Hagen–Rubens relation

In optics, the Hagen–Rubens relation (or Hagen–Rubens formula) is a relation between the coefficient of reflection and the conductivity for materials that are good conductors.^[1] The relation states that for solids where the contribution of the dielectric constant to the index of refraction is negligible, the reflection coefficient can be written as (in SI Units):^[2]

${\displaystyle R\approx 1-2{\sqrt {\frac {2\epsilon _{0}\omega }{\sigma }}}}$

where ${\displaystyle \omega }$ is the frequency of observation, ${\displaystyle \sigma }$ is the conductivity, and ${\displaystyle \epsilon _{0}}$ is the vacuum permittivity. For metals, this relation holds for frequencies (much) smaller than the Drude relaxation rate, and in this case the otherwise frequency-dependent conductivity ${\displaystyle \sigma }$ can be assumed frequency-independent and equal to the dc conductivity.

The relation is named after German physicists Ernst Bessel Hagen and Heinrich Rubens who discovered it in 1903.^[3][4]