29 Jul 2025
Dielectric and Optical Properties
Polarization, local fields, Langevin-Debye theory, normal and anomalous dispersion, Cauchy and Sellmeier relations, complex permittivity, and optical attenuation.
Polarization is electric dipole moment per unit volume. In a linear isotropic dielectric,
\[\mathbf D=\epsilon_0\mathbf E+\mathbf P =\epsilon_0\epsilon_r\mathbf E,\]so
\[\boxed{\mathbf P=\epsilon_0(\epsilon_r-1)\mathbf E =\epsilon_0\chi_e\mathbf E}.\]$\mathbf P$ has unit $\mathrm{C\,m^{-2}}$; $\epsilon_r$ and $\chi_e$ are dimensionless.
Polarizability and local field
If one molecule develops dipole $\mathbf p=\alpha\mathbf E_{\rm loc}$, then $\alpha$ has unit $\mathrm{C\,m^2V^{-1}}$. For number density $N$,
\[\mathbf P=N\alpha\mathbf E_{\rm loc}.\]In an isotropic cubic dielectric, a spherical Lorentz cavity gives
\[\boxed{\mathbf E_{\rm loc}=\mathbf E+\frac{\mathbf P}{3\epsilon_0}}.\]Substitution yields
\[P=N\alpha\left(E+\frac{P}{3\epsilon_0}\right).\]Using $P=\epsilon_0(\epsilon_r-1)E$ and solving,
\[\boxed{\frac{\epsilon_r-1}{\epsilon_r+2} =\frac{N\alpha}{3\epsilon_0}},\]the Clausius-Mossotti relation.
Langevin-Debye equation
A permanent dipole $p_0$ at angle $\theta$ to the local field has energy
\[U=-p_0E_{\rm loc}\cos\theta.\]With $x=p_0E_{\rm loc}/(k_BT)$, its orientational partition integral is
\[Z=2\pi\int_0^\pi e^{x\cos\theta}\sin\theta\,d\theta =4\pi\frac{\sinh x}{x}.\]Therefore
\[\langle\cos\theta\rangle =\frac{d\ln Z}{dx}=\coth x-\frac1x\equiv L(x).\]For $x\ll1$, $L(x)=x/3+O(x^3)$, and
\[P_{\rm or}=Np_0L(x) \simeq\frac{Np_0^2}{3k_BT}E_{\rm loc}.\]If $\alpha_i$ is the induced electronic-plus-ionic polarizability, the effective weak-field polarizability is
\[\alpha=\alpha_i+\frac{p_0^2}{3k_BT}.\]The local-field result becomes the Langevin-Debye equation
\[\boxed{ \frac{\epsilon_r-1}{\epsilon_r+2} =\frac{N}{3\epsilon_0}\left(\alpha_i+\frac{p_0^2}{3k_BT}\right) }.\]Complex dielectric constant
Use the time convention $E(t)=\operatorname{Re}[E_0e^{-i\omega t}]$. A bound charge $q$ of mass $m$ obeys
\[m\ddot x+m\gamma\dot x+m\omega_0^2x=qE.\]For $x=x_0e^{-i\omega t}$,
\[x_0=\frac{qE_0/m}{\omega_0^2-\omega^2-i\gamma\omega}.\]With $N$ oscillators per unit volume and $P=Nqx$,
\[\boxed{ \epsilon_r(\omega)=\epsilon_\infty+ \frac{\Omega^2}{\omega_0^2-\omega^2-i\gamma\omega}}, \qquad \Omega^2=\frac{Nq^2}{\epsilon_0m}.\]Write $\epsilon_r=\epsilon^{\prime}+i\epsilon^{\prime\prime}$. Multiplying by the complex conjugate of the denominator gives
\[\boxed{ \epsilon'=\epsilon_\infty+ \frac{\Omega^2(\omega_0^2-\omega^2)} {(\omega_0^2-\omega^2)^2+\gamma^2\omega^2}},\] \[\boxed{ \epsilon''=\frac{\Omega^2\gamma\omega} {(\omega_0^2-\omega^2)^2+\gamma^2\omega^2}}.\]$\epsilon^{\prime\prime}>0$ represents loss. The mean absorbed power density is
\[\boxed{\langle p\rangle=\frac12\omega\epsilon_0\epsilon''\lvert E_0\rvert^2}\]in $\mathrm{W\,m^{-3}}$.
Normal and anomalous dispersion
Away from resonance, damping is negligible and a transparent nonmagnetic material has $n^2\simeq\epsilon_r$. With several resonances,
\[n^2(\omega)=1+\sum_j\frac{A_j}{\omega_j^2-\omega^2}.\]Below a resonance, increasing $\omega$ normally increases $n$: $dn/d\omega>0$, equivalently $dn/d\lambda<0$. Close to an absorption resonance the slope can reverse, giving anomalous dispersion $dn/d\omega<0$.
Using $\omega=2\pi c/\lambda$ and defining $C_j=(2\pi c/\omega_j)^2$ gives the Sellmeier form
\[\boxed{n^2(\lambda)=1+\sum_j\frac{B_j\lambda^2}{\lambda^2-C_j}}.\]Far from resonance, $C_j/\lambda^2\ll1$ and
\[\frac{\lambda^2}{\lambda^2-C_j} =\frac1{1-C_j/\lambda^2} \simeq1+\frac{C_j}{\lambda^2}+\frac{C_j^2}{\lambda^4}+\cdots.\]Taking the square root and collecting constants produces Cauchy’s transparent-region relation
\[\boxed{n(\lambda)=A+\frac{B}{\lambda^2}+\frac{C}{\lambda^4}+\cdots}.\]Complex refractive index and extinction
Let the complex refractive index be
\[\widetilde n=n+i\kappa,\]where $\kappa$ is the extinction coefficient. For a nonmagnetic solid,
\[(n+i\kappa)^2=\epsilon'+i\epsilon'',\]so
\[\boxed{n^2-\kappa^2=\epsilon'}, \qquad \boxed{2n\kappa=\epsilon''}.\]A wave propagating along $z$ contains
\[e^{i\widetilde n\omega z/c-i\omega t} =e^{-\kappa\omega z/c}e^{i(n\omega z/c-\omega t)}.\]Its intensity therefore obeys $I(z)=I_0e^{-\alpha z}$ with
\[\boxed{\alpha=\frac{2\omega\kappa}{c}=\frac{4\pi\kappa}{\lambda}}.\]At normal incidence from vacuum, the optical reflectance is
\[\boxed{R=\left\lvert\frac{\widetilde n-1}{\widetilde n+1}\right\rvert^2 =\frac{(n-1)^2+\kappa^2}{(n+1)^2+\kappa^2}}.\]Thus $n$ controls phase velocity and refraction, while $\kappa$ controls attenuation; both follow from the same complex dielectric response.
Discussion