Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth

Assume the elementary inertial-rest branch of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch in the laboratory region $U\cong I\times \mathrm{\Omega }$ 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, in its isotropic scalar case: the material is at rest in the laboratory frame, $D,H$ are defined by Polarization, magnetization, bound and free sources, and the fields D and H(d), and the constitutive relations of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch(b) hold with constant scalars $\epsilon >0$ (F$$m${}^{-1}$), $\mu >0$ (H$$m${}^{-1}$) and $\sigma \ge 0$ (S$$m${}^{-1}$), so that $D=\epsilon E$, $B=\mu H$ and the free current is $J_f=\sigma E+J_{\mathrm{e}\mathrm{x}\mathrm{t}}$ for a declared external free current $J_{\mathrm{e}\mathrm{x}\mathrm{t}}$. Write $c>0$ for the exact vacuum speed of Framework-local SI units, the vacuum constants and the exact/measured status of their values and let $\omega >0$ (s${}^{-1}$) be an angular frequency. Define:

(a) [Lossless parameters] with $\sqrt{\cdot }$ the nonnegative square root of Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$, $v:=(\epsilon \mu {)}^{-1\mathrm{/}2}(\mathrm{m}{\mathrm{s}}^{-1}),Z:=(\frac{\mu }{\epsilon }{)}^{1\mathrm{/}2}(\mathrm{\Omega }),n:=\frac{c}{v}=c(\epsilon \mu {)}^{1\mathrm{/}2}(\text{dimensionless}),k(\omega ):=\frac{\omega }{v};$ $v$ is the phase speed, $Z$ the wave impedance, $n$ the refractive index relative to vacuum and $k(\omega )$ the real wavenumber of the lossless case.

(b) [Complex propagation constant] put $a:=\mu \epsilon {\omega }^2>0$, $b:=\mu \sigma \omega \ge 0$, $r:=\sqrt{a^2+b^2}$ and $p(\omega ):=\sqrt{\frac{r+a}{2}},q(\omega ):=\sqrt{\frac{r-a}{2}},\kappa (\omega ):=p(\omega )+iq(\omega )\mathrm{.}$ Then $\kappa (\omega )$ is the unique $k\in \mathbb{C}$ with $\mathrm{Re}k\ge 0$, $\mathrm{Im}k\ge 0$ and $k^2=\epsilon \mu {\omega }^2+i\mu \sigma \omega =a+ib$; in the lossless case $\sigma =0$ one has $b=0$, $q=0$ and $\kappa (\omega )=k(\omega )=p(\omega )$ real. (The explicit formula is the one of Every complex number has a square root, by an explicit Cartesian formula; the verification $p^2-q^2=a$, $2pq=b$ and the uniqueness of the root with $\mathrm{Re}k\ge 0$, $\mathrm{Im}k\ge 0$ are verified in the well-definedness paragraph below.)

(c) [Skin depth] call $\delta (\omega ):=1\mathrm{/}q(\omega )$ (m) the skin depth if $\sigma >0$, and set $\delta (\omega ):=+\mathrm{\infty }$ if $\sigma =0$: for $\kappa (\omega )=p+iq$ and $x_3\in \mathbb{R}$ the modulus of $e^{i(\kappa x_3-\omega t)}$ equals $e^{-q{x}_3}$ ($\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$), so $\delta$ is the distance in which that modulus decays by the factor $1\mathrm{/}e$ in the direction of increasing $x_3$; for $\sigma =0$ there is no attenuation and $\delta$ is infinite rather than a division by zero.

(d) [Status and scope] This item fixes objects, units, branch conventions and terminology only. It asserts no field equation, no dispersion relation for a solution, no attenuation of any concrete field and no material validity. The parameters are declared for the isotropic scalar constant-coefficient branch at rest in the laboratory frame; for $\epsilon \mu <c^{-2}$ the branch is, by Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch(b), a declared finite-band effective phase-response model with no universal front-speed claim. An exact broadband interpretation must be separately adopted and requires $\epsilon \mu \ge c^{-2}$; that inequality alone does not adopt a broadband interpretation or establish material validity, and a finite-band interpretation remains possible when it holds. No measured value, precision or empirical statement is included.

Well-definedness and conventions

The material assumptions, the rest-frame clause, the isotropic scalar case and the finite-band caveat are those of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch(a)-(b) read with the polarization and magnetization definitions; the unit system and the exact constant $c$ are those of Framework-local SI units, the vacuum constants and the exact/measured status of their values; the laboratory chart is that of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields.

The only content beyond notation is the well-definedness of $\kappa (\omega )$ in (b). With $a>0$, $b\ge 0$ and $r=\sqrt{a^2+b^2}$ one has $r+a>0$ and $(r-a)(r+a)=b^2\ge 0$, so $r-a\ge 0$ and both defining radicands are nonnegative. If $b>0$, the same identity gives $r-a>0$, hence $q>0$; if $b=0$, uniqueness of nonnegative roots gives $r=a$ and $q=0$. Also $p>0$ because $r+a>0$. Thus $p^2-q^2=(r+a)\mathrm{/}2-(r-a)\mathrm{/}2=a$ and $(2pq{)}^2=4\cdot (r+a)(r-a)\mathrm{/}4=r^2-a^2=b^2$, so $2pq=b$ because $p,q\ge 0$ (Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$); hence ${\kappa }^2=p^2-q^2+2ipq=a+ib$, by the field arithmetic of The complex numbers as $\mathbb{R}[x]\mathrm{/}(x^2+1)$, with the real embedding and imaginary unit $i$. The square roots of $a+ib$ are exactly $\pm \kappa$: if $z^2={\kappa }^2$, then $(z-\kappa )(z+\kappa )=0$, and the complex field property ($\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$) gives $z=\kappa$ or $z=-\kappa$. If $k=u+iv$ with $u,v\ge 0$ and $k^2=a+ib$, then $u^2-v^2=a$, $2uv=b$ and $u^2+v^2=\mathrm{\mid }k{\mathrm{\mid }}^2=\mathrm{\mid }{k}^2\mathrm{\mid }=\sqrt{a^2+b^2}=r$, where $\mathrm{\mid }{k}^2\mathrm{\mid }=\mathrm{\mid }k{\mathrm{\mid }}^2$ is the multiplicativity of the modulus (Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive) and $\mathrm{\mid }a+ib\mathrm{\mid }=\sqrt{a^2+b^2}$ is the modulus of Real and imaginary parts, complex conjugation, and modulus; solving gives $u^2=(r+a)\mathrm{/}2=p^2$ and $v^2=(r-a)\mathrm{/}2=q^2$, and since $u,v,p,q\ge 0$ the uniqueness of the nonnegative square root (Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$) gives $u=p$ and $v=q$, so $k=\kappa$. In the lossless case $\sigma =0$ one has $b=0$, $r=a$, $q=0$, $p=\sqrt{a}=\omega (\epsilon \mu {)}^{1\mathrm{/}2}=k(\omega )$ by Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$. All divisions are by strictly positive real numbers, $v$ and $q$ are nonzero exactly where defined, and no choice principle is used.