Plane waves in matter: dispersion relations, skin depth, and the complex refractive index

Intuition Beginner

When an electromagnetic wave enters a material, three things can happen: it slows down, it gets absorbed, and different frequencies travel at different speeds. These three effects are described by the refractive index $n$, the absorption coefficient, and the dispersion relation.

In vacuum, all electromagnetic waves travel at the speed of light $c$. In a transparent material like glass, the wave slows down to $v=c/n$ where $n$ is the refractive index (about 1.5 for glass). This slowing is what bends light at the surface of a lens — the basis of all optics.

In a conductor like copper, the wave does not just slow down — it gets absorbed. The field decays exponentially with distance from the surface, over a characteristic length called the skin depth. For copper at 60 Hz, the skin depth is about 8.5 mm; at 1 GHz, it is only 2 micrometres. This is why high-frequency circuits need very thin conductors: the current only flows in a thin skin at the surface.

Dispersion means that different frequencies travel at different speeds. Red light and blue light have slightly different refractive indices in glass, so a prism separates white light into a rainbow. Dispersion is what makes a pulse of light spread out as it travels through a fiber optic cable — different frequency components arrive at different times.

Visual Beginner

Medium	Wave speed	Attenuation	Example
Vacuum	$c$	None	Space
Dielectric	$c/n$	Weak	Glass, water
Conductor	Very slow	Strong	Copper, seawater

Worked example Beginner

The skin depth in copper ($\sigma =5.8\times 10^7$ S/m) at frequency $f=1$ MHz:

$\delta =\sqrt{2/(\omega {\mu }_0\sigma )}=\sqrt{2/(2\pi \times 10^6\times 4\pi \times 10^{-7}\times 5.8\times 10^7)}$

$=\sqrt{2/(4.59\times 10^8)}=\sqrt{4.36\times 10^{-9}}=6.6\times 10^{-5}$ m $=66\mu$m.

At 1 MHz, electromagnetic waves only penetrate 66 micrometres into copper. This is why AM radio signals (around 1 MHz) do not penetrate metal buildings, and why the shielding on coaxial cables only needs to be a thin layer of copper.

Check your understanding Beginner

Formal definition Intermediate+

Plane wave in a dielectric. In a linear, isotropic, non-conducting medium with permittivity $ϵ$ and permeability $\mu$, the wave equation for $\mathbf{E}$ is:

$${\nabla }^2\mathbf{E}=\mu ϵ\frac{{\partial }^2\mathbf{E}}{\partial t^2}$$

The plane-wave solution $\mathbf{E}={\mathbf{E}}_0{e}^{i(kz-\omega t)}$ has the dispersion relation:

$$k^2=\mu ϵ{\omega }^2⟹k=n\frac{\omega }{c}$$

where $n=c\sqrt{\mu ϵ}=\sqrt{{\mu }_r{ϵ}_r}$ is the refractive index and $v=c/n=1/\sqrt{\mu ϵ}$ is the phase velocity.

Complex wave number in a conductor. In a conducting medium ($\sigma >0$), the Maxwell equations give the modified dispersion relation:

$${\tilde{k}}^2=\mu ϵ{\omega }^2+i\mu \sigma \omega$$

The complex wave number $\tilde{k}=k+i\kappa$ where:

$$k=\omega \sqrt{\frac{\mu ϵ}{2}}{\left(\sqrt{1+{\left(\frac{\sigma }{\omega ϵ}\right)}^2}+1\right)}^{1/2}$$ $$\kappa =\omega \sqrt{\frac{\mu ϵ}{2}}{\left(\sqrt{1+{\left(\frac{\sigma }{\omega ϵ}\right)}^2}-1\right)}^{1/2}$$

For a good conductor ($\sigma \gg \omega ϵ$):

$$k\approx \kappa \approx \sqrt{\frac{\mu \sigma \omega }{2}}$$

and the skin depth is:

$$\delta =\frac{1}{\kappa }=\sqrt{\frac{2}{\mu \sigma \omega }}$$

The field inside the conductor decays as $e^{-z/\delta }$.

Complex refractive index. Define $\tilde{n}=n+i\kappa$ (complex refractive index, where $\kappa$ is the extinction coefficient). Then $\tilde{k}=\tilde{n}\omega /c$ and:

$$\mathbf{E}={\mathbf{E}}_0{e}^{i(\tilde{k}z-\omega t)}={\mathbf{E}}_0{e}^{-\kappa \omega z/c}{e}^{i(n\omega z/c-\omega t)}$$

The real part $n$ determines the phase velocity $v_p=c/n$. The imaginary part $\kappa$ determines the attenuation length $\ell =c/(\kappa \omega )$.

Group velocity. When the refractive index depends on frequency ($n=n(\omega )$), a wave packet travels at the group velocity:

$$v_g=\frac{d\omega }{dk}=\frac{c}{n+\omega dn/d\omega }$$

Information and energy propagate at $v_g$, not $v_p$. In regions of normal dispersion ($dn/d\omega >0$), $v_g<v_p<c$. In regions of anomalous dispersion ($dn/d\omega <0$), $v_g$ can exceed $c$ but no information travels faster than light.

