ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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An oblique plane wave with a nonzero longitudinal electric component

Physical scope

Exact plane-wave example in the smooth, source-free vacuum Maxwell model and the propagating cone of the ideal angular-spectrum model. It illustrates the longitudinal component required by Maxwell's equations for its specified transverse polarization; it asserts no measured field, focal apparatus, or torque.

Example

Let $0 < κ \leq δ k_{0}$ , $q = (κ, 0)$ , $k_{z} : = \sqrt{k_{0}^{2} - κ^{2}}$ , and choose a real $E_{0} \neq 0$ in $V m^{- 1}$ . Put $e : = (E_{0}, 0, - κ E_{0} / k_{z}), b : = (0, \frac{k_{0}^{2} E_{0}}{ω k_{z}}, 0), ϕ (x, t) : = κ x + k_{z} z - ω t .$ The real fields $E (x, t) : = Re (e e^{i ϕ (x, t)})$ and $B (x, t) : = Re (b e^{i ϕ (x, t)})$ are an exact, forward, source-free vacuum plane-wave mode. In particular, $e_{z} = - \frac{κ E_{0}}{\sqrt{k_{0}^{2} - κ^{2}}} \neq 0, \frac{∣ e_{z} ∣}{∣ E_{0} ∣} = \frac{κ}{\sqrt{k_{0}^{2} - κ^{2}}} \leq \frac{δ}{\sqrt{1 - δ^{2}}} .$ Thus setting $e_{z} = 0$ while keeping this transverse amplitude would not give an exact Maxwell mode.

Example references

Forward angular-spectrum beam

Paraxial phase and longitudinal-field bounds

Facts & Assumptions

Given: A fixed SI laboratory chart, $c > 0$ , $ω > 0$ , $k_{0} = ω / c$ , $0 < δ < 1$ , $0 < κ \leq δ k_{0}$ , and real $E_{0} \neq 0$ in $V m^{- 1}$ .

[F1]

The SI chart has spatial coordinates in metres and time in seconds, $k_{0} = ω / c$ , and $E$ and $B$ have units $V m^{- 1}$ and tesla. The electric and magnetic fields as fixed-time spatial vector fields 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 Forward angular-spectrum beam

[F2]

The plane-wave convention uses the real part of complex amplitudes times $e^{i (K \cdot x - ω t)}$ . Complex plane-wave fields: real parts of complex exponentials

[A1]

The adopted smooth source-free vacuum Maxwell equations, with $ρ = 0$ and $J = 0$ , are $div E = div B = 0$ , $curl E = - \partial_{t} B$ , and $curl B = c^{- 2} \partial_{t} E$ ; $ρ$ and $J$ are the charge and current densities of the source definition, and divergence and curl are the coordinate operators of the fixed spatial chart. The microscopic vacuum Maxwell equations: smooth and weak branches Charge parameter, frame charge and current densities, volume charge and flux, and the four-current Divergence and curl of a $C^{1}$ vector field

[F3]

The complex exponential satisfies $\exp^{'} = \exp$ and the chain rule; real parts commute with these real partial derivatives. The plane-wave fields are therefore $C^{1}$ . Use the Euclidean dot and cross products on the spatial vectors. 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: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ The cross product in $R^{3}$

[F4]

For an exact mode with these $q, k_{z}$ , the longitudinal relation and cone bound are as stated in the paraxial error theorem. Paraxial phase and longitudinal-field bounds

Verification

1.1givenF1algebra

Since $κ \leq δ k_{0} < k_{0}$ , $k_{z} = \sqrt{k_{0}^{2} - κ^{2}}$ is positive. The definitions give $κ^{2} + k_{z}^{2} = k_{0}^{2}$ , $κ E_{0} + k_{z} e_{z} = 0$ , and $ω b_{y} = k_{0}^{2} E_{0} / k_{z}$ . Thus $K = (κ, 0, k_{z})$ is a forward wavevector of length $k_{0}$ , and $e$ is transverse to it.

2.1step 1.1F1F2F3A1algebra

By [F3], for each spatial coordinate $x_{j}$ , $\partial_{x_{j}} E = Re (i K_{j} e e^{i ϕ})$ componentwise, and $\partial_{t} E = Re (- i ω e e^{i ϕ})$ ; the same formulas hold with $B, b$ in place of $E, e$ . From [step 1.1], $K \cdot e = κ E_{0} + k_{z} e_{z} = 0, K \cdot b = 0,$ $K \times e = (0, k_{z} E_{0} - κ e_{z}, 0) = (0, \frac{k_{0}^{2} E_{0}}{k_{z}}, 0) = ω b,$ and $K \times b = (- k_{z} b_{y}, 0, κ b_{y}) = (- \frac{k_{0}^{2} E_{0}}{ω}, 0, \frac{κ k_{0}^{2} E_{0}}{ω k_{z}}) = - \frac{ω}{c^{2}} e,$ where the last equality uses $k_{0} = ω / c$ . Summing the spatial derivatives therefore yields $div E = Re (i (K \cdot e) e^{i ϕ}) = 0, div B = Re (i (K \cdot b) e^{i ϕ}) = 0,$ $curl E = Re (i (K \times e) e^{i ϕ}) = - \partial_{t} B, curl B = Re (i (K \times b) e^{i ϕ}) = c^{- 2} \partial_{t} E .$ Thus all four source-free Maxwell equations in [A1] hold pointwise, so these real fields are an exact mode.

3.1givenstep 1.1F4algebra∎

The constructed transverse amplitude is $(E_{0}, 0)$ . Applying [F4] gives $e_{z} = - q \cdot e_{⊥} / k_{z} = - κ E_{0} / k_{z}$ and $∣ e_{z} ∣ / ∣ E_{0} ∣ \leq δ / \sqrt{1 - δ^{2}}$ . Since $κ > 0$ , $E_{0} \neq 0$ , and $k_{z} > 0$ , $e_{z} \neq 0$ . Hence the alternative $e_{z} = 0$ with this same transverse amplitude is incompatible with source-free Maxwell transversality.

Remarks

For the illustrative case $δ = 3 / 4$ and $κ / k_{0} = 1 / 2$ , the normalized x-z projections are $K / k_{0} = (1 / 2, \sqrt{3} / 2)$ and $e / E_{0} = (1, - 1 / \sqrt{3})$ ; their dot product is zero. The arrows use these separate dimensionless normalizations, and the figure is schematic only. The Maxwell verification is the component proof in step 2.1, not the drawing.

\begin{tikzpicture}[x=2.5cm,y=2cm,>=stealth]
  \draw[->] (-0.12,0) -- (1.25,0) node[right] {$x$ component};
  \draw[->] (0,-0.8) -- (0,1.15) node[above] {$z$ component};
  \draw[dashed,gray] (0,0) -- (0.5,0);
  \draw[dashed,gray] (0.5,0) -- (0.5,{sqrt(3)/2});
  \draw[very thick,->] (0,0) -- (0.5,{sqrt(3)/2})
    node[above right] {$\mathbf K/k_0$};
  \draw[very thick,->,blue] (0,0) -- (1,{-1/sqrt(3)})
    node[below right] {$\mathbf e/E_0$};
  \node[below] at (0.5,0) {$\kappa/k_0=1/2$};
  \node[right] at (0.5,0.53) {$k_z/k_0=\sqrt3/2$};
\end{tikzpicture}

An oblique plane wave with a nonzero longitudinal electric component

Example

Example references

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources