Electrodynamics

Electromagnetic waves in matter

Material electric and magnetic response changes wave speed, wavelength, polarization, and attenuation.

In a homogeneous linear medium, D = εE and B = μH. Losses can be represented by complex ε and μ; dispersion means different frequencies propagate differently.

k² = ω²με, n = ck/ω, S = E×H

Definition: Quantities and model

In a lossless medium, a plane wave has k = ω√(με), phase speed v = 1/√(με), and impedance η = √(μ/ε). For a nonmagnetic medium, n ≈ √ε_r.

Adjust parameters and rotate the view to inspect field structure; this is illustrative, not a general Maxwell solver.

Interpretation and consequences

At an interface, frequency is conserved while wavelength changes with speed. Boundary conditions on E and H determine reflection and refraction (Snell’s law for isotropic media).

Example: Quantitative example

Glass with n = 1.50 gives v = c/n = 2.00×10⁸ m/s and wavelength 400 nm for 600 nm vacuum light; frequency is unchanged.

Solution

Substitute into the stated relation, keep SI units consistent, and check the result dimensionally.

With phase convention $e^{i(kz-\omega t)}$ , an absorbing medium has complex wavenumber $k=k'+i\kappa$ : amplitude falls as $e^{-\kappa z}$ and intensity as $e^{-2\kappa z}$ . The dispersion relation is complex and frequency dependent; causality links dispersion and absorption through the Kramers–Kronig relations. A single real refractive index therefore cannot fully describe a material across an absorption band.

The wave impedance $\eta=\sqrt{\mu/\varepsilon}$ sets the amplitude ratio $E/H$ . At normal incidence on a boundary between lossless media, field continuity gives the electric-field reflection coefficient $r=(\eta_2-\eta_1)/(\eta_2+\eta_1)$ when there is no surface current. Matched impedances eliminate reflection; a refractive-index contrast can still redirect an obliquely incident wave.

In a dispersive medium, group velocity can differ from phase velocity; near resonance, strong absorption and dispersion require care when using constant material parameters. Away from resonances, weak dispersion often permits a useful local refractive-index approximation.

Quick check

Which relation is correct in the idealized situation described?

What should be checked first when applying a field formula?

References

Max Born, Emil Wolf (1999). Principles of Optics