Key derivation Intermediate+

Derivation (Skin depth in a good conductor).

Theorem. For a good conductor ($\sigma \gg \omega ϵ$), a plane wave of frequency $\omega$ decays exponentially with skin depth $\delta =\sqrt{2/(\mu \sigma \omega )}$. The wave number is $k\approx 1/\delta$ and the phase velocity is $v_p=\omega \delta =\sqrt{2\omega /(\mu \sigma )}$.

Proof. In a conductor, Maxwell's equations with Ohm's law ($\mathbf{J}=\sigma \mathbf{E}$) give:

$$\nabla \times \mathbf{B}=\mu \sigma \mathbf{E}+\mu ϵ\frac{\partial \mathbf{E}}{\partial t}$$

For a plane wave $\mathbf{E}=E_0{e}^{i(kz-\omega t)}\hat{\mathbf{x}}$, the curl equation gives $ik{B}_y=\mu (\sigma -i\omega ϵ)E_x$. From Faraday's law: $ik{E}_x=i\omega B_y$. Combining:

$$k^2=\mu \omega (\sigma -i\omega ϵ)\cdot \omega =\mu ϵ{\omega }^2+i\mu \sigma \omega$$

For $\sigma \gg \omega ϵ$: $k^2\approx i\mu \sigma \omega$. Writing $i=e^{i\pi /2}$: $k\approx \sqrt{\mu \sigma \omega }{e}^{i\pi /4}=(1+i)\sqrt{\mu \sigma \omega /2}$. So $k=\kappa =\sqrt{\mu \sigma \omega /2}$ and the skin depth is $\delta =1/\kappa =\sqrt{2/(\mu \sigma \omega )}$. $\square$

Bridge. The skin depth formula builds toward the design of all high-frequency electromagnetic devices. The foundational insight is that in a good conductor the displacement current is negligible compared to the conduction current, so the wave equation reduces to a diffusion equation. The central message is that the skin depth decreases as $1/\sqrt{\omega }$, confining high-frequency currents to thin surface layers. This is exactly the reason why radio-frequency shielding works and why microwave circuits use thin conductors. Putting these together, the skin depth determines the wall losses in waveguides 10.04.04, the complex refractive index generalises to the Kramers-Kronig relations 10.08.05, and the dispersion relation in matter modifies the propagation of all electromagnetic signals through material media.

Exercises Intermediate+

Lean formalization Intermediate+

Mathlib has the wave equation and basic complex analysis but does not contain the complex refractive index, the dispersion relation for waves in conductors, the skin depth, the Fresnel equations, or the Drude model. The theory of complex-valued solutions to the Helmholtz equation with complex coefficients is not formalised. lean_status: none.

Advanced results Master

The Fresnel equations at oblique incidence. For a plane wave incident at angle ${\theta }_i$ on a boundary between media with indices $n_1$ and $n_2$, the reflection and transmission coefficients depend on the polarisation:

$$r_s=\frac{n_1\cos {\theta }_i-n_2\cos {\theta }_t}{n_1\cos {\theta }_i+n_2\cos {\theta }_t},r_p=\frac{n_2\cos {\theta }_i-n_1\cos {\theta }_t}{n_2\cos {\theta }_i+n_1\cos {\theta }_t}$$

where $s$ and $p$ denote the two linear polarisations (TE and TM respectively). At the Brewster angle ${\theta }_B=\arctan (n_2/n_1)$, $r_p=0$ and the p-polarised wave is fully transmitted. At the critical angle ${\theta }_c=\arcsin (n_2/n_1)$ (for $n_1>n_2$), total internal reflection occurs.

Anomalous dispersion and resonant absorption. Near an atomic resonance at frequency ${\omega }_0$, the permittivity has a resonant structure:

$$ϵ(\omega )={ϵ}_0\left(1+\frac{{\omega }_p^2}{{\omega }_0^2-{\omega }^2-i\gamma \omega }\right)$$

where $\gamma$ is the damping rate. Below resonance, $n$ increases with $\omega$ (normal dispersion). Above resonance, $n$ decreases with $\omega$ (anomalous dispersion). At resonance, the absorption is maximum (the imaginary part of $n$ peaks). The Kramers-Kronig relations 10.08.05 connect the real and imaginary parts of $ϵ(\omega )$ through a causal convolution integral.

Metamaterials and negative refraction. Materials with both $ϵ<0$ and $\mu <0$ simultaneously have a negative refractive index $n=-\sqrt{|ϵ\mu |}/({ϵ}_0{\mu }_0)$. In such materials, the phase velocity is opposite to the group velocity, and Snell's law gives a negative refraction angle. Veselago (1968) predicted this, and Pendry (2000) showed that a flat slab of $n=-1$ material acts as a perfect lens. The first experimental demonstration was by Shelby, Smith, and Schultz (2001) using a metamaterial structure.

