Thermal radiation of Er doped dielectric crystals: Probing the range of   applicability of the Kirchhoff law
  The Kirchhoff law of thermal radiation, relating emissivity {\epsilon} and absorptance {\alpha}, has been originally formulated for opaque bodies in thermodynamic equilibrium with the environment. However, in many systems of practical importance, both assumptions are often not satisfied. In this work, we revisit the century-old law and examine the limits of its applicability in an example of Er:YAG and Er:YLF dielectric crystals, potential radiation converters for thermophotovoltaic applications. In our experiments, the (80 at.%) Er:YAG crystal was opaque between 1.45 {\mu}m and 1.64 {\mu}m. In this spectral range, its absorptance {\alpha}({\lambda}) is spectrally flat and differentiates from unity only by a small amount of reflection. The shape of the emissivity spectrum {\epsilon}({\lambda}) closely matches that of absorptance {\alpha}({\lambda}), suggesting that the Kirchhoff law can adequately describe thermal radiation of opaque bodies, even if the requirement of thermodynamic equilibrium is not satisfied. The(20 at.%) Er:YLF crystal had smaller size, lower concentration of Er ions, and it was not opaque. Nevertheless, its spectrum of emissivity {\epsilon}({\lambda}) had almost the same shape (between 1.45 {\mu}m and 1.62 {\mu}m) as the spectrum of absorptance {\alpha}({\lambda}) derived from the transmission measurements. This observation is in line with our prediction that the spectra of emissivity and absorptance should have identical shapes in optically thin slabs. We, thus, show that the Kirchhoff law of thermal radiation can be extended (with caution) even to not-opaque bodies away from the thermodynamic equilibrium. 
Ekembu K. Tanyi, Brandi T. Burton, Evgenii E. Narimanov, and M. A.   Noginov
physics.optics