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Phase and group speeds are not causal fronts

Physical scope

An exact narrowband dispersion calculation for one passive, damped Lorentz medium at a specified frequency. It shows that the local derivative of the real wave-number phase can be negative in this lossy model; it assigns no signal-front speed, energy velocity, empirical observation or universal group-velocity bound.

Statement

Take the finite Lorentz response with $ε_{\infty} = ε_{0}$ , $μ = μ_{0}$ , one oscillator $a = ε_{0} ω_{0}^{2}$ in F m $^{- 1}$ s $^{- 2}$ , and damping $γ = ω_{0} / 2$ in s $^{- 1}$ , where $ω_{0} > 0$ is an angular frequency in s $^{- 1}$ . For angular frequency $ω > 0$ in s $^{- 1}$ , let $ν = ω / ω_{0}$ ; its relative absolute permittivity is

$Z (ν) : = \frac{ε (ω)}{ε_{0}} = 1 + \frac{1}{1 - ν^{2} - i ν / 2} .$

For each $ν > 0$ , define $n (ν)$ to be the dimensionless square root of $Z (ν)$ with positive real and positive imaginary parts, and define the complex wave number $k (ω) : = (ω / c) n (ω / ω_{0})$ in m $^{- 1}$ for a harmonic field with spatial factor $e^{i k z}$ , where $z$ is in m, $c$ is in m s $^{- 1}$ , and $+ z$ is the propagation direction. Use standard right-handed Cartesian coordinates $(\hat{x}, \hat{y}, \hat{z})$ and let $Ω = R^{3}$ , $I = R$ . Fix a real amplitude $E_{0} > 0$ in V m $^{- 1}$ and, with the real-part plane-wave convention, explicitly extended here to complex $k$ , define the fields

$E_{ω} (x, y, z, t) = Re ⁣ (E_{0} \hat{x} e^{i (k (ω) z - ω t)}), B_{ω} (x, y, z, t) = Re ⁣ (\frac{k (ω) E_{0}}{ω} \hat{y} e^{i (k (ω) z - ω t)}) .$

These are real $C^{\infty}$ maps $Ω \times I \to R^{3}$ . Set $H_{ω} = B_{ω} / μ_{0}$ and

$D_{ω} (x, y, z, t) = Re ⁣ (ε (ω) E_{0} \hat{x} e^{i (k (ω) z - ω t)}),$

the harmonic particular displacement response specified by the Lorentz susceptibility theorem. This is a formal source-free transverse bulk mode; no boundary condition or finite-energy property is asserted. The imaginary part of $k$ gives attenuation along $+ z$ . Define the narrowband group-delay derivative per unit length to be $d Re k / d ω$ in s m $^{- 1}$ ; it is a local derivative of phase with respect to angular frequency. For a real temporal signal $f : R \to R$ , define its support by $supp f : = \overline{{t \in R : f (t) \neq 0}}$ ; if this support is nonempty and bounded below, its front time is $\inf supp f$ . The nonzero harmonic histories at each fixed spatial point are eternal and have support $R$ , hence have no front time. At $ν = 1$ , the group-delay derivative is negative. This derivative does not by itself determine a pulse's support front, which is a property of its full time-dependent signal.

Statement references

Analytic susceptibility and dispersion restrictions

Real plane-wave fields encoded by complex amplitudes

Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch

The macroscopic Maxwell equations in terms of D, H and the free sources

Framework-local SI units, the vacuum constants and the exact/measured status of their values

Facts & Assumptions

Given: Countable Choice and Dependent Choice, the one-oscillator Lorentz closure with the parameter choices in the Statement, the SI relation $ε_{0} μ_{0} = c^{- 2}$ , and the source-free homogeneous transverse Maxwell branch. Use complex amplitudes only as the standard encoding of real harmonic fields.

[F3]

Countable Choice and Dependent Choice are carried by the analytic-response supplier used for the passive Lorentz susceptibility (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, Analytic susceptibility and dispersion restrictions).

[A1]

In the material rest frame, the absolute oscillator susceptibility is $χ_{a b s} (ω) = a / (ω_{0}^{2} - ω^{2} - i γ ω)$ and the absolute permittivity is $ε (ω) = ε_{\infty} + χ_{a b s} (ω)$ ; with $γ > 0$ the oscillator response is causal and has the stated analytic transfer under the integrable-kernel hypotheses (Analytic susceptibility and dispersion restrictions).

[A2]

For zero free sources, the smooth macroscopic Maxwell equations are $div D = 0$ , $div B = 0$ , $curl E = - \partial_{t} B$ and $curl H = \partial_{t} D$ , with the constitutive quantities of the material model (The macroscopic Maxwell equations in terms of D, H and the free sources, clause (b); Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch).

[F1]

The SI constants satisfy $ε_{0} μ_{0} = c^{- 2}$ with $ε_{0}, μ_{0}, c > 0$ (Framework-local SI units, the vacuum constants and the exact/measured status of their values, clause (a)).

[F2]

Every positive real number has a unique positive square root; the product rule and total-derivative chain rule hold under the displayed smooth rational functions (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ , 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 for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F4]

The coordinate identity $k \times (k \times e) = ⟨ k, e ⟩ k - ⟨ k, k ⟩ e$ holds for real vectors and extends to complex vectors because both sides are polynomial in their coordinates; all complex dot and cross products here are extended complex-bilinearly (The vector triple product identity $u \times (v \times w) = ⟨ u, w ⟩ v - ⟨ u, v ⟩ w$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F5]

The complex exponential is defined by its everywhere-convergent power series and has derivative itself; the chain rule differentiates its real-linear spatial and temporal phase (The complex exponential by its power series, The complex exponential is entire and its complex derivative is itself, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F6]

The plane-wave definition uses the real-part encoding $E = Re (e e^{i (k \cdot x - ω t)})$ , $B = Re (b e^{i (k \cdot x - ω t)})$ on $R^{3} \times R$ ; this Statement specifies its extension to complex $k$ , defines the resulting real field maps, and uses complex-bilinear dot and cross products (Real plane-wave fields encoded by complex amplitudes).

Verification

technique · derive the homogeneous transverse wave symbol, then differentiate the explicitly selected passive square-root branch at its carrier frequency

1.1givenA1A2F1F3F4F5F6algebra

At a real frequency $ω > 0$ , set $k = k \hat{z}$ and substitute the temporal factor $e^{- i ω t}$ and spatial factor $e^{i k z}$ of the Statement into [A2]; [F5] differentiates these phases, and [F6] fixes the real-field encoding, extended to this complex wavevector. The two curl equations give $k \times e = ω b$ and $k \times h = - ω ε (ω) e$ , with $b = μ_{0} h$ . For a transverse amplitude $k \cdot e = 0$ , [F4] gives $(k \cdot k) e = μ_{0} ε (ω) ω^{2} e$ . Since $e \neq 0$ , $k^{2} = μ_{0} ε (ω) ω^{2}$ ; [F1] and the definition of $Z$ imply $k = (ω / c) n$ on the selected square-root branch. Conversely, for this root take $e = E_{0} \hat{x}$ , $b = (k E_{0} / ω) \hat{y}$ and $h = b / μ_{0}$ . Then $k \times e = ω b$ and $k \times h = - k^{2} e / (μ_{0} ω) = - ω ε (ω) e$ . Also $k \cdot e = k \cdot b = 0$ , so the divergence amplitudes of $D$ and $B$ vanish. Thus the fields in the Statement realize the source-free transverse harmonic mode. As $Im k > 0$ , $e^{i k z} = e^{- Im (k) z} e^{i Re (k) z}$ attenuates along $+ z$ .

1.2givenA1F2algebra

At $ν = 1$ , $1 - ν^{2} - i ν / 2 = - i / 2$ , hence $Z (1) = 1 + 2 i$ . Write the selected root as $n (1) = p + i q$ with $p, q > 0$ . Equating real and imaginary parts gives $p^{2} - q^{2} = 1$ and $2 p q = 2$ , hence $(p^{2} + q^{2})^{2} = (p^{2} - q^{2})^{2} + (2 p q)^{2} = 5$ and $p^{2} + q^{2} = \sqrt{5}$ . For general $ν > 0$ , write $Z (ν) = x (ν) + i y (ν)$ ; rationalization gives $y (ν) = (ν / 2) / ((1 - ν^{2})^{2} + ν^{2} / 4) > 0$ . Put $r (ν) = \sqrt{x (ν)^{2} + y (ν)^{2}}$ . Then $r (ν)^{2} - x (ν)^{2} = y (ν)^{2} > 0$ , so $r (ν) > ∣ x (ν) ∣$ and both $(r + x) / 2$ and $(r - x) / 2$ are positive. Define $p (ν) = \sqrt{(r + x) / 2}$ and $q (ν) = \sqrt{(r - x) / 2}$ . These satisfy $p^{2} - q^{2} = x$ and $(2 p q)^{2} = r^{2} - x^{2} = y^{2}$ ; since $p, q, y > 0$ , $2 p q = y$ , so $(p + i q)^{2} = Z$ . For any other root $p_{0} + i q_{0}$ with $p_{0}, q_{0} > 0$ , the same equations imply $(p_{0}^{2} + q_{0}^{2})^{2} = x^{2} + y^{2} = r^{2}$ , hence $p_{0}^{2} + q_{0}^{2} = r$ and $p_{0}^{2} = (r + x) / 2$ , $q_{0}^{2} = (r - x) / 2$ ; positivity gives $p_{0} = p$ , $q_{0} = q$ . The rational functions $x, y$ are continuous for $ν > 0$ ; the positive square root is continuous at each positive $u_{0}$ , since $∣ \sqrt{u} - \sqrt{u_{0}} ∣ = ∣ u - u_{0} ∣ / (\sqrt{u} + \sqrt{u_{0}}) \leq ∣ u - u_{0} ∣ / \sqrt{u_{0}}$ for $u > 0$ . Applying this to $r$ and to the positive arguments $(r \pm x) / 2$ proves that $p, q$ , hence $n$ , are continuous near $1$ . Therefore $n (ν) + n (1) \to 2 n (1) \neq 0$ , and the difference quotient identity $(n (ν) - n (1)) / (ν - 1) = (Z (ν) - Z (1)) / ((ν - 1) (n (ν) + n (1)))$ gives $n^{'} (1) = Z^{'} (1) / (2 n (1))$ . Set $D (ν) = 1 - ν^{2} - i ν / 2$ . Viewing $C$ as $R^{2}$ , the real product rule applied to $D D^{- 1} = 1$ gives $Z^{'} (1) = - D^{'} (1) / D (1)^{2} = (2 + i / 2) / (- i / 2)^{2} = - 8 - 2 i$ . Thus $n^{'} = (- 4 - i) / n$ at $1$ , and $1 / n = (p - i q) / \sqrt{5}$ gives $Re (n + n^{'}) = ((\sqrt{5} - 4) p - q) / \sqrt{5} < 0$ , since $p, q > 0$ and $\sqrt{5} < 4$ .

2.1step 1.2F2algebra∎

From $k = (ω / c) n (ν)$ and $ν = ω / ω_{0}$ , the chain and product rules give $\frac{d}{d ω} Re k = \frac{1}{c} Re (n + ν n^{'})$ . At $ν = 1$ , this is negative by step 1.2 and $c > 0$ . Thus the conventional narrowband group-delay derivative is negative for this lossy causal model at that frequency. Its value does not give the support-front time of a pulse, which depends on the full time-domain signal; no front-speed or universal group-speed claim follows.

Phase and group speeds are not causal fronts

Statement

Statement references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources