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Dispersion, transversality and the two polarization conditions for plane-wave solutions

Physical scope

Conditional classification of ideal whole-space plane-wave solutions of the smooth source-free vacuum Maxwell branch: amplitude equations, dispersion relation, transversality, orthogonality and the magnitude relation $∣ B ∣ = ∣ E ∣ / c$ , in a named inertial chart with SI constants. Excluded: non-plane-wave fields, sources, boundaries, finite total energy, material media and any empirical claim.

Statement

Let $k \in R^{3} ∖ {0}$ , $ω > 0$ and $e, b \in C^{3}$ , and let $(E, B)$ be the plane-wave fields of Real plane-wave fields encoded by complex amplitudes on the whole-space region $Ω = R^{3}$ , $I = R$ , with the terminology and phase convention of that definition, the cross product and inner product of The cross product in $R^{3}$ and The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ extended to $C^{3}$ componentwise, the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values and $\hat{k} = k / ∣ k ∣$ . 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 $ρ \equiv 0$ and $J \equiv 0$ on $R^{3} \times R$ if and only if

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

(b) [Dispersion, nonzero waves, two polarizations] The amplitude equations admit a nonzero solution $(e, b) \neq (0, 0)$ if and only if $∣ k ∣^{2} = ω^{2} / c^{2}$ ; 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 $∣ k ∣ = ω / c$ , the solutions are exactly the pairs $(e, b)$ with $e \in {v \in C^{3} : k \cdot v = 0}$ arbitrary and $b = k \times e / ω$ : a complex two-dimensional family; $e = 0$ gives the zero wave, and for every nonzero solution $e \neq 0$ and $b \neq 0$ .

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

(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.

Facts & Assumptions

Given: The wavevector $k \neq 0$ , the angular frequency $ω > 0$ , the complex amplitudes $e, b \in C^{3}$ , the plane-wave fields $(E, B)$ of Real plane-wave fields encoded by complex amplitudes on $Ω = R^{3}$ , $I = R$ with phase $θ (x, t) = k \cdot x - ω t$ , and the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values with $ε_{0} = (μ_{0} c^{2})^{- 1}$ .

[F1]

The fields are given by $E (x, t) = Re (e \exp (i θ (x, t)))$ and $B (x, t) = Re (b \exp (i θ (x, t)))$ componentwise; they are of class $C^{\infty}$ on $R^{3} \times R$ , and $(E, B)$ is the identically zero pair exactly when $e = b = 0$ (Real plane-wave fields encoded by complex amplitudes).

[F2]

The smooth branch of the vacuum Maxwell postulate holds for jointly $C^{1}$ fields with the additional derivatives required by each equation and continuous sources; on the whole-space region $Ω = R^{3}$ , $I = R$ with the identically zero source fields $ρ = 0$ , $J = 0$ it consists of the four equations $div E = 0$ , $div B = 0$ , $curl E = - \partial_{t} B$ , $curl B = c^{- 2} \partial_{t} E$ (The microscopic vacuum Maxwell equations: smooth and weak branches, Framework-local SI units, the vacuum constants and the exact/measured status of their values).

[F3]

On $C^{3}$ the inner product and cross product are the componentwise extensions of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and The cross product in $R^{3}$ : $u \cdot v = \sum_{j} u_{j} v_{j}$ and $u \times v$ given by the coordinate differences of products. The identities of The cross product is bilinear, alternating, and orthogonal to both factors are polynomial identities with integer coefficients and therefore hold for complex entries as well: the cross product is bilinear and alternating, $⟨ u \times v, w ⟩ = \det [u v w]$ , and $u \times v$ is orthogonal to $u$ and to $v$ . Moreover $∣ u \times v ∣^{2} = ∣ u ∣^{2} ∣ v ∣^{2} - (u \cdot v)^{2}$ for real $u, v$ (The squared cross-product norm is the Gram determinant of two vectors).

[F4]

The operators: divergence and curl of a $C^{1}$ field are those of Divergence and curl of a $C^{1}$ vector field; partial derivatives are those of Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ; the Laplacian is that of The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field. The complex exponential is entire with derivative itself (The complex exponential is entire and its complex derivative is itself), the chain rule for total derivatives applies (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ), complex differentiability is equivalent to real total differentiability with a complex-linear derivative (Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with $\partial_{\overset{ˉ}{z}} f = 0$ , or with the Cauchy–Riemann equations), and the real part, imaginary part and conjugation are those of Real and imaginary parts, complex conjugation, and modulus, so that $Re$ is $R$ -linear and every $W \in C^{3}$ has a unique decomposition $W = A + i B$ with $A, B \in R^{3}$ .

[F5]

The source-free wave-equation theorem Source-free vacuum Maxwell fields satisfy the wave equation holds for jointly $C^{2}$ fields on a source-free open region of the smooth branch, with $Δ E = c^{- 2} \partial_{t}^{2} E$ and $Δ B = c^{- 2} \partial_{t}^{2} B$ componentwise.

[F6]

Complex field arithmetic and Euler's formula: $C$ is a field, so multiplication is distributive and $i^{2} = - 1$ ( $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ ); for every real $θ$ , $\exp (i θ) = \cos θ + i \sin θ$ (Euler's formula: $\exp (i θ) = \cos θ + i \sin θ$ for every real $θ$ ); and $\cos (π / 2) = 0$ , $\sin (π / 2) = 1$ , $\cos π = - 1$ , $\sin π = 0$ (Quarter-turn values and shifts by pi/2 and pi).

Proof

1.1F1F3F4givenalgebra

For the fields of [F1] with phase $θ (x, t) = k \cdot x - ω t$ , differentiation through the real part gives componentwise $\partial_{x_{j}} E = Re (i k_{j} e \exp (i θ))$ and $\partial_{t} E = Re (- i ω e \exp (i θ))$ , hence $div E = Re (i (k \cdot e) \exp (i θ))$ and $curl E = Re (i (k \times e) \exp (i θ))$ ; replacing $e$ by $b$ gives the same four displays with $B$ and $b$ . Indeed each component of $(x, t) \mapsto e_{j} \exp (i θ (x, t))$ is a composition of the entire exponential with the real-linear maps $(x, t) \mapsto i θ (x, t)$ and multiplication by the constant $e_{j}$ , so the chain rule [F4] gives $\partial_{x_{j}} (e_{j} \exp (i θ)) = e_{j} \exp (i θ) i k_{j}$ and $\partial_{t} (e_{j} \exp (i θ)) = e_{j} \exp (i θ) (- i ω)$ ; the derivative of the real part equals the real part of the derivative because $Re$ is $R$ -linear [F4], and summing coordinates with the coordinate formulas for divergence and curl [F4] and the cross-product formula [F3] gives the displays.

1.2F3F4F6givenalgebra

Let $W \in C^{3}$ satisfy $Re (W \exp (i θ)) = 0$ for every real $θ$ . Writing $W = A + i B$ with $A, B \in R^{3}$ [F4] and using Euler's formula and the field arithmetic [F6] gives $W \exp (i θ) = (A \cos θ - B \sin θ) + i (A \sin θ + B \cos θ)$ , so $Re (W \exp (i θ)) = A \cos θ - B \sin θ$ ; evaluating at $θ = π / 2$ gives $- B = 0$ and at $θ = π$ gives $- A = 0$ , using the quarter-turn values of [F6]; hence $A = B = 0$ and $W = 0$ . Moreover, for every $(x, t)$ the phase attains every real value: given a real $θ$ and a time $t$ , choose a coordinate $j$ with $k_{j} \neq 0$ and let $x$ have $x_{j} = (θ + ω t) / k_{j}$ and $x_{i} = 0$ for $i \neq j$ ; then $k \cdot x = θ + ω t$ by the coordinate form of the inner product [F3], so $θ (x, t) = θ$ . Consequently a relation $Re (W \exp (i θ (x, t))) = 0$ valid for all $(x, t) \in R^{3} \times R$ forces $W = 0$ .

2.1step 1.1step 1.2F2algebra

By [F2] and step 1.1, the four equations $div E = 0$ , $div B = 0$ , $curl E = - \partial_{t} B$ , $curl B = c^{- 2} \partial_{t} E$ hold on $R^{3} \times R$ exactly when $Re (i (k \cdot e) \exp (i θ)) = 0$ , $Re (i (k \cdot b) \exp (i θ)) = 0$ , $Re (i (k \times e - ω b) \exp (i θ)) = 0$ and $Re (i (k \times b + (ω / c^{2}) e) \exp (i θ)) = 0$ for all $(x, t)$ ; by step 1.2 each of these is equivalent to the vanishing of the complex vector it multiplies, that is to $k \cdot e = 0$ , $k \cdot b = 0$ , $k \times e = ω b$ and $k \times b = - (ω / c^{2}) e$ . This proves the equivalence of (a).

3.1step 2.1F3algebra

Assume $(e, b)$ satisfies the four amplitude equations of step 2.1 and $(e, b) \neq (0, 0)$ . If $e = 0$ , then $k \times e = ω b$ gives $b = 0$ , a contradiction; hence $e \neq 0$ . Since $k \cdot e = 0$ , the triple-product expansion $k \times (k \times e) = k (k \cdot e) - ∣ k ∣^{2} e = - ∣ k ∣^{2} e$ (the coordinate identity of [F3]) shows that $k \times e = 0$ would force $∣ k ∣^{2} e = 0$ and hence $e = 0$ ; therefore $k \times e \neq 0$ and $b = k \times e / ω \neq 0$ . Crossing $k \times b = - (ω / c^{2}) e$ with $k$ gives $k \times (k \times b) = - (ω / c^{2}) (k \times e) = - (ω^{2} / c^{2}) b$ , while the same triple-product identity and $k \cdot b = 0$ give $k \times (k \times b) = k (k \cdot b) - ∣ k ∣^{2} b = - ∣ k ∣^{2} b$ ; hence $(∣ k ∣^{2} - ω^{2} / c^{2}) b = 0$ and, since $b \neq 0$ and $∣ k ∣^{2}, c^{2}$ are real with $c > 0$ , $∣ k ∣^{2} = ω^{2} / c^{2}$ . Equivalently, if $∣ k ∣^{2} \neq ω^{2} / c^{2}$ then the only solution is $e = b = 0$ , because any nonzero solution would have $∣ k ∣^{2} = ω^{2} / c^{2}$ by the argument just given.

3.2step 2.1F3algebra

Conversely assume $∣ k ∣^{2} = ω^{2} / c^{2}$ . For any $e \in C^{3}$ with $k \cdot e = 0$ , put $b : = k \times e / ω$ ; then $k \cdot b = (k \cdot (k \times e)) / ω = 0$ because $⟨ k \times e, k ⟩ = \det [k e k] = 0$ by alternation [F3], and $k \times b = (k (k \cdot e) - ∣ k ∣^{2} e) / ω = - (ω^{2} / c^{2}) e / ω = - (ω / c^{2}) e$ , so $(e, b)$ satisfies the amplitude equations; by step 2.1 the corresponding plane wave lies in the source-free smooth Maxwell branch. If $e = 0$ then $b = k \times e / ω = 0$ , the zero wave; if $e \neq 0$ then $k \times e \neq 0$ , since $k \times e = 0$ together with $k \cdot e = 0$ would give $k \times (k \times e) = k (k \cdot e) - ∣ k ∣^{2} e = - ∣ k ∣^{2} e = 0$ and hence $e = 0$ , so $b \neq 0$ . Conversely, every solution with $∣ k ∣ = ω / c$ satisfies $k \cdot e = 0$ and $b = k \times e / ω$ by the amplitude equations, so on the dispersion relation the solutions are exactly the pairs $(e, k \times e / ω)$ with $k \cdot e = 0$ ; since $k \neq 0$ , the condition $k \cdot e = 0$ is one nontrivial complex-linear equation in $C^{3}$ , whose solution space is a complex two-dimensional subspace.

