Real plane-wave fields encoded by complex amplitudes

Fix a wavevector $k\in {\mathbb{R}}^3\setminus \{0\}$ and a positive angular frequency $\omega >0$; here $k$ carries the unit ${\mathrm{m}}^{-1}$ and $\omega$ the unit ${\mathrm{s}}^{-1}$ in the SI conventions of Framework-local SI units, the vacuum constants and the exact/measured status of their values. Fix complex amplitudes $e,b\in {\mathbb{C}}^3$, where $e$ carries the unit $\mathrm{V}{\mathrm{m}}^{-1}$ and $b$ the unit T of the electric and magnetic fields of The electric and magnetic fields as fixed-time spatial vector fields. Let $k\cdot x$ denote the Euclidean inner product of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, and let $\exp$ and $\mathrm{Re}$ be those of the encoding definition Complex plane-wave fields: real parts of complex exponentials, whose convention that physical fields are the real parts of complex expressions is used throughout. Define, for each $t\in \mathbb{R}$, maps $E(\cdot ,t),B(\cdot ,t):{\mathbb{R}}^3\to {\mathbb{R}}^3$ by

$E(x,t):=\mathrm{Re}(e\exp (i(k\cdot x-\omega t))),B(x,t):=\mathrm{Re}(b\exp (i(k\cdot x-\omega t))),$

componentwise, and put $\mathrm{\mid }k\mathrm{\mid }:=\sqrt{⟨k,k⟩}$, $\widehat{k}:=k\mathrm{/}\mathrm{\mid }k\mathrm{\mid }$ and $T:=2\pi \mathrm{/}\omega$. Terminology: $k$ is the wavevector, $\widehat{k}$ the propagation direction, $\mathrm{\mid }k\mathrm{\mid }$ the wavenumber, $\omega$ the angular frequency, $T$ the period, $e,b$ the complex amplitudes, and $\theta (x,t):=k\cdot x-\omega t$ the phase. Then:

(a) [Objects, domain and regularity] Both maps are of class $C^{\mathrm{\infty }}$ on ${\mathbb{R}}^3\times \mathbb{R}$, as verified below; on the whole-space laboratory region $\mathrm{\Omega }={\mathbb{R}}^3$, $I=\mathbb{R}$ of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields they are an electric and a magnetic field of The electric and magnetic fields as fixed-time spatial vector fields of every finite class $C^k$, $k\ge 1$, read in the frame basis $(e_1,e_2,e_3)$.

(b) [Phase and periodicity] At fixed $x$ the map $t\mapsto \theta (x,t)$ is affine with derivative $-\omega <0$, so the phase decreases in time (the $e^{-i\omega t}$ convention); $E$ and $B$ depend on $(x,t)$ only through $\theta$ and are $T$-periodic in $t$.

(c) [Zero and nonzero waves] $(E,B)$ is the identically zero pair on ${\mathbb{R}}^3\times \mathbb{R}$ if and only if $e=b=0$; call the wave nonzero otherwise. The electric field alone determines $e$ and the magnetic field alone determines $b$, because $\mathrm{Re}(w{e}^{i\theta })=0$ for all real $\theta$ forces $w=0$ (evaluate at $\theta =0$ and at $\theta =\pi \mathrm{/}2$).

(d) [Status] This item fixes objects, conventions and terminology only. It asserts no field equation, no dispersion relation, no transversality, no energy claim and no polarization classification; those are the content of Dispersion, transversality and the two polarization conditions for plane-wave solutions and Polarization of a transverse plane wave: linear, circular and elliptical traces, and handedness. The units and the propagation constant $c$ are those of Framework-local SI units, the vacuum constants and the exact/measured status of their values, and no empirical realization or measurement is claimed.

Well-definedness and conventions

For every $(x,t)\in {\mathbb{R}}^3\times \mathbb{R}$ the phase $\theta (x,t)$ is a real number, so $i\theta (x,t)$ is a complex number and $\exp (i\theta (x,t))$ is defined by the everywhere-convergent exponential series of Complex plane-wave fields: real parts of complex exponentials; multiplying by the complex number $e_j$ or $b_j$ and taking real parts is likewise defined, so each $E_j$ and $B_j$ is a well-defined real-valued function. Since $\omega >0$ and $k\mathrm{\ne }0$, $T=2\pi \mathrm{/}\omega >0$ and $\mathrm{\mid }k\mathrm{\mid }>0$, so $\widehat{k}$ is defined. The joint map $(x,t)\mapsto (E(x,t),B(x,t))$ has the open set ${\mathbb{R}}^3\times \mathbb{R}$ as its domain. For regularity, let $q=(k_1,k_2,k_3,-\omega )$ and $y=(x_1,x_2,x_3,t)$. The exponential has derivative itself (The complex exponential is entire and its complex derivative is itself), its complex derivative is also its real total derivative (Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with ${\mathrm{\partial }}_{\bar{z}}f=0$, or with the Cauchy–Riemann equations), and the real-linear map $\mathrm{Re}$ commutes with differentiation by The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$. Thus for every coordinate word $(j_1,\dots ,j_m)$ and component $\mathrm{\ell }$, ${\mathrm{\partial }}_{y_{j_m}}\cdots {\mathrm{\partial }}_{y_{j_1}}{E}_{\mathrm{\ell }}=\mathrm{Re}(e_{\mathrm{\ell }}{\prod }_{r=1}^m(i{q}_{j_r})\exp (iq\cdot y))$, by induction using the same chain rule; the identical formula holds with $b_{\mathrm{\ell }}$ for $B_{\mathrm{\ell }}$. These expressions are continuous for every finite word, proving the $C^{\mathrm{\infty }}$ assertion in (a). The equivalence in (c) is the elementary injectivity of $w\mapsto \mathrm{Re}(w{e}^{i\cdot })$ on ${\mathbb{C}}^3$: if $\mathrm{Re}(w{e}^{i\theta })=0$ for every real $\theta$, then taking $\theta =0$ gives $\mathrm{Re}w=0$ and taking $\theta =\pi \mathrm{/}2$ gives $\mathrm{Re}(iw)=-\mathrm{Im}w=0$, so $w=0$, and the converse is immediate. The complex notation is an encoding of real fields and of real linear equations, not a complex physical field; the symbols $e$ and $b$ are constant complex amplitudes, not fields. No electromagnetic equation, no matter model and no physical interpretation is introduced by this definition; in particular the complex amplitudes need not satisfy the Maxwell amplitude equations, and the plane-wave expressions above are defined for every $e,b\in {\mathbb{C}}^3$.