ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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A closed lossless one-port has unit reflection modulus

Physical scope

An algebraic consequence for a closed, lossless one-channel scattering model with no unlisted output. It gives the reflected-amplitude modulus and shows that power conservation alone leaves its phase undetermined; it does not assert an experimental result or the realization of an arbitrary frequency-dependent phase law.

Example

Assume ${A C}_{ω}$ . Let $I \subset (0, \infty)$ be a set of admissible angular frequencies at which a closed lossless one-channel finite port model is defined, with no unlisted outgoing power. At each $ω \in I$ , assume the solution satisfies all hypotheses of Passivity gives a contractive scattering matrix: the source-free macroscopic Maxwell equations with $ρ_{f} = J_{f} = 0$ , a smooth $T$ -periodic real solution and material states on one of the stated homogeneous finite Lorentz or Drude branches, and the stated passive-wall and outgoing-power bookkeeping. Here $T = 2 π / ω$ ; material, wall, and unlisted-output losses vanish for every input. Write the scalar scattering matrix at each frequency as $S (ω) = [r (ω)]$ . Then $∣ r (ω) ∣ = 1$ for each such frequency. This fixes reflected power but leaves the phase unconstrained by power conservation: any unit-modulus scalar has the same power factor. For any real-valued function $ϕ : I \to R$ , $r_{ϕ} (ω) = e^{i ϕ (ω)}$ has unit modulus; when this circle-valued function is nonconstant, its phase varies. If $I$ contains two distinct frequencies, such a varying family exists: set $ϕ$ equal to $0$ and $π$ at those two frequencies, respectively, and set $ϕ = 0$ elsewhere. This is an algebraic freedom allowed by the power constraint, not a claim that every such phase law is realized by a Maxwell device.

Statement references

The Axiom of Countable Choice ( ${A C}_{ω}$ )

Finite propagating-port scattering map

Passivity gives a contractive scattering matrix

Facts & Assumptions

Given: ${A C}_{ω}$ ; one incoming and one outgoing channel; at each frequency the finite port solution satisfies the smooth periodic Maxwell and homogeneous Lorentz/Drude hypotheses of the contraction theorem with $ρ_{f} = J_{f} = 0$ ; and material, wall, and unlisted-channel losses vanish.

[A1]

${A C}_{ω}$ is carried by the finite scattering theorem and its bounded-domain mathematical supplier. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

For complex matrices, an overline is componentwise conjugation, a superscript $T$ is ordinary transpose, $A^{†} : = {\overline{A}}^{T}$ is the conjugate transpose, and $I_{1} = [1]$ is the identity. In this power-normalized one-channel model, $S = [r]$ and the input and output powers are $∣ a ∣^{2}$ and $∣ r a ∣^{2}$ . Under the stated smooth periodic Lorentz/Drude hypotheses, the contraction theorem gives $S^{†} S = I_{1}$ when material, wall, and unlisted-output losses vanish for every input. Finite propagating-port scattering map Passivity gives a contractive scattering matrix

Verification

1.1givenA1F1algebra

Choose any nonzero incoming amplitude $a \in C$ . By [F1], $∣ r a ∣^{2} = ∣ a ∣^{2}$ , so $∣ r ∣^{2} ∣ a ∣^{2} = ∣ a ∣^{2}$ . Divide by $∣ a ∣^{2} > 0$ to obtain $∣ r ∣^{2} = 1$ , hence $∣ r ∣ = 1$ .

2.1step 1.1algebra∎

For every $ω \in I$ , $∣ e^{i ϕ (ω)} ∣^{2} = e^{i ϕ (ω)} e^{- i ϕ (ω)} = 1$ . If $I$ contains distinct $ω_{1}, ω_{2}$ , define $ϕ (ω_{1}) = 0$ , $ϕ (ω_{2}) = π$ , and $ϕ = 0$ elsewhere; then $r_{ϕ}$ takes the distinct values $1$ and $- 1$ , while both obey step 1.1. Thus the modulus condition leaves phase variation algebraically possible on a multi-frequency set.

A closed lossless one-port has unit reflection modulus

Example

Statement references

Facts & Assumptions

Verification

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