\pi x_3)$; the pairing $\int G_k(x-y)d\alpha (y)$ is therefore the integral of a bounded continuous complex function against a finite complex measure and is defined by Integration against a signed or complex measure, and the class L^1(nu) = L^1(|nu|), with $\mathrm{\mid }u(x)\mathrm{\mid }\le \mathrm{\parallel }\alpha {\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}\mathrm{/}(4\pi x_3)$ in the total variation of The total variation |nu|(E) from countable measurable partitions. The model permits the compact closed-radius support $\mathrm{\mid }y\mathrm{\mid }\le a$, including aperture corners: the integral, off-support regularity and closed-radius estimates at $r\ge 2a$ are established for finite measures supported in $\overline{B_a(0)}$ by Outgoing source Helmholtz field: fundamental solution, Fraunhofer and Fresnel error terms. The consuming diffraction theorem also supplies direct kernel estimates at that same closed radius.

The integral against a finite complex measure, the real-field reading and the squared field $\frac{1}{2}\mathrm{\mid }u{\mathrm{\mid }}^2$ constitute the whole mathematical content adopted here; the physical reading uses the aperture plane and spatial region of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, the relation $k=\omega \mathrm{/}c$ and the units of Framework-local SI units, the vacuum constants and the exact/measured status of their values. Neither the smooth nor the weak branch of The microscopic vacuum Maxwell equations: smooth and weak branches implies this scalar source model, and the finite-conductivity and material branches of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch contain no ideal aperture or screen model. The item uses no choice principle beyond those carried by the cited mathematical suppliers; the Countable Choice hypothesis is inherited verbatim from the kernel and boundary-representation suppliers and this postulate adds no choice use of its own. No derivation of the adopted data from the underlying postulates, and no empirical validation of the model, is claimed or implied by this item.