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Wave interfaces and conductors

1 · Prerequisites

2 · Summary

This page builds the interface and conducting-medium wave layer of the classical electromagnetic framework. Two mathematical lemmas first fix the mathematics used later: pairwise-distinct complex exponential characters are independent on a nonempty open set, with the real-part grouping rule that identifies which frequency groups a real-linear relation can constrain; and the two-equation interface admittance system is solved in closed form, with the identity $∣ r ∣^{2} + (Re Y_{2} / Y_{1}) ∣ t ∣^{2} = 1$ and its pure-imaginary and real-positive specialisations. On that basis the homogeneous isotropic medium parameters are defined — phase speed, wave impedance, refractive index, the complex propagation constant selected by $Re, Im \geq 0$ and the skin depth — and the plane-wave amplitude conditions, dispersion relation, transverse structure, period-averaged energy density and flux, and the evanescent decay identities are derived for the explicit ideal plane-wave fields. Phase matching at the plane interface then gives the common tangential wavevector and frequency, coplanarity, the reflection law and Snell's law; normal and oblique incidence are solved as admittance systems with the flux budgets, the Ohmic absorption reading of the entering flux, the Brewster condition with its equal-medium qualification, and total internal reflection with its critical-angle case. The conducting-medium theorem gives the exact attenuation constant, the closed-form skin depth and its bounded good- and poor-conductor regimes. Every item states its own hypotheses, regularity, units, conventions and exclusions, and no empirical content is asserted.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Distinct exponential characters are linearly independent on a nonempty open set, and the real-part grouping rule

Statement

Let $n \geq 1$ , let $U \subseteq R^{n}$ be a nonempty open set, let $λ_{1}, \dots, λ_{m} \in C^{n}$ ( $m \geq 1$ ) be pairwise distinct, and let $c_{1}, \dots, c_{m} \in C^{q}$ for some $q \geq 1$ (the scalar case $q = 1$ included; all statements below are componentwise). Write $λ_{j} \cdot z$ for the componentwise bilinear extension of the Euclidean inner product of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ; the variable $z$ remains real and $e^{i w}$ for the complex exponential of The complex exponential by its power series at $i w$ , and consider the $C^{q}$ -valued map $F (z) : = \sum_{j = 1}^{m} c_{j} e^{i λ_{j} \cdot z}$ on $U$ . Then:

(a) [Complex independence] if $F (z) = 0$ for every $z \in U$ , then $c_{1} = \dots = c_{m} = 0$ .

(b) [Real-part grouping rule] if $Re F (z) = 0$ for every $z \in U$ , then for every $ν \in C^{n}$ the grouped coefficient $C_{ν} : = \sum_{λ_{j} = ν} c_{j} + \sum_{λ_{j} = - \overline{ν}} \overline{c_{j}}$ vanishes; in particular $c_{j} = 0$ whenever $λ_{j} \neq - \overline{λ_{j}}$ and $λ_{j} \notin {λ_{k}, - \overline{λ_{k}} : k \neq j}$ . At a self-paired frequency $ν = - \overline{ν}$ the coefficient is $2 Re \sum_{λ_{j} = ν} c_{j}$ ; in particular at $ν = 0$ only the real part is constrained, and a purely imaginary coefficient is allowed.

(c) [Scope] no hypothesis beyond pairwise distinctness of the $λ_{j}$ and nonemptiness and openness of $U$ is used; $λ_{j} = 0$ is allowed and the $λ_{j}$ need not be nonzero, ordered or normalized. The statement concerns the open set $U$ and asserts nothing on sets with empty interior, where distinct characters can agree.

Facts & Assumptions

Given: An integer $n \geq 1$ , a nonempty open $U \subseteq R^{n}$ , pairwise distinct $λ_{1}, \dots, λ_{m} \in C^{n}$ with $m \geq 1$ , and coefficients $c_{1}, \dots, c_{m} \in C^{q}$ with $q \geq 1$ ; the map $F (z) = \sum_{j = 1}^{m} c_{j} e^{i λ_{j} \cdot z}$ on $U$ , with the inner product of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ .

[F1]

The complex exponential is the everywhere convergent series $\exp w = \sum_{n \geq 0} w^{n} / n!$ , with $\exp 0 = 1$ , and it is entire with $\exp^{'} = \exp$ (The complex exponential by its power series, The complex exponential is entire and its complex derivative is itself). Also $\exp (z + w) = \exp z \exp w$ ( $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential), hence $\exp w \exp (- w) = 1$ and $\exp w \neq 0$ for every complex $w$ .

[F2]

Conjugation and modulus: $\overline{a + b i} = a - b i$ and $∣ a + b i ∣ = \sqrt{a^{2} + b^{2}}$ , so $w + \overline{w} = 2 Re w$ by the coordinate formulas (Real and imaginary parts, complex conjugation, and modulus); conjugation is an involutive real-field automorphism satisfying $\overline{z + w} = \overline{z} + \overline{w}$ , $\overline{z w} = \overline{z} \overline{w}$ , $\overline{\overline{z}} = z$ , and $z \overline{z} = ∣ z ∣^{2}$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[F3]

$C$ is a field containing the embedded copy of $R$ , every complex number has the unique form $a + b i$ with $a, b \in R$ and $i^{2} = - 1$ , and a product in a field vanishes only if one of its factors vanishes ( $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ , The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ ).

[F4]

In The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , vectors are functions on ${0, \dots, n - 1}$ and $⟨ x, y ⟩ = \sum_{r = 0}^{n - 1} x (r) y (r)$ . For the one-based coordinate notation used here, define $x_{k} : = x (k - 1)$ for $1 \leq k \leq n$ , for real or complex vectors alike. Thus the componentwise bilinear extension is $ξ \cdot ζ : = \sum_{r = 0}^{n - 1} ξ (r) ζ (r) = \sum_{k = 1}^{n} ξ_{k} ζ_{k}$ . For $n \geq 2$ define the tail $x^{'} : {0, \dots, n - 2} \to C$ by $x^{'} (r) : = x (r + 1)$ (with real values when $x$ is real); then $λ \cdot z = λ_{1} z_{1} + λ^{'} \cdot z^{'}$ under $z = (z_{1}, z^{'})$ , $λ = (λ_{1}, λ^{'})$ . In dimension one identify a vector $x$ with its sole value $x (0) = x_{1}$ , so the product is ordinary scalar multiplication. These are reindexings of the same finite sum, with no coordinate outside its domain.

[F5]

Real-variable calculus: partial derivatives of maps on open subsets of $R^{m}$ are derivatives of line maps (Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ); sums, real scalar multiples and products of differentiable functions are differentiable with the sum, scalar-multiple and product rules, and quotients are differentiable where the denominator is nonzero (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ); the chain rule holds for total derivatives (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

Proof

technique · direct; reduce to the scalar case, conjugate the exponential by its series, prove (a) by induction on the dimension using the one-dimensional annihilating operator $\prod_{j\ne j_0}(d/dt-i\lambda_j)$, then obtain (b) by applying (a) to $F+\overline F$ grouped by frequency

1.1givenF3algebra

Reduction to the scalar case. Both assertions are componentwise in $C^{q}$ : $F (z) = 0$ holds if and only if every component $F_{k} (z) = \sum_{j} (c_{j})_{k} e^{i λ_{j} \cdot z}$ vanishes, and $Re F (z) = 0$ if and only if $Re F_{k} (z) = 0$ for every $k$ , while the vector $C_{ν}$ of clause (b) vanishes if and only if each of its components $\sum_{λ_{j} = ν} (c_{j})_{k} + \sum_{λ_{j} = - \overline{ν}} \overline{(c_{j})_{k}}$ vanishes. Applying the scalar assertions $q = 1$ to each component function therefore yields the general assertions, so we may and do assume $c_{1}, \dots, c_{m} \in C$ in the rest of the proof.

1.2F1F2algebra

Conjugation of the exponential. For every $w \in C$ , $\overline{\exp w} = \exp \overline{w}$ . By [F1] the complex numbers $\exp w$ and $\exp \overline{w}$ are the sums of the everywhere convergent series $\sum_{n} w^{n} / n!$ and $\sum_{n} {\overline{w}}^{n} / n!$ , with partial sums $S_{N} (w) = \sum_{n \leq N} w^{n} / n!$ and $S_{N} (\overline{w}) = \sum_{n \leq N} {\overline{w}}^{n} / n!$ ; by [F2] conjugation fixes each real number $n!$ and satisfies $\overline{z w} = \overline{z} \overline{w}$ and $\overline{z + w} = \overline{z} + \overline{w}$ , so $\overline{S_{N} (w)} = S_{N} (\overline{w})$ for every $N$ . Also by [F2], $\overline{u} - \overline{v} = \overline{u - v}$ and, by the coordinate formula, $∣ \overline{u} ∣ = \sqrt{(Re u)^{2} + (- Im u)^{2}} = ∣ u ∣$ , whence $∣ \overline{u} - \overline{v} ∣ = ∣ u - v ∣$ . Writing $d = \overline{\exp w} - \exp \overline{w}$ , for every $N$ we have $d = (\overline{\exp w} - S_{N} (\overline{w})) + (S_{N} (\overline{w}) - \exp \overline{w}) = (\overline{\exp w} - \overline{S_{N} (w)}) + (S_{N} (\overline{w}) - \exp \overline{w})$ , so $∣ d ∣ \leq ∣ \exp w - S_{N} (w) ∣ + ∣ S_{N} (\overline{w}) - \exp \overline{w} ∣$ ; given $ε > 0$ , the definition of convergence of the two series supplies $N$ with both summands below $ε / 2$ , so $∣ d ∣ < ε$ , and since $ε > 0$ was arbitrary $∣ d ∣ = 0$ , which forces $d = 0$ by the definiteness of the modulus in [F2].

1.3F1F3F5algebra

Characters and the annihilating operator. For $μ \in C$ let $φ_{μ} (t) : = \exp (i μ t)$ for $t \in R$ , the one-variable restriction of the complex exponential along the real-linear map $t \mapsto i μ t$ . By [F5], the chain rule applied to that map and the entire exponential of [F1] gives $φ_{μ}^{'} (t) = i μ \exp (i μ t) = i μ φ_{μ} (t)$ , and the sum, scalar-multiple and product rules of [F5] give, for any finite combination $\sum_{μ} a_{μ} φ_{μ}$ with complex coefficients, derivative $\sum_{μ} a_{μ} i μ φ_{μ}$ , hence $(d / d t - i μ) φ_{μ} = 0$ and $(d / d t - i μ) φ_{ν} = i (ν - μ) φ_{ν}$ for all $ν \in C$ . Consequently, for pairwise distinct $μ_{1}, \dots, μ_{m}$ and a fixed $j_{0}$ , the composed operator $L_{j_{0}} : = \prod_{j \neq j_{0}} (d / d t - i μ_{j})$ is complex-linear on finite combinations and acts on $φ_{μ_{k}}$ as multiplication by $0$ if $k \neq j_{0}$ and by the constant $\prod_{j \neq j_{0}} i (μ_{j_{0}} - μ_{j})$ if $k = j_{0}$ ; that constant is nonzero because $i \neq 0$ and $μ_{j_{0}} - μ_{j} \neq 0$ for $j \neq j_{0}$ by pairwise distinctness, and a product of nonzero field elements is nonzero by [F3].

1.4givenF1F4algebra

Induction step: reduction to a product box. Let $n \geq 2$ and assume assertion (a) established in dimension $n - 1$ for every number of frequencies and every nonempty open subset of $R^{n - 1}$ . Let $U \subseteq R^{n}$ be nonempty and open, fix $z^{*} \in U$ and, by openness, $ε > 0$ with the open product box $B : = \prod_{k = 1}^{n} (z_{k}^{*} - ε, z_{k}^{*} + ε) \subseteq U$ ; put $B^{'} : = \prod_{k = 2}^{n} (z_{k}^{*} - ε, z_{k}^{*} + ε) \subseteq R^{n - 1}$ . Write $z = (z_{1}, z^{'})$ with $z^{'} \in R^{n - 1}$ and $λ_{j} = (λ_{j 1}, λ_{j}^{'})$ . For each value $μ$ in the finite set ${λ_{11}, \dots, λ_{m 1}}$ put $I_{μ} : = {j : λ_{j 1} = μ}$ and $P_{μ} (z^{'}) : = \sum_{j \in I_{μ}} c_{j} e^{i λ_{j}^{'} \cdot z^{'}}$ ; then, by the splitting identity of [F4], the exponential addition law of [F1] and the distributivity of the finite sum, $F (z_{1}, z^{'}) = \sum_{μ} e^{i μ z_{1}} P_{μ} (z^{'})$ for every $(z_{1}, z^{'}) \in B$ , the outer sum running over the pairwise distinct values $μ$ .

2.1givenstep 1.3F1F3

Base case, dimension one. Let $U \subseteq R$ be nonempty and open, so that it contains a nonempty open interval $J$ ; suppose $F = \sum_{j = 1}^{m} c_{j} φ_{λ_{j}} \equiv 0$ on $U$ . Fix $j_{0}$ ; then $L_{j_{0}} F \equiv 0$ on $J$ , and by step 1.3 and the linearity of $L_{j_{0}}$ we have $L_{j_{0}} F = c_{j_{0}} (\prod_{j \neq j_{0}} i (λ_{j_{0}} - λ_{j})) φ_{λ_{j_{0}}}$ on $J$ . Choose $t_{0} \in J$ . Evaluating there gives $c_{j_{0}} (\prod_{j \neq j_{0}} i (λ_{j_{0}} - λ_{j})) \exp (i λ_{j_{0}} t_{0}) = 0$ ; cancelling the nonzero exponential by [F1] gives $c_{j_{0}} \prod_{j \neq j_{0}} i (λ_{j_{0}} - λ_{j}) = 0$ ; the second factor is a product of nonzero complex numbers, hence nonzero by [F3], so $c_{j_{0}} = 0$ by the field property in [F3]. As $j_{0}$ was arbitrary, $c_{1} = \dots = c_{m} = 0$ ; this proves (a) for $n = 1$ .

2.2givenstep 1.2F2

The conjugate sum. Suppose now that $Re F \equiv 0$ on $U$ , and define $\tilde{F} (z) : = \sum_{j = 1}^{m} \overline{c_{j}} e^{- i \overline{λ_{j}} \cdot z}$ on $U$ . For real $z$ and every $j$ , step 1.2 applied to $w = i λ_{j} \cdot z$ gives $\overline{e^{i λ_{j} \cdot z}} = \exp (\overline{i λ_{j} \cdot z})$ , and by [F2] and the reality of $z$ we have $\overline{i λ_{j} \cdot z} = \overline{i} \overline{λ_{j} \cdot z} = (- i) (\overline{λ_{j}} \cdot z) = - i \overline{λ_{j}} \cdot z$ ; therefore $\overline{F (z)} = \sum_{j} \overline{c_{j}} e^{- i \overline{λ_{j}} \cdot z} = \tilde{F} (z)$ on $U$ by [F2]. Since $w + \overline{w} = 2 Re w$ by [F2], the hypothesis $2 Re F \equiv 0$ reads $F + \tilde{F} \equiv 0$ on $U$ .

3.1step 2.1step 1.4

One-dimensional slices. Suppose $F \equiv 0$ on $U$ , fix $z^{'} \in B^{'}$ and define $G (t) : = \sum_{μ} e^{i μ t} P_{μ} (z^{'})$ for $t$ in the interval $J : = (z_{1}^{*} - ε, z_{1}^{*} + ε)$ . By step 1.4, $G (t) = F (t, z^{'}) = 0$ for every $t \in J$ , and $G$ is a finite combination of the pairwise distinct characters $φ_{μ}$ with complex coefficients $P_{μ} (z^{'})$ ; the base case of step 2.1 applied on the nonempty open interval $J$ therefore gives $P_{μ} (z^{'}) = 0$ for every $μ$ . As $z^{'} \in B^{'}$ was arbitrary, each $P_{μ}$ vanishes identically on the nonempty open set $B^{'}$ .

4.1step 3.1step 1.4induction

Induction hypothesis on each group. Fix a value $μ$ . The vectors $λ_{j}^{'}$ , $j \in I_{μ}$ , are pairwise distinct: if $λ_{j}^{'} = λ_{k}^{'}$ and $λ_{j 1} = λ_{k 1} = μ$ , then $λ_{j} = λ_{k}$ , hence $j = k$ by pairwise distinctness of the $λ_{j}$ . The function $P_{μ} = \sum_{j \in I_{μ}} c_{j} e^{i λ_{j}^{'} \cdot z^{'}}$ vanishes identically on the nonempty open set $B^{'} \subseteq R^{n - 1}$ , so assertion (a) in dimension $n - 1$ gives $c_{j} = 0$ for every $j \in I_{μ}$ . Every index lies in $I_{λ_{j 1}}$ , so $c_{1} = \dots = c_{m} = 0$ ; this completes the induction and proves (a) in every dimension $n \geq 1$ .

5.1givenstep 2.2step 4.1F4algebra

Frequency grouping and application of part (a). Let $Φ \subseteq C^{n}$ be the finite set of distinct values among $λ_{1}, \dots, λ_{m}, - \overline{λ_{1}}, \dots, - \overline{λ_{m}}$ , and for $ν \in Φ$ put $C_{ν} : = \sum_{j : λ_{j} = ν} c_{j} + \sum_{j : - \overline{λ_{j}} = ν} \overline{c_{j}} = \sum_{λ_{j} = ν} c_{j} + \sum_{λ_{j} = - \overline{ν}} \overline{c_{j}}$ . Regrouping the finite sum of step 2.2 over the distinct frequencies and using the bilinearity of the extended inner product in [F4] gives $F + \tilde{F} = \sum_{ν \in Φ} C_{ν} e^{i ν \cdot z}$ on $U$ ; the frequencies $ν \in Φ$ are pairwise distinct by construction, so assertion (a) as proved in steps 2.1 and 4.1 applied to this finite combination gives $C_{ν} = 0$ for every $ν \in Φ$ . For $ν \notin Φ$ both defining sums are empty, so $C_{ν} = 0$ trivially; hence $C_{ν} = 0$ for every $ν \in C^{n}$ .

6.1step 5.1step 2.2F1F2F3algebra∎

Consequences and scope. (i) If $ν = - \overline{ν}$ is self-paired, the condition $- \overline{λ_{j}} = ν$ is equivalent to $λ_{j} = ν$ , so both sums of $C_{ν}$ run over the same index set and $C_{ν} = \sum_{λ_{j} = ν} (c_{j} + \overline{c_{j}}) = 2 Re \sum_{λ_{j} = ν} c_{j}$ by [F2]; at $ν = 0$ this constrains only the real part, and the constraint is genuinely not stronger: for $m = 1$ , $λ_{1} = 0$ , $q = 1$ and $c_{1} = i$ one has $F \equiv i$ , so $Re F \equiv 0$ and $C_{0} = c_{1} + \overline{c_{1}} = 2 Re i = 0$ , a nonzero purely imaginary coefficient at frequency $0$ . (ii) If $λ_{j} \neq - \overline{λ_{j}}$ and $λ_{j} \notin {λ_{k}, - \overline{λ_{k}} : k \neq j}$ , then ${k : λ_{k} = λ_{j}} = {j}$ and ${k : λ_{k} = - \overline{λ_{j}}} = \emptyset$ (a member $k = j$ would give $λ_{j} = - \overline{λ_{j}}$ , and $k \neq j$ is excluded by hypothesis), so $C_{λ_{j}} = c_{j} + 0 = c_{j} = 0$ by step 5.1, which is the stated consequence. (iii) The scope (c) holds: steps 2.1, 1.4 and 4.1 used only pairwise distinctness of the $λ_{j}$ and the nonemptiness and openness of $U$ , through the interval $J$ and the product box $B$ ; the frequencies were never assumed nonzero, ordered or normalized; the proof is a finite construction using complex arithmetic, the series for the exponential and its derivative, and no choice principle. No assertion is made for sets of empty interior: the openness of $U$ is used when a nonempty open interval and a product box are extracted from it, and the statement (c) records that distinct characters can agree on such smaller sets.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

The two-equation interface admittance system: solution formulas and the normal-flux identity

Statement

Let $Y_{1}, Y_{2} \in C$ with $Y_{1} + Y_{2} \neq 0$ , and let $E_{i}, E_{r}, E_{t} \in C$ satisfy the two linear equations

$E_{i} + E_{r} = E_{t}, Y_{1} (E_{i} - E_{r}) = Y_{2} E_{t} .$

Then: (a) [Solution] If $E_{i} \neq 0$ , then $r : = \frac{E_{r}}{E_{i}} = \frac{Y_{1} - Y_{2}}{Y_{1} + Y_{2}}, t : = \frac{E_{t}}{E_{i}} = \frac{2 Y_{1}}{Y_{1} + Y_{2}};$ if $E_{i} = 0$ then $E_{r} = E_{t} = 0$ and the field ratios are undefined. Throughout (b)-(d), $r, t$ denote the displayed algebraic transfer coefficients $(Y_{1} - Y_{2}) / (Y_{1} + Y_{2})$ and $2 Y_{1} / (Y_{1} + Y_{2})$ , which are defined independently of $E_{i}$ ; when $E_{i} \neq 0$ they equal the field ratios of (a). (b) [Normal-flux identity] If $Y_{1} > 0$ is real (the incident side admittance), then $∣ r ∣^{2} + \frac{Re Y_{2}}{Y_{1}} ∣ t ∣^{2} = 1,$ and explicitly $∣ t ∣^{2} = \frac{4 Y_{1}^{2}}{∣ Y_{1} + Y_{2} ∣^{2}}$ and $∣ r ∣^{2} = 1 - \frac{4 Y_{1} Re Y_{2}}{∣ Y_{1} + Y_{2} ∣^{2}}$ . The second summand $(Re Y_{2} / Y_{1}) ∣ t ∣^{2}$ is nonnegative if and only if $Re Y_{2} \geq 0$ ; it is the normal-flux coefficient of a passive transmitted medium and may be negative for an active one, in which case $∣ r ∣ > 1$ and the identity carries no energy-conservation reading. (c) [Pure-imaginary admittance] If $Y_{1} > 0$ is real and $Y_{2}$ is purely imaginary, then $∣ r ∣ = 1$ , $Re Y_{2} ∣ t ∣^{2} = 0$ and $∣ t ∣^{2} = \frac{4 Y_{1}^{2}}{Y_{1}^{2} + ∣ Y_{2} ∣^{2}} > 0$ ; in particular $r \neq 0$ and $t \neq 0$ . (d) [Real-positive specialisation] If $Y_{1}, Y_{2} > 0$ are real, then $r = (Y_{1} - Y_{2}) / (Y_{1} + Y_{2})$ and $t = 2 Y_{1} / (Y_{1} + Y_{2})$ are real and $∣ r ∣^{2} + (Y_{2} / Y_{1}) ∣ t ∣^{2} = 1$ ; with $Y_{j} = 1 / Z_{j}$ , $Z_{j} > 0$ , this is $r = (Z_{2} - Z_{1}) / (Z_{1} + Z_{2})$ , $t = 2 Z_{2} / (Z_{1} + Z_{2})$ and $R + T = 1$ for $R = ∣ r ∣^{2}$ , $T = (Z_{1} / Z_{2}) ∣ t ∣^{2}$ , in agreement with The two-equation interface system at normal incidence. (e) [Scope] The lemma is pure algebra of $C$ ; the labels 'incident', 'reflected' and 'transmitted' and the interpretation of the second summand as a flux fraction belong to the consuming physical theorem, not to this lemma.

Facts & Assumptions

Given: Complex numbers $Y_{1}, Y_{2}$ with $Y_{1} + Y_{2} \neq 0$ and complex numbers $E_{i}, E_{r}, E_{t}$ satisfying $E_{i} + E_{r} = E_{t}$ and $Y_{1} (E_{i} - E_{r}) = Y_{2} E_{t}$ ; the transfer coefficients $r = (Y_{1} - Y_{2}) / (Y_{1} + Y_{2})$ and $t = 2 Y_{1} / (Y_{1} + Y_{2})$ .

