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Vacuum waves and polarization — Examples

1 · Prerequisites

2 · Summary

Objective. The companion collects the four leaf items of the pair: explicit polarization cases, the infinite-energy contrast for the ideal plane wave, a conditional radiation-pressure computation, and the deterministic two-beam fringe geometry with its measurement-model qualifications.

Polarization cases. With $k = K e_{3}$ , $ω = c K$ and $b = k \times e / ω$ , the example verifies six amplitude choices (Linear, circular and elliptical amplitude choices): the linear case $e = E_{0} e_{1}$ with trace the segment $[- E_{0}, E_{0}] e_{1}$ ; the circular cases $e = E_{0} (e_{1} \pm i e_{2})$ of radius $E_{0}$ with the two signs of the helicity $H$ ; the elliptical case $e = E_{0} (e_{1} + 2 i e_{2})$ with semiaxes $E_{0}$ and $2 E_{0}$ ; the zero wave; and the dependent-amplitude linear edge case $e = E_{0} (1 + i) e_{1}$ , which the classification covers without requiring $a$ and $d$ to be orthogonal. Every choice satisfies the amplitude equations and $∣ B ∣ = ∣ E ∣ / c$ pointwise.

Infinite energy of the ideal plane wave. For the explicit configuration $E = E_{0} \cos θ e_{1}$ , $B = (E_{0} / c) \cos θ e_{2}$ the density is $u = ε_{0} E_{0}^{2} \cos^{2} θ$ , positive on average and vanishing only on the nodal planes; the box integrals are bounded below by $π ε_{0} E_{0}^{2} R^{2} / K \to \infty$ along the sequence $R = N P$ with $P = 2 π / K$ (A nonzero whole-space plane wave has infinite total energy). In contrast, the compactly supported branch of the constrained initial-value theorem has compact spatial support at each time and hence finite instantaneous energy: the ideal whole-space plane wave is not a finite-energy configuration.

Radiation pressure under stated idealisations. The thought experiment adopts an ideal absorber or an ideal reflector at $x_{3} = 0$ together with an explicit pillbox boundary-force law (Ideal absorber and ideal reflector momentum transfer (radiation pressure)). For the incident wave of intensity $I = ε_{0} c E_{0}^{2} / 2$ the period-averaged normal stress is $- I / c$ near the surface, giving pressure $I / c$ on the absorber, and the standing wave formed by the incident and reflected fields has averaged normal stress $- 2 I / c$ at the surface, giving pressure $2 I / c$ on the reflector. The vanishing-field idealisations and the force law are adopted model assumptions, not derived material laws; no measured pressure is reported.

Two-beam fringes. For the two crossing waves with $k_{1, 2} = K (\pm \sin α e_{1} + \cos α e_{3})$ and amplitudes $E_{0} e_{2}$ , $E_{0} e^{i φ} e_{2}$ , the averaged energy density is $ε_{0} E_{0}^{2} [1 + \cos^{2} α \cos (2 K \sin α x_{1} - φ)]$ , with fringe period $λ / (2 \sin α)$ and spatial visibility $\cos^{2} α = \frac{1}{2} (1 + {\hat{k}}_{1} \cdot {\hat{k}}_{2})$ (Ideal deterministic two-wave interference and its fringe geometry). The copropagating equal-amplitude law $⟨ u ⟩ \propto ∣ 1 + e^{i Δ} ∣^{2}$ is the form used in the content model's double-slit discussion, and phase-scan visibility is a different quantity from spatial visibility. The item reports no observation and no finite-sample visibility estimate: a measured fringe pattern would additionally require a declared detector, binning, background, integration and sampling model, and classical waves interfere, so the prediction alone does not discriminate between classical and quantum descriptions.

All four items are conditional computations inside the framework's declared models; none reports an observation, and none asserts an empirical conclusion.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passOpen item page →

Linear, circular and elliptical amplitude choices

Physical scope

Explicit ideal plane-wave amplitude choices on $R^{3} \times R$ in a named inertial frame with SI units: linear, circular (both handedness signs), elliptical with orthogonal axes, the zero wave and one dependent-amplitude linear edge case. Excluded: partially polarized light, superposition with other waves, and any empirical or measurement claim.

Example

Use the frame basis $(e_{1}, e_{2}, e_{3})$ and the SI conventions of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields and Framework-local SI units, the vacuum constants and the exact/measured status of their values. Fix $K > 0$ , put $ω : = c K$ and $k : = K e_{3}$ , so that $∣ k ∣ = K = ω / c$ and $\hat{k} = e_{3}$ , and let $E_{0} > 0$ carry the unit $V m^{- 1}$ . For each amplitude choice below put $b : = k \times e / ω$ ; then the amplitude conditions of Dispersion, transversality and the two polarization conditions for plane-wave solutions hold and the corresponding fields $(E, B)$ of Real plane-wave fields encoded by complex amplitudes are smooth source-free vacuum Maxwell solutions on $R^{3} \times R$ , with $θ (x, t) = K x_{3} - ω t$ .

(i) [Linear] $e : = E_{0} e_{1}$ , $b = (E_{0} / c) e_{2}$ . Here $a = E_{0} e_{1}$ and $d = 0$ , the trace is the segment $[- E_{0}, E_{0}] e_{1} = {s e_{1} : - E_{0} \leq s \leq E_{0}}$ traversed twice per period, the polarization is linear and $H = ⟨ a \times d, \hat{k} ⟩ = 0$ ; moreover $E = E_{0} \cos θ e_{1}$ , $B = (E_{0} / c) \cos θ e_{2}$ and $∣ B ∣ = ∣ E ∣ / c$ .

(ii) [Circular] $e : = E_{0} (e_{1} + i e_{2})$ , $b = (E_{0} / c) (e_{2} - i e_{1})$ . Here $a = E_{0} e_{1}$ and $d = E_{0} e_{2}$ , so $∣ a ∣ = ∣ d ∣ = E_{0}$ and $⟨ a, d ⟩ = 0$ : the polarization is circular of radius $E_{0}$ , with $E = E_{0} (\cos θ e_{1} - \sin θ e_{2})$ , $B = (E_{0} / c) (\sin θ e_{1} + \cos θ e_{2})$ , $∣ E ∣ = E_{0}$ and $∣ B ∣ = E_{0} / c$ everywhere, and $H = ⟨ a \times d, \hat{k} ⟩ = E_{0}^{2} > 0$ , so this polarization is left-handed in the convention of Polarization of a transverse plane wave: linear, circular and elliptical traces, and handedness. The reversed choice $e = E_{0} (e_{1} - i e_{2})$ gives $H = - E_{0}^{2} < 0$ and is right-handed.

(iii) [Elliptical] $e : = E_{0} (e_{1} + 2 i e_{2})$ . Here $a = E_{0} e_{1}$ and $d = 2 E_{0} e_{2}$ are orthogonal with unequal lengths, so by clause (b)(iii) of Polarization of a transverse plane wave: linear, circular and elliptical traces, and handedness the trace is the axis-aligned ellipse $E_{0} \cos θ e_{1} - 2 E_{0} \sin θ e_{2}$ with semiaxes $E_{0}$ and $2 E_{0}$ along $e_{1}$ and $e_{2}$ ; $∣ E ∣^{2} = E_{0}^{2} (\cos^{2} θ + 4 \sin^{2} θ)$ varies between $E_{0}^{2}$ and $4 E_{0}^{2}$ ; and $b = (E_{0} / c) (e_{2} - 2 i e_{1})$ , $H = 2 E_{0}^{2} > 0$ .

