Power-normalized scattering and passivity

Fix a real angular frequency $\omega >0$ in ${\mathrm{s}}^{-1}$ and use the $e^{-i\omega t}$ convention and SI field units of the fixed-frequency reciprocal-phasor model. Let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be a bounded device region with $C^1$ boundary in one stationary material-rest frame. Its continuous phasor coefficient maps are $\epsilon ,\mu :\bar{\mathrm{\Omega }}\to {\mathbb{C}}^{3\times 3}$ in $\mathrm{F}{\mathrm{m}}^{-1}$ and $\mathrm{H}{\mathrm{m}}^{-1}$, respectively. The complex phasor fields $\tilde{E},\tilde{H}\in C^1(\bar{\mathrm{\Omega }};{\mathbb{C}}^3)$ have units $\mathrm{V}{\mathrm{m}}^{-1}$ and $\mathrm{A}{\mathrm{m}}^{-1}$. They satisfy the fixed-frequency Maxwell equations of the reciprocal-phasor data class, with no impressed volume current or charge in $\mathrm{\Omega }$. The corresponding real fields are $E(x,t)=\mathrm{Re}(\tilde{E}(x)e^{-i\omega t})$ and $H(x,t)=\mathrm{Re}(\tilde{H}(x)e^{-i\omega t})$, with period $T=2\pi \mathrm{/}\omega$. Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ as defined by the countable-choice axiom when the reciprocal boundary identity below is used.

An admissible port consists of a bounded connected $C^1$ cross-section ${\mathrm{\Sigma }}_j$ in a plane perpendicular to a fixed unit guide axis ${\widehat{z}}_j$, together with a uniform lossless straight reference guide extending from that section. The outward normal $n_j$ on the port points from $\mathrm{\Omega }$ into the guide. The reference-guide phasor coefficients and wall conditions are independent of distance along ${\widehat{z}}_j$. A propagating channel mode is a pair of $C^1$ complex transverse profiles $(e_{j\alpha },h_{j\alpha })$ whose associated source-free phasor fields in the guide solve the fixed-frequency Maxwell equations and have longitudinal factor $e^{\pm i{\beta }_{\alpha }{s}_j}$ with real ${\beta }_{\alpha }>0$ in ${\mathrm{m}}^{-1}$. The profiles are normalized per unit amplitude in ${\mathrm{W}}^{1\mathrm{/}2}$, so the resulting fields have units $\mathrm{V}{\mathrm{m}}^{-1}$ and $\mathrm{A}{\mathrm{m}}^{-1}$. The plus and minus signs are assigned according to whether energy travels toward or away from $\mathrm{\Omega }$. At $\omega$, assume a finite list of $N$ propagating mode channels in total and that this list is complete for the propagating channels represented by the finite interface. Any outgoing radiation continuum or omitted propagating channel is separately recorded as nonnegative outward power $P_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}}$. Evanescent modes may be required in the exact near-port matching field, but are not entries of the propagating amplitude vectors; no finite truncation of such matching modes is asserted to be exact.

Let $a=(a_1,\dots ,a_N{)}^T\in {\mathbb{C}}^N$ and $b=(b_1,\dots ,b_N{)}^T\in {\mathbb{C}}^N$ be the incoming and outgoing amplitudes. Choose the mode shapes and amplitude convention so that each coefficient has units ${\mathrm{W}}^{1\mathrm{/}2}$, the Hermitian norm is $\mathrm{\parallel }a{\mathrm{\parallel }}_2^2={\sum }_{\alpha =1}^N\mathrm{\mid }{a}_{\alpha }{\mathrm{\mid }}^2$ in W, and the total time-averaged outward power through the port sections is

$P_{\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{s}}=\mathrm{\parallel }b{\mathrm{\parallel }}_2^2-\mathrm{\parallel }a{\mathrm{\parallel }}_2^2\mathrm{.}$

This equality, including the vanishing of cross terms in the real-power pairing, is a port-mode normalization hypothesis. For complex phasor fields the time-averaged Poynting vector is $\mathbf{\Pi }=\frac{1}{2}\mathrm{Re}(\tilde{E}\times {\tilde{H}}^{\ast })$ in $\mathrm{W}{\mathrm{m}}^{-2}$, where $\ast$ denotes componentwise conjugation before the bilinear cross product. This Hermitian power pairing is distinct from the unconjugated Lorentz bilinear pairing.

On the non-port wall ${\mathrm{\Gamma }}_w$, the model prescribes either a smooth PEC condition $E_{\tau }(t)=0$, giving zero outward wall power, or a passive wall condition with explicitly specified nonnegative cycle-averaged outward power $P_{\mathrm{\partial }}\ge 0$. Here $E_{\tau }(t)=E(t)-(n\cdot E(t))n$ and $H_{\tau }(t)=H(t)-(n\cdot H(t))n$ are the classical tangential restrictions of the smooth real fields. A smooth impedance wall is one such condition: for a real scalar surface admittance $Y\ge 0$ in siemens and $H_{\tau }(t)=Y(n\times E_{\tau }(t))$, its instantaneous outward power density is $n\cdot (E_{\tau }(t)\times H_{\tau }(t))=Y\mathrm{\mid }{E}_{\tau }(t){\mathrm{\mid }}^2\ge 0$, so its cycle-averaged outward power is nonnegative. For phasor amplitude ${\tilde{E}}_{\tau }$, that average is $\frac{1}{2}Y\mathrm{\mid }{\tilde{E}}_{\tau }{\mathrm{\mid }}^2$. This is a smooth-field model assumption; no $L^2$ trace is inferred from membership in $H(\mathrm{curl})$.

The total cycle-averaged outward boundary power is the port contribution $\mathrm{\parallel }b{\mathrm{\parallel }}_2^2-\mathrm{\parallel }a{\mathrm{\parallel }}_2^2$, plus $P_{\mathrm{\partial }}$ and any unlisted outgoing power $P_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}}$. No unlisted incoming power is included.

For each $a$, assume the adopted linear source-free Maxwell boundary problem has a unique time-harmonic solution and a unique outgoing vector $b$. The model assumes no incoming wave in unlisted channels. By linearity of the phasor equations and uniqueness, the map $a\mapsto b$ is complex-linear: a linear combination of two solutions solves the problem for the same linear combination of their incoming data, and uniqueness identifies its outgoing data. The finite scattering matrix is the unique matrix $S(\omega )\in {\mathbb{C}}^{N\times N}$ such that $b=S(\omega )a$. The definition assumes this well-posed solution map; it does not prove its existence, uniqueness, modal completeness, or absence of resonances.

For reciprocal use, the incoming and outgoing modal phases are additionally chosen so that for every pair of source-free solutions with incoming vectors $a_1,a_2$ and outgoing vectors $b_1,b_2$, the Lorentz boundary pairing reduces to

${\int }_{\mathrm{\partial }\mathrm{\Omega }}n\cdot ({\tilde{E}}_1\times {\tilde{H}}_2-{\tilde{E}}_2\times {\tilde{H}}_1)dS=a_1^T{b}_2-b_1^T{a}_2\mathrm{.}$

This is an explicit reciprocal-normalization condition, using the complex-bilinear products of the reciprocal-phasor model and the bounded-$C^1$ reciprocity identity of the Lorentz reciprocity theorem. Power normalization alone does not imply this bilinear pairing for arbitrary complex reference impedances. Rephasing a port is allowed only when the declared reciprocal pairing is preserved.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Fixed-frequency reciprocal tensor medium

Reciprocity in reciprocal linear media

Energy balance and zero-data forward uniqueness for admissible weak Maxwell fields

The arrows show the sign convention for two ports: incoming amplitudes point toward the device and outgoing amplitudes point away. The drawing records orientation only; the flux identities in the definition, not the drawing, determine the proofs.

The rectangular-guide suppliers on the prerequisite page provide mode-profile results only under their stated rectangular, lossless hypotheses; their completeness theorem expressly does not prove an open-end scattering map. The finite channel basis and the unique scattering solution above therefore remain explicit interface assumptions. For an open radiating device, radiation power must be included in $P_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}}$; finite-channel unitarity is not concluded while that term is omitted.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Fix the finite port model at a real angular frequency $\omega >0$ and period $T=2\pi \mathrm{/}\omega$, with bounded $C^1$ device region $\mathrm{\Omega }$ and scattering matrix $S(\omega )$. For every incoming amplitude $a\in {\mathbb{C}}^N$, let ${\tilde{E}}_a,{\tilde{H}}_a\in C^1(\bar{\mathrm{\Omega }};{\mathbb{C}}^3)$ be that model's unique phasor solution with outgoing amplitude $b=S(\omega )a$, and set $E(x,t)=\mathrm{Re}({\tilde{E}}_a(x)e^{-i\omega t})$ and $H(x,t)=\mathrm{Re}({\tilde{H}}_a(x)e^{-i\omega t})$. Assume for every such input that these real fields, $D,B:\bar{\mathrm{\Omega }}\times \mathbb{R}\to {\mathbb{R}}^3$, and the material states are jointly $C^{\mathrm{\infty }}$ and $T$-periodic, satisfy the smooth source-free macroscopic Maxwell equations with ${\rho }_f=0$ and $J_f=0$, and obey the same one of the following two homogeneous material branches with coefficients independent of the input. The units of $E,H,D,B$ are respectively $\mathrm{V}{\mathrm{m}}^{-1}$, $\mathrm{A}{\mathrm{m}}^{-1}$, $\mathrm{C}{\mathrm{m}}^{-2}$, and $\mathrm{T}$.

For real vectors $v,w\in {\mathbb{R}}^3$, the Euclidean product is $v\cdot w:={\sum }_{r=1}^3{v}_r{w}_r$ and $\mathrm{\mid }v{\mathrm{\mid }}^2:=v\cdot v$.

The port model uses incoming and outgoing vectors $a,b\in {\mathbb{C}}^N$ with amplitude units ${\mathrm{W}}^{1\mathrm{/}2}$; for either vector $z$, $\mathrm{\parallel }z{\mathrm{\parallel }}_2^2:={\sum }_{\alpha =1}^N\mathrm{\mid }{z}_{\alpha }{\mathrm{\mid }}^2$ is its power in W, and the net outward port power is $\mathrm{\parallel }b{\mathrm{\parallel }}_2^2-\mathrm{\parallel }a{\mathrm{\parallel }}_2^2$.

Finite Lorentz branch. For a finite $M\ge 1$, each $P_{\mathrm{\ell }}:\bar{\mathrm{\Omega }}\times \mathbb{R}\to {\mathbb{R}}^3$ has units $\mathrm{C}{\mathrm{m}}^{-2}$, $D={\epsilon }_{\mathrm{\infty }}E+{\sum }_{\mathrm{\ell }=1}^M{P}_{\mathrm{\ell }}$, $B=\mu H$, and ${\mathrm{\partial }}_t^2{P}_{\mathrm{\ell }}+{\gamma }_{\mathrm{\ell }}{\mathrm{\partial }}_t{P}_{\mathrm{\ell }}+{\omega }_{\mathrm{\ell }}^2{P}_{\mathrm{\ell }}=a_{\mathrm{\ell }}E$, where ${\epsilon }_{\mathrm{\infty }},\mu ,a_{\mathrm{\ell }},{\omega }_{\mathrm{\ell }}>0$ and ${\gamma }_{\mathrm{\ell }}\ge 0$.

Drude branch. $P,V:\bar{\mathrm{\Omega }}\times \mathbb{R}\to {\mathbb{R}}^3$ have units $\mathrm{C}{\mathrm{m}}^{-2}$ and $\mathrm{A}{\mathrm{m}}^{-2}$, respectively; $D={\epsilon }_{\mathrm{\infty }}E+P$, $B=\mu H$, ${\mathrm{\partial }}_{t}P=V$, and ${\mathrm{\partial }}_{t}V+\gamma V=aE$, where ${\epsilon }_{\mathrm{\infty }},\mu ,a>0$ and $\gamma \ge 0$.

The constants have the SI units of the respective adopted closure: ${\epsilon }_{\mathrm{\infty }}$ in $\mathrm{F}{\mathrm{m}}^{-1}$, $\mu$ in $\mathrm{H}{\mathrm{m}}^{-1}$, each $a_{\mathrm{\ell }}$ or $a$ in $\mathrm{F}{\mathrm{m}}^{-1}{\mathrm{s}}^{-2}$, and each ${\omega }_{\mathrm{\ell }}$ or damping rate in ${\mathrm{s}}^{-1}$. In the Lorentz branch define the loss-density map $\mathrm{\ell }:\bar{\mathrm{\Omega }}\times \mathbb{R}\to [0,\mathrm{\infty })$ by $\mathrm{\ell }={\sum }_{\mathrm{\ell }=1}^M({\gamma }_{\mathrm{\ell }}\mathrm{/}{a}_{\mathrm{\ell }})\mathrm{\mid }{\mathrm{\partial }}_t{P}_{\mathrm{\ell }}{\mathrm{\mid }}^2$; in the Drude branch define it by $\mathrm{\ell }=(\gamma \mathrm{/}a)\mathrm{\mid }V{\mathrm{\mid }}^2$. These have units $\mathrm{W}{\mathrm{m}}^{-3}$. For the solution with the fixed input $a$, set $P_{\mathrm{m}\mathrm{a}\mathrm{t}}(a):=T^{-1}{\int }_0^T{\int }_{\mathrm{\Omega }}\mathrm{\ell }dVdt$. Let $P_{\mathrm{\partial }}(a)\ge 0$ be its cycle-averaged outward power at passive physical walls, and let $P_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}}(a)\ge 0$ be all its outgoing radiation or propagating-channel power not represented in $b$, as defined by the port model. For brevity, in the proof write these three quantities as $P_{\mathrm{m}\mathrm{a}\mathrm{t}}$, $P_{\mathrm{\partial }}$, and $P_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}}$. Then

$\mathrm{\parallel }a{\mathrm{\parallel }}_2^2-\mathrm{\parallel }S(\omega )a{\mathrm{\parallel }}_2^2=P_{\mathrm{m}\mathrm{a}\mathrm{t}}(a)+P_{\mathrm{\partial }}(a)+P_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}}(a)\ge 0.$

For complex vectors and matrices, an overline denotes componentwise complex conjugation and a superscript ${}^{\mathsf{T}}$ denotes ordinary transpose; set $z^{†}:={\bar{z}}^{\mathsf{T}}$ for $z\in {\mathbb{C}}^N$ and $A^{†}:={\bar{A}}^{\mathsf{T}}$ for $A\in {\mathbb{C}}^{N\times N}$. A matrix is Hermitian when $A^{†}=A$, and is unitary when $A^{†}A=A{A}^{†}=I_N$, where $I_N$ is the identity matrix on ${\mathbb{C}}^N$.

Consequently $S(\omega )$ is contractive: $\mathrm{\parallel }Sa{\mathrm{\parallel }}_2\le \mathrm{\parallel }a{\mathrm{\parallel }}_2$ for every $a$. If $P_{\mathrm{m}\mathrm{a}\mathrm{t}}(a)=P_{\mathrm{\partial }}(a)=P_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}}(a)=0$ for every incoming vector $a$, then $S^{†}S=I_N$; because the defined map is square, $S$ is unitary. The exact equality and unitarity are conditional deductions of this model, not existence claims about arbitrary scatterers.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Finite propagating-port scattering map

The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$

The divergence and curl of a cross product

Finite Lorentz oscillator closure

Drude current closure and collisionless limit

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

Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$

Divergence on a bounded C1 Euclidean domain

Fubini's theorem for L^1 functions on a sigma-finite product

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ and the finite port model at a fixed real $\omega >0$. Let $S(\omega )\in {\mathbb{C}}^{N\times N}$ be its unique scattering matrix. Suppose the continuous device phasor coefficient maps $\epsilon ,\mu :\bar{\mathrm{\Omega }}\to {\mathbb{C}}^{3\times 3}$ obey $\epsilon (x{)}^T=\epsilon (x)$ and $\mu (x{)}^T=\mu (x)$ pointwise, and for every pair of incoming vectors the corresponding source-free scattering solutions lie in $C^1(\bar{\mathrm{\Omega }};{\mathbb{C}}^3)$ and have reciprocal port modes satisfying the exact bilinear pairing stipulated in the model definition. Then

$S(\omega {)}^T=S(\omega )\mathrm{.}$

For $A\in {\mathbb{C}}^{N\times N}$, define its conjugate transpose by $A^{†}:={\bar{A}}^{\mathsf{T}}$, where the overline is componentwise conjugation and a superscript $\mathsf{T}$ is ordinary transpose; $A$ is Hermitian when $A^{†}=A$ and unitary when $A^{†}A=A{A}^{†}=I_N$, with $I_N$ the identity on ${\mathbb{C}}^N$. The transpose relation above uses the unconjugated Lorentz pairing; the unitary relation below uses the separate conjugated Hermitian power pairing. No time-reversal claim follows.

For each incoming vector $a$, define the operational net cycle-averaged material power transferred from the fields in this source-free phasor model by

$P_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(a):=\frac{\omega }{2}{\int }_{\mathrm{\Omega }}\mathrm{Im}({\tilde{E}}^{†}\epsilon \tilde{E}+{\tilde{H}}^{†}\mu \tilde{H})dV\mathrm{.}$

Here $v^{†}={\bar{v}}^T$ for a complex column vector, and this power has units W. It is a signed net transfer, not an assumption of local passivity or a definition of a material realization. Let $P_{\mathrm{\partial }}(a)$ and $P_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}}(a)$ be the nonnegative outward wall and unlisted-channel powers of the same finite-port model, whose total outward boundary power is $\mathrm{\parallel }Sa{\mathrm{\parallel }}_2^2-\mathrm{\parallel }a{\mathrm{\parallel }}_2^2+P_{\mathrm{\partial }}(a)+P_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}}(a)$. Then

$\mathrm{\parallel }a{\mathrm{\parallel }}_2^2-\mathrm{\parallel }Sa{\mathrm{\parallel }}_2^2=P_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(a)+P_{\mathrm{\partial }}(a)+P_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}}(a)\mathrm{.}$

Consequently, if all three powers vanish for every incoming vector $a\in {\mathbb{C}}^N$, then $S^{†}S=S{S}^{†}=I_N$, so $S$ is symmetric and unitary. Pointwise Hermitian $\epsilon$ and $\mu$ suffice to make $P_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(a)=0$ for every input, since their quadratic forms are real; Hermitian symmetry is distinct from the ordinary transpose symmetry required for reciprocity. No constitutive branch beyond the stated phasor model is required for this operational zero-power conclusion.

The following full finite-port contraction hypotheses give additional Lorentz and Drude sufficient cases on this same port map. For every incoming vector, its real field maps $E,H,D,B:\bar{\mathrm{\Omega }}\times \mathbb{R}\to {\mathbb{R}}^3$ and its material-state maps on that domain are jointly $C^{\mathrm{\infty }}$ and $T$-periodic, where $T=2\pi \mathrm{/}\omega$, and satisfy the smooth source-free macroscopic Maxwell equations with ${\rho }_f=J_f=0$.

The material closure is one of these two homogeneous, constant-coefficient branches. In the finite Lorentz branch, choose finite $M\ge 1$ and maps $P_{\mathrm{\ell }}:\bar{\mathrm{\Omega }}\times \mathbb{R}\to {\mathbb{R}}^3$ for $1\le \mathrm{\ell }\le M$. They satisfy

$D={\epsilon }_{\mathrm{\infty }}E+{\sum }_{\mathrm{\ell }=1}^M{P}_{\mathrm{\ell }},B=\mu H,{\mathrm{\partial }}_t^2{P}_{\mathrm{\ell }}+{\gamma }_{\mathrm{\ell }}{\mathrm{\partial }}_t{P}_{\mathrm{\ell }}+{\omega }_{\mathrm{\ell }}^2{P}_{\mathrm{\ell }}=a_{\mathrm{\ell }}E,$

with constant ${\epsilon }_{\mathrm{\infty }},\mu ,a_{\mathrm{\ell }},{\omega }_{\mathrm{\ell }}>0$ and ${\gamma }_{\mathrm{\ell }}\ge 0$. In the Drude branch, take maps $P,V:\bar{\mathrm{\Omega }}\times \mathbb{R}\to {\mathbb{R}}^3$. They satisfy

