Vacuum waves and polarization

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 $e,b\in {\mathbb{C}}^3$.

Assume the smooth branch of the vacuum Maxwell postulate The microscopic vacuum Maxwell equations: smooth and weak branches on the laboratory region $U\cong I\times \mathrm{\Omega }$ of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, with the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values and ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$, the divergence and curl of Divergence and curl of a $C^1$ vector field, the partial derivatives of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ and the Laplacian $\mathrm{\Delta }$ of The Laplacian of a $C^2$ function and of a $C^2$ vector field. Let $E,B:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ be jointly $C^2$ fields of The electric and magnetic fields as fixed-time spatial vector fields, let $(\rho ,J)$ be the source fields of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current, and let $W\subseteq \mathrm{\Omega }\times I$ be open with $\rho =0$ and $J=0$ on $W$ (a source-free region). Then on $W$, componentwise,

$\mathrm{\Delta }E=c^{-2}{\mathrm{\partial }}_t^{2}E,\mathrm{\Delta }B=c^{-2}{\mathrm{\partial }}_t^{2}B;$

equivalently each of the six Cartesian components of $E$ and of $B$ satisfies the scalar wave equation $c^{-2}{\mathrm{\partial }}_t^{2}v=\mathrm{\Delta }v$ of speed $c$ on $W$.

(a) [Hypotheses actually used] The derivation differentiates the Faraday and Ampère equations in time, commutes the time derivative with $\mathrm{curl}$ (Clairaut's theorem, available for jointly $C^2$ fields), uses the curl-curl identity of The curl of a curl is the gradient of the divergence minus the Laplacian, and uses the two Gauss equations $\mathrm{div}E=0$, $\mathrm{div}B=0$ on $W$. The two evolution equations alone are not sufficient for the conclusion without the constraint equations.

(b) [One-directional] No converse is asserted. The two vector wave equations do not imply the four source-free Maxwell equations: the constraints $\mathrm{div}E=\mathrm{div}B=0$ and the two curl relations are not recovered, and solutions of the six uncoupled scalar wave equations need not lie in a source-free smooth Maxwell branch. The correctly constrained initial-value problem is a separate result on this page.

(c) [Scope] Only the smooth branch is covered. No decay, boundedness, compact support, finite-energy or boundary hypothesis is imposed or needed; the conclusion is local on $W$, asserts no value of any field or derivative at a point of singular source support, and does not cover the weak branch of The microscopic vacuum Maxwell equations: smooth and weak branches. The result is conditional on the adopted postulates and makes no empirical claim.

The cited treatments derive the source-free vacuum wave equation by exactly this route: differentiate the two evolution equations, commute derivatives, apply the curl-curl identity and use $\mathrm{div}E=\mathrm{div}B=0$ in the source-free region. The textbook statements are the motivation for the claim; the deduction above uses only the imported divergence, curl, Laplacian, Clairaut and curl-curl interfaces together with the adopted smooth-branch postulate. The one-directional warning is the standard observation that solutions of the wave equation need not satisfy the constraints.

Let $k\in {\mathbb{R}}^3\setminus \{0\}$, $\omega >0$ and $e,b\in {\mathbb{C}}^3$, and let $(E,B)$ be the plane-wave fields of Real plane-wave fields encoded by complex amplitudes on the whole-space region $\mathrm{\Omega }={\mathbb{R}}^3$, $I=\mathbb{R}$, with the terminology and phase convention of that definition, the cross product and inner product of The cross product in ${\mathbb{R}}^3$ and The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ extended to ${\mathbb{C}}^3$ componentwise, the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values and $\widehat{k}=k\mathrm{/}\mathrm{\mid }k\mathrm{\mid }$. Then:

(a) [Amplitude equations] $(E,B)$ satisfies the four smooth-branch vacuum Maxwell equations of The microscopic vacuum Maxwell equations: smooth and weak branches with $\rho \equiv 0$ and $J\equiv 0$ on ${\mathbb{R}}^3\times \mathbb{R}$ if and only if

$k\cdot e=0,k\cdot b=0,k\times e=\omega b,k\times b=-\frac{\omega }{c^2}e\mathrm{.}$

(b) [Dispersion, nonzero waves, two polarizations] The amplitude equations admit a nonzero solution $(e,b)\mathrm{\ne }(0,0)$ if and only if $\mathrm{\mid }k{\mathrm{\mid }}^2={\omega }^2\mathrm{/}{c}^2$; off this dispersion relation the only solution is $e=b=0$. On the dispersion relation both zero and nonzero solutions occur; a particular solution is nonzero exactly when its amplitudes are not both zero, in the sense of Real plane-wave fields encoded by complex amplitudes(c). If $\mathrm{\mid }k\mathrm{\mid }=\omega \mathrm{/}c$, the solutions are exactly the pairs $(e,b)$ with $e\in \{v\in {\mathbb{C}}^3:k\cdot v=0\}$ arbitrary and $b=k\times e\mathrm{/}\omega$: a complex two-dimensional family; $e=0$ gives the zero wave, and for every nonzero solution $e\mathrm{\ne }0$ and $b\mathrm{\ne }0$.

(c) [Transversality, orthogonality, magnitude relation and the wave equation] For every solution with $\mathrm{\mid }k\mathrm{\mid }=\omega \mathrm{/}c$ and every $(x,t)\in {\mathbb{R}}^3\times \mathbb{R}$: $k\cdot E(x,t)=k\cdot B(x,t)=0$, $E(x,t)\cdot B(x,t)=0$ and $B(x,t)=\widehat{k}\times E(x,t)\mathrm{/}c$; in particular $\mathrm{\mid }B(x,t)\mathrm{\mid }=\mathrm{\mid }E(x,t)\mathrm{\mid }\mathrm{/}c$. Each component of $E$ and of $B$ is a $C^{\mathrm{\infty }}$ solution of the wave equation $\mathrm{\Delta }v=c^{-2}{\mathrm{\partial }}_t^{2}v$ of Source-free vacuum Maxwell fields satisfy the wave equation on ${\mathbb{R}}^3\times \mathbb{R}$.

(d) [Scope] The equivalence in (a) is between the amplitude equations and membership in the source-free smooth Maxwell branch; it says nothing about fields that solve the wave equations but not the amplitude equations (see Source-free vacuum Maxwell fields satisfy the wave equation(b)). The item covers ideal whole-space plane waves only, makes no finite-energy, boundary, material or empirical claim, and is conditional on the adopted postulate.

The cited treatments obtain the amplitude equations by substituting the monochromatic ansatz, read the dispersion relation and transversality off them, and state the relation between the magnetic and electric amplitudes; they also stress that solutions of the wave equation need not satisfy the full Maxwell system. The deduction above uses the plane-wave definition, the smooth-branch postulate and the imported complex, cross-product and vector-calculus interfaces; the vanishing fact of step 1.2 is the elementary statement that the real parts of $W\exp (i\theta )$ for all phases determine $W$, and the products of amplitudes are never decomposed as if they were additive.

Let $k\in {\mathbb{R}}^3\setminus \{0\}$ and $\omega >0$, let $e,b\in {\mathbb{C}}^3$, and let $(E,B)$ be the plane-wave pair of Real plane-wave fields encoded by complex amplitudes on the whole-space region $\mathrm{\Omega }={\mathbb{R}}^3$, $I=\mathbb{R}$, with phase $\theta (x,t)=k\cdot x-\omega t$. Suppose the amplitude conditions of Dispersion, transversality and the two polarization conditions for plane-wave solutions hold, so that $k\cdot e=0$ and $b=k\times e\mathrm{/}\omega$. By that theorem, if the pair is nonzero then it also satisfies the dispersion relation $\mathrm{\mid }k\mathrm{\mid }=\omega \mathrm{/}c$, while the zero pair is permitted for every $k\mathrm{\ne }0$ and $\omega >0$, both on and off this dispersion relation. This item classifies the polarization of that class of transverse plane waves; it adds no field-theoretic content.

Write $e=a+id$ with the real vectors $a=\mathrm{Re}e$ and $d=\mathrm{Im}e$ of Real and imaginary parts, complex conjugation, and modulus, applied componentwise in ${\mathbb{C}}^3$; since $k$ is real and $k\cdot e=0$, the real and imaginary parts of this complex equation give $⟨k,a⟩=⟨k,d⟩=0$. Put $\widehat{k}=k\mathrm{/}\mathrm{\mid }k\mathrm{\mid }$, and let $⟨\cdot ,\cdot ⟩$, $\mathrm{\mid }\cdot \mathrm{\mid }$ and $\times$ be the inner product, norm and cross product of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and The cross product in ${\mathbb{R}}^3$. Then:

(a) [Trace] At every fixed $x$ one has $E(x,t)=a\cos \theta -d\sin \theta$ with $\theta =\theta (x,t)$. The curve $\gamma :\mathbb{R}\to {\mathbb{R}}^3$, $\gamma (\theta ):=a\cos \theta -d\sin \theta$, lies in the transverse plane $k^{\perp }=\{v\in {\mathbb{R}}^3:⟨k,v⟩=0\}$; it is the image of the unit circle $S^1=\{(u_1,u_2)\in {\mathbb{R}}^2:u_1^2+u_2^2=1\}$ under the $\mathbb{R}$-linear map $(u_1,u_2)\mapsto u_{1}a-u_{2}d$; it is $2\pi$-periodic in the phase parameter $\theta$ and satisfies $\gamma (\theta +\pi )=-\gamma (\theta )$, while the map $t\mapsto E(x,t)$ has time period $T=2\pi \mathrm{/}\omega$. The curve $\gamma$ is called the polarization trace at $x$; it does not depend on $x$, so the classification below is the same at every point.

(b) [Classification] Exactly one of the following three cases holds.

(i) Linear polarization: ${\dim }_{\mathbb{R}}\mathrm{span}\{a,d\}\le 1$. When $\dim \mathrm{span}\{a,d\}=1$, the trace is a nontrivial closed segment in that span, traversed twice per period; when $a=d=0$, the trace is the single point $\{0\}$ and $(E,B)$ is the zero wave.

(ii) Circular polarization: $\mathrm{\mid }a\mathrm{\mid }=\mathrm{\mid }d\mathrm{\mid }>0$ and $⟨a,d⟩=0$. The trace is the circle of radius $\mathrm{\mid }a\mathrm{\mid }$ in $k^{\perp }$, and $\mathrm{\mid }E(x,t)\mathrm{\mid }=\mathrm{\mid }a\mathrm{\mid }$ for all $(x,t)\in {\mathbb{R}}^3\times \mathbb{R}$.

(iii) Elliptical (nondegenerate, noncircular) polarization: $\dim \mathrm{span}\{a,d\}=2$ and not (ii). Here a nondegenerate ellipse centred at $0$ in a real plane $P$ means the set $L(S^1)$ for a real-linear isomorphism $L:{\mathbb{R}}^2\to P$. The trace is such an ellipse in $\mathrm{span}\{a,d\}$, and the origin lies in the relative interior of the enclosed region $L(\{u\in {\mathbb{R}}^2:\mathrm{\mid }u\mathrm{\mid }\le 1\})$ in that plane. If in addition $⟨a,d⟩=0$, the parametrization is along principal axes and the semiaxis lengths are $\mathrm{\mid }a\mathrm{\mid }$ and $\mathrm{\mid }d\mathrm{\mid }$; the general principal-axis computation is not asserted by this item.