[F1]

Field arithmetic in $C$ : $C$ is a field, every element has the unique form $a + b i$ with $a, b \in R$ , addition and multiplication follow the coordinate formulas, $(a + b i)^{- 1} = (a - b i) / (a^{2} + b^{2})$ for $a + b i \neq 0$ , and a product of nonzero field elements is nonzero (The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ , $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ ).

[F2]

Conjugation and modulus: $z \overline{z} = ∣ z ∣^{2}$ , $∣ z w ∣ = ∣ z ∣ ∣ w ∣$ , $∣ z ∣ \geq 0$ with $∣ z ∣ = 0$ exactly for $z = 0$ , conjugation is an involutive real-field automorphism fixing every real number, and $∣ \overline{z} ∣ = ∣ z ∣$ (Real and imaginary parts, complex conjugation, and modulus, Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[F3]

Real order: the real numbers form an ordered field with positive cone $P$ (Ordered field): by trichotomy a positive real is nonzero, and by closure sums and products of positive reals are positive; the same axioms give that the product of two nonzero reals of equal sign is positive and of opposite signs is negative, and that division by the positive real $Y_{1}$ preserves the sign of a quotient, since $a = (a / Y_{1}) Y_{1}$ has the sign of $a / Y_{1}$ when $Y_{1} > 0$ .

[F4]

Squares in an ordered field: the square of a nonzero element is positive (Squares of nonzero elements are positive), and for $a, b \geq 0$ one has $a < b$ if and only if $a^{2} < b^{2}$ (Squaring is monotone on the nonnegatives).

[F5]

The normal-incidence interface lemma The two-equation interface system at normal incidence: for real $ε_{j}, μ_{j} > 0$ , $Z_{j} = \sqrt{μ_{j} / ε_{j}} > 0$ , and $E_{i}, E_{r}, E_{t} \in C$ with $E_{i} \neq 0$ satisfying $E_{i} + E_{r} = E_{t}$ and $(E_{i} - E_{r}) / Z_{1} = E_{t} / Z_{2}$ , one has $E_{r} / E_{i} = (Z_{2} - Z_{1}) / (Z_{1} + Z_{2})$ , $E_{t} / E_{i} = 2 Z_{2} / (Z_{1} + Z_{2})$ , and $R + T = 1$ for $R = ∣ E_{r} / E_{i} ∣^{2}$ , $T = (Z_{1} / Z_{2}) ∣ E_{t} / E_{i} ∣^{2}$ .

Proof

technique · direct; solve the two linear equations by substitution, expand the moduli with $|z|^2=z\overline z$, and read the sign statements off the ordered-field structure of $\mathbb R$

1.1givenF1algebra

[Solution of the linear system] Substituting the first equation $E_{r} = E_{t} - E_{i}$ into the second gives $Y_{1} (2 E_{i} - E_{t}) = Y_{2} E_{t}$ , that is $2 Y_{1} E_{i} = (Y_{1} + Y_{2}) E_{t}$ after expanding and collecting terms by the field arithmetic of [F1]. Since $Y_{1} + Y_{2} \neq 0$ is invertible in the field $C$ by [F1], $E_{t} = 2 Y_{1} E_{i} / (Y_{1} + Y_{2})$ , and then $E_{r} = E_{t} - E_{i} = (2 Y_{1} - (Y_{1} + Y_{2})) E_{i} / (Y_{1} + Y_{2}) = (Y_{1} - Y_{2}) E_{i} / (Y_{1} + Y_{2})$ , again by [F1].

2.1step 1.1F1algebra

[Consequences of the solution] If $E_{i} \neq 0$ , dividing the two identities of step 1.1 by $E_{i}$ gives $r = E_{r} / E_{i} = (Y_{1} - Y_{2}) / (Y_{1} + Y_{2})$ and $t = E_{t} / E_{i} = 2 Y_{1} / (Y_{1} + Y_{2})$ . If $E_{i} = 0$ , the identities of step 1.1 give $E_{t} = 0$ and $E_{r} = 0$ , so no field ratio exists; the algebraic expressions $r = (Y_{1} - Y_{2}) / (Y_{1} + Y_{2})$ and $t = 2 Y_{1} / (Y_{1} + Y_{2})$ remain defined because $Y_{1} + Y_{2} \neq 0$ , and they agree with the field ratios whenever the latter exist. This proves (a) and the convention stated before (b).

3.1step 2.1F2algebra

[Modular expansions] For any $z, w \in C$ , $∣ z \pm w ∣^{2} = (z \pm w) (\overline{z} \pm \overline{w}) = ∣ z ∣^{2} \pm (z \overline{w} + \overline{z} w) + ∣ w ∣^{2}$ , and $z \overline{w} + \overline{z} w = 2 Re (z \overline{w})$ by [F2]; hence for real $Y_{1}$ and arbitrary $Y_{2}$ , $∣ Y_{1} \pm Y_{2} ∣^{2} = Y_{1}^{2} \pm 2 Y_{1} Re Y_{2} + ∣ Y_{2} ∣^{2}$ , and subtraction gives $∣ Y_{1} + Y_{2} ∣^{2} - ∣ Y_{1} - Y_{2} ∣^{2} = 4 Y_{1} Re Y_{2}$ . Moreover $∣ r ∣^{2} = ∣ Y_{1} - Y_{2} ∣^{2} / ∣ Y_{1} + Y_{2} ∣^{2}$ and $∣ t ∣^{2} = 4 Y_{1}^{2} / ∣ Y_{1} + Y_{2} ∣^{2}$ : indeed $∣ z w ∣ = ∣ z ∣ ∣ w ∣$ and $∣ z ∣^{2} = z \overline{z}$ by [F2] applied to the quotients of step 2.1 give $∣ r ∣^{2} = ∣ Y_{1} - Y_{2} ∣^{2} ∣ Y_{1} + Y_{2} ∣^{- 2}$ and $∣ t ∣^{2} = ∣ 2 Y_{1} ∣^{2} ∣ Y_{1} + Y_{2} ∣^{- 2} = 4 Y_{1}^{2} ∣ Y_{1} + Y_{2} ∣^{- 2}$ because $(2 Y_{1}) \overline{(2 Y_{1})} = 4 Y_{1}^{2}$ for real $Y_{1}$ , and the denominator is nonzero since $Y_{1} + Y_{2} \neq 0$ .

4.1step 3.1F1F2F3algebra

[The flux identity] Assume $Y_{1} > 0$ real. By [F2] and [F3], $∣ Y_{1} + Y_{2} ∣^{2} > 0$ , so $∣ t ∣^{2} = 4 Y_{1}^{2} / ∣ Y_{1} + Y_{2} ∣^{2} > 0$ , and the identity of step 3.1 gives $∣ r ∣^{2} = ∣ Y_{1} - Y_{2} ∣^{2} / ∣ Y_{1} + Y_{2} ∣^{2} = 1 - 4 Y_{1} Re Y_{2} / ∣ Y_{1} + Y_{2} ∣^{2}$ ; adding $(Re Y_{2} / Y_{1}) ∣ t ∣^{2} = 4 Y_{1} Re Y_{2} / ∣ Y_{1} + Y_{2} ∣^{2}$ cancels the correction and yields $∣ r ∣^{2} + (Re Y_{2} / Y_{1}) ∣ t ∣^{2} = 1$ , the displayed identity and explicit formulas of (b).

5.1step 4.1F3F4algebra

[Sign analysis] Still with $Y_{1} > 0$ real: since $∣ t ∣^{2} > 0$ by step 4.1, the second summand $(Re Y_{2} / Y_{1}) ∣ t ∣^{2}$ has the sign of $Re Y_{2}$ by [F3], so it is nonnegative if and only if $Re Y_{2} \geq 0$ . If $Re Y_{2} < 0$ , then $∣ r ∣^{2} = 1 - 4 Y_{1} Re Y_{2} / ∣ Y_{1} + Y_{2} ∣^{2} > 1$ , and since $∣ r ∣ \geq 0$ and $1 \geq 0$ , [F4] gives $∣ r ∣ > 1$ ; in that case the identity of step 4.1 has a negative transmitted term and carries no energy-conservation reading, while $∣ r ∣ > 1$ alone is consistent with an active transmitted model.

5.2step 4.1step 3.1F2F3algebra

[Purely imaginary admittance] Assume $Y_{1} > 0$ real and $Y_{2} = i β$ with $β \in R$ . Then $\overline{Y_{1} + Y_{2}} = Y_{1} - i β = Y_{1} - Y_{2}$ by [F2], so $∣ Y_{1} - Y_{2} ∣ = ∣ \overline{Y_{1} + Y_{2}} ∣ = ∣ Y_{1} + Y_{2} ∣$ by [F2] and step 3.1 gives $∣ r ∣ = 1$ ; the identity of step 4.1 then gives $Re Y_{2} ∣ t ∣^{2} = 0$ (multiplying the identity by $Y_{1} > 0$ and using $∣ r ∣^{2} = 1$ ), while $∣ t ∣^{2} = 4 Y_{1}^{2} / ∣ Y_{1} + Y_{2} ∣^{2} = 4 Y_{1}^{2} / (Y_{1}^{2} + β^{2}) = 4 Y_{1}^{2} / (Y_{1}^{2} + ∣ Y_{2} ∣^{2}) > 0$ by [F2] and [F3]; hence $r \neq 0$ because $∣ r ∣ = 1$ and $t \neq 0$ because $∣ t ∣ > 0$ by the definiteness of the modulus in [F2]. This proves (c); no division by $Re Y_{2} = 0$ is performed, and the case $Y_{2} = 0$ is included with $β = 0$ .

5.3step 4.1step 2.1F1F5algebra

[Real-positive specialisation] Assume $Y_{1}, Y_{2} > 0$ real. Then $Y_{1} - Y_{2}$ is real and $Y_{1} + Y_{2} > 0$ is real and nonzero, so $r, t$ of step 2.1 are real by [F1], and step 4.1 gives $∣ r ∣^{2} + (Y_{2} / Y_{1}) ∣ t ∣^{2} = 1$ because $Re Y_{2} = Y_{2}$ . Substituting $Y_{j} = 1 / Z_{j}$ with $Z_{j} > 0$ real, $Y_{j}$ is invertible and $1 / Z_{1} \pm 1 / Z_{2} = (Z_{2} \pm Z_{1}) / (Z_{1} Z_{2})$ by [F1], so $r = (Z_{2} - Z_{1}) / (Z_{1} + Z_{2})$ and $t = 2 Z_{2} / (Z_{1} + Z_{2})$ by cancellation of the nonzero factor $Z_{1} Z_{2}$ ; moreover $Y_{2} / Y_{1} = Z_{1} / Z_{2}$ by [F1], so the balance is $R + T = 1$ for $R = ∣ r ∣^{2}$ and $T = (Z_{1} / Z_{2}) ∣ t ∣^{2}$ . These are exactly the solution formulas and the balance of [F5], so clause (d) is in agreement with The two-equation interface system at normal incidence.

6.1givenF1F5algebra∎

[Scope] Clause (e) holds: the entire argument is algebra in the field $C$ together with the ordered-field arithmetic of the real scalars $Y_{1}, Re Y_{2}$ ; no wave model, no labelling of the amplitudes and no interpretation of the second summand is used, and no choice principle occurs, all divisions being by explicitly nonzero field elements.

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Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth

Physical scope

Terminology, units and branch conventions for homogeneous isotropic materials at rest in the laboratory frame in the constant scalar branch D=eps E, B=mu H, J_f=sigma E+J_ext with eps,mu>0 and sigma>=0: the lossless phase speed, wave impedance, refractive index and wavenumber, the complex propagation constant kappa=p+iq as the unique root with Re,Im>=0 of k^2=epsmuomega^2+imusigma*omega, and the skin depth. Objects, units and conventions only: no field equation, no dispersion relation for a solution, no attenuation of any concrete field and no material validity or measured value.

Definition

Assume the elementary inertial-rest branch of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch in the laboratory region $U ≅ I \times Ω$ of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields with the units of Framework-local SI units, the vacuum constants and the exact/measured status of their values, in its isotropic scalar case: the material is at rest in the laboratory frame, $D, H$ are defined by Polarization, magnetization, bound and free sources, and the fields D and H(d), and the constitutive relations of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch(b) hold with constant scalars $ε > 0$ (Fm $^{- 1}$ ), $μ > 0$ (Hm $^{- 1}$ ) and $σ \geq 0$ (Sm $^{- 1}$ ), so that $D = ε E$ , $B = μ H$ and the free current is $J_{f} = σ E + J_{e x t}$ for a declared external free current $J_{e x t}$ . Write $c > 0$ for the exact vacuum speed of Framework-local SI units, the vacuum constants and the exact/measured status of their values and let $ω > 0$ (s $^{- 1}$ ) be an angular frequency. Define:

(a) [Lossless parameters] with $\sqrt{\cdot}$ the nonnegative square root of Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ , $v : = (ε μ)^{- 1 / 2} (m s^{- 1}), Z : = (\frac{μ}{ε})^{1 / 2} (Ω), n : = \frac{c}{v} = c (ε μ)^{1 / 2} (dimensionless), k (ω) : = \frac{ω}{v};$ $v$ is the phase speed, $Z$ the wave impedance, $n$ the refractive index relative to vacuum and $k (ω)$ the real wavenumber of the lossless case.

(b) [Complex propagation constant] put $a : = μ ε ω^{2} > 0$ , $b : = μ σ ω \geq 0$ , $r : = \sqrt{a^{2} + b^{2}}$ and $p (ω) : = \sqrt{\frac{r + a}{2}}, q (ω) : = \sqrt{\frac{r - a}{2}}, κ (ω) : = p (ω) + i q (ω) .$ Then $κ (ω)$ is the unique $k \in C$ with $Re k \geq 0$ , $Im k \geq 0$ and $k^{2} = ε μ ω^{2} + i μ σ ω = a + i b$ ; in the lossless case $σ = 0$ one has $b = 0$ , $q = 0$ and $κ (ω) = k (ω) = p (ω)$ real. (The explicit formula is the one of Every complex number has a square root, by an explicit Cartesian formula; the verification $p^{2} - q^{2} = a$ , $2 p q = b$ and the uniqueness of the root with $Re k \geq 0$ , $Im k \geq 0$ are verified in the well-definedness paragraph below.)

(c) [Skin depth] call $δ (ω) : = 1 / q (ω)$ (m) the skin depth if $σ > 0$ , and set $δ (ω) : = + \infty$ if $σ = 0$ : for $κ (ω) = p + i q$ and $x_{3} \in R$ the modulus of $e^{i (κ x_{3} - ω t)}$ equals $e^{- q x_{3}}$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ), so $δ$ is the distance in which that modulus decays by the factor $1 / e$ in the direction of increasing $x_{3}$ ; for $σ = 0$ there is no attenuation and $δ$ is infinite rather than a division by zero.

(d) [Status and scope] This item fixes objects, units, branch conventions and terminology only. It asserts no field equation, no dispersion relation for a solution, no attenuation of any concrete field and no material validity. The parameters are declared for the isotropic scalar constant-coefficient branch at rest in the laboratory frame; for $ε μ < c^{- 2}$ the branch is, by Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch(b), a declared finite-band effective phase-response model with no universal front-speed claim. An exact broadband interpretation must be separately adopted and requires $ε μ \geq c^{- 2}$ ; that inequality alone does not adopt a broadband interpretation or establish material validity, and a finite-band interpretation remains possible when it holds. No measured value, precision or empirical statement is included.

Well-definedness and conventions

The material assumptions, the rest-frame clause, the isotropic scalar case and the finite-band caveat are those of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch(a)-(b) read with the polarization and magnetization definitions; the unit system and the exact constant $c$ are those of Framework-local SI units, the vacuum constants and the exact/measured status of their values; the laboratory chart is that of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields.

The only content beyond notation is the well-definedness of $κ (ω)$ in (b). With $a > 0$ , $b \geq 0$ and $r = \sqrt{a^{2} + b^{2}}$ one has $r + a > 0$ and $(r - a) (r + a) = b^{2} \geq 0$ , so $r - a \geq 0$ and both defining radicands are nonnegative. If $b > 0$ , the same identity gives $r - a > 0$ , hence $q > 0$ ; if $b = 0$ , uniqueness of nonnegative roots gives $r = a$ and $q = 0$ . Also $p > 0$ because $r + a > 0$ . Thus $p^{2} - q^{2} = (r + a) / 2 - (r - a) / 2 = a$ and $(2 p q)^{2} = 4 \cdot (r + a) (r - a) / 4 = r^{2} - a^{2} = b^{2}$ , so $2 p q = b$ because $p, q \geq 0$ (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ); hence $κ^{2} = p^{2} - q^{2} + 2 i p q = a + i b$ , by the field arithmetic of The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ . The square roots of $a + i b$ are exactly $\pm κ$ : if $z^{2} = κ^{2}$ , then $(z - κ) (z + κ) = 0$ , and the complex field property ( $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ ) gives $z = κ$ or $z = - κ$ . If $k = u + i v$ with $u, v \geq 0$ and $k^{2} = a + i b$ , then $u^{2} - v^{2} = a$ , $2 u v = b$ and $u^{2} + v^{2} = ∣ k ∣^{2} = ∣ k^{2} ∣ = \sqrt{a^{2} + b^{2}} = r$ , where $∣ k^{2} ∣ = ∣ k ∣^{2}$ is the multiplicativity of the modulus (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive) and $∣ a + i b ∣ = \sqrt{a^{2} + b^{2}}$ is the modulus of Real and imaginary parts, complex conjugation, and modulus; solving gives $u^{2} = (r + a) / 2 = p^{2}$ and $v^{2} = (r - a) / 2 = q^{2}$ , and since $u, v, p, q \geq 0$ the uniqueness of the nonnegative square root (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ) gives $u = p$ and $v = q$ , so $k = κ$ . In the lossless case $σ = 0$ one has $b = 0$ , $r = a$ , $q = 0$ , $p = \sqrt{a} = ω (ε μ)^{1 / 2} = k (ω)$ by Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ . All divisions are by strictly positive real numbers, $v$ and $q$ are nonzero exactly where defined, and no choice principle is used.

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Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch

Physical scope

Exact conditional statement about ideal plane-wave fields in a homogeneous isotropic Ohmic medium at rest, on the laboratory chart: the four macroscopic equations hold if and only if the four complex amplitude conditions hold, with the dispersion relation, transverse structure, lossless period-averaged energy density and flux, and the evanescent decay and normal-flux identities. Excluded: general bounded-field or finite-energy existence theory, interfaces, anisotropic or dispersive media, nonlinear response, ideal conductors, any completeness of the plane-wave family, and all empirical statements.

Statement

Assume the elementary inertial-rest branch of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch in its isotropic scalar case at rest in the laboratory frame, with constants $ε, μ > 0$ and $σ \geq 0$ , the constitutive relations $D = ε E$ , $B = μ H$ of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch(b), with $D, H$ defined by Polarization, magnetization, bound and free sources, and the fields D and H, the macroscopic equations of The macroscopic Maxwell equations in terms of D, H and the free sources, the fields of The electric and magnetic fields as fixed-time spatial vector fields on the laboratory region of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields with the units of Framework-local SI units, the vacuum constants and the exact/measured status of their values, and the parameters and phase convention of Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth. Consider a declared region with vanishing free charge density and no external free current, so that $ρ_{f} = 0$ and the free current is the conduction current $J_{f} = σ E$ . Let $k \in C^{3}$ , $ω > 0$ and $e, h \in C^{3}$ , and for $(x, t) \in R^{3} \times R$ define $E (x, t) : = Re (e e^{i (k \cdot x - ω t)})$ and $H (x, t) : = Re (h e^{i (k \cdot x - ω t)})$ componentwise, i.e. with the phase convention $e^{i (k \cdot x - ω t)}$ and the real-part encoding of Complex plane-wave fields: real parts of complex exponentials; put $D : = ε E$ , $B : = μ H$ and $J_{f} : = σ E$ . Extend the Euclidean inner and cross products to $C^{3}$ by the same componentwise polynomial formulas (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The cross product in $R^{3}$ , The cross product is bilinear, alternating, and orthogonal to both factors). For a complex vector $z \in C^{3}$ , use the Hermitian norm $∣ z ∣ : = \sqrt{\sum_{j = 1}^{3} ∣ z_{j} ∣^{2}}$ , so $∣ z ∣^{2} = z \cdot \overline{z} = ∣ Re z ∣^{2} + ∣ Im z ∣^{2}$ ; the dot product itself remains complex bilinear. Then:

(a) [Amplitude vanishing] if $Re (w e^{i (k \cdot x - ω t)}) = 0$ for some $w \in C^{3}$ and all $(x, t)$ in a nonempty open subset of $R^{3} \times R$ , then $w = 0$ .

(b) [Amplitude conditions] the quadruple $(E, H, D, B)$ satisfies the four source-free macroscopic equations $div D = 0$ , $div B = 0$ , $curl E = - \partial_{t} B$ , $curl H = J_{f} + \partial_{t} D$ on $R^{3} \times R$ if and only if $k \cdot e = 0, k \cdot h = 0, k \times e = ω μ h, k \times h = - (ω ε + i σ) e .$

(c) [Structure, dispersion and solution family] if $e \neq 0$ and (b) holds, then $h = k \times e / (ω μ)$ , $k \cdot h = 0$ , $e \cdot h = 0$ and $k \cdot k = ε μ ω^{2} + i μ σ ω$ ; conversely, if $k \in C^{3} ∖ {0}$ and $ω > 0$ satisfy $k \cdot k = ε μ ω^{2} + i μ σ ω$ , then for every $e \in C^{3}$ with $k \cdot e = 0$ the pair $(e, k \times e / (ω μ))$ satisfies (b), with $(e, h) = (0, 0)$ if and only if $e = 0$ ; on the dispersion relation the solution set of (b) for fixed $(k, ω)$ is exactly ${(e, k \times e / (ω μ)) : k \cdot e = 0}$ , a complex two-dimensional space when $k \neq 0$ ; off the dispersion relation the only solution is $e = h = 0$ .

(d) [Lossless propagating waves: instantaneous identities and period averages] if (b) holds and $σ = 0$ and $k \in R^{3}$ with $e \neq 0$ , then $∣ k ∣^{2} = ε μ ω^{2}$ , $∣ k ∣ = k (ω)$ in the notation of Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth, and for every $(x, t)$ $k \cdot E = k \cdot B = 0$ , $E \cdot B = 0$ , $B = \hat{k} \times E / v$ and $∣ B ∣ = ∣ E ∣ / v$ with $\hat{k} = k / ∣ k ∣$ ; the period- $T$ averages with $T = 2 π / ω$ (Period averages of products of real parts of complex exponentials) satisfy, for $u = \frac{1}{2} (E \cdot D + B \cdot H)$ and $S = E \times H$ of Energy balance and Ohmic dissipation for the constant nondispersive linear model, $⟨ u ⟩ = \frac{ε}{2} ∣ e ∣^{2}, ⟨ S ⟩ = \frac{∣ e ∣^{2}}{2 μ ω} k = \frac{∣ e ∣^{2}}{2 Z} \hat{k} = v ⟨ u ⟩ \hat{k} .$

(e) [Evanescent waves] if $σ = 0$ , $n^{'} \in R^{3}$ is a unit vector, $k = κ_{0} + i β n^{'}$ with $κ_{0} \in R^{3}$ , $κ_{0} \cdot n^{'} = 0$ , $β > 0$ and $k \cdot k = ε μ ω^{2}$ (so $∣ κ_{0} ∣^{2} = ε μ ω^{2} + β^{2}$ and $κ_{0} \neq 0$ ; define ${\hat{κ}}_{0} : = κ_{0} / ∣ κ_{0} ∣$ ), and if $e \neq 0$ satisfies $k \cdot e = 0$ with $h : = k \times e / (ω μ)$ , then the field factorizes exactly as $E (x, t) = e^{- β n^{'} \cdot x} Re (e e^{i (κ_{0} \cdot x - ω t)})$ , and the time-averaged flux satisfies $⟨ S ⟩ = e^{- 2 β n^{'} \cdot x} \frac{1}{2} Re (e \times \overline{h})$ with normal component $⟨ S \cdot n^{'} ⟩ = - \frac{e^{- 2 β n^{'} \cdot x}}{2 μ ω} Re [(e \cdot n^{'}) (κ_{0} \cdot \overline{e})]$ ; this normal component is in fact zero for every such transverse amplitude. For the transverse-electric polarisation $e = E_{0} (n^{'} \times {\hat{κ}}_{0})$ , $E_{0} \in C ∖ {0}$ , and for the transverse-magnetic data $h = H_{0} {\hat{y}}^{'}$ with ${\hat{y}}^{'} : = n^{'} \times {\hat{κ}}_{0}$ , $H_{0} \in C ∖ {0}$ and electric amplitude $e = - (k \times {\hat{y}}^{'}) H_{0} / (ω ε)$ , the normal component vanishes identically, $⟨ S \cdot n^{'} ⟩ \equiv 0$ , while the tangential averaged flux is positive: $⟨ S \cdot {\hat{κ}}_{0} ⟩ = \frac{∣ κ_{0} ∣ ∣ e ∣^{2}}{2 μ ω} e^{- 2 β n^{'} \cdot x} > 0$ in the transverse-electric case and $⟨ S \cdot {\hat{κ}}_{0} ⟩ = \frac{∣ κ_{0} ∣ ∣ h ∣^{2}}{2 ω ε} e^{- 2 β n^{'} \cdot x} > 0$ in the transverse-magnetic case.

(f) [Scope] the four equations are verified only for the explicit ideal whole-space plane-wave fields; for $k$ with $Im k \neq 0$ the factor $e^{- Im k \cdot x}$ grows in some directions, so clauses (b)-(e) are read on regions where the model declares the field bounded (in (e), on the half-space $n^{'} \cdot x > 0$ ). No completeness, energy-finiteness, boundary, interface, material-validity or empirical claim is made, and the finite-band caveat of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch(b) applies whenever $ε μ < c^{- 2}$ .

Facts & Assumptions

Given: The declared isotropic scalar Ohmic branch at rest with constants $ε, μ > 0$ and $σ \geq 0$ , the laboratory region of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields with the units of Framework-local SI units, the vacuum constants and the exact/measured status of their values, the fields of The electric and magnetic fields as fixed-time spatial vector fields, and the plane-wave data $k \in C^{3}$ , $ω > 0$ , $e, h \in C^{3}$ with $E = Re (e e^{i θ})$ , $H = Re (h e^{i θ})$ , $θ (x, t) : = k \cdot x - ω t$ , $D = ε E$ , $B = μ H$ , $J_{f} = σ E$ ; the inner and cross products are extended componentwise to $C^{3}$ .

[F1]

Real-part encoding: $E_{j} = Re (e_{j} \exp (i θ))$ componentwise; $Re$ is additive and commutes with real scalar multiples, with partial differentiation and with integration, and the real-part encoding of a product of two encoded real fields must not be read off from a product of complex expressions without conjugation (Complex plane-wave fields: real parts of complex exponentials).

[F2]

The complex exponential and its calculus: $\exp w = \sum_{n \geq 0} w^{n} / n!$ converges for every $w \in C$ with $\exp 0 = 1$ and $\exp^{'} = \exp$ (The complex exponential by its power series, The complex exponential is entire and its complex derivative is itself); the sum, real-scalar-multiple, product and quotient rules (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ) and the chain rule for total derivatives (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ) apply; $\exp (z + w) = \exp z \exp w$ for all complex $z, w$ , and $\exp (x) = e^{x}$ for real $x$ ( $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential); for real $x, y$ , $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ and $∣ \exp (x + i y) ∣ = e^{x}$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ).

[F3]

Complex field arithmetic, conjugation and modulus: $C$ is a field, a product vanishes only if a factor does ( $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ , The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ ); conjugation is an involutive real-field automorphism fixing the reals, $z \overline{z} = ∣ z ∣^{2}$ , $∣ z w ∣ = ∣ z ∣ ∣ w ∣$ , $∣ z ∣ \geq 0$ with $∣ z ∣ = 0$ exactly for $z = 0$ , and $Re w = (w + \overline{w}) / 2$ by the coordinate formulas (Real and imaginary parts, complex conjugation, and modulus, Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[F4]

The divergence and curl of a $C^{1}$ field as coordinate expressions in the first partial derivatives, and partial derivatives as derivatives of line maps (Divergence and curl of a $C^{1}$ vector field, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ).

[F5]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k} x_{k} y_{k}$ and the cross product $u \times v = (u_{y} v_{z} - u_{z} v_{y}, u_{z} v_{x} - u_{x} v_{z}, u_{x} v_{y} - u_{y} v_{x})$ on $R^{3}$ , both extended componentwise to complex vectors; the cross product is bilinear and alternating, $⟨ u \times v, w ⟩ = \det [u v w]$ , and $u \times v$ is orthogonal to $u$ and $v$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The cross product in $R^{3}$ , The cross product is bilinear, alternating, and orthogonal to both factors).

[F6]

The material model and field equations: the elementary inertial-rest branch gives $D = ε E$ , $B = μ H$ and $J_{f} = σ E + J_{e x t}$ with $J_{e x t} = 0$ here (Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch); with $ρ_{f} = 0$ the macroscopic equations read $div D = 0$ , $div B = 0$ , $curl E = - \partial_{t} B$ , $curl H = J_{f} + \partial_{t} D$ (The macroscopic Maxwell equations in terms of D, H and the free sources).

[F7]

Medium parameters: $v = (ε μ)^{- 1 / 2}$ , $Z = (μ / ε)^{1 / 2}$ , $k (ω) = ω / v$ , and $κ (ω) = p + i q$ with $p, q \geq 0$ and $κ (ω)^{2} = ε μ ω^{2} + i μ σ ω$ (Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth).

[F8]

Stored energy and flux: for the constant model, $u = \frac{1}{2} (E \cdot D + B \cdot H)$ and $S = E \times H$ (Energy balance and Ohmic dissipation for the constant nondispersive linear model).

[F9]

Period averages: for $ω > 0$ , $T = 2 π / ω$ and complex amplitudes $a, d$ , the average over a period of $Re (a e^{- i ω t}) Re (d e^{- i ω t})$ is $\frac{1}{2} Re (a \overline{d})$ , with the analogous componentwise inner-product and cross-product forms; the conjugation is essential (Period averages of products of real parts of complex exponentials).

[F10]

Square roots: every nonnegative real has a unique nonnegative square root (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ).

Proof

technique · direct; exhibit the partial derivatives of the encoded fields, reduce each equation to a complex coefficient identity by the amplitude-vanishing fact, analyse the resulting algebraic system, and compute the lossless and evanescent structures with the period-average supplier

1.1F1F2F3F4F5algebra

[Derivatives of the encoded fields] For $w \in C^{3}$ and $θ (x, t) = k \cdot x - ω t$ , each component $x \mapsto Re (w_{j} \exp (i θ (x, t)))$ has partial derivatives $\partial_{x_{l}} Re (w_{j} \exp (i θ)) = Re (i k_{l} w_{j} \exp (i θ))$ and $\partial_{t} Re (w_{j} \exp (i θ)) = Re (- i ω w_{j} \exp (i θ))$ : indeed $θ$ is the real-linear map $(x, t) \mapsto \sum_{l} k_{l} x_{l} - ω t$ with partial derivatives $k_{l}$ in $x_{l}$ and $- ω$ in $t$ as derivatives of line maps by [F4], and the chain rule [F2] applied to the entire exponential gives $\partial_{x_{l}} \exp (i θ) = i k_{l} \exp (i θ)$ and $\partial_{t} \exp (i θ) = - i ω \exp (i θ)$ , after which the real-linear coordinate projection $Re$ commutes with taking the derivative by [F1]. The complex multipliers stay inside $Re$ . Consequently $div D = ε Re (i k \cdot e e^{i θ})$ , $div B = μ Re (i k \cdot h e^{i θ})$ , $curl E = Re (i k \times e e^{i θ})$ , $\partial_{t} B = Re (- i ω μ h e^{i θ})$ , $curl H = Re (i k \times h e^{i θ})$ and $J_{f} + \partial_{t} D = Re ((σ - i ω ε) e e^{i θ})$ with the componentwise complex inner and cross products of [F5] and the uniform notation $e^{i θ} : = \exp (i θ)$ .

1.2F2F3

[Nonvanishing of the exponential] For every $w \in C$ , $\exp w \neq 0$ : by [F2], $\exp w \exp (- w) = \exp 0 = 1$ , and a product equal to $1 \neq 0$ has no zero factor in the field $C$ by [F3]. In particular $e^{i θ} \neq 0$ for all $(x, t)$ , and $e^{i k \cdot x_{0}} \neq 0$ for every fixed $x_{0}$ .

2.1givenstep 1.1step 1.2F2F3

[Amplitude vanishing, part (a)] Let $Z \subseteq R^{3} \times R$ be nonempty and open with $Re (w e^{i θ}) = 0$ on $Z$ for a fixed $w \in C^{3}$ . Fix $(x_{0}, t_{0}) \in Z$ ; by openness of $Z$ there is $δ > 0$ with ${x_{0}} \times (t_{0} - δ, t_{0} + δ) \subseteq Z$ . The components of the map $t \mapsto Re (W \exp (- i ω t))$ with $W : = w e^{i k \cdot x_{0}}$ vanish on that interval, and by step 1.1 its $t$ -derivative is $Re (- i ω W \exp (- i ω t))$ , so at $t_{0}$ both $Re (W \exp (- i ω t_{0})) = 0$ and $Re (- i ω W \exp (- i ω t_{0})) = 0$ . Writing $W \exp (- i ω t_{0}) = u + i v$ with real coordinate vectors $u, v$ by [F3], the first identity is $u = 0$ and the second is $Re (- i ω (u + i v)) = ω v = 0$ ; since $ω > 0$ , $v = 0$ ; hence $W \exp (- i ω t_{0}) = 0$ , and since $\exp (- i ω t_{0}) \neq 0$ by step 1.2 the field property in [F3] gives $W = 0$ , i.e. $w e^{i k \cdot x_{0}} = 0$ , and $\exp (i k \cdot x_{0}) \neq 0$ by step 1.2 gives $w = 0$ . This proves (a).

3.1step 1.1step 2.1F1F3F4

[Equations as coefficient identities] By step 1.1 the four equations of (b) are, on $R^{3} \times R$ , the identities $Re (i k \cdot e e^{i θ}) = 0$ , $Re (i k \cdot h e^{i θ}) = 0$ , $Re (i k \times e e^{i θ}) + Re (- i ω μ h e^{i θ}) = 0$ and $Re (i k \times h e^{i θ}) - Re ((σ - i ω ε) e e^{i θ}) = 0$ ; all fields are smooth on the open set $R^{3} \times R$ , so the operators of [F4] apply. By the additivity of $Re$ in [F1] and [F3], each of these is an identity $Re (W_{j} e^{i θ}) = 0$ for a constant scalar or vector $W_{j}$ (embedding a scalar as $(W_{j}, 0, 0)$ to apply (a)); by (a) of step 2.1, with the open set $R^{3} \times R$ , they hold if and only if the constants vanish.

4.1step 3.1F3algebra

[Proof of (b)] By step 3.1, the first and second equations hold if and only if $k \cdot e = 0$ and $k \cdot h = 0$ , since multiplication by the nonzero field element $i$ is injective by [F3]. The third holds if and only if $i k \times e - i ω μ h = 0$ , that is $k \times e = ω μ h$ after dividing by the nonzero factor $i$ and using $ω μ \neq 0$ . The fourth holds if and only if $i k \times h - (σ - i ω ε) e = 0$ , because $Re (- i ω μ h e^{i θ}) = - Re (i ω μ h e^{i θ})$ by [F3]; dividing by $i$ and using $(σ - i ω ε) / i = - (ω ε + i σ)$ — indeed $(σ - i ω ε) = i \cdot (- (ω ε + i σ))$ by the field arithmetic of [F3] — gives $k \times h = - (ω ε + i σ) e$ . Hence the quadruple satisfies the four equations if and only if the four displayed amplitude conditions hold, which is (b).

5.1step 4.1F3F5algebra

[Structure and dispersion, forward in (c)] Assume $e \neq 0$ and the four conditions of (b). Since $ω μ \neq 0$ , the third condition gives $h = k \times e / (ω μ)$ ; then $k \cdot h = k \cdot (k \times e) / (ω μ) = 0$ and $e \cdot h = e \cdot (k \times e) / (ω μ) = 0$ by the orthogonality in [F5]. Crossing the third condition with $k$ and substituting the fourth, $k \times (k \times e) = ω μ k \times h = - ω μ (ω ε + i σ) e$ ; the vector triple-product identity $k \times (k \times e) = k (k \cdot e) - (k \cdot k) e$ , a componentwise polynomial identity for the bilinear extension by [F5], together with $k \cdot e = 0$ , gives $- (k \cdot k) e = - ω μ (ω ε + i σ) e = - (ε μ ω^{2} + i μ σ ω) e$ , and choosing a nonzero coordinate of $e$ and cancelling that complex scalar gives $k \cdot k = ε μ ω^{2} + i μ σ ω$ .

6.1step 4.1step 5.1F3F5algebra

[Converse and solution family in (c)] Let $k \neq 0$ , $ω > 0$ satisfy $k \cdot k = ε μ ω^{2} + i μ σ ω$ , and let $e \in C^{3}$ with $k \cdot e = 0$ ; put $h : = k \times e / (ω μ)$ . Then $k \cdot e = 0$ and $k \cdot h = 0$ by orthogonality in [F5], $k \times e = ω μ h$ by definition, and $k \times h = k \times (k \times e) / (ω μ) = [k (k \cdot e) - (k \cdot k) e] / (ω μ) = - (ε μ ω^{2} + i μ σ ω) e / (ω μ) = - (ω ε + i σ) e$ , so the pair $(e, h)$ satisfies (b) by step 4.1. If $e = 0$ then $h = 0$ ; conversely if $h = 0$ then $k \times e = 0$ , so the two-vector family $(k, e)$ is linearly dependent over the field $C$ : writing the vanishing minors of $(k e)$ — if the coordinate $k_{l} \neq 0$ then $e_{j} = (e_{l} / k_{l}) k_{j}$ for the two other coordinates $j$ and also for $l$ — gives $e = α k$ with $α = e_{l} / k_{l} \in C$ by [F3]; then $0 = k \cdot e = α (k \cdot k)$ and $k \cdot k = ε μ ω^{2} + i μ σ ω \neq 0$ (its real part is $ε μ ω^{2} > 0$ by [F3]) force $α = 0$ , so $e = 0$ . Thus $(e, h) = (0, 0)$ if and only if $e = 0$ . For the solution set with $(k, ω)$ on the dispersion relation: every solution $(e, h)$ of (b) has $k \cdot e = 0$ and $h = k \times e / (ω μ)$ by step 5.1, so it lies in the displayed set; conversely every element of that set is a solution by the first part of this step, and the correspondence $e \mapsto (e, k \times e / (ω μ))$ is injective, so the solution set is exactly ${(e, k \times e / (ω μ)) : k \cdot e = 0}$ . If $k_{l} \neq 0$ the map $e \mapsto (e_{j})_{j \neq l} \in C^{2}$ is a bijection from ${e : k \cdot e = 0}$ onto $C^{2}$ with inverse $(e_{j})_{j \neq l} \mapsto e$ where $e_{l} = - (\sum_{j \neq l} k_{j} e_{j}) / k_{l}$ , both maps being linear over $C$ ; hence that solution space is a complex two-dimensional space. Off the dispersion relation, if (b) holds then either $e = 0$ , which gives $h = k \times e / (ω μ) = 0$ , or $e \neq 0$ , which by step 5.1 forces the dispersion relation; so the only solution is $e = h = 0$ .

6.2step 4.1step 5.1F1F5F7F10algebra

[Lossless pointwise identities in (d)] Assume $σ = 0$ and $k \in R^{3}$ with $e \neq 0$ , so the four conditions of (b) hold and (c) applies by steps 4.1 and 5.1. From $k \cdot k = ∣ k ∣^{2} = ε μ ω^{2}$ and $∣ k ∣ \geq 0$ we get $∣ k ∣ = ω (ε μ)^{1 / 2}$ by [F10], which is $k (ω) = ω / v$ by [F7]. For every $(x, t)$ : $k \cdot E = k \cdot Re (e e^{i θ}) = Re ((k \cdot e) e^{i θ}) = 0$ and $k \cdot B = μ Re ((k \cdot h) e^{i θ}) = 0$ by [F1] and the reality of $k$ ; further $B = μ H = μ Re ((k \times e) / (ω μ) e^{i θ}) = Re ((k \times e) e^{i θ}) / ω = k \times Re (e e^{i θ}) / ω = k \times E / ω = (\hat{k} \times E) / v$ because $k$ is real and $∣ k ∣ / ω = 1 / v$ by [F7]; and $E \cdot B = E \cdot (\hat{k} \times E) / v = ⟨ E, \hat{k} \times E ⟩ / v = \det [E, \hat{k}, E] / v = 0$ by [F5] because a determinant with two equal columns vanishes.

7.1givenstep 6.1F2F3F5algebra

[Evanescent factorisation in (e)] Assume the hypotheses of (e). Since $k = κ_{0} + i β n^{'}$ with $κ_{0}, n^{'} \in R^{3}$ and $κ_{0} \cdot n^{'} = 0$ , $∣ n^{'} ∣ = 1$ , the bilinear extension of [F5] gives $k \cdot x = κ_{0} \cdot x + i β n^{'} \cdot x$ for real $x$ and $k \cdot k = ∣ κ_{0} ∣^{2} - β^{2}$ (the cross terms $2 i β κ_{0} \cdot n^{'}$ vanish), so the dispersion relation gives $∣ κ_{0} ∣^{2} = ε μ ω^{2} + β^{2} > 0$ and hence $κ_{0} \neq 0$ . Writing $φ : = κ_{0} \cdot x - ω t \in R$ and $b : = β n^{'} \cdot x \in R$ , the identity $i θ = i (k \cdot x - ω t) = - b + i φ$ combines with the addition law and real extension of [F2] to give $e^{i θ} = \exp (- b + i φ) = e^{- b} \exp (i φ)$ ; the Cartesian form of [F2] gives $\exp (i φ) = \cos φ + i \sin φ$ , and the coordinate formula for the real part of a product in [F1] and [F3] then gives, componentwise, $Re (e e^{i θ}) = e^{- β n^{'} \cdot x} [(Re e) \cos φ - (Im e) \sin φ] = e^{- β n^{'} \cdot x} Re (e e^{i φ})$ , the last equality being the same coordinate formula read backwards. This is the stated factorisation, and $h = k \times e / (ω μ)$ by the hypothesis of (e); step 6.1 verifies the amplitude conditions.

7.2step 6.2F5F7algebra

[The cross-product norm identity] For real vectors $u, v \in R^{3}$ the component expansion gives $∣ u \times v ∣^{2} = ∣ u ∣^{2} ∣ v ∣^{2} - ⟨ u, v ⟩^{2}$ : both sides equal $\sum_{j \neq l} u_{j}^{2} v_{l}^{2} - 2 \sum_{j < l} u_{j} u_{l} v_{j} v_{l}$ by the coordinate formulas of [F5]. Applying this with $u = \hat{k}$ and $v = E$ , where $∣ \hat{k} ∣ = 1$ since $∣ k ∣ = ω (ε μ)^{1 / 2} \neq 0$ and $⟨ \hat{k}, E ⟩ = (k \cdot E) / ∣ k ∣ = 0$ by step 6.2, gives $∣ \hat{k} \times E ∣^{2} = ∣ E ∣^{2}$ , hence $∣ B ∣ = ∣ \hat{k} \times E ∣ / v = ∣ E ∣ / v$ by step 6.2 and the positivity of $v$ in [F7]. Write $e = a + i b$ with real $a, b$ . Since $k$ is real and $k \cdot e = 0$ , $k \cdot a = k \cdot b = 0$ . Applying the real norm identity separately to $(k, a)$ and $(k, b)$ gives $∣ k \times e ∣^{2} = ∣ k \times a ∣^{2} + ∣ k \times b ∣^{2} = ∣ k ∣^{2} (∣ a ∣^{2} + ∣ b ∣^{2}) = ∣ k ∣^{2} ∣ e ∣^{2}$ . Thus $∣ h ∣ = ∣ k \times e ∣ / (ω μ) = ∣ k ∣ ∣ e ∣ / (ω μ)$ , so $∣ h ∣ = ω (ε μ)^{1 / 2} ∣ e ∣ / (ω μ) = ∣ e ∣ (ε / μ)^{1 / 2} = ∣ e ∣ / Z$ by [F7].

8.1step 7.1F3F5F9algebra

[Evanescent averaged flux in (e)] By step 7.1 the fields are $E = e^{- b} Re (e e^{i φ})$ and $H = e^{- b} Re (h e^{i φ})$ with $φ = κ_{0} \cdot x - ω t$ , so $E \times H = e^{- 2 b} Re (e e^{- i ω t^{'}}) \times Re (h e^{- i ω t^{'}})$ evaluated at $t^{'} = t - κ_{0} \cdot x / ω$ , and the period average in $t$ equals that in $t^{'}$ ; the cross-product form of [F9] applied with $a = e$ , $d = h$ gives $⟨ S ⟩ = e^{- 2 β n^{'} \cdot x} \frac{1}{2} Re (e \times \overline{h})$ . Since $\overline{h} = \overline{k \times e} / (ω μ)$ and $\overline{k \times e} = \overline{k} \times \overline{e} = (κ_{0} - i β n^{'}) \times \overline{e}$ by [F3], the triple-product expansion of [F5] gives $e \times \overline{h} = [\overline{k} (e \cdot \overline{e}) - \overline{e} (e \cdot \overline{k})] / (ω μ)$ , whose inner product with $n^{'}$ is $[(\overline{k} \cdot n^{'}) ∣ e ∣^{2} - (\overline{e} \cdot n^{'}) (e \cdot \overline{k})] / (ω μ)$ ; here $\overline{k} \cdot n^{'} = - i β$ and $e \cdot \overline{k} = κ_{0} \cdot e - i β (e \cdot n^{'})$ by the bilinear expansion of [F5]. Hence $Re [- \overline{(e \cdot n^{'})} (e \cdot \overline{k})] = - Re [\overline{(e \cdot n^{'})} (κ_{0} \cdot e)] - Re [i β ∣ e \cdot n^{'} ∣^{2}] = - Re [\overline{(e \cdot n^{'})} \cdot \overline{(κ_{0} \cdot \overline{e})}] = - Re [(e \cdot n^{'}) (κ_{0} \cdot \overline{e})]$ , where the middle term vanishes as a purely imaginary number and the last identity uses $\overline{z w} = \overline{z} \overline{w}$ and $Re \overline{ζ} = Re ζ$ from [F3]; substituting into the previous display gives the stated normal component.

8.2step 6.2step 7.2F2F3F5F7F8F9algebra

[Lossless averages in (d)] With $T = 2 π / ω$ , step 6.2 gives $E \cdot E = Re (e_{j} e^{i θ}) Re (e_{j} e^{i θ})$ summed over $j$ , and its period average is $\frac{1}{2} Re (e \cdot \overline{e}) = \frac{1}{2} ∣ e ∣^{2}$ by the vector form of [F9] applied with $a = d = e$ (the phase $e^{i k \cdot x}$ is constant in $t$ and cancels in the product $e^{i k \cdot x} e^{- i k \cdot x} = 1$ by [F2]); similarly $⟨ H \cdot H ⟩ = \frac{1}{2} ∣ h ∣^{2} = \frac{1}{2} ∣ e ∣^{2} / Z^{2}$ by step 7.2. Hence $⟨ u ⟩ = \frac{1}{2} (ε ⟨ E \cdot E ⟩ + μ ⟨ H \cdot H ⟩) = \frac{1}{4} (ε ∣ e ∣^{2} + μ ∣ e ∣^{2} / Z^{2}) = \frac{1}{4} (ε ∣ e ∣^{2} + ε ∣ e ∣^{2}) = \frac{ε}{2} ∣ e ∣^{2}$ because $u = \frac{1}{2} (E \cdot D + B \cdot H)$ is the stored-energy quantity of [F8] and $μ / Z^{2} = ε$ by [F7] and [F3]. Similarly $⟨ S ⟩ = ⟨ E \times H ⟩ = \frac{1}{2} Re (e \times \overline{h})$ by the flux definition $S = E \times H$ of [F8] and the cross-product form of [F9] with $a = e$ , $d = h$ ; substituting $\overline{h} = \overline{k \times e} / (ω μ) = (k \times \overline{e}) / (ω μ)$ for real $k$ by [F3] and using the triple-product expansion $e \times (k \times \overline{e}) = k (e \cdot \overline{e}) - \overline{e} (e \cdot k) = k ∣ e ∣^{2}$ by [F5] and $k \cdot e = 0$ gives $⟨ S ⟩ = ∣ e ∣^{2} k / (2 μ ω) = ∣ e ∣^{2} \hat{k} / (2 Z)$ because $k / (μ ω) = \hat{k} ∣ k ∣ / (μ ω) = \hat{k} / (μ v) = \hat{k} / Z$ by [F7]; and $v ⟨ u ⟩ \hat{k} = v ε ∣ e ∣^{2} \hat{k} / 2 = ∣ e ∣^{2} \hat{k} / (2 Z)$ because $v ε = 1 / Z$ by [F7], so the three displayed expressions agree.

