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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.

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

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources