Einstein Relations Connecting Broadband Emission and Absorption Spectra

Relations described by Einstein connecting the rates of spontaneous emission, stimulated emission, and absorption of radiation by an atomic system in free space are generalized to apply to broadband spectra of quantized systems dilutely distributed in a dielectric medium. Although the gross features of the broadband emission and absorption spectra can be qualitatively different (especially at low temperatures), the various spectra are connected at any specific frequency by simple expressions. For a two-level system imbedded in a medium at temperature $T$, typical equations connecting stimulated-emission and absorption cross sections $\ensuremath{\sigma}(\ensuremath{\omega})$ to the rate $f(\ensuremath{\omega})$ of spontaneous emission of photons per unit solid angle per unit frequency interval are: ${\ensuremath{\sigma}}_{e}(\ensuremath{\omega})={\ensuremath{\sigma}}_{a}(\ensuremath{\omega})\mathrm{exp}[\frac{\ensuremath{\hbar}(\ensuremath{\mu}\ensuremath{-}\ensuremath{\omega})}{\mathrm{kT}}]=f(\ensuremath{\omega}){(\frac{2\ensuremath{\pi}c}{\ensuremath{\omega}n})}^{2}$, where $\ensuremath{\hbar}\ensuremath{\mu}$ is a temperature-dependent excitation potential and $n$ is the index of refraction of the host material.