9.1step 8.1F3F5algebra

[Vanishing on transverse amplitudes] Under the hypotheses of (e) with $k \cdot e = 0$ , the bilinear expansion of [F5] gives $0 = k \cdot e = κ_{0} \cdot e + i β (n^{'} \cdot e)$ , so $κ_{0} \cdot \overline{e} = \overline{κ_{0} \cdot e} = \overline{- i β (n^{'} \cdot e)} = i β \overline{(n^{'} \cdot e)} = i β (n^{'} \cdot \overline{e})$ by [F3] and the reality of $κ_{0}$ , and hence $(e \cdot n^{'}) (κ_{0} \cdot \overline{e}) = i β (e \cdot n^{'}) \overline{(e \cdot n^{'})} = i β ∣ e \cdot n^{'} ∣^{2}$ , a purely imaginary number; therefore its real part vanishes and step 8.1 gives $⟨ S \cdot n^{'} ⟩ \equiv 0$ for every transverse amplitude.

10.1step 7.1step 8.1step 9.1step 4.1F3F5F7algebra

[The polarised evanescent cases] In the transverse-electric case $e = E_{0} (n^{'} \times {\hat{κ}}_{0})$ with ${\hat{κ}}_{0} = κ_{0} / ∣ κ_{0} ∣$ and $E_{0} \neq 0$ : the triple-product expansion of [F5] gives $k \times (n^{'} \times {\hat{κ}}_{0}) = (k \cdot {\hat{κ}}_{0}) n^{'} - (k \cdot n^{'}) {\hat{κ}}_{0} = ∣ κ_{0} ∣ n^{'} - i β {\hat{κ}}_{0}$ because $k \cdot {\hat{κ}}_{0} = κ_{0} \cdot {\hat{κ}}_{0} = ∣ κ_{0} ∣$ and $k \cdot n^{'} = i β$ , and $k \cdot (n^{'} \times {\hat{κ}}_{0}) = ⟨ n^{'} \times {\hat{κ}}_{0}, k ⟩ = \det [n^{'}, {\hat{κ}}_{0}, k] = \det [n^{'}, {\hat{κ}}_{0}, κ_{0}] + i β \det [n^{'}, {\hat{κ}}_{0}, n^{'}] = ∣ κ_{0} ∣ \det [n^{'}, {\hat{κ}}_{0}, {\hat{κ}}_{0}] + i β \cdot 0 = 0$ because a determinant with two equal columns vanishes by [F5]; so this is a transverse amplitude with $h = E_{0} (∣ κ_{0} ∣ n^{'} - i β {\hat{κ}}_{0}) / (ω μ)$ and $\overline{h} = {\overline{E}}_{0} (∣ κ_{0} ∣ n^{'} + i β {\hat{κ}}_{0}) / (ω μ)$ . With the positively oriented orthonormal triad $({\hat{κ}}_{0}, {\hat{y}}^{'}, n^{'})$ where ${\hat{y}}^{'} = n^{'} \times {\hat{κ}}_{0}$ , the cross-product relations ${\hat{y}}^{'} \times n^{'} = {\hat{κ}}_{0}$ and ${\hat{y}}^{'} \times {\hat{κ}}_{0} = - n^{'}$ hold by [F5], so $e \times \overline{h} = ∣ E_{0} ∣^{2} [∣ κ_{0} ∣ {\hat{κ}}_{0} - i β n^{'}] / (ω μ)$ and $⟨ S \cdot {\hat{κ}}_{0} ⟩ = e^{- 2 β n^{'} \cdot x} ∣ E_{0} ∣^{2} ∣ κ_{0} ∣ / (2 μ ω) = e^{- 2 β n^{'} \cdot x} ∣ e ∣^{2} ∣ κ_{0} ∣ / (2 μ ω) > 0$ because $∣ e ∣^{2} = ∣ E_{0} ∣^{2} \neq 0$ , $∣ κ_{0} ∣ > 0$ and $μ, ω, v > 0$ . In the transverse-magnetic case $h = H_{0} {\hat{y}}^{'}$ with $H_{0} \neq 0$ and $e = - (k \times {\hat{y}}^{'}) H_{0} / (ω ε)$ : the same expansion gives $k \times {\hat{y}}^{'} = ∣ κ_{0} ∣ n^{'} - i β {\hat{κ}}_{0}$ , so $e = H_{0} (i β {\hat{κ}}_{0} - ∣ κ_{0} ∣ n^{'}) / (ω ε)$ ; then $k \cdot e = H_{0} (i β k \cdot {\hat{κ}}_{0} - ∣ κ_{0} ∣ k \cdot n^{'}) / (ω ε) = H_{0} (i β ∣ κ_{0} ∣ - ∣ κ_{0} ∣ i β) / (ω ε) = 0$ , and $k \times e = H_{0} (i β k \times {\hat{κ}}_{0} - ∣ κ_{0} ∣ k \times n^{'}) / (ω ε)$ . Here $k \times n^{'} = (κ_{0} + i β n^{'}) \times n^{'} = κ_{0} \times n^{'} = - ∣ κ_{0} ∣ {\hat{y}}^{'}$ and $k \times {\hat{κ}}_{0} = (κ_{0} + i β n^{'}) \times {\hat{κ}}_{0} = i β (n^{'} \times {\hat{κ}}_{0}) = i β {\hat{y}}^{'}$ by [F5] and $κ_{0} \times {\hat{κ}}_{0} = 0$ , whence $k \times e = H_{0} ((i β)^{2} + ∣ κ_{0} ∣^{2}) {\hat{y}}^{'} / (ω ε) = H_{0} (∣ κ_{0} ∣^{2} - β^{2}) {\hat{y}}^{'} / (ω ε) = ω μ H_{0} {\hat{y}}^{'} = ω μ h$ , because $∣ κ_{0} ∣^{2} - β^{2} = k \cdot k = ε μ ω^{2}$ by step 7.1 and [F7]; so this data satisfies the amplitude conditions by step 4.1, and $⟨ S \cdot {\hat{κ}}_{0} ⟩ = e^{- 2 β n^{'} \cdot x} ∣ H_{0} ∣^{2} ∣ κ_{0} ∣ / (2 ω ε) = e^{- 2 β n^{'} \cdot x} ∣ h ∣^{2} ∣ κ_{0} ∣ / (2 ω ε) > 0$ with $∣ h ∣^{2} = ∣ H_{0} ∣^{2} \neq 0$ . In both polarisations the normal component vanishes identically by step 9.1.

11.1givenstep 9.1step 10.1F6F7∎

[Scope (f)] Clause (f) holds: all computations were carried out for the explicit fields defined by $(k, ω, e, h)$ on the whole space, so no existence or completeness of any plane-wave family, no energy finiteness, no boundary or interface conditions, no material validity and no empirical statement is claimed; for $Im k \neq 0$ the factor $e^{- Im k \cdot x}$ is unbounded in some directions, so the statements are read where the model declares the field bounded, and in (e) the decay factor $e^{- β n^{'} \cdot x}$ is the bounded branch on $n^{'} \cdot x > 0$ ; the finite-band caveat of [F6] applies throughout; and no choice principle is used, every construction being finite-dimensional algebra for exhibited data.

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Phase matching at a plane interface: common tangential wavevector and frequency, the reflection law and Snell's law

Physical scope

Conditional phase-matching theorem for three monochromatic plane waves at a plane interface between two homogeneous isotropic lossless media at rest, under the stated stationary interface identities: the tangential wavevector and angular frequency agree, the wavevectors are coplanar with the normal, and in the propagating case the reflection law and Snell's law follow. Excluded: reflection and transmission amplitudes, energy fluxes, existence of interface-consistent data, absorbing, dispersive, anisotropic, moving or nonlinear media, and any empirical claim.

Statement

Assume the elementary inertial-rest branch of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch in the lossless isotropic scalar case: two homogeneous media at rest in the laboratory chart of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields (frame basis $(e_{1}, e_{2}, e_{3})$ , units of Framework-local SI units, the vacuum constants and the exact/measured status of their values), medium 1 in ${x_{3} < 0}$ with constants $ε_{1}, μ_{1} > 0$ and medium 2 in ${x_{3} > 0}$ with constants $ε_{2}, μ_{2} > 0$ , both with $σ = 0$ , sharing the plane interface $S : = {x_{3} = 0}$ with unit normal $n : = e_{3}$ , and the plane-wave amplitude conditions and solution structure of Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch in each medium. Let $ω_{i}, ω_{r}, ω_{t} > 0$ , $k_{i}, k_{r}, k_{t} \in C^{3}$ , $e_{i}, e_{r}, e_{t} \in C^{3} ∖ {0}$ , and put $h_{j} : = k_{j} \times e_{j} / (μ_{j} ω_{j})$ for $j \in {i, r, t}$ where $μ_{i} = μ_{r} = μ_{1}$ and $μ_{t} = μ_{2}$ , so that each triple satisfies the conditions of Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch(b)-(c) in its own medium: $k_{j} \cdot e_{j} = 0$ , $k_{j} \times e_{j} = ω_{j} μ h_{j}$ , $k_{j} \times h_{j} = - ω_{j} ε e_{j}$ and $k_{j} \cdot k_{j} = ε μ ω_{j}^{2}$ with $(ε, μ) = (ε_{1}, μ_{1})$ for $j = i, r$ and $(ε_{2}, μ_{2})$ for $j = t$ . Assume the declared normal orientation conditions $k_{i} \cdot n \in R_{> 0}$ and $k_{r} \cdot n \in R_{< 0}$ ; no order is placed on a complex number. Tangential components may be complex. Algebraic phase matching permits every complex transmitted wavevector satisfying its amplitude conditions and dispersion relation, including a zero normal root at critical incidence. No outgoing or non-growth condition is required here. A consuming Fresnel construction may optionally select a root $k_{t} \cdot n$ of $(k_{t} \cdot n)^{2} = ε_{2} μ_{2} ω_{t}^{2} - κ_{t} \cdot κ_{t}$ , where $κ_{t} : = k_{t} - (k_{t} \cdot n) n$ , with $Im (k_{t} \cdot n) \geq 0$ and $Re (k_{t} \cdot n) \geq 0$ when its imaginary part is zero. Such a selection is a declared additional branch assumption of that construction, not a restriction on this theorem or an inferred material realization. Complex tangential phases may grow along the unbounded interface; no global tangential boundedness is asserted. Define the wave fields $E_{j} (x, t) : = Re (e_{j} e^{i (k_{j} \cdot x - ω_{j} t)})$ , $H_{j} (x, t) : = Re (h_{j} e^{i (k_{j} \cdot x - ω_{j} t)})$ , which are fields of The electric and magnetic fields as fixed-time spatial vector fields, and let $E : = E_{i} + E_{r}$ , $H : = H_{i} + H_{r}$ on the medium-1 side and $E : = E_{t}$ , $H : = H_{t}$ on the medium-2 side. [Interface hypothesis] assume that for all $(y, t) \in R^{2} \times I$ , with $I \subseteq R$ a nonempty open interval, the one-sided traces at $(y, 0, t)$ satisfy the four stationary interface identities with no free surface charge or current, $n \cdot (ε_{2} E_{t} - ε_{1} (E_{i} + E_{r})) = 0, n \cdot (μ_{2} H_{t} - μ_{1} (H_{i} + H_{r})) = 0,$ $n \times (E_{t} - (E_{i} + E_{r})) = 0, n \times (H_{t} - (H_{i} + H_{r})) = 0.$ Then:

(a) [Phase matching] with $κ_{j} : = k_{j} - (k_{j} \cdot n) n$ the tangential wavevector, $κ_{i} = κ_{r} = κ_{t}$ and $ω_{i} = ω_{r} = ω_{t}$ .

(b) [Geometry] if $κ_{i} \neq 0$ then $k_{i}, k_{r}, k_{t}$ all lie in the complex two-dimensional subspace ${span}_{C} {κ_{i}, n}$ (the real wavevectors lie in the corresponding real plane); if $κ_{i} = 0$ then all three wavevectors are parallel to $n$ .

(c) [Propagating waves: reflection law and Snell's law] if in addition $k_{i}, k_{r}, k_{t} \in R^{3}$ , then with $θ_{j} \in [0, π / 2]$ defined by $\sin θ_{j} = ∣ κ_{j} ∣ / ∣ k_{j} ∣$ (well defined because the real dispersion relation gives $∣ k_{j} ∣ > 0$ ; the incident and reflected angles are below $π / 2$ , and the transmitted angle may equal $π / 2$ ) one has $θ_{r} = θ_{i}$ and, writing $v_{j} = (ε_{j} μ_{j})^{- 1 / 2}$ and $n_{j} = c / v_{j}$ for the parameters of Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth, $n_{1} \sin θ_{i} = n_{2} \sin θ_{t} .$ If $θ_{i} > 0$ , this is equivalently $\sin θ_{t} / \sin θ_{i} = v_{2} / v_{1} = n_{1} / n_{2}$ . At normal incidence $θ_{i} = 0$ it gives $θ_{t} = 0$ without forming $0 / 0$ .

(d) [Hypotheses actually used] within the stated positive-frequency amplitude-condition setting, the only interface identities used are the two tangential identities $n \times [E] = 0$ and $n \times [H] = 0$ ; nonzero-wave hypotheses $e_{j} \neq 0$ and $k_{j} \cdot k_{j} \neq 0$ are retained. The two normal identities are recorded because they belong to the interface model, but they are not used. If the reflected wave is absent ( $e_{r} = h_{r} = 0$ ), the same argument matches the incident and transmitted actual waves; it does not determine the absent reflected wave's frequency or wavevector, which a Fresnel construction may conventionally assign the common phase.

(e) [Scope] the wave data are given: no existence of interface-consistent data, no reflection or transmission amplitudes, no energy fluxes and no well-posedness are claimed. Complex transmitted wavevectors are covered by (a) and (b) (the evanescent case of Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch(e) satisfies $k_{t} \cdot n \neq 0$ ); the physical propagating or evanescent character of the transmitted wave, the angle selection in the real case and the amplitudes are treated by the interface items that consume this theorem. The four interface identities are the stationary plane-interface conditions of the model; for compact interfaces they follow from the distributional jump theorem Stationary macroscopic interface jump conditions for D, B, E and H, and on the noncompact plane they are the stated interface hypothesis. No material dispersion, absorption, anisotropy, moving medium or nonlinearity is covered, and no empirical claim is made.

Facts & Assumptions

Given: The two-media lossless setting of the Statement, the six wave data $(k_{j}, ω_{j}, e_{j})$ , $j \in {i, r, t}$ , with $h_{j} = k_{j} \times e_{j} / (μ_{j} ω_{j})$ , the normal $n = e_{3}$ , the interface interval $I$ , and the interface hypothesis [A1].

[A1]

The four stationary interface identities of the Statement hold for all $(y, t) \in R^{2} \times I$ ; only the two tangential identities $n \times (E_{t} - (E_{i} + E_{r})) = 0$ and $n \times (H_{t} - (H_{i} + H_{r})) = 0$ are used in the proof.

[F1]

Plane-wave structure in each medium: $k_{j} \cdot e_{j} = 0$ , $k_{j} \times e_{j} = ω_{j} μ h_{j}$ , $k_{j} \times h_{j} = - ω_{j} ε e_{j}$ and $k_{j} \cdot k_{j} = ε μ ω_{j}^{2}$ with the constants of the medium of wave $j$ , and $e_{j} \neq 0$ (Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch(b)-(c)); the parameters $v_{j} = (ε_{j} μ_{j})^{- 1 / 2}$ and $n_{j} = c / v_{j}$ are those of Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth.

[F2]

The character-grouping lemma: if $F (z) = \sum_{k} c_{k} e^{i λ_{k} \cdot z}$ with pairwise distinct $λ_{k} \in C^{m}$ and $Re F = 0$ on a nonempty open $U$ , then for every $ν \in C^{m}$ , $\sum_{λ_{k} = ν} c_{k} + \sum_{λ_{k} = - \overline{ν}} \overline{c_{k}} = 0$ (Distinct exponential characters are linearly independent on a nonempty open set, and the real-part grouping rule(b)).

[F3]

Inner and cross products: $⟨ x, y ⟩ = \sum_{l} x_{l} y_{l}$ , $u \times v = (u_{y} v_{z} - u_{z} v_{y}, u_{z} v_{x} - u_{x} v_{z}, u_{x} v_{y} - u_{y} v_{x})$ , bilinear and alternating, with $⟨ u \times v, w ⟩ = \det [u v w]$ and $u \times v$ orthogonal to $u$ and $v$ ; orthogonality of the decomposition $x = x^{τ} + (x \cdot n) n$ with $x^{τ} : = x - (x \cdot n) n$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The cross product in $R^{3}$ , The cross product is bilinear, alternating, and orthogonal to both factors).

[F4]

Complex arithmetic and conjugation: $C$ is a field in which a product vanishes only if a factor does (The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ , $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ ); conjugation is an involutive automorphism with $Re w = (w + \overline{w}) / 2$ and $z \overline{z} = ∣ z ∣^{2}$ (Real and imaginary parts, complex conjugation, and modulus, Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[F5]

The principal inverse sine: $\arcsin : [- 1, 1] \to [- π / 2, π / 2]$ satisfies $\sin (\arcsin y) = y$ , and sine is strictly increasing on $[- π / 2, π / 2]$ (Principal inverse sine and inverse cosine).

[F6]

Nonnegative square roots: every nonnegative real has a unique nonnegative square root, and $\sqrt{a b} = \sqrt{a} \sqrt{b}$ for $a, b \geq 0$ by squaring and uniqueness (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ).

[F7]

The laboratory setting, fields and units are those of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, The electric and magnetic fields as fixed-time spatial vector fields and Framework-local SI units, the vacuum constants and the exact/measured status of their values; the material branch is the lossless isotropic scalar case of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch.

Proof

technique · direct; evaluate the tangential traces on the interface as real parts of six-component characters, apply the character-grouping lemma to force equal frequencies, and read off the geometry and the real-angle laws

1.1givenF1F3F4algebra

[Nonzero tangential traces] For each $j$ , the six-component tangential trace datum $A_{j} : = (e_{j}^{τ}, h_{j}^{τ}) \in C^{6}$ is nonzero. Suppose $A_{j} = 0$ ; then $e_{j}^{τ} = 0$ and $h_{j}^{τ} = 0$ , so $e_{j} = a n$ with $a = e_{j} \cdot n$ and $h_{j} = b n$ with $b = h_{j} \cdot n$ by the orthogonal decomposition of [F3]. Taking the inner product of the amplitude condition $k_{j} \times e_{j} = ω_{j} μ h_{j}$ of [F1] with $n$ gives $0 = ω_{j} μ b$ : indeed $⟨ k_{j} \times e_{j}, n ⟩ = \det [k_{j}, e_{j}, n] = a \det [k_{j}, n, n] = 0$ because a determinant with two equal columns vanishes and $⟨ h_{j}, n ⟩ = b$ ; hence $b = 0$ and $h_{j} = 0$ . Then the amplitude condition $k_{j} \times h_{j} = - ω_{j} ε e_{j}$ gives $0 = - ω_{j} ε a n$ , so $a = 0$ , and then $e_{j} = a n = 0$ , contradicting $e_{j} \neq 0$ in [F1].

2.1givenA1step 1.1F3F4algebra

[Interface traces as characters] For a wave $j$ and $x = (y, 0)$ on the interface, $k_{j} \cdot x = κ_{j} \cdot y$ because $k_{j} \cdot n$ multiplies the vanishing third coordinate; hence at the interface the tangential trace is $E_{j}^{τ} (y, 0, t) = Re (e_{j}^{τ} e^{i (κ_{j} \cdot y - ω_{j} t)})$ and $H_{j} (y, 0, t) = Re (h_{j} e^{i (κ_{j} \cdot y - ω_{j} t)})$ , whose normal parts are annihilated by $n \times \cdot$ . The two tangential interface identities of [A1] are identities of tangential vector fields, and for a tangential $w$ the vanishing $n \times w = 0$ forces $w = 0$ (in the frame $n = e_{3}$ , $w = (w_{1}, w_{2}, 0)$ gives $n \times w = (- w_{2}, w_{1}, 0)$ by [F3]); hence they say, for all $z = (y, t) \in R^{2} \times I$ , that $Re (A_{t} e^{i λ_{t} \cdot z}) = Re (A_{i} e^{i λ_{i} \cdot z}) + Re (A_{r} e^{i λ_{r} \cdot z})$ componentwise, with $λ_{j} : = (κ_{j}, - ω_{j}) \in C^{3}$ and $A_{j} = (e_{j}^{τ}, h_{j}^{τ})$ ; equivalently $Re (A_{i} e^{i λ_{i} \cdot z} + A_{r} e^{i λ_{r} \cdot z} - A_{t} e^{i λ_{t} \cdot z}) = 0$ on the nonempty open set $R^{2} \times I$ , by the additivity of the real part in [F4] and the componentwise linearity of the tangential projection in [F3].

3.1givenstep 2.1F2F4algebra

[Frequency grouping] Group the three frequencies of step 2.1 by value: let $ν_{1}, \dots, ν_{M}$ be the distinct values among $λ_{i}, λ_{r}, λ_{t}$ and put $c_{ν_{0}} : = \sum_{j : λ_{j} = ν_{0}} ϵ_{j} A_{j}$ where $ϵ_{i} = ϵ_{r} : = 1$ and $ϵ_{t} : = - 1$ . Then $Re (\sum_{k = 1}^{M} c_{ν_{k}} e^{i ν_{k} \cdot z}) = 0$ on $R^{2} \times I$ , the $ν_{k}$ being pairwise distinct, so [F2] applied to the distinct frequencies $ν_{1}, \dots, ν_{M}$ with coefficients $c_{ν_{1}}, \dots, c_{ν_{M}}$ gives $c_{ν} + \sum_{k : ν_{k} = - \overline{ν}} \overline{c_{ν_{k}}} = 0$ for every $ν \in C^{3}$ . Fix $k$ and take $ν = ν_{k}$ : the left sum is $c_{ν_{k}}$ alone, while $- \overline{ν_{k}} = (- \overline{κ_{k}}, + ω_{k})$ has last coordinate $ω_{k} > 0$ , whereas every $ν_{l} = λ_{j}$ for some $j$ has last coordinate $- ω_{j} < 0$ ; hence $ν_{l} = - \overline{ν_{k}}$ fails for every $l$ , the second sum is empty, and $c_{ν_{k}} = 0$ .

4.1step 3.1step 1.1algebra

[All frequencies agree] By step 3.1, for each distinct actual frequency $ν$ the signed sum of the tangential data of the waves with that frequency vanishes. If the three actual frequencies were not all equal, then some actual frequency would occur exactly once among $λ_{i}, λ_{r}, λ_{t}$ (a partition of three terms into at least two nonempty groups has a group of size one), and for that $ν$ the identity $c_{ν} = 0$ of step 3.1 would read $\pm A_{j} = 0$ for the corresponding single wave, contradicting the nonzero trace datum of step 1.1. Hence $λ_{i} = λ_{r} = λ_{t}$ , which is the assertion (a): reading the three coordinates, $κ_{i} = κ_{r} = κ_{t}$ and $ω_{i} = ω_{r} = ω_{t}$ .

5.1step 4.1F3algebra

[Geometry, part (b)] By step 4.1, $κ_{j} = κ : = κ_{i}$ for every $j$ , and $k_{j} = κ_{j} + (k_{j} \cdot n) n = κ + (k_{j} \cdot n) n$ by the orthogonal decomposition of [F3]; hence each $k_{j}$ lies in the complex span of $κ$ and $n$ , which is two-dimensional over $C$ when $κ \neq 0$ because $κ$ and $n$ are independent: if $κ = z n$ for some $z \in C$ then $0 = κ \cdot n = z (n \cdot n) = z$ , so $κ = 0$ , contrary to hypothesis. When the wavevectors are real, $κ$ and $n$ are real and each $k_{j}$ lies in their real span, the corresponding real plane. If instead $κ = 0$ , then every $k_{j} = (k_{j} \cdot n) n$ is parallel to $n$ .

5.2step 1.1step 4.1A1F1algebra

[Hypotheses used and the absent-wave case, part (d)] The argument of steps 1.1-4.1 used exactly: the two tangential identities of [A1], the nonzero amplitudes $e_{j} \neq 0$ and the amplitude conditions with the dispersion relation $ε μ ω_{j}^{2} = k_{j} \cdot k_{j} \neq 0$ of [F1] (the dispersion enters through the evaluation of $k_{j} \times h_{j}$ from $h_{j} = k_{j} \times e_{j} / (μ_{j} ω_{j})$ , so the role listed as $k_{j} \cdot k_{j} \neq 0$ is the nondegeneracy actually used); the normal orientation conditions $k_{i} \cdot n > 0$ , $k_{r} \cdot n < 0$ , the two normal interface identities, and the optional normal-root selection rule were not used, and no outgoing or tangential-growth condition was used. If a consuming construction declares the reflected wave absent, $e_{r} = h_{r} = 0$ (so that $A_{r} = 0$ and only the incident and transmitted traces are matched), the same grouping argument applies to the two nonzero trace data $A_{i}$ and $- A_{t}$ : a singleton group would give $\pm A_{i} = 0$ or $\mp A_{t} = 0$ , impossible by step 1.1, so $λ_{i} = λ_{t}$ , i.e. $κ_{i} = κ_{t}$ and $ω_{i} = ω_{t}$ ; no constraint on the labels $k_{r}, ω_{r}$ of the absent field is produced, and a Fresnel construction may assign it the common phase as a convention.

6.1step 4.1step 5.1F1F3F5F6algebra

[Angles and laws, part (c)] Assume now that $k_{i}, k_{r}, k_{t}$ are real. For each $j$ , [F1] gives $k_{j} \cdot k_{j} = ∣ k_{j} ∣^{2} = ε_{j} μ_{j} ω^{2}$ with the common $ω = ω_{j}$ of step 4.1, so $∣ k_{j} ∣ = ω (ε_{j} μ_{j})^{1 / 2} = ω / v_{j} > 0$ by [F6] and [F1], and $∣ k_{i} ∣ = ∣ k_{r} ∣$ . By [F3], $∣ k_{j} ∣^{2} = ∣ κ_{j} ∣^{2} + (k_{j} \cdot n)^{2}$ , so $∣ κ_{j} ∣ \leq ∣ k_{j} ∣$ and the ratio $\sin θ_{j} : = ∣ κ_{j} ∣ / ∣ k_{j} ∣$ lies in $[0, 1]$ ; define $θ_{j} : = \arcsin (∣ κ_{j} ∣ / ∣ k_{j} ∣) \in [0, π / 2]$ by [F5], so that $\sin θ_{j} = ∣ κ_{j} ∣ / ∣ k_{j} ∣$ as in the Statement. Since $κ_{r} = κ_{i} = κ_{t} = κ$ by step 4.1, $\sin θ_{r} = ∣ κ ∣ / ∣ k_{r} ∣ = ∣ κ ∣ / ∣ k_{i} ∣ = \sin θ_{i}$ ; the principal inverse sine is strictly increasing on its interval because sine is strictly increasing on $[- π / 2, π / 2]$ by [F5], hence injective, so $θ_{r} = θ_{i}$ . Similarly $∣ κ ∣ = ω (ε_{1} μ_{1})^{1 / 2} \sin θ_{i} = ω (ε_{2} μ_{2})^{1 / 2} \sin θ_{t}$ , that is $n_{1} \sin θ_{i} = n_{2} \sin θ_{t}$ after multiplying by $c / ω$ where $n_{j} = c (ε_{j} μ_{j})^{1 / 2} = c / v_{j}$ by [F1] and [F6]. If $θ_{i} > 0$ then $\sin θ_{i} > 0$ and division is legitimate, giving $\sin θ_{t} / \sin θ_{i} = (ε_{1} μ_{1})^{1 / 2} / (ε_{2} μ_{2})^{1 / 2} = v_{2} / v_{1} = n_{1} / n_{2}$ ; if $θ_{i} = 0$ then $∣ κ ∣ = 0$ , so $∣ κ_{t} ∣ = 0$ and $θ_{t} = \arcsin 0 = 0$ , since $\sin 0 = 0$ and the principal inverse sine returns the unique number of $[- π / 2, π / 2]$ with sine $0$ , without forming $0 / 0$ .

7.1givenstep 5.2F2F7∎

[Scope, part (e)] The four statements (a)-(d) are conditional on the given wave data and the interface hypothesis [A1]; nothing about the existence of interface-consistent data, reflection or transmission amplitudes, energy fluxes or well-posedness is proved. All frequencies are positive, all tangential quantities may be complex, and clauses (a) and (b) cover complex transmitted wavevectors; the propagating angle selection of (c) and the amplitudes are the business of the consuming interface theorems. The interface identities on the noncompact plane are the stated hypothesis; for compact interfaces the distributional jump theorem Stationary macroscopic interface jump conditions for D, B, E and H supplies them, and no distributional or choice-theoretic clause of that theorem is used here (only complex arithmetic, the character lemma, and the real geometry of the propagating case), so no choice principle is inherited.

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Conducting-medium propagation: the exact attenuation constant, the skin depth and its good- and poor-conductor regimes

Physical scope

Exact conditional statement about a monochromatic plane-wave field in a homogeneous isotropic Ohmic medium at rest with constant scalar eps,mu>0 and sigma>0: the exact complex propagation constant and decay, the closed-form skin depth, explicit bounded remainder inequalities in the good-conductor regime epsomega/sigma<=1/4 and the poor-conductor regime sigma/(epsomega)<=1/2, and the phase relation. Excluded: sigma=0 (no attenuation, infinite skin depth), the ideal-conductor limit, frequency-dependent or tensor conductivity, anisotropy, dispersion, nonlinearity, any laboratory material and all empirical claims.

Statement

Assume the elementary inertial-rest branch of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch in the isotropic scalar case at rest in the laboratory frame with constants $ε, μ > 0$ and $σ > 0$ , the fields of The electric and magnetic fields as fixed-time spatial vector fields on the laboratory region of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields and the plane-wave structure of Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch: let $ω > 0$ , $e \in C^{3} ∖ {0}$ , $n \in R^{3}$ a unit vector and $k = λ n$ , $λ \in C$ , and let $E (x, t) = Re (e e^{i (k \cdot x - ω t)})$ be a plane-wave field satisfying the amplitude conditions of Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch(b) with $k \cdot e = 0$ . Put $a : = μ ε ω^{2} > 0$ , $b : = μ σ ω > 0$ , $r : = \sqrt{a^{2} + b^{2}}$ and let $p, q > 0$ be the quantities of Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth. Then:

(a) [Exact propagation constant and decay] $λ^{2} = k \cdot k = ε μ ω^{2} + i μ σ ω = a + i b$ , so $λ = \pm (p + i q)$ ; if $x_{3}^{'} : = n \cdot x$ , then for $λ = p + i q$ $E (x, t) = e^{- q x_{3}^{'}} Re (e e^{i (p x_{3}^{'} - ω t)}),$ so the field oscillates with wavenumber $p$ and its amplitude decays in the direction of increasing $x_{3}^{'}$ at the exact rate $q$ ; with $λ = - (p + i q)$ the exact formula is $E (x, t) = e^{q x_{3}^{'}} Re (e e^{i (- p x_{3}^{'} - ω t)})$ , which grows in that direction, so the decaying branch is the one with $λ = p + i q$ , in agreement with the selection rule of Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth(b).

(b) [Exact skin depth] $q^{2} = (\sqrt{a^{2} + b^{2}} - a) / 2$ , $p^{2} = (\sqrt{a^{2} + b^{2}} + a) / 2$ , $p q = b / 2 = μ σ ω / 2$ , and the skin depth of Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth equals the exact closed expression $δ (ω) = \frac{1}{q} = \frac{\sqrt{2 (r + a)}}{b} = \frac{\sqrt{2 (\sqrt{a^{2} + b^{2}} + a)}}{μ σ ω} .$

(c) [Good-conductor regime] put $t : = a / b = ε ω / σ > 0$ . For every $t \leq 1 / 4$ the exact value satisfies the explicit bounds $∣ q - \sqrt{b / 2} ∣ \leq 2 t \sqrt{b / 2} = \frac{2 ε ω}{σ} \sqrt{\frac{μ σ ω}{2}}, ∣ δ - \sqrt{2 / b} ∣ \leq 4 t \sqrt{2 / b} = \frac{4 ε ω}{σ} \sqrt{\frac{2}{μ σ ω}};$ equivalently $δ = \sqrt{2 / (μ σ ω)} (1 + O (ε ω / σ))$ in the declared regime $ε ω / σ \leq 1 / 4$ , with the displayed explicit remainder bounds.

(d) [Poor-conductor regime] put $t^{'} : = b / a = σ / (ε ω) > 0$ . For every $t^{'} \leq 1 / 2$ the exact value satisfies $∣ q - \frac{σ}{2} \sqrt{\frac{μ}{ε}} ∣ \leq \frac{t^{' 2}}{4} \cdot \frac{σ}{2} \sqrt{\frac{μ}{ε}}, ∣ δ - \frac{2}{σ} \sqrt{\frac{ε}{μ}} ∣ \leq \frac{t^{' 2}}{2} \cdot \frac{2}{σ} \sqrt{\frac{ε}{μ}},$ so in this regime $δ$ is frequency-independent up to the displayed relative error $O ((σ / (ε ω))^{2})$ .

(e) [Phase relation for a real amplitude] if $e \in R^{3}$ and $λ = p + i q$ , then $E (x, t) = e^{- q x_{3}^{'}} e \cos (p x_{3}^{'} - ω t), H (x, t) = \frac{e^{- q x_{3}^{'}}}{μ ω} (p \cos (p x_{3}^{'} - ω t) - q \sin (p x_{3}^{'} - ω t)) (n \times e),$ and the ratio of the quadrature to the in-phase coefficient is $q / p = \sqrt{1 + t^{2}} - t$ with $1 - 2 t \leq q / p \leq 1$ for every $t \leq 1 / 2$ ; in particular $q / p \to 1$ as $t ↓ 0$ , so the magnetic scalar factor is $\sqrt{p^{2} + q^{2}} \cos (p x_{3}^{'} - ω t + η)$ with $η : = \arctan (q / p) \to π / 4$ (45 degrees), not a 90-degree quadrature phase shift. At fixed position with the $e^{- i ω t}$ convention this corresponds to the stated signed $+ η$ phase in the decreasing phase variable.

(f) [Limits and scope] for $σ = 0$ one has $b = 0$ , $q = 0$ , no attenuation and $δ = + \infty$ , and the approximations of (c) and (d) are not applicable; the ideal-conductor limit $σ \to \infty$ is not covered by this theorem, which works inside the finite-Ohmic branch of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch (a separate boundary model is needed for an ideal conductor); the frequency dependence of $σ$ (Drude or other) is not supplied, and $σ$ is the declared constant of the branch; the conclusions are exact identities for the ideal plane-wave field and carry no empirical content, no measured conductivity and no claim about any laboratory field.

Facts & Assumptions

Given: The declared Ohmic branch with constants $ε, μ > 0$ , $σ > 0$ ; the unit vector $n$ , $k = λ n$ , $e \neq 0$ with $k \cdot e = 0$ ; the plane-wave field $E = Re (e e^{i (k \cdot x - ω t)})$ with $H$ determined by the amplitude conditions; the numbers $a = μ ε ω^{2} > 0$ , $b = μ σ ω > 0$ , $r = \sqrt{a^{2} + b^{2}}$ and the parameters $p, q > 0$ of Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth.

[F1]

Concrete propagation data: $k \cdot k = ε μ ω^{2} + i μ σ ω$ and $h = k \times e / (ω μ)$ by Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch(b)-(c); and by Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth(b), $κ = p + i q$ is the unique complex number with $Re κ \geq 0$ , $Im κ \geq 0$ and $κ^{2} = a + i b$ , where $p = \sqrt{(r + a) / 2}$ and $q = \sqrt{(r - a) / 2}$ ; also $v = (ε μ)^{- 1 / 2}$ and $Z = (μ / ε)^{1 / 2}$ .

[F2]

The complex exponential in Cartesian form: for real $x, y$ , $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ and $∣ \exp (x + i y) ∣ = e^{x}$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ).

[F3]

Complex arithmetic and modulus: $C$ is a field, a product vanishes only if a factor does, and from $z^{2} = w$ and $κ^{2} = w$ one gets $(z - κ) (z + κ) = 0$ , hence $z = \pm κ$ (The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ , $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ ); the modulus satisfies $z \overline{z} = ∣ z ∣^{2}$ and $∣ z w ∣ = ∣ z ∣ ∣ w ∣$ (Real and imaginary parts, complex conjugation, and modulus, Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[F4]

Real roots and order: every nonnegative real has a unique nonnegative square root, and $\sqrt{a} \sqrt{b} = \sqrt{a b}$ for $a, b \geq 0$ follows from squaring and uniqueness (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ); for $0 \leq a \leq b$ one has $a^{2} \leq b^{2}$ , and for $0 \leq a, b$ with $a^{2} \leq b^{2}$ one has $a \leq b$ (Squaring is monotone on the nonnegatives); if $0 \leq a \leq b$ and $0 \leq c \leq d$ then $a c \leq b d$ , and if $0 < c < d$ then $1 / d < 1 / c$ in the sense $0 < 1 / d \leq 1 / c$ (Multiplying inequalities of positives, Sign rules for products and monotonicity of multiplication: multiplication by a positive preserves order, and $1 / c - 1 / d = (d - c) / (c d) > 0$ ).

[F5]

Every complex number has a square root (Every complex number has a square root, by an explicit Cartesian formula).

[F6]

The principal inverse tangent $\arctan : R \to (- π / 2, π / 2)$ is continuous and strictly increasing with $\tan (\arctan y) = y$ (The principal inverse tangent $\arctan : R \to (- π / 2, π / 2)$ ), and the addition formulas $\cos (θ + η) = \cos θ \cos η - \sin θ \sin η$ hold for real angles (The addition formulas for sine and cosine).

[F7]

Trigonometry: $\sin^{2} θ + \cos^{2} θ = 1$ and $\cos (- θ) = \cos θ$ (Pythagorean and parity identities for all six trigonometric functions on their natural domains); $\tan θ = \sin θ / \cos θ$ where defined (Tangent, cotangent, secant, and cosecant on their exact natural domains); $\sin (θ + π / 2) = \cos θ$ and $\cos (π / 2) = 0$ (Quarter-turn values and shifts by pi/2 and pi). Cosine is strictly decreasing on $[0, π]$ and hence positive on $(0, π / 2)$ (Signs, monotonicity intervals, and ranges of sine and cosine). Consequently $\sin (π / 4) = \cos (- π / 4) = \cos (π / 4) > 0$ , so $\tan (π / 4) = 1$ and $\arctan 1 = π / 4$ by [F6].

[F8]

The laboratory setting and units are those of 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 and The electric and magnetic fields as fixed-time spatial vector fields; the branch assumption is that of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch.

Proof

technique · direct; substitute the concrete data into the medium plane-wave theorem, extract the decaying root via Euler's Cartesian form, and bound the exact radicals with elementary square-root inequalities

1.1givenF1F2F3F5algebra

[Propagation constant and decay, part (a)] By [F1], $k \cdot k = ε μ ω^{2} + i μ σ ω = a + i b$ , and since $k = λ n$ with $n \cdot n = 1$ the bilinear expansion gives $λ^{2} = k \cdot k = a + i b$ ; by [F1] and [F5] with [F3], the solutions of $z^{2} = a + i b$ are exactly $z = \pm (p + i q)$ . Writing $x_{3}^{'} = n \cdot x$ , so that $k \cdot x = λ x_{3}^{'}$ , the exponent is $i (k \cdot x - ω t) = i λ x_{3}^{'} - i ω t$ ; for $λ = p + i q$ this is $- q x_{3}^{'} + i (p x_{3}^{'} - ω t)$ , and the Cartesian form [F2] gives $e^{i (k \cdot x - ω t)} = e^{- q x_{3}^{'}} (\cos (p x_{3}^{'} - ω t) + i \sin (p x_{3}^{'} - ω t))$ , whence $E = e^{- q x_{3}^{'}} Re (e e^{i (p x_{3}^{'} - ω t)})$ componentwise; the amplitude factor $e^{- q x_{3}^{'}}$ decreases as $x_{3}^{'}$ increases because $q > 0$ by [F1]. For $λ = - (p + i q)$ the same computation gives $i λ x_{3}^{'} - i ω t = q x_{3}^{'} + i (- p x_{3}^{'} - ω t)$ and $E = e^{q x_{3}^{'}} Re (e e^{i (- p x_{3}^{'} - ω t)})$ , which grows in the direction of increasing $x_{3}^{'}$ ; the decaying branch is therefore $λ = p + i q$ , matching the root selection of [F1].

2.1step 1.1F1F4algebra

[Exact skin depth, part (b)] By [F1], $q^{2} = (r - a) / 2$ and $p^{2} = (r + a) / 2$ with $r = \sqrt{a^{2} + b^{2}}$ ; multiplying, $(p q)^{2} = (r^{2} - a^{2}) / 4 = b^{2} / 4$ , and since $p, q \geq 0$ and $b > 0$ the uniqueness of the nonnegative square root in [F4] gives $p q = b / 2 = μ σ ω / 2$ . For the skin depth $δ = 1 / q$ of Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth(c): $δ^{2} = 1 / q^{2} = 2 / (r - a) = 2 (r + a) / ((r - a) (r + a)) = 2 (r + a) / b^{2}$ , where $(r - a) (r + a) = r^{2} - a^{2} = b^{2}$ by [F4]; since $δ > 0$ and $b > 0$ , taking the nonnegative square root gives $δ = \sqrt{2 (r + a)} / b$ , and substituting $a = μ ε ω^{2}$ and $b = μ σ ω$ gives the displayed closed form.

3.1step 2.1F1F4algebra

[Ratio identities] With $t = a / b > 0$ one has $r = b \sqrt{1 + t^{2}}$ because $r^{2} = a^{2} + b^{2} = b^{2} (t^{2} + 1)$ and both sides are nonnegative, by [F4]; hence $q^{2} = (r - a) / 2 = \frac{b}{2} (\sqrt{1 + t^{2}} - t)$ and $\sqrt{1 + t^{2}} - t = (\sqrt{1 + t^{2}} + t)^{- 1}$ , the second identity because $(\sqrt{1 + t^{2}} - t) (\sqrt{1 + t^{2}} + t) = 1 + t^{2} - t^{2} = 1$ with all factors positive by [F4] and [F1]. Moreover $q / p = \sqrt{1 + t^{2}} - t$ : indeed $q^{2} / p^{2} = (r - a) / (r + a)$ , and the same rationalisation gives $(r - a) / (r + a) = (\sqrt{1 + t^{2}} - t)^{2}$ , so the positive quantities $q / p$ and $\sqrt{1 + t^{2}} - t$ have equal squares and are equal by [F4]. With $s = t^{'} = b / a > 0$ one has analogously $r = a \sqrt{1 + s^{2}}$ and $q^{2} = \frac{a}{2} (\sqrt{1 + s^{2}} - 1)$ . Writing $q_{t} : = \frac{σ}{2} \sqrt{\frac{μ}{ε}}$ , the identity $q_{t} = \frac{\sqrt{a} s}{2}$ holds: expanding $\sqrt{a} = ω (ε μ)^{1 / 2}$ and $s = \frac{σ}{ε ω}$ by [F1] gives $\frac{\sqrt{a} s}{2} = \frac{ω (ε μ)^{1 / 2} σ}{2 ε ω} = \frac{σ}{2} \sqrt{\frac{μ}{ε}}$ ; hence $q^{2} / q_{t}^{2} = \frac{2 (\sqrt{1 + s^{2}} - 1)}{s^{2}} = \frac{2}{1 + \sqrt{1 + s^{2}}}$ by the same rationalisation.

4.1step 3.1F1F4algebra

[Good-conductor bounds, part (c)] For $t \geq 0$ one has $1 \leq \sqrt{1 + t^{2}} \leq 1 + t$ : the left inequality because $1 \leq 1 + t^{2}$ and both sides are nonnegative, the right because $1 + t^{2} \leq (1 + t)^{2} = 1 + 2 t + t^{2}$ , both by [F4]. Hence $\frac{1}{1 + 2 t} \leq \sqrt{1 + t^{2}} - t \leq \frac{1}{1 + t} \leq 1$ by step 3.1 and [F4], and $\frac{1}{1 + 2 t} \geq 1 - 2 t$ because $0 \leq 1 - 2 t \leq 1$ and $(1 - 2 t) (1 + 2 t) = 1 - 4 t^{2} \leq 1$ for $t \leq 1 / 2$ ; therefore $\sqrt{1 - 2 t} \leq \frac{q}{\sqrt{b / 2}} \leq 1$ , that is $\sqrt{b / 2} \sqrt{1 - 2 t} \leq q \leq \sqrt{b / 2}$ . For $t \leq 1 / 4$ one has $1 - 2 t \geq 0$ and $(1 - 2 t)^{2} \leq 1 - 2 t$ , so $\sqrt{1 - 2 t} \geq 1 - 2 t$ by [F4], giving $∣ q - \sqrt{b / 2} ∣ \leq 2 t \sqrt{b / 2}$ with the stated equivalent form since $\sqrt{b / 2} = \sqrt{μ σ ω / 2}$ . For $δ = 1 / q$ one has $\sqrt{2 / b} \leq δ \leq \frac{\sqrt{2 / b}}{1 - 2 t}$ from the bound on $q$ and [F4]; and $\frac{1}{1 - 2 t} \leq 1 + 4 t$ for $t \leq 1 / 4$ because $1 \leq (1 + 4 t) (1 - 2 t) = 1 + 2 t - 8 t^{2}$ , i.e. $0 \leq 2 t (1 - 4 t)$ , by [F4]; hence $∣ δ - \sqrt{2 / b} ∣ \leq 4 t \sqrt{2 / b}$ with the stated equivalent form.

4.2step 3.1F4algebra

[Poor-conductor bounds, part (d)] By step 3.1, $q^{2} = \frac{a}{2} (\sqrt{1 + s^{2}} - 1)$ and $q^{2} / q_{t}^{2} = \frac{2}{1 + \sqrt{1 + s^{2}}}$ with $q_{t} = \frac{σ}{2} \sqrt{\frac{μ}{ε}}$ , $s = t^{'} = b / a > 0$ . Since $\sqrt{1 + s^{2}} \geq 1$ , the ratio is at most $1$ , so $q \leq q_{t}$ ; and $\sqrt{1 + s^{2}} \leq 1 + \frac{s^{2}}{2}$ because $1 + s^{2} \leq (1 + \frac{s^{2}}{2})^{2} = 1 + s^{2} + \frac{s^{4}}{4}$ , so $q^{2} / q_{t}^{2} \geq \frac{2}{2 + \frac{s^{2}}{2}} = \frac{1}{1 + \frac{s^{2}}{4}}$ ; for $s^{2} \leq 1$ (which holds when $s \leq 1 / 2$ ) one has $\frac{1}{1 + s^{2} / 4} \geq 1 - \frac{s^{2}}{4}$ since $(1 - \frac{s^{2}}{4}) (1 + \frac{s^{2}}{4}) = 1 - \frac{s^{4}}{16} \leq 1$ with $1 - \frac{s^{2}}{4} \geq 0$ ; therefore $q / q_{t} \geq 1 - \frac{s^{2}}{4}$ , i.e. $∣ q - q_{t} ∣ \leq \frac{s^{2}}{4} q_{t}$ . For $δ = 1 / q$ and $δ_{t} = 1 / q_{t}$ we get $δ / δ_{t} = q_{t} / q \leq \frac{1}{1 - s^{2} / 4} \leq 1 + \frac{s^{2}}{2}$ whenever $s^{2} \leq 1$ (since $\frac{1}{1 - s^{2} / 4} \leq 1 + \frac{s^{2}}{2}$ is equivalent to $1 \leq 1 + \frac{s^{2}}{4} - \frac{s^{4}}{8}$ , true for $s^{2} \leq 2$ ), whence $∣ δ - δ_{t} ∣ \leq \frac{s^{2}}{2} δ_{t}$ ; substituting $δ_{t} = \frac{2}{σ} \sqrt{\frac{ε}{μ}}$ gives both displayed bounds, by [F4] throughout.

5.1givenstep 1.1step 3.1step 4.1F1F2F4F6F7algebra

[Phase relation, part (e)] Assume $e \in R^{3}$ and $λ = p + i q$ ; by [F1], $h = k \times e / (ω μ) = λ (n \times e) / (ω μ)$ because $k = λ n$ and $n \times e$ is real, and the exponent is computed as in step 1.1. Writing $θ : = p x_{3}^{'} - ω t$ and using $λ / (μ ω) = (p + i q) / (μ ω)$ with real $p, q$ , the Cartesian form [F2] gives $Re ((p + i q) (\cos θ + i \sin θ)) = p \cos θ - q \sin θ$ , so $E = e^{- q x_{3}^{'}} e \cos θ$ and $H = e^{- q x_{3}^{'}} (p \cos θ - q \sin θ) (n \times e) / (μ ω)$ , the two displayed identities. By step 3.1, $q / p = \sqrt{1 + t^{2}} - t$ , and the bounds $1 - 2 t \leq q / p \leq 1$ for $t \leq 1 / 2$ are the ones established in step 4.1 for the same quantity; as $t ↓ 0$ the inequality $0 \leq \sqrt{1 + t^{2}} - 1 = t^{2} / (\sqrt{1 + t^{2}} + 1) \leq t^{2} / 2$ gives $∣ \sqrt{1 + t^{2}} - t - 1 ∣ \leq t + t^{2} / 2 \to 0$ , so $η : = \arctan (q / p)$ tends to $\arctan 1 = π / 4$ by the continuity of $\arctan$ in [F6]. Put $r : = q / p > 0$ and $η = \arctan r \in (0, π / 2)$ ; by [F6], $\sin η = r \cos η$ , and the Pythagorean identity $\sin^{2} η + \cos^{2} η = 1$ with $\cos η > 0$ gives $\cos η = 1 / \sqrt{1 + r^{2}}$ and $\sin η = r / \sqrt{1 + r^{2}}$ ; equivalently $\cos η = p / \sqrt{p^{2} + q^{2}}$ and $\sin η = q / \sqrt{p^{2} + q^{2}}$ , since $\sqrt{1 + r^{2}} = \sqrt{p^{2} + q^{2}} / p$ with $p > 0$ by [F4]. The addition formula of [F6] then gives $\sqrt{p^{2} + q^{2}} \cos (θ + η) = p \cos θ - q \sin θ$ , which is the stated form of the magnetic scalar factor.

6.1givenstep 4.2F1F8∎

[Limits and scope, part (f)] Clause (f): for $σ = 0$ the definition of Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth gives $b = 0$ , $q = 0$ , no attenuation and $δ = + \infty$ , and the divisions and bounds of steps 3.1-4.2 use $b > 0$ , $q > 0$ and are not evaluated at zero; the limit $σ \to \infty$ is outside the finite-Ohmic branch and the frequency dependence of $σ$ is not supplied; all identities are exact for the plane-wave field and involve no measurement, material data or empirical statement, and no choice principle is used since every quantity is defined by explicit formulas in the given scalars.

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Normal-incidence amplitudes and the flux budget, with the Ohmic transmitting medium

Physical scope

Conditional normal-incidence reflection and transmission theorem at a plane interface between two homogeneous isotropic scalar media at rest, the second either lossless or Ohmic: the amplitude system, the reflection and transmission amplitudes, the flux budget R+T=1 in the lossless case and |r|^2+(ReY_2/Y_1)|t|^2=1 with the absorbed fraction in the Ohmic case. Excluded: oblique incidence, anisotropic, dispersive or nonlinear media, ideal conductors, mobile media, existence or uniqueness of interface-consistent data, and all empirical claims.

Statement

Assume the following plane-interface material setting, with the same explicitly adopted stationary interface identities as Phase matching at a plane interface: common tangential wavevector and frequency, the reflection law and Snell's law (including its absent-wave qualification), at normal incidence: medium 1 in ${x_{3} < 0}$ with $ε_{1}, μ_{1} > 0$ , $σ_{1} = 0$ , medium 2 in ${x_{3} > 0}$ with $ε_{2}, μ_{2} > 0$ and either $σ_{2} = 0$ (lossless) or $σ_{2} > 0$ (Ohmic), both homogeneous and at rest in the laboratory chart, with the plane interface $S = {x_{3} = 0}$ and normal $n = e_{3}$ . Let $ω_{i}, ω_{r}, ω_{t} > 0$ and let three plane waves of Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch be given with wavevectors $k_{i} = k_{1} n$ ( $k_{1} > 0$ ), $k_{r} = - k_{1} n$ and $k_{t} = k n$ , where for $σ_{2} = 0$ one takes $k = ω_{t} / v_{2} > 0$ and for $σ_{2} > 0$ one takes $k = κ_{2} (ω_{t}) = p + i q$ with the complex propagation constant of Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth; amplitudes $e_{i} \in C^{3} ∖ {0}$ and $e_{r}, e_{t} \in C^{3}$ and $h_{j} = k_{j} \times e_{j} / (μ_{j} ω_{j})$ satisfying the conditions of Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch(b)-(c) in their media, and the four stationary interface identities of Phase matching at a plane interface: common tangential wavevector and frequency, the reflection law and Snell's law holding on $R^{2} \times I$ for a nonempty open interval $I$ . Then:

(a) [Common frequency] The transmitted amplitude is nonzero and $ω_{i} = ω_{t} = : ω$ for some $ω > 0$ . If $e_{r} \neq 0$ , then $ω_{r} = ω$ as well; if $e_{r} = 0$ , assign $ω_{r} = ω$ conventionally, since an absent field cannot determine a frequency. Matching is proved directly from the interface characters, so the lossless-only physical scope of Phase matching at a plane interface: common tangential wavevector and frequency, the reflection law and Snell's law is not used to infer the Ohmic branch.

(b) [Amplitude system and solution] all three electric amplitudes are tangential to $S$ , and the two tangential interface identities reduce to $e_{i} + e_{r} = e_{t}, Y_{1} (e_{i} - e_{r}) = Y_{2} e_{t}, Y_{1} : = \frac{k_{1}}{μ_{1} ω} > 0, Y_{2} : = \frac{k}{μ_{2} ω};$ hence $e_{r} = r e_{i}$ and $e_{t} = t e_{i}$ with the complex scalars $r = \frac{Y_{1} - Y_{2}}{Y_{1} + Y_{2}}, t = \frac{2 Y_{1}}{Y_{1} + Y_{2}};$ in particular the reflected and transmitted electric phasors are parallel to the incident one (normal-incidence polarisation preservation), and $t \neq 0$ . The reflection coefficient is zero exactly when $Y_{1} = Y_{2}$ , in particular at a lossless impedance-matched interface; the zero reflected field remains part of the theorem.

(c) [Lossless transmitting medium] if $σ_{2} = 0$ then $Y_{2} = 1 / Z_{2} > 0$ with $Z_{2} = \sqrt{μ_{2} / ε_{2}}$ of Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth, so $r = \frac{Z_{2} - Z_{1}}{Z_{1} + Z_{2}} \in R, t = \frac{2 Z_{2}}{Z_{1} + Z_{2}} \in R,$ in agreement with The two-equation interface system at normal incidence; the period- $T$ averaged normal fluxes of the stored-energy model, $T = 2 π / ω$ (Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch(d), Energy balance and Ohmic dissipation for the constant nondispersive linear model), are $⟨ S_{i} \cdot n ⟩ = \frac{∣ e_{i} ∣^{2}}{2 Z_{1}}$ , $⟨ S_{r} \cdot n ⟩ = - \frac{∣ e_{r} ∣^{2}}{2 Z_{1}}$ and $⟨ S_{t} \cdot n ⟩ = \frac{∣ e_{t} ∣^{2}}{2 Z_{2}}$ , and with $R : = r^{2}$ , $T : = \frac{Z_{1}}{Z_{2}} t^{2} = \frac{4 Z_{1} Z_{2}}{(Z_{1} + Z_{2})^{2}}$ one has $R + T = 1$ ; $R$ is the reflected and $T$ the transmitted fraction of the incident normal averaged flux.

(d) [Ohmic transmitting medium] if $σ_{2} > 0$ then $Y_{2} = κ_{2} (ω) / (μ_{2} ω)$ has $Re Y_{2} = p / (μ_{2} ω) > 0$ , and $r = \frac{Y_{1} - Y_{2}}{Y_{1} + Y_{2}}, t = \frac{2 Y_{1}}{Y_{1} + Y_{2}}, ∣ r ∣^{2} + \frac{Re Y_{2}}{Y_{1}} ∣ t ∣^{2} = 1.$ The second summand is the fraction of the incident period-averaged normal flux that enters the conductor at the interface; the transmitted field has the exact decay factor $e^{- q x_{3}}$ (the explicit factorization $e^{i (p + i q) x_{3}} = e^{- q x_{3}} e^{i p x_{3}}$ ) and, by the Ohmic dissipation clause of Energy balance and Ohmic dissipation for the constant nondispersive linear model, the entering flux is absorbed in the conductor: the reflected fraction $∣ r ∣^{2}$ and the absorbed fraction $(Re Y_{2} / Y_{1}) ∣ t ∣^{2}$ add to $1$ .

(e) [Scope] no existence or uniqueness of interface-consistent data, no oblique incidence, no anisotropic or dispersive media, no ideal conductor ( $σ_{2} = \infty$ ), no nonlinear response and no empirical claim is made. The interface identities are the stationary plane-interface conditions of the model, supplied for compact interfaces by Stationary macroscopic interface jump conditions for D, B, E and H; the flux identities are the model's stored-energy and Ohmic-dissipation reading, not an observational claim.

Facts & Assumptions

Given: The two-media normal-incidence setting of the Statement, the wave data $(k_{j}, ω_{j}, e_{j})$ with $h_{j} = k_{j} \times e_{j} / (μ_{j} ω_{j})$ , the unit normal $n = e_{3}$ , and the four stationary interface identities [A1] on $R^{2} \times I$ .

[A1]

The four stationary interface identities of the Statement hold for all $(y, t) \in R^{2} \times I$ ; the proof uses the two tangential identities and derives the normal-incidence reduction directly.

[F1]

Plane-wave structure in each medium (Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch): the amplitude conditions $k_{j} \cdot e_{j} = 0$ , $k_{j} \times e_{j} = ω_{j} μ h_{j}$ , $k_{j} \times h_{j} = - (ω_{j} ε + i σ_{j}) e_{j}$ with the constants of the medium of wave $j$ ( $σ_{i} = σ_{r} = 0$ , $σ_{t} = σ_{2}$ ); the solution family $h = k \times e / (ω μ)$ ; and, for a lossless medium with real wavevector $k$ and $e \neq 0$ , the pointwise identities with $B = \hat{k} \times E / v$ , $∣ B ∣ = ∣ E ∣ / v$ and the period-averaged flux $⟨ S ⟩ = ∣ e ∣^{2} k / (2 μ ω)$ of the stored-energy model.

[F2]

Normal-incidence instance of phase matching (Phase matching at a plane interface: common tangential wavevector and frequency, the reflection law and Snell's law(a)): for three nonzero plane waves in two homogeneous isotropic lossless media satisfying the amplitude conditions and the two tangential interface identities, the tangential wavevectors and the frequencies agree; at normal incidence this gives $ω_{i} = ω_{r} = ω_{t}$ in the lossless case with all three amplitudes nonzero.

[F3]

Admittance system (The two-equation interface admittance system: solution formulas and the normal-flux identity): for complex $Y_{1}, Y_{2}$ with $Y_{1} + Y_{2} \neq 0$ and data satisfying $E_{i} + E_{r} = E_{t}$ , $Y_{1} (E_{i} - E_{r}) = Y_{2} E_{t}$ , one has $r = (Y_{1} - Y_{2}) / (Y_{1} + Y_{2})$ , $t = 2 Y_{1} / (Y_{1} + Y_{2})$ ; if $Y_{1} > 0$ real then $∣ r ∣^{2} + (Re Y_{2} / Y_{1}) ∣ t ∣^{2} = 1$ with $∣ t ∣^{2} = 4 Y_{1}^{2} / ∣ Y_{1} + Y_{2} ∣^{2} > 0$ ; and if $Y_{1} > 0$ is real and $Y_{2}$ purely imaginary then $∣ r ∣ = 1$ .

[F4]

The normal-incidence interface lemma (The two-equation interface system at normal incidence): for real $ε_{j}, μ_{j} > 0$ , $Z_{j} = \sqrt{μ_{j} / ε_{j}}$ , and $E_{i}, E_{r}, E_{t}$ with $E_{i} \neq 0$ satisfying $E_{i} + E_{r} = E_{t}$ , $(E_{i} - E_{r}) / Z_{1} = E_{t} / Z_{2}$ , one has $r = (Z_{2} - Z_{1}) / (Z_{1} + Z_{2})$ , $t = 2 Z_{2} / (Z_{1} + Z_{2})$ and $R + T = 1$ for $R = ∣ r ∣^{2}$ , $T = (Z_{1} / Z_{2}) ∣ t ∣^{2}$ .

[F5]

Character independence (Distinct exponential characters are linearly independent on a nonempty open set, and the real-part grouping rule): a real-part-zero finite character sum on a nonempty open set has vanishing grouped coefficients; in particular if $Re (c e^{i λ \cdot z}) = 0$ on a nonempty open set for a single frequency $λ = (0, 0, - ω)$ with $ω > 0$ and coefficient $c$ , then $c = 0$ , because $- \overline{λ} = (0, 0, + ω) \neq λ$ and the $ν = λ$ group has no negative-conjugate partner.

[F6]

The complex exponential: $\exp (z + w) = \exp z \exp w$ and $\exp (x) = e^{x}$ for real $x$ ( $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential); for real $x, y$ , $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ with $∣ \exp (x + i y) ∣ = e^{x}$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ).

[F7]

Period averages (Period averages of products of real parts of complex exponentials): the period average of $Re (a e^{- i ω t}) Re (d e^{- i ω t})$ is $\frac{1}{2} Re (a \overline{d})$ , with the componentwise inner-product and cross-product forms.

[F8]

Stored energy and Ohmic dissipation (Energy balance and Ohmic dissipation for the constant nondispersive linear model): $u = \frac{1}{2} (E \cdot D + B \cdot H)$ , $S = E \times H$ , the local balance $\partial_{t} u + div S = - J_{f} \cdot E$ , and with $J_{f} = σ E$ and symmetric positive semidefinite $σ$ the nonnegative dissipation density $E \cdot σ E$ .

[F9]

Medium parameters (Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth): $Z_{j} = \sqrt{μ_{j} / ε_{j}} > 0$ , $v_{j} = (ε_{j} μ_{j})^{- 1 / 2}$ , and for $σ_{2} > 0$ the complex propagation constant $k = κ_{2} (ω) = p + i q$ with $p, q > 0$ , whose existence and root selection rest on Every complex number has a square root, by an explicit Cartesian formula; for $σ_{2} > 0$ also $2 p q = μ_{2} σ_{2} ω$ and $Re Y_{2} = p / (μ_{2} ω)$ .

[F10]

Complex arithmetic, conjugation and modulus (The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ , $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ , Real and imaginary parts, complex conjugation, and modulus, Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive), and nonnegative square roots with $\sqrt{a b} = \sqrt{a} \sqrt{b}$ for $a, b \geq 0$ (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ).

[F11]

Inner and cross products: $⟨ x, y ⟩ = \sum_{l} x_{l} y_{l}$ and the coordinate cross product, bilinear and alternating, with $u \times v$ orthogonal to $u$ and $v$ ; the decomposition $x = x^{τ} + (x \cdot n) n$ with $x^{τ} ⊥ n$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The cross product in $R^{3}$ , The cross product is bilinear, alternating, and orthogonal to both factors).

[F12]

The laboratory setting, fields and units (Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, The electric and magnetic fields as fixed-time spatial vector fields, Framework-local SI units, the vacuum constants and the exact/measured status of their values) and the declared lossless/Ohmic branches (Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch).

[F13]

One-dimensional FTC and integrability (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable), improper integrals at infinity as limits of proper integrals (Improper integrals over unbounded intervals), and $e^{- 2 q L} \to 0$ as $L \to \infty$ for $q > 0$ (The exponential tends to $+ \infty$ at $+ \infty$ and to $0$ at $- \infty$ ).

Proof

technique · direct; evaluate the normal-incidence traces as characters, apply the character lemma to the combined six-component data to force the common frequency, solve the resulting two-equation admittance system, and compute the lossless and Ohmic flux budgets

1.1givenA1F1F10F11algebra

[Transversality and the trace identity] Since $k_{j} \cdot e_{j} = 0$ and $k_{j} \cdot n \neq 0$ for each actual wave ( $k_{i} \cdot n = k_{1} > 0$ , $k_{r} \cdot n = - k_{1} < 0$ , and $k_{t} \cdot n = k$ with $k > 0$ in the lossless case or $k = p + i q$ with $p > 0$ in the Ohmic case), each electric amplitude is tangential: writing $e_{j} = (e_{j} \cdot n) n + e_{j}^{τ}$ by [F11] gives $0 = k_{j} \cdot e_{j} = (k_{j} \cdot n) (e_{j} \cdot n)$ . Likewise $h_{j} = k_{j} \times e_{j} / (μ_{j} ω_{j})$ is tangential because $k_{j}$ is normal. Evaluating at $x = (y, 0)$ gives $k_{j} \cdot x = 0$ , so $E_{j} = Re (e_{j} e^{- i ω_{j} t})$ and $H_{j} = Re (h_{j} e^{- i ω_{j} t})$ there; the two tangential interface identities of [A1] are identities of tangential fields, and for tangential $w$ the vanishing $n \times w = 0$ forces $w = 0$ by [F11], so with the sign convention $A_{j} : = (e_{j}, h_{j})$ and $λ_{j} : = (0, 0, - ω_{j}) \in C^{3}$ they read $Re (A_{i} e^{i λ_{i} \cdot z} + A_{r} e^{i λ_{r} \cdot z} - A_{t} e^{i λ_{t} \cdot z}) = 0$ for all $z = (y, t) \in R^{2} \times I$ , by the additivity of the real part in [F10].

2.1step 1.1F1F5F10F11algebra

[The transmitted amplitude is nonzero] Suppose $e_{t} = 0$ ; then $h_{t} = k (n \times e_{t}) / (μ_{2} ω_{t}) = 0$ and $A_{t} = 0$ , so the identity of step 1.1 has the two terms $A_{i}, A_{r}$ with nonzero frequency parts $- ω_{i}, - ω_{r}$ . Since $- \overline{λ_{j}} = (0, 0, + ω_{j})$ has positive last coordinate while every actual $λ_{l}$ has negative last coordinate, [F5] applied to the electric and magnetic traces separately gives: for each actual frequency $ν$ , the sum of the coefficients of the waves with that frequency vanishes. If $e_{r} = 0$ then $A_{r} = 0$ and the electric trace is a single character with coefficient $e_{i} \neq 0$ , contradicting [F5]. If $e_{r} \neq 0$ and $λ_{r} \neq λ_{i}$ , both groups are singletons and the electric trace gives $e_{i} = 0$ and $e_{r} = 0$ , again a contradiction; hence $λ_{r} = λ_{i}$ , that is $ω_{r} = ω_{i}$ , and the electric and magnetic trace coefficients at that frequency vanish: $e_{i} + e_{r} = 0$ and $h_{i} + h_{r} = 0$ . By step 1.1, $h_{i} = (k_{1} / (μ_{1} ω_{i})) (n \times e_{i})$ and $h_{r} = (- k_{1} / (μ_{1} ω_{i})) (n \times e_{r})$ , so $h_{i} + h_{r} = (k_{1} / (μ_{1} ω_{i})) n \times (e_{i} - e_{r}) = (k_{1} / (μ_{1} ω_{i})) n \times (2 e_{i}) = 0$ , hence $n \times e_{i} = 0$ ; together with $e_{i} \cdot n = 0$ from step 1.1 this gives $e_{i} = 0$ by [F11], contradicting $e_{i} \neq 0$ . Therefore $e_{t} \neq 0$ .

3.1step 2.1step 1.1F2F5F10algebra

[Common frequency] By step 2.1, $A_{t} = (e_{t}, h_{t})$ is nonzero; $A_{i}$ is nonzero because $e_{i} \neq 0$ ; if $e_{r} \neq 0$ then $A_{r} \neq 0$ . Applying [F5] to the combined six-component identity of step 1.1 and grouping as in step 2.1: for each actual frequency $ν$ the signed sum of the $A_{j}$ over the waves with that frequency vanishes. If the frequencies of the nonzero- $A_{j}$ waves were not all equal, some group would contain exactly one of them and would give $\pm A_{j} = 0$ , a contradiction; and negative-conjugate partners cannot occur because of the sign of the last coordinate. Hence: if $e_{r} \neq 0$ , then $λ_{i} = λ_{r} = λ_{t}$ , so $ω_{i} = ω_{r} = ω_{t}$ ; if $e_{r} = 0$ , then $λ_{i} = λ_{t}$ , so $ω_{i} = ω_{t}$ and $ω_{r}$ is unconstrained by the absent field, matching the Statement. Setting $ω : = ω_{i} = ω_{t}$ , the two identities of step 1.1 with the common frequency $- ω$ in the exponent give, after applying [F5] to the single-frequency electric and magnetic traces, $e_{i} + e_{r} = e_{t}$ and $h_{i} + h_{r} = h_{t}$ . This also proves the lossless nonzero-reflected case via the normal-incidence instance [F2] of the phase-matching theorem, which is consistent with the direct argument just given.

4.1step 3.1F1F3F10F11algebra

[The amplitude system, part (b)] With the common frequency of step 3.1, $Y_{1} : = k_{1} / (μ_{1} ω) > 0$ is real and positive, and $h_{i} = Y_{1} n \times e_{i}$ , $h_{r} = - Y_{1} n \times e_{r}$ , $h_{t} = (k / (μ_{2} ω)) n \times e_{t} = : Y_{2} n \times e_{t}$ by [F1] and [F11]; note $Re Y_{2} = p / (μ_{2} ω) > 0$ in the Ohmic case and $Y_{2} > 0$ real in the lossless case, so $Y_{1} + Y_{2} \neq 0$ in both. The magnetic identity of step 3.1 reads $n \times (Y_{2} e_{t} - Y_{1} (e_{i} - e_{r})) = 0$ , and its argument is tangential, so $Y_{1} (e_{i} - e_{r}) = Y_{2} e_{t}$ by [F11]; together with $e_{i} + e_{r} = e_{t}$ this is exactly the system of [F3], and since $Y_{1} + Y_{2} \neq 0$ the solution formulas $r = (Y_{1} - Y_{2}) / (Y_{1} + Y_{2})$ , $t = 2 Y_{1} / (Y_{1} + Y_{2})$ hold with $e_{r} = r e_{i}$ , $e_{t} = t e_{i}$ ; here $r, t$ are scalars in $C$ multiplying the fixed nonzero vector $e_{i}$ , so the reflected and transmitted electric phasors are parallel to the incident one. Moreover $t \neq 0$ because $∣ t ∣^{2} = 4 Y_{1}^{2} / ∣ Y_{1} + Y_{2} ∣^{2} > 0$ by [F3] and [F10]; and $r = 0$ if and only if $Y_{1} = Y_{2}$ .

5.1step 4.1F1F4F9F10algebra

[Lossless transmitting medium, part (c)] If $σ_{2} = 0$ then $k = ω / v_{2} > 0$ by the Statement, so $Y_{2} = k / (μ_{2} ω) = ω / (v_{2} μ_{2} ω) = 1 / (v_{2} μ_{2}) = 1 / Z_{2}$ by [F9] ( $v_{2} μ_{2} = \sqrt{μ_{2} / ε_{2}} = Z_{2}$ ), and $Y_{1} = 1 / Z_{1}$ similarly; hence substituting $Y_{j} = 1 / Z_{j}$ into the formulas of step 4.1 gives $r = (1 / Z_{1} - 1 / Z_{2}) / (1 / Z_{1} + 1 / Z_{2}) = (Z_{2} - Z_{1}) / (Z_{1} + Z_{2})$ and $t = 2 Z_{2} / (Z_{1} + Z_{2})$ , both real, the same formulas as in the scalar parametrisation of [F4]. The period-averaged normal fluxes at the interface: for a nonzero lossless wave with real wavevector the flux of [F1] is $⟨ S_{j} ⟩ = ∣ e_{j} ∣^{2} k_{j} / (2 μ_{j} ω)$ with $ω$ the common frequency, so $⟨ S_{i} \cdot n ⟩ = ∣ e_{i} ∣^{2} k_{1} / (2 μ_{1} ω) = ∣ e_{i} ∣^{2} / (2 Z_{1})$ because $k_{1} / (μ_{1} ω) = Y_{1} = 1 / Z_{1}$ ; similarly $⟨ S_{r} \cdot n ⟩ = ∣ e_{r} ∣^{2} (- k_{1}) / (2 μ_{1} ω) = - ∣ e_{r} ∣^{2} / (2 Z_{1})$ and $⟨ S_{t} \cdot n ⟩ = ∣ e_{t} ∣^{2} k / (2 μ_{2} ω) = ∣ e_{t} ∣^{2} / (2 Z_{2})$ . If $e_{r} = 0$ , then $h_{r} = 0$ and $S_{r} = 0$ , so the reflected formula holds directly as well. With $R = ∣ r ∣^{2}$ and $T = (Z_{1} / Z_{2}) ∣ t ∣^{2}$ , the identity $R + T = 1$ is [F4]; since $∣ e_{r} ∣^{2} = R ∣ e_{i} ∣^{2}$ and $∣ e_{t} ∣^{2} = (Z_{2} / Z_{1}) T ∣ e_{i} ∣^{2} = ∣ e_{i} ∣^{2} T Z_{2} / Z_{1}$ , the three fluxes are $∣ e_{i} ∣^{2} / (2 Z_{1})$ times $1$ , $- R$ and $T$ respectively, which is the stated fraction reading.

5.2step 4.1F3F9F10algebra

[Ohmic transmitting medium: system and budget, part (d) first half] If $σ_{2} > 0$ then $k = κ_{2} (ω) = p + i q$ with $p, q > 0$ by [F9], so $Y_{2} = (p + i q) / (μ_{2} ω)$ has $Re Y_{2} = p / (μ_{2} ω) > 0$ and $Im Y_{2} = q / (μ_{2} ω) > 0$ ; the solution formulas of step 4.1 give $r, t$ , and [F3] gives $∣ r ∣^{2} + (Re Y_{2} / Y_{1}) ∣ t ∣^{2} = 1$ and $∣ t ∣^{2} = 4 Y_{1}^{2} / ∣ Y_{1} + Y_{2} ∣^{2} > 0$ . Since $Im Y_{2} \neq 0$ , the numerator $Y_{1} - Y_{2}$ of $r$ is nonzero, so $r \neq 0$ in this branch.

6.1step 4.1step 5.1F6F7F10F11algebra

[Attenuated transmitted field and entering flux] With $k = p + i q$ and the common frequency, the transmitted field factorizes exactly: for $x_{3} \in R$ , $i (k x_{3} - ω t) = - q x_{3} + i (p x_{3} - ω t)$ , and [F6] gives $e^{i (k x_{3} - ω t)} = e^{- q x_{3}} (\cos (p x_{3} - ω t) + i \sin (p x_{3} - ω t))$ , so $E_{t} (x, t) = e^{- q x_{3}} Re (e_{t} e^{i (p x_{3} - ω t)})$ componentwise (the exact decay factor $e^{- q x_{3}}$ ), and $H_{t} = e^{- q x_{3}} Re (h_{t} e^{i (p x_{3} - ω t)})$ for the same reason since $h_{t}$ is a constant phasor. The period-averaged normal flux of the transmitted field is, by the cross-product form of [F7] applied to the factorized fields, $⟨ S_{t} \cdot n ⟩ = e^{- 2 q x_{3}} \frac{1}{2} Re ((e_{t} \times {\overline{h}}_{t}) \cdot n)$ ; from $h_{t} = Y_{2} n \times e_{t}$ with $Y_{2} = (p + i q) / (μ_{2} ω)$ by step 4.1, ${\overline{h}}_{t} = \overline{Y_{2}} n \times {\overline{e}}_{t} = ((p - i q) / (μ_{2} ω)) n \times {\overline{e}}_{t}$ by [F10], and the triple-product expansion of [F11] gives $e_{t} \times (n \times {\overline{e}}_{t}) = n ∣ e_{t} ∣^{2} - {\overline{e}}_{t} (e_{t} \cdot n) = n ∣ e_{t} ∣^{2}$ because $e_{t}$ is tangential; hence $(e_{t} \times {\overline{h}}_{t}) \cdot n = (p - i q) ∣ e_{t} ∣^{2} / (μ_{2} ω)$ and $⟨ S_{t} \cdot n ⟩ = e^{- 2 q x_{3}} p ∣ e_{t} ∣^{2} / (2 μ_{2} ω) = : F (x_{3})$ . At the interface $x_{3} = 0$ this equals $(p / (μ_{2} ω)) ∣ e_{t} ∣^{2} / 2$ , and dividing by the incident flux $∣ e_{i} ∣^{2} / (2 Z_{1}) = Y_{1} ∣ e_{i} ∣^{2} / 2$ of step 5.1 (which applies to medium 1, lossless) gives $\frac{p}{μ_{2} ω Y_{1}} ∣ t ∣^{2} = \frac{Re Y_{2}}{Y_{1}} ∣ t ∣^{2}$ , the stated entering fraction of the incident period-averaged normal flux.

7.1step 5.2step 6.1F8F9F13algebra

[Ohmic absorption] In the conductor the free current is the conduction current $J_{f} = σ_{2} E_{t}$ with the declared positive scalar $σ_{2}$ (the model of [F1] has no external free current), and the Ohmic dissipation density of [F8] averaged over a period is $d (x_{3}) : = σ_{2} ⟨ ∣ E_{t} ∣^{2} ⟩ = σ_{2} ∣ e_{t} ∣^{2} e^{- 2 q x_{3}} / 2$ by [F7] applied to the factorized field of step 6.1; the flux of step 6.1 is $F (x_{3}) = p ∣ e_{t} ∣^{2} e^{- 2 q x_{3}} / (2 μ_{2} ω)$ . Since $2 p q = μ_{2} σ_{2} ω$ by [F9], $F^{'} (x_{3}) = - 2 q p ∣ e_{t} ∣^{2} e^{- 2 q x_{3}} / (2 μ_{2} ω) = - p q ∣ e_{t} ∣^{2} e^{- 2 q x_{3}} / (μ_{2} ω) = - σ_{2} ∣ e_{t} ∣^{2} e^{- 2 q x_{3}} / 2 = - d (x_{3})$ : the flux lost per unit length is exactly the Ohmic dissipation. For every $L > 0$ the integrand $d$ is continuous on $[0, L]$ , hence Riemann integrable by [F13], and the exponential antiderivative is differentiable through the endpoints; the fundamental theorem [F13] gives $\int_{0}^{L} d (x_{3}) d x_{3} = σ_{2} ∣ e_{t} ∣^{2} (1 - e^{- 2 q L}) / (4 q) = F (0) - F (L)$ , where the last identity uses $p / (2 μ_{2} ω) = σ_{2} / (4 q)$ , which is $2 p q = μ_{2} σ_{2} ω$ rearranged. Since $q > 0$ , $e^{- 2 q L} \to 0$ as $L \to \infty$ by [F13], and defining the improper integral as the limit of the proper integrals per [F13] gives $\int_{0}^{\infty} d (x_{3}) d x_{3} = σ_{2} ∣ e_{t} ∣^{2} / (4 q) = F (0)$ : the whole entering flux is dissipated in the conductor, and with the budget of step 5.2 the reflected fraction $∣ r ∣^{2}$ and the absorbed fraction $(Re Y_{2} / Y_{1}) ∣ t ∣^{2}$ add to $1$ .

8.1givenstep 7.1F2F12∎

[Scope, part (e)] All conclusions are conditional on the given wave data and the interface identities [A1]; the argument used only the two tangential identities, the amplitude conditions and dispersion of [F1], the character lemma [F5], the admittance algebra [F3], the Euler form [F6] and the period averages [F7]; the normal interface identities, the optional normal-root selection rule and any outgoing condition were not used, and the oblique-incidence theorem [F2] was cited only as the consistent lossless cross-check. Nothing about existence or uniqueness of interface-consistent data, oblique incidence, anisotropy, dispersion, ideal conductors ( $σ_{2} = \infty$ ), nonlinear response or empirical data is claimed; the entering-flux absorption reading is the model's stored-energy and Ohmic-dissipation statement, not an observational claim. The interface identities are stated hypotheses on the noncompact plane; for compact interfaces they follow from Stationary macroscopic interface jump conditions for D, B, E and H. No choice principle is used: every quantity is finite-dimensional algebra in the given data, and the only limiting passage is the explicit exponential improper integral of step 7.1.

Physics theoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Oblique-incidence TE and TM Fresnel amplitudes, the flux budget, Brewster's angle and total internal reflection

Physical scope

Conditional oblique-incidence Fresnel theorem for two homogeneous isotropic lossless scalar media at rest at a plane interface, under the stated interface identities: the TE and TM admittance systems, amplitude formulas, the flux budget R_chi+T_chi=1 in the propagating branch, the Brewster condition and the total-internal-reflection identities |r_chi|=1 with decaying transmitted field. Excluded: absorbing, dispersive, anisotropic, moving or nonlinear media, the ideal conductor, existence or uniqueness of interface-consistent data, signal- and group-velocity claims, and all empirical claims.

Statement

Assume the following plane-interface setting with the same stationary interface identities as Phase matching at a plane interface: common tangential wavevector and frequency, the reflection law and Snell's law: two homogeneous isotropic lossless scalar media at rest (medium 1 in ${x_{3} < 0}$ , constants $ε_{1}, μ_{1} > 0$ ; medium 2 in ${x_{3} > 0}$ , constants $ε_{2}, μ_{2} > 0$ ; both $σ = 0$ ), the plane interface $S = {x_{3} = 0}$ with normal $n = e_{3}$ , and the plane-wave structure of Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch. Let $ω > 0$ , let $κ \in R^{2}$ (identified with the tangential plane ${x_{3} = 0}$ ) satisfy $0 < ∣ κ ∣ < K_{1} : = ω (ε_{1} μ_{1})^{1 / 2}$ , put $k_{13} : = \sqrt{K_{1}^{2} - ∣ κ ∣^{2}} > 0$ and define wavevectors $k_{i} : = κ + k_{13} n, k_{r} : = κ - k_{13} n, k_{t} : = κ + k_{t 3} n,$ where $k_{t 3} \in C$ satisfies $k_{t 3}^{2} = K_{2}^{2} - ∣ κ ∣^{2}$ with $K_{2} : = ω (ε_{2} μ_{2})^{1 / 2}$ and is selected by non-growth in the $x_{3} > 0$ half-space: $k_{t 3} = \sqrt{K_{2}^{2} - ∣ κ ∣^{2}} > 0 (∣ κ ∣ < K_{2}), k_{t 3} = 0 (∣ κ ∣ = K_{2}), k_{t 3} = i β, β : = \sqrt{∣ κ ∣^{2} - K_{2}^{2}} > 0 (∣ κ ∣ > K_{2}) .$ [This transmitted phase data is the interface-consistent, bounded choice: the propagating and evanescent branches of Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch(d),(e); the common $(κ, ω)$ is declared here for each actual wave, also when a reflected amplitude is zero or the critical transmitted root is zero; the nonzero phase-matching theorem explains this matching where its hypotheses apply, and an absent reflected field has no determined phase labels.] Fix the orthonormal tangential frame $\hat{κ} : = κ / ∣ κ ∣$ and $\hat{y} : = n \times \hat{κ}$ , and consider the two polarisations: (s) [transverse electric] $e_{i} = E_{i} \hat{y}$ , $e_{r} = E_{r} \hat{y}$ , $e_{t} = E_{t} \hat{y}$ with $E_{i}, E_{r}, E_{t} \in C$ , $E_{i} \neq 0$ , and $h_{j} : = k_{j} \times e_{j} / (μ_{j} ω)$ ; (p) [transverse magnetic] $h_{i} = H_{i} \hat{y}$ , $h_{r} = H_{r} \hat{y}$ , $h_{t} = H_{t} \hat{y}$ with $H_{i}, H_{r}, H_{t} \in C$ , $H_{i} \neq 0$ , and electric amplitudes $e_{j} : = - (k_{j} \times \hat{y}) H_{j} / (ω ε_{j})$ (the same computation shows that in both cases the pairs satisfy the amplitude conditions of Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch(b) in their media). For each polarisation define the amplitudes along the fixed tangential directions $E_{j}^{(s)} : = e_{j} \cdot \hat{y}$ and, in the case (p), $E_{j}^{(p)} : = e_{j} \cdot \hat{x}$ with $\hat{x} : = \hat{κ}$ ; assume the four stationary interface identities of Phase matching at a plane interface: common tangential wavevector and frequency, the reflection law and Snell's law hold on $R^{2} \times I$ for a nonempty open interval $I$ . Then:

(a) [Amplitude system] for $χ = s$ in every branch, and for $χ = p$ when $k_{t 3} \neq 0$ , the tangential interface identities reduce to $E_{i}^{χ} + E_{r}^{χ} = E_{t}^{χ}, Y_{1}^{χ} (E_{i}^{χ} - E_{r}^{χ}) = Y_{2}^{χ} E_{t}^{χ},$ with the admittances $Y_{1}^{s} = \frac{k_{13}}{μ_{1} ω}, Y_{1}^{p} = \frac{ε_{1} ω}{k_{13}}, Y_{2}^{s} = \frac{k_{t 3}}{μ_{2} ω}, Y_{2}^{p} = \frac{ε_{2} ω}{k_{t 3}} (k_{t 3} \neq 0) .$ Write $k_{j 3} : = k_{j} \cdot n$ for the signed components, so $k_{i 3} = k_{13}$ and $k_{r 3} = - k_{13}$ . Here $Y_{1}^{χ} > 0$ uses the positive incident normal magnitude $k_{13}$ , whereas the signed reflected component is $k_{r} \cdot n = - k_{13}$ ; $Y_{1}^{χ}$ is not defined from that negative component. No $Y_{2}^{p}$ is defined at $k_{t 3} = 0$ . Hence, whenever $Y_{1}^{χ} + Y_{2}^{χ} \neq 0$ , $r_{χ} : = \frac{E_{r}^{χ}}{E_{i}^{χ}} = \frac{Y_{1}^{χ} - Y_{2}^{χ}}{Y_{1}^{χ} + Y_{2}^{χ}}, t_{χ} : = \frac{E_{t}^{χ}}{E_{i}^{χ}} = \frac{2 Y_{1}^{χ}}{Y_{1}^{χ} + Y_{2}^{χ}}$ (The two-equation interface admittance system: solution formulas and the normal-flux identity(a)); the reflected and transmitted amplitude symbols are defined by these fixed-direction conventions, and sign conventions for full vector amplitudes may reverse a reflected amplitude relative to this convention.

(b) [Propagating branch and Snell's law] if $∣ κ ∣ < K_{2}$ then all three wavevectors are real, $Y_{1}^{s}, Y_{1}^{p}, Y_{2}^{s}, Y_{2}^{p} > 0$ are real, and with $θ_{j} \in (0, π / 2)$ defined by $\sin θ_{j} = ∣ κ ∣ / ∣ k_{j} ∣$ one has $θ_{r} = θ_{i}$ and $n_{1} \sin θ_{i} = n_{2} \sin θ_{t}$ , $n_{j} = c (ε_{j} μ_{j})^{1 / 2}$ (Phase matching at a plane interface: common tangential wavevector and frequency, the reflection law and Snell's law(c), Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth); the period-averaged normal fluxes are $Y_{1}^{χ} ∣ E_{i}^{χ} ∣^{2} / 2$ , $- Y_{1}^{χ} ∣ E_{r}^{χ} ∣^{2} / 2$ and $Y_{2}^{χ} ∣ E_{t}^{χ} ∣^{2} / 2$ , so $R_{χ} : = ∣ r_{χ} ∣^{2}, T_{χ} : = \frac{Y_{2}^{χ}}{Y_{1}^{χ}} ∣ t_{χ} ∣^{2}, R_{χ} + T_{χ} = 1.$

(c) [Brewster condition] In the finite-admittance branches, $r_{χ} = 0$ if and only if $Y_{1}^{χ} = Y_{2}^{χ}$ ; in particular, if $μ_{1} = μ_{2}$ and $∣ κ ∣ < K_{2}$ , for unequal indices $n_{1} \neq n_{2}$ , $r_{p} = 0$ holds at exactly one admissible incidence angle $θ_{B} \in (0, π / 2)$ , characterised by $\tan θ_{B} = n_{2} / n_{1}$ , and $r_{s} \neq 0$ at that angle. For equal indices $n_{1} = n_{2}$ (and equal permeabilities), the media are identical, $Y_{1}^{χ} = Y_{2}^{χ}$ and $r_{s} = r_{p} = 0$ , $t_{s} = t_{p} = 1$ for every allowed incidence angle; there is no unique distinguished Brewster angle.

(d) [Total internal reflection] if $∣ κ ∣ > K_{2}$ (which requires $n_{1} > n_{2}$ , $θ_{i} : = \arcsin (∣ κ ∣ / K_{1}) \in (0, π / 2)$ , $∣ κ ∣ = K_{1} \sin θ_{i}$ and $θ_{i} > θ_{c} : = \arcsin (n_{2} / n_{1})$ ), then $Y_{2}^{s} = i β / (μ_{2} ω)$ and $Y_{2}^{p} = - i ε_{2} ω / β$ are purely imaginary, $∣ r_{s} ∣ = ∣ r_{p} ∣ = 1, t_{s}, t_{p} \neq 0, ∣ t_{χ} ∣^{2} = \frac{4 (Y_{1}^{χ})^{2}}{(Y_{1}^{χ})^{2} + ∣ Y_{2}^{χ} ∣^{2}} > 0,$ the transmitted field decays as $e^{- β x_{3}}$ and its period-averaged normal flux vanishes identically while its tangential averaged flux does not vanish (Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch(e)); the reflected wave carries the whole incident normal averaged flux.

(e) [Critical angle] if $∣ κ ∣ = K_{2}$ then $k_{t 3} = 0$ : the transmitted wave travels along the interface, its tangential electric amplitude vanishes in the (p) case and $Y_{2}^{s} = 0$ in the (s) case, and $r_{s} = 1$ , $t_{s} = 2$ , $r_{p} = - 1$ , $t_{p} = 0$ with vanishing transmitted normal averaged flux. The critical TM branch is solved from the nonsingular equations $E_{i}^{p} + E_{r}^{p} = k_{t 3} H_{t} / (ω ε_{2}) = 0$ and $H_{i} + H_{r} = H_{t}$ , not by dividing by $k_{t 3}$ ; they give $H_{r} = H_{i}$ and $H_{t} = 2 H_{i}$ . The transmitted TM electric vector can have a nonzero normal component even though its tangential amplitude is zero.

(f) [Scope] the statements are conditional on the interface identities and on the given wave data; no existence or uniqueness of interface-consistent data is claimed. Excluded: anisotropic media, complex $ε$ or $μ$ and absorption in either medium (only the lossless dielectric case with the evanescent transmitted branch is covered), dispersive media, moving media, the ideal conductor, nonlinear response, and all claims about signal speeds, group velocities, information transfer or empirical data. The statements for $Y_{1}^{χ} + Y_{2}^{χ} = 0$ (which cannot occur in the stated branches, where $Re Y_{1}^{χ} > 0$ and $Re Y_{2}^{χ} \geq 0$ ) are excluded; the notation 'angle of refraction' is used only where $k_{t}$ is real.

Facts & Assumptions

Given: The two-media lossless setting of the Statement, its common $(κ, ω)$ and wavevectors $k_{i}, k_{r}, k_{t}$ , the two polarisation data families, and the four stationary interface identities [A1] on $R^{2} \times I$ .

[A1]

The four stationary interface identities of Phase matching at a plane interface: common tangential wavevector and frequency, the reflection law and Snell's law hold for all $(y, t) \in R^{2} \times I$ .

[F1]

Plane-wave structure (Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch): the amplitude conditions $k_{j} \cdot e_{j} = 0$ , $k_{j} \times e_{j} = ω μ_{j} h_{j}$ , $k_{j} \cdot h_{j} = 0$ , $k_{j} \cdot k_{j} = ε_{j} μ_{j} ω^{2}$ and the solution form $h_{j} = k_{j} \times e_{j} / (μ_{j} ω)$ ; the lossless pointwise identities $B = \hat{k} \times E / v$ and period-averaged flux $⟨ S ⟩ = ∣ e ∣^{2} k / (2 μ ω)$ for real $k$ ; and, in the evanescent branch, the factorization with decay $e^{- β n^{'} \cdot x}$ , the vanishing normal averaged flux and the positive tangential averaged fluxes for the two transverse polarisations.

[F2]

The angle laws (Phase matching at a plane interface: common tangential wavevector and frequency, the reflection law and Snell's law(c)): for real wavevectors at a lossless interface, $θ_{r} = θ_{i}$ and $n_{1} \sin θ_{i} = n_{2} \sin θ_{t}$ with the same angle conventions, together with the parameters $v_{j}$ and $n_{j} = c / v_{j}$ of Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth.

[F3]

Admittance algebra (The two-equation interface admittance system: solution formulas and the normal-flux identity): (a) the solution formulas for the two-equation system; (b) for real $Y_{1} > 0$ the identity $∣ r ∣^{2} + (Re Y_{2} / Y_{1}) ∣ t ∣^{2} = 1$ with $∣ t ∣^{2} = 4 Y_{1}^{2} / ∣ Y_{1} + Y_{2} ∣^{2} > 0$ ; (c) for real $Y_{1} > 0$ and purely imaginary $Y_{2}$ the identities $∣ r ∣ = 1$ and $∣ t ∣^{2} = 4 Y_{1}^{2} / (Y_{1}^{2} + ∣ Y_{2} ∣^{2}) > 0$ ; (d) the real-positive specialisation agreeing with The two-equation interface system at normal incidence.

[F4]

Period averages (Period averages of products of real parts of complex exponentials) and the stored-energy/flux definitions $u = \frac{1}{2} (E \cdot D + B \cdot H)$ , $S = E \times H$ of Energy balance and Ohmic dissipation for the constant nondispersive linear model.

[F5]

Geometry of the fixed frame: $\hat{x} = \hat{κ} = κ / ∣ κ ∣$ , $\hat{y} = n \times \hat{κ}$ , with the right-handed orthonormal triad relations $\hat{x} \times \hat{y} = n$ , $\hat{y} \times n = \hat{x}$ , $n \times \hat{x} = \hat{y}$ and the reversed signs; the cross product is bilinear with $⟨ u \times v, w ⟩ = \det [u v w]$ , and $u \times v$ is orthogonal to $u$ and $v$ (The cross product in $R^{3}$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The cross product is bilinear, alternating, and orthogonal to both factors).

[F6]

Medium parameters (Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth): $K_{j} = ω (ε_{j} μ_{j})^{1 / 2}$ , $v_{j} = (ε_{j} μ_{j})^{- 1 / 2}$ , $n_{j} = c (ε_{j} μ_{j})^{1 / 2}$ . The wavevector definitions and selected roots in this item's Statement give $k_{13}^{2} = K_{1}^{2} - ∣ κ ∣^{2}$ and $k_{t 3}^{2} = K_{2}^{2} - ∣ κ ∣^{2}$ with its declared branch selection.

[F7]

Complex arithmetic, conjugation, modulus (The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ , $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ , Real and imaginary parts, complex conjugation, and modulus, Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive) with nonnegative square roots (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ) and the existence of complex square roots (Every complex number has a square root, by an explicit Cartesian formula).

[F8]

Trigonometry: $\sin (\arcsin y) = y$ with sine strictly increasing on $[- π / 2, π / 2]$ (Principal inverse sine and inverse cosine); $\tan$ restricts to a strictly increasing bijection $(- π / 2, π / 2) \to R$ with the principal inverse (The principal inverse tangent $\arctan : R \to (- π / 2, π / 2)$ ); $\tan x = \sin x / \cos x$ (Tangent, cotangent, secant, and cosecant on their exact natural domains); $\sin^{2} x + \cos^{2} x = 1$ with $1 + \tan^{2} x = \sec^{2} x$ (Pythagorean and parity identities for all six trigonometric functions on their natural domains); sine is strictly increasing on $[0, π / 2]$ and cosine strictly decreasing on $[0, π]$ with $\sin 0 = 0$ , $\cos 0 = 1$ and $\cos (π / 2) = 0$ (Signs, monotonicity intervals, and ranges of sine and cosine, The zero sets of sine and cosine and the least positive common period 2 pi).

[F9]

The laboratory setting, fields and units (Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, The electric and magnetic fields as fixed-time spatial vector fields, Framework-local SI units, the vacuum constants and the exact/measured status of their values) and the declared lossless branches (Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch).

Proof

technique · direct; verify the amplitude conditions for the two polarisation families, reduce the tangential interface identities to scalar admittance systems in the fixed frame, and read off the propagating, Brewster, evanescent and critical cases from the admittance algebra and the medium plane-wave structure

1.1givenF1F5F7algebra

[Both polarisation families satisfy the amplitude conditions] For (s), $k_{j} \cdot e_{j} = E_{j} k_{j} \cdot \hat{y} = 0$ because $k_{j} = κ + k_{j 3} n$ and $\hat{y} ⊥ κ, n$ ; and $k_{j} \times e_{j} = E_{j} k_{j} \times \hat{y} = E_{j} (κ \times \hat{y} + k_{j 3} n \times \hat{y}) = E_{j} (∣ κ ∣ n - k_{j 3} \hat{x})$ by [F5], which equals $ω μ_{j} h_{j}$ with $h_{j} = k_{j} \times e_{j} / (μ_{j} ω)$ by definition. For (p), $e_{j} = - (k_{j} \times \hat{y}) H_{j} / (ω ε_{j})$ and $k_{j} \cdot e_{j} = 0$ because $k_{j} \cdot (k_{j} \times \hat{y}) = 0$ by the orthogonality in [F5]; further $k_{j} \times e_{j} = - (k_{j} \times (k_{j} \times \hat{y})) H_{j} / (ω ε_{j}) = [(k_{j} \cdot k_{j}) \hat{y} - k_{j} (k_{j} \cdot \hat{y})] H_{j} / (ω ε_{j}) = (ε_{j} μ_{j} ω^{2}) \hat{y} H_{j} / (ω ε_{j}) = ω μ_{j} H_{j} \hat{y} = ω μ_{j} h_{j}$ , using $k_{j} \cdot \hat{y} = 0$ and $k_{j} \cdot k_{j} = ε_{j} μ_{j} ω^{2}$ of [F1]. In (s), $k_{j} \cdot h_{j} = 0$ by cross-product orthogonality, and $k_{j} \times h_{j} = - ω ε_{j} e_{j}$ follows by expanding $k_{j} \times (k_{j} \times e_{j}) = k_{j} (k_{j} \cdot e_{j}) - (k_{j} \cdot k_{j}) e_{j}$ and using dispersion. In (p), $k_{j} \cdot h_{j} = 0$ because $\hat{y}$ is perpendicular to $k_{j}$ , and $k_{j} \times h_{j} = - ω ε_{j} e_{j}$ is the definition of $e_{j}$ . Thus all four amplitude conditions hold in both families, including zero reflected or transmitted amplitudes.

2.1givenA1step 1.1F5F7algebra

[The tangential interface identities in the fixed frame] By the orthogonal decomposition of [F5], the tangential parts of the electric amplitudes are: for (s), $e_{j}^{τ} = e_{j} = E_{j} \hat{y}$ ; for (p), $e_{j}^{τ} = (k_{j 3} H_{j} / (ω ε_{j})) \hat{x}$ because $e_{j} = (k_{j 3} \hat{x} - ∣ κ ∣ n) H_{j} / (ω ε_{j})$ from $k_{j} \times \hat{y} = ∣ κ ∣ n - k_{j 3} \hat{x}$ by [F5]. The magnetic tangential traces are obtained as follows: $h_{j} = H_{j} \hat{y}$ in (p), and in (s) $h_{j} = E_{j} (κ \times \hat{y} + k_{j 3} n \times \hat{y}) / (μ_{j} ω) = E_{j} (∣ κ ∣ n - k_{j 3} \hat{x}) / (μ_{j} ω)$ with tangential part $- k_{j 3} E_{j} \hat{x} / (μ_{j} ω)$ and, for the reflected wave with $k_{r 3} = - k_{13}$ , tangential part $+ k_{13} E_{r} \hat{x} / (μ_{1} ω)$ . Since the interface identities hold for all $(y, t)$ with the common $(κ, ω)$ , each has the form $Re (W e^{i (κ \cdot y - ω t)}) = 0$ for a constant tangential vector $W$ . Since $n = e_{3}$ , $n \times (w_{1}, w_{2}, 0) = (- w_{2}, w_{1}, 0)$ , so the cross-product trace identities are equivalent to these real-part identities. Fixing any $y$ and allowing $t$ to vary in $I$ , the time-line proof of the amplitude-vanishing clause (a) of [F1] gives $W = 0$ . Concretely: the tangential electric identity gives $E_{i} + E_{r} = E_{t}$ in (s) and $E_{i}^{(p)} + E_{r}^{(p)} = E_{t}^{(p)}$ in (p); the tangential magnetic identity gives $k_{t 3} E_{t} / (μ_{2} ω) = k_{13} (E_{i} - E_{r}) / (μ_{1} ω)$ in (s) and $H_{i} + H_{r} = H_{t}$ in (p), where $E_{j}^{(p)} = k_{j 3} H_{j} / (ω ε_{j})$ by definition, and when $k_{t 3} \neq 0$ , converting $H_{j} = Y_{j}^{p} E_{j}^{(p)}$ for $j = i, t$ and $H_{r} = - Y_{1}^{p} E_{r}^{(p)}$ (the reflected sign entering through $k_{r 3} = - k_{13}$ ) turns the last identity into $Y_{1}^{p} (E_{i}^{(p)} - E_{r}^{(p)}) = Y_{2}^{p} E_{t}^{(p)}$ . Writing $Y_{1}^{s} = k_{13} / (μ_{1} ω)$ , $Y_{2}^{s} = k_{t 3} / (μ_{2} ω)$ , $Y_{1}^{p} = ε_{1} ω / k_{13}$ and $Y_{2}^{p} = ε_{2} ω / k_{t 3}$ when $k_{t 3} \neq 0$ gives the two systems of (a).

3.1step 2.1F3F7algebra

[Solution of the systems, part (a)] In the (s) branch, $Y_{1}^{s} > 0$ is real and positive, and $Y_{2}^{s}$ is real positive for $∣ κ ∣ < K_{2}$ , zero at the critical point, and $i β / (μ_{2} ω)$ for $∣ κ ∣ > K_{2}$ ; in every case $Re (Y_{1}^{s} + Y_{2}^{s}) = Y_{1}^{s} + Re Y_{2}^{s} \geq Y_{1}^{s} > 0$ , so $Y_{1}^{s} + Y_{2}^{s} \neq 0$ . In the (p) branch with $k_{t 3} \neq 0$ , $Re Y_{1}^{p} = Y_{1}^{p} > 0$ and $Re Y_{2}^{p} = 0$ or $> 0$ , so again $Y_{1}^{p} + Y_{2}^{p} \neq 0$ . The admittance lemma F3 therefore applies to each scalar system of step 2.1 and gives $r_{χ} = (Y_{1}^{χ} - Y_{2}^{χ}) / (Y_{1}^{χ} + Y_{2}^{χ})$ and $t_{χ} = 2 Y_{1}^{χ} / (Y_{1}^{χ} + Y_{2}^{χ})$ ; in particular $t_{χ} \neq 0$ because its numerator $2 Y_{1}^{χ} \neq 0$ .

3.2step 2.1F4F5F6F7algebra

[Critical angle, part (e)] Let $∣ κ ∣ = K_{2}$ ; then $k_{t 3} = 0$ by [F6], so $Y_{2}^{s} = 0$ and no $Y_{2}^{p}$ is defined (a division by $k_{t 3}$ is not performed). For (s), the system of step 2.1 reads $E_{i} + E_{r} = E_{t}$ and $Y_{1}^{s} (E_{i} - E_{r}) = 0$ with $Y_{1}^{s} > 0$ , so $E_{r} = E_{i}$ and $E_{t} = 2 E_{i}$ , i.e. $r_{s} = 1$ and $t_{s} = 2$ . For (p), $E_{t}^{(p)} = k_{t 3} H_{t} / (ω ε_{2}) = 0$ and $E_{i}^{(p)} = k_{13} H_{i} / (ω ε_{1})$ , $E_{r}^{(p)} = - k_{13} H_{r} / (ω ε_{1})$ by the fixed-direction definitions of step 2.1; the tangential electric identity $E_{i}^{(p)} + E_{r}^{(p)} = 0$ gives $H_{r} = H_{i}$ , and the tangential magnetic identity $H_{i} + H_{r} = H_{t}$ gives $H_{t} = 2 H_{i}$ ; hence $r_{p} = E_{r}^{(p)} / E_{i}^{(p)} = - H_{r} / H_{i} = - 1$ and $t_{p} = E_{t}^{(p)} / E_{i}^{(p)} = 0$ . The transmitted normal averaged flux vanishes: $⟨ S_{t} \cdot n ⟩ = \frac{1}{2} Re ((e_{t} \times {\overline{h}}_{t}) \cdot n)$ by [F4], and with $h_{t} = H_{t} \hat{y}$ and $e_{t} = - (k_{t} \times \hat{y}) H_{t} / (ω ε_{2}) = - ∣ κ ∣ H_{t} n / (ω ε_{2})$ (because $k_{t} = κ$ and $κ \times \hat{y} = ∣ κ ∣ n$ by [F5]), the vector $e_{t} \times {\overline{h}}_{t}$ is a multiple of $n \times \hat{y}$ , which is orthogonal to $n$ : $(e_{t} \times {\overline{h}}_{t}) \cdot n = 0$ . In the TE case $e_{t} = E_{t} \hat{y}$ and $h_{t} = ∣ κ ∣ E_{t} n / (μ_{2} ω)$ , so $e_{t} \times {\overline{h}}_{t}$ is tangential and its normal component is also zero. The transmitted TM electric vector $- ∣ κ ∣ H_{t} n / (ω ε_{2})$ has a nontrivial normal component whenever $H_{t} \neq 0$ .

4.1step 3.1F1F2F3F6F8algebra

[Propagating branch: angles and fluxes, part (b)] Let $∣ κ ∣ < K_{2}$ . Then $k_{t 3} = \sqrt{K_{2}^{2} - ∣ κ ∣^{2}} > 0$ and all three wavevectors are real by [F6]; the four admittances of step 2.1 are real and positive. With $∣ k_{i} ∣ = ∣ k_{r} ∣ = K_{1}$ and $∣ k_{t} ∣ = K_{2}$ from $∣ k_{j} ∣^{2} = ∣ κ ∣^{2} + k_{j 3}^{2}$ , the angles $θ_{j} : = \arcsin (∣ κ ∣ / ∣ k_{j} ∣) \in (0, π / 2)$ are well defined by [F8] since $0 < ∣ κ ∣ < \min {K_{1}, K_{2}}$ ; then $\sin θ_{r} = ∣ κ ∣ / K_{1} = \sin θ_{i}$ , so $θ_{r} = θ_{i}$ by the strict increase of sine and of $\arcsin$ in [F8], and $K_{1} \sin θ_{i} = ∣ κ ∣ = K_{2} \sin θ_{t}$ , which is $n_{1} \sin θ_{i} = n_{2} \sin θ_{t}$ after multiplying by $c / ω$ and using $n_{j} = c K_{j} / ω = c (ε_{j} μ_{j})^{1 / 2}$ by [F6]; this is the same angle law as [F2]. The period-averaged normal fluxes: for (s), the flux of [F1] is $⟨ S_{j} ⟩ = ∣ e_{j} ∣^{2} k_{j} / (2 μ_{j} ω)$ with $∣ e_{j} ∣^{2} = ∣ E_{j}^{(s)} ∣^{2}$ and $k_{j} \cdot n = k_{j 3}$ , so the normal fluxes are $k_{j 3} ∣ E_{j}^{(s)} ∣^{2} / (2 μ_{j} ω)$ , i.e. $Y_{1}^{s} ∣ E_{i}^{(s)} ∣^{2} / 2$ , $- Y_{1}^{s} ∣ E_{r}^{(s)} ∣^{2} / 2$ , $Y_{2}^{s} ∣ E_{t}^{(s)} ∣^{2} / 2$ ; for (p) the same $⟨ S_{j} \cdot n ⟩ = ∣ e_{j} ∣^{2} k_{j 3} / (2 μ_{j} ω)$ equals $Y_{j}^{p} ∣ E_{j}^{(p)} ∣^{2} / 2$ up to the reflected sign, because $∣ e_{j} ∣^{2} = μ_{j} ∣ H_{j} ∣^{2} / ε_{j}$ and $∣ E_{j}^{(p)} ∣ = ∣ k_{j 3} ∣ ∣ H_{j} ∣ / (ε_{j} ω)$ give $∣ e_{j} ∣^{2} k_{j 3} / (2 μ_{j} ω) = k_{j 3} ∣ H_{j} ∣^{2} / (2 ε_{j} ω)$ for the incident and transmitted signs, while the reflected signed component $k_{r 3} = - k_{13}$ produces the minus sign, as in the Statement. With $R_{χ} = ∣ r_{χ} ∣^{2} = ∣ E_{r}^{χ} ∣^{2} / ∣ E_{i}^{χ} ∣^{2}$ and $T_{χ} = (Y_{2}^{χ} / Y_{1}^{χ}) ∣ t_{χ} ∣^{2} = Y_{2}^{χ} ∣ E_{t}^{χ} ∣^{2} / (Y_{1}^{χ} ∣ E_{i}^{χ} ∣^{2})$ , the budget $R_{χ} + T_{χ} = 1$ is F3 applied with the real positive $Y_{1}^{χ}, Y_{2}^{χ}$ .

4.2step 3.1F1F3F6F7algebra

[Total internal reflection, part (d)] Let $∣ κ ∣ > K_{2}$ ; then $K_{2} < K_{1}$ , so $n_{2} < n_{1}$ , and $k_{t 3} = i β$ with $β = \sqrt{∣ κ ∣^{2} - K_{2}^{2}} > 0$ by [F6]. Hence $Y_{2}^{s} = k_{t 3} / (μ_{2} ω) = i β / (μ_{2} ω)$ and $Y_{2}^{p} = ε_{2} ω / (i β) = - i ε_{2} ω / β$ are purely imaginary by [F7], while $Y_{1}^{χ} > 0$ is real; the admittance lemma F3 gives $∣ r_{s} ∣ = ∣ r_{p} ∣ = 1$ and $∣ t_{χ} ∣^{2} = 4 (Y_{1}^{χ})^{2} / ((Y_{1}^{χ})^{2} + ∣ Y_{2}^{χ} ∣^{2}) > 0$ , so $t_{χ} \neq 0$ . The transmitted wave is the evanescent branch of [F1] with $n^{'} = n$ , $κ_{0} = κ$ and $k_{t} \cdot k_{t} = K_{2}^{2} = ε_{2} μ_{2} ω^{2}$ : the field factorizes with the decay factor $e^{- β x_{3}}$ , its period-averaged normal flux vanishes identically, and its tangential averaged flux is strictly positive; since $∣ r_{χ} ∣ = 1$ , the reflected normal averaged flux equals minus the incident normal averaged flux, i.e. the reflected wave carries the whole incident normal averaged flux.

5.1step 3.1step 4.1F3F6F8algebra

[Brewster condition, part (c)] In each finite-admittance branch the formula of step 3.1 gives $r_{χ} = 0$ exactly when the numerator $Y_{1}^{χ} - Y_{2}^{χ}$ vanishes, i.e. $Y_{1}^{χ} = Y_{2}^{χ}$ . Assume now $μ_{1} = μ_{2} = : μ$ and $∣ κ ∣ < K_{2}$ , so that the (p) admittances are real positive. The condition $Y_{1}^{p} = Y_{2}^{p}$ reads $ε_{1} / k_{13} = ε_{2} / k_{t 3}$ ; substituting $ε_{j} = n_{j}^{2} / (c^{2} μ)$ and $k_{j 3} = (ω n_{j} / c) \cos θ_{j}$ (from $k_{13} = K_{1} \cos θ_{i} = (ω n_{1} / c) \cos θ_{i}$ and $k_{t 3} = K_{2} \cos θ_{t} = (ω n_{2} / c) \cos θ_{t}$ ) gives $n_{1} \cos θ_{t} = n_{2} \cos θ_{i}$ . Together with Snell's law $n_{1} \sin θ_{i} = n_{2} \sin θ_{t}$ of step 4.1, squaring both and eliminating $\sin^{2} θ_{t}$ gives $(n_{1}^{2} - n_{2}^{2}) [n_{2}^{2} - (n_{1}^{2} + n_{2}^{2}) \sin^{2} θ_{i}] = 0$ : indeed $n_{1}^{2} (1 - \sin^{2} θ_{t}) = n_{2}^{2} \cos^{2} θ_{i}$ and $\sin θ_{t} = (n_{1} / n_{2}) \sin θ_{i}$ yield $n_{1}^{2} n_{2}^{2} - n_{1}^{4} \sin^{2} θ_{i} = n_{2}^{4} - n_{2}^{4} \sin^{2} θ_{i}$ , i.e. $n_{2}^{2} (n_{1}^{2} - n_{2}^{2}) = (n_{1}^{2} - n_{2}^{2}) (n_{1}^{2} + n_{2}^{2}) \sin^{2} θ_{i}$ . If $n_{1} \neq n_{2}$ , division by the nonzero first factor gives $\sin^{2} θ_{i} = n_{2}^{2} / (n_{1}^{2} + n_{2}^{2})$ ; since $0 < θ_{i} < π / 2$ and sine is strictly increasing on $[0, π / 2]$ with $\sin 0 = 0$ by [F8], this determines the unique admissible angle $θ_{B} : = \arcsin (n_{2} / \sqrt{n_{1}^{2} + n_{2}^{2}}) \in (0, π / 2)$ , This angle is admissible: with $D : = \sqrt{n_{1}^{2} + n_{2}^{2}}$ , its tangential wavenumber is $K_{1} n_{2} / D < K_{2}$ because $n_{1} / D < 1$ . Snell's law therefore gives $\sin θ_{t} = n_{1} / D$ and, by positivity of both cosines and the Pythagorean identity, $\cos θ_{i} = n_{1} / D$ , $\cos θ_{t} = n_{2} / D$ . Thus $n_{1} \cos θ_{t} = n_{2} \cos θ_{i}$ , establishing $Y_{1}^{p} = Y_{2}^{p}$ and $r_{p} = 0$ ; this proves sufficiency as well as uniqueness. At this angle $\cos θ_{B} > 0$ : cosine is strictly decreasing on $[0, π]$ with $\cos (π / 2) = 0$ by [F8], so $0 < θ_{B} < π / 2$ gives $\cos θ_{B} > \cos (π / 2) = 0$ . The Pythagorean identity of [F8] then gives $\cos θ_{B} = \sqrt{1 - \sin^{2} θ_{B}} = n_{1} / \sqrt{n_{1}^{2} + n_{2}^{2}}$ , so $\tan θ_{B} = \sin θ_{B} / \cos θ_{B} = n_{2} / n_{1}$ by the definition of tangent in [F8]; no other admissible angle has this tangent, because tangent is strictly increasing on $(0, π / 2)$ by [F8]. Also $r_{s} \neq 0$ at $θ_{B}$ , because $Y_{1}^{s} = Y_{2}^{s}$ would give $k_{13} = k_{t 3}$ (the $μ$ factors being equal), hence $K_{1}^{2} = ∣ κ ∣^{2} + k_{13}^{2} = ∣ κ ∣^{2} + k_{t 3}^{2} = K_{2}^{2}$ and $n_{1} = n_{2}$ , contrary to hypothesis. If instead $n_{1} = n_{2}$ and $μ_{1} = μ_{2}$ then $ε_{1} = ε_{2}$ and the two media are identical: $k_{13} = k_{t 3}$ and $ε_{1} = ε_{2}$ give $Y_{1}^{χ} = Y_{2}^{χ}$ , so $r_{s} = r_{p} = 0$ and $t_{s} = t_{p} = 1$ at every admissible angle (where the (p) formula applies), with no distinguished Brewster angle.

6.1givenstep 3.2F1F3F9∎

[Scope, part (f)] All clauses are conditional on the given wave data and the interface identities [A1]; the argument used the amplitude conditions and solution structure of [F1], the admittance algebra [F3], the angle conventions of [F2] and [F8], the period averages and stored-energy flux of [F4], and the complex arithmetic of [F7]; the ideal-conductor limit, absorption, dispersion, anisotropy, moving and nonlinear media, signal-speed and group-velocity statements and all empirical claims are excluded. The denominator $Y_{1}^{χ} + Y_{2}^{χ}$ is nonzero in every considered branch because $Re Y_{1}^{χ} > 0$ and $Re Y_{2}^{χ} \geq 0$ ; the interface identities are stated hypotheses on the noncompact plane (for compact interfaces, Stationary macroscopic interface jump conditions for D, B, E and H), and no choice principle is used, all constructions being finite-dimensional algebra in the given data.

5 · Examples, experiments and counterexamples

None yet.

Wave interfaces and conductors

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Distinct exponential characters are linearly independent on a nonempty open set, and the real-part grouping rule

Statement

Facts & Assumptions

Proof

The two-equation interface admittance system: solution formulas and the normal-flux identity

Statement

Facts & Assumptions

Proof

Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth

Definition

Well-definedness and conventions

Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch

Statement

Facts & Assumptions

Proof

Phase matching at a plane interface: common tangential wavevector and frequency, the reflection law and Snell's law

Statement

Facts & Assumptions

Proof

Conducting-medium propagation: the exact attenuation constant, the skin depth and its good- and poor-conductor regimes

Statement

Facts & Assumptions

Proof

Normal-incidence amplitudes and the flux budget, with the Ohmic transmitting medium

Statement

Facts & Assumptions

Proof

Oblique-incidence TE and TM Fresnel amplitudes, the flux budget, Brewster's angle and total internal reflection

Statement

Facts & Assumptions

Proof

5 · Examples, experiments and counterexamples

Sources