Forward angular-spectrum beam

Fix a named positively oriented inertial laboratory chart with coordinates $(x,y,z,t)$, spatial domain ${\mathbb{R}}^3$, and time domain $\mathbb{R}$. Adopt SI units, a vacuum speed $c>0$, an angular frequency $\omega >0$, and $k_0:=\omega \mathrm{/}c$ (so $[k_0]={\mathrm{m}}^{-1}$). Let ${\mathbf{x}}_{\perp }:=(x,y)\in {\mathbb{R}}^2$. Use the unitary two-dimensional Fourier transform ${\mathcal{F}}_2$ whose Schwartz-space convention is $\widehat{f}(\mathit{\xi })={\int }_{{\mathbb{R}}^2}f({\mathbf{x}}_{\perp })e^{-2\pi i{\mathbf{x}}_{\perp }\cdot \mathit{\xi }}{d}^2{\mathbf{x}}_{\perp },$ and whose inverse is its unitary $L^2$ extension. The variable $\mathit{\xi }$ is in cycles per metre; its associated angular spatial wavevector is $\mathbf{q}:=2\pi \mathit{\xi }$ in ${\mathrm{m}}^{-1}$, with $\mathrm{\mid }\mathbf{q}{\mathrm{\mid }}^2:=q_x^2+q_y^2$.

Choose $0<\delta <1$ and $u_0\in L^2({\mathbb{R}}^2;\mathbb{C})$ with ${\mathcal{F}}_2{u}_0$ supported, up to null sets, in $D_{\delta }:=\{\mathit{\xi }:\mathrm{\mid }2\pi \mathit{\xi }\mathrm{\mid }\le \delta k_0\}$. For $\mathit{\xi }\in D_{\delta }$ put $k_z(\mathit{\xi }):=\sqrt{k_0^2-\mathrm{\mid }2\pi \mathit{\xi }{\mathrm{\mid }}^2},$ which is real and strictly positive there. For each $z\in \mathbb{R}$, define the exact multiplier on all of ${\mathbb{R}}^2$ by $m_{\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}}(z,\mathit{\xi }):=\{\begin{array}{ll}{e}^{iz{k}_z(\mathit{\xi })},& \mathit{\xi }\in D_{\delta },\\ 0,& \mathit{\xi }\mathrm{\notin }{D}_{\delta },\end{array}$ and define the paraxial multiplier on all of ${\mathbb{R}}^2$ by $m_{\mathrm{p}\mathrm{a}\mathrm{r}}(z,\mathit{\xi }):=e^{i{k}_{0}z}{e}^{-iz\mathrm{\mid }2\pi \mathit{\xi }{\mathrm{\mid }}^2\mathrm{/}(2k_0)}\mathrm{.}$ The exact scalar and paraxial scalar propagators are the $L^2$ equivalence classes $u_{\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}}(z):={\mathcal{F}}_2^{-1}\text{\hspace{0.17em}⁣}[m_{\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}}(z,\mathit{\xi }){\mathcal{F}}_2{u}_0(\mathit{\xi })],u_{\mathrm{p}\mathrm{a}\mathrm{r}}(z):={\mathcal{F}}_2^{-1}\text{\hspace{0.17em}⁣}[m_{\mathrm{p}\mathrm{a}\mathrm{r}}(z,\mathit{\xi }){\mathcal{F}}_2{u}_0(\mathit{\xi })]\mathrm{.}$ The associated real scalar fields are the real parts of these phasors multiplied by $e^{-i\omega t}$. If $u_0$ carries $\mathrm{V}{\mathrm{m}}^{-1}$, both propagated fields carry the same unit. “Exact” here names the exact positive axial dispersion multiplier inside this scalar model; it does not assert exact Maxwell validity. Forward propagation is $z\ge 0$; the formulas also define the mathematical continuation to negative $z$.

At a fixed $\mathit{\xi }\in D_{\delta }$, write $\mathbf{K}(\mathit{\xi }):=(2\pi {\xi }_x,2\pi {\xi }_y,k_z(\mathit{\xi }))$. For complex amplitudes $\mathbf{e},\mathbf{b}\in {\mathbb{C}}^3$ at frequency $\omega$, the real-field encoding assigns the maps $\mathbf{E},\mathbf{B}:{\mathbb{R}}^3\times \mathbb{R}\to {\mathbb{R}}^3$ by $\mathbf{E}(\mathbf{x},t):=\mathrm{Re}\text{\hspace{0.17em}⁣}\left(\mathbf{e}{e}^{i(\mathbf{K}\cdot \mathbf{x}-\omega t)}\right),\mathbf{B}(\mathbf{x},t):=\mathrm{Re}\text{\hspace{0.17em}⁣}\left(\mathbf{b}{e}^{i(\mathbf{K}\cdot \mathbf{x}-\omega t)}\right)\mathrm{.}$ In the SI field interpretation, $\mathbf{e}$ has units $\mathrm{V}{\mathrm{m}}^{-1}$ and $\mathbf{b}$ has units tesla. The candidate is an exact source-free vacuum plane-wave mode precisely when these real fields are jointly $C^1$ and satisfy, pointwise on ${\mathbb{R}}^3\times \mathbb{R}$, $\mathrm{div}\mathbf{E}=0,\mathrm{div}\mathbf{B}=0,\mathrm{curl}\mathbf{E}=-{\mathrm{\partial }}_t\mathbf{B},\mathrm{curl}\mathbf{B}=c^{-2}{\mathrm{\partial }}_t\mathbf{E},$ the zero-source specialization of the adopted smooth vacuum Maxwell branch. The longitudinal electric coefficient is $e_z$, and ${\mathbf{e}}_{\perp }:=(e_x,e_y)$ has norm $\mathrm{\mid }{\mathbf{e}}_{\perp }{\mathrm{\mid }}^2:=\mathrm{\mid }{e}_x{\mathrm{\mid }}^2+\mathrm{\mid }{e}_y{\mathrm{\mid }}^2$. No scalar/vector polarization factorization is part of this definition.

Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields

Framework-local SI units, the vacuum constants and the exact/measured status of their values

Fourier inversion on Schwartz space

Plancherel theorem

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

The complex exponential by its power series

Real and imaginary parts, complex conjugation, and modulus

The adopted scalar aperture-diffraction model

Classical transverse Fourier amplitudes

The microscopic vacuum Maxwell equations: smooth and weak branches

Complex plane-wave fields: real parts of complex exponentials

The reals are complete

Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$

Squaring is monotone on the nonnegatives

$\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$

Uniform phase-multiplier bound for the square-root dispersion law

The following normalized cross-section uses $Q=q_x\mathrm{/}{k}_0$ and $K_z=k_z\mathrm{/}{k}_0$, sets $q_y=0$, and sketches the illustrative value $\delta =1\mathrm{/}2$. The exact admissible region in the Definition is the disk $\mathrm{\mid }q\mathrm{\mid }\le \delta k_0$; the diagram is only a cross-section and is not used in any proof.

\begin{tikzpicture}[x=2.2cm,y=1.7cm]
  \draw[->] (-1.15,0) -- (1.2,0) node[right] {$Q$};
  \draw[->] (0,0) -- (0,1.2) node[above] {$K_z$};
  \draw[thick,domain=-1:1,samples=80]
    plot (\x,{sqrt(1-\x*\x)});
  \draw[dashed,thick,domain=-0.5:0.5,samples=40]
    plot (\x,{1-\x*\x/2});
  \draw[dotted] (-0.5,0) -- (-0.5,1.08);
  \draw[dotted] (0.5,0) -- (0.5,1.08);
  \node[above right] at (0.56,0.84) {$1-Q^2/2$};
  \node[above left] at (-0.88,0.47) {$\sqrt{1-Q^2}$};
  \node[below] at (-0.5,0) {$-\delta$};
  \node[below] at (0.5,0) {$\delta$};
\end{tikzpicture}