(iv) [Degenerate and edge cases] $e = 0$ gives the zero wave, for which $a = d = 0$ and the trace is the single point ${0}$ . For $e : = E_{0} (1 + i) e_{1}$ one has $a = d = E_{0} e_{1}$ , so the polarization is linear with trace the segment $[- \sqrt{2} E_{0}, \sqrt{2} E_{0}] e_{1}$ and $H = 0$ ; the classification (b) of Polarization of a transverse plane wave: linear, circular and elliptical traces, and handedness covers this case without requiring $a$ and $d$ to be orthogonal.

(v) [Verification and scope] In every case $b = k \times e / ω$ with $∣ k ∣ = ω / c$ , so the amplitude equations hold and the fields are smooth source-free solutions by Dispersion, transversality and the two polarization conditions for plane-wave solutions; the amplitudes and the semiaxis lengths carry the unit $V m^{- 1}$ , and $∣ B ∣ = ∣ E ∣ / c$ holds pointwise. The examples assert no empirical realization and no measured polarization.

Facts & Assumptions

Given: The frame basis $(e_{1}, e_{2}, e_{3})$ , the constants $c > 0$ and $K > 0$ , $ω = c K$ , $k = K e_{3}$ , the amplitude $E_{0} > 0$ in $V m^{- 1}$ , the six amplitude choices $e \in {E_{0} e_{1}, E_{0} (e_{1} + i e_{2}), E_{0} (e_{1} - i e_{2}), E_{0} (e_{1} + 2 i e_{2}), E_{0} (1 + i) e_{1}, 0}$ with the corresponding $b = k \times e / ω$ , and the phase $θ (x, t) = K x_{3} - ω t$ .

[F1]

The polarization definition fixes the trace $E = a \cos θ - d \sin θ$ with $a = Re e$ , $d = Im e$ , the linear, circular and elliptical clauses with their criteria, and the helicity sign $H = ⟨ a \times d, \hat{k} ⟩$ , with $H = 0$ exactly in the linear case, right-handed meaning $H < 0$ and left-handed meaning $H > 0$ (Polarization of a transverse plane wave: linear, circular and elliptical traces, and handedness).

[F2]

For a plane wave with $k \neq 0$ , $ω > 0$ , if $∣ k ∣ = ω / c$ and $e$ satisfies $k \cdot e = 0$ then $b = k \times e / ω$ makes the amplitude equations hold, so the pair lies in the source-free smooth Maxwell branch; all solution fields are $C^{\infty}$ and satisfy $∣ B (x, t) ∣ = ∣ E (x, t) ∣ / c$ pointwise (Dispersion, transversality and the two polarization conditions for plane-wave solutions).

[F3]

The plane-wave pair is $E = Re (e \exp (i θ))$ , $B = Re (b \exp (i θ))$ with $θ = k \cdot x - ω t$ , $b$ the complex magnetic amplitude, and $\hat{k} = k / ∣ k ∣$ (Real plane-wave fields encoded by complex amplitudes).

[F4]

The cross product has the coordinate formula $u \times v = (u_{2} v_{3} - u_{3} v_{2}, u_{3} v_{1} - u_{1} v_{3}, u_{1} v_{2} - u_{2} v_{1})$ on the standard basis (The cross product in $R^{3}$ ).

[F5]

The Euclidean inner product is $⟨ u, v ⟩ = \sum_{k} u_{k} v_{k}$ , additive and real-homogeneous in each argument, so that $∣ u_{1} a - u_{2} d ∣^{2} = u_{1}^{2} ∣ a ∣^{2} + u_{2}^{2} ∣ d ∣^{2} - 2 u_{1} u_{2} ⟨ a, d ⟩$ for real $u_{1}, u_{2}$ and $a, d \in R^{3}$ ; in particular the frame basis is orthonormal (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F6]

The SI conventions provide the propagation constant $c > 0$ with its unit and the units of $k$ , $ω$ , $E$ and $B$ (Framework-local SI units, the vacuum constants and the exact/measured status of their values).

[F7]

The laboratory region carries the frame basis $(e_{1}, e_{2}, e_{3})$ in which components are read (Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields).

Verification

technique · direct

1.1givenF3F4F5F7algebra

Common data. With $k = K e_{3}$ one has $∣ k ∣ = K$ , $∣ k ∣ = ω / c$ because $ω = c K$ , and $\hat{k} = e_{3}$ ; by [F4] and the coordinate formula, $e_{3} \times e_{1} = e_{2}$ and $e_{3} \times e_{2} = - e_{1}$ . Hence for each choice $b = k \times e / ω = (K / ω) (e_{3} \times e)$ : for $e = E_{0} e_{1}$ , $b = (E_{0} / c) e_{2}$ ; for $e = E_{0} (e_{1} + i e_{2})$ , $b = (E_{0} / c) (e_{2} - i e_{1})$ ; for $e = E_{0} (e_{1} - i e_{2})$ , $b = (E_{0} / c) (e_{2} + i e_{1})$ ; for $e = E_{0} (e_{1} + 2 i e_{2})$ , $b = (E_{0} / c) (e_{2} - 2 i e_{1})$ ; and for $e = E_{0} (1 + i) e_{1}$ , $b = (E_{0} / c) (1 + i) e_{2}$ . Also $k \cdot e = K e_{3} \cdot e = 0$ for each of these amplitudes, since only the first two coordinates are nonzero.

2.1step 1.1F1F2F3F4algebra

Case (i), linear. For $e = E_{0} e_{1}$ one has $a = E_{0} e_{1}$ , $d = 0$ , so $\dim span {a, d} = 1$ and clause (b)(i) of [F1] gives a linear trace, the segment $[- E_{0}, E_{0}] e_{1}$ traversed twice per period; $a \times d = E_{0} e_{1} \times 0 = 0$ by [F4], so $H = 0$ . The fields are $E = Re (E_{0} e_{1} e^{i θ}) = E_{0} \cos θ e_{1}$ and $B = (E_{0} / c) \cos θ e_{2}$ , whence $∣ B ∣ = ∣ E ∣ / c$ pointwise; this also matches $b = (E_{0} / c) e_{2}$ from step 1.1 and clause (c) of [F2].

2.2step 1.1F1F2F3F4F5algebra

Case (ii), circular. For $e = E_{0} (e_{1} + i e_{2})$ one has $a = E_{0} e_{1}$ , $d = E_{0} e_{2}$ , so $∣ a ∣ = ∣ d ∣ = E_{0} > 0$ and $⟨ a, d ⟩ = E_{0}^{2} ⟨ e_{1}, e_{2} ⟩ = 0$ by the orthonormality of [F5]; clause (b)(ii) of [F1] gives a circular polarization of radius $E_{0}$ . The trace is $E = E_{0} (\cos θ e_{1} - \sin θ e_{2})$ with $∣ E ∣^{2} = E_{0}^{2} (\cos^{2} θ + \sin^{2} θ) = E_{0}^{2}$ by [F5], and $B = Re ((E_{0} / c) (e_{2} - i e_{1}) e^{i θ}) = (E_{0} / c) (\sin θ e_{1} + \cos θ e_{2})$ with $∣ B ∣ = E_{0} / c$ ; finally $H = ⟨ a \times d, e_{3} ⟩ = E_{0}^{2} ⟨ e_{1} \times e_{2}, e_{3} ⟩ = E_{0}^{2} > 0$ by [F4] and [F5], so this polarization is left-handed by [F1]. For the reversed choice $e = E_{0} (e_{1} - i e_{2})$ one has $d = - E_{0} e_{2}$ and $H = E_{0}^{2} ⟨ e_{1} \times (- e_{2}), e_{3} ⟩ = - E_{0}^{2} < 0$ , which is right-handed.

2.3step 1.1F1F2F3F4F5algebra

Case (iii), elliptical. For $e = E_{0} (e_{1} + 2 i e_{2})$ one has $a = E_{0} e_{1}$ , $d = 2 E_{0} e_{2}$ , which are orthogonal by [F5] and have unequal lengths, and $\dim span {a, d} = 2$ ; since $∣ a ∣ = E_{0} \neq 2 E_{0} = ∣ d ∣$ , clause (b)(ii) fails, so clause (b)(iii) of [F1] applies and, because $⟨ a, d ⟩ = 0$ , the trace is the axis-aligned ellipse with semiaxes $E_{0}$ and $2 E_{0}$ along $e_{1}$ and $e_{2}$ . Explicitly $E = E_{0} \cos θ e_{1} - 2 E_{0} \sin θ e_{2}$ and $∣ E ∣^{2} = E_{0}^{2} \cos^{2} θ + 4 E_{0}^{2} \sin^{2} θ$ by [F5], which lies between $E_{0}^{2}$ and $4 E_{0}^{2}$ ; $b = (E_{0} / c) (e_{2} - 2 i e_{1})$ by step 1.1, and $H = ⟨ E_{0} e_{1} \times 2 E_{0} e_{2}, e_{3} ⟩ = 2 E_{0}^{2} > 0$ .

2.4step 1.1givenF1F3F4F5algebra

Case (iv), degenerate and edge cases. For $e = 0$ the fields vanish identically by [F3] (the zero wave), $a = d = 0$ , and clause (b)(i) of [F1] covers the trace ${0}$ ; for $e = E_{0} (1 + i) e_{1}$ one has $a = d = E_{0} e_{1}$ , so $\dim span {a, d} = 1$ , the polarization is linear, and the trace is $E = E_{0} (\cos θ - \sin θ) e_{1}$ , whose coefficient $\cos θ - \sin θ = \sqrt{2} \cos (θ + π / 4)$ ranges over $[- \sqrt{2}, \sqrt{2}]$ ; the trace is therefore the segment $[- \sqrt{2} E_{0}, \sqrt{2} E_{0}] e_{1}$ , and $H = ⟨ a \times a, e_{3} ⟩ = 0$ by alternation of the coordinate formula [F4]. No orthogonality of $a$ and $d$ is used, matching the covering clause of [F1]; the corresponding $b$ is the one computed in step 1.1.

3.1step 1.1F2F6given∎

Scope check (v). In every case step 1.1 gives $b = k \times e / ω$ with $k \cdot e = 0$ and $∣ k ∣ = ω / c$ , so by clause (b) of [F2] the amplitude equations hold and the corresponding fields are smooth source-free Maxwell solutions; the amplitudes $e$ and the semiaxis lengths $E_{0}$ , $2 E_{0}$ carry $V m^{- 1}$ by [F6], and the pointwise relation $∣ B ∣ = ∣ E ∣ / c$ is clause (c) of [F2]. The item computes only ideal field quantities and asserts no measurement or empirical realization.

Remarks

Classification, not evidence. The six choices illustrate clauses of Polarization of a transverse plane wave: linear, circular and elliptical traces, and handedness; they neither predict nor report any laboratory measurement, and the handedness names are the conventional labels fixed there, not physical claims.

ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passOpen item page →

A nonzero whole-space plane wave has infinite total energy

Physical scope

An exact computation for one explicit ideal plane-wave configuration on $R^{3} \times R$ : the positivity and uniformity of the period-averaged energy density, the divergence of the box integrals, and the contrast with compactly supported finite-energy data. No measured quantity, no finite beam and no empirical claim.

Example

Assume Countable Choice ( ${A C}_{ω}$ , The Axiom of Countable Choice ( ${A C}_{ω}$ )) for the Lebesgue-energy and compact-source comparison branches (ii), (iii), as required by their measure and wave-Cauchy suppliers. The explicit plane-wave fields of (i) use no choice.

Fix $K > 0$ , $ω : = c K$ , $k : = K e_{3}$ , $E_{0} > 0$ (unit $V m^{- 1}$ ), the amplitude $e : = E_{0} e_{1}$ and $b : = k \times e / ω = (E_{0} / c) e_{2}$ , and let $(E, B)$ be the corresponding plane-wave solution of Dispersion, transversality and the two polarization conditions for plane-wave solutions (Real plane-wave fields encoded by complex amplitudes) on $R^{3} \times R$ with energy density $u$ of Vacuum electromagnetic energy density and energy flux (the Poynting vector); write $θ (x, t) = K x_{3} - ω t$ . Then:

(i) [Explicit density] $E = E_{0} \cos θ e_{1}$ , $B = (E_{0} / c) \cos θ e_{2}$ and $u (x, t) = ε_{0} E_{0}^{2} \cos^{2} θ (x, t)$ , a nonnegative function vanishing exactly on the nodal planes $\cos θ = 0$ ; the period average at every fixed $x$ is $⟨ u ⟩ = ε_{0} E_{0}^{2} / 2 > 0$ , uniformly in $x$ and $t$ (Period-averaged energy density, Poynting vector and momentum flux of a plane wave(a)).

(ii) [Unbounded box integrals] Fix $t \in R$ and let $R \geq (ω ∣ t ∣ + π / 4) / K$ . On the sub-box $Q_{R} : = [- R, R]^{2} \times [(ω t - π / 4) / K, (ω t + π / 4) / K]$ of the box $[- R, R]^{3}$ one has $\cos^{2} θ \geq 1 / 2$ , hence $u \geq ε_{0} E_{0}^{2} / 2$ there; both boxes are product boxes with finite volume (Axis-parallel rectangles in $R^{m}$ and their volume), $u$ is continuous on them and therefore Riemann integrable (A continuous real function on a compact Jordan measurable set is Riemann integrable over that set), and monotonicity of the multidimensional Darboux integral (The lower and upper Darboux integrals over a nondegenerate rectangle in $R^{m}$ , The Riemann integral of a bounded function over a bounded Jordan measurable set) gives $\int_{[- R, R]^{3}} u (x, t) d x \geq \int_{Q_{R}} u (x, t) d x \geq \frac{ε_{0} E_{0}^{2}}{2} v o l (Q_{R}) = \frac{π ε_{0} E_{0}^{2}}{K} R^{2} ⟶ + \infty (R \to \infty) .$ Consequently the box integrals have no finite limit as $R \to \infty$ : the ideal plane wave has infinite total energy in the box-exhaustion sense, in contrast to the total energy of a single wave 'mode' being finite on any bounded box.

(iii) [Contrast with compactly supported data] If instead the constrained data of Constrained smooth vacuum Maxwell initial-value theory, finite-energy conservation and finite propagation vanish off a compact set, then by clause (d) of that theorem the fields have compact spatial support at each time, and the instantaneous total energy is the integral of the continuous density $u$ over a bounding closed box containing its support, hence finite. So finite total energy is available for compactly supported data, while the ideal plane wave of (i)-(ii) is not a finite-energy configuration.

(iv) [Scope] The example is an exact computation for the ideal plane wave; it reports no measurement, and "infinite total energy" means the divergence statement (ii), not a claim about any laboratory field or finite beam.