$D={\epsilon }_{\mathrm{\infty }}E+P,B=\mu H,{\mathrm{\partial }}_{t}P=V,{\mathrm{\partial }}_{t}V+\gamma V=aE,$

with constant ${\epsilon }_{\mathrm{\infty }},\mu ,a>0$ and $\gamma \ge 0$. Their SI units are those specified in the finite-port contraction model.

For each input, require the real fields and material states to have only this frequency: let $\tilde{E},\tilde{H},\tilde{D},\tilde{B}:\bar{\mathrm{\Omega }}\to {\mathbb{C}}^3$ be their phasor amplitudes with $E(x,t)=\mathrm{Re}(\tilde{E}(x)e^{-i\omega t})$, and analogously for $H,D,B$. In the Lorentz branch also write $P_{\mathrm{\ell }}(x,t)=\mathrm{Re}({\tilde{P}}_{\mathrm{\ell }}(x)e^{-i\omega t})$ with ${\tilde{P}}_{\mathrm{\ell }}:\bar{\mathrm{\Omega }}\to {\mathbb{C}}^3$; in the Drude branch write $P,V$ in the same form with amplitudes $\tilde{P},\tilde{V}:\bar{\mathrm{\Omega }}\to {\mathbb{C}}^3$. Substitution into the closure equations gives

${\epsilon }_{\mathrm{L}}(\omega )=({\epsilon }_{\mathrm{\infty }}+{\sum }_{\mathrm{\ell }=1}^M\frac{a_{\mathrm{\ell }}}{{\omega }_{\mathrm{\ell }}^2-{\omega }^2-i{\gamma }_{\mathrm{\ell }}\omega })I_3,{\epsilon }_{\mathrm{D}}(\omega )=({\epsilon }_{\mathrm{\infty }}-\frac{a}{{\omega }^2+i\gamma \omega })I_3,$

where every Lorentz denominator is required nonzero, $I_3$ is the $3\times 3$ identity, and the magnetic coefficient is $\mu I_3$ in either branch. Require these harmonic coefficient maps to be the continuous coefficients of this same reciprocal-phasor port model. The matrices are symmetric under ordinary transpose. The branch loss density is $\mathrm{\ell }={\sum }_{j=1}^M({\gamma }_j\mathrm{/}{a}_j)\mathrm{\mid }{\mathrm{\partial }}_t{P}_j{\mathrm{\mid }}^2$ in the Lorentz branch and $\mathrm{\ell }=(\gamma \mathrm{/}a)\mathrm{\mid }V{\mathrm{\mid }}^2$ in the Drude branch, using the real vector norm $\mathrm{\mid }v{\mathrm{\mid }}^2=v\cdot v$. For the solution with incoming vector $a$, define $P_{\mathrm{m}\mathrm{a}\mathrm{t}}(a):=T^{-1}{\int }_0^T{\int }_{\mathrm{\Omega }}\mathrm{\ell }dVdt$, and let $P_{\mathrm{\partial }}(a)$ and $P_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}}(a)$ be its nonnegative passive-wall and unlisted-outgoing powers under the finite-port model's bookkeeping. The field, material-state, and parameter units are exactly those specified in that model.

In either harmonic branch, $P_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}(a)=P_{\mathrm{m}\mathrm{a}\mathrm{t}}(a)$. Thus $P_{\mathrm{m}\mathrm{a}\mathrm{t}}(a)=P_{\mathrm{\partial }}(a)=P_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}}(a)=0$ for every incoming vector $a$ is a sufficient case of the operational zero-power criterion, also supplied by the contraction result. These branch hypotheses are sufficient cases, not restrictions on the general phasor power identity above.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Finite propagating-port scattering map

Reciprocity in reciprocal linear media

Passivity gives a contractive scattering matrix

The divergence and curl of a cross product

Divergence on a bounded C1 Euclidean domain