Synthesis. The theory of plane waves in matter unifies the propagation, attenuation, and dispersion of electromagnetic waves in all material media. The foundational insight is that the complex wave number $\tilde{k}$ encodes all three effects: its real part gives the wavelength and phase velocity, its imaginary part gives the attenuation (skin depth), and its frequency dependence gives the dispersion. The central message is that the material response ($ϵ$, $\mu$, $\sigma$) modifies the vacuum wave equation in a way captured entirely by the complex refractive index. Putting these together, the skin depth determines the losses in waveguides 10.04.04 and cavities 10.04.05, the dispersion relation constrains signal propagation in optical fibers, and the Kramers-Kronig relations 10.08.05 provide the fundamental link between dispersion and absorption through causality.

Full proof set Master

Proposition (Wave equation in a conducting medium). In a linear, isotropic medium with conductivity $\sigma$, the electric field satisfies the damped wave equation:

$${\nabla }^2\mathbf{E}=\mu ϵ\frac{{\partial }^2\mathbf{E}}{\partial t^2}+\mu \sigma \frac{\partial \mathbf{E}}{\partial t}$$

Proof. Start from the Maxwell-Ampere law in a conductor: $\nabla \times \mathbf{B}=\mu \sigma \mathbf{E}+\mu ϵ\partial \mathbf{E}/\partial t$. Take the curl of Faraday's law $\nabla \times \mathbf{E}=-\partial \mathbf{B}/\partial t$:

$$\nabla \times (\nabla \times \mathbf{E})=-\frac{\partial }{\partial t}(\nabla \times \mathbf{B})=-\mu \sigma \frac{\partial \mathbf{E}}{\partial t}-\mu ϵ\frac{{\partial }^2\mathbf{E}}{\partial t^2}$$

Using the vector identity $\nabla \times (\nabla \times \mathbf{E})=\nabla (\nabla \cdot \mathbf{E})-{\nabla }^2\mathbf{E}$ and Gauss's law ($\nabla \cdot \mathbf{E}=0$ for a neutral conductor with no free charge):

$${\nabla }^2\mathbf{E}=\mu ϵ\frac{{\partial }^2\mathbf{E}}{\partial t^2}+\mu \sigma \frac{\partial \mathbf{E}}{\partial t}\square$$

The $\sigma$ term is the damping term that converts field energy into Joule heat, producing the exponential attenuation characterised by the skin depth.

Connections Master

• EM waves in vacuum 10.04.02 are the special case $\sigma =0$, $ϵ={ϵ}_0$, $\mu ={\mu }_0$.
• Maxwell equations in matter 10.03.04 provide the constitutive relations ($\mathbf{D}=ϵ\mathbf{E}$, $\mathbf{J}=\sigma \mathbf{E}$) used to derive the wave equation.
• Waveguides 10.04.04 use the skin depth to calculate wall losses and the refractive index contrast for confinement.
• Cavities 10.04.05 have quality factors limited by the skin-depth losses on the cavity walls.
• Kramers-Kronig relations 10.08.05 connect the real and imaginary parts of the refractive index through a causal integral transform.

Historical & philosophical context Master

The concept of dispersion was studied experimentally by Cauchy (1836), who derived the empirical formula $n(\lambda )=A+B/{\lambda }^2+C/{\lambda }^4$ describing the wavelength dependence of the refractive index in transparent materials. The Sellmeier equation (1871) provided a more accurate form based on a resonance model, correctly predicting anomalous dispersion near absorption lines.

The skin effect was first observed by Hughes (1885) and explained theoretically by Rayleigh (1886), who derived the exponential decay of AC currents in conductors. The practical importance grew with the development of radio technology in the early 20th century, where skin depth determines the efficiency of antennas and transmission lines.

The Drude model (1900) was one of the first applications of the newly discovered electron to explain optical properties of metals. It correctly predicted that metals become transparent above the plasma frequency, a result confirmed experimentally in the 1930s for alkali metals.

The discovery of negative refraction (Shelby et al., 2001) opened the field of metamaterials and transformation optics, demonstrating that engineered materials can achieve electromagnetic properties not found in nature.

Bibliography Master

• Sellmeier, W., "Zur Erklaerung der abnormen Farbenfolge im Spectrum einiger Substanzen," Ann. Phys. 219, 272-282 (1871).
• Drude, P., "Zur Elektronentheorie der Metalle," Ann. Phys. 306, 566-613 (1900).
• Veselago, V. G., "The electrodynamics of substances with simultaneously negative values of $ϵ$ and $\mu$," Sov. Phys. Usp. 10, 509-514 (1968).
• Shelby, R. A., Smith, D. R., and Schultz, S., "Experimental Verification of a Negative Index of Refraction," Science 292, 77-79 (2001).
• Jackson, J. D., Classical Electrodynamics, 3rd ed. (Wiley, 1999).
• Griffiths, D. J., Introduction to Electrodynamics, 4th ed. (Cambridge, 2017).