Facts & Assumptions

Given: Countable Choice ${A C}_{ω}$ for clauses (ii), (iii); The constants $K > 0$ , $c > 0$ , $ω = c K$ , $k = K e_{3}$ , $E_{0} > 0$ in $V m^{- 1}$ , the amplitude $e = E_{0} e_{1}$ , the magnetic amplitude $b = k \times e / ω = (E_{0} / c) e_{2}$ , the phase $θ (x, t) = K x_{3} - ω t$ , the plane-wave fields $(E, B)$ , and the energy density $u$ .

[F1]

Plane-wave data (Real plane-wave fields encoded by complex amplitudes, Dispersion, transversality and the two polarization conditions for plane-wave solutions): the fields are $E = Re (e \exp (i θ))$ , $B = Re (b \exp (i θ))$ , and with $∣ k ∣ = ω / c$ , $k \cdot e = 0$ , $b = k \times e / ω$ the pair lies in the source-free smooth Maxwell branch.

[F2]

Energy density (Vacuum electromagnetic energy density and energy flux (the Poynting vector)): $u = \frac{ε_{0}}{2} ∣ E ∣^{2} + \frac{1}{2 μ_{0}} ∣ B ∣^{2} \geq 0$ is $C^{\infty}$ and $ε_{0} = (μ_{0} c^{2})^{- 1}$ .

[F3]

Plane-wave averages (Period-averaged energy density, Poynting vector and momentum flux of a plane wave(a)): for this class of plane waves $⟨ u ⟩ = \frac{ε_{0}}{2} ∣ e ∣^{2}$ , independent of $x$ and of the starting time, where the average is taken over the period $T = 2 π / ω$ ; in particular the real-amplitude identity $\frac{1}{T} \int_{0}^{T} \cos^{2} (ϕ_{0} - ω s) d s = \frac{1}{2}$ is the scalar case of Period averages of products of real parts of complex exponentials.

[F4]

Integration over boxes and Jordan sets (Axis-parallel rectangles in $R^{m}$ and their volume, The lower and upper Darboux integrals over a nondegenerate rectangle in $R^{m}$ , The Riemann integral of a bounded function over a bounded Jordan measurable set, A continuous real function on a compact Jordan measurable set is Riemann integrable over that set): a product box $[a, b] = \prod_{j} [a_{j}, b_{j}]$ has volume $\prod_{j} (b_{j} - a_{j})$ ; a continuous function on a compact Jordan measurable set is Riemann integrable there; and the Darboux integral of a nonnegative function is monotone under inclusion of product boxes.

[F5]

Compact-support propagation (Constrained smooth vacuum Maxwell initial-value theory, finite-energy conservation and finite propagation(d)): for initially constraint-satisfying $C^{\infty}$ data vanishing off a compact $K \subseteq R^{3}$ with $K \subseteq {\overline{B}}_{ρ} (0)$ , the solution fields satisfy $E (x, t) = B (x, t) = 0$ whenever $∣ x ∣ > ρ + c (t - t_{0})$ .

[F6]

Euclidean structure (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ): the norm and inner product are the Euclidean ones, so for real vectors $v$ , $∣ v ∣^{2} = ⟨ v, v ⟩$ .

Verification

technique · direct

1.1givenF1F2F6algebra

Explicit fields and density. Since $k \times e = (K e_{3}) \times (E_{0} e_{1}) = K E_{0} e_{2}$ , one has $b = (E_{0} / c) e_{2}$ ; by [F1] and Euler's formula in the real-part encoding, $E = Re (E_{0} e_{1} e^{i θ}) = E_{0} \cos θ e_{1}$ and $B = Re ((E_{0} / c) e_{2} e^{i θ}) = (E_{0} / c) \cos θ e_{2}$ . By [F2] and [F6], $u = \frac{ε_{0}}{2} E_{0}^{2} \cos^{2} θ + \frac{1}{2 μ_{0}} \frac{E_{0}^{2}}{c^{2}} \cos^{2} θ = ε_{0} E_{0}^{2} \cos^{2} θ$ because $1 / (μ_{0} c^{2}) = ε_{0}$ ; the function is nonnegative and vanishes exactly where $\cos θ = 0$ , the union of the nodal planes $K x_{3} - ω t = π / 2 + π Z$ .

2.1step 1.1F3algebra

Period average. At fixed $x$ , $u (x, \cdot)$ is the $\cos^{2}$ profile $ε_{0} E_{0}^{2} \cos^{2} (ϕ_{0} - ω s)$ with $ϕ_{0} = K x_{3}$ ; the scalar case of [F3] gives $⟨ u ⟩ = ε_{0} E_{0}^{2} \cdot \frac{1}{2} = \frac{ε_{0}}{2} E_{0}^{2}$ , which equals $\frac{ε_{0}}{2} ∣ e ∣^{2}$ and is strictly positive, independent of $x$ and of the starting time.

2.2step 1.1F4algebra

Unbounded box integrals. Fix $t \in R$ and $R \geq (ω ∣ t ∣ + π / 4) / K$ , and let $Q_{R}$ be the sub-box of $[- R, R]^{3}$ with $x_{3} \in [(ω t - π / 4) / K, (ω t + π / 4) / K]$ and $x_{1}, x_{2} \in [- R, R]$ . Each point of $Q_{R}$ satisfies $∣ K x_{3} - ω t ∣ \leq π / 4$ , so $\cos^{2} θ \geq 1 / 2$ and $u \geq ε_{0} E_{0}^{2} / 2$ there, while $Q_{R} \subseteq [- R, R]^{3}$ because $∣ (ω t \pm π / 4) / K ∣ \leq R$ ; both are product boxes with finite volume by [F4], and $u$ is continuous on the compact boxes, hence Riemann integrable by [F4], so monotonicity of the Darboux integral gives $\int_{[- R, R]^{3}} u \geq \int_{Q_{R}} u \geq \frac{ε_{0} E_{0}^{2}}{2} v o l (Q_{R})$ . The volume is $v o l (Q_{R}) = (2 R)^{2} \cdot \frac{π}{2 K} = \frac{2 π R^{2}}{K}$ by [F4], so $\int_{[- R, R]^{3}} u \geq \frac{π ε_{0} E_{0}^{2}}{K} R^{2} \to + \infty$ as $R \to \infty$ ; hence the box integrals are unbounded and the box-exhaustion total energy $\sup_{R > 0} \int_{[- R, R]^{3}} u$ is $+ \infty$ .

3.1step 2.2F4F5algebra

Compactly supported data. Suppose the constrained $C^{\infty}$ data of [F5] satisfy $div E_{0} = div B_{0} = 0$ and vanish off a compact $K \subseteq {\overline{B}}_{ρ} (0)$ , and let $(E, B)$ be the corresponding solution; for fixed $t$ the fields vanish off the closed ball ${\overline{B}}_{ρ + c (t - t_{0})} (0)$ , a compact set, so $u (\cdot, t)$ is a continuous function with compact support there; choose a nondegenerate closed box containing that ball. By [F4] the continuous density is Riemann integrable on the box with a finite integral, and it vanishes outside the box. All larger box integrals have this same finite value. Thus the instantaneous total energy of the compactly supported configuration is finite, in contrast with the divergence of step 2.2 for the ideal plane wave.

4.1step 1.1step 2.1step 2.2step 3.1given∎

Conclusion and scope. Step 1.1 is clause (i) except for the average, which is step 2.1; step 2.2 is clause (ii) including the divergence of the box integrals; step 3.1 is clause (iii); and clause (iv) is a scope qualification: all quantities are exact ideal-field quantities of the specified plane wave, no laboratory measurement or finite beam is described, and the phrase "infinite total energy" refers to the divergence statement of step 2.2 in the box-exhaustion sense, not to any finite measured energy.

Remarks

Idealisation, not evidence. The computation shows that the ideal whole-space plane wave is not a finite-energy configuration; it makes no claim about any laboratory field, and the compact-support branch of Constrained smooth vacuum Maxwell initial-value theory, finite-energy conservation and finite propagation is the source of the finite-energy contrast.

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Ideal absorber and ideal reflector momentum transfer (radiation pressure)

Physical scope

Adopted idealisations of a plane beam at normal incidence on an infinite plane ideal absorber or ideal reflector, with the pillbox force law as an explicitly stated model assumption; the conclusion is the conditional pressure $I / c$ or $2 I / c$ along the propagation direction. Excluded: oblique incidence, finite beams and pulses, real material response, radiation reaction, measurements, and any empirical claim; ${A C}_{ω}$ is retained as an assumed premise.

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ , The Axiom of Countable Choice ( ${A C}_{ω}$ )). Idealised setup: the vacuum region is ${x_{3} < 0}$ and a material body occupies ${x_{3} > 0}$ with plane surface $x_{3} = 0$ ; an incident plane wave of Real plane-wave fields encoded by complex amplitudes with $k = K e_{3}$ ( $K > 0$ , $ω = c K$ ), $e = E_{0} e_{1}$ ( $E_{0} > 0$ , unit $V m^{- 1}$ ) and $b = (E_{0} / c) e_{2}$ propagates in the $+ e_{3}$ direction. By Period-averaged energy density, Poynting vector and momentum flux of a plane wave its period-averaged Poynting flux is $⟨ S ⟩ = I \hat{k}$ with $I = \frac{ε_{0} c}{2} E_{0}^{2}$ , and its period-averaged normal Maxwell stress is $⟨ σ_{33} ⟩ = - I / c$ (Momentum density, Maxwell stress, momentum flux and orbital angular density in vacuum for $σ$ ). All period-averaged quantities below are independent of time. Let $A > 0$ be the area (in $m^{2}$ ) of a finite rectangular surface patch, let $F \in R^{3}$ be its period-averaged force (in N), and let $p \in R$ be the signed normal force per unit area (in $N m^{- 2}$ ). The adopted boundary-force law below sets $F = A p e_{3}$ ; these are patch quantities in the uniform infinite-plane model. Two idealised models of the surface:

(i) [Ideal absorber] Adopt (A1) no reflected vacuum field and a zero back-face stress for an ideal absorption layer, and (A2) the force on each surface patch is given by the pillbox limit of the momentum balance, i.e. its signed pressure is the difference $⟨ σ_{33} ⟩_{b a c k} - ⟨ σ_{33} (0^{-}) ⟩$ of the back-face and incident-side normal stresses, with the back-face stress set to zero by (A1). This is an exterior boundary-force model: the incident vacuum field is not asserted to extend by zero across a sharp interface as a global Maxwell solution. A real absorption layer must supply a compatible material-field solution; no such realization is claimed here. The incident-wave average is constant throughout the vacuum region by Period-averaged energy density, Poynting vector and momentum flux of a plane wave(c), so its one-sided surface limit is $⟨ σ_{33} (0^{-}) ⟩ = - I / c$ . Then the force per unit area is $p = I / c$ and $F / A = (I / c) \hat{k}$ : the wave pushes the absorber along its direction of propagation with pressure $I / c$ .

(ii) [Ideal reflector] Adopt instead (A1') the field on the material side vanishes identically while the vacuum field is the sum of the incident wave and the reflected plane wave with $k^{'} = - k$ , $e^{'} = - e$ (so that the tangential electric field vanishes on $x_{3} = 0$ ; the sum is a source-free solution by Superposition for prescribed Maxwell fields and sources), together with the same boundary-force law (A2) with zero material-side stress. Compute the averaged normal stress of the standing wave directly: the real fields satisfy $E = E_{0} [\cos (K x_{3} - ω t) - \cos (K x_{3} + ω t)] e_{1} = 2 E_{0} \sin (K x_{3}) \sin (ω t) e_{1}$ and $B = (2 E_{0} / c) \cos (K x_{3}) \cos (ω t) e_{2}$ , whose cross terms cancel identically in $σ_{33} = - (ε_{0} / 2) ∣ E ∣^{2} - (1 / (2 μ_{0})) ∣ B ∣^{2}$ ; with $⟨ \sin^{2} ⟩ = ⟨ \cos^{2} ⟩ = 1 / 2$ this gives $⟨ σ_{33} (0^{-}) ⟩ = - ε_{0} E_{0}^{2} = - 2 I / c$ (the incident and reflected waves contribute $- I / c$ each). Hence $p = 2 I / c$ and $F / A = (2 I / c) \hat{k}$ : twice the absorbing pressure, because the momentum of the reflected wave is reversed.

(iii) [Status] The absorber back-face and reflector vanishing-field idealisations (A1)/(A1') and the boundary-force law (A2) are adopted model assumptions of this thought experiment, not derived material laws and not observations; the deduction from them with the computed averaged stresses is complete and conditional. The setup idealises an infinite plane beam at normal incidence on an infinite plane surface; oblique incidence, finite beams, real absorbers and any magnetisation/current response of a material require a material model and are excluded. No measured pressure is reported and no empirical or statistical claim is made.

Facts & Assumptions

Given: The Axiom of Countable Choice ${A C}_{ω}$ ; the frame basis $(e_{1}, e_{2}, e_{3})$ ; $K > 0$ , $ω = c K$ , $k = K e_{3}$ , $E_{0} > 0$ in $V m^{- 1}$ ; the incident amplitude $e = E_{0} e_{1}$ ; the constants $c, μ_{0} > 0$ , $ε_{0} = (μ_{0} c^{2})^{- 1}$ ; the vacuum region ${x_{3} < 0}$ and the surface $x_{3} = 0$ ; a patch area $A > 0$ , signed pressure $p$ and force vector $F = A p e_{3}$ .

[F1]

Plane waves and their amplitude conditions (Real plane-wave fields encoded by complex amplitudes, Dispersion, transversality and the two polarization conditions for plane-wave solutions): for $j$ indexing the incident and reflected waves, $∣ k_{j} ∣ = ω / c$ , $k_{j} \cdot e_{j} = 0$ , $b_{j} = k_{j} \times e_{j} / ω$ imply that the pair lies in the source-free smooth Maxwell branch with $E_{j} = Re (e_{j} \exp (i (k_{j} \cdot x - ω t)))$ and $B_{j} = Re (b_{j} \exp (i (k_{j} \cdot x - ω t)))$ .

[F2]

Plane-wave averages (Period-averaged energy density, Poynting vector and momentum flux of a plane wave): for a transverse plane wave with amplitude $e$ , wavevector $k$ with $∣ k ∣ = ω / c$ and $b = k \times e / ω$ , one has $⟨ S ⟩ = I \hat{k}$ , $I = \frac{ε_{0} c}{2} ∣ e ∣^{2}$ , and componentwise $⟨ σ_{i j} ⟩ = - ⟨ u ⟩ {\hat{k}}_{i} {\hat{k}}_{j}$ with $⟨ u ⟩ = I / c$ , so in particular $⟨ σ_{33} ⟩ = - I / c$ for $k = K e_{3}$ .

[F4]

Momentum density and stress (Momentum density, Maxwell stress, momentum flux and orbital angular density in vacuum): $g = S / c^{2}$ and $σ_{i j} = ε_{0} (E_{i} E_{j} - \frac{1}{2} δ_{i j} ∣ E ∣^{2}) + \frac{1}{μ_{0}} (B_{i} B_{j} - \frac{1}{2} δ_{i j} ∣ B ∣^{2})$ .

[F5]

Linear superposition (Superposition for prescribed Maxwell fields and sources): a finite linear combination with real coefficients of smooth source-free solutions is again a smooth source-free solution on the same region.

[F6]

Period averages of squares (Period averages of products of real parts of complex exponentials): for $T = 2 π / ω$ , $\frac{1}{T} \int_{0}^{T} \sin^{2} (ω s) d s = \frac{1}{T} \int_{0}^{T} \cos^{2} (ω s) d s = \frac{1}{2}$ .

[F9]

SI relation (Framework-local SI units, the vacuum constants and the exact/measured status of their values): $ε_{0} = (μ_{0} c^{2})^{- 1}$ , so $ε_{0} E_{0}^{2} = 2 I / c$ because $I = \frac{ε_{0} c}{2} E_{0}^{2}$ .

[A1]

Adopted ideal-absorber model: no reflected vacuum field and zero back-face stress; the force per unit area on the material is $p = ⟨ σ_{33} ⟩_{b a c k} - ⟨ σ_{33} (0^{-}) ⟩$ , with $F / A = p e_{3}$ , a pillbox boundary-force law, not a derived material law.

[A2]

Adopted ideal-reflector model: the field on the material side vanishes identically, the vacuum field is the sum of the incident wave and the reflected plane wave with $k^{'} = - k$ , $e^{'} = - e$ , and the same boundary-force law with zero material-side stress applies.

Proof

technique · direct

1.1givenF1F2F4algebra

Incident wave data. For the incident wave $k \times e = (K e_{3}) \times (E_{0} e_{1}) = K E_{0} e_{2}$ , so $b = (E_{0} / c) e_{2}$ , and $∣ k ∣ = ω / c$ , $k \cdot e = 0$ ; hence by [F1] the pair is a smooth source-free solution, with $E = E_{0} \cos (K x_{3} - ω t) e_{1}$ and $B = (E_{0} / c) \cos (K x_{3} - ω t) e_{2}$ , fields independent of $x_{1}, x_{2}$ and $T$ -periodic in $t$ with $T = 2 π / ω$ . By [F2] the averaged Poynting flux is $⟨ S ⟩ = I \hat{k}$ with $I = \frac{ε_{0} c}{2} E_{0}^{2}$ , and the averaged normal stress of the incident wave is the constant $⟨ σ_{33} ⟩ = - ⟨ u ⟩ = - I / c$ everywhere.

2.1step 1.1F2A1

Absorber: by [A1] the vacuum field is the incident wave alone. Step 1.1 gives $⟨ σ_{33} (x_{3}) ⟩ = - I / c$ for every $x_{3} < 0$ , independently of $x_{3}$ . Its limit as $x_{3} ↑ 0$ is therefore $⟨ σ_{33} (0^{-}) ⟩ = - I / c$ .

2.2step 1.1F1F4F5F6F9A2algebra

Reflector: the standing-wave averaged stress. Adopt [A2]; the reflected data satisfy $k^{'} \cdot e^{'} = (- k) \cdot (- e) = 0$ , $∣ k^{'} ∣ = ∣ k ∣ = ω / c$ and $b^{'} = k^{'} \times e^{'} / ω = (- K e_{3}) \times (- E_{0} e_{1}) / ω = (E_{0} / c) e_{2} = b$ , so by [F1] the reflected pair is a plane-wave solution and by [F5] the sum $(E, B) = (E_{1}, B_{1}) + (E^{'}, B^{'})$ is a smooth source-free solution on the vacuum side; explicitly $E = E_{0} [\cos (K x_{3} - ω t) - \cos (K x_{3} + ω t)] e_{1} = 2 E_{0} \sin (K x_{3}) \sin (ω t) e_{1}$ and $B = (E_{0} / c) [\cos (K x_{3} - ω t) + \cos (K x_{3} + ω t)] e_{2} = (2 E_{0} / c) \cos (K x_{3}) \cos (ω t) e_{2}$ by the sum-to-product identities, so the tangential electric field vanishes on $x_{3} = 0$ . Since $E_{3} = B_{3} = 0$ , [F4] gives $σ_{33} = - \frac{ε_{0}}{2} ∣ E ∣^{2} - \frac{1}{2 μ_{0}} ∣ B ∣^{2} = - ε_{0} \cdot 2 E_{0}^{2} \sin^{2} (K x_{3}) \sin^{2} (ω t) - \frac{1}{2 μ_{0}} \cdot \frac{4 E_{0}^{2}}{c^{2}} \cos^{2} (K x_{3}) \cos^{2} (ω t)$ ; period-averaging with [F6] and using $\frac{E_{0}^{2}}{μ_{0} c^{2}} = ε_{0} E_{0}^{2}$ gives $⟨ σ_{33} (x_{3}) ⟩ = - ε_{0} E_{0}^{2} [\sin^{2} (K x_{3}) + \cos^{2} (K x_{3})] = - ε_{0} E_{0}^{2}$ , and $ε_{0} E_{0}^{2} = 2 I / c$ by [F9]; in particular $⟨ σ_{33} (0^{-}) ⟩ = - 2 I / c$ .

3.1step 2.1A1algebra

Absorber force. By [A1] the back-face stress is zero, so the signed pressure is $p = ⟨ σ_{33} ⟩_{b a c k} - ⟨ σ_{33} (0^{-}) ⟩ = 0 - (- I / c) = I / c$ , so $F / A = (I / c) e_{3} = (I / c) \hat{k}$ , the propagation direction of the incident beam: the wave pushes the absorber with pressure $I / c$ .

3.2step 2.2A2algebra

Reflector force. By [A2] the material-side stress is zero, so the signed pressure is $p = 0 - ⟨ σ_{33} (0^{-}) ⟩ = 2 I / c$ , so $F / A = (2 I / c) e_{3} = (2 I / c) \hat{k}$ : twice the absorbing pressure; the momentum of the reflected wave being reversed.

4.1step 3.1step 3.2A1A2given∎

Status and scope. The two conclusions are conditional on the adopted idealisations [A1]/[A2]: neither the vanishing of the reflected or transmitted fields nor the pillbox force law is derived from a material model, and the incident vacuum field is not asserted to extend by zero across the sharp interface as a global Maxwell solution; a real absorption layer or reflector would require a compatible material-field solution. The deduction from the adopted assumptions and the computed averaged stresses is complete, the geometry is the ideal infinite plane beam at normal incidence, and oblique incidence, finite beams, material response, radiation reaction and measurements are excluded; no measured pressure, statistical statement or empirical claim is made.

Source notes

The cited treatments give the momentum flux $I / c$ of the incident beam and state the absorber pressure $I / c$ and reflector pressure $2 I / c$ for perfect absorbers and reflectors; the deduction above makes the required boundary assumptions explicit (A1)/(A1')/(A2) and computes both averaged normal stresses from the framework's definitions, so that the textbook statements are the motivation rather than premises. No material law, absorbed or reflected amplitude, or observed force is asserted.

