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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.

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

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources