Paraxial phase and longitudinal-field bounds

Given: $k_0>0$, $0<\delta <1$, $z\in \mathbb{R}$, scalar data $u_0$ supported in the angular cone, and an exact source-free monochromatic vacuum plane-wave mode at $\mathrm{\mid }\mathbf{q}\mathrm{\mid }\le \delta k_0$.

The exact and paraxial fields are the $L^2$ Fourier multipliers $e^{iz{k}_z(\mathit{\xi })}$ and $e^{i{k}_{0}z}{e}^{-iz\mathrm{\mid }2\pi \mathit{\xi }{\mathrm{\mid }}^2\mathrm{/}(2k_0)}$ on the support disk; the ${\mathcal{F}}_2$ convention is $e^{-2\pi ix\cdot \xi }$. Forward angular-spectrum beam

Under countable choice, ${\mathcal{F}}_2$ is unitary on complex $L^2$ and preserves its norm. Plancherel theorem The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

For $0\le s\le {\delta }^2$, the difference between phases $z{k}_0\sqrt{1-s}$ and $z{k}_0(1-s\mathrm{/}2)$ has multiplier modulus at most $\mathrm{\mid }z\mathrm{\mid }{k}_0{s}^2\mathrm{/}[8(1-{\delta }^2{)}^{3\mathrm{/}2}]$. Uniform phase-multiplier bound for the square-root dispersion law

A real plane wave is $E(\mathbf{x},t)=\mathrm{Re}(\mathbf{e}{e}^{i(\mathbf{K}\cdot \mathbf{x}-\omega t)})$; the exponential is differentiable and its phase derivative follows the chain rule. Complex plane-wave fields: real parts of complex exponentials The complex exponential is entire and its complex derivative is itself The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$ Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$

In the smooth source-free vacuum branch, the adopted Maxwell equations include $\mathrm{div}E=0$. The microscopic vacuum Maxwell equations: smooth and weak branches

The Euler formula is $e^{i\theta }=\cos \theta +i\sin \theta$. The quarter-turn theorem gives $\sin (\pi \mathrm{/}2)=1$, $\cos (\pi \mathrm{/}2)=0$, and the shifts $\sin (x+\pi )=-\sin x$, $\cos (x+\pi )=-\cos x$. Taking $x=-\pi \mathrm{/}2$ in the shift identities yields $\sin (-\pi \mathrm{/}2)=-1$ and $\cos (-\pi \mathrm{/}2)=0$, hence $e^{-i\pi \mathrm{/}2}=-i$. $\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$ Quarter-turn values and shifts by pi/2 and pi

Complex modulus is the Euclidean modulus of real and imaginary parts, and the real transverse Euclidean norm is induced by the dot product; extend a real dot product bilinearly when its second argument is complex. Order of squares reflects order for nonnegative real numbers. For real $r>0$ and $w=a+ib$, $\mathrm{\mid }w\mathrm{/}r\mathrm{\mid }=\sqrt{(a\mathrm{/}r{)}^2+(b\mathrm{/}r{)}^2}=\mathrm{\mid }w\mathrm{\mid }\mathrm{/}r$. Real and imaginary parts, complex conjugation, and modulus The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ Squaring is monotone on the nonnegatives

In the fixed SI chart, $z$ is measured in metres, $k_0=\omega \mathrm{/}c$ in ${\mathrm{m}}^{-1}$, and $\delta$ is dimensionless. Forward angular-spectrum beam Framework-local SI units, the vacuum constants and the exact/measured status of their values

The root $k_z=\sqrt{k_0^2-\mathrm{\mid }\mathbf{q}{\mathrm{\mid }}^2}$ is nonnegative, and $\sqrt{k_0^2(1-{\delta }^2)}=k_0\sqrt{1-{\delta }^2}$ is nonnegative. Since $\mathrm{\mid }\mathbf{q}\mathrm{\mid }\le \delta k_0$, their squares satisfy $k_z^2\ge k_0^2(1-{\delta }^2)$; monotonicity of squaring on nonnegative reals therefore gives $k_z\ge k_0\sqrt{1-{\delta }^2}>0$. Forward angular-spectrum beam 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