3.3step 2.1F1F5

By step 2.1 the pair $(E, B)$ satisfies the four equations of the source-free smooth branch on $R^{3} \times R$ , and the fields are $C^{\infty}$ by [F1]; the source-free wave-equation theorem [F5] applied on the source-free region $W = R^{3} \times R$ therefore gives $Δ E = c^{- 2} \partial_{t}^{2} E$ and $Δ B = c^{- 2} \partial_{t}^{2} B$ componentwise on $R^{3} \times R$ .

4.1step 2.1step 3.2F3F4algebra

For a solution with $∣ k ∣ = ω / c$ and any $(x, t)$ , the equalities $k \cdot e = 0$ and $k \cdot b = 0$ of step 2.1 give $k \cdot E (x, t) = \sum_{j} k_{j} Re (e_{j} \exp (i θ)) = Re ((k \cdot e) \exp (i θ)) = 0$ and likewise $k \cdot B (x, t) = 0$ , using the $R$ -linearity of $Re$ and the coordinate form of the inner product [F3, F4].

4.2step 3.2F1F3algebra

For a solution with $∣ k ∣ = ω / c$ one has $b = k \times e / ω = (∣ k ∣ / ω) (\hat{k} \times e) = \hat{k} \times e / c$ , so componentwise taking real parts and using that $\hat{k} \times \cdot$ is a real-linear combination of coordinates gives $B (x, t) = Re (b \exp (i θ)) = \hat{k} \times Re (e \exp (i θ)) / c = \hat{k} \times E (x, t) / c$ . Consequently $E (x, t) \cdot B (x, t) = E \cdot (\hat{k} \times E) / c = \det [E \hat{k} E] / c = 0$ for every $(x, t)$ , by the determinant identity and alternation of [F3], the last determinant having two equal columns.

5.1step 4.1step 4.2F3algebra

For a solution with $∣ k ∣ = ω / c$ , step 4.2 gives $∣ B (x, t) ∣^{2} = ∣ \hat{k} \times E (x, t) ∣^{2} / c^{2}$ , and the Gram determinant identity of [F3] together with $\hat{k} \cdot E = 0$ from step 4.1 and $∣ \hat{k} ∣^{2} = ⟨ k, k ⟩ / ∣ k ∣^{2} = 1$ gives $∣ \hat{k} \times E ∣^{2} = ∣ \hat{k} ∣^{2} ∣ E ∣^{2} - (\hat{k} \cdot E)^{2} = ∣ E ∣^{2}$ ; hence $∣ B (x, t) ∣^{2} = ∣ E (x, t) ∣^{2} / c^{2}$ and, both sides being nonnegative and $c > 0$ , $∣ B (x, t) ∣ = ∣ E (x, t) ∣ / c$ .

5.2step 3.1step 3.2step 4.1step 4.2given

The equivalence in (a) is between the amplitude equations and membership in the source-free smooth Maxwell branch, and it asserts nothing about fields that solve the wave equations but not the amplitude equations; [F5] records the one-directional character of the wave-equation implication for general fields. 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.

6.1step 2.1step 3.1step 3.2step 3.3step 4.1step 4.2step 5.1step 5.2given∎

Steps 1.1 and 1.2 supply the derivative formulas and the phase-injectivity fact; step 2.1 proves the amplitude-equation equivalence (a); steps 3.1 and 3.2 prove the dispersion alternative, the nonzero-solution criterion and the complex two-dimensional solution family (b); step 3.3 proves the wave-equation clause and steps 4.1, 4.2 and 5.1 prove transversality, orthogonality, the relation $B = \hat{k} \times E / c$ and the magnitude relation (c); step 5.2 records the scope (d).

Source notes

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 $W \exp (i θ)$ for all phases determine $W$ , and the products of amplitudes are never decomposed as if they were additive.

Dispersion, transversality and the two polarization conditions for plane-wave solutions

Statement

Facts & Assumptions

Proof

Source notes

Depends on

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