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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.

Ideal deterministic two-wave interference and its fringe geometry

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

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