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Ideal deterministic two-wave interference and its fringe geometry

Physical scope

A deterministic interference computation for two explicit ideal coherent plane waves of equal frequency and amplitude in a named inertial frame with SI units: averaged energy density, fringe period, visibility and the copropagating amplitude law. Excluded: real sources and detectors, finite coherence and bandwidth, sampling and noise models, finite-sample statistics, and any empirical claim; no discrimination between classical and quantum descriptions is asserted.

Example

Let $K > 0$ , $ω : = c K$ , let $α \in (0, π / 2)$ (half the angle between the beams) and $φ \in R$ , and take the two plane waves of Real plane-wave fields encoded by complex amplitudes with $k_{1} = K (\sin α e_{1} + \cos α e_{3}), k_{2} = K (- \sin α e_{1} + \cos α e_{3}),$ $e_{1}^{a m p} = E_{0} e_{2}, e_{2}^{a m p} = E_{0} e^{i φ} e_{2},$ where $E_{0} > 0$ carries $V m^{- 1}$ and both amplitudes are polarized along $e_{2}$ , perpendicular to both wavevectors; with $b_{j} = k_{j} \times e_{j}^{a m p} / ω$ and $∣ k_{j} ∣ = ω / c$ both pairs satisfy the amplitude equations of Dispersion, transversality and the two polarization conditions for plane-wave solutions, so the superposition is a source-free smooth Maxwell solution (Superposition intensity and the coherent interference cross term). Write $δ (x) : = (k_{2} - k_{1}) \cdot x = - 2 K \sin α x_{1}$ . Then:

(i) [Averaged energy density] $⟨ u ⟩ (x) = ε_{0} E_{0}^{2} [1 + \cos^{2} α \cos (2 K \sin α x_{1} - φ)]$ for every $x$ , by clause (b) of Superposition intensity and the coherent interference cross term with $e_{1}^{a m p} \cdot \overline{e_{2}^{a m p}} = E_{0}^{2} e^{- i φ}$ and $({\hat{k}}_{1} \times e_{1}^{a m p}) \cdot ({\hat{k}}_{2} \times \overline{e_{2}^{a m p}}) = E_{0}^{2} e^{- i φ} \cos 2 α$ .

(ii) [Fringes and visibility] At fixed $x_{2}, x_{3}$ the averaged energy density is periodic in $x_{1}$ with period $π / (K \sin α) = λ / (2 \sin α)$ , where $λ = 2 π / K$ ; its maximum and minimum are $ε_{0} E_{0}^{2} (1 \pm \cos^{2} α)$ , so the visibility $V : = \frac{⟨ u ⟩_{\max} - ⟨ u ⟩_{\min}}{⟨ u ⟩_{\max} + ⟨ u ⟩_{\min}}$ equals $\cos^{2} α = \frac{1}{2} (1 + {\hat{k}}_{1} \cdot {\hat{k}}_{2})$ . In particular $V \to 1$ as $α ↓ 0$ through crossing beams, and $V \to 0$ as $α \to π / 2$ . At $α = 0$ with fixed relative phase the profile is spatially constant: its spatial visibility is $0$ when its value is positive, and the displayed ratio is undefined when the field vanishes.

(iii) [The promised copropagating amplitude law] For $k_{1} = k_{2} = k$ with $∣ k ∣ = K$ and $k \cdot e_{2} = 0$ , retain the common linear polarization $e_{2}$ and amplitudes $E_{0} e_{2}$ , $E_{0} e^{i Δ} e_{2}$ (so that the sum is the single plane wave with vector amplitude $E_{0} (1 + e^{i Δ}) e_{2}$ ), $⟨ u ⟩ = ε_{0} E_{0}^{2} (1 + \cos Δ) = 2 ε_{0} E_{0}^{2} \cos^{2} (Δ / 2) \propto ∣ 1 + e^{i Δ} ∣^{2}$ , the form used in the double-slit discussion of the content model. Scanning $Δ$ gives phase-scan visibility $1$ , distinct from spatial visibility at fixed phase.

(iv) [Statistical qualifications] This is a deterministic prediction of the classical wave model for the specified coherent two-wave preparation, at every point of the ideal plane-wave field; a measured fringe pattern or detector histogram would additionally require a declared measurement model (detector response, finite bin width, background, integration time, sampling) and a statistical analysis. The example reports no observation, no finite-sample claim and no visibility estimate from data; in particular classical electromagnetic waves interfere, so the prediction alone does not discriminate between classical and quantum descriptions of light. All quantities are idealized field quantities of the framework.

Facts & Assumptions

Given: $K > 0$ , $ω = c K$ , $α \in (0, π / 2)$ , $φ \in R$ , $E_{0} > 0$ in $V m^{- 1}$ , the wavevectors $k_{1}, k_{2}$ and amplitudes $e_{1}^{a m p}, e_{2}^{a m p}$ above, the associated $b_{j} = k_{j} \times e_{j}^{a m p} / ω$ , and the frame basis $(e_{1}, e_{2}, e_{3})$ .

[F1]

Coherent interference (Superposition intensity and the coherent interference cross term): for two same-frequency transverse plane waves with amplitudes $e_{1}, e_{2}$ , real coefficients $a_{1}, a_{2}$ and $δ (x) = (k_{2} - k_{1}) \cdot x$ , the sum is a smooth source-free solution and $⟨ u ⟩ = \frac{ε_{0}}{2} (a_{1}^{2} ∣ e_{1} ∣^{2} + a_{2}^{2} ∣ e_{2} ∣^{2}) + \frac{ε_{0}}{2} a_{1} a_{2} Re (W)$ with $W = (e_{1} \cdot {\overline{e}}_{2} + ({\hat{k}}_{1} \times e_{1}) \cdot ({\hat{k}}_{2} \times {\overline{e}}_{2})) e^{- i δ}$ ; if $k_{1} = k_{2} = : k$ the sum is the single plane wave with amplitude $a_{1} e_{1} + a_{2} e_{2}$ and $⟨ u ⟩ = \frac{ε_{0}}{2} ∣ a_{1} e_{1} + a_{2} e_{2} ∣^{2}$ .

[F2]

Plane-wave solutions (Dispersion, transversality and the two polarization conditions for plane-wave solutions): for $∣ k ∣ = ω / c$ , $k \cdot e = 0$ and $b = k \times e / ω$ the pair lies in the source-free smooth Maxwell branch.

[F3]

Plane-wave encoding (Real plane-wave fields encoded by complex amplitudes): the fields are the real parts of $e \exp (i (k \cdot x - ω t))$ and $b \exp (i (k \cdot x - ω t))$ , and the complex amplitudes are constant.

[F4]

Euclidean structure (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ): the frame basis is orthonormal, $⟨ e_{i}, e_{j} ⟩ = δ_{i j}$ , and $∣ v ∣^{2} = ⟨ v, v ⟩$ ; in particular ${\hat{k}}_{j} = k_{j} / K$ satisfies $∣ {\hat{k}}_{j} ∣ = 1$ .

[F5]

Cross product (The cross product in $R^{3}$ ): $u \times v$ is given by the coordinate differences of products in the positively oriented basis $(e_{1}, e_{2}, e_{3})$ .

[F6]

SI conventions (Framework-local SI units, the vacuum constants and the exact/measured status of their values): $ε_{0}$ and $c$ are the framework's positive constants with $ε_{0} = (μ_{0} c^{2})^{- 1}$ .

Verification

technique · direct

1.1givenF2F3F4F5F6algebra

Setup and amplitude conditions. Since $\sin^{2} α + \cos^{2} α = 1$ , one has $∣ k_{1} ∣ = ∣ k_{2} ∣ = K = ω / c$ and ${\hat{k}}_{1} = \sin α e_{1} + \cos α e_{3}$ , ${\hat{k}}_{2} = - \sin α e_{1} + \cos α e_{3}$ ; both $e_{j}^{a m p}$ are multiples of $e_{2}$ , which is orthogonal to $e_{1}$ and $e_{3}$ , so $k_{j} \cdot e_{j}^{a m p} = 0$ by [F4], and with $b_{j} = k_{j} \times e_{j}^{a m p} / ω$ the amplitude conditions of [F2] hold for both pairs; moreover $δ (x) = (k_{2} - k_{1}) \cdot x = (- 2 K \sin α e_{1}) \cdot x = - 2 K \sin α x_{1}$ , and ${\hat{k}}_{1} \cdot {\hat{k}}_{2} = - \sin^{2} α + \cos^{2} α = \cos 2 α$ by [F4].

2.1step 1.1F1F4F5algebra

Clause (i). Apply [F1] with $a_{1} = a_{2} = 1$ and the amplitudes $e_{1}^{a m p} = E_{0} e_{2}$ , $e_{2}^{a m p} = E_{0} e^{i φ} e_{2}$ : then $∣ e_{j}^{a m p} ∣^{2} = E_{0}^{2}$ and $e_{1}^{a m p} \cdot \overline{e_{2}^{a m p}} = E_{0}^{2} e^{- i φ}$ , while $k_{j} \times e_{2}$ computed coordinatewise by [F5] gives ${\hat{k}}_{1} \times e_{2} = \sin α e_{3} - \cos α e_{1}$ and ${\hat{k}}_{2} \times e_{2} = - \sin α e_{3} - \cos α e_{1}$ (using $∣ k_{j} ∣ = K$ ), hence $({\hat{k}}_{1} \times e_{1}^{a m p}) \cdot ({\hat{k}}_{2} \times \overline{e_{2}^{a m p}}) = E_{0}^{2} e^{- i φ} (\cos^{2} α - \sin^{2} α) = E_{0}^{2} e^{- i φ} \cos 2 α$ by [F4]; the bracket is therefore $E_{0}^{2} e^{- i φ} (1 + \cos 2 α) = 2 E_{0}^{2} \cos^{2} α e^{- i φ}$ , so $W = 2 E_{0}^{2} \cos^{2} α e^{- i (δ + φ)}$ , $Re (W) = 2 E_{0}^{2} \cos^{2} α \cos (2 K \sin α x_{1} - φ)$ , and $⟨ u ⟩ = \frac{ε_{0}}{2} (E_{0}^{2} + E_{0}^{2}) + \frac{ε_{0}}{2} Re (W) = ε_{0} E_{0}^{2} [1 + \cos^{2} α \cos (2 K \sin α x_{1} - φ)]$ by [F1] and $\cos (- δ - φ) = \cos (δ + φ)$ ; this is clause (i).

3.1step 2.1F4algebra

Clause (ii). By step 2.1, at fixed $x_{2}, x_{3}$ the function $x_{1} \mapsto ⟨ u ⟩ (x)$ is $ε_{0} E_{0}^{2} [1 + \cos^{2} α \cos (2 K \sin α x_{1} - φ)]$ ; since $\cos^{2} α > 0$ for $α \in (0, π / 2)$ and $\sin α > 0$ , its oscillation has period $2 π / (2 K \sin α) = π / (K \sin α) = λ / (2 \sin α)$ with $λ = 2 π / K$ , maximum $ε_{0} E_{0}^{2} (1 + \cos^{2} α)$ at $\cos (2 K \sin α x_{1} - φ) = 1$ and minimum $ε_{0} E_{0}^{2} (1 - \cos^{2} α)$ at $\cos (2 K \sin α x_{1} - φ) = - 1$ ; hence $V = \cos^{2} α$ , and $\cos^{2} α = \frac{1}{2} (1 + \cos 2 α) = \frac{1}{2} (1 + {\hat{k}}_{1} \cdot {\hat{k}}_{2})$ by step 1.1. As $α ↓ 0$ with $α > 0$ the value $\cos^{2} α \to 1$ , and as $α \to π / 2$ it tends to $0$ . At $α = 0$ one has $k_{1} = k_{2}$ and the profile is $ε_{0} E_{0}^{2} [1 + \cos (- 2 K \sin α x_{1} - φ)]$ evaluated at $α = 0$ , which is the spatial constant $ε_{0} E_{0}^{2} [1 + \cos φ] = 2 ε_{0} E_{0}^{2} \cos^{2} (φ / 2)$ : if this value is positive then max $=$ min and the displayed ratio is $0$ , while if it vanishes (that is, $φ \equiv π m o d 2 π$ ) the ratio is the undefined expression $0 / 0$ .

3.2step 1.1step 2.1F1algebra

Clause (iii). If $k_{1} = k_{2} = : k$ with equal amplitudes and relative phase $Δ$ , then by [F1] (equal-wavevector clause) the sum is the single plane wave with amplitude $E_{0} (1 + e^{i Δ}) e_{2}$ , and $⟨ u ⟩ = \frac{ε_{0}}{2} ∣ E_{0} (1 + e^{i Δ}) e_{2} ∣^{2} = \frac{ε_{0}}{2} E_{0}^{2} ∣ 1 + e^{i Δ} ∣^{2} = ε_{0} E_{0}^{2} (1 + \cos Δ) = 2 ε_{0} E_{0}^{2} \cos^{2} (Δ / 2)$ using $∣ 1 + e^{i Δ} ∣^{2} = 2 + 2 \cos Δ$ ; scanning $Δ$ over $R$ gives maximum $2 ε_{0} E_{0}^{2}$ at $Δ \equiv 0$ and minimum $0$ at $Δ \equiv π$ modulo $2 π$ , so the phase-scan visibility $(\max - \min) / (\max + \min)$ equals $1$ , a different quantity from the spatial visibility at fixed phase.

4.1step 2.1step 3.1step 3.2given∎

Clause (iv) and conclusion. Clause (i) is step 2.1, clause (ii) is step 3.1 and clause (iii) is step 3.2; clause (iv) records that all quantities above are deterministic field quantities of the stated ideal coherent preparation, that a measured fringe pattern or histogram would require a declared detector response, finite bin width, background, integration time and sampling model together with a statistical analysis, and that no observation or finite-sample visibility estimate is reported; in particular classical electromagnetic waves interfere, so the deterministic prediction alone does not discriminate between classical and quantum descriptions of light.

Remarks

Prediction, not measurement. The example computes the ideal averaged energy density of the specified two-wave field and its fringe geometry; a finite detector record is a different object requiring its own measurement and sampling model, and no agreement or disagreement with data is claimed here.

Vacuum waves and polarization — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

Linear, circular and elliptical amplitude choices

Example

Facts & Assumptions

Verification

Remarks

A nonzero whole-space plane wave has infinite total energy

Example

Facts & Assumptions

Verification

Remarks

Ideal absorber and ideal reflector momentum transfer (radiation pressure)

Statement

Facts & Assumptions

Proof

Source notes

Ideal deterministic two-wave interference and its fringe geometry

Example

Facts & Assumptions

Verification

